Discuss events which are simultaneous in one frame?

[neopolitan]

JesseM,

I will get back to you with respect to the diagrams.

With respect to the post responding to my long post where I point out that I never said anything about the

QUOTE expanding hypersphere idea in GR UNQUOTE

My whole point is that I am not deliberately using this idea which I was not familiar with, even if you think that they are similar. Thanks for the links, I will look at them later.

You bring up the triangles again. I thought we put those aside. Oh well, apparently not.

Take a flat surface in two spacelike dimensions, and wrap it around a sphere in three spacelike dimensions and you will end up with triangles whose internal angles sum to greater than 180 degrees. Take a flat volume in three spacelike dimensions and wrap it around a "hypersphere" in three spacelike dimensions and one timelike dimension ... what happens to the angles then?

I'll get back the rest of that when I have had time to absorb it and have time to devote to a response (if a response is warranted).

By the way, pointing to the "complete unfoundedness of (my) diagrams" and then, in the next sentence, admitting that you don't comprehend the diagrams doesn't bode well. Intelligent designers use that all the time "evolution is a complete load of bollocks, I don't understand it". If they took the time to understand then they might not think it is a load of bollocks. Similarly, just vaguely possibly, if you took the time to understand then it might work out that I am not just talking bollocks (and I did humbly ask for a critique based on understanding what I am trying to say, not abuse followed by an admission that you don't understand what I am trying to say). It's possible that I am talking bollocks, I accept that.

Anyway, I presented my diagrams in response to a request from you, JesseM, to show how the equations for time dilation and length contraction could be derived. I have shown that. Is the maths correct? If it isn't there is no point in going further into explaining the physical significance of my diagrams, is there?

cheers,

neopolitan

 

[neopolitan]

In addition, the comment "to the extent of attributing to me comments to which I am responding" also comes across as some kind of dig at my ability to pay attention to what you write, and I'm pretty sure this accusation has little or no factual basis--can you point out occasions when I've done this?

Sure, but it was you who was connecting your model to the expanding hypersphere idea in GR

No, you connected my model to the expanding hypersphere idea in GR. I had tried numerous times previously to restrict the discussion to SR and only discussed SR (#228, #221, #202, #200 and #198). The need for this began in post #189 when you brought up GR, I responded in #190 and it seems that since then you have thought I am talking about GR - I can see that in #190 I perhaps should have said "I am pretty sure that the model I have in mind is no more inconsistent with the equations of GR than SR is." Then there is the whole triangle thing, where I was arguing against the relevance of it after a post by DrGreg.

But I don't have time for this. I assure you that it wasn't meant as a criticism. Pointing out your abuse and immediate admission of a lack of understanding was meant as criticism.

cheers,

neopolitan

 

[JesseM]

Take a flat surface in two spacelike dimensions, and wrap it around a sphere in three spacelike dimensions and you will end up with triangles whose internal angles sum to greater than 180 degrees. Take a flat volume in three spacelike dimensions and wrap it around a "hypersphere" in three spacelike dimensions and one timelike dimension ... what happens to the angles then?

I'm sorry, but this is gibberish. A spacelike surface cannot be "wrapped around" a surface which has one timelike dimension. Anyway, if your hyperspheres were supposed to be surfaces of simultaneity, they are by definition spacelike surfaces and do not have a timelike dimension (for a single surface to 'have a timelike dimension' it must include points which have a timelike separation, i.e. points which lie within one another's light cones and thus cannot be simultaneous in any valid coordinate systems).

By the way, pointing to the "complete unfoundedness of (my) diagrams" and then, in the next sentence, admitting that you don't comprehend the diagrams doesn't bode well.

"Unfoundedness" just refers to what you have presented so far. If Einstein just presented minkowski diagrams in 1905 without any explanation of their physical significance (why different frames have lines of simultaneity that are tilted relative to one another in the diagrams, for example) then this would be "unfounded" too, even if he had the derivation in his head. But Einstein would have known this would be a silly way to present his ideas, and that if he were in the place of the physicists being shown some diagrams with no explanation of their physical/mathematical basis, he would reject them as unfounded too. You seem to expect the rest of us to see your diagrams as somehow meaningful, which suggests you don't have any idea of how physicists think about spacetime diagrams, and are imagining that the diagram can come before the physical explanation, which is absurd.

If you have some secret answer to the question of what in tarnation is the physical meaning of your lines of simultaneity and lengths in terms of readings on actual physical rulers and clocks, what the lines of simultaneity would look like for different observers in motion relative to one another, what equations would give the coordinate transformation between different observer's coordinate systems (from which you could derive their different lines of simultaneity), then by all means present these answers. But if you don't already have clear answers to these questions in your head, then your diagrams are indeed unfounded, period.

And actually, regardless of whether you have answers to these questions, please be sure to answer my previous question first before posting anything further on this subject:

And as I said in my second-to-last post, if your "model" requires something other than flat spacetime (which I think is what belliott meant by Minkowski space-time) in order to "explain" Lorentz contraction, then your model is incompatible with SR and GR, and should be discussed in the "Independent Research" forum or by email, not here (please address this issue of whether you do or do not require spacetime to be curved in order to explain Lorentz contraction as soon as possible, and if your answer is 'yes' I hope you see that this discussion must come to an end).

Anyway, I presented my diagrams in response to a request from you, JesseM, to show how the equations for time dilation and length contraction could be derived. I have shown that.

No, you haven't. You've just tried to show that if we compare a line parallel to one of the surfaces of simultaneity to another line that appears horizontal in the diagram, then the visual difference between lengths of the two lines is given by the Lorentz factor (I don't know if this actually works, and it's pointless to check until you answer questions about how these visual lengths are in any way related to physics). Why is the second line horizontal and not, say, at 33.7592 degrees, which would be just as arbitrary? What do the visual lengths of lines on a diagram have to do with the measured lengths of physical rulers in motion relative to another, which is what the Lorentz formula deals with? Like I said, unless you have clear answers to these questions in your mind, this is a meaningless game with pictures, and even if you do have answers, for you to expect that others would find your diagrams interesting or relevant before you present the answers shows a basic lack of understanding of how people think in physics.

 

[JesseM]

In addition, the comment "to the extent of attributing to me comments to which I am responding" also comes across as some kind of dig at my ability to pay attention to what you write, and I'm pretty sure this accusation has little or no factual basis--can you point out occasions when I've done this?

Sure, but it was you who was connecting your model to the expanding hypersphere idea in GR

No, you connected my model to the expanding hypersphere idea in GR. I had tried numerous times previously to restrict the discussion to SR and only discussed SR (#228, #221, #202, #200 and #198).

But this isn't a matter of me "attributing to you comments to which you are responding" through careless reading, it's a matter of me being genuinely confused by your ideas or your way of expressing your ideas, perhaps because your ideas are themselves confused. Even if I had read all those statements over ten times with utmost care, I still would have come away with the clear impression that you were "connecting your model to the expanding hypersphere idea in GR". And it seems that you are in fact fairly confused about the difference between SR and GR, because you don't seem to realize that if spacetime is curved (as you suggest it is in one of the diagrams and in other comments), then by definition this means you have left the realm of SR which deals only with flat spacetime of zero curvature everywhere. Perhaps recognizing that my thinking was informed by this fact will help you understood why I assumed you couldn't be talking about pure SR.

But I don't have time for this. I assure you that it wasn't meant as a criticism. Pointing out your abuse and immediate admission of a lack of understanding was meant as criticism.

Saying your ideas as expressed are incoherent and lacking any kind of physical/mathematical foundation that would allow others to make sense of them is not "abuse", it is my sincere evaluation of your postings on this subject, one which I feel confident that anyone else who is knowledgeable about relativity and reading this thread from beginning to end would share. The point is that even if you have secret foundations in your head, you are not doing the work of communicating them, so if you had any understanding of the way that people think in physics you wouldn't expect others to be able to make the slightest bit of sense of your ideas based on what you've presented to us so far.

 

[Mentz114]

Neopolitan,
I have to agree with JesseM that you aren't getting across whatever you hold in your head in a way that makes it a physical theory. I can't understand what you are saying.
Explaining a theory is just like story-telling. You have to draw up a scenario, add players that act therein and then logically develop to make your point. I don't know what is the point of your 'theory'.

M

 

[belliott4488]

neopolitan:

I'm not trying to be abusive, so please don't get snippy with me. I have been genuinely interested in trying to understand your picture of space-time, but so far I haven't found a clear explanation. It also was not "laziness", as you so blithely suggest, that prevents my reading all (now 250+) posts on this thread, but the fact that it has wandered down a variety of paths and so it is difficult to see which ones are worth following and which lead nowhere.

I have, in fact, read all the ones where you have presented your "onion" diagrams, and while I first thought I had simply missed some more careful explanation, I am now convinced that the explanation still has not been given.

My appeal to Minkowski space-time was not due to a "fixation", but simply to the fact that this has been the de facto standard for discussing flat space-time for over 3/4 of a century, so I thought it would be a good point of common understanding for us to start (to start logically, that is - I'm not suggesting that you start this entire discussion all over again). If that is not a good point of departure from the standard picture of SR for you, then I guess you'll have to start at some prior point of common understanding, as I suppose you've tried to do in your attempts to derive the Lorentz transformations in your onions.

I cannot make sense of these alleged derivations, however, for the same reason that JesseM cannot: you still have not explained how to interpret these diagrams in terms of basic physical observations. Since this attempt to connect some accepted principles of SR to your diagrams has not yet been successful, however, I think you might have to go back even further, i.e. to the postulates themselves, the principles of relativity and of the constancy of the speed of light, and show how to get from them to your diagrams.

One way or the other, you have to start somewhere where we all agree on our terms, and then build from that point. What you have done so far seems closer to presenting a finished end-product and saying, "see? There it is - don't you get it?" (I know it's not a finish end-product - I just mean you haven't started at the beginning.) Please, please - just back up and start at the beginning.

 

[neopolitan]

belliott,

My apologies for any offense from my outburst at the end of last week. It was less you and more my general circumstances that led to it. So far you have indicated genuine interest and a willingness to engage in a constructive dialog and I am a little ashamed to have punished you for it. My implication of laziness was completely unfair and I can understand in retrospect where you were coming from with regard to Minkowski space.

Not in self-defence, but more in explanation, I did say a long time ago in this thread that I came at this originally from an unorthodox direction, even if I think that what I arrived at is consistent with SR. That means that I didn't start with Minkowski space and so to discuss it and how it relates to my model would be to give a wrong impression, which I want to avoid.

I do intend to try to explain how I came to the model which the diagrams are an attempt to portray. This will be a personal history, so there will be no Minkowski space considerations, just the issues that I was dealing with and how I resolved them. Please bear with me while I do that. Once you understand where I was coming from, we can look back and see how what I arrived at relates to Minkowski space (with full understanding that it wasn't an original consideration of mine).

First however, I feel that I need to address some of JesseM's concerns. Especially in light of his recent comment (and a few earlier similar comments):

please address (the) issue of whether you do or do not require spacetime to be curved in order to explain Lorentz contraction as soon as possible, and if your answer is 'yes' I hope you see that this discussion must come to an end

Again, sorry, I will get back to you. I did say JesseM is a busy guy, but that also applies to me, not because I am a distracted child but because I am currently managing a multi-million dollar project and have a lot on my mind. If I appear snippy, it is more likely because of that and less likely to be a result of anything you have said or done.

cheers,

neopolitan

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

I'm sorry, but this is gibberish. A spacelike surface cannot be "wrapped around" a surface which has one timelike dimension. Anyway, if your hyperspheres were supposed to be surfaces of simultaneity, they are by definition spacelike surfaces and do not have a timelike dimension (for a single surface to 'have a timelike dimension' it must include points which have a timelike separation, i.e. points which lie within one another's light cones and thus cannot be simultaneous in any valid coordinate systems).

Hi JesseM,

I am going to interpret "... this is gibberish." as meaning "I don't understand what you are saying here." and also assume that "Could you please explain what you mean better?" was implied.

A spacelike surface cannot be "wrapped around" a surface which has one timelike dimension.

I didn't say that. I said (bold and underline added):

Take a flat volume in three spacelike dimensions and wrap it around a "hypersphere" in three spacelike dimensions and one timelike dimension.

This may still be gibberish to you but what you paraphrased me as saying sounded like gibberish even to me :)

if your hyperspheres were supposed to be surfaces of simultaneity

Only the surface of the hypersphere. To me a sphere has volume, it includes not just the surface but also all which is bounded by that surface. So a ball bearing is a sphere which consists of metal, not a sphere with metal inside it.

Since your introductory clause has problems (at least from my perspective) I won't address the rest.

It is possible that this is also where the misunderstanding about flat and curved spacetime comes in.

I have said many times now, space is flat in terms of three dimensions and curved in terms of four dimensions (one of the dimensions being timelike) and spacetime is flat (in terms of four dimensions, one of the dimensions being timelike).

I emphatically do not require spacetime to be curved in order to explain Lorentz contraction.

Any triangle in the "onion" will consist of three points which must be expressed in terms of x,y,z and t. If you try to label a point (x,y,z), you will be describing a line from the centre of the "onion" outwards with any and all values of t. The sum of the internal angles of the triangle defined by those three points will sum to 180 degree. This is because spacetime (in the model) is flat.

Space in terms of 3 dimensions is also flat. The idea of drawing an apparent triangle on a 2D surface mapped onto a sphere, but in reality drawing three joined arcs, and then trying to measure the angles on surface of the 3D sphere, is misleading. The higher sum of internal angles is due to our using a curve drawn in 3D rather than drawing lines directly from point to point (three chords, which will produce a genuine triangle). Another way of looking at it is to consider any point on the surface of the sphere as being a point on the radial out from the centre of the sphere. Then get yourself a 2d surface (imagine a flat piece of paper) and manoeuvre it so that it is tangential to the sphere and intersects all three radials. The resultant triangle will have angles which sum to 180 degrees. Really though, it is the triangle defined by the three chords that matters, since the lengths of those chords are directly proportional to the lengths of the arcs.

Another way is to think of a projection of the 2D triangle onto the the surface of a sphere. The triangle is a genuine triangle with a sum of internal angles of 180 degrees. In terms of the 3D sphere though, you may not see a triangle, but rather three joined arcs. But this is an illusion due to our efforts in putting the 2D triangle onto the surface of a 3D object. In 2D it remains a triangle, but yes, in 3D it is now a pseudotriangle with a sum of angles of greater than 180 degrees.

You may not agree with any of these conceptualisations.

The next step is to explain my "horizontal lines". But that will have to wait.

cheers,

neopolitan

 

[JesseM]

A spacelike surface cannot be "wrapped around" a surface which has one timelike dimension.

I didn't say that. I said (bold and underline added):

Take a flat volume in three spacelike dimensions and wrap it around a "hypersphere" in three spacelike dimensions and one timelike dimension.

This may still be gibberish to you but what you paraphrased me as saying sounded like gibberish even to me :)

You misunderstood me, when I said a spacelike surface cannot be wrapped around a surface which has one timelike dimension, of course I didn't mean that the timelike dimension was its only dimension. I meant that you can't wrap a spacelike surface around a surface that contains any timelike dimensions.

Only the surface of the hypersphere. To me a sphere has volume, it includes not just the surface but also all which is bounded by that surface. So a ball bearing is a sphere which consists of metal, not a sphere with metal inside it.

I assumed that when you talk about "wrapping a surface around another", you meant mapping points in the first surface to points on the second surface. If you're only mapping points in flat space to points on what you call the "surface" of the hypersphere, and that surface is purely spacelike, then I would say you're wrapping the flat space around a purely spacelike surface. I guess what you're saying is because the "inside" of this spacelike surface contains points at different times, you're wrapping it around a 4D region (not a 3D surface) with a timelike dimension and three spacelike ones, so that's where I misunderstood. But I hope you will agree that if the surface of the hypersphere is a surface of simultaneity then it must be purely spacelike, it does not contain a timelike dimension.

I have said many times now, space is flat in terms of three dimensions and curved in terms of four dimensions (one of the dimensions being timelike) and spacetime is flat (in terms of four dimensions, one of the dimensions being timelike).

When you say space is "curved in terms of four dimensions", do you just mean that on your illustrations it is represented as a sphere? Would you say this is just a curvature in the coordinate system being used, rather than a genuine "physical" curvature that would cause angles of a triangle in space to add up to something other than 180? As an analogy, we can project the surface of the globe onto a flat plane in Mercator projection and this distorts the apparent distance between different points (for example, Greenland appears much larger than it actually is in comparison to continents near the equator), but we understand that the visual distances on the projection are different from the "real" distances.

If the wrapping of flat space around the sphere is supposed to be something more than just a coordinate representation--if you're claiming that the physical distance between points on the sphere as measured by rulers would actually be proportional to the geodesic distance between those points on the sphere--then that would mean that space really has a spherical geometry in a physical sense, not just a coordinate sense, which would be inconsistent with your claim above that space is still flat in each surface of simultaneity in your diagram. On the other hand, if you're not claiming that the spherical geometry is physical, then the statement "space is curved in terms of four dimensions" cannot really be physical either, but just a feature of the coordinate system you're using, with the surface of simultaneity appearing curved because you're using a coordinate system where the coordinate distance is not proportional to actual physical distance as measured by rulers.

And if you are indeed just using a coordinate system where a regular inertial surface of simultaneity (i.e. all the points x,y,z for some single value of t in one of the inertial coordinate systems given by the Lorentz transformation) is projected onto a sphere, then you really need to present the equations that define how the mapping works. For example, are you saying all the infinite space on the surface of simultaneity when represented in Minkowski coordinate is projected onto a finite sphere in the new coordinate system, a bit like like how the Riemann sphere contains the entire infinite complex plane? Or are you just taking some finite slice of an inertial surface of simultaneity and projecting it onto a sphere?

The equations would perhaps be easier to write down if we imagine that there are only two spatial dimensions which are described in terms of polar coordinates (r,phi), and then we want to map those onto the surface of a 3D sphere with some radius R, where different points on the surface can be described in spherical coordinates of r', phi' and theta' (with r'=R', a constant, for all points on the surface...the value of R' for a particular surface of simultaneity would presumably depend on the time coordinate t of that surface). For example, to map the entire infinite 2D plane (r going from 0 to infinity, phi going from 0 to 2pi) onto the entire sphere (phi' going from 0 to 2pi, theta' going from 0 to pi) we could use the transform:

r' = R'
phi' = phi
theta' = pi/(r + 1)

This would ensure that every point (r, phi) in the plane was mapped to a point r', phi', theta' on the surface of a 3D sphere with radius R'.

Similarly, if we just wanted to map a circular subsection of the 2D plane, limited to points where r was between 0 and the radius of the circle R (with phi still varying from 0 to 2pi), and we wanted to map this region onto the surface of the 3D sphere with radius R', we could use this transformation:

r'=R'
phi' = phi
theta' = pi*r/R

You may not wish to use either of these transforms, but if the spherical surfaces of simultaneity in your onion diagram are just supposed to be a coordinate representation of flat space rather than a physically curved space, you need to specify how this coordinate representation works, exactly--what equations would map points in flat space as described in terms of some standard coordinate system (for 2D flat space, the usual ones would be either cartesian coordinates x,y or polar coordinates r,phi) onto points on a given spherical surface. But without some clearly specified mapping there is no way that your diagrams can be understood as sufficiently well-defined to be physically meaningful.

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

As an analogy, we can project the surface of the globe onto a flat plane in Mercator projection and this distorts the apparent distance between different points (for example, Greenland appears much larger than it actually is in comparison to continents near the equator), but we understand that the visual distances on the projection are different from the "real" distances.

I talked about the reverse of this later in my earlier post, when I talked about projecting the triangle onto a sphere. I thought of mentioning cartography myself, but held back.

For example, are you saying all the infinite space on the surface of simultaneity when represented in Minkowski coordinate is projected onto a finite sphere in the new coordinate system, a bit like like how the Riemann sphere contains the entire infinite complex plane? Or are you just taking some finite slice of an inertial surface of simultaneity and projecting it onto a sphere?

I am not saying space is infinite. I am saying it is unbounded, while being finite and not being curved (in 3d) - which would make space a bit like as in the classic Asteroids game, which you sort of mentioned before (although I think you actually referred to the game Space Invaders). I don't think that SR relies on space being infinite, does it?

Note the following:

http://www.Newton.dep.anl.gov/askasci/ast99/ast99547.htm
http://cosmos.phy.tufts.edu/~zirbel/ast21/sciam/IsSpaceFinite.pdf
http://www.space.com/scienceastronomy/universe_soccer_031008.html (This is from 2003, so may be outdated.)

In any event, if I don't consider the universe to be infinite, does that invalidate my model? If it doesn't, do I have prove something that I don't hold to be true and isn't actually necessary? I would have to think about it for a while to see if I could do so.

While your writings on polar coordinates are interesting, they indicate a misunderstanding.

2D spatial coordinates couldn't be described in terms of r and phi. They would be described in terms of phi and theta (yes I know that doesn't quite make sense without the r).

A change in radius represents a delta-t. Remember I said that everything is expanding. For example a ruler (a conceptual length not a physical wooden stick) at t=1 would subtend 1 second of arc and would subtend 1 second of arc at t=2 as well (and t=10^100 and t=10^-100). If two identical rulers at right angles subtend 1 second of arc at t=1 and they would also do so at t=2 (etc etc). The locations would be given by the arcs subtended in your two reference directions. These locations would be unaffected by the radius. If you were a god and could be outside the universe and unaffected by time, then in one sense the rulers would be bigger as t increases - but relative to all measuring devises in the universe, they would remain the same size.

Remember the horizonal line you asked about? Well, that was just a ruler (the conceptual one, not the wooden one). Yes, in "reality" the ruler would be projected onto the surface of the hypersphere, so I could have drawn it as an arc but I chose instead to draw it as a tangential line - it's not horizontal so much as in a plane which is tangential to the surface of the hypersphere at the point where Observer C is (which happens to make it horizontal).

Something that I thought about on the weekend was interesting (to me). If you take a small section of the universe as onion, a local section, if you like, and call that the visible universe, what does it look like, if you just map that. Think of the section as being an array of circles subtending the same angle on each hypersurface of simultaneity. Don't you end up with the single ended football? I personally think you do.

cheers,

neopolitan

 

[JesseM]

While your writings on polar coordinates are interesting, they indicate a misunderstanding.

2D spatial coordinates couldn't be described in terms of r and phi. They would be described in terms of phi and theta (yes I know that doesn't quite make sense without the r).

Huh? When I talked about the surface of the sphere, I did use phi' and theta' to represent different points on the surface. And when I used r and phi, I wasn't talking about the sphere at all, but points in ordinary Euclidean 2D space prior to being mapped onto the sphere. Do you agree that for ordinary Euclidean 2D space, we can describe points either in terms of cartesian coordinates x,y, or we can describe points with 2D polar coordinates r,phi? (polar coordinates in 2D are normally written in terms of r,theta, but I chose phi instead because it makes it less confusing when converting to spherical coordinates where the 'horizontal' angle is phi and the 'vertical' angle is theta). For example, a point at cartesian coordinates x=0, y=3 in a 2D plane would be at polar coordinates r=3, phi=pi/2, assuming the phi=0 axis is the same as the +x axis. In general, the relation between these two ways of describing points on a 2D plane will be x=r*cos(phi) and y=r*sin(phi).

A change in radius represents a delta-t.

Yes, on the sphere, but that's a change in the r' coordinate I used above, not a change in the r coordinate which was for polar coordinates in the 2D plane. That's why I said:

The equations would perhaps be easier to write down if we imagine that there are only two spatial dimensions which are described in terms of polar coordinates (r,phi), and then we want to map those onto the surface of a 3D sphere with some radius R, where different points on the surface can be described in spherical coordinates of r', phi' and theta' (with r'=R', a constant, for all points on the surface...the value of R' for a particular surface of simultaneity would presumably depend on the time coordinate t of that surface)

Remember I said that everything is expanding. For example a ruler (a conceptual length not a physical wooden stick) at t=1 would subtend 1 second of arc and would subtend 1 second of arc at t=2 as well (and t=10^100 and t=10^-100). If two identical rulers at right angles subtend 1 second of arc at t=1 and they would also do so at t=2 (etc etc).

Yes, all this would already be true of the spatial coordinate transformations I gave in the last post, assuming that different times in the ordinary unprimed coordinate system (where different points in 2D space at a single time are identified with r, phi) map to different radii r' in the primed spherical coordinate system (where different points in space at a single time are identified with phi', theta'). I think you misunderstood what was going on in those coordinate transforms, please read them over again.

If you were a god and could be outside the universe and unaffected by time, then in one sense the rulers would be bigger as t increases - but relative to all measuring devises in the universe, they would remain the same size.

Again, this is not a physical fact at all, but just a feature of the coordinate system we are using. We could equally well use a coordinate system where successively smaller spheres corresponded to later times, so that the coordinate length of objects was progressively shrinking. Neither of these are in any sense "physical" truths, if you think they are you're taking a mere coordinate representation too seriously.

Remember the horizonal line you asked about? Well, that was just a ruler (the conceptual one, not the wooden one). Yes, in "reality" the ruler would be projected onto the surface of the hypersphere, so I could have drawn it as an arc but I chose instead to draw it as a tangential line - it's not horizontal so much as in a plane which is tangential to the surface of the hypersphere at the point where Observer C is (which happens to make it horizontal).

I don't see how this makes sense, because if the hyperspheres are surfaces of simultaneity, then a line which didn't lie along the surface would cut through multiple surfaces, so it would have nothing to do with physical length which is supposed to be the distance between ends of the object at a single moment in time. Perhaps you're saying that just as different frames have different surfaces of simultaneity in Minkowski coordinates, the same would be true in your new coordinate system, so that a horizontal line would actually lie along the surface of simultaneity for a different frame; I doubt this will work the way you want it to though, not if we actually calculate the equations for mapping different points in spacetime as represented in Minkowski coordinates (usually x,y,t, but as I said I've been using r,phi,t instead, with x=r*cos(phi) and y=r*sin(phi)) to points in your new coordinate system where the surfaces of simultaneity for one frame look like concentric spheres. So please, look over the coordinate transformations I gave in the last post to clear up your misunderstandings, then tell me what coordinate transformation you want to use for mapping spatial coordinates r,phi at a constant time t in some frame in Minkowski coordinates to coordinates, phi',theta' on the surface of a sphere with constant r' that corresponds to that time.

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

Perhaps I misunderstand. Yes, you can represent cartesian coordinates in polar coordinate form and vice versa with equations like "x=r*cos(phi) and y=r*sin(phi)".

This requires that you pick a null-point as your origin of r and your standard null-direction, where phi=0. (The equivalent of two null-points x=0 and y=0 in cartesian nomenclature.)

If you then map this onto a sphere, then the null-point and the null-direction would be a point on the surface of the sphere and a direction on the surface of that sphere. The radius of the sphere (R?) at any instant would have a constant relationship to the radius of the cartesian coordinate (r?) (R/r(R)=a constant).

In my model, R, the radius of the sphere, is the time coordinate.

Progressively shrinking spheres don't provide an explanation for why we have a speed limitation in our universe, so such a coordinate system is not just as good.

tell me what coordinate transformation you want to use for mapping spatial coordinates r,phi at a constant time t in some frame in Minkowski coordinates to coordinates, phi',theta' on the surface of a sphere with constant r' that corresponds to that time

If you must have r and phi, in vanilla flat space, then in the onion model, I want to have R, r and phi. This requires two null-points, one in the centre of the sphere (which has the radius R) plus one on the surface of the sphere and a null-direction on the sphere.

I actually would be more happy to have cartesian coordinates in flat space, and then map them onto the onion using something like R, R*sin(phi) and R*sin(theta). (Here you have a null-point in the centre of the sphere, and two collocated null-directions, which can be thought of two intersecting planes in which the angles phi and theta are measured from the shared null-direction.)

I think we need to sort this out before we get back to the horizontal line.

cheers,

neopolitan

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

belliott,

I did say I would get back to you. Since JesseM is still working through other things, I have nothing to respond to on that strand at the moment, so I can devote a little time to addressing questions you had.

I cannot make sense of these alleged derivations, however, for the same reason that JesseM cannot: you still have not explained how to interpret these diagrams in terms of basic physical observations. Since this attempt to connect some accepted principles of SR to your diagrams has not yet been successful, however, I think you might have to go back even further, i.e. to the postulates themselves, the principles of relativity and of the constancy of the speed of light, and show how to get from them to your diagrams.

One way or the other, you have to start somewhere where we all agree on our terms, and then build from that point. What you have done so far seems closer to presenting a finished end-product and saying, "see? There it is - don't you get it?" (I know it's not a finish end-product - I just mean you haven't started at the beginning.) Please, please - just back up and start at the beginning.

I am not sure if the previous posts help with this address the issue you had in the first paragraph above.

I will try to go back to the beginning and see if it helps you understand where I was coming from and how I got to the diagrams I posted earlier.

The issue I had in mind many many years ago was the two spaceship, two torches scenario. You have two spaceships, A and B, traveling towards each other, both at 0.75c. (It wasn't originally in my conceptualisation, but for clarification's sake I point out that I was inherently assuming a third observer in whose frames these 0.75c speeds are measured, call this guy Q.)

The torches on each spaceship are used to fire photons at the other. Everyone measures the all photons traveling at c in their own frame:

1. Photons from A, measured by A, travel at c.
2. Photons from B, measured by A, travel at c.
3. Photons from A, measured by B, travel at c.
4. Photons from B, measured by B, travel at c.
5. Photons from A, measured by Q, travel at c.
6. Photons from B, measured by Q, travel at c.

Additionally, the relative velocities of A and B will be less than c even though the relative velocities between both A and Q and B and Q will be 0.75c.

I didn't just say "that's just relativity for you" but tried to think of how it could be so.

The answer I came up with was vector addition.

THIS IS AN INTERMEDIATE STEP FOR CLARIFICATION

Say you have two ships at sea (A and B) both heading for an island, Q. Both travel at "company speed" according to regulations set by the board, notated as "0.75c" (75% of the company max, a speed which is only used with light loads and so is referred by some captains as "lightspeed"). First assume that they are not approaching the island from opposite directions, so that relative to Q, the velocities are not equal and opposite even if the speeds are equal.

A and B will be approaching each other a speed with is less than 0.75c + 0.75c.

This gets us part of the way, but not all of the way, since there is no reason to assume that these two ships can't be steered towards each other making their speed of approach 0.75c + 0.75c. Or is there?

End of intermediate step

We will now consider two ships moving towards each other, which is equivalent to the two spaceships, two torches scenario.

The two ships are not approaching each other directly. The are on the surface of an ocean which is wrapped around the surface of a big sphere we call the earth. This is another approximation of the two spaceships, two torches scenario because the angle of approach decreases progressively as the two ships get nearer, until they are effectively approaching each other directly (and because it is due to spatial curvature - such spatial curvature is not in my model and would cause problems as others have pointed out before).

So that is still not enough.

But what if the surface of the Earth (and the surface of the ocean) was expanding?

Now the total velocity of the ships would a combination of motion due to the exansion and the velocity on the ocean's surface. There would be vector addition again, rather than the ships traveling directly at each other and such vector addition would be unaffected by the proximity of the ships.

What would the rate of expansion of the Earth have to be to prevent the ships from ever approaching each other at more than c? It would have to be c (so dr/dt=c). (Note that I am not saying that the ship's owners make the Earth expand to limit the relative speeds that the ships can reach.)

Now we can apply this to the whole universe and the two spaceships, two torches scenario.

If the whole universe is expanding, and the spatially flat universe is mapped onto a hypersphere in flat spacetime, and objects in the universe follow straight lines in spacetime, then the only way to have two objects move in the same plane is to have them share the same velocity (speed and direction). Otherwise they will move in different planes at an angle which is dependent on the rate of universal expansion and their relative 3-velocities (velocities in space). The greater their relative velocities, the greater the angle, which means that even if A and B travel at 0.99999c relative to Q then, their relative velocities - relative to each other - will be less than c.

This all implies that the universe expands at a particular rate. That rate is c.

I said earlier that the Earth would have to expand at a rate of dr/dt=c to limit the ship's relative speeds to c, but in the situation where we are considering the universe, we are not trying to limit the relative speed of the spaceships, just explaining the limit.

Is there any other reason to consider the rate of expansion of the universe to be c?

I think there is. If the universe is expanding, then you can (conceptually) think of all instants as being hypersurfaces of simultaneity relative to the rest frame of any inertial observer you wish to nominate. Say we are fussy and take only selection of these hypersurfaces, one per second. The rate of change between each hypersurface as measured on a clock at rest in that rest previously nominated, is one second per second.

If we want to get this in common terms, since the other dimensions are measured in metres, we would have to use the conversion we use all the time with our SI units.

1 metre = (1/c) seconds

or ... one second per second = c

This wording might not be the class answer, but it is why the time axis is often given as the ct axis.

Anyway, this is how I started on my journey to the diagrams I posted earlier.

Perhaps I should stop there and let you ask questions. Then we can go further and explain what is in the diagrams later, if you are still interested.

cheers,

neopolitan

 

[belliott4488]

neo -
Thanks for the long and thorough response. I think it goes a long way toward clarifying up your ideas. Not surprisingly, I have many responses, but I'll try to keep them within reasonable limits.

The issue I had in mind many many years ago was the two spaceship, two torches scenario ... <details snipped> ... the relative velocities of A and B will be less than c even though the relative velocities between both A and Q and B and Q will be 0.75c.

I didn't just say "that's just relativity for you" but tried to think of how it could be so.

The answer I came up with was vector addition.

I'm sure you know that the relativistic velocity addition formula gives you the answer. Do you object to that result, i.e. do you believe you have a more accurate way to predict the relative velocity between A and B, or are you offering an alternate method for deriving the same result, which method you find preferable for understanding "how it could be so"?

... Say you have two ships at sea (A and B) both heading for an island, Q ... A and B will be approaching each other a speed with is less than 0.75c + 0.75c.

This gets us part of the way, but not all of the way, since there is no reason to assume that these two ships can't be steered towards each other making their speed of approach 0.75c + 0.75c. Or is there?

Okay, I think I'm with you so far ...

We will now consider two ships moving towards each other, which is equivalent to the two spaceships, two torches scenario.

The two ships are not approaching each other directly. The are on the surface of an ocean which is wrapped around the surface of a big sphere we call the earth. This is another approximation of the two spaceships, two torches scenario because the angle of approach decreases progressively as the two ships get nearer, until they are effectively approaching each other directly (and because it is due to spatial curvature - such spatial curvature is not in my model and would cause problems as others have pointed out before).

Okay, I think it's worth pointing something out here. Yes, in our 3-d world we recognize that the paths taken by the ships are not straight lines, but are most likely arcs of great circles (assuming they're taking the shortest paths between start and finish points). From their point of view, however, they are moving in a 2-d world, where they need only two coordinates to specify their locations exactly, i.e. lat and lon. Moreover, as far as they are concerned, they are following straight paths, since those paths are the shortest distances between points. This is a perfectly valid point of view and makes sense for them. Moreover, they can do geometry in these coordinates. Specifically, they can do spherical geometry, which is most definitely the geometry of a curved space - which is no surprise to us, of course. The only point I want to make here is that in all likelihood they define their speed in terms of these coordinates. I'll return to this later.

But what if the surface of the Earth (and the surface of the ocean) was expanding?

Now the total velocity of the ships would a combination of motion due to the exansion and the velocity on the ocean's surface. There would be vector addition again, rather than the ships traveling directly at each other and such vector addition would be unaffected by the proximity of the ships.

Note that from the ships' perspectives, no distances are changing, since they use lat/lon to measure distances, and those do not change with a change in R. (Yes, the sizes of rigid objects like ships and rulers should not change, but that's a shortcoming of the analogy. If you want this to be comparable to the expansion of space-time, you should really have all points on the surface expanding uniformly, so that measuring rods grow, too - it's space that's expanding, and it takes matter along with it for the ride.

Now, the ship captains will still notice that something has changed, since you are not doing anything to change the rates of their clocks. They will notice now that it takes them longer to get from one lat/lon location to another, so to them this will appear as an apparent decrease in their speed, despite their using the same amount of fuel, etc.

(This is in contrast to our expanding universe, where it is not just the spatial, but also the time axis that expands, thus our measurements of velocities don't change, except for the speed of light. We now see that it takes longer for light to reach us from Star X then it did in the past, which we interpret as expansion. We do not, however, notice that it takes longer for the Earth to revolve around the Sun, i.e the solar year is not getting longer.)

What would the rate of expansion of the Earth have to be to prevent the ships from ever approaching each other at more than c? It would have to be c (so dr/dt=c). (Note that I am not saying that the ship's owners make the Earth expand to limit the relative speeds that the ships can reach.)

This is an important point. If their speed limit of c is voluntarily imposed, i.e. they maintain their speeds such that they always observe it be less than c, then their maximum speed in our 3-d space increases, since their measurement of 1 ct unit grows along with their coordinate system.

If, on the other hand, you mean that their speed is physically limited to c in our 3-d sense, then yes, they will find that their maximum speed is getting slower and slower, perhaps reasoning that their universe is expanding, until they find themselves unable to "outrun" the expasion and therefore unable to reach their destinations.

Now we can apply this to the whole universe and the two spaceships, two torches scenario.

If the whole universe is expanding, and the spatially flat universe is mapped onto a hypersphere in flat spacetime, and objects in the universe follow straight lines in spacetime, then the only way to have two objects move in the same plane is to have them share the same velocity (speed and direction).

Okay, this is where I think the big problem resides, and I'm pretty sure my objection here is equivalent to JesseM's. I don't believe what you just stated makes mathematical sense. If you take a flat space (i.e. a Euclidean space, a space with the N-d identity matrix for a metric, however you want to describe it) and map it to a hypersphere, the resulting space is either not a vector space at all (e.g. a projection, as JesseM suggested), or it is a vector space with curvature > 0. There's just no way around it. It's as if you said, take a spatially flat 2-d space and map it to the surface of a globe (i.e. a 2-sphere) - the target space is curved, for the same reason that you can't wrap a globe with a flat piece of paper without wrinkling it.

You could do the projection trick, but then you get distortions in distances (like with the Mercator projection mentioned earlier, except in the other direction, i.e. flat space to spherical), and it is not a metric space. That means that your discussions of distances and velocities all go out the door.

Otherwise they will move in different planes at an angle which is dependent on the rate of universal expansion and their relative 3-velocities (velocities in space). The greater their relative velocities, the greater the angle, which means that even if A and B travel at 0.99999c relative to Q then, their relative velocities - relative to each other - will be less than c.

All of this, by appealing to geometry, i.e. angles between vectors and so forth, implies that this space is a physical space, which means it must be curved if it in fact resides on the surface of a hypersphere.

I have one further problem, which is that you've essentially swept under the rug the fact that your ships-on-a-globe analogy is constructed entirely with spatial coordinates. You haven't really addressed how the time axis in space-time gets mapped in your hypersphere picture. That's clearly critical, but I'll leave it for now, since I think you have more fundamental hurdles to cross first, i.e. how exactly you're doing this mapping. I think what you need is an explicit recipe for going from point (t,x,y,z) to point (r,a1 a2,a3), where the a's are the angles - or other coordinates - that you choose to parametrize your 3-sphere.

I'm afraid that I'm suspicious that your approach, whatever its aesthetic appeal might be, is not going to turn out to be mathematically viable. It wouldn't be the first appealing physical theory to suffer this fate, so you'd be in good company. You really have to work out the details to find out if it will survive, though.

[I'm deferring any other comments until later, since I think these issues need to be addressed first.]

 

[JesseM]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

Perhaps I misunderstand. Yes, you can represent cartesian coordinates in polar coordinate form and vice versa with equations like "x=r*cos(phi) and y=r*sin(phi)".

This requires that you pick a null-point as your origin of r and your standard null-direction, where phi=0. (The equivalent of two null-points x=0 and y=0 in cartesian nomenclature.)

If you then map this onto a sphere, then the null-point and the null-direction would be a point on the surface of the sphere and a direction on the surface of that sphere. The radius of the sphere (R?) at any instant would have a constant relationship to the radius of the cartesian coordinate (r?) (R/r(R)=a constant).

Sure, that was exactly how it worked in the two mappings I proposed. Since you seem to be interested in mapping a finite region of flat space onto the sphere rather than an entire infinite region, let's just look at the second of the two mappings I suggested:

if we just wanted to map a circular subsection of the 2D plane, limited to points where r was between 0 and the radius of the circle R (with phi still varying from 0 to 2pi), and we wanted to map this region onto the surface of the 3D sphere with radius R', we could use this transformation:

r'=R'
phi' = phi
theta' = pi*r/R

Here you can see that if we vary the radius r in the flat space, this varies the theta' coordinate on the sphere, with larger values of r corresponding to larger values of theta'. r=0 maps to theta'=0, which in spherical coordinates (see the diagram on the wikipedia page) would be the "north pole" of the sphere. r=R, the edge of the circular disc in flat space, corresponds to theta' = pi, which would be the "south pole" of the sphere. r=R/4 would be theta'=pi/4, one fourth of the way from the north pole to the south pole; r=R/2 would be theta'=pi/2, one half the way from the north pole to the south pole (on the equator of the sphere), and so forth. So, you can see that if you have two rods placed along the radial axis inside the disc that we're looking at in flat space, then the ratio of their lengths in flat space would be the same as the ratio of their arc-length on the sphere.

Note that in the above transformation I just mapped a circular disc cut out of a single surface of simultaneity in an inertial coordinate system in flat spacetime to a single spherical surface of simultaneity with fixed radius R' in the new coordinate system. If we wanted to include time in there, then if we have an event with spatial coordinates r and phi and time coordinate t in our normal inertial coordinate system, then we could map the event into the new spherical coordinate system (which uses coordinates r', phi', and theta', with the meanings of each being the same as in the diagram on the wikipedia page) with a transformation like this:

r' = t
phi' = phi
theta' = pi*r/R

This would ensure that all events on the same surface of simultaneity in the inertial coordinate system also lie on a sphere with the same radius, but events at different times in the inertial coordinate systems are on spheres with different radii, with later times corresponding to larger spheres (just as this mapping ignores all events that lie outside a disc of radius R in the inertial coordinate system, it also ignores events that happened before time t=0 in the inertial coordinate system).

In my model, R, the radius of the sphere, is the time coordinate.

Yes, I already said this in previous posts. As I said, I was just looking at the mapping for a particular surface of simultaneity in the inertial coordinate system to a particular sphere of fixed radius R', but perhaps the second coordinate mapping I introduced above, which explicitly shows that different time coordinates in the inertial system map to spheres of different radii r', makes this more clear.

Progressively shrinking spheres don't provide an explanation for why we have a speed limitation in our universe, so such a coordinate system is not just as good.

But do you understand that any "explanation" that is coordinate-dependent cannot really be a physical explanation, but only a sort of aide to intuition?

tell me what coordinate transformation you want to use for mapping spatial coordinates r,phi at a constant time t in some frame in Minkowski coordinates to coordinates, phi',theta' on the surface of a sphere with constant r' that corresponds to that time

If you must have r and phi, in vanilla flat space, then in the onion model, I want to have R, r and phi. This requires two null-points, one in the centre of the sphere (which has the radius R) plus one on the surface of the sphere and a null-direction on the sphere.

I don't understand--are you proposing a geometrically different type of coordinate system than the spherical system shown on the wikipedia page (where varying r corresponds to varying the radius, varying phi corresponds to varying the longitude like moving in the east-west direction on a globe, and varying theta corresponds to varying the latitude like moving in the north-south direction on a globe), or are you still using the same type of spherical coordinates but just relabeling theta as r? If it's just a relabeling, I would rather use the standard notation for spherical coordinates, I already understand that varying theta in this coordinate system corresponds to varying the r coordinate in the polar coordinate system for flat space (as I discussed earlier in this post). Assume for the sake of the argument that you have the coordinates of events in an inertial coordinate system and you want your computer to graphically represent where these events would appear in terms of your 3D onion diagram, but the computer only knows how to plot 3D points using either 3D cartesian coordinates x',y',z' or 3D spherical coordinates r',phi',theta'.

I actually would be more happy to have cartesian coordinates in flat space, and then map them onto the onion using something like R, R*sin(phi) and R*sin(theta). (Here you have a null-point in the centre of the sphere, and two collocated null-directions, which can be thought of two intersecting planes in which the angles phi and theta are measured from the shared null-direction.)

If you're using cartesian coordinates, those would usually be denoted x and y rather than phi and theta. If we wanted to map points (x,y,t) in our inertial coordinate system into the new coordinate system which uses the standard spherical coordinates r', phi', and theta', then the coordinate transform I wrote above could be rewritten as:

r&#039; = t
\phi&#039; = tan^{-1} (y/x)
\theta&#039; = \pi *\sqrt{x^2 + y^2} /R

The reason this works is that when converting from cartesian coordinates x,y to polar coordinates r,phi in flat space, the conversion would be:

r = \sqrt{x^2 + y^2}
\phi = tan^{-1} (y/x)

But if this isn't the mapping you want, please provide your own equations for mapping a point (x,y,t) in an inertial coordinate system to a point (r', phi', theta') in the spherical coordinate system for your onion diagram.

By the way, it might be simpler to reduce the number of spatial dimensions by one, so that we just have points (x,t) in an inertial coordinate system, and we want to map events that lie in some finite spatial interval 0 \leq x &lt; x_1 and which happen after t=0 onto a 2D "onion diagram" where each surface of simultaneity is just a circle, with points in this diagram being identified using polar coordinates r' and phi'. In this case we could use the simple mapping:

r&#039; = t
\phi&#039; = 2\pi * x / x_1

So, if a single surface of simultaneity in our inertial system (x,t) contained two 1D rods of unequal lengths, the ratio of their lengths would be equal to their arc-length on the circle of radius t.

Let me know if you want to use different mappings for either the 2D case or the 1D case or both, and if so, what mappings you'd choose. Once we have the mappings represented mathematically, we can go backwards and look at what a horizontal line in the "onion diagram" would correspond to in the original inertial coordinate system.

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.

<snip>you seem to be interested in mapping a finite region of flat space onto the sphere rather than an entire infinite region<snip>

I am not saying space is infinite. I am saying it is unbounded, while being finite and not being curved (in 3d) <snip> I don't think that SR relies on space being infinite, does it?

Note the following:

http://www.Newton.dep.anl.gov/askasci/ast99/ast99547.htm
http://cosmos.phy.tufts.edu/~zirbel/ast21/sciam/IsSpaceFinite.pdf
http://www.space.com/scienceastronomy/universe_soccer_031008.html (This is from 2003, so may be outdated.)

In any event, if I don't consider the universe to be infinite, does that invalidate my model? If it doesn't, do I have prove something that I don't hold to be true and isn't actually necessary?

I don't see any benefit in trying to rephrase what I have already said. It makes some sections of your post which followed redundant (since you say "Note that in the above transformation I just mapped a circular disc cut out of a single surface of simultaneity" which isn't what I am doing in my model).

But do you understand that any "explanation" that is coordinate-dependent cannot really be a physical explanation, but only a sort of aide to intuition?

Hm, well did I say my explanation is coordinate dependent? Is it coordinate dependent? Playing around with co-ordinates wasn't originally my idea, I was trying to oblige you. Here is what I was responding to:

We could equally well use a coordinate system where successively smaller spheres corresponded to later times, so that the coordinate length of objects was progressively shrinking.

Here is what you demanded earlier.

without some clearly specified mapping there is no way that your diagrams can be understood as sufficiently well-defined to be physically meaningful.

Damned if I do, damned if I don't?

I don't understand--are you proposing a geometrically different type of coordinate system than the spherical system shown on the wikipedia page (where varying r corresponds to varying the radius, varying phi corresponds to varying the longitude like moving in the east-west direction on a globe, and varying theta corresponds to varying the latitude like moving in the north-south direction on a globe), or are you still using the same type of spherical coordinates but just relabeling theta as r? If it's just a relabeling, I would rather use the standard notation for spherical coordinates, I already understand that varying theta in this coordinate system corresponds to varying the r coordinate in the polar coordinate system for flat space (as I discussed earlier in this post). Assume for the sake of the argument that you have the coordinates of events in an inertial coordinate system and you want your computer to graphically represent where these events would appear in terms of your 3D onion diagram, but the computer only knows how to plot 3D points using either 3D cartesian coordinates x',y',z' or 3D spherical coordinates r',phi',theta'.

I very much appreciate that you say that you don't understand and ask for clarification.

If in 2d you use polar coordinates r and phi, then you can label any position on a plane uniquely, with the assumption of a point at which r=0 and a line from that point along which phi=0. Correct?

Then after mapping to the surface of a 3d shape, you could abandon r and phi entirely and no longer measure distances from a point on the surface, but rather use R to measure distance from the centre of the shape (R=0) and two new angles, theta and squiggle. With the introduction of theta and squiggle you need two more lines out from R=0, one for which theta=0 and one for which squiggle=0. Those two line can run parallel if you like, which makes them indistinguishable, since they share at least one point (where R=0).

Alternatively, you could keep r and phi, and, so long as you are mapping to the surface of a sphere, you can then use a constant value of R to indicate that the positions you are identifying lie on the the surface of that sphere.

I understand that you normally use one r and as many angles as you need to uniquely locate a position, but you don't have to.

If you're using cartesian coordinates, those would usually be denoted x and y rather than phi and theta.

Yes, I know this. I would prefer to keep x and y too. But you seem to want to bring in polar coordinates. So what I am saying is that x and y have a direct correspondence with the angles they subtend from a point on the sphere's surface where x=y=0 and theta=squiggle=0 (initially I wrote x=y=theta=squiggle=0, but you would complain that this is wrong, which it is because the units don't match). Then you have a distance between the null point and the location being described, what the value of r is open to discussion, do we use arc length or chord length? For me it doesn't matter, since any length you measure will be measured in multiples and/or subdivisions of rulers, and as long as you don't try to measure chords with arc-rulers or arcs with chord-rulers, then you'll be ok.

So you can express x and y as r*sin(theta) and r*sin(squiggle). It's up to you.

But if this isn't the mapping you want, please provide your own equations for mapping a point (x,y,t) in an inertial coordinate system to a point (r', phi', theta') in the spherical coordinate system for your onion diagram.

Well, I did indicate that I didn't want to use r, phi and theta. I am happy to, if you want to.

I am not overly happy about using primed values of r, phi and theta. It is bound to muddy the waters in a forum about special and general relativity. Since you are proposing going from (x,y,t) to polar coordinates, what possible need is there to prime anything?

I don't like the use of primes before either when you were converting from what was effectively (r,phi,t) to something like (R,theta,squiggle).

Note further that you will confuse people here, the transformations are more accurately:

t,x,y -> R,theta,squiggle

and

t,r,phi -> R,theta,squiggle

since t is R (or if you want to be more precise, R is ct).

Let me know if you want to use different mappings for either the 2D case or the 1D case or both, and if so, what mappings you'd choose. Once we have the mappings represented mathematically, we can go backwards and look at what a horizontal line in the "onion diagram" would correspond to in the original inertial coordinate system.

The horizontal line was an approximation. Take a sufficiently small length in the universe and is approximates a tangential line. Being horizontal was a consequence of taking a tangent at the top of the circle. Please don't read too much into it being horizontal, since this is due most to limitations in the program I was using to create the diagram.

Use whatever mapping or coordinate system you want. It doesn't matter for me.

Just be aware that in my model a person living in flatland will experience a plane, not the curved surface of a sphere. Their standard ruler is of length L, and while for them it might lie flat, it could be thought of as being mapped to an arc but even so any two rulers which share the same rest frame will have the same length - irrespective of whether that ruler is flat or curved. (Shall I rephrase the "rest frame" part, or do you understand that I understand? - "if each ruler is at rest in the rest frame of the other ruler", is that good enough? Can we quit it with the semantics, both in this thread and in other threads?)

cheers,

neopolitan

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.


Hi belliott,

It is refreshing to engage in a proper discussion. I don't mind being told I am wrong by someone who understands what I am saying, and you are making a real effort to understand. Makes me feel worse about being "snippy" with you earlier.

I will try my best to address all the issues you have brought up so far.

I'm sure you know that the relativistic velocity addition formula gives you the answer. Do you object to that result, i.e. do you believe you have a more accurate way to predict the relative velocity between A and B, or are you offering an alternate method for deriving the same result, which method you find preferable for understanding "how it could be so"?

It's alternate derivation of the same thing. The thing that I find unsatisfying about the second postulate is that there is no explanation for why it should be so. My derivation gives an explanation (which is probably already there in standard derivations, just not so overtly).

Why can't anything go faster than the speed of light? Why this particular speed?

In my model, if you have a hypothetical tachyon which goes faster than the speed of light relative to an inertial observer (I don't say they exist, I am just using "a hypothetical tachyon" as a more useful substitute for the word "something") then that tachyon will zip away at a rate which is faster than the expansion of the universe. It will leave the universe, in my conception.

My two postulates would be the standard first postulate about laws of physics (physical processes, if you prefer) being invariant in all inertial frames and a postulate that the universe is expanding at some rate. This is why I used Q earlier, rather than c, for the rate of expansion of the universe. I can derive an equation with Q, the rate of the expansion of the universe, then what Q comes out to be can either be found by experiment or can be reasoned out. The fastest anything can move in our universe without leaving it, is Q. Nothing with mass can move as fast as Q (simplistically we could use the argument that you would need infinite energy to accelerate even the smallest mass to Q) but it could be so that massless energy not only could, but would, move at such a speed (again simplistically, because it can), relative to any observer at the instant when the observer, or the observer's instrumentation, and the massless energy interact. But all forms (frequencies) of such massless energy would be limited to this maximum universal speed (in a vaccuum). Do we have any examples of this? Yes, electromagnetic radiation, all of which travels at one speed.

Naturally I am a bit behind here, since we already know about the speed of light. But I didn't want to discover it, I just wanted to work out why.

Okay, I think it's worth pointing something out here. Yes, in our 3-d world we recognize that the paths taken by the ships are not straight lines, but are most likely arcs of great circles (assuming they're taking the shortest paths between start and finish points). From their point of view, however, they are moving in a 2-d world, where they need only two coordinates to specify their locations exactly, i.e. lat and lon. Moreover, as far as they are concerned, they are following straight paths, since those paths are the shortest distances between points. This is a perfectly valid point of view and makes sense for them. Moreover, they can do geometry in these coordinates. Specifically, they can do spherical geometry, which is most definitely the geometry of a curved space - which is no surprise to us, of course. The only point I want to make here is that in all likelihood they define their speed in terms of these coordinates. I'll return to this later.

I did consider mentioning global navigation, complete with a mention of Great Circles (lines of longitude, the equator and any circle you effectively would create by pointing yourself in one direction and walking around the globe). I thought it would cause a distraction. When I talked about how the ships are approaching each other, I meant how they are approaching each other in 3d, not the rate at which the section of the great line defined by their separation (which changes all the time) decreases.

To permit a slight deviation from the main topic, I do have experience with navigation. When plotting habour approaches, straight lines on a mercator are used and the pilot/navigator merely compensates to get the ship back on track when it strays. Currents and wind will actually play a bigger part in pushing the ship off track than any inaccuracy due to the mercator projection at the scale of the chart used in this situation. When planning a pilotage from one continent to another, blank charts are used and the navigator will plot a Great Circle. Why? Probably for a bunch of historical reasons, but today it is probably mostly because we can mass produce blank charts with vertical and horizontal lines, and label them with longitude and lattitude as required. If we wanted to use another projection, different charts would be required at different lattitudes. My experience is military where they still do chart their pilotages by hand. I understand that the merchant marine may well rely a lot more on automation of such mundane tasks.

Note that from the ships' perspectives, no distances are changing, since they use lat/lon to measure distances, and those do not change with a change in R. (Yes, the sizes of rigid objects like ships and rulers should not change, but that's a shortcoming of the analogy. If you want this to be comparable to the expansion of space-time, you should really have all points on the surface expanding uniformly, so that measuring rods grow, too - it's space that's expanding, and it takes matter along with it for the ride.

My model doesn't have rigid objects like ships and rulers, so you did indeed identify the shortcoming of the analogy. It is why I had the bolded sections.

Now, the ship captains will still notice that something has changed, since you are not doing anything to change the rates of their clocks. They will notice now that it takes them longer to get from one lat/lon location to another, so to them this will appear as an apparent decrease in their speed, despite their using the same amount of fuel, etc.

(This is in contrast to our expanding universe, where it is not just the spatial, but also the time axis that expands, thus our measurements of velocities don't change, except for the speed of light. We now see that it takes longer for light to reach us from Star X then it did in the past, which we interpret as expansion. We do not, however, notice that it takes longer for the Earth to revolve around the Sun, i.e the solar year is not getting longer.)

I think you are in line with me here. In the shortfall plagued analogy, the ship's captain would have problems if his ship didn't expand along with his fuel and the physics of that fuel. (I made the rash assumption that 10 kgs of expanded fuel will drive one expanded ship one expanded furlong at one expanded knot, if 10 kgs of unexpanded fuel drives one unexpanded ship one unexpanded furlong at one unexpanded knot. This is in line with my use of the first postulate. Being expanded makes no change to physical processes.)

The explanation for why it takes longer for light to get to us from Star X than it used to is not part of this SR only explanation. I can explain it and it is consistent, but it will have to come later.

This is an important point. If their speed limit of c is voluntarily imposed, i.e. they maintain their speeds such that they always observe it be less than c, then their maximum speed in our 3-d space increases, since their measurement of 1 ct unit grows along with their coordinate system.

If, on the other hand, you mean that their speed is physically limited to c in our 3-d sense, then yes, they will find that their maximum speed is getting slower and slower, perhaps reasoning that their universe is expanding, until they find themselves unable to "outrun" the expasion and therefore unable to reach their destinations.

I think that if you look at what I wrote before, you might want to revise this. I see simple speeds as invariant. I say simple because I am not thinking of looking at a third velocity from a second frame which is not at rest relative to me (ie relative to my rest frame). If my speed compared to you is v then your speed compared to me is also v. This invariance is not affected by expansion.

Now we can apply this to the whole universe and the two spaceships, two torches scenario.

If the whole universe is expanding, and the spatially flat universe is mapped onto a hypersphere in flat spacetime, and objects in the universe follow straight lines in spacetime, then the only way to have two objects move in the same plane is to have them share the same velocity (speed and direction).

Okay, this is where I think the big problem resides, and I'm pretty sure my objection here is equivalent to JesseM's. I don't believe what you just stated makes mathematical sense. If you take a flat space (i.e. a Euclidean space, a space with the N-d identity matrix for a metric, however you want to describe it) and map it to a hypersphere, the resulting space is either not a vector space at all (e.g. a projection, as JesseM suggested), or it is a vector space with curvature > 0. There's just no way around it. It's as if you said, take a spatially flat 2-d space and map it to the surface of a globe (i.e. a 2-sphere) - the target space is curved, for the same reason that you can't wrap a globe with a flat piece of paper without wrinkling it.

You could do the projection trick, but then you get distortions in distances (like with the Mercator projection mentioned earlier, except in the other direction, i.e. flat space to spherical), and it is not a metric space. That means that your discussions of distances and velocities all go out the door.

Ok, here I was attempting too much too quickly. So I will go back a bit.

Flatme stands on the surface of his sphere of simultaneity and looks around in flat 2d space. The plane I am thinking of is tangential to the surface. I don't see any of the stuff that is around me of course, since everything on that plane (apart from me) is what I will see in the future. What I see is the stuff from the past that is only now reaching me.

Say another inertial flatlander comes traveling towards Flatme (Flatyou). That inertial flatlander has his own frame with his own sphere of simultaneity and his own tangential plane. When those two flatlanders are coincident, the planes will be at an angle.

In my conception, the relative motion of Flatyou, relative to Flatme, is in the (spacetime) direction of his tangential plane.

Now, if you introduce a third flatlander, Flatfred, also in motion relative to Flatme and additionally in motion relative to Flatyou. Flatfred will have his own tangential plane, and will move in the (spacetime) direction that tangential plan, relative to Flatme.

The angles between those tangential frames will be where the relative speed limitation comes in. I am not sure this makes things easier. But hopefully you can nut it out.

About mapping flat paper onto a globe. I still think this is a two spacelike to three spacelike projection which is not equivalent to a three spacelike to three spacelike and one timelike projection. I can't point to the future. (I will have to give serious though to refuting this concern since it comes up time and time again.

All of this, by appealing to geometry, i.e. angles between vectors and so forth, implies that this space is a physical space, which means it must be curved if it in fact resides on the surface of a hypersphere.

