Is Event Sequencing Relative in the Theory of Manifolds?

[JesseM]

Do you have a reference for that? My calculations show that your claim is false. The line of simultaneity keeps rotating wrt the x-axis in the Minkowski diagram as the speed varies but it never crosses over itself, even at the inflection points.

It depends on how fast the acceleration is, i.e. how long it takes the ship to go from 0.8c to rest in the Earth frame. For example, suppose the ship is traveling towards Earth at 0.8c until it is 10 light years from Earth at t=0 years in the Earth frame (so if the Earth is at x=0 light years, the ship could be at x=-10 light years), then it accelerates for a year until it is at rest at t=1 year the Earth frame. In that case, in the ship's inertial rest frame at the moment it begins to accelerate, the moment of its beginning to accelerate is simultaneous with an event that occurs on Earth at t=8 years in the Earth frame (since in the Earth's frame we have dx=10 and dt=8 between these events, meaning in the ship's frame dt'=gamma*(dt - v*dx/c^2)=0), but in the ship's inertial rest frame at the moment it stops accelerating, the moment it stops accelerating is simultaneous with an event that occurs on Earth at t=1 year in the Earth frame.

 

[starthaus]

It depends on how fast the acceleration is, i.e. how long it takes the ship to go from 0.8c to rest in the Earth frame. For example, suppose the ship is traveling towards Earth at 0.8c until it is 10 light years from Earth at t=0 years in the Earth frame (so if the Earth is at x=0 light years, the ship could be at x=-10 light years), then it accelerates for a year until it is at rest at t=1 year the Earth frame. In that case, in the ship's inertial rest frame at the moment it begins to accelerate, the moment of its beginning to accelerate is simultaneous with an event that occurs on Earth at t=8 years in the Earth frame (since in the Earth's frame we have dx=10 and dt=8 between these events, meaning in the ship's frame dt'=gamma*(dt - v*dx/c^2)=0), but in the ship's inertial rest frame at the moment it stops accelerating, the moment it stops accelerating is simultaneous with an event that occurs on Earth at t=1 year in the Earth frame.

I don't think that the above is correct. The issue in discussion was whether acceleration can invert the ordering of events. I will repeat the proof I gave for inertial frames and I will generalize to accelerated frames:

dt'=\gamma(\tau-vx/c^2)

dt'=\gamma (d\tau-vdx/c^2)

dt'=\gamma d\tau (1-v/c^2*\frac{dx}{d\tau})

Since v \frac{dx}{d\tau}<c^2 it follows that dt' and d\tau always have the same sign. If you are ok with the above, I can post the generalization to accelerated frames.

 

[JesseM]

I don't think that the above is correct. The issue in discussion was whether acceleration can invert the ordering of events. I will repeat the proof I gave for inertial frames and I will generalize to accelerated frames:

dt'=\gamma(\tau-vx/c^2)

dt'=\gamma (d\tau-vdx/c^2)

dt'=\gamma d\tau (1-v/c^2*\frac{dx}{d\tau})

How would you define the non-inertial frame for a non-inertial observer? There's no single "correct" way, but I think the most common type of non-inertial frame used by physicists in SR would be one that has the following properties:

1. For point on the non-inertial observer's worldline, the coordinate time is just equal to the observer's proper time at that point

2. At any point on the observer's worldline, a line of simultaneity in the non-inertial coordinate system which goes through that point would be identical to a line of simultaneity in the inertial frame where the observer has an instantaneous velocity of zero at that point

3. For two events which lie on a single line of simultaneity in the non-inertial coordinate system, the coordinate distance between them should be the same as the coordinate distance in the inertial frame which also has a line of simultaneity going through both events

4. The non-inertial observer has a coordinate position that doesn't change with coordinate time in the non-inertial system

If you define the non-inertial coordinate system in this way, then (3) implies the system's judgments about simultaneity always agree with those of the observer's instantaneous inertial rest frame, so this was the basis for my comments above (although I think it would actually be more common to just say the coordinate system "ends" at the point where lines of simultaneity would cross over one another, as with Rindler coordinates which don't extend past the Rindler horizon). I think Al68 was also assuming this sort of coordinate system when he commented about the dead rising from the grave in post #30. If you're assuming a different type of non-inertial coordinate system, you'll have to explain how it's defined for an observer with a non-inertial worldline that has a known parametrization x(t) relative to some inertial frame (for an observer experiencing constant proper acceleration a, their x(t) in an inertial frame where they started at rest would be x(t) = (c^2/a) (sqrt[1 + (at/c)^2] - 1) according to the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html ).

