Solving Einstein Field Equations for Minkowski Space with CTC

[JesseM]

Topology is irrelevant that allows you to have Deutsch-Politzer spacetime in GR with CTCs.

But we weren't talking about a Deutsch-Politzer spacetime where only two finite spacelike strips cause one to go back in time (as seen in http://plato.stanford.edu/entries/time-machine/figure2.html ), rather about a spacetime where you have two infinite (and unavoidable; all timelike worldlines cross them) spacelike surfaces that are identified with one another, as if the time axis were the circumferential axis of an infinite cylinder.

The fact is that here the choice of coordinates is messy!

Mentz114 said the choice of coordinates was just the first thing that came to mind--I think it would be much easier just to use a standard inertial frame where the metric is $$ds^2 = dt^2 - dx^2 - dy^2 - dz^2$$, and then add the condition that the time coordinate can lie in the range T₀ ≤ t < T₁. Presumably it would be possible to find the coordinate transformation between this system and the coordinate system suggested by Mentz114, both should just be different coordinate systems on the same spacetime.

ds^2=0[/tex] hold. I guess there is no misunderstanding on my side in this case!

What's your reasoning that an object which started out at rest would ever have ds² = 0? I haven't looked at Mentz114's coordinate system in detail, but even if this is true it would presumably just be an odd feature of the coordinate system that some lightlike worldline could have a coordinate velocity of 0 at some point, you should be able to transform into an inertial frame where light always moves at c. Assuming that's the case, there'd be nothing physically unusual going on here.

Oh wait a sec! This is a very poor definition of the Deutsch-Politzer spacetime where time resets after you walk around the cylinder-like spacetime.

Huh? I never said anything about time resetting "after you walk around the cylinder-like spacetime", and that isn't true in a Deutsch-Politzer spacetime either. I was contrasting the "groundhog day spacetime", where space is infinite but time is finite, with a different flat spacetime where time is infinite (so there is never any time reset) but space is finite like a cylinder (so you can 'walk around' it and return to the same point in space, but at a different time).

The problem is that staying at rest (i.e. spatial coordinates do not change) in this spacetime is equivalent to starting to move at the speed of light whenever the observer's clock ticks any multiple of pi seconds!

Again, that would seem to have nothing to do with any real feature of the spacetime itself, just a coordinate system on that spacetime. Similarly in Rindler coordinates one would have to move at the speed of light to stay at rest at position x=0, but this is just because of a coordinate singularity at that point, it has nothing to do with the underlying spacetime geometry which is just ordinary Minkowski spacetime.

See, for example, http://arxiv.org/abs/gr-qc/9803020v1

This paper only deals with singularities in the Deutsch-Politzer spacetime where there is a finite region of spacetime where CTCs are possible, I don't think the conclusions would apply to a groundhog day spacetime where two infinite spacelike surfaces are identified and there are no worldlines which avoid them.

 

[yuiop]

Could someone just make clear to me something about this "groundhog day" metric. Does it actually violate the second law of thermodynamics? Would a wineglass that is shattered reform itself periodically or would an organism that grows and dies periodically be reborn? If not, is it just a mathematical oddity using contrived clocks? I suspect the latter, but I would be interested in any different viewpoints.

 

[JesseM]

Could someone just make clear to me something about this "groundhog day" metric. Does it actually violate the second law of thermodynamics? Would a wineglass that is shattered reform itself periodically or would an organism that grows and dies periodically be reborn?

Yes, it really is an "eternal return" in this sense (although one might imagine everything is always at maximum entropy so there is no violation of the 2nd law). The finiteness of time is just as physically real as the finiteness of space in the flat spacetimes with weird topologies that I mentioned.

 

[Rasalhague]

http://books.google.co.uk/books?id=...ce=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

Some pages of Gott's book can be read here. See also pp. 139-140. Does his mention of the presence of energy density and pressure in "cylindrical Groundhog Day spacetime" make it something different from simply Minkowski space with circular/wraparound/looping time?