I have one further problem, which is that you've essentially swept under the rug the fact that your ships-on-a-globe analogy is constructed entirely with spatial coordinates. You haven't really addressed how the time axis in space-time gets mapped in your hypersphere picture. That's clearly critical, but I'll leave it for now, since I think you have more fundamental hurdles to cross first, i.e. how exactly you're doing this mapping. I think what you need is an explicit recipe for going from point (t,x,y,z) to point (r,a1 a2,a3), where the a's are the angles - or other coordinates - that you choose to parametrize your 3-sphere.

I'm afraid that I'm suspicious that your approach, whatever its aesthetic appeal might be, is not going to turn out to be mathematically viable. It wouldn't be the first appealing physical theory to suffer this fate, so you'd be in good company. You really have to work out the details to find out if it will survive, though.

[I'm deferring any other comments until later, since I think these issues need to be addressed first.]

You said "I think what you need is an explicit recipe for going from point (t,x,y,z) to point (r,a1 a2,a3), where the a's are the angles - or other coordinates - that you choose to parametrize your 3-sphere".

r is t, or rather delta-r is delta-t (since we always measure time from a nominally null-point, not from any absolute zero). I agree that conceptually x,y and z are effectively measurements of angles, such that a ruler along the x-axis of length x could be said to subtend r*sin (arcsin (x/(2pi*r)) so you could use:

(ct,x,y,z)->(r=ct,arcsin (x/(2pi*r),arcsin (y/(2pi*r),arcsin (z/(2pi*r))

I do think it is just as simple to use ct, x,y and z, though.

I don't know how clear I can make the fact that I don't see there being a unitary time axis. The only way you could validly talk about there being a unitary time axis would be as an approximation where you cut out a section of the surface hypersphere which is sufficiently small to be approximately flat in terms of 3 dimensions as well as in terms of 4 dimensions. If this section was circular it would be basically the same as JesseM's American football model.

Just as you can't pick one direction and say "that is the third dimension", you can't find anyone direction which is universally parallel with the time axis. The closest you can possibly get to saying this is (in terms of my model) to say that I experience the passage time as being perpendicular to all spatial dimensions when and where I am. And that isn't particularly helpful anyway.

That will have to do for today.

cheers,

neopolitan

 

[JesseM]

<snip>you seem to be interested in mapping a finite region of flat space onto the sphere rather than an entire infinite region<snip>

I am not saying space is infinite. I am saying it is unbounded, while being finite and not being curved (in 3d) <snip> I don't think that SR relies on space being infinite, does it?

Note the following:

http://www.Newton.dep.anl.gov/askasc...9/ast99547.htm
http://cosmos.phy.tufts.edu/~zirbel/...paceFinite.pdf
http://www.space.com/scienceastronom...er_031008.html (This is from 2003, so may be outdated.)

In any event, if I don't consider the universe to be infinite, does that invalidate my model? If it doesn't, do I have prove something that I don't hold to be true and isn't actually necessary?

I don't see any benefit in trying to rephrase what I have already said. It makes some sections of your post which followed redundant (since you say "Note that in the above transformation I just mapped a circular disc cut out of a single surface of simultaneity" which isn't what I am doing in my model).

I understood that you were talking about a finite flat space--but were you not talking about mapping this finite flat space onto a sphere? Mapping a disc in flat space onto a sphere is the only way I could think of to ensure that two line segments along the radial direction would map to two arcs on the sphere in such a way that the ratio between lengths would be equal to the ratio between arc-lengths. By the way, note that you don't actually have to assume that the finite region is disc-shaped, only that the disc contains the finite region--remember that I mentioned earlier that flat space can be finite if you pick some region with edges like a square, and map the edges to each other, like the asteroids video game. This is in fact the only way that space can be both finite and flat, and it's what's being discussed in the second two of the three links you posted above. In this case, one can model this by taking the infinite flat space assumed by SR and filling it with a quilt of interlocking copies of the same finite region. Look again at the article I posted earlier, specifically the paragraph that begins 'Alternatively, we can visualize the the compact space by gluing together identical copies of the fundamental cell edge-to-edge' (you could also take a look at http://www.etsu.edu/physics/etsuobs/starprty/120598bg/section7.htm which pictures the CMBR sphere as possibly being larger than a finite cube-shaped universe). So in this case the same point in space will have multiple sets of coordinates, and if you take a disc that contains the finite square-shaped region, it will also contain multiple copies of certain points in space, but it will contain every point in your finite region at least once.

But look, if you want some other mapping onto the sphere that preserves the length-in-flat-space-to-arc-length-on-sphere ratios, that's fine--just give me the specific equations, if you aren't using specific equations then your onion diagrams are indeed too ill-defined to be meaningful.

Hm, well did I say my explanation is coordinate dependent? Is it coordinate dependent? Playing around with co-ordinates wasn't originally my idea, I was trying to oblige you. Here is what I was responding to:

We could equally well use a coordinate system where successively smaller spheres corresponded to later times, so that the coordinate length of objects was progressively shrinking.

Here is what you demanded earlier.

without some clearly specified mapping there is no way that your diagrams can be understood as sufficiently well-defined to be physically meaningful.

Damned if I do, damned if I don't?

Huh? How are you "damned if you do"? Do you consider it "damning" for me to say that your onion diagrams just represent a remapping of flat space (i.e. a coordinate change) rather than actual physical curvature? Or do you imagine there is some third alternative beyond either 1) space being genuinely curved, or 2) space being flat but being represented as a curved sphere due to a coordinate shift? If you think there's a third alternative, I suspect that once again the problem is that you think and argue in vague verbal terms which don't correspond to any well-defined mathematical ideas, like your statement eariler that "I am thinking of flat space which has been wrapped around a hypersphere so the whole of it is curved, but only in terms of 4 dimensions, not in terms of 3dimensions. I have said that a few times." There is simply no physical sense in which it is meaningful to say that space is flat, spacetime is flat, but space is "curved in terms of 4 dimensions"--the only way I can interpret a statement like this is as a statement about a coordinate representation where flat spatial surfaces of simultaneity from a flat spacetime appear curved. But if "curvature" can't be represented in intrinsic differential-geometry terms using a line element as I discussed in post #194, if it only appears in an embedding diagram of curved space or spacetime, then it simply cannot correspond to anything that can actually be physically measured.

So we really need to be clear on this. If you think that both space and spacetime can be physically flat, and yet your onion-diagrams are supposed to represent a physical reality that goes beyond just a coordinate remapping of flat surfaces of simultaneity, then I think you're just confused about the relationship between visual diagrams and actual mathematical physics. If you disagree, then you need to explain what the curvature is supposed to represent using mathematics, not just fuzzy english phrases that don't mean anything to me (or anyone else reading this thread, I'd wager) like "flat space which has been wrapped around a hypersphere so the whole of it is curved, but only in terms of 4 dimensions, not in terms of 3dimensions".

I don't understand--are you proposing a geometrically different type of coordinate system than the spherical system shown on the wikipedia page (where varying r corresponds to varying the radius, varying phi corresponds to varying the longitude like moving in the east-west direction on a globe, and varying theta corresponds to varying the latitude like moving in the north-south direction on a globe), or are you still using the same type of spherical coordinates but just relabeling theta as r? If it's just a relabeling, I would rather use the standard notation for spherical coordinates, I already understand that varying theta in this coordinate system corresponds to varying the r coordinate in the polar coordinate system for flat space (as I discussed earlier in this post). Assume for the sake of the argument that you have the coordinates of events in an inertial coordinate system and you want your computer to graphically represent where these events would appear in terms of your 3D onion diagram, but the computer only knows how to plot 3D points using either 3D cartesian coordinates x',y',z' or 3D spherical coordinates r',phi',theta'.

I very much appreciate that you say that you don't understand and ask for clarification.

If in 2d you use polar coordinates r and phi, then you can label any position on a plane uniquely, with the assumption of a point at which r=0 and a line from that point along which phi=0. Correct?

Sure, then any point can be labeled by its distance from the origin r, and the angle between the line from that point to the origin and the phi=0 line.

Then after mapping to the surface of a 3d shape, you could abandon r and phi entirely and no longer measure distances from a point on the surface, but rather use R to measure distance from the centre of the shape (R=0) and two new angles, theta and squiggle. With the introduction of theta and squiggle you need two more lines out from R=0, one for which theta=0 and one for which squiggle=0. Those two line can run parallel if you like, which makes them indistinguishable, since they share at least one point (where R=0).

Why are you using "squiggle" rather than phi? And I don't understand what you mean when you say that lines for the two angular coordinates can "run parallel". The way I'd describe it is that you define coordinates using a point R=0, a plane which contains the phi = 0 axis but which also corresponds to theta = pi/2 in radians (or 90 degrees), and then a theta = 0 axis. Then for an arbitrary point in space, you find its distance from the origin R, and imagine a sphere of radius R centered on R=0 and containing that point in space, with the "equator" of the sphere being where it intersects the theta = pi/2 plane, and the "poles" of the sphere being where it intersects the theta = 0 axis. Then you can draw curved lines of longitude latitude on the sphere (with one of the 'lines of latitude' being the equator), with the point at which the phi = 0 axis intersects the equator corresponding to 0 longitude. On any given line of latitude, longitude varies from 0 to 2pi as you travel around it, and meanwhile latitude varies from 0 to pi as you travel along a line of longitude from the "north pole" to the "south pole". This gives you a clear way of identifying the coordinates of any point on the surface of a sphere with a given radius R (the diagram at the top of http://www.stuif.com/confluencing.html may be helpful in visualizing this).

Are suggesting some different 3D coordinate system that conceptually does not correspond to the sort of visual picture I describe above, namely finding a spherical surface that contains a point at a certain radius, and then identifying the angular distance along "latitude" and "longitude" lines from a point of 0 latitude (the sphere's 'north pole') and a curved line of 0 longitude? If you are trying to suggest a totally different type of 3D coordinate system, then please be more specific, your above comments about "squiggle" and "thata" lines don't really make clear how you want to identify the coordinates of arbitrary points in 3D space.

Alternatively, you could keep r and phi, and, so long as you are mapping to the surface of a sphere, you can then use a constant value of R to indicate that the positions you are identifying lie on the the surface of that sphere.

Do you mean use a type of coordinate system where r corresponds to the distance along a geodesic from some origin point on the sphere to the point you're interested in, and phi is the angle of the geodesic from the origin to that point relative to some other geodesic line on the sphere's surface? Note that if we take the origin point r=0 to be the "north pole" of our sphere, then any geodesic from the north pole to some other point will be a line of longitude, and we could take the phi coordinate to be the angle of the line of longitude from the pole to our point relative to the line of 0 longitude. In this case the coordinate system you're describing would be almost identical to spherical coordinates, except instead of a coordinate measuring the angular distance theta from the pole to your point along a line of longitude, you're measuring the actual distance to the point along that same line of longitude, which would just be theta times the radius of the sphere R.

If you're using cartesian coordinates, those would usually be denoted x and y rather than phi and theta.

Yes, I know this. I would prefer to keep x and y too. But you seem to want to bring in polar coordinates.

No, it doesn't matter to me what coordinates you use for the inertial system, cartesian coordinates might be a little clearer although they make the coordinate transform a little more complicated.

So what I am saying is that x and y have a direct correspondence with the angles they subtend from a point on the sphere's surface where x=y=0 and theta=squiggle=0

x=y=0 would correspond to the north pole of the sphere (theta=0, phi=irrelevant since it won't change the point we're talking about when theta=0, just like the longitude of the north pole is irrelevant) in the cartesian-to-polar transformation I gave earlier:

R = t
\phi = tan^{-1} (y/x)
\theta = \pi *\sqrt{x^2 + y^2} /R

(note that the distance of a point x,y from the origin in flat space is just \sqrt{x^2 + y^2}, so I have made this proportional to the angular distance from the 'north pole' along a line of longitude; likewise, note that in flat space the angle between a line from the origin to a point x,y and the y=0 line is tan^{-1} (y/x), so I've made this proportional to the longitude)

Then you have a distance between the null point and the location being described, what the value of r is open to discussion, do we use arc length or chord length?

If you use arc length, then again, it seems to me that what you are calling r is identical to what I am calling theta, and that what you're saying is not meaningfully different from how spherical coordinates always work. Are you sure you're clear on how spherical coordinates are supposed to work?

So you can express x and y as r*sin(theta) and r*sin(squiggle). It's up to you.

Where are you getting these equations? Are you suggesting a new mapping for points in the x-y plane onto the sphere, different from the one I give above? Or are you talking about switching between spherical coordinates r, theta, phi (I'm going to assume this is what you meant by squiggle, I wish you wouldn't introduce new terms without defining them) to cartesian coordinates x,y,z in the same 3D space? The 3D cartesian to 3D spherical transformation is given here if you're interested, x = r*sin(theta) would be true when phi=0 (on the equator), and y=r*sin(phi) would be true when theta = pi/2 radians or 90 degrees (on a line of longitude at right angles to the line of 0 longitude), but these aren't correct in general. On the other hand, if you're suggesting a new mapping between x and y coordinates in flat space and coordinates theta and phi on a sphere of radius r, different from the one I gave above, your mapping would be problematic because it would no longer be true that if you have two line segments on the x-axis or the y-axis (or any other radial axis), then the ratio between their lengths in 2D would be equal to the ration between the arc-lengths of the mapped points on the sphere. For example, if one mapped line extended from theta = 30 to theta = 60 degrees, and another extended from theta = 90 to theta = 120 degrees, they'd both have the same arc length, but sin(60) - sin(30) = 0.366 and sin(90) - sin(120) = 0.134.

Well, I did indicate that I didn't want to use r, phi and theta. I am happy to, if you want to.

I am not overly happy about using primed values of r, phi and theta. It is bound to muddy the waters in a forum about special and general relativity. Since you are proposing going from (x,y,t) to polar coordinates, what possible need is there to prime anything?

My original reason was that I was identifying points in flat space identified in polar coordinates r and phi, and since spherical coordinates also have an r and a phi, that could be confusing. But if you want to describe the original flat space using x and y, then no problem, we can then use unprimed R, phi, theta for spherical coordinates without confusion.

The horizontal line was an approximation. Take a sufficiently small length in the universe and is approximates a tangential line. Being horizontal was a consequence of taking a tangent at the top of the circle. Please don't read too much into it being horizontal, since this is due most to limitations in the program I was using to create the diagram.

Use whatever mapping or coordinate system you want. It doesn't matter for me.

But obviously it does matter somewhat, because you want the mapping to have certain properties like the ratio between lengths in flat space being proportional to the length of arc-lengths in the onion diagram. Can we at least agree on a mapping for the simplest 1D case where our original inertial system just has x and t coordinates, and space has a finite topology so it only extends from x=0 to x=x_1 before repeating? In this case, what do you think of the simple mapping into polar coordinates for the onion diagram that I proposed?

r = t
\phi = 2\pi * x / x_1

(perhaps it would be better to write r = ct so the units work out)

You can see that the ratio of lengths in the inertial coordinate system to arc-lengths on the circles will work with this transformation--for example, a line extending 1/4 of the way from 0 to x_1 will also extend 1/4 of the way around the circle from phi = 0 to phi = pi/2, and a line exactly filling up the whole space from 0 to x_1 will exactly extend around a full circle from phi = 0 to phi = 2pi (remember that 2pi radians = 360 degrees).

Just be aware that in my model a person living in flatland will experience a plane, not the curved surface of a sphere. Their standard ruler is of length L, and while for them it might lie flat, it could be thought of as being mapped to an arc but even so any two rulers which share the same rest frame will have the same length - irrespective of whether that ruler is flat or curved. (Shall I rephrase the "rest frame" part, or do you understand that I understand? - "if each ruler is at rest in the rest frame of the other ruler", is that good enough? Can we quit it with the semantics, both in this thread and in other threads?)

I never objected to your talking about the rest frame of a particular object like a ruler, what I objected to was your talking as if it is standard procedure when approaching a relativity problem to select one frame as "the" rest frame for that particular problem. These are conceptually distinct notions, the first is about the rest frames for objects being analyzed in the problem, the second is about the guy who's analyzing the problem picking some frame which he will call "nominally at rest" (to use your phrase from previous posts) throughout the course of the problem. The second is not something that is actually normally done when approaching problems in relativity, normally we could in the course of a single problem talk about how things work in the rest frame of object A and how they work in the rest frame of object B without defining either as "nominally at rest".

 

[neopolitan]

Hi JesseM,

I don't think we are getting anywhere with these walls of text. In an effort to be open and honest, I admit that I am getting irritated and I accept that this may be causing me to interpret your attempts to help as slights - such as when you pointed out that you were using radians which was obvious to me. I have tried to avoid irritating you in the same way, but probably failed in the opposite direction because I don't make explicit that which I feel is obvious. I work on the basis that you fully understand polar coordinates and other possible coordinate systems. You (seem to) work on the basis that I don't have a clue. Perhaps your approach is ultimately better, since it should avoid confusion. It hasn't so far though.

Can we address the coordinates issue and be done with it? I will explain my understanding. I am not teaching you, I am hopefully showing you that I understand not just the squiggles on paper (equations) but what they represent.

Coordinates in one dimension - you have a line and a reference point. You can describe any point on the line as a separation from that reference point. Let's call that separation both r and x, such that r=x. Note that the line now has two directions, since the line is bisected by the reference point, so r and x being positive means the separation is on direction and r and x being negative means the separation is in the other direction.

Coordinates in two dimensions - you have a plane, a reference point, and two directions now. Here you have a choice between a variation of cartesian coordinates or polar coordinates. I say a variation of cartesian coordinates because we are so used to (x,y) notation that we assume that our axes have to be at 90 degrees [pi/2 radians], but they don't. It's not impossible to have a coordinate system which has two axes which are separated by another angle (excepting where the angle is either 0 or 180 degrees [0 or pi radians]) and it is not invalid to do so. It's just damn inconvenient. So damn inconvenient that I didn't initially think it was worth pointing out the possibility.

So let's use standard cartesian coordinates, (x,y) where the x- and y-axes are orthogonal. (0,0) is your reference point, and your directions are x>0 and y>0. In same way as above, we really have two directions per axis, positive and negative, which we handle the same way.

Alternatively we could use polar coordinates, which are in a way much simpler and more flexible. You have a reference point, r=0, a linear direction (r>0,phi=0) and an angular direction (clockwise or anticlockwise, with one corresponding to phi>0 and the other corresponding to phi<0). Yet again you can use negatives if you like, but in polar coordinates it is not necessary. A "problem" with polar coordinates is that can express a single location in different ways, for example (r,0)=(-r,pi)=(-r,-pi)=(r,2pi). Also the reference point is not unique (0,0)=(0,phi) where phi can take any value you like. In cartesian coordinates, the reference point (0,0) is nothing but (0,0).

Coordinates in three dimensions - we now have three options.

"Pure" cartesian coordinates (x,y,z) where the three axes are orthogonal (but don't have to be, it is just more convenient that way). I don't want to labour something so simple, but I do want to point out that if all the points you wanted to discuss were in the same plane, for example, you could choose your x-, y- and z-axes such that z=0 for all three points and treat the points as if they were in two dimensions. This is equivalent to some spatial simultaneity at z=0. If you don't like that terminology, swap out z for t.

"Pure" polar coordinates (R,theta,squiggle) where you have a reference point (R=0) and two linear directions (R>0,theta=0) and (R>0,squiggle=0) and two angular directions (clockwise or anticlockwise, in two planes). Now, squiggle is an angle which has nothing to do with theta or phi. I thought that would be obvious in context, if a little irreverent, but I make it explicit to avoid confusion.

Note that I say that the two directions are in two planes but do not stipulate that they have to be orthogonal. Again, it is not impossible to have a coordinate system in which those planes are not orthogonal, it's not invalid, it's just inconvenient. I assumed we would use convenient directions but I will explicit again. If I am going to describe locations in polar coordinates, I will use two "reference planes" for theta and squiggle (or theta and "some other greek letter which not phi in order to avoid confusion, but still maintain the convention of using greek letters even though it really has nothing, absolutely nothing, to do with the fundamental mathematics") which are orthogonal.

I wish to point out that there is no automatic correspondence between r and R. For there to be a correspondence you have to accept a limitation to your selection of the reference point r=0. This might not be clear, so I will try to explain.

Pick a plane, any plane. Ok, if you have done that, does your plane include the reference point R=0? If, so that is a special case, pick another plane.

Now try to eliminate one of the coordinate values, as we did in the cartesian coordinate example above, by choosing our axes so that z=0. You can't do it. You can only do it if the plane includes the reference point R=0 and you align the plane related to one of the angular directions so that is lies along the plane you selected. And that is a special case and in that special case r=R - if and only if you select a reference point such that r=R=0. You don't have to do that, although it is certainly convenient if you do.

Bastardised blend of cartesian and polar coordinates - (x,y,doodle) where you have two orthogonal axes (x,y), a linear direction (doodle=0 and for convenience either x>0 or y>0, but really it doesn't matter if you choose any direction on the x,y plane) and an angular direction (clockwise or anticlockwise, with one corresponding to doodle>0 and the other corresponding to doodle<0). If you think of "pure" polar coordinates in three dimensions as corresponding to a sphere (the value of R changes the size of the sphere and the angles move your pointer around on the surface of the sphere to point to the location you want to identify), this system corresponds to a cylinder where you use the angle to swing the x,y plane so that the location you want to indentify is on that plane, and then you just use cartesian coordinates the finalise the location (it might be easier to think of x giving you the radius of the cylinder, y giving you the height of the cylinder and doodle giving you how far around the cylinder you have to go).

(Why am I using "doodle"? Because there is not sufficient correspondence between phi, theta, squiggle and "doodle" to justify reusing the same term and putting a prime on it. It also amuses me. Oh, and "doodle" is an angle.)

Bastardised version of polar coordinates - (R,r,splodge) where you have a reference point (R=0), an axis (R>0, r=0), a linear direction (orthogonal to the axis, r>0, splodge=0) and an angular direction (splodge=0). Effectively you end up with the same as above even if here I have more of an image of an upside down L with adjustable lengths and being swung around the vertical axis, rather than a map being swung around a similar vertical axis.

Now, I know this does nothing to answer the mapping question. But can you accept that the problem does not lie in my not knowing what polar coordinates are about?

And if so, can we move on?

cheers,

neopolitan

 

[neopolitan]

In an attempt to move on I ask this question:

Say I am inertial such that I could refer to a frame in which I am at rest and there are a few other things at rest in that frame in which I am at rest.

Say I measure the distance between myself and an ancient, highly durable artifact at rest in the frame in which I am at rest. Say that distance is 10m.

Note that I never specified when I measured the distance.

What is the spatial distance between me today and that ancient, highly durable artifact 10,000 years ago (noting that we are both at rest relative to each other and assuming that has always been the case)?

I think it is either 10m or approximately 95x10^15 kilometres. It all depends on whether you can think that space is flat in 3+1 dimensions or not. I think it is, so I prefer the first option. But I can understand the other answer also (oh alright, let's just call it a nice round 10,000 lightyears to make it easier to comprehend) - but I don't think it is a purely spatial distance.

cheers,

neopolitan

 

[JesseM]

Now, I know this does nothing to answer the mapping question. But can you accept that the problem does not lie in my not knowing what polar coordinates are about?

And if so, can we move on?

I'll respond at greater length later, but just to be clear, would you agree that none of the 3D coordinate systems you described above correspond to the spherical coordinates that I have been using?

Also, would it be fair to say that what you refer to above as a "Bastardised version of polar coordinates" is really identical to what are normally referred to as cylindrical coordinates? Part of the reason I am inclined to explain certain things is because you sometimes don't seem familiar with standard terminology in math and physics, which makes me less confident that you'll understand what I mean when I use that terminology without explaining the meaning in detail. For me it also makes communication difficult when you introduce your own idiosyncratic terminology without explaining in detail what you mean by it, as you did with "squiggle" in a previous post--since I had been talking about spherical coordinates, I thought you were too, and were just using squiggle instead of phi to avoid confusion with the phi I'd been using to describe the angular coordinate in 2D polar coordinates.

 

[neopolitan]

I'll respond at greater length later, but just to be clear, would you agree that none of the 3D coordinate systems you described above correspond to the spherical coordinates that I have been using?

Also, would it be fair to say that what you refer to above as a "Bastardised version of polar coordinates" is really identical to what are normally referred to as cylindrical coordinates? Part of the reason I am inclined to explain certain things is because you sometimes don't seem familiar with standard terminology in math and physics, which makes me less confident that you'll understand what I mean when I use that terminology without explaining the meaning in detail. For me it also makes communication difficult when you introduce your own idiosyncratic terminology without explaining in detail what you mean by it, as you did with "squiggle" in a previous post--since I had been talking about spherical coordinates, I thought you were too, and were just using squiggle instead of phi to avoid confusion with the phi I'd been using to describe the angular coordinate in 2D polar coordinates.

Actually, I think the "pure" polar coordinate system I describe is essentially identical to the http://en.wikipedia.org/wiki/Spherical_coordinates" . If there is something I have added or something crucial that I have omitted, I can't see it.

Yes, the "bastardised version of polar coordinates" is really cylindrical coordinates, but so is the "bastardised blend of cartesian and polar coordinates". They are just different conceptualisations of the same thing.

Yes, I admit to being idiosyncratic. I just assumed it was bleeding obvious what "squiggle" was. It seems you worked it out so I wasn't far off being right.

Can we be clear that you understand that I used this terminology in order to make bleeding obvious the fact that the phi you used in 2D polar coordinates is not the same as the phi you introduced for 3D polar coordinates? And that I used r and R distinctly whereas you at one point wrote " r'=R' " when setting up a transformation for mapping 2D polar coordinates to 3D polar coordinates (a transformation which seems to assume that there is a prexisting 2D shape being mapped and so is not consistent with what I have in mind).

I am still unclear as to whether space has to be infinite under SR. If you have already said and I didn't pick it up, I apologise.

cheers,

neopolitan

 

[belliott4488]

Hi, guys ... I haven't responded to Neopolitan's last response to me because he and JesseM are so deep into a rigorous discussion that I'd rather see how that plays out than to add distractions.

Overall, I think my position is very similar to JesseM's, so I'm more than content to let him guide the discussion to a conclusion. If I still have to something to add then, I will.

In the mean time - carry on! I'm curious to see where all this will lead. ;-)

 

[JesseM]

Actually, I think the "pure" polar coordinate system I describe is essentially identical to the http://en.wikipedia.org/wiki/Spherical_coordinates" . If there is something I have added or something crucial that I have omitted, I can't see it.

I didn't actually find your description of "pure polar coordinates" very clear, but since you didn't distinguish between the two planes theta and squiggle, I assumed you were having them work the same way. What I would guess is that if we take the two planes to be orthogonal, and both contain the point R = 0, then if we want to assign a theta coordinate to a point that doesn't actually lie in the original theta-plane that contains R=0, we just move the theta plane along a direction orthogonal to it, until the plane contains the point; then we just assign an angle to the point in the plane in the usual way, in terms of the angular difference between some reference line theta=0 and the line from the point to the central point (which was formerly R=0 before we moved the plane). Then, I imagine that if the point doesn't lie in the squiggle-plane which contains R=0, we move that plane in a direction orthogonal to itself in the same way, until it does contain the point, and assign a squiggle angle to it in the same way. Finally, the R of the point is just the distance between the point and R=0.

If this isn't how you are imagining assigning coordinates to a point, please elaborate, using the type of explanation I give above which tells us how we move or reorient a plane so that it contains the point we're interested in and we can assign the point an angle using the usual polar coordinate method.

Yes, the "bastardised version of polar coordinates" is really cylindrical coordinates, but so is the "bastardised blend of cartesian and polar coordinates". They are just different conceptualisations of the same thing.

OK, I was picturing the "bastardised blend of cartesian and polar coordinates" a little differently, but now I think I see what you mean. Correct me if I'm wrong, but I think you're saying we have an xy plane, we have an orthoganal doodle plane, and say for the sake of simplicity we can say that the central point of the polar doodle coordinates is the same as x=0 and y=0, and that the doodle plane intersects the xy plane along the x-axis, and this also corresponds to the doodle = 0 axis in the doodle plane. In this case, if our point already lies in the xy plane, we assign it x and y coordinates in the usual way, and doodle=0; but if it doesn't lie in the xy plane, we rotate the xy plane about the y-axis until it does contain our point, and the angle we had to rotate it from its original orientation is doodle, and then we assign it x and y coordinates in the usual way. This then is just like R, r, splodge, with splodge = doodle, and R = x, and r = y; it's also like the cylindrical coordinates r, theta, z shown here, with their theta corresponding to your splodge/doodle, their r corresponding to your R/x, and their z corresponding to your r/y.

Yes, I admit to being idiosyncratic. I just assumed it was bleeding obvious what "squiggle" was. It seems you worked it out so I wasn't far off being right.

Well, I thought originally that squiggle was just the phi in spherical coordinates, but then your description of "pure polar coordinates" above seems different from spherical coordinates, though I may be interpreting it incorrectly. Please clarify whether my picture of how the planes need to be moved to contain the point we're assigning coordinates to is correct.

I am still unclear as to whether space has to be infinite under SR. If you have already said and I didn't pick it up, I apologise.

No, it can be finite, that's what I was talking about with the stuff about topology and space being like an "asteroids" game. As I said though, a finite space can also be described using the ordinary coordinates of SR going from -infinity to +infinity by taking the finite region and using it to tile an infinite space (the finite region must be a shape that it can be used to tile an infinite flat space, like a square or triangle in 2D, or a cube in 3D) so that all objects just repeat like a hall of mirrors; this will just mean that the same points of the flat space gets assigned multiple sets of coordinates, but you can apply all the standard rules of SR to this hall-of-mirrors universe.