 

[starthaus]

How would you define the non-inertial frame for a non-inertial observer?

I'll show you next. First off, do you agree that inertial motion cannot reverse the order of events between frames? Yes or no?

 

[JesseM]

I'll show you next. First off, do you agree that inertial motion cannot reverse the order of events between frames? Yes or no?

Can you clarify what you mean by that? It is certainly true that for events with a space-like separation (|dx| > |c*dt| in any given inertial frame), you can find a pair of inertial frames which disagree on their order, are you denying that? If you don't deny it, why doesn't this qualify as an example of "inertial motion reversing the order of events between frames" as you define this phrase?

 

[yossell]

dt'=\gamma(\tau-vx/c^2)

dt'=\gamma (d\tau-vdx/c^2)

dt'=\gamma d\tau (1-v/c^2*\frac{dx}{d\tau})

Since v \frac{dx}{d\tau}<c^2 it follows that dt' and d\tau always have the same sign. If you are ok with the above, I can post the generalization to accelerated frames.

Your question is whether inertial motion can invert the order of events.

I do not think you have shown this claim in its full generality, but I think you have shown a weaker one, that inertial motion cannot invert the order of timelike events.

First, an aside: what's going on in step 1 to 2 - why the replacement of x and \tau with dx and d\tau? Just notational?

Secondly, I take the strategy to be this: dt' represents the difference in time between two events A and B in one inertial frame; d\tau and dx the difference in time and the difference in x-coordinate between these two events in another inertial frame. You then try to show that d\tau and dt' have the same sign, and therefore that any two frames must agree on the temporal order of the two events.

The trouble is that dx/d\tau is not necessarily a velocity of anything - it is *just* a ratio. The events may be spacelike separated. In which case, the ratio is not less than c. In such cases, your proof can be used to show that the temporal order can be inverted.

However, when attention is restricted to events which are timelike separated, then your proof seems to go through.

 

[starthaus]

Your question is whether inertial motion can invert the order of events.

More precisely, if order of timelike events (see the OP) is maintained in SR. We aren't talking about spacelike events.

I do not think you have shown this claim in its full generality, but I think you have shown a weaker one, that inertial motion cannot invert the order of timelike events.

...which is precisely what I have been talking about at posts 12 and 13. This answer is for both you and JesseM. In mathematical terms, as already shown twice, the proof is for \frac{dx}{d\tau}<1 (no "faster than light signalling"). I am addressing the OP, not the case of spacelike events.

First, an aside: what's going on in step 1 to 2 - why the replacement of x and \tau with dx and d\tau? Just notational?

No, this is standard differentiation.

Secondly, I take the strategy to be this: dt' represents the difference in time between two events A and B in one inertial frame; d\tau and dx the difference in time and the difference in x-coordinate between these two events in another inertial frame. You then try to show that d\tau and dt' have the same sign, and therefore that any two frames must agree on the temporal order of the two events.

yes

The trouble is that dx/d\tau is not necessarily a velocity of anything - it is *just* a ratio. The events may be spacelike separated.

No, see above.

However, when attention is restricted to events which are timelike separated, then your proof seems to go through.

It does not "seem", it does. It is just a repeat of the one at post 12. Same conditions, same outcome.

 

[yossell]

I still don't understand what you're calling differentiation or what you're doing from 1 - 2.

From line 2 onwards we're agreed that it shows that, for any timelike related (and I emphasise this because you've not been explicitly putting in this rider) events, all inertial observers agree about the events' temporal order ---- as I in fact wrote way back in post 5.

Great.

Now, I'm interested in the generalisation you've mentioned, to deal with Al68's point, that a rocket ship can, beginning at space time point A and in an inertial frame where events A and space time point B are simultaneous, travel to a spacetime point C and an inertial frame where C is simultaneous with an event that lies in B's past.

 

[starthaus]

I still don't understand what you're calling differentiation or what you're doing from 1 - 2.

Because this is a well known operation in calculus. It is important that you understand it for the next step, accelerated motion. How familiar are you with differential calculus?

 

[JesseM]

...which is precisely what I have been talking about at posts 12 and 13. This answer is for both you and JesseM. In mathematical terms, as already shown twice, the proof is for \frac{dx}{d\tau}<1 (no "faster than light signalling")

OK, so when you said "First off, do you agree that inertial motion cannot reverse the order of events between frames?" were you only talking about pairs of events for which |dx/dt| <= c? If so, I agree that different inertial frames won't disagree on their order.

