Solving Einstein Field Equations for Minkowski Space with CTC

[edgepflow]

For Minkowski spacetime, the metric is:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2

I have read there is a solution when the time dimension is "rolled" into a cylinder forming a closed timelike curve. So the BC is t -> [0,T] with t = 0 identical with t = T.

The Field Equation is:

Rab - 1/2 gab R = 8 Pi G Tab.

With gab = diag [-1,1,1,1]

Could I determine the Ricci Tensor & Scalar and solve for the stress energy tensor?

Please point me in the direction to perform this solution or let me know where I can find it.

 

[bcrowell]

The field equations only depend on the local properties, not the global topology. The curvature is zero, and so is the stress-energy tensor.

 

[edgepflow]

The field equations only depend on the local properties, not the global topology. The curvature is zero, and so is the stress-energy tensor.

I noticed the curvature is zero just after I posted this. If the time dimension of space is rolled into a cylinder forming a closed timelike curve, what would be the metric?

 

[Mentz114]

This is the 'groundhog day' metric where time goes in circles - but it is still flat spacetime.


ds² = -sin(t)²dt² + dx² + dy² + dz²

 

[edgepflow]

This is the 'groundhog day' metric where time goes in circles - but it is still flat spacetime.


ds² = -sin(t)²dt² + dx² + dy² + dz²

Thank you !

 

[K^2]

This is the 'groundhog day' metric where time goes in circles - but it is still flat spacetime.


ds² = -sin(t)²dt² + dx² + dy² + dz²

And what happens to proper time of a massive particle at a turning point? That's right, it diverges. The time might be looped here, but no particle will ever reach the turning point by its own clock, so no Groundhog Day.

Topology and geometry of space-time are two entirely different topics. Topology gives you boundary conditions and is a given when you solve Einstein Field Equations. If you need closed time, just set up a periodic boundary condition in R⁴. The result will be equivalent to TxR³.

 

[George Jones]

This is the 'groundhog day' metric where time goes in circles - but it is still flat spacetime.ds² = -sin(t)²dt² + dx² + dy² + dz²

Interesting, I have never seen the groundhog day metric. I think that t is a poor choice of coordinate label.

And what happens to proper time of a massive particle at a turning point? That's right, it diverges.

I have been playing around a bit, and I don't think so.

The time might be looped here, but no particle will ever reach the turning point by its own clock, so no Groundhog Day.

The t component of of the 4-velocity of a particularly natural family of observers diverges, not proper time for the observers.

Topology and geometry of space-time are two entirely different topics. Topology gives you boundary conditions and is a given when you solve Einstein Field Equations. If you need closed time, just set up a periodic boundary condition in R⁴. The result will be equivalent to TxR³.

By T, do you mean S^1? I think that the metric that Mentz114 gave is intended for the spacetime after (or in combination with) identification, i.e., for S^1 \times \mathbf{R}^3.

More later.

 

[Mentz114]

ds² = -sin(t)²dt² + dx² + dy² + dz²

Solving the equations of motion with the initial condition t=0 \rightarrow \tau=0 gives

<br /> \tau=1-\cos(t)<br />

 

[George Jones]

ds² = -sin(t)²dt² + dx² + dy² + dz²

Solving the equations of motion with the initial condition t=0 \rightarrow \tau=0 gives

<br /> \tau=1-\cos(t)<br />

For and an inertial observer with constant x, y, and z, this is what I get as well. The components of such an observer's 4-velocity are

\left( u_t , u_x , u_y , u_z \right) = \left( \frac{1}{\sin t} , 0 , 0, 0 \right).

 

[PAllen]

For and an inertial observer with constant x, y, and z, this is what I get as well. The components of such an observer's 4-velocity are

\left( u_t , u_x , u_y , u_z \right) = \left( \frac{1}{\sin t} , 0 , 0, 0 \right).

Ah, and the norm of this 4 velocity - using the given metric - is identically 1, as it should be. Tricky to use this metric (for me at least), but all looks valid and consistent.

 

[PAllen]

I see that the cause of the 4 velocity's divergent coordinate representation is that it is actually periodically lightlike. Light paths take the form (in X T plane): x=k-cos(t), so any constant position path is periodically tangent to a light path. This is physically dubious - it should not be possible for a real particle's world line to ever be tangent to null geodesic.

So, I think we have a problem if there are no world lines that are everwhere timelike. My initial take is that, in fact, there are no such world lines.

Reactions?

 

[Altabeh]

Ah, and the norm of this 4 velocity - using the given metric - is identically 1, as it should be. Tricky to use this metric (for me at least), but all looks valid and consistent.

This is a very poor metric. It certainly has uncountably many coordinate singularities and so it is not a suitable case for Minkowski spacetime! Taking T=\cos(t), we have -dT^2=-sin^2(t)dt^2. But this only works if 0=&lt;t&lt;\pi/2 which means time is bounded from above!

Very bizarre!

 

[JesseM]

This is a very poor metric. It certainly has uncountably many coordinate singularities and so it is not a suitable case for Minkowski spacetime! Taking T=\cos(t), we have -dT^2=-sin^2(t)dt^2. But this only works if 0=&lt;t&lt;\pi/2 which means time is bounded from above!

Very bizarre!

It's not that time is bounded from above, it's that spacetime has an unusual topology such that when you reach what would be t=pi/2 and x=X, y=Y, z=Z, you are actually back at the point in spacetime that was already labeled with coordinates t=0 and x=X, y=Y, z=Z. Think of the analogous case where space is finite but unbounded due to a weird topology (as discussed in this article), here it would also be the case that at least one of the spatial coordinates could only have a finite extent before "resetting" (assuming you want a well-behaved coordinate system that doesn't assign multiple sets of coordinates to the same physical point in spacetime).

 

[Altabeh]

It's not that time is bounded from above, it's that spacetime has an unusual topology such that when you reach what would be t=pi/2 and x=X, y=Y, z=Z, you are actually back at the point in spacetime that was already labeled with coordinates t=0 and x=X, y=Y, z=Z. Think of the analogous case where space is finite but unbounded due to a weird topology (as discussed in this article), here it would also be the case that at least one of the spatial coordinates could only have a finite extent before "resetting" (assuming you want a well-behaved coordinate system that doesn't assign multiple sets of coordinates to the same physical point in spacetime).

In fact I don't think GR accepts such a nonsense! Because if time here is not bounded, you can stand at someplace say (x_0,y_2,z_0) to only observe you getting accelerated to the speed of light as your hand-watch shows t=k\pi.

AB

 

[JesseM]

In fact I don't think GR accepts such a nonsense!

Topology is irrelevant, GR only demands that the field equations are satisfied at every point on the manifold.

Because if time here is not bounded, you can stand at someplace say (x_0,y_2,z_0) to only observe you getting accelerated to the speed of light as your hand-watch shows t=k\pi.

Huh? There is no "acceleration at the speed of light", the point t=0 really is right nearby the point t=pi/2 - epsilon (where epsilon is some very small number). And of course, the choice of where to put t=0 is an arbitrary coordinate choice with no physical significance, you could easily do a coordinate transformation on the same spacetime and put it somewhere different.

Did you read the article about how an unusual topology can allow a flat spacetime to have a finite but unbounded amount of space? If we put a coordinate system on such a spacetime where at least one of the spatial coordinates "resets" in a similar way, do you think you experience an acceleration when you reach the reset point? Just think of a cylinder, if you wanted to put a coordinate system on that then one of the coordinates would have to reset, but nothing special would happen to an ant walking around the cylinder when it reached the reset point, and again it is an arbitrary coordinate choice where the coordinates reset.

 

[edgepflow]

I attempted to compute the Ricci Tensor for this Groundhog Day metric and found all terms equal zero. Is this correct?

I know it is still considered flat spacetime, but I have read that there is a strange energy momentum tensor to bend time like this.

 

[JesseM]

I attempted to compute the Ricci Tensor for this Groundhog Day metric and found all terms equal zero. Is this correct?

I know it is still considered flat spacetime, but I have read that there is a strange energy momentum tensor to bend time like this.

