Do Light Cones Truly Tip Over in General Relativity?

[pervect]

Just because a dimension d₁ is orthogonal to a dimension d₂ does not imply that d₁ lies in d₂'s complex plane. It could be, but only if defined as such.

Feel free to demonstrate why Lorentz invariance implies such a connection.

Maybe it's because it's late at night, and I ought to get some sleep, but I'm not following you at all.

Basically, the problem here is that we have gone beyond the realm of talking about specific experiments to some sort definitional discussion or philospohical discussion, and I don't quite see where you are coming from and why you are making the statements you are and what you are asking for when you talk about physical interpretations.

Here is the way I see things.

An object or a person's path through space-time can be described by a time-like worldline. This worldline can be parameterized by a single parameter, the 'age' of the person (or object). The change in age of a person is given by the Lorentz interval between two nearby points on his worldline.

If we have a time-like worldline that intersects itself (which means that the worldline goes to the same location in space AND time for two different values of the "age" parameter, we have pretty much, by defintion, time travel. For instance, if the self-intersecting worldine is that of a person, a time-like worldline that intersects itself represents an "older" you meeting a "younger" you. Of course we probably don't want the worldlines to exactly intersect, just pass close to each other, so that they are nearly at the same place at the same time for such a meeting.

A closed timelike curve is a little more pathological than this. If we imagine a CTC that's a person, he would never be born, and would never die either. He'd just sort of exist - perhaps like the movie "Groundhog day".

However, if we assume Novikov self consistency (not required, perhaps, but I think it makes the most sense - this is the assumption in the billiard ball paper, for example), such a person would not be able to remember events from previous cycles (as he does in the fictional movie I mentioned), and would not even be aware that he was in a loop, much less be able to escape it.

Depending on the exact dynamics, it will probably be pretty easy to perturb a true CTC into a less-pathological but more recongnizable form of time travel, the self-intersecting timelike curve. You might also be able to have "multi-loop" CTC's, depending on the exact dynamics, which would probably best be thought of in terms of a phase space.

 

[MeJennifer]

Basically, the problem here is that we have gone beyond the realm of talking about specific experiments to some sort definitional discussion or philospohical discussion, and I don't quite see where you are coming from and why you are making the statements you are and what you are asking for when you talk about physical interpretations.

No it is simply mathematics.
Complexify each dimension, then are you still convinced that the loop is closed?

Wick rotations generally do not work in curved spacetime.

 

[pervect]

No it is simply mathematics.
Complexify each dimension, then are you still convinced that the loop is closed?

Wick rotations generally do not work in curved spacetime.

I don't recall ever suggesting that they did, or reading anyone else in the thread who suggested that they did.

In fact, if you re-read my remarks, you will see that I suggest that the ict formalism, which is what I assume you are referring to by "Wick rotations", is not generally used in GR. One says "goodbye to ict" and deals with only real numbers.

So I'm feeling like there is straw all over the floor.

(Straw all over the floor? What does that mean? It means that a strawman argument has been totally demolished.)

Since it appears to be your strawman, MeJennifer, (it certainly isn't mine!) I suggest you sweep up his poor remains ...

 

[MeJennifer]

You are effectively producing some kind of rotation over some plane, that causes the complex axis of the complex space coordinate plane to map over the real axis of the complex time coordinate plane. Perhaps my usage of the term Wick rotation is a bit to wide for this kind of rotation but nevertheless it is a rotation.

Hopefully it is uncontested that rotations over a plane consisting of the real axis from one complex plane and the imaginary axis of another plane in a curved manifold with a Lorentzian signature is at least "fishy".

As soon as the space coordinate becomes imaginary in the van Stockum dust solution you have to ask in which direction the curve is heading. Obviously it is heading in the orthogonal direction of the space coordinate axis. But is the orthogonal direction of the space coordinate axis the time coordinate axis? If you think the answer is yes you should ask yourself why you reason as such, since there is nothing in the theory of relativity that either implies or excludes that.

 

[Chris Hillman]

"Tipping", Wick rotation, &c.

Hi all,

This has been an amazingly confusing thread, but I sense that at least some readers with less experience working with gtr might be clearing up some misconceptions from reading some of the comments by those with more experience, so forging ahead, I have some comments on points I haven't yet addressed (I plead exhaustion):

Yep, keep in mind that "r" and "t" are just coordinates, they don't necessarilly have any global meaning. In particular, t certainly isn't purely timelike for all observers. The original motivation for my current work is all the existing EFE solutions which don't yet have much physical interpretation.

I agree with the main point here, that a local coordinate chart on some region (homeomorphic to ordinary R^4) in a Lorentzian manifold is associated with four almost arbitrary monotonic functions; hence, in general, coordinates are arbitrary labels lacking any physical interpretation.

