Do Light Cones Truly Tip Over in General Relativity?

[MeJennifer]

"Light cones tipping over"

A common phrase used to show alleged time travel solutions in GR.
Even a person like Kip Thorne uses it.

But my question is, is that an accurate representation of GR in strong gravitational fields?

The Schwarzschild metric expressed using the Eddington-Finkelstein coordinates show those "light cones tipping over", and eventually the radial and time coordinates reverse.

But this reversal, and even the tipping over seems to me a peculiarity of the choice of coordinates. It seems to me that it is assumed that there is a particular relationship between the radial and time coordinate.
That seems to be a rather liberal interpretation of the metric.

Afteral nothing in the Minkowskian metric of ds² = dt² - dr² implies that if dt² becomes negative it must be considered space and if dr² becomes negative it must be considered time.

Any comments or explanations?

 

[Aether]

A common phrase used to show alleged time travel solutions in GR.
Even a person like Kip Thorne uses it.

But my question is, is that an accurate representation of GR in strong gravitational fields?

The Schwarzschild metric expressed using the Eddington-Finkelstein coordinates show those "light cones tipping over", and eventually the radial and time coordinates reverse.

But this reversal, and even the tipping over seems to me a peculiarity of the choice of coordinates. It seems to me that it is assumed that there is a particular relationship between the radial and time coordinate.
That seems to be a rather liberal interpretation of the metric.

Afteral nothing in the Minkowskian metric of ds² = dt² - dr² implies that if dt² becomes negative it must be considered space and if dr² becomes negative it must be considered time.

Any comments or explanations?

Minkowski space-time, at least according to one of my textbooks, is specifically defined as a 4-D real linear space. Neither dt² nor dr² can ever become negative while dr and dt are real numbers. That wouldn't apply to a complex linear space though.

 

[robphy]

http://casa.colorado.edu/~ajsh/schwp.html may have useful diagrams for this question. Physically, the tipping of the light cones [which are traced out by null geodesics (a coordinate invariant idea)] is an indication that timelike observers traveling across the horizon cannot escape back out beyond the horizon.

 

[Thrice]

http://casa.colorado.edu/~ajsh/schwp.html may have useful diagrams for this question. Physically, the tipping of the light cones [which are traced out by null geodesics (a coordinate invariant idea)] is an indication that timelike observers traveling across the horizon cannot escape back out beyond the horizon.

Yeah the tipping just means the future part of the light cones point into a point in space (the black hole).

 

[cesiumfrog]

if dt² becomes negative it must be considered space and if dr² becomes negative it must be considered time.

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.

 

[MeJennifer]

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.

But even in coordinate space, what is the compelling argument for calling this a change from one to the other? Would that not imply that t and r are both related to each others imaginary plane? So that if t² becomes negative it becomes space and that if r² becomes negative it becomes time?
Would that not be a rather liberal interpretation of the Minkowski metric?

In plain Schwarzschild the light cones get narrower the closer one gets to the event horizon and eventually becomming zero. I would think it much more logical to see them get an imaginary size past the event horizon than to assume that, magically, the time and space coordinates get switched.

You say to keep in mind that coordinate space is not observer space is of course true. But, people use the tipping of light cones, which by the way is only happening when you use the Eddington-Finkelstein coordinates, to "explain" closed time curves. That again is a rather premature conclusion IMHO.

But on the other hand we see that people like Kip Thorne and more surprisingly Roger Penrose use it as well. Penrose who at one point proposed the use of a complex compactified Minkowski spacetime. Obviously this spacetime does not satisfy the condition that t and r are both related to each others imaginary plane.
So go figure, what's the real story here?

 

[cesiumfrog]

..if r² becomes negative..

Normally you would leave r as simply a real valued coordinate; it's the metric components (eg. g_{rr} vs. g_{tt}) that may change in sign.

In plain Schwarzschild the light cones get narrower the closer one gets to the event horizon and eventually becomming zero. ...

I don't know that it makes any sense to say that (by what additional metric are you measuring the cones?)..

In the plain old (ie. Boyer-Lindquist) coordinates, the light cones "tip" as you approach the event horizon, which means light rays begin to traverse greater intervals of r coordinate (note we're implicity comparing to a euclidean metric over the same chart). From points on the horizon (neglecting that the chart is actually invalid just there), the cone would be rotated such that the future cone is centered towards the negative r direction, and the past cone in the direction of increasing r coordinate.

But, people use the tipping of light cones, which by the way is only happening when you use the Eddington-Finkelstein coordinates, to "explain" closed time curves.

The tipping happens in general, unless you specifically choose a coordinate system to avoid it. The relation to causality violation can occur for example if null cones in a coordinate space tip such that the \phi coordinate becomes everywhere timelike AND if there is also some kind of closed geometry such that having zero \phi coordinate be identified with having \phi coordinate equal to 2\pi.

 

[MeJennifer]

I don't know that it makes any sense to say that (by what additional metric are you measuring the cones?)..

Using standard Schwarzschild coordinates light slows down for decreasing values of r. That is why the cones get narrower. But remember that r is nothing but a coordinate, it does not represent the physical distance from the center of gravity.

The tipping happens in general, unless you specifically choose a coordinate system to avoid it.

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.

Anyway we can do a lot of juggling with coordinates and then demonstrating all kinds of effects that are completely non physical but only due to the usage of certain coordinate systems.

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.

 

[cesiumfrog]

But remember that r is nothing but a coordinate, it does not represent the physical distance from the center of gravity.

Using standard Schwarzschild coordinates light cones do not tip over.

By saying "standard", you are referring to "Boyer-Lindquist", right?
In any case, say you draw a t vs. r plot (standard coords) depicting the null cones for observers at different positions.. would you not describe them as "tipping"?

Anyway we can do a lot of juggling with coordinates and then demonstrating all kinds of effects that are completely non physical but only due to the usage of certain coordinate systems.

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.

Hmm. I would really avoid saying "time can become like space" or vice-versa, because I don't think it means anything. Physically, to an observer journeying across the event horizon, nothing ever happens of local significance. It's a confusing coincidence that the original Schwarzschild coordinates have undesirable features exactly where the event horizon is... Just that when we try to draw some set of coordinate axis lines (initially chosen somewhat arbitrarily), we sometimes find one of those lines is timelike in one region, and spacelike in another.

 

[MeJennifer]

By saying "standard", you are referring to "Boyer-Lindquist", right?

Boyer-Lindquist is used for a rotating mass not for a non rotating mass.

Using Schwarzschild coordinates there is only one r coordinate. Eddington-Finkelstein introduces an additional r* called the tortoise coordinate. With this in place we get tilting cones.
And if one additional coordinate is not enough we can always use the Krusal coordinates that introduces two coordinates that relate to t and r. In Krusal coordinates the cones are always straight up and also 45 degrees (so no narrowing).

In any case, say you draw a t vs. r plot (standard coords) depicting the null cones for observers at different positions.. would you not describe them as "tipping"?

A very good question!
For starters it would not be applicable for Schwarzschild coordinates since this is a view of the situation from the perspective of a distant observer far removed from the gravitational field. So the point is that in Schwarzschild coordinates the observer is fixed.

But with regards to a observer in free fall who is approaching the event horizon and continuing towards the singularity, the speed of light will remain c, at least locally, so that implies that the angle of the cone would remain constant. However we can hardly speak of a cone from the observer's perspective, it would seem clear that since the curvature for this observer is so strong that a construction of an orthonormal coordinate system would show anything but a cone for incoming and outgoing light rays except for a very small local region. Past the event horizon the observer would still measure the speed of light at c, again only locally, but because the curvature is getting so strong here that the closure between the observer and the center of mass is faster than the width angle of the observer's light cone would allow. Now some seem to interpret this as time becoming space and vice versa, which is the whole point of this topic.

It's a confusing coincidence that the original Schwarzschild coordinates have undesirable features exactly where the event horizon is...

To me that is not a coincidence at all. Remember that the Schwarzschild model is a view from the perspective of an observer far removed from the gravitational influence.

 

[cesiumfrog]

A very good question!
For starters it would not be applicable...

So your answer is no?

If so I disagree. Regardless of whether you use Schwarzschild or Ingoing (Eddington-Finkelstein) coordinates, the tilting is what you see when you draw light "cones" on a t vs. r (or t* vs. r*) plot. It's a coordinate effect.

Since coordinates are nothing physical, and you can't ever speak of shape or orientation of a light cone from the observer's perspective, why make a big deal of the tilting at all?

As for time travel, that occurs when the tilting is such that one's future light cone intersects one's past light cone. It isn't an issue in a black hole (the future cone is directed at the singularity, closed timelike loops don't become possible), it only means that from our perspective the gravity of a black hole sucks things away too strongly for their rockets or even their radio messages to approach us.

 

[MeJennifer]

If so I disagree. Regardless of whether you use Schwarzschild or Ingoing (Eddington-Finkelstein) coordinates, the tilting is what you see when you draw light "cones" on a t vs. r (or t* vs. r*) plot. It's a coordinate effect.

Well I am quite sure that you would not accept from me that using Schwarzschild coordinates light cones do not tip, and that instead you need Eddington-Finkelstein coordinates for that.
Perhaps from someone else you will.

 

[cesiumfrog]

I accept I may be wrong (you haven't yet motivated me to draw the plot myself) but my reasoning is that in Schwarzschild coordinates: far from the black hole, light tends to travel in the time direction (the cone spreading equally in +/- radial directions); inside the even horizon light must always travel radially inward (presumably also spreading in the t coordinate directions); if anything Ingoing coordinates would demonstrate less tilt (limited to 45 degrees).

Regardless, if you believe the tilting appears only in one of two equally valid coordinate systems, do you agree that "space becoming time" is not a physical effect (in black holes)?

 

[MeJennifer]

I accept I may be wrong (you haven't yet motivated me to draw the plot myself) but my reasoning is that in Schwarzschild coordinates: far from the black hole, light tends to travel in the time direction (the cone spreading equally in +/- radial directions); inside the even horizon light must always travel radially inward (presumably also spreading in the t coordinate directions); if anything Ingoing coordinates would demonstrate less tilt (limited to 45 degrees).

The whole issue IMHO with cones is that they work well with SR but miserably fail with GR.

In flat spacetime we have cones, but not in curved spacetime. Yes of course if we take a very small region that we can consider flat then we have a mini cone but in curved spacetime we cannot speak about a cone at all. Sure we can insist on cones by picking the proper coordinate system, but it does not mean anything, for all intents and purposes we might as well create a coordinate system that shows cones as passa doblé steps.

Regardless, if you believe the tilting appears only in one of two equally valid coordinate systems, do you agree that "space becoming time" is not a physical effect (in black holes)?

I agree that such coordinate representations are more or less meaningless from a physical perspective.

But the problem is that tipping of cones is shown as theoretical evidence for things like closed time loops. Even people like Kip Thorne, Roger Penrose and Stephen Hawking seem run away with it and write books that "clearly shows" what is going on. In my understanding at least it seem that they should know better, but clearly I just don't seem to understand why it is obvious that time and space can flip inside the event horizon. Any takers on a simple explanation?

The main issue that I wanted to bring up in this topic is that suggestion, the suggestion that time becomes like space and vice versa. I don't see any indication for that, except for when we make a particular interpretation of the time and space relationship in the Minkowski metric.

 

[Haelfix]

The interpretation is natural though since asymptotically that's the preffered coordinate system.

The choice of tipping is indeed a coordinate artifact, but it makes good sense from a physical standpoint to compare it with what we know best. Eg a rockets journey starts out with the obvious Minkowski coordinates system, and as you get close and pass through the black holes event horizon it serves as an illustration how either you have to abandon your choice of coordinates, or you have to admit that ones notion of time/space are going to warp and switch places.

And there's nothing wrong with writing down lightcones locally. Indeed we specifically choose not to pick coordinates where they are torus's or something like that, b/c no one has any intuition whatseover about that and indeed most calculational strategies vanish with such a stupid choice of local coordinates.

 

[MeJennifer]

The choice of tipping is indeed a coordinate artifact, but it makes good sense from a physical standpoint to compare it with what we know best. Eg a rockets journey starts out with the obvious Minkowski coordinates system, and as you get close and pass through the black holes event horizon it serves as an illustration how either you have to abandon your choice of coordinates, or you have to admit that ones notion of time/space are going to warp and switch places.

So demonstrate to me how they switch place!
How do two separate dimensions get intertwined on a Riemann surface?

 

[robphy]

The whole issue IMHO with cones is that they work well with SR but miserably fail with GR.

In flat spacetime we have cones, but not in curved spacetime. Yes of course if we take a very small region that we can consider flat then we have a mini cone but in curved spacetime we cannot speak about a cone at all. Sure we can insist on cones by picking the proper coordinate system, but it does not mean anything, for all intents and purposes we might as well create a coordinate system that shows cones as passa doblé steps.

In Minkowski spacetime M, we have two ways to think of the "light cone of an event p"...

It is the "set of events in M" that can only reach or be reached by the vertex event p by a light ray. It also forms the boundary between events that can be causally connected to p from those that cannot.

It is also a "set of directions [set of vectors] in T_pM (the tangent space at p)" that are tangent to lightlike paths through event p (i.e., "set of lightlike tangent vectors at p"). It also forms the boundary between the spacelike and non-spacelike (i.e. causal) tangent vectors. T_pM is a vector space with a Minkowski metric.

In a general curved spacetime, the notion of "light cone of an event p" usually means the "set of lightlike tangent vectors at p".

Nothing I have said above depends on any choice of coordinates.

To draw these light cones in the spacetime diagram of a spacetime, curved or otherwise, one must identify all of the lightlike geodesics (a coordinate invariant notion). At a particular event, its light cone is determined by the tangents to these geodesics.

Depending on your choice of coordinates, the image of these geodesics in your coordinate chart may trace out all sorts of crazy looking paths (akin to the distortions one gets from various map projections of the earth). In some cases, the image of these light cones may look tipped or distorted relative to the images of other light cones in your coordinate chart.

Regardless of appearances in your chart, the physics is determined by the lightlike geodesics, essentially telling you which events are in the causal future [and causal past] of events in spacetime (i.e. the causal connectivity of events). The worldlines of observers are bounded by these lightlike geodesics.

It may turn out that the null geodesics tell you that you might have closed causal curves. Or it may turn out that certain sets of events have causal futures that don't extend to spatial or null infinity. Or something other feature that one doesn't see in Minkowski spacetime.

Depending on your choice of chart (and thus the images of the light cones), it may be easier or harder to tell the story of what is going on physically. Depending on the particular aspect of the story you want to tell, some charts are better suited than others.

But the problem is that tipping of cones is shown as theoretical evidence for things like closed time loops.

More correctly, these show that closed time loops are mathematically possible, given the constraints imposed in the situation. In other words, saying that one has a 4-manifold with a Lorentzian-signature metric places some restrictions on what "physics" is possible. However, by themselves, they don't restrict the possibility of closed time loops or other pathologies. Even imposing the field equations might still allow pathologies. That is why one is led to the notion of "causality conditions" and the study of "causal structure", which were developed using "global methods" (i.e. geometric, coordinate-free methods). One may also impose other conditions like "energy conditions", "asymptotic conditions", etc...

When I find the time, I'll try to address your concern about "switching".

 

[MeJennifer]

To draw these light cones in the spacetime diagram of a spacetime, curved or otherwise, one must identify all of the lightlike geodesics (a coordinate invariant notion). At a particular event, its light cone is determined by the tangents to these geodesics.

Sure I follow what you say here.

Depending on your choice of coordinates, the image of these geodesics in your coordinate chart may trace out all sorts of crazy looking paths (akin to the distortions one gets from various map projections of the earth). In some cases, the image of these light cones may look tipped or distorted relative to the images of other light cones in your coordinate chart.

Sure in certain coordinates.

Regardless of appearances in your chart, the physics is determined by the lightlike geodesics, essentially telling you which events are in the causal future [and causal past] of events in spacetime (i.e. the causal connectivity of events). The worldlines of observers are bounded by these lightlike geodesics.

Completely agree!

More correctly, these show that closed time loops are mathematically possible, given the constraints imposed in the situation. In other words, saying that one has a 4-manifold with a Lorentzian-signature metric places some restrictions on what "physics" is possible. However, by themselves, they don't restrict the possibility of closed time loops or other pathologies. Even imposing the field equations might still allow pathologies. That is why one is led to the notion of "causality conditions" and the study of "causal structure", which were developed using "global methods" (i.e. geometric, coordinate-free methods). One may also impose other conditions like "energy conditions", "asymptotic conditions", etc...

Well at one point the time part of the geodesic has to connect to another time part of the same geodesic while the spatial parts are irrelevant. Apart from a closed spacetime or a wormhole I do not see how that can be the case. Can you?

When I find the time, I'll try to address your concern about "switching"

Please do, to me it makes absolutely no sense.

 

[Thrice]

So demonstrate to me how they switch place!

Ok you guys are much better than I am at this, but I thought I had a handle on it. Let's 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?

 

[coalquay404]

In case anyone's still interested in this, the clearest pedagogical explanation of this is (unsurprisingly) in MTW. Check out pp. 823-826 for a discussion that uses the Schwarzschild geometry and the surface at r=2M as an example.

 

[MeJennifer]

See this posting in the https://www.physicsforums.com/showpost.php?p=1146536&postcount=21" topic for a comment that I think is applicable to this topic as well.

For a large, ideal non-rotationg black hole, which is a valid solution to the equations of GR, both our senses and GR work fine within the horizon. In particular, time and space don't "swap" inside the horizon. What does happen is that: a poor choice of labels is used; spacetime becomes non-stationary.

Here is an analogy. In some city, imagine that you are driving East on Bridge Street East. After the street makes a sharp left, you are driving almost north on Bridge Street East. East and North did not interchange, it is just that the labellng system has become poor.

In the same manner as street names are convenient labels that humans assign to positions in cities, spacetime coordinates (like r) are just labels assigned by humans to spacetime events. Inside the event horizon, r is a timelike coordinate, so it would make more sense to chanlge the name of the human-assigned label r to something more descriptive. For (partially) historical reasons, this isn't done.

Similarly, in the above city, after the left, it would make sense to change the human-assigned label Bridge Street East to something like Bridge Street North. This hasn't been (and won't be) done, because the city's inhabitants have been calling it Bridge Street East since before anyone can remember.

 

[Chris Hillman]

Appearance of light cones in curved spacetimes

Hi again, Jennifer,

A common phrase used to show alleged time travel solutions in GR. Even a person like Kip Thorne uses it.

But my question is, is that an accurate representation of GR in strong gravitational fields?

It is when Kip Thorne uses it! :-/ I know that because he can provide a correct figure which conforms to this informal description, as I can verify using my own computations.

I prefer to be more specific about this "tipping". I guess you are talking about light cones in the Boyer-Lindquist chart for the Kerr vacuum solution in gtr, which does feature closed timelike curves in the interior region, or light cones in the Goedel lambdadust solution, which also features closed timelike curves (see for example the beautiful figures in Hawking and Ellis, Large Scale Structure of Space-Time, for both of these examples).

But let's study an even simpler example:

The Schwarzschild metric expressed using the Eddington-Finkelstein coordinates show those "light cones tipping over", and eventually the radial and time coordinates reverse.

Specifically, consider the advanced (infalling) Eddington chart, in which the line element takes the form
ds^2 = -(1-2 m/r) \, du^2 + 2 \, du \, dr + r^2 \, \left( d\theta^2 + \sin(\theta) \, d\phi^2 \right),
-\infty < u < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
We can write down a "frame field" consisting of four orthonormal vector fields, a timelike unit vector
\vec{e}_1 = \partial_u - m/r \, \partial_r
plus three spacelike unit vectors
\vec{e}_2 = \partial_u - (1-m/r) \, \partial_r
\vec{e}_3 = 1/r \, \partial_\theta
\vec{e}_4 = 1/r/\sin(\theta) \, \partial_\phi
You can use these to draw the light cones. If you do it right, they will all be tangent to the null vector field \partial_r and as r decreases, they lean inwards, until at r=2 m they are also tangent to \partial_u.

eventually the radial and time coordinates reverse.

Many people, even some who ought to know better, do talk that way, and invariably they wind up confusing everyone, including themselves. What they should really say is that the vectors \partial_u are timelike outside the horizon, null at the horizon, and spacelike inside the horizon. Nothing "reverses"; in particular, the frame vectors given above are unambiguously timelike throughout (for the first) or spacelike throughout (for the remaining three).

George Jones is completely correct: of course "time and space" do not "swap roles" inside the horizon, that would be nonsense!

To elaborate on one point he alluded to, the coordinate basis vector field \partial_u happens to be a Killing vector field; that is, the Schwarzschild vacuum is invariant under time translation. Similarly, the coordinate basis vector \partial_\phi is a spacelike Killing vector whose integral curves are circles; that is, the Schwarzschild vacuum is invariant under rotation about the axis r=0.

The fact that in the exterior we have an irrotational timelike Killing vector and a spacelike Killing vector (whose integral curves are circles) means that the exterior region is static and axisymmetric. (This is also true of the Kerr vacuum solution.) Inside, we have two spacelike Killing vectors, but no timelike Killing vector; the solution is NOT static inside the horizon. Of course not, since otherwise an observer could use his rocket engine to hover at some Schwarzschild radius 0 < r < 2m.

But this reversal, and even the tipping over seems to me a peculiarity of the choice of coordinates. It seems to me that it is assumed that there is a particular relationship between the radial and time coordinate.

Not sure I understand that, but it sounds like you did correctly recognize that the coordinate basis vector \partial_u changes character at the horizon.

Hope this helps,

Chris Hillman

 

[cesiumfrog]

...the Kerr vacuum solution in gtr, which does feature closed timelike curves in the interior region...

Could you elaborate a little on exactly where those closed timelike curves are in the Kerr solution?

 

[Chris Hillman]

CTC's in the Kerr vacuum

Hi, Cesium,

Could you elaborate a little on exactly where those closed timelike curves are in the Kerr solution?

The system still seems (as least to me) to be quite unstable (but then I've only been here for a few days), so I daren't try to write very much (having lost quite a bit of work here in the past few days), but briefly, the CTCs in the Kerr vacuum are all located in the "deep interior" blocks (referring to the usual Carter-Penrose block or conformal diagram). The no-hair theorems do not imply that the -interior- geometry prefers to be Kerr-like, and there are various considerations which suggest that it should not be, quite apart from our natural desire to avoid predicting CTCs even in places where, even if a physicist should experience such weirdness, he'd be unable to report this to his colleages in the exterior.

The Taub-NUT vacuum (Misner's "counterexample to everything") and Goedel lambdadust also exhibit some startling causal structure. In fact, the best example to become familiar with CTC's is probably the Goedel lambdadust solution. As it happens, I just came across a spanking new arXiv eprint which offers an extensive and well illustrated discussion; see http://www.arxiv.org/abs/gr-qc/0611093

Chris Hillman

 

[Los Bobos]

Regarding the "tipping of lightcones" can someone point a coordinate system for Schwarzschild black hole, where this does not happen? At least with the usual suspects this seems to happen ("the nature of dx -> dt" in the usual way or the relationship of the light cones and the tangent of fwo-path in Kruskal coordinates).

 

[George Jones]

Could you elaborate a little on exactly where those closed timelike curves are in the Kerr solution?

Let me elaborate a bit on what Chris said.

O'Neill, in his book The Geometry of Kerr Black Holes, proves:

there is a closed timelike curve through any event inside the inner (Cauchy) horizon, i.e., through any event for which r < r-.

Carroll gives the following simple example. Consider a curve for which \phi varies, and for which t, r, \theta are held constant. Because of periodicity with respect to \phi, any such curve is closed.

Now, the timelike part.

Take r &lt; 0 with |r| small, and \theta = \pi/2. Note r is a coordinate, not a radial distance, and negative r is part of (extended) Kerr. Because 0 = dt = dr = d \theta, the line element along the curve is

<br /> ds^2 = \left( r^2 + a^2 + \frac{2Mr a^2}{r^2} \right) d\phi^2<br />

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

 

[Chris Hillman]

Utility of light cones, plus a Vaidya thought experiment

Hi, Jennifer,

The whole issue IMHO with cones is that they work well with SR but miserably fail with GR.

I am coming into this thread very late, so this might not do much good, but I would say that to the contrary, drawing "infinitesimal" light cones in some coordinate chart is one of the most important things you can do when you are trying to understand the local geometry.

As so often happens in this subject, there is room for confusion here since some authors also use "light cone" (which when used without qualification generally means the concept defined at the level of a tangent space) to refer to the "absolute future" and "absolute past" of some event. For example, in discussions of Cauchy horizons or cosmological horizons.

But the problem is that tipping of cones is shown as theoretical evidence for things like closed time loops. Even people like Kip Thorne, Roger Penrose and Stephen Hawking seem run away with it and write books that "clearly shows" what is going on.

I think it DOES show clearly what is going on, and I can't imagine how one could understand gtr without appealing to this kind of imagery.

In my understanding at least it seem that they should know better, but clearly I just don't seem to understand why it is obvious that time and space can flip inside the event horizon.

Ah! I think this the key misconception, and I entirely agree with you that anyone who says "time and space can flip inside the event horizon" either doesn't understand how gtr treats the notion of a black hole, or else is being terribly negligent in attempting to give an informal verbal description of the mathematics. Forgive me if you already mentioned this somewhere, but can I ask what book(s) you are reading? If one of them is MTW, I hope that you will soon come to appreciate why anyone saying that "time and space flip roles inside a black hole" is speaking nonsense. On the other hand, if we use a coordinate chart, such as the Eddington chart
ds^2 = -(1-2m/r) \, du^2 + 2 \, du \, dr + r^2 \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
-\infty &lt; u &lt; \infty, \; 0 &lt; r &lt; \infty, \; 0 &lt; \theta &lt; \pi, \; -\pi &lt; \phi &lt; \pi
then the coordinate basis vector field \partial_u is timelike outside the horizon but null on the horizon and spacelike inside the horizon.

Since \partial_u is also a Killing vector field, the fact that it is null on the horizon means that in the Schwarzschild vacuum solution, the event horizon happens to be a "Killing horizon", i.e. it has a local characterization (it is the locus where our Killing vector field is null). But in general, the event horizon has no such description. A very similar model which slightly generalizes this is the Vaidya null dust (to obtain this, just consider m to be a monotonically increasing function of u in the Eddington chart above!) is a very good model to study in detail in order to understand the "teleological" global nature of the event horizon. The Vaidya null dust is not static (if m is increasing with u), \partial_u is no longer a Killing vector field. In addition, if we allow a spherical shell of incoherent radiation to fall into the hole, thus increasing its mass and Schwarzschild radius, the event horizon smoothly increases, and the apparent horizon is not consistently defined during this increase. Even more startling, if the shell of incoming radiation carries a sufficient amount of energy, an observer hovering just outside the horizon could actually be inside the event horizon even BEFORE the radiation reaches him. Since it is moving at the speed of light, he cannot possibly obtain warning in time to hastily increase his distance from the hole. See the discussion in Frolov and Novikov, Black Hole Physics, for more detail plus some nice if microscopic pictures.

Just one more reason why the immediate vicinity of a black hole is probably not a very safe place to visit!

Chris Hillman

 

[MeJennifer]

I am coming into this thread very late, so this might not do much good, but I would say that to the contrary, drawing "infinitesimal" light cones in some coordinate chart is one of the most important things you can do when you are trying to understand the local geometry.

As so often happens in this subject, there is room for confusion here since some authors also use "light cone" (which when used without qualification generally means the concept defined at the level of a tangent space) to refer to the "absolute future" and "absolute past" of some event. For example, in discussions of Cauchy horizons or cosmological horizons.

Well but a cone drawn at one particular point in curved space-time does not represent the complete past or the future since in GR past and future events are related within the confines of space-time curvature, clearly a picture that is hardly resembling anything close to a conical shape.

I think it DOES show clearly what is going on, and I can't imagine how one could understand gtr without appealing to this kind of imagery.

Well to me, but surely I am mistaken, a cone drawn at a particular point in curved space-time shows its past and future connections assuming that space-time is flat. Since space-time is curved the shape of the "cone" is anything but a cone.

Ah! I think this the key misconception, and I entirely agree with you that anyone who says "time and space can flip inside the event horizon" either doesn't understand how gtr treats the notion of a black hole, or else is being terribly negligent in attempting to give an informal verbal description of the mathematics.

Well I am glad I find someone who can relate to this.

But then, given that you agree with me they can't flip, how could some people even argue that some given space-time can form a closed time loop?

In addition, if we allow a spherical shell of incoherent radiation to fall into the hole, thus increasing its mass and Schwarzschild radius, the event horizon smoothly increases, and the apparent horizon is not consistently defined during this increase.

True, but then the Schwarzschild solution would no longer apply since the space-time is no longer static right?

 

[Chris Hillman]

CTC's again

Hi, Jennifer

a cone drawn at one particular point in curved space-time does not represent the complete past or the future since in GR past and future events are related within the confines of space-time curvature, clearly a picture that is hardly resembling anything close to a conical shape.

That is actually one of the points I was hinting at. Of course, this doesn't mean that the absolute future of some event in a curved spacetime is not a valuable concept, even when (as in pp-wave models, for example) this concept breaks down globally.

But then, given that you agree with me they can't flip, how could some people even argue that some given space-time can form a closed time loop?

You mean closed timelike curves (CTCs), which is something different (but sharing the same basic idea): in some Lorentzian manifolds, at least in some regions there exist closed timelike curves. For example, suppose we just solved the EFE to obtain the metric tensor written in something like cylindrical coordinates notices.

In fact, for concreteness let's take a specific example, the van Stockum dust solution (1937), which was historically the first solution in which CTCs were noticed, by Van Stockum (see the article "Willem Jakob van Stockum at http://en.wikipedia.org/wiki/User:Hillman/Archive#Contributors_to_general_relativity)
ds^2 = -\left( dt -a\, r^2 d\phi \right)^2 + \exp(-a^2 \, r^2/2) \, \left( dz^2 + dr^2 \right) + r^2 \, d\phi^2,
-\infty &lt; t &lt; \infty, \; 0 &lt; r &lt; \infty, \; 0 &lt; \theta &lt; \pi, \; -\pi &lt; \phi &lt; \pi
Multiplying out, we find that g_{\phi \phi} = \left( 1-a^2 \, r^2 \right) \, r^2, so that the coordinate basis vector \partial_\phi is spacelike on 0&lt; r &lt; 1/a, null at r=1/a, and timelike on 1/a &lt; r &lt; \infty. This means that the circles t=t_0, \; z=z_0, \; r=r_0 are spacelike curves when r_0 &lt; 1/a but TIMELIKE when r_0 &gt; 1/a.

At this point, you might wish to jump to the article "van Stockum dust" archived at http://en.wikipedia.org/wiki/User:Hillman/Archive#Contributors_to_general_relativity
Note in particular the pictures of how the (infinitesimal) light cones change in appearance as you increase radial coordinate in this spacetime.

Chris Hillman

 

[MeJennifer]

Thanks Chris you gave me something to digest.

 

[MeJennifer]

Multiplying out, we find that g_{\phi \phi} = \left( 1-a^2 \, r^2 \right) \, r^2, so that the coordinate basis vector \partial_\phi is spacelike on 0&lt; r &lt; 1/a, null at r=1/a, and timelike on 1/a &lt; r &lt; \infty. This means that the circles t=t_0, \; z=z_0, \; r=r_0 are spacelike curves when r_0 &lt; 1/a but TIMELIKE when r_0 &gt; 1/a.

Well, unquestionably, 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?

 

[tehno]

Well, unquestionably, 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?

Certainly not.
It's just funny of mathematical model used (rather abusive one).
Make an experiment. That's the only way you will know for sure .

 

[quantum123]

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

 

[pervect]

Well, unquestionably, 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?

Given that the proper time experienced by an observer moving along a path is equal equal to the intergal of the Lorentz interval along that path (using a +--- sign convention), how could a closed path in space-time that was everywhere timelike not warrant such a physical interpretation?

 

[MeJennifer]

Given that the proper time experienced by an observer moving along a path is equal equal to the intergal of the Lorentz interval along that path (using a +--- sign convention), how could a closed path in space-time that was everywhere timelike not warrant such a physical interpretation?

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.

 

[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} &lt; 0, \; g^{ww} &gt; 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} &lt; 0, \; g^{ww} &gt; 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

<br /> ds^2 = \left( r^2 + a^2 + \frac{2Mr a^2}{r^2} \right) d\phi^2<br />

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...