It should be correct, it's just flat spacetime with a weird global topology. My understanding is that the issue of the "strange energy momentum tensor" only arises when you want to have closed timelike curves in a finite region of an infinite space (with a 'Cauchy horizon' as the boundary between the region where CTCs are possible and the region where they're not), using something like a wormhole...see here:

A more fundamental objection to time travel schemes based on rotating cylinders or cosmic strings has been put forward by Stephen Hawking, who proved a theorem showing that according to general relativity it is impossible to build a time machine of a special type (a "time machine with the compactly generated Cauchy horizon") in a region where the weak energy condition is satisfied, meaning that the region contains no matter with negative energy density (exotic matter). Solutions such as Tipler's assume cylinders of infinite length, which are easier to analyze mathematically, and although Tipler suggested that a finite cylinder might produce closed timelike curves if the rotation rate were fast enough,[38] he did not prove this. But Hawking points out that because of his theorem, "it can't be done with positive energy density everywhere! I can prove that to build a finite time machine, you need negative energy."[39] This result comes from Hawking's 1992 paper on the chronology protection conjecture, where he examines "the case that the causality violations appear in a finite region of spacetime without curvature singularities" and proves that "[t]here will be a Cauchy horizon that is compactly generated and that in general contains one or more closed null geodesics which will be incomplete. One can define geometrical quantities that measure the Lorentz boost and area increase on going round these closed null geodesics. If the causality violation developed from a noncompact initial surface, the averaged weak energy condition must be violated on the Cauchy horizon."[40]

 

[PAllen]

Any reactions to my conclusion that there seem to be no extended 'everywhere timelike' world lines with this metric. That seems to make it not physically plausible.

 

[JesseM]

I see that the cause of the 4 velocity's divergent coordinate representation is that it is actually periodically lightlike. Light paths take the form (in X T plane): x=k-cos(t), so any constant position path is periodically tangent to a light path. This is physically dubious - it should not be possible for a real particle's world line to ever be tangent to null geodesic.

Why should a real particle's path be a "constant position path" though? Perhaps in these coordinates timelike paths must have changing position coordinate. Might help to figure out the coordinate transformation between these coordinates and inertial coordinates.

Maybe a naive question, but I don't really get the point in using the type of non-inertial coordinate system suggested by Mentz114 in the first place--why not just use an inertial coordinate system with a t-coordinate that can only range over a certain set of values? Then the metric would just be the regular Minkowski metric.

 

[George Jones]

First, a little more about the construction of the spacetime. Choose a particular inertial coordinate system (T, x) for 2-dimensional Minkowski spacetime. I use T instead of t because Mentzt114 used t for a completely different coordinate, hence

I think that t is a poor choice of coordinate label.

Using this inertial coordinate system, make the indentification (T, x) ~ (T + 2 , x). This turns spacetime into a cylinder with the the T coordinate being circular. S, we can restrict ourselves to the strip of spacetime where x can take on any value, and 0 \leq T &lt; 2.

Now, implicitly define a new coordinate t by T = 1 - \cos t. Consequently, 0 \leq t &lt; \pi. What is the metric in the non-inertial (t, x) coordinate system?

Light paths take the form (in X T plane): x=k-cos(t), so any constant position path is periodically tangent to a light path.

I don't think so. Certainly, if (t, x) = (t, 1 - cos(t)) and (t, 0) (observer at spatial origin) are both plotted, the curves look tangent at t = 0, but here we're dealing with parametrized curves. For parametrized curves, the notion of tangency depends not on the image of the curve (the events in spacetime on the the curve), but also on choice of parameter (how "fast" the events are traversed).

It might be thought that the components of the tangent vectors to the above two curves are (1, sin(t)) = d/dt (t, 1 - cos(t)) and (1, 0) = d/dt (t, 0), which are the same at t = 0. Certainly, (1, sin(t)) is a lightlike vector, but (1, 0) is not the tangent the tangent vector to a worldline parametrized by proper time. What is needed is

\frac{d}{d \tau} \left( t \left( \tau \right), 0 \right) = \frac{dt}{d \tau} \frac{d}{dt} \left( t \left( \tau \right), 0 \right) = \left( \frac{dt}{d \tau} , 0 \right) = \left( \frac{1}{\sin t} , 0 \right).

This is not lightlike "at" t = 0, so the observer worldline is not tangent to a light path.

 

[PAllen]

Why should a real particle's path be a "constant position path" though? Perhaps in these coordinates timelike paths must have changing position coordinate. Might help to figure out the coordinate transformation between these coordinates and inertial coordinates.

I didn't show my work, but some work I did (not complete and rigorous, though) seemed to show there could not be any extended everywhere timelike world lines at all. I wasn't assuming only constant position. Thus I was hoping those more facile with these calculations would contradict or verify my hypothesis.

 

[George Jones]

PAllen, I'm not sure if you saw them (sorry to be a nag) since we moved to a new page, but I posted some thoughts in post #20.

