# Discuss events which are simultaneous in one frame?

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 travelling 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

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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 travelled 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

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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 travelled 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.

neopolitan said:
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.

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.

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 spacial separations are enormous). Does that not mean that spacetime is flat?

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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 modelled, 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

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

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