At the same time, though (no pun intended), it is important to recognize that some charts feature coordinates which do have a notable geometric or coordinate-free interpretation! In particular, consider the Schwarzschild time coordinate used in the interior or exterior Schwarzschild charts in the Schwarzschild vacuum (which cover an interior or an exterior region respectively). Here, the coordinate vector field $\partial_t$ is also a Killing vector field. In fact, it is uniquely determined by the stipulation that we choose the Killing vector field on the full Schwarzschild vacuum which is timelike in the asymptotically flat regions and also tends to unit length near "spatial infinity". This condition is independent of which coordinate chart we use.

One of the many important advances which occurred in gtr during the Golden Age (say c. 1959-1979, covering Bondi radiation theory through the positive energy theorem) was that it became common for researchers to use this kind of coordinate-free thinking to construct charts "adapted" to the symmetries they were assuming in constructing some solution. See Stephani et al., Exact Solutions of Einstein's Field Equations, 2nd. Ed., Cambridge University Press, 2001, for some fine expository chapters on coordinate-free methods and for many examples of this kind of construction of solutions.

cesiumfrog added: "all the existing EFE solutions which don't yet have much physical interpretation"; I am probably just misreading this, but I wouldn't want anyone to get the impression that "no known exact solutions possesses a reasonable physical interpretation", for this is certainly not true! To the contrary, there are important classes of solutions, such as the static spherically symmetric stellar models, which not only have an unobjectionable interpretation but are also useful in modeling real astrophysical objects.

Examples of exact solutions with clear physical interpretations (including a clear understanding of the limits on their applications to realistic physical scenarios) include plane wave solutions, some null dust solutions such as the Vaidya null dust, many cosmological models such as the FRW models and various generalizations, colliding plane wave (CPW) models, etc. Then there are solutions which have clear interpretations in that it is clear what one is trying to describe, but which on closer inspection have physically objectionable features; these include Weyl vacuum solutions with "struts", the Van Stockum "rotating" cylindrically symmetric dust, Robinson-Trautman vacuums with "pipes", and so on.

The Kerr vacuum is unobjectionable and realistic (for black hole models) in the exterior regions, and unobjectionable but perhaps unrealistic (for black hole models) in the "shallow interior" regions, but as several commentators have mentioned, it is objectionable in the "deep interior" regions, since it there admits closed timelike curves (CTCs), as does the Goedel lambdadust. These CTCs are problematical.

Using standard Schwarzschild coordinates light cones do not tip over.
They also do not tip using Kruskal coordinates. However they do tip over using the Eddington-Finkelstein coordinates.

I know what you mean, but be careful, since of course in terms of the intrinsic geometry of a spacetime model, infinitesimal light cones do not really becoming "sheared" (although they appear that way when we draw them in the Painleve chart), or "stretched temporally and squeezed radially" (although in the exterior region, they appear that way when we draw them in the exterior Schwarzschild chart), or "rescaled without changing shape" (although they appear that way when we draw them in the Kruskal chart, or other "conformal" charts).

The properties you are referring to can be read off the components $g_{ab}, \; g^{ab}$ of the metric tensor as expressed in the given chart. If w is the one of the coordinates, it is associated with a coordinate vector field $\partial_w$ and a "dual" coordinate covector field $dw$, which is a simple or Darboux rank zero) one-form. Think of a covector as a "hyperplane element". Assuming without loss of generality -+++ signature, at some event E, we can partially classify the geometric nature of the coordinate w as follows:

1. $g_{ww} < 0, \; g^{ww} > 0$ means that $dw$ cuts the light cone at E ("-++ signature hyperplane element") and $\partial_w$ is timelike at E.

2. $g_{ww} = 0, \; g^{ww} = 0$ means that $dw$ is tangent to the light cone at E ("null hyperplane element") and $\partial_w$ is null is E.

3. $g_{ww} < 0, \; g^{ww} > 0$ means that $dw$ is transverse to the light cone ("+++ signature hyperplane element") and $\partial_w$ is spacelike.

It is quite possible to give charts for Minkowski vacuum which exhibit various combinations of these alternative behaviors for the four coordinates (not entirely independently, of course).

The above "classification" is incomplete, and a complete classification of the local causal properties of all possible coordinate charts (local in sense of "local neighborhood") involves considerations I haven't mentioned. It turns out there are 199 types in the Coll-Morales scheme, which uses 4+6+4=14 parameters, including the signs noted above. See http://www.arxiv.org/abs/gr-qc/0507121

I wonder who first interpreted this "tipping of light cone" and r becomes time.
Was it Schwarzschild himself?