There are some nice illustrations, including some of flat spacetime exactly like Minkowski space except that two spacelike hypersurfaces are identified, in these lecture notes for a course on the http://ls.poly.edu/~jbain/philrel/[/url : lectures 15 and 16, Time Travel Parts I & II.

 

[PAllen]

There are some nice illustrations, including some of flat spacetime exactly like Minkowski space except that two spacelike hypersurfaces are identified, in these lecture notes for a course on the http://ls.poly.edu/~jbain/philrel/[/url : lectures 15 and 16, Time Travel Parts I & II.

I see that using overlapping coordinate patches, all inertial minkowski, removes many superficial difficulties with this spacetime. However, I wonder what the proposed physical interpretation of a smooth timelike worldline (actually, it is an always inertial world line) spiralling through:

(T,x)=(0,0), (0,1),(0,2),(0,3), ...

(assumption T=2 worldsheet has been joined with T=0 worldsheet; T only needs values [0,2) on 'standard' coordinate patch).

It suggests that at any T, a particle is in a possibly infinite number of positions.

 

[edgepflow]

I think you misunderstood, my point was that with a "folded cylinder" there would be no "finite region" where CTCs were possible, instead they would be possible from any possible point in spacetime. Thus, Hawking's theorem about the need for exotic matter wouldn't apply here.

Acknowledged. Does anyone know where a GR solution to this spacetime is written down?

 

[Altabeh]

But we weren't talking about a Deutsch-Politzer spacetime where only two finite spacelike strips cause one to go back in time (as seen in http://plato.stanford.edu/entries/time-machine/figure2.html ), rather about a spacetime where you have two infinite (and unavoidable; all timelike worldlines cross them) spacelike surfaces that are identified with one another, as if the time axis were the circumferential axis of an infinite cylinder.

Well, I guess you have no idea what a Deutsch-Politzer spacetime is! You should take a look at this paper http://ls.poly.edu/~jbain/philrel/philrellectures/15.TimeMachines.pdf

In the case of a DP spacetime, time axis is bent over in such a way that the neighbourhood of +infinity meets that of -infinity to form the circumferential axis of an infinitely extended cylinder. Yet, I don't understand if there is any difference between this groundhog day spacetime and DP, what it would be!

Mentz114 said the choice of coordinates was just the first thing that came to mind--I think it would be much easier just to use a standard inertial frame where the metric is $$ds^2 = dt^2 - dx^2 - dy^2 - dz^2$$, and then add the condition that the time coordinate can lie in the range T₀ ≤ t < T₁. Presumably it would be possible to find the coordinate transformation between this system and the coordinate system suggested by Mentz114, both should just be different coordinate systems on the same spacetime.

I had not heard of "groundhog day" spacetime in my entire life until yesterday! But definitely the choice of coordinate system is fully wrong if there is no identification of bounds for the time-component!

What's your reasoning that an object which started out at rest would ever have ds² = 0? I haven't looked at Mentz114's coordinate system in detail, but even if this is true it would presumably just be an odd feature of the coordinate system that some lightlike worldline could have a coordinate velocity of 0 at some point, you should be able to transform into an inertial frame where light always moves at c. Assuming that's the case, there'd be nothing physically unusual going on here.

Starting at $$t=0$$ means starting from a spacelike surface and thus traveling faster than light! (in the notation Mentz114 uses $$ds^2>0$$ means FTL!) If one is at rest at t=0, so $$dx=dy=dz=0$$ and this basically says that $$ds^2=0$$ at $$(0,x_0,y_0_z_0)$$. Is there any ambiguity about this simple calculus? Either the coordinate system is very badly chosen or I fail to make you understand what really goes on!

Huh? I never said anything about time resetting "after you walk around the cylinder-like spacetime", and that isn't true in a Deutsch-Politzer spacetime either.

I said that you don't have any idea about DP spacetime! There we have CTCs and if you study that paper I cited above, everything will be clear to you. After you walk around the cylider along one of the cross-sectional surfaces you get to where you were at with having your time the same as it was at the start point! Indeed, your time-line starts at some T>0, and then goes to infinity and then jumps to -infinity and then reaches 0 to only be reset when it becomes T! This is all behind the name "time-machine" given to this example!

I was contrasting the "groundhog day spacetime", where space is infinite but time is finite, with a different flat spacetime where time is infinite (so there is never any time reset) but space is finite like a cylinder (so you can 'walk around' it and return to the same point in space, but at a different time).

The idea of time-reset is way over the top in being unrealistic! All in all, with that coordinate system defined by Mentz114, I think we are forced to bond time to avoid singularities! If you want time to be infinite, you have to study DP spacetime but yet in their picture singularities turn out to be irremovable because actually one needs to cut off two balls centered at, say, origin in each of the spacelike hypersurfaces (t=0 and t=1) to get it! Though you can smooth out such a spacetime, one cannot ever suss out the problem of having a smooth Lorentz metric on the smooth Politzer manifold whose answer has been unveiled to be negative by Chamblin, Gibbons and Steif.

Again, that would seem to have nothing to do with any real feature of the spacetime itself, just a coordinate system on that spacetime. Similarly in Rindler coordinates one would have to move at the speed of light to stay at rest at position x=0, but this is just because of a coordinate singularity at that point, it has nothing to do with the underlying spacetime geometry which is just ordinary Minkowski spacetime.

Well, then none of the efforts on explaining why DP spacetime can be a time-machine is meaningless because after all doing a transformation reverts your metric back to the original one! Sometimes it is not the case, JeeseM. CTCs do not exist in a spacetime transformed from Minkowskin spacetime by a coordinate transformation! If they exist, they all seem to not exist for real because nothing has changed the physical nature of spacetime! True, but then does GR have this capability to address its own defects? I guess no!

This paper only deals with singularities in the Deutsch-Politzer spacetime where there is a finite region of spacetime where CTCs are possible, I don't think the conclusions would apply to a groundhog day spacetime where two infinite spacelike surfaces are identified and there are no worldlines which avoid them.

You don't think what? Do you know what a DP spacetime is? Please read about it and then I'm going to continue this discussion! Otherwise we are wasting time over here!

AB

 

[Mentz114]

I had not heard of "groundhog day" spacetime in my entire life until yesterday! But definitely the choice of coordinate system is fully wrong if there is no identification of bounds for the time-component!

Me neither. It is badly specified. Everyone agrees on that.

Starting at $$t=0$$ means starting from a spacelike surface and thus traveling faster than light! (in the notation Mentz114 uses $$ds^2>0$$ means FTL!) If one is at rest at t=0, so $$dx=dy=dz=0$$ and this basically says that $$ds^2=0$$ at $$(0,x_0,y_0,z_0)$$. Is there any ambiguity about this simple calculus? Either the coordinate system is very badly chosen or I fail to make you understand what really goes on!
AB

True. Singularity at t=0.

 

[JesseM]

Well, I guess you have no idea what a Deutsch-Politzer spacetime is! You should take a look at this paper http://ls.poly.edu/~jbain/philrel/philrellectures/15.TimeMachines.pdf

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.

In the case of a DP spacetime, time axis is bent over in such a way that the neighbourhood of +infinity meets that of -infinity to form the circumferential axis of an infinitely extended cylinder.

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:

as illustrated by the (1 + 1)-dimensional version of Deutsch-Politzer spacetime[15] (see Figure 2), which is constructed from two-dimensional Minkowski spacetime by deleting the points p1–p4 and then gluing together the strips as shown ... The deletion of the points p1–p4 means that the Deutsch-Politzer spacetime is singular in the sense that it is geodesically incomplete.

Mentz114 said the choice of coordinates was just the first thing that came to mind--I think it would be much easier just to use a standard inertial frame where the metric is $$ds^2 = dt^2 - dx^2 - dy^2 - dz^2$$, and then add the condition that the time coordinate can lie in the range T₀ ≤ t < T₁. Presumably it would be possible to find the coordinate transformation between this system and the coordinate system suggested by Mentz114, both should just be different coordinate systems on the same spacetime.