 

[PAllen]

I don't think so. Certainly, if (t, x) = (t, 1 - cos(t)) and (t, 0) (observer at spatial origin) are both plotted, the curves look tangent at t = 0, but here we're dealing with parametrized curves. For parametrized curves, the notion of tangency depends not on the image of the curve (the events in spacetime on the the curve), but also on choice of parameter (how "fast" the events are traversed).

It might be thought that the components of the tangent vectors to the above two curves are (1, sin(t)) = d/dt (t, 1 - cos(t)) and (1, 0) = d/dt (t, 0), which are the same at t = 0. Certainly, (1, sin(t)) is a lightlike vector, but (1, 0) is not the tangent the tangent vector to a worldline parametrized by proper time. What is needed is

\frac{d}{d \tau} \left( t \left( \tau \right), 0 \right) = \frac{dt}{d \tau} \frac{d}{dt} \left( t \left( \tau \right), 0 \right) = \left( \frac{dt}{d \tau} , 0 \right) = \left( \frac{1}{\sin t} , 0 \right).

This is not lightlike "at" t = 0, so the observer worldline is not tangent to a light path.

I'm still not convinced. Obviously(?) d/d tau is undefined along a light path. Your (1/sin(t),0) is undefined at precisely the points where I think the world line is lightlike. This seems consistent with my claim.

 

[George Jones]

Sorry, I have to run for my bus, and I don't know if I'll have a chance to get to this tonight or tomorrow. Try transforming the curves (T, x) = (T, 0) and (T, x) = (T, T) to t-x coordinates. Also transform the tangent vectors.

 

[Mentz114]

Maybe a naive question, but I don't really get the point in using the type of non-inertial coordinate system suggested by Mentz114 in the first place--why not just use an inertial coordinate system with a t-coordinate that can only range over a certain set of values? Then the metric would just be the regular Minkowski metric.

I never intended that metric to be taken seriously - it's rubbish.
I'm sure a much better 'ground-hog' metric can be found.

 

[K^2]

ds² = -sin(t)²dt² + dx² + dy² + dz²

Solving the equations of motion with the initial condition t=0 \rightarrow \tau=0 gives

<br /> \tau=1-\cos(t)<br />

Except that what you want is the t(\tau), which does diverge. It does end up periodic, though, so yeah, I made a mistake. But I still don't like the divergence where the sign flips.

 

[edgepflow]

It should be correct, it's just flat spacetime with a weird global topology. My understanding is that the issue of the "strange energy momentum tensor" only arises when you want to have closed timelike curves in a finite region of an infinite space (with a 'Cauchy horizon' as the boundary between the region where CTCs are possible and the region where they're not), using something like a wormhole...see here:

Yes, I read that the energy momentum tensor would need exotic matter (maybe an unphysical tensor).

I wanted to try to use my pea brain and calculate this energy momentum tensor with exotic matter. Anyone have any suggestions how to do this?

 

[JesseM]

Yes, I read that the energy momentum tensor would need exotic matter (maybe an unphysical tensor).

I wanted to try to use my pea brain and calculate this energy momentum tensor with exotic matter. Anyone have any suggestions how to do this?

In what spacetime? In Minkowski spacetime (even one with a weird topology) there is no exotic matter, since if there was it would make the curvature nonzero.

 

[Altabeh]

Topology is irrelevant, GR only demands that the field equations are satisfied at every point on the manifold.

Topology is irrelevant that allows you to have Deutsch-Politzer spacetime in GR with CTCs. The fact is that here the choice of coordinates is messy! By "messy" I mean if det g = 0 at some point(s) in spacetime, then either you have chosen a bad coordinate system, or you have to bond coordinates in such a way that spacetime gets bounded to avoid having singularities!

Huh? There is no "acceleration at the speed of light", the point t=0 really is right nearby the point t=pi/2 - epsilon (where epsilon is some very small number). And of course, the choice of where to put t=0 is an arbitrary coordinate choice with no physical significance, you could easily do a coordinate transformation on the same spacetime and put it somewhere different.

You better read carefully! You're at rest and suddenly you're accelerated to the speed of light, so as to have ds^2=0 hold. I guess there is no misunderstanding on my side in this case!

Did you read the article about how an unusual topology can allow a flat spacetime to have a finite but unbounded amount of space? If we put a coordinate system on such a spacetime where at least one of the spatial coordinates "resets" in a similar way, do you think you experience an acceleration when you reach the reset point? Just think of a cylinder, if you wanted to put a coordinate system on that then one of the coordinates would have to reset, but nothing special would happen to an ant walking around the cylinder when it reached the reset point, and again it is an arbitrary coordinate choice where the coordinates reset.

