
#19
Nov106, 09:22 PM

P: 232





#20
Nov306, 09:06 PM

P: 218

In case anyone's still interested in this, the clearest pedagogical explanation of this is (unsurprisingly) in MTW. Check out pp. 823826 for a discussion that uses the Schwarzschild geometry and the surface at [tex]r=2M[/tex] as an example.




#21
Nov406, 08:50 AM

P: 2,043

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




#22
Nov1906, 06:56 PM

Sci Advisor
P: 2,341

Hi again, Jennifer,
I prefer to be more specific about this "tipping". I guess you are talking about light cones in the BoyerLindquist 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 SpaceTime, for both of these examples). But let's study an even simpler example: [itex]ds^2 = (12 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  (1m/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]. 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]. Hope this helps, Chris Hillman 



#23
Nov1906, 08:01 PM

P: 2,050





#24
Nov2106, 02:55 AM

Sci Advisor
P: 2,341

Hi, Cesium,
The TaubNUT 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/grqc/0611093 Chris Hillman 



#25
Nov2106, 04:18 AM

P: 33

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 fwopath in Kruskal coordinates).




#26
Nov2106, 10:01 AM

Mentor
P: 6,044

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. 



#27
Nov2706, 06:42 PM

Sci Advisor
P: 2,341

Hi, Jennifer,
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. [itex]ds^2 = (12m/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 



#28
Nov2706, 07:18 PM

P: 2,043

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



#29
Nov2706, 09:36 PM

Sci Advisor
P: 2,341

Hi, Jennifer
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( 1a^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 



#30
Nov2806, 12:22 AM

P: 2,043

Thanks Chris you gave me something to digest.




#31
Jan307, 10:17 AM

P: 2,043

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



#32
Jan307, 11:06 AM

P: 363

It's just funny of mathematical model used (rather abusive one). Make an experiment. That's the only way you will know for sure . 



#33
Jan307, 07:37 PM

P: 312

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



#34
Jan307, 07:48 PM

Emeritus
Sci Advisor
P: 7,445





#35
Jan407, 12:58 AM

P: 2,043

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



#36
Jan407, 03:34 AM

Emeritus
Sci Advisor
P: 7,445

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 spacetime can be described by a timelike 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 timelike 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 selfintersecting worldine is that of a person, a timelike 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 lesspathological but more recongnizable form of time travel, the selfintersecting timelike curve. You might also be able to have "multiloop" CTC's, depending on the exact dynamics, which would probably best be thought of in terms of a phase space. 


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