I had not heard of "groundhog day" spacetime in my entire life until yesterday! But definitely the choice of coordinate system is fully wrong if there is no identification of bounds for the time-component!

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

Starting at $$t=0$$ means starting from a spacelike surface and thus traveling faster than light!

Again, are you talking about Mentz114's coordinate system or mine? In my coordinate system no object is actually traveling along the spacelike surface t=T₀ in my example, the spacelike surface is simply the set of points which we wish to topologically identify with a different spacelike surface, just as the spacelike segments L+ and L- are identified in the diagram of DP spacetime from the paper you linked to. And just like in the second diagram on p. 7 of that paper in the section 'Prediction I', the worldines of actual particles are never parallel to these surfaces, rather the worldlines hit one surface at an angle (between 45 degrees and 90 degrees in a spacetime diagram) and reappear at the corresponding point on the other surface, at the same angle relative to that surface.

If one is at rest at t=0, so $$dx=dy=dz=0$$ and this basically says that $$ds^2=0$$ at $$(0,x_0,y_0_z_0)$$. Is there any ambiguity about this simple calculus?

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 you don't have any idea about DP spacetime! There we have CTCs and if you study that paper I cited above, everything will be clear to you. After you walk around the cylider along one of the cross-sectional surfaces you get to where you were at with having your time the same as it was at the start point!

What page of that paper are you looking at? I see no "cylinder" in the discussion of the DP spacetime. P. 7 introduces the DP spacetime, and they explain it is just a Minkowksi spacetime but with the spacelike strips L+ and L- identified, they even say "Deutsch-Politzer spacetime only differs from Minkowski spacetime on these two strips". In Minkowski spacetime you can go infinitely far in any spatial direction and never return to your point of origin, the same would be true here.

I just looked back over earlier parts of the paper and realized that probably you mistakenly think that the "rolled-up Minkowski spacetime" on p. 1 is a DP spacetime. But that's wrong, they never use the words "Deutsch-Politzer spacetime" to describe that spacetime, they only introduce this term on p. 7 to describe the different spacetime illustrated there.

Indeed, your time-line starts at some T>0, and then goes to infinity and then jumps to -infinity and then reaches 0 to only be reset when it becomes T! This is all behind the name "time-machine" given to this example!

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".

All in all, with that coordinate system defined by Mentz114, I think we are forced to bond time to avoid singularities!

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.

Well, then none of the efforts on explaining why DP spacetime

No one is talking about DP spacetime but you, and it seems like you have gotten the "rolled-up Minkowski spacetime" on p.1 of that paper confused with the DP spacetime on p.7, they are not the same thing at all. Googling for "rolled-up Minkowski" in quotes seems to indicate that this is the standard term for what this thread is calling "groundhog day spacetime".

Sometimes it is not the case, JeeseM. CTCs do not exist in a spacetime transformed from Minkowskin spacetime by a coordinate transformation!

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.

If they exist, they all seem to not exist for real because nothing has changed the physical nature of spacetime!

The 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.

 

[PAllen]

I see that using overlapping coordinate patches, all inertial minkowski, removes many superficial difficulties with this spacetime. However, I wonder what the proposed physical interpretation of a smooth timelike worldline (actually, it is an always inertial world line) spiralling through:

(T,x)=(0,0), (0,1),(0,2),(0,3), ...

(assumption T=2 worldsheet has been joined with T=0 worldsheet; T only needs values [0,2) on 'standard' coordinate patch).

It suggests that at any T, a particle is in a possibly infinite number of positions.

A few general comments first: George Jones demonstrates what JesseM has been saying: all the singular features of Mentz metric are pure coordinate issues; this metric can be obtained by coordinate transform from the simple anomaly free coordinates of identifying two spacelike slices of Minkowski space.

Also I see the points where ds^2=0 on timelike curves correspond to points where the metric determinant is zero - that suggests normal interpretation at these points is invalid. If you just remove these points from the coordinate patch, it is well behaved elsewhere, and limits approaching these points produce physically consistent results even for these poor coordinates.

