Solving Einstein Field Equations for Minkowski Space with CTC

[PAllen]

Here's another way of looking at this. Imagine 'unrolling' a wrapped time spacetime, repeating history over and over. This should be identical, as far as physical law to the wrapped spacetime. Literally, assuming physical law was consistent everywhere, and you took the initial (actually, any ) state of the wrapped universe and recreated it in a normal universe (assuming classical determinism), the result should be cyclic repeat of states of the universe.

Thus a time-cylinder universe allows only physical states / solutions with the property that throughout the universe, following natural law, the state repeats. I don't think there is any physical law that prevents this, it is just effectively zero probability. Such states are the only ones allowed in a time-cylinder universe.

 

[PAllen]

Here's another way of looking at this. Imagine 'unrolling' a wrapped time spacetime, repeating history over and over. This should be identical, as far as physical law to the wrapped spacetime. Literally, assuming physical law was consistent everywhere, and you took the initial (actually, any ) state of the wrapped universe and recreated it in a normal universe (assuming classical determinism), the result should be cyclic repeat of states of the universe.

Thus a time-cylinder universe allows only physical states / solutions with the property that throughout the universe, following natural law, the state repeats. I don't think there is any physical law that prevents this, it is just effectively zero probability. Such states are the only ones allowed in a time-cylinder universe.

[EDIT] Note, if there ever were a normal topology universe with a repeating state, it seems it would be indistinguishable from a time cylinder universe. This answers the entropy question: a time cylinder universe with standard local physical laws only admits states that entropically unusual, e.g. where scattered billiard balls can re-form the starting triangle.

 

[Altabeh]

Um, why do you think "I have no idea" what it is? Perhaps you could point out a specific flaw in my comments instead of just making unproductive derogatory comments like this.

There is one big hole in our comments here: the lack of scientific percision. If you really think that we can talk about, in relation to the topic here, a new spacetime with CTCs whatever it is, please provide us with the mathematical construction. From the beginning you've just insisted on telling a story about something you think might be true! Okay, now what is the mathematics? I think people here have confused science fiction with physics! These are not things that we can sort out by just doing a little conversation. I'm waiting for a mathematical version of your spacetime with the asscociated Penrose diagram!

Not according to the http://ls.poly.edu/~jbain/philrel/philrellectures/15.TimeMachines.pdf from the Stanford Encyclopedia of Philosophy, where the two finite strips are labeled P1/P4 and P2/P3, and the article says that it is specifically the deletion and identification of these points at the boundaries of the strips that make the spacetime singular:

My mistake! Now that I look back at my old notes, I see Rolled-Up somewhere there! In fact I have no idea about DP, now!

Are you referring to Mentz114's choice of coordinate system, or my alternate choice in the paragraph you were replying to above?

I strongly suggest you to take a look at post https://www.physicsforums.com/showpost.php?p=3069017&postcount=43". I think it answers your call!

Yes, in Mentz114's coordinate system since sin(0)=0, then if dx=dy=dz=0 it will be true that ds^2=0. But as I said this is merely a sign of a badly-behaved coordinate system, a coordinate system where it would be impossible for any timelike worldline to actually be "at rest" at t=0, and where the t-axis is lightlike rather than timelike at this point (Rindler coordinates have the same problem at the Rindler horizon at x=0...no timelike observer could be 'at rest' there). Coordinate issues like this don't mean there is any physical problem with the spacetime, for example an actual timelike worldline will never be accelerated to the speed of light. Would you be willing to discuss my coordinate system rather than Mentz114's?

I said that this idea is leaky that a topological change in, say, Minkowski, spacetime would make it impossible to see any change in physics! If this is not the case, then people would not try to discuss possibility of time traveling in such spacetimes because that is just a mathematical construction! I don't believe it! Indeed, Rindler metric has also a Rindler obsever to whom physics doesn't change! To another observer, why not? Back to this, even a wormhole is a topological structure but when constructing a ring using lots of hese structures, as in the example of a Roman ring, (just a change in topology of spacetime) energy conditions seem not to be broken in contrary to the material feature of the original one-wormhole spacetime where E<0.

They do say that the "rolled up Minkowski spacetime" (not the DP spacetime) is obtained by identifying +infinity and -infinity on ordinary Minkowski spacetime. But I think they just mean that this is how you can look at the rolled up Minkowski time topologically, not that a given timelike worldline must actually experience an infinite proper time before performing a single "loop".

No, this is what you exactly do to time axis! But doing so seems to have a topological effect in spacetime since yet the metric is a solution to the vacuum field equations of GR. I guess this is because maybe they are doing a coordinate transformation. I've never seen a explicit metrical form assigned to this spacetime, have you?

Are you talking about coordinate singularities (like the event horizon of a black hole) or physical singularities (which cause a spacetime to be geodesically incomplete because worldlines simply "end" when they hit the singularity)? It might be that there if we plotted timelike worldlines in Mentz114's system we'd find some coordinate singularities where a worldline would take an infinite coordinate time to reach a boundary, but this would not be proof of a physical singularity in the spacetime itself.

I think I answered this above. If still questionable, we can discuss it!

No one said anything about obtaining CTCs by a "coordinate transformation", the idea is to perform a topological identification of what would be two different spacelike surfaces in Minkowski spacetime, producing a spacetime with a different topology but the same curvature.

I don't know about no-one but topic says something else!

he curvature has not changed but the topology has; GR simply doesn't say anything about global topology, although it would be easy to make a slight modification to it by adding some rule about topology, and the resulting theory would be just as consistent with empirical observations. While the idea of producing CTCs by topology change isn't something anyone thinks is likely to be true in reality, there are some physicists who investigate the idea that the universe may have a nontrivial spatial topology which could make it finite in spatial extent even if the curvature is zero or negative, see here for example.