Tragically, Schwarzschild was actually at the front during WWI when he wrote his paper, and died (the exact cause is a bit mysterious) before he was able to exploit his solutions. Others struggled to make sense of his vacuum solution; the first good coordinate chart was introduced by Painleve in 1921, but Einstein and Painleve seem to have been distracted by the desire to reconcile the European nations and prevent further wars, and when they met in Berlin, according the diary of Count Kessler, their conversations seem to have been mostly concerned with the urgent need to ensure world peace. Unfortunately, despite Painleve's status as a former cabinet minister for one of the former combatants, their efforts came to naught.

MTW has a good overview of the other well-known systems, including the Eddington chart (rediscovered by Finkelstein) and the Kruskal-Szekeres chart (discovered independently by Martin Kruskal and Peter Szekeres). Synge is sometimes mentioned as the first to have understood the global geometry of the maximal analytic extension of the Kerr vacuum solution, but his writings were not entirely clear, so many tend to credit this to Kruskal and Szekeres.

More important is the critical review of the suggestion that time can become like space and space can become like time. I know for some that is a done deal and I simply do not "understand" it but to me it is not a done deal at all.

Referring to the above, this kind of statement is just a (terribly misleading!) shorthand for "$\partial_w$ is spacelike in region A and timelike in region B and null at their interface".

Lets look at this in the Schwarzschild coordinates. Is it sufficient to show t becomes imaginary & that a decrease in r becomes as inevitable as going forward in time?

Arghghgh! Of course t does not become imaginary; we are discussing real coordinates here.

One can discuss complex coordinates, but these don't mean what you probably think, and this thread is already confused enough!

the line element along the curve is

$$ds^2 = \left( r^2 + a^2 + \frac{2Mr a^2}{r^2} \right) d\phi^2$$

For $r$ negative and small. the last term, whcih is negative, dominates, and thus $ds^2$ is the line element for a timilike curve.

Probably everyone realized that the last + in the expression for the metric (restricted to the curve) should be a -...

if we define the term 'timelike' as any coordinate that contributes positively (assuming the + - - - convention) to g it follows that this would indeed be a closed timelike loop.

But does this warrant a physical interpretation or, even stronger, a conclusion that a test mass traveling on this CTC meets its own past?

Just because a dimension d₁ is orthogonal to a dimension d₂ does not imply that d₁ lies in d₂'s complex plane. It could be, but only if defined as such.

I also am finding it hard to follow what MeJennifer is asking here. But perhaps what I said above will help...

Wick rotations generally do not work in curved spacetime.

For those who don't know what a Wick rotation is, in this context it is best to say that:

Write out the source-free Maxwell field equations in flat spacetime in their full partial differential glory; the independent variables are t,x,y,z and the dependent variables are the components of the electric and magnetic fields.

Now compute the Lie algebra of the point symmetry group of this system according to the method given by Sophus Lie. You obtain a large Lie algebra which includes the 15 dimensional algebra so(2,4) (the Lie algebra of the conformal group on Minkowski spacetime), plus the generator of scalar multiplications of the dependent variables,
$$E^x \, \partial_{E^x} + E^y \, \partial_{E^y} + E^z \, \partial_{E^z} + B^x \, \partial_{B^x} + B^y \, \partial_{B^y} + B^z \, \partial_{B^z}$$
plus generators arising from the freedom to add any solution (so as with any linear system, the point symmetry group is technically infinite dimensional), plus the generator
$$B^x \, \partial_{E^x} + B^y \, \partial_{E^y} + B^z \, \partial_{E^z} - E^x \, \partial_{B^x} - E^y \, \partial_{B^y} - E^z \, \partial_{B^z}$$
which generates the one parameter subgroup of "Wick rotations". These effect only the dependent variables and as you can see can be considered a "rotation" in a six dimensional space (the space of components of the two vector fields).

You are effectively producing some kind of rotation over some plane, that causes the complex axis of the complex space coordinate plane to map over the real axis of the complex time coordinate plane. Perhaps my usage of the term Wick rotation is a bit to wide for this kind of rotation but nevertheless it is a rotation.

I hope it's clear that this rotation involves the dependent variables only.

Hopefully it is uncontested that rotations over a plane consisting of the real axis from one complex plane and the imaginary axis of another one in a curved manifold with a Lorentzian signature is at least "fishy".

As soon as the space coordinate becomes imaginary in the van Stockum dust solution you have to ask in which direction the curve is heading. Obviously it is heading in the orthogonal direction of the space coordinate axis. But is the orthogonal direction of the space coordinate axis the time coordinate axis? If you think the answer is yes you should ask yourself why you reason as such, since there is nothing in the theory of relativity that either implies or excludes that.

Oh shoot, got to run...