 

[starthaus]

OK, so when you said "First off, do you agree that inertial motion cannot reverse the order of events between frames?" were you only talking about pairs of events for which |dx/dt| <= c? If so, I agree that different inertial frames won't disagree on their order.

Excellent, let's wait for yossell, see if he's comfortable with some heavy duty differential calculus required for the case of accelerated frames.

Do you think that the above holds for accelerated frames?

 

[JesseM]

Do you think that the above holds for accelerated frames?

It depends what you allow to qualify as an accelerated frame. You can write down a coordinate transformation which relates a non-inertial coordinate system to an inertial one, with the equations of the transformation implying that lines of simultaneity in the accelerated frame will actually cross when plotted in the inertial frame, but some authors would say this is not a well-behaved frame since it doesn't assign unique coordinates to each event, so they'd require that the accelerated frame only be defined up to the point where lines of simultaneity would cross rather than throughout all of spacetime (this is what's done with Rindler coordinates, which only cover a region of spacetime to one side of the Rindler horizon)

Also, it depends on whether you impose the rule that the space coordinate must actually be spacelike at all times (i.e. every surface of constant time is a spacelike surface, meaning path in that surface is spacelike) and the time coordinate must be timelike everywhere (every line of constant position coordinate is a timelike worldline). If you don't impose that rule, then non-inertial frames can disagree on the order of events with a timelike separation even without any lines of simultaneity crossing in any of the non-inertial frames.

 

[Al68]

I asked you for a valid reference to your claim that acceleration can produce event order reversal. So, you provided no reference. Not only that, you got the wrong answer.

LOL. That's not the claim I made and you know it. I said that events could be in reverse order in an accelerated reference frame. Slight difference there. And I provided the only reference I used: the lorentz transformations.

And I got the right answer. All you need to do to see how obvious it is is to imagine a clock (clock C) at rest with Earth local to the ship's (almost instantaneous) deceleration, that is synched with Earth's clock in Earth's frame. Prior to deceleration, Earth's clock is ahead of clock C by 8 years in the ship's frame. After deceleration, Earth's clock matches clock C in the ship's frame. Clock C is (almost) local to the ship before and after decel and has (almost) the same reading before and after decel. Simple lorentz transformations. No kawcoolus required.

Why would an experimental physicist have so much trouble understanding such a simple scenario? What do you think the right answer is?

 

[starthaus]

LOL. That's not the claim I made and you know it. I said that events could be in reverse order in an accelerated reference frame.

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

 

[JesseM]

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

What about the accelerated frame I defined in post #53 with properties 1-4? Do you disagree that lines of simultaneity can cross in such a frame, causing the frame to label timelike-separated events with a reversed order from the order in inertial frames? Or do you think my definition wasn't clear?

 

[Al68]

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

LOL. I can only assume my claim to be perfectly correct given that an experimental physicist such as yourself has failed to show it to be erroneous in any way. Thanks for the confirmation! :!)

BTW, my claim was that (in an accelerated frame) events could be assigned time coordinates in reverse order by the SR simultaneity convention, not that they actually occur or are observed in reverse order. Unless someone was so pedantic as to take my "dead rising from the grave" statement seriously.

 

[yossell]

What about the accelerated frame I defined in post #53 with properties 1-4? Do you disagree that lines of simultaneity can cross in such a frame, causing the frame to label timelike-separated events with a reversed order from the order in inertial frames? Or do you think my definition wasn't clear?

I have a question about non-inertial frames: In the kinds of frames that you're talking about, one and the same space-time point will be assigned two different coordinates - e.g. the point of intersection of the lines of simultaneity. But does the concept of a frame allow for this to happen? The time of an event will be multivalued in such a frame. Do people know if this is ok?

Let me say that I don't see anything deeply physical going on here - I recognise that non-inertial frames are a somewhat artificial concept. it's just an interesting (to me) question about the definition of a frame.

 

[Ken Natton]

Wow. I can’t believe the pace that this thread has gone at. Just one day away and I’ve had four pages of challenging stuff to read and try to understand. I hate to act as a weight and drag the pace down, but my hope is that if anyone does answer me, that doesn’t necessarily have to impinge on the conversation between the rest of you.

I’m still on the difference between special and general relativity.