As for my question above (quoted), I think my confusion is focusing on coordinate time. With proper time, a spiraling world line advances in proper time continuously; it happens to go through the different spacetime events (0,0),(0,1),(0,2), etc. but these are distinct spacetime events, reached at different proper times along the worldline.

A really strange situation is a CTC where a baby grows up and the adult meets the baby in a crib. Since the adult doesn't coincide with the baby on the timeline after the meeting either the adult would be 'forced' to step away, or would simply vanish at that point.

Interestingly, it seems that CTC curves occupied by matter divide the spacetime into 'uncrossable' regions. If some object wasn't already on the CTC, then the normal way these are interpreted says they can never cross certain positions.

 

[JesseM]

A really strange situation is a CTC where a baby grows up and the adult meets the baby in a crib. Since the adult doesn't coincide with the baby on the timeline after the meeting either the adult would be 'forced' to step away, or would simply vanish at that point.

I don't get it, why would the adult be forced to step away? It's not like if the adult reached down and picked up the baby the worldlines of their constituent particles would somehow merge, any more than you'd expect this to happen for any other random adult holding the baby.

 

[PAllen]

I don't get it, why would the adult be forced to step away? It's not like if the adult reached down and picked up the baby the worldlines of their constituent particles would somehow merge, any more than you'd expect this to happen for any other random adult holding the baby.

The adult is standing exactly where the crib and baby 'will be'. If the don't step away or vanish, they would physically overlap the baby; since that isn't the state of the world line 'before', they must either step away (just enough not to occupy the same space), or they have to vanish.

 

[PAllen]

Interestingly, it seems that CTC curves occupied by matter divide the spacetime into 'uncrossable' regions. If some object wasn't already on the CTC, then the normal way these are interpreted says they can never cross certain positions.

I realize this comment is silly, it comes from simplification of only one spatial dimension. You don't need CTCs to have trouble with plausible physics in one spatial dimension.

 

[JesseM]

The adult is standing exactly where the crib and baby 'will be'. If the don't step away or vanish, they would physically overlap the baby; since that isn't the state of the world line 'before', they must either step away (just enough not to occupy the same space), or they have to vanish.

I still don't understand, does the adult know that the crib and the baby "will be" at the adult's current position based on some historical records? If not, this is no different from ordinary Minkowski spacetime, where if you take a God's-eye-view of 4D spacetime as a whole, and know that there is a particular point in space where a crib "will be" at a time T, then if and an earlier time there is someone standing there then even without looking at the details of their future worldline you know they "must" move before time T (and they certainly aren't going to 'vanish' if the same laws of physics are respected throughout this spacetime). If the adult does know that the crib and baby are supposed to be at the position they're standing in the future, this just shows how CTC spacetimes could violate our intuitive sense of free will, but even so I don't see what the significance is of talking about the adult's own younger self as opposed to any other random object that the adult knows from historical records is supposed to soon occupy the position they are currently standing.

 

[PAllen]

I still don't understand, does the adult know that the crib and the baby "will be" at the adult's current position based on some historical records? If not, this is no different from ordinary Minkowski spacetime, where if you take a God's-eye-view of 4D spacetime as a whole, and know that there is a particular point in space where a crib "will be" at a time T, then if and an earlier time there is someone standing there then even without looking at the details of their future worldline you know they "must" move before time T (and they certainly aren't going to 'vanish' if the same laws of physics are respected throughout this spacetime). If the adult does know that the crib and baby are supposed to be at the position they're standing in the future, this just shows how CTC spacetimes could violate our intuitive sense of free will, but even so I don't see what the significance is of talking about the adult's own younger self as opposed to any other random object that the adult knows from historical records is supposed to soon occupy the position they are currently standing.

The adult and baby are not relevant, just a specific example I found amusing. It could be as simple as not being able to stand where a piece of furniture is 'about' to be. I guess it all boils down to the requirement that there is one state of any spatial slice, and in the case where the same slice is in your past and future, the consequences are counter-intuitive.

 

[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).