I don't think physicists are sure about saying something unknown is likely to happen or not! They just predict what would be the most possible "true guess" on a issue like time machines! Of course one thing is obvious here and it is how serious a topological change can be for a physicist to identify it as a source of CTCs thus a physical possibility!

AB

 

[JesseM]

There is one big hole in our comments here: the lack of scientific percision. If you really think that we can talk about, in relation to the topic here, a new spacetime with CTCs whatever it is, please provide us with the mathematical construction.

Altabeh, it rather seems that you don't understand what "topology" is. A mathematical description of a topology change is simply a description of which points are to be identified, and I already did that--I said you should take an inertial coordinate system in ordinary Minkoswki spacetime, then identify points on one spacelike surface t=T₀ with points at the same position coordinates on another spacelike surface at t=T₁ (this 'identification' is not just a matter of the coordinates we assign, it's a topological identification where they really become the same physical points in the spacetime). If you think a topology change should involve any change to the metric equation whatsoever you're confused, the metric only describes local curvature so the metric in this spacetime with altered topology is still just ds^2 = dt^2 - dx^2 - dy^2 - dz^2.

From the beginning you've just insisted on telling a story about something you think might be true! Okay, now what is the mathematics? I think people here have confused science fiction with physics! These are not things that we can sort out by just doing a little conversation. I'm waiting for a mathematical version of your spacetime with the asscociated Penrose diagram!

The above is a mathematical description, and I already gave it to you before. There isn't really any standard convention in Penrose diagrams for identifying topology changes, but in topology itself a common convention is to draw arrows on the edge of a surface that define which edges are to be identified and in which direction, called a "gluing diagram", as illustrated on http://www.ornl.gov/sci/ortep/topology/orbfld2.html :

[PLAIN]http://www.ornl.gov/sci/ortep/topology/topo5.gif

So, presumably one could use the same type of convention for a Penrose diagram, drawing arrows on an upper and lower spacelike surface to show they are identified.

I said that this idea is leaky that a topological change in, say, Minkowski, spacetime would make it impossible to see any change in physics!

There would be no change in local physics, but certainly you would notice if waiting long enough took you into your own past and allowed you to meet your younger self! Similarly in a different spacetime where a topology change has caused space to become finite (while time is still infinite), it would be a change that if you traveled far enough in a straight line you would return to your planet of origin (at a later date than when you left).

Back to this, even a wormhole is a topological structure but when constructing a ring using lots of hese structures, as in the example of a Roman ring, (just a change in topology of spacetime) energy conditions seem not to be broken in contrary to the material feature of the original one-wormhole spacetime where E<0.

Wormholes and Roman rings both involve changes to the curvature of spacetime (at least 'realistic' wormholes do), they are not only a topological change where the curvature remains Minkowski everywhere.

No, this is what you exactly do to time axis! But doing so seems to have a topological effect in spacetime since yet the metric is a solution to the vacuum field equations of GR. I guess this is because maybe they are doing a coordinate transformation. I've never seen a explicit metrical form assigned to this spacetime, have you?

As I said, the metrical form is just that of ordinary Minkowski spacetime, ds^2 = dt^2 - dx^2 - dy^2 - dz^2. Again I think you need to read up a little on the meaning of "topology" and how it is different from curvature if you think a change in topology would involve any change to the metric.

Are you talking about coordinate singularities (like the event horizon of a black hole) or physical singularities (which cause a spacetime to be geodesically incomplete because worldlines simply "end" when they hit the singularity)? It might be that there if we plotted timelike worldlines in Mentz114's system we'd find some coordinate singularities where a worldline would take an infinite coordinate time to reach a boundary, but this would not be proof of a physical singularity in the spacetime itself.

I think I answered this above. If still questionable, we can discuss it!

I don't see any previous post where you distinguish between physical and coordinate singularities and say if you think there are any physical singularities in this spacetime. Can I take it that your answer is "no, there are no physical singularities here"?

No one said anything about obtaining CTCs by a "coordinate transformation", the idea is to perform a topological identification of what would be two different spacelike surfaces in Minkowski spacetime, producing a spacetime with a different topology but the same curvature.

I don't know about no-one but topic says something else!

What do you mean, "says something else"? The original post doesn't say that a "coordinate transformation" can create CTCs.

 

[edgepflow]

JesseM, that was an informative post.

Just to confirm: we are dealing with a topology change only to flat Minkowski spacetime. Thus, the Ricci Tensor and stress energy tensor are always zero.

The question of how to calculate a stress energy tensor to bend spacetime into a time cylinder does not apply.

Let me know if I got this straight.

 

[JesseM]

JesseM, that was an informative post.

Just to confirm: we are dealing with a topology change only to flat Minkowski spacetime. Thus, the Ricci Tensor and stress energy tensor are always zero.

The question of how to calculate a stress energy tensor to bend spacetime into a time cylinder does not apply.

Right. I'm not sure that any stress energy tensor can "bend spacetime into a time cylinder" under the simplest topology (unlike with space, where space is bent into a finite sphere for the simplest choice of topology in the FLRW metric with positive spatial curvature), although it depends on exactly what you mean by the phrase "time cylinder" (I think a good definition would be that you can slice the complete 4D spacetime up into a series of 3D spacelike surfaces where you have a 'first' and 'last' surface' that are identified topologically, such that a timelike curve only takes a finite proper time to get from the first to last...I don't think this description would fit either the Godel spacetime or the Tipler cylinder spacetime though I'm not sure about this).