Is Event Sequencing Relative in the Theory of Manifolds?

[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.