Here was what I wrote about this earlier in post #268, if you haven't looked at the links I posted I recommend at least looking at the first one:

Mapping a disc in flat space onto a sphere is the only way I could think of to ensure that two line segments along the radial direction would map to two arcs on the sphere in such a way that the ratio between lengths would be equal to the ratio between arc-lengths. By the way, note that you don't actually have to assume that the finite region is disc-shaped, only that the disc contains the finite region--remember that I mentioned earlier that flat space can be finite if you pick some region with edges like a square, and map the edges to each other, like the asteroids video game. This is in fact the only way that space can be both finite and flat, and it's what's being discussed in the second two of the three links you posted above. In this case, one can model this by taking the infinite flat space assumed by SR and filling it with a quilt of interlocking copies of the same finite region. Look again at the article I posted earlier, specifically the paragraph that begins 'Alternatively, we can visualize the the compact space by gluing together identical copies of the fundamental cell edge-to-edge' (you could also take a look at http://www.etsu.edu/physics/etsuobs/starprty/120598bg/section7.htm which pictures the CMBR sphere as possibly being larger than a finite cube-shaped universe). So in this case the same point in space will have multiple sets of coordinates, and if you take a disc that contains the finite square-shaped region, it will also contain multiple copies of certain points in space, but it will contain every point in your finite region at least once.

You can also see this article on finite universes with unusual topologies that I linked to in post #246.

 

[neopolitan]

I didn't actually find your description of "pure polar coordinates" very clear, but since you didn't distinguish between the two planes theta and squiggle, I assumed you were having them work the same way. What I would guess is that if we take the two planes to be orthogonal, and both contain the point R = 0, then if we want to assign a theta coordinate to a point that doesn't actually lie in the original theta-plane that contains R=0, we just move the theta plane along a direction orthogonal to it, until the plane contains the point; then we just assign an angle to the point in the plane in the usual way, in terms of the angular difference between some reference line theta=0 and the line from the point to the central point (which was formerly R=0 before we moved the plane). Then, I imagine that if the point doesn't lie in the squiggle-plane which contains R=0, we move that plane in a direction orthogonal to itself in the same way, until it does contain the point, and assign a squiggle angle to it in the same way. Finally, the R of the point is just the distance between the point and R=0.

If this isn't how you are imagining assigning coordinates to a point, please elaborate, using the type of explanation I give above which tells us how we move or reorient a plane so that it contains the point we're interested in and we can assign the point an angle using the usual polar coordinate method.

I do think we are thinking the same thing but just have different aspects that we hold to be more important. For instance, you hold the planes which theta and squiggle vary (so one in which theta is variable and squiggle is fixed and one in which theta is fixed and squiggle is variable) to be very important. For me, the null points and null angles are more important.

As far as I can tell you and I are both describing spherical coordinates in our own ways (and I accept that your way is most likely the standard way). Perhaps I take the term "spherical coordinates" too literally, since I see it as expanding out the surface of conceptual sphere until it contains the location we want to describe (thus setting R) then swinging a pointer around to the location. That pointer will then be at an angle theta from the axis in one plane and an angle squiggle from the same axis in another plane (thus setting theta and squiggle in one fell swoop). But the point is that you can do it in which order you feel more comfortable with. You can set the angles first and then R (as you did) - or one angle, then R and then the other angle - or a variation of what I did but do my second step in two phases with planes (in a manner similar to what you did). The end result is the same.

OK, I was picturing the "bastardised blend of cartesian and polar coordinates" a little differently, but now I think I see what you mean. Correct me if I'm wrong, but I think you're saying we have an xy plane, we have an orthoganal doodle plane, and say for the sake of simplicity we can say that the central point of the polar doodle coordinates is the same as x=0 and y=0, and that the doodle plane intersects the xy plane along the x-axis, and this also corresponds to the doodle = 0 axis in the doodle plane. In this case, if our point already lies in the xy plane, we assign it x and y coordinates in the usual way, and doodle=0; but if it doesn't lie in the xy plane, we rotate the xy plane about the y-axis until it does contain our point, and the angle we had to rotate it from its original orientation is doodle, and then we assign it x and y coordinates in the usual way. This then is just like R, r, splodge, with splodge = doodle, and R = x, and r = y; it's also like the cylindrical coordinates r, theta, z shown here, with their theta corresponding to your splodge/doodle, their r corresponding to your R/x, and their z corresponding to your r/y.

Well, I thought originally that squiggle was just the phi in spherical coordinates, but then your description of "pure polar coordinates" above seems different from spherical coordinates, though I may be interpreting it incorrectly. Please clarify whether my picture of how the planes need to be moved to contain the point we're assigning coordinates to is correct.

I did say "Yes, the 'bastardised version of polar coordinates' is really cylindrical coordinates, but so is the 'bastardised blend of cartesian and polar coordinates'" so I don't think we have any meaningful disagreement here. In one variant I thought about moving a cartesian plane around inside a conceptual cylinder (with infinite radius) until the location lies on the plane, thus setting doodle. The cartesian plane would be attached to an axis running up the centre of that cylinder. In the other variant I thought about moving a circle up or down that same axis until the location lies on a plane parallel to the cirle and orthogonal to the axis (setting R), then expanding the circle out from that axis until the location we are describing lies on the cirle (setting r), and then moving a pointer around (setting splodge).

Again the order in which you conceptually take the readings makes no difference whatsoever, so yes as long as your axes and reference points are chosen the right way, R corresponds to x or y, r corresponds to y or x and splodge corresponds to doodle.

No, it can be finite, that's what I was talking about with the stuff about topology and space being like an "asteroids" game. As I said though, a finite space can also be described using the ordinary coordinates of SR going from -infinity to +infinity by taking the finite region and using it to tile an infinite space (the finite region must be a shape that it can be used to tile an infinite flat space, like a square or triangle in 2D, or a cube in 3D) so that all objects just repeat like a hall of mirrors; this will just mean that the same points of the flat space gets assigned multiple sets of coordinates, but you can apply all the standard rules of SR to this hall-of-mirrors universe.

Here was what I wrote about this earlier in post #268, if you haven't looked at the links I posted I recommend at least looking at the first one:

You can also see this article on finite universes with unusual topologies that I linked to in post #246.

My gut reaction is to not like the "asteroids" topoology. But I recognise it as a gut reaction, not the consequence of reasoning and careful analysis.

I have been thinking a lot about the mapping issue, since it was not something I concerned myself with initially (see my post directed to belliott to see what I was concerning myself with initially).

I understand that you wish me to concern myself with it though. If I may, I would like to read the links you gave more carefully, absorb them and then explain what I have in mind right now, if I feel it is still valid after further thought. Suffice it to say that despite disliking the "asteroids" topology, I am being forced in that direction.

cheers,

neopolitan

 

[neopolitan]

Ok, I had another look at the links.

I am still not completely comfortable with the idea of a patchwork universe with all the patches (effectively?) being the same patch.

I am however comfortable with the idea of the universe being "compact", with no sharp edges or discontinuities.

It is entirely possible that you will not like what I am about to suggest. That's ok, since I am not totally comfortable with it either.

You wanted to know what mapping regime I had in mind. As I have pointed out I didn't concern myself with that initially, but now I have thought it through and cannot justify the projection of a plane onto the surface of a sphere or a volume to the hypersurface of a hyperspere. But I can justify the projection of a plane onto the surface of a hemisphere or a volume to the hypersurface of a hyperhemisphere (hemihypersphere?)

This unfortunately, from my perspective, then demands the sort of patchwork arrangement discussed in the links you sent so that anything moving past the border of the hemisphere (let's stick with 2+1 to make it simpler) would appear on the other side of the universe traveling along the same line (or arc).

Each one of us would perceive the universe as a plane stretching out tangentially from the surface of the sphere, effectively out to infinity. But that effective infinity is in terms of metres right now. What is infinity today won't necessarily be infinity tomorrow. (Yes, I don't like this either.)

Take a look at the diagram now. I will try to show what I mean graphically since words seem to fail me here.

Location A can be thought of as lying on the plane but that version of the location is in different time from the one we are "in". It's in the future. The version that is on our surface of simultaneity is closer and that is the one that really matters. Note that we cannot "see" either, since photons have to get to us.

The same applies to Location B. If you take a line like the one to Location B and increase the angle of it from the top of the hemicircle until it nears pi/2, then you can see that the plane effectively stretches out to infinity. But when that version of the location lies on the same surface of simultaneity as me, it won't be infinitely distant (admittedly though, it might be at an infinitely distant time).

Anyway, it is this plane (flat in 2d) that I want mapped onto the surface of simultaneity.

To the best of my knowledge the transformation would be something like:

(t*tan(theta),t*tan(phi)) -> (t,theta,phi) ...or... (x,y) -> (t,arctan(x/t),arctan(y/t))

I don't think this schema is bad locally, but I really would not want to be fiddling around at the edges.

I did say I wasn't totally comfortable, didn't I?

cheers,

neopolitan

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.


I suppose that I should clarify that I don't see the tangential plane as being "the real universe". That is just the perception of the universe with which we are most familiar, possibly because we find it difficult to grasp that something, ie space, can be flat and curved at the same time (flat in terms of the dimensions in question, so 3d flat in terms of three dimensions, curved in terms of spacetime, in terms of four dimensions).

The idea of grabbing a piece of paper and trying to make a sphere of it is misleading, at the very least because a piece of paper and the resultant crinkly sphere are both static. Additionally, a better analogy would be to have a sphere from the start and look at projections from the surface of that sphere to a plane (not to try to cut the surface and spread it out to get a contiguous, flat plane).

In my model the hypersphere is expanding over time but you can also think of there being different layers each with its own "time" index, and this makes a difference. A tangential plane would intersect future instants in which rulers would be longer than today. I bring this up in part because of the whole "triangle" issue that keeps resurfacing.

A pseudo-triangle drawn on the surface of a sphere has a sum of internal angles (SIA) which is greater than 180 degrees (with the exception of special case "flat pseudo-triangles" for which one side has a length of zero units - these will have a SIA of 180 degrees). But these are pseudo-triangles since there not lines joining the vertices but rather curves. The real triangle joining three vertices will cut right through the sphere, taking the shortest path (in three dimensions), and the SIA for that triangle will be 180 degrees.

I did ask a question before which has been ignored, so I will ask it again.

Say I am inertial such that I could refer to a frame in which I am at rest and there are a few other things at rest in that frame in which I am at rest.

Say I measure the distance between myself and an ancient, highly durable artifact at rest in the frame in which I am at rest. Say that distance is 10m.

Note that I never specified when I measured the distance.

What is the spatial distance between me today and that ancient, highly durable artifact 10,000 years ago (noting that we are both at rest relative to each other and assuming that has always been the case)?

I think it is either 10m or approximately 95x10^15 kilometres. It all depends on whether you can think that space is flat in 3+1 dimensions or not. I think it is, so I prefer the first option. But I can understand the other answer also (oh alright, let's just call it a nice round 10,000 lightyears to make it easier to comprehend) - but I don't think it is a purely spatial distance.

Say you pick two ancient, highly durable artifacts (at rest in the frame in which I am at rest) - Artifact A and Artifact B - and measure the spatial distance between me, them and each other, where the selected events are:

me now,

Artifact A 10,000 years ago (ie, 10,000 years before the event which is Artifact A simultaneous with my now, according to me in the frame in which I am at rest), and

Artifact B 10,000 years in the future (ie, 10,000 years ater the event which is Artifact B simultaneous with my now, according to me in the frame in which I am at rest).

What is the sum of the internal angles of the triangle defined by these events? How will I measure the angle between me-Artifact A(-10,000 years) and me-Artifact B(+10,000 years), given that I know that all three of us are at rest relative to each other, and conceptually have always been and will always be.

In my model, a tangential plane would actually have "me", Artifact A in the future and Artifact B in the future. But we can select any time indices we like, so long as the three points remain at rest relative to each other.

--

Anyway, I see a unbounded but finite universe mapped onto an infinite plane. How do we interpret this? Think about a photon released from us today and aimed at the outer reaches of the universe (which is the same as "release a photon" since what seem to us to be the outer reaches of the universe lie around us in all directions).

If the universe is expanding as I suggest, then when does the photon reach the edge of the universe? If it traveled along a plane it would never get there, because that edge is expanding out.

However, I suggest that everything moves tangentially to the hypersurface of simultaneity inhabited. I also suggest a certain graininess to the universe, specifically at the Planck level.

So, in one unit of Planck time, a photon moves one unit of Planck length and is then in a new hypersurface of simultaneity, with a very very slight change in angle and very very slight change of position (which means that even though the edge of the universe is still effectively infinitely distant, it is now a different edge, including a thin section that would otherwise have been in the opposite direction).

The upshot is that a photon can reach a position that was previously infinitely distant, but that position is then no longer on the edge of the universe. At that "time", the photon's origin will be infinitely distant (and on the edge of the universe in the opposite direction to the photon's velocity).

How is this possible? Well, my rough explanation would be that a photon effectively travels with infinite speed (time "experienced" by a photon while the universe apparently zips past ... zero, 1/0=undefined, asymptotically infinite) but the graininess of the universe limits the speed we measure it having. Anything that has mass will never reach a speed necessary to reach the edge of the universe, which means that effectively the universe does have an edge, it is effectively bounded and effectively infinite but in actuality it is unbounded and finite.

Note, however, that this is all just my interpretation. I am not saying it is the way things are, but it might be worth pondering it before discarding the idea.

I fully understand that my interpretation seems riddled with paradoxes. I guess what I am doing is organising the paradoxes so they make sense, to me if no-one else.

(And note that there are other existing paradoxes, such as if the universe is infinite, and Copernican, then it should have infinite mass, and anything with infinite mass, infinite mass, should be collapsed in on itself - no matter how much space it fills, or whether it is expanding or not - begging the question, what would cause an infinite mass to expand out anyway, is this not representative of infinite kinetic energy? However, if the universe were infinite then, no matter how much mass was in it, the average density would be zero, which would satisfy the Copernican principle if the universe was empty, but that the average density where we are and in all the universe we can observe is a little over that.

I firmly believe that if you present any argument against this, you will be either sweeping the paradox under the mat or shifting the question back one level, akin to the religious solution - Where did the universe come from? God made it. Where did God come from? He was always here. Why can't the universe have always been here? Don't be silly, nothing comes from nothing, something must have started the universe. What started God? I am going to start persecuting you if you don't stop asking inconvenient questions.

Dealing with the paradoxes might not be a silly idea.)

cheers,

neopolitan

 

[JesseM]

I do think we are thinking the same thing but just have different aspects that we hold to be more important. For instance, you hold the planes which theta and squiggle vary (so one in which theta is variable and squiggle is fixed and one in which theta is fixed and squiggle is variable) to be very important. For me, the null points and null angles are more important.

As far as I can tell you and I are both describing spherical coordinates in our own ways (and I accept that your way is most likely the standard way). Perhaps I take the term "spherical coordinates" too literally, since I see it as expanding out the surface of conceptual sphere until it contains the location we want to describe (thus setting R) then swinging a pointer around to the location. That pointer will then be at an angle theta from the axis in one plane and an angle squiggle from the same axis in another plane (thus setting theta and squiggle in one fell swoop). But the point is that you can do it in which order you feel more comfortable with. You can set the angles first and then R (as you did) - or one angle, then R and then the other angle - or a variation of what I did but do my second step in two phases with planes (in a manner similar to what you did). The end result is the same.

Actually you misunderstand me a little. What I described, where there is a theta plane and a squiggle plane and each is varied by moving them from their "original" positions (which contains R=0) along an axis orthogonal to themselves, is how I understood your description of "pure polar coordinates", but it is not the same as spherical coordinates. In spherical coordinates we could start with a phi plane and a theta plane which are orthogonal and which contain R=0, and if the point we want to assign coordinates to is outside the phi plane we do move it along an axis orthogonal to itself until it contains the point, then assign it a phi-coordinate in the usual 2D polar way, but if the point we want to assign coordinates to is outside the theta plane, instead of moving it along an axis orthogonal to itself we rotate it around an axis in the theta plane which goes through R=0 and is orthogonal to the phi plane, similar to how I suggested we rotate the xy plane around the y-axis in the "bastardised blend of cartesian and polar coordinates".

Suppose we look at a sphere of constant R, and we call the intersection of the sphere's surface with the phi plane the "equator" of the sphere, then the axis which we rotate the theta plane in will be the one that goes from the "north pole" of the sphere to the "south pole", and the intersection of the theta plane with the sphere's surface will be two lines of longitude on opposite sides of the sphere. So if we fix R and move the phi plane up and down orthogonal to itself, its intersections with the sphere as it moves creates a series of lines of latitude expanding from one pole to the equator and then contracting to the other pole; if we fix the angle phi in the plane, then this corresponds to a fixed angle on each line of latitude, so the collection of all points with a fixed R and fixed phi gives a line of longitude. Likewise, if we fix R and rotate the theta plane around the axis from pole to pole, its intersections with the sphere create a series of paired lines of longitude which each go from one pole to the other; if we fix the angle theta in the plane, this corresponds to a fixed angle on each line of longitude, so the collection of all points with a fixed R and fixed theta gives a line of latitude.

In contrast, in the "pure polar coordinates" as I described them, if we say the intersection of the squiggle plane with a sphere is the sphere's equator and the intersection of the theta plane with the sphere is two lines of longitude on opposite sides, then if we allow the squiggle plane to move in a direction orthogonal to itself its intersections with the sphere give a series of lines of latitude expanding from one pole to the equator and then contracting to the other pole, so fixed R and fixed squiggle means a pair of lines of longitude from one pole to the other. But if we also allow the theta plane to move in a direction to itself, this creates a series of lines of pseudo-latitude like if you turned a globe on its side, which expand from a point on the equator and then contract to a point on the equator on the opposite side; so if you fix R and fix theta, that means a pair of lines of pseudo-longitude going from one point on the equator to the opposite point on the equator. So you can see this is really a rather different coordinate system from spherical coordinates.

If you haven't encountered spherical coordinates before and done math problems using them, then I don't blame you for getting a little confused about how they work, it can be a little subtle. But I wish you wouldn't get offended at me for trying to explain them in detail, trying to avoid these sort of subtle confusions is exactly why I did so.

In any case, the spherical coordinates thing is a bit of a sidetrack from this discussion. As I said earlier, if we're talking about a mapping, I think it's sufficient to map the coordinates of an inertial frame with one spatial dimension x and one time dimension t onto a set of polar coordinates r and theta (with varying r corresponding to varying time, and varying theta corresponding to varying x). You're free to map the finite section of the x-axis corresponding to a finite universe onto just a section of the circle (relating to your 'hemisphere' comments above) rather than the whole circle, it doesn't matter to me. But even before we get into the issue of a specific mapping, I really think it's vital that we clear up this issue from post #268 which you never addressed:

How are you "damned if you do"? Do you consider it "damning" for me to say that your onion diagrams just represent a remapping of flat space (i.e. a coordinate change) rather than actual physical curvature? Or do you imagine there is some third alternative beyond either 1) space being genuinely curved, or 2) space being flat but being represented as a curved sphere due to a coordinate shift? If you think there's a third alternative, I suspect that once again the problem is that you think and argue in vague verbal terms which don't correspond to any well-defined mathematical ideas, like your statement eariler that "I am thinking of flat space which has been wrapped around a hypersphere so the whole of it is curved, but only in terms of 4 dimensions, not in terms of 3dimensions. I have said that a few times." There is simply no physical sense in which it is meaningful to say that space is flat, spacetime is flat, but space is "curved in terms of 4 dimensions"--the only way I can interpret a statement like this is as a statement about a coordinate representation where flat spatial surfaces of simultaneity from a flat spacetime appear curved. But if "curvature" can't be represented in intrinsic differential-geometry terms using a line element as I discussed in post #194, if it only appears in an embedding diagram of curved space or spacetime, then it simply cannot correspond to anything that can actually be physically measured.

So we really need to be clear on this. If you think that both space and spacetime can be physically flat, and yet your onion-diagrams are supposed to represent a physical reality that goes beyond just a coordinate remapping of flat surfaces of simultaneity, then I think you're just confused about the relationship between visual diagrams and actual mathematical physics. If you disagree, then you need to explain what the curvature is supposed to represent using mathematics, not just fuzzy english phrases that don't mean anything to me (or anyone else reading this thread, I'd wager) like "flat space which has been wrapped around a hypersphere so the whole of it is curved, but only in terms of 4 dimensions, not in terms of 3dimensions".

In more recent posts you have continued to make comments that make it sound like you think your "mapping" represents some real physical truth rather than just a new coordinate system for describing the same flat spacetime as in SR, like your comments in post #277:

If the universe is expanding as I suggest, then when does the photon reach the edge of the universe? If it traveled along a plane it would never get there, because that edge is expanding out.

However, I suggest that everything moves tangentially to the hypersurface of simultaneity inhabited. I also suggest a certain graininess to the universe, specifically at the Planck level.

If the statement "if the universe is expanding as I suggest" is supposed to mean that you think you are offering a physical hypothesis about the universe rather than just an interesting new coordinate system, I think there's a problem here, both because I don't think you've really offered any meaningful statement of what your diagrams could mean physically (you claim that neither the spacelike surfaces nor spacetime are 'really' curved, for example), and also because new physical hypotheses belong in the Independent Research forum, not here.

And to address your more recent question:

Say I am inertial such that I could refer to a frame in which I am at rest and there are a few other things at rest in that frame in which I am at rest.

Say I measure the distance between myself and an ancient, highly durable artifact at rest in the frame in which I am at rest. Say that distance is 10m.

Note that I never specified when I measured the distance.

What is the spatial distance between me today and that ancient, highly durable artifact 10,000 years ago (noting that we are both at rest relative to each other and assuming that has always been the case)?

For this question to be well-defined, you really need to give a physical definition of what you mean by "spatial distance", the question is meaningless otherwise. Normally in SR, each inertial observer has their own set of inertial rulers at rest with respect to themselves, and the spatial distance between two events can be found just by noting the position of the first event on the rulers, and then noting the position of the second event on the rulers, and using the pythagorean theorem \sqrt{x^2 + y^2 + z^2} to find the spatial distance. In this case, the answer to your question will just depend on how you and the artifact are moving in the observer's frame. If you are both at rest in the observer's frame, then the distance is just 10m; but if you're moving at 0.7c in the observer's frame, the distance would be close to 7,000 light-years.

 

[neopolitan]

Note: JesseM believes that there are serious problems with the model I am discussing below. I think it is entirely consistent with SR and will attempt to prove that, but please take the words of science advisors and PF mentors more seriously than mine.


Yes, the 3d polar coordinates/spherical coordinates discussion is off track. Suffice it to say that I didn't think of moving the theta and squiggle planes. The planes to me were merely where the theta and squiggle "pointers" had freedom of movement from nominated null angle directions. You can nominate a cartesian axis as a null direction and it certainly makes it easier, but you don't have to. If you don't then I agree, strictly speaking, you can't call the result "spherical coordinates". The fundamental idea is the same, but the execution is different.

In more recent posts you have continued to make comments that make it sound like you think your "mapping" represents some real physical truth rather than just a new coordinate system for describing the same flat spacetime as in SR, like your comments in post #277:

If the statement "if the universe is expanding as I suggest" is supposed to mean that you think you are offering a physical hypothesis about the universe rather than just an interesting new coordinate system, I think there's a problem here, both because I don't think you've really offered any meaningful statement of what your diagrams could mean physically (you claim that neither the spacelike surfaces nor spacetime are 'really' curved, for example), and also because new physical hypotheses belong in the Independent Research forum, not here.

The thing is that I am not convinced that what I am saying represents any new physical hypotheses. As far as I know all the equations work out the same in my model. It's an interpretation of what those equations are telling us that may vary (albeit I did come at it from the opposite direction). As for my claim that "neither the spacelike surfaces nor spacetime are 'really' curved", that is what I am getting at in the question you addressed below.

And to address your more recent question:

Say I am inertial such that I could refer to a frame in which I am at rest and there are a few other things at rest in that frame in which I am at rest.

Say I measure the distance between myself and an ancient, highly durable artifact at rest in the frame in which I am at rest. Say that distance is 10m.

Note that I never specified when I measured the distance.

What is the spatial distance between me today and that ancient, highly durable artifact 10,000 years ago (noting that we are both at rest relative to each other and assuming that has always been the case)?

For this question to be well-defined, you really need to give a physical definition of what you mean by "spatial distance", the question is meaningless otherwise. Normally in SR, each inertial observer has their own set of inertial rulers at rest with respect to themselves, and the spatial distance between two events can be found just by noting the position of the first event on the rulers, and then noting the position of the second event on the rulers, and using the pythagorean theorem \sqrt{x^2 + y^2 + z^2} to find the spatial distance. In this case, the answer to your question will just depend on how you and the artifact are moving in the observer's frame. If you are both at rest in the observer's frame, then the distance is just 10m; but if you're moving at 0.7c in the observer's frame, the distance would be close to 7,000 light-years.

Don't you already have a definition for spatial distance? I am happy to use yours.

Note that once again you brought in a new observer who I didn't invite. I am at rest in the frame in which I am at rest, and the artifact is at rest in the frame in which I am at rest and I measure the distance between me and the artifact. I never invited another observer and, for the purposes of the question I asked, I don't care what any other observer thinks.

However, mea culpa, I was inaccurate in my phrasing and you called me on it. So I will rephrase:

What is the spatial distance between me today and that ancient, highly durable artifact 10,000 years ago (noting that we are both at rest relative to each other and assuming that has always been the case) - measured in the frame in which both I and the artifact are at rest?

The answer is therefore inequivocably 10m, yes?

Then, can you address the question I asked in a later post, which is obliquely addressing the triangle issue, which seems so central to whether or not space is curved.

Here is the question again (note the total and complete absence of any observer other than "me", I have even removed the word "you" from this slight editing, which was a linguistic inaccuracy in the original):

Say I pick two ancient, highly durable artifacts (at rest in the frame in which I am at rest) - Artifact A and Artifact B - and I measure (in terms of the frame in which I and the artifacts are at rest) the spatial distance between me and each of the artifacts and between the two artifacts, where the selected events are:

me "now",

Artifact A 10,000 years ago (ie, 10,000 years before the event which is Artifact A simultaneous with my now, according to me in the frame in which I am at rest), and

Artifact B 10,000 years in the future (ie, 10,000 years ater the event which is Artifact B simultaneous with my now, according to me in the frame in which I am at rest).

What is the sum of the internal angles of the triangle defined by these events? How will I measure the angle between me-Artifact A(-10,000 years) and me-Artifact B(+10,000 years), given that I know that all three of us are at rest relative to each other, and conceptually they always have been and always will be at rest relative to each other.

If the sum of the internal angles, space-wise, is 180 degrees, is not space flat? If the sum of the internal angles, space-wise, is not 180 degrees, how would we measure it? Note that if we follow our sphere analogy, we would be measuring the angle between two curves with a time component.

If we were to work out the sum of the internal angles spacetime wise, we would also find that they sum to 180 degrees (the angles with be close enough to 0, 0 and 180 degrees for government work, unless the spatial separations are enormous). Does that not mean that spacetime is flat?

----------------------------------

Why do I think that my model is nothing new physically?

While we have covered a lot of ground in this thread, and brought in a lot of different issues, some of which I have possibly not been as careful with as I could have been, I have tried to be very consistent about how I talk about dimensions. I didn't talk about going from 2 dimensions to 3 dimensions, or from 3 to four. I have tried to always talk about it in terms of 2 dimensions to 2+1 dimensions, or 3 to 3+1.

I have done this on purpose. The reason for it is that while we can nominate an x, y and z axis at random, or select axes which are most convenient for us, we can't do that with time.

You have done the same, at least effectively. You remove a dimension to make it easier to grasp what is being modeled, but you only ever take away a spacelike dimension, never the timelike dimension.

You can't take an inertial perspective (an inertial frame) and choose your four axes at random. There are three dimensions in which you can select axes however you like and one which is inviolate. Say you and I are at rest relative to each other. There is also a television in our frame, at rest relative to both of us and not lying on the line defined by our two positions. I could choose me-TV as my x axis, with myself as the origin. You could chose you-TV as your x-axis with the television as the origin. We could then assign internally consistent orthogonal y and z axes that are not common to each other. Your x, y and z axes would be a blend of my x, y and z axes. What we would be extremely unlikely to do is chose axes such that your x, y and z axes correspond to a blend of my x, y, x and t axes. If we did, then everything would have to be moving in order to stay still in this strange coordinate system. Can you see that is a problem?

So, what I am saying is that time is special, you have to treat it specially.

Now if time could be represented by just another othogonal plane, you could look at it from another perspective and end up with the problem of having blended spacelike and timelike axes.

If the timelike dimension has more of a circular (really hyperspherical) nature then, no matter what perspective you took, the timelike dimension would be unaffected. Yes, your altered perspective would affect the spacelike dimensions, making my x-axis a blend of your x,y and z axes. But our timelike dimension would be unaffected.

Now this might be something completely new, but I sincerely doubt it. I am probably just using clumsy almost physics-like terminology to express something that is already accepted. In any event, this is the physical aspect of what I am discussing. It leads to "an interesting coordinate system" but I think that coordinate system does make sense, even if it may be difficult to grasp.

cheers,

neopolitan

 

[neopolitan]

Is there a chance that either JesseM or Belliott could address the questions in the previous post?

While I am posting, I would like to clarify something about the second last paragraph in that post:

If the timelike dimension has more of a circular (really hyperspherical) nature then, no matter what perspective you took, the timelike dimension would be unaffected. Yes, your altered perspective would affect the spacelike dimensions, making my x-axis a blend of your x,y and z axes. But our timelike dimension would be unaffected.

This paragraph relates to the scenario described in the fifth last paragraph:

You can't take an inertial perspective (an inertial frame) and choose your four axes at random. There are three dimensions in which you can select axes however you like and one which is inviolate. Say you and I are at rest relative to each other. There is also a television in our frame, at rest relative to both of us and not lying on the line defined by our two positions. I could choose me-TV as my x axis, with myself as the origin. You could chose you-TV as your x-axis with the television as the origin. We could then assign internally consistent orthogonal y and z axes that are not common to each other. Your x, y and z axes would be a blend of my x, y and z axes. What we would be extremely unlikely to do is chose axes such that your x, y and z axes correspond to a blend of my x, y, x and t axes. If we did, then everything would have to be moving in order to stay still in this strange coordinate system. Can you see that is a problem?

We are at rest with respect to each other in this scenario. If we were not at rest with respect to each other - which would be a completely different scenario - then my x-axis would indeed be a blend of your x, y, z and t axes (although we normally would make it simple by eliminating our y and z axes from consideration by means of careful framing of the scenario).

Note that, other than the request for a reply, the only question in this post is in a quote box from the previous post. Please address the previous post.

thanks,

neopolitan

 

[neopolitan]

Would it be presumptive of me to assume that the three weeks of resounding silence indicate that there are no arguments against what I have to say (at the very least in light of my last two clarifying posts)?

cheers,

neopolitan