Special relativity mathematically takes place against a background of Minkowski space. General relativity takes place against a background of 4 dimensional Lorentzian manifolds of which Minkwoski space is one (special) example. The motivation for this is that choosing Lorentzian manifolds other than Minkowski space allows gravity to be modeled relativistically.

Okay, but like Cartesian co-ordinate systems and Euclidian geometry, these metrics are just intellectual constructs, tools of analysis, rather aspects of physical reality. Yes, the reality of the curvature of spacetime is part of general relativity theory, but that is always the true physical reality, a body isn’t actually transformed from being in curved spacetine to being in flat spacetime when it ceases to accelerate and starts to travel at a constant speed. An inertial reference frame is really just an idealised state to make the concept easier to understand, as is the case for Newton’s first law, when we are asked to imagine a body in a situation without gravity and without friction. The truth, is it not, is that all real bodies, from subatomic particles to galaxies, exist in permanent non-inertial reference frames?

There are those that argue that special relativity is superseded and made redundant by general relativity. The only defence against that is that special relativity is the route into an understanding. Special relativity is more graspable for those coming to relativity for the first time, and once you have got that, it becomes easier to extend the principle to cover all reference frames.

Yes I know Einstein was specifically interested in the equivalence of gravitational freefall and acceleration under some other force, and that this led him to the idea that mass actually warps spacetime and that this is the actual explanation for gravity. But the concept of equivalence is extendable to the equivalence of all accelerating reference frames, is it not? It seems unlikely to me that special relativity describes a physical reality that only exists in idealised conditions. Surely, the physical reality is always the same. Special relativity just covers a special case of it, general relativity generalises that principle.

So I suppose, to bring it back to the original subject of the thread, what you are telling me is that within the idealised constraints of an inertial reference frame, changing the sequence of events is not possible. Physical reality does not actually impose those constraints, and thus relativity of sequence is always possible. Once again, we have the undermining of the notion of cause and effect that worried me.

 

[Rasalhague]

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

Is the issue here: (1) "Are there pairs of non-identical events with no defined coordinate time order or with multiple coordinate time orders in one single given accelerated coordinate system?" Al68: "Yes." starthaus: "No."

At first I thought the issue was: (2) "Does Al68's scenario correctly exemplify the fact that spacelike separated events don't have a frame-independent natural time order. (First simultaneity judgement in the instantaneous comoving inertial rest frame of the ship, 10 light years from the earth, traveling towards the Earth at 0.8c relative to the earth. Second simultaneity judgement in the instantaneous comoving inertial rest frame of the ship after it decelerates till it's at rest with respect to the earth, i.e. in the rest frame of the earth.)"

I thought starthaus imagined Al68 was talking about a pair of timelike separated events. I thought you were both just talking at cross purposes, particularly as starthaus's proof consisted of using the Lorentz boost formulas to show that timelike separated events can't change order under a boost.

But if the issue is actually (1), then I would have thought the answer was no because of the definition of "coordinate system", "reference frame", "chart", specifically the requirement that the coordinate functions be, as their name suggests, functions, i.e. single-valued. Events in spacetime for which a coordinate system doesn't behave this way, I'm thinking, wouldn't belong to the domain of that particular coordinate system. In this case, if we have an accelerated coordinate system something like Rindler coordinates (as JesseM says in the final paragraph of #53), events on Earth just aren't covered by these coordinates. Or, if they were, then, by definition of a chart, there'd have to be one single time order specified, even though it needen't necessarily agree with the time order of another coordinate system, and--at least for some pairs of events--not every possible coordinate system will agree.

Is that anywhere near the mark?

 

[yossell]

Okay, but like Cartesian co-ordinate systems and Euclidian geometry, these metrics are just intellectual constructs, tools of analysis, rather aspects of physical reality.

Since the rest of this paragraph looks right to me, there's probably nothing wrong with your understanding, but I would be careful about this way of putting it. The metrics of spacetime themselves aren't normally thought of as mere intellectual constructs. The metric is quite unlike a choice of coordinate system, and in fact conveys coordinate independent information about space-time itself, such as whether space is curved, and the coordinate-independent Minkowski `length' of a space-time curve.

There have been those (I think Poincare was one) who thought that the choice of geometry itself was as much a convention as choice of coordinate system. But I wouldn't say that this was a mainstream idea today.

The truth, is it not, is that all real bodies, from subatomic particles to galaxies, exist in permanent non-inertial reference frames?

Again, this is probable pit-nicking, but accelerated bodies can be analysed from the point of view of an inertial frame. What's (I think) true is that, the existence of gravity, inertial frames as understood in SR no longer exist globally. Rather, one can apply SR at a local level - working on a small scale, not extending your t and x coordinates too far, objects still behave approximately as SR says they do.

But the concept of equivalence is extendable to the equivalence of all accelerating reference frames, is it not?

I tentatively believe it's only to all *freely* falling frames.

It seems unlikely to me that special relativity describes a physical reality that only exists in idealised conditions. Surely, the physical reality is always the same. Special relativity just covers a special case of it, general relativity generalises that principle.

Since gravity is pervasive, all space time is warped a little - so I think in that sense SR does only describe a physical reality that exists in idealised conditions. At least, it's not clear to me that there's anywhere where the idealised conditions obtain. Do you see this as a problem? The way in which SR approximates GR is mathematically well defined and well understood.

Physical reality does not actually impose those constraints, and thus relativity of sequence is always possible. Once again, we have the undermining of the notion of cause and effect that worried me.

I don't think this was quite the lesson - though as you can see, there was controversy and...maybe something more...

I would summarise the main points as: (a) the temporal order of two space-like events is dependent on inertial frame, but, in standard interpretations of SR*, this has no causal implications as such events lie outside the light cone and thus are causally independent; (b) very crazy/artificial non-inertial frames may be constructed on which a later event (my death) has a smaller coordinate time than an earlier event (my birth); but such frames are so artificial - really little more than a choice of labelling or giving coordinates to distant events - that nobody should try and read off the causal story or physical story off the numbers of the resulting chart; (c) some very strange solutions of GR are possible, which allow a kind of circular causation, but these solutions seem removed from reality and, at least locally, there's an event by event causal story - it's just that it loops around on itself; Even in this model, though, this circular causal chain is not a frame dependent matter.

tl;dr
I don't think you should worry.

*e.g. no tachyons.

 

[seto6]

if a happens first and causes b to happen later in all frame we would see a then b. the events must have their order in all frame.

 

[bcrowell]

if a happens first and causes b to happen later in all frame we would see a then b. the events must have their order in all frame.

It would be nice if relativity worked that way, but it doesn't. The field equations of GR admit solutions with closed timelike curves. In a spacetime with CTCs, you can't even define an ordering, much less ensure that it's coordinate-independent.

Re the discussion of accelerating frames, I'm skeptical that any of it has any significance. Changing frames of reference is just a change of coordinates. By a change in coordinates, I can always trivially reorder events in the sense of reversing the numerical ordering of their time coordinates. For example, I can simply do the coordinate transformation t \rightarrow -t. The only thing that has a coordinate-independent significance is a closed timelike curve.

 

[JesseM]

It would be nice if relativity worked that way, but it doesn't. The field equations of GR admit solutions with closed timelike curves. In a spacetime with CTCs, you can't even define an ordering, much less ensure that it's coordinate-independent.

This is true, but one thing to note about this is that a lot of the CTC solutions require an infinite universe that has some "unrealistic" properties throughout, like a dense rotating cylinder of infinite length (the Tipler cylinder) or for the entire universe to have some nonzero rotation (the Godel metric, discussed here). If you want to create a finite region where CTCs are allowed in an otherwise "normal" universe, like time travel based on a traversable wormhole, a result by Hawking proved that you must use exotic matter which violates the "weak energy condition" (see third paragraph here), and at least in the case of wormholes some other energy conditions need to be violated too (see here, and note that quantum effects like the Casimir effect may not be sufficient). It's not known whether matter or fields that violate all these energy conditions are actually allowed by the fundamental laws of nature, so GR solutions involving them may not correspond to anything that could be realized in nature, even in principle (and this is before we get into the issue of whether CTC solutions might be one where GR's predictions would depart significantly from those of a theory of quantum gravity--some analysis suggests that in semiclassical gravity the energy density of quantum fields would always go to infinity on the boundary between the CTC region and the non-CTC region, which would indicate this is a situation where semiclassical gravity breaks down and a full theory of quantum gravity is needed)

 

[bcrowell]

@JesseM: It seems like we have two parallel threads going on here, one on SR and one on GR, which is making it hard to keep the discussion coherent. The SR posts are swamping the GR posts, and therefore a lot of us posting on GR are repeating ourselves or repeating each other. I'm going to start a separate thread for the GR stuff, and I hope you don't mind if I quote your (very interesting!) post #73 there in full.

 

[JesseM]

@JesseM: It seems like we have two parallel threads going on here, one on SR and one on GR, which is making it hard to keep the discussion coherent. The SR posts are swamping the GR posts, and therefore a lot of us posting on GR are repeating ourselves or repeating each other. I'm going to start a separate thread for the GR stuff, and I hope you don't mind if I quote your (very interesting!) post #73 there in full.

Sure, go for it.

 

[yuiop]

So I suppose, to bring it back to the original subject of the thread, what you are telling me is that within the idealised constraints of an inertial reference frame, changing the sequence of events is not possible. Physical reality does not actually impose those constraints, and thus relativity of sequence is always possible. Once again, we have the undermining of the notion of cause and effect that worried me.

In the strictly flat space SR context, it is possible for two inertial observers to have a different opinion about the order of two events, if the two events do not share the same light cone. In this special case it is not possible for one of these events to be the cause of the other, so this case has no sigificance on the notion of cause and effect. If the events are both located in a common light cone and so one event could in principle be the cause of the other, then all inertial observers will agree on which event came first.

Things are a bit different in GR. Things are lot more complicated and I am sure you will find some experts say CTCs, time travel, hyperspace jumping to distant galaxies (or even other universes) via black holes and worm holes is possible and another group of experts that would disagree.

 

[yuiop]

Yes. If you simply take Minkowski space and identify the surface t=t_1 with the surface t=t_2, then you have a spacetime that has CTCs and zero intrinsic curvature everywhere.

Can someone explain how the above works? How is it possible to have CTCs (i.e. return to a time in the past) in flat space?

Is just a case of relabelling time coordinates so that you effectively call tomorrow, yesterday, but no actual time travel or reversal of causality has really occured?

 

[yossell]

Can someone explain how the above works? How is it possible to have CTCs (i.e. return to a time in the past) in flat space?

My guess is that it's like the construction of a cylinder from a flat piece of paper by identifying the lines y = 0 and y = 1, except that its two lines of simultaneity that are identified. This is only a change of topology - intrinsically, the cylinder is still a flat surface, the Euclidean distances are still pythagorean. So the construction is compatible with the metric being flat.

I've not seen it before though - very interesting - so I'm not immediately sure whether it works in Minkowski space-time, or whether there's some hidden problem.

 

[yossell]

Can someone explain how the above works? How is it possible to have CTCs (i.e. return to a time in the past) in flat space?

The reason this is not obviously coherent to me (bracketing the CTCs) is this: the surfaces picked out are dependent on a frame. In different frames, different lines of simultaneity. The simultaneity lines that are not parallel to this surface will, it seems, repeatedly curl around this surface, as the x-coordinate is not bounded, and the t-lines will loop around oddly too.

Is this just odd, or is it somehow in conflict with SR?

 

[JesseM]

The reason this is not obviously coherent to me (bracketing the CTCs) is this: the surfaces picked out are dependent on a frame. In different frames, different lines of simultaneity. The simultaneity lines that are not parallel to this surface will, it seems, repeatedly curl around this surface, as the x-coordinate is not bounded, and the t-lines will loop around oddly too.

Is this just odd, or is it somehow in conflict with SR?

A universe with a closed spatial topology (so if you travel far enough in any direction you return to your place of origin) can have what seems to be a preferred global frame even if in any small region of spacetime the laws of physics work the same in any frame (see this thread), so I think the same would apply here. I don't really see this as a conflict with SR but I guess it depends on how you define "SR".

 

[Al68]

I have a question about non-inertial frames: In the kinds of frames that you're talking about, one and the same space-time point will be assigned two different coordinates - e.g. the point of intersection of the lines of simultaneity. But does the concept of a frame allow for this to happen? The time of an event will be multivalued in such a frame. Do people know if this is ok?

A single event could have several time coordinates in an accelerated frame. If the dead can rise from the grave once, why not several times?

 

[Rasalhague]

A single event could have several time coordinates in an accelerated frame. If the dead can rise from the grave once, why not several times?

The definitions I've read of a manifold include the idea of "coordinate functions" which each associate each point in their domain with (in the case of a real manifold) a single real number. A function (map, mapping) is usually defined to exclude multi-valuedness. So if I've understood this right, a single event could have no more than one time coordinate in a single, given frame (chart, coordinate system), although there might be any number of charts that include that event in their domain, and they won't in general agree on its time coordinate.

http://mathworld.wolfram.com/CoordinateChart.html

I think the other side of the Rindler horizon is like the north pole in this example, not part of U, the domain of the chart.