Discuss events which are simultaneous in one frame?

[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