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. Yet, this doesn't solve anything about the problem we have here! 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! Even starting at t=0 is nonsense because there you'll have either a speed greater-than the speed of light or the same feeling that a photon has when traveling at speed c! Indeed, the null geodesic generators of Deutsch-Politzer spacetime are (not all of them) those discussed here! Most important thing here is that this is not even because of a poor choice of metric, but irremovablity of its singularities!

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

AB

 

[edgepflow]

In what spacetime? In Minkowski spacetime (even one with a weird topology) there is no exotic matter, since if there was it would make the curvature nonzero.

I must admit I am an engineering graduate trying to teach myself some GR. So when you ask "what spacetime" I would ask: what are my choices? Are there other non-Minkowski metrics that describe the time dimension folded into a cylinder I could use to find a stress-momentum tensor with exotic matter?

 

[JesseM]

I must admit I am an engineering graduate trying to teach myself some GR. So when you ask "what spacetime" I would ask: what are my choices? Are there other non-Minkowski metrics that describe the time dimension folded into a cylinder I could use to find a stress-momentum tensor with exotic matter?

If spacetime is "folded into a cylinder" wouldn't that mean that you can travel in a closed timelike curve from any point in spacetime? Again, the point of Hawking's result was that if you have a finite region where CTCs are possible, in a larger spacetime where they are not, then there must be exotic matter on the "Cauchy horizon" which forms the boundary between the two. There are various non-Minkowski spacetimes where CTCs are possible everywhere (though not because time is folded into a cylinder), like the Godel metric and the Tipler cylinder, but because these don't meet Hawking's conditions above there does not need to be any exotic matter present in these spacetimes.

 

[PAllen]

Sorry, I have to run for my bus, and I don't know if I'll have a chance to get to this tonight or tomorrow. Try transforming the curves (T, x) = (T, 0) and (T, x) = (T, T) to t-x coordinates. Also transform the tangent vectors.

I see that your coordinate transform #20 generates the sin^2 metric from a glued minkowski metric. I note the glued metric raises a few questions:

Is the total proper time from T=0 to T=4 zero, two or four? What about from T=0 to T=4-epsilon ?

This ambiguity seems to get cast as coordinate divergence after the transform. Altabeh has noted the metric determinant is 0 when t is a multiple of pi (including t=0). This seems problematic. Maybe you really need to restrict yourself to the open interval (0,pi) for t.

Anyway, I continue to find all timelike curves in the x, T coordinates transform to curves with ds^2=0 periodically along them (in the x,t) coordinates (I had derived these curves independently before, directly from the sin^2 metric). I have learned that ds^2=0 implies a lightlike character. Maybe it is valid to say that because it is only at one point, and won't affect integration of interval along the transformed timelike curves, it is fine (?). I certainly do find, as noted in my post #10 that tangent vectors of timelike curves have well defined norm of 1 for all except singular points. So is it ok just fill in these points with their limit?

 

[edgepflow]

If spacetime is "folded into a cylinder" wouldn't that mean that you can travel in a closed timelike curve from any point in spacetime? Again, the point of Hawking's result was that if you have a finite region where CTCs are possible, in a larger spacetime where they are not, then there must be exotic matter on the "Cauchy horizon" which forms the boundary between the two. There are various non-Minkowski spacetimes where CTCs are possible everywhere (though not because time is folded into a cylinder), like the Godel metric and the Tipler cylinder, but because these don't meet Hawking's conditions above there does not need to be any exotic matter present in these spacetimes.

Thanks JesseM. I see what you are saying. Yes, I was interested in this folded cylinder (finite region with CTC) with exotic matter.

If anyone has the popular book "Time Travel in Einstein's Universe" by Gott, please see Figure 17 on Page 135. It shows a "Groundhog Day Vacuum" I was hoping to figure out the stress-momentun tensor to do this in a finite region of space.

Anyway, I hope my question is not pointless.

 

[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?

 

[JesseM]

If spacetime is "folded into a cylinder" wouldn't that mean that you can travel in a closed timelike curve from any point in spacetime?

Thanks JesseM. I see what you are saying. Yes, I was interested in this folded cylinder (finite region with CTC) with exotic matter.

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.

 

[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&gt;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&gt;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.