Re: Extensions of black-hole spacetimes
- From: tessel@xxxxxx
- Date: Tue, 22 Nov 2005 18:29:04 +0000 (UTC)
On Sat, 19 Nov 2005, David Baker wrote:
Whenever I see a diagram for the gravitational collapse of the star, it lacks an extended region like you get in Kruskal spacetime.
To be specific, you can find such diagrams sketched in ASCII art at
http://math.ucr.edu/home/baez/RelWWW/history.html
and (unfortunately rather small) genuine diagrams in Frolov & Novikov, Black Hole Physics, Kluwer, 1998.
Here is a Carter-Penrose diagram for the maximal extension of Schwarzschild vacuum:
future
singularity 888888888888 i^+ ("future timelike infinity")
/\ /\
/ \ future / \
/ \ int. / \
/ \ / \ scri^+ ("future null infinity")
/ second \ / first \
/ exterior \/ exterior \
\ region /\ region / i^0 ("spatial infinity", r = infty)
\ / \ /
\ / \ /
\ / past \ / scri^- ("past null infinity")
\ / int. \ /
\/ \/
888888888888 i^- ("past timelike infinity") past
singularityAnd here is a Carter-Penrose diagram for the Oppenheimer-Snyder model of a collapsing star:
singularity
888888 i^+ ("future timelike infinity")
|* /\
|** / \ scri^+ ("future null infinity")
|**/ \
|** \ i^0 ("spatial infinity", r = infty)
|*** /
r = 0 |*** /
|** /
|** /
|** /
|* / scri^- ("past null infinity")
|* /
|*/
|/ i^- ("past timelike infinity")In the OS model, a shrinking ball of FRW dust (locally flat hyperslices orthogonal to the world lines of the dust particles, which form a contracting timelike geodesic congruence) is matched across a shrinking spherical surface to a Schwarzschild vacuum region (i.e. part of the maximally extended spacetime depicted above).
In the ASCII art diagram, the asterisks suggest where the dust is. You can think of this region as an ordinary shrinking ball in E^3, cut out and pasted across the world sheet of the shrinking spherical surface to the vacuum exterior.
In both diagrams above, we have a family of nested spheres, each represented by a point. Points near the right edges represent spheres with surface areas tending to infinity; points near the left edge represent spheres with surface areas tending to zero. In the second diagram, the world sheet of dust particles on the surface of the collapsing dust ball is represented as a curve which runs through the event horizon and into the developing black hole. The event horizon forms at the center, propagates outward, and eventually reaches the point i^+ in the diagram. At left, the ball exists in a perfectly flat region of E^3, and -locally- the dust particle in the center is in no way distinguished, as you'd expect from the homogeneity and isotropy properties of FRW dust. (Globally, of course, the existence of the stellar surface breaks these symmetries.)
I haven't looked at the details, but does anyone know if these spacetimes are actually maximally extended?
I claim that -both- are maximally extended, in particular
Or do they admit of Kruskal-style extensions (which I suppose would involve another star collapsing into a black hole in the extended region)?
you can't extend the dust ball of the OS model to more than one exterior region, since its only got one surface across which you can match and we've already used that up. That's the major global geometrical difference between these models: the OS model only has one exterior region. The major physical difference is that the OS initial/boundary conditions make good sense, but the maximal Schwarzschild vacuum conditions probably do not. Cf. discussion in another current thread ("gravity wave" [sic]) of Feynman's famous discussion of a spherical shell of EM radiation shrinking onto a charge you are holding and wiggling it while your arm hapens to jerk, versus using your arm to wiggle a charge, thus creating an expanding spherical shell of EM radiation. (There is a nice illustration of such an expanding shell in MTW, btw.)
The world lines of the dust particles -do- all wind up striking the strong scalar spacelike singularity (at top, indicated by "hazard eights", the perverted cousins of "lazy eights"), but they cannot be extended beyond that place, at least not in a smooth manifold, so these are "inextensible" geodesic curves. Infalling particles can pass under the horizon and they too strike the singularity (probably to the right of the dust) in finite proper time after passing the horizon.
Frolov and Novikov offer an -excellent- discussion of the Vaidya null dust, an exact solution which is a very fruitful source of thought experiments in gtr. If you've never heard of the Vaidya null dust, fear not: you obtain it by letting the mass parameter in the Schwarzschild solution (written in an ingoing or outgoing Eddington chart) depend on time in an -arbitrary- way! This is probably the only example of an exact solution which was discovered by adding "(t)" after a constant! Then you can study infalling spherical shells of massless radiation, pretty much like Feynman's thought experiment.
A particularly striking thought experiment of this kind has an initially Minkowksi region (everything below below the diagonal periods in the diagram below). Suddenly a shrinking spherical shell of radiation arrives (periods in the diagram below) and collapses to a point, forming (uh oh!) a black hole. Nearby observers notice the shell's energy as it passes, and then they begin to fall after the shrinking shell. With dismay they realize they are trapped in a black hole!! Aiiiaiaii!
8888888
|.. /\
| .. / \ scri^+
| .. \
| A/.. / i^0
| / .. /
|/ ./
| /
| /
| / scri^-
| /
| /
|/i^-
The most interesting thing about their predicament is that the horizon has already expanded beyond their location -before the radiation arrived- in the first place! The could have avoided a grisly fate if they'd known the shell was coming, but it is coming at the speed of light and they can't possibly know it is coming before it passes them. (See event A.) This shows in a very striking way the crucial distinction between local and global structure. The mass energy of the infalling radiation, which shows up locally in the suddenly nonvanishing tensor field T^(ab), is a local and immediately observable quantity. The horizon is a global concept. Are we currently inside a black hole? As this experiment shows, we -could- be, since in general there is no way to detect the possibility that an event horizon has formed and expanded to include us, at least not until it is too late! Not that I'd lose any sleep over this possibility, but it's something to think about next time you are caught in traffic!
If you start with the outgoing Eddington chart and let m depend on time, you get the time reversed case, i.e. an outgoing Vaidya model. In principle, this can be matched to a fluid with radial heat flow, with spherical wavefronts of massless radiation expanding from the shrinking surface of the "star". The mass of star decreases because some of it is being converted into the outgoing radiation. That is, the idea is try to create a highly idealized model of a -shining- star, which is a cool thing to do. Student model builders should try this, in fact, since unexpected things will probably happen :-/ If you get stuck, try preprints like
gr-qc/0504045 gr-qc/0310080 gr-qc/0209035
for inspiration--- and you-are-not-alone comfort :-)
Another fun thing to try: negative energy shells. What do you expect the quantum inequalities (see the current thread on wormholes) to imply?
Obviously you won't get a white hole in the past, but can you still get another external region connected to the black hole event horizon?
Not in the classic OS model. But there are many more possibilities for you to play with, some of which should please you.
In the past we've discussed an "inside out" OS model in which a Schwarzchild vacuumm -interior- is matched to an FRW dust -exterior-. This represents a kind of bubble immersed in an FRW universe.
Another suggestion: throw in a dS term to get a Schwarzschild-de Sitter lambdavac solutions, aka Koettler solution. This has more than one maximal extension! See these nice preprints by Brill
http://xxx.lanl.gov/abs/gr-qc/9507019 http://xxx.lanl.gov/abs/gr-qc/9501023
The simplest Carter-Penrose diagram in this case is
B
____8888____
|\ /\ /\ /|
| \/ \/ \/ |
| /\ /\ /\ |
|/__\/ \/__\|
8888
AIf you've seen Carter-Penrose diagrams for de Sitter, hopefully you can see this is like a white hole/black hole pair with two exterior regions, each asympotically approaching dS rather than Mink spacetime at large distances. This is an example of a spacetime which is "asymptotically simple" but not "asympotically flat".
In the right hand exterior region, there is a vertical line segment going from A to B. You can take this to represnt the world sheet of a spherical shell of test particles which can "hover" at a certain "radius" because the lambda term just balances the gravitational attraction of the hole. (This shell of test particles is unstable to small perturbations, of course.)
Next, of course, you should try to match to a collapsing dust ball to create an OS model with lambda term. (Don't forget to include the lambda term on the RHS of the EFE.) Then try to create a bubble model. Then look at Brill's preprints for some bigger maximal extensions of Schwarzschild-de Sitter, and repeat. I think you'll like what you see! :-)
The Robinson-Trautman family of spacetimes, which can be interpreted as null dusts (or vacuums in special cases) which generalize the Vaidya null dust (Schwarzschild vacuum), and also includes the famous Kinnersley-Walker photon rocket, also have very interesting Carter-Penrose diagrams. Like Vaidya, these come in ingoing and outgoing versions which can be interpreted as modeling ingoing or outgoing radiation, with wavefronts which are distorted contracting or expanding spheres. See for example the fