"Light cones tipping over"

MeJennifer

Sure I follow what you say here.

Sure in certain coordinates.

Completely agree!

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?

Please do, to me it makes absolutely no sense.

Thrice

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

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 [tex]r=2M[/tex] as an example.

MeJennifer

See this posting in the Black Hole question topic for a comment that I think is applicable to this topic as well.

Chris Hillman

Hi again, Jennifer,

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:

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

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 [tex]\partial_u[/tex] 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 [tex]\partial_u[/tex] happens to be a Killing vector field; that is, the Schwarzschild vacuum is invariant under time translation. Similarly, the coordinate basis vector [tex]\partial_\phi[/tex] 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 [tex]0 < r < 2m[/tex].

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

Hope this helps,

Chris Hillman

cesiumfrog

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

Chris Hillman

Hi, Cesium,

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

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 [itex]\phi[/itex] varies, and for which [itex]t[/itex], [itex]r[/itex], [itex]\theta[/itex] are held constant. Because of periodicity with respect to [itex]\phi[/itex], any such curve is closed.

Now, the timelike part.

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

[tex]
ds^2 = \left( r^2 + a^2 + \frac{2Mr a^2}{r^2} \right) d\phi^2
[/tex]

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

Chris Hillman

Hi, Jennifer,

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.

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.

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
[itex]ds^2 = -(1-2m/r) \, du^2 + 2 \, du \, dr + r^2 \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right), [/itex]
[itex]-\infty < u < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi[/itex]
then the coordinate basis vector field [tex]\partial_u[/tex] is timelike outside the horizon but null on the horizon and spacelike inside the horizon.

Since [tex]\partial_u[/tex] 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), [tex]\partial_u[/tex] 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

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.


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.


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?

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

Chris Hillman

Hi, Jennifer

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.

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:Hi...ral_relativity)
[itex]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, [/itex]
[itex]-\infty < t < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi[/itex]
Multiplying out, we find that [tex] g_{\phi \phi} = \left( 1-a^2 \, r^2 \right) \, r^2 [/tex], so that the coordinate basis vector [tex]\partial_\phi[/tex] is spacelike on [tex]0< r < 1/a[/tex], null at [tex]r=1/a[/tex], and timelike on [tex]1/a < r < \infty[/tex]. This means that the circles [tex]t=t_0, \; z=z_0, \; r=r_0[/tex] are spacelike curves when [tex]r_0 < 1/a[/tex] but TIMELIKE when [tex]r_0 > 1/a[/tex].

At this point, you might wish to jump to the article "van Stockum dust" archived at http://en.wikipedia.org/wiki/User:Hi...ral_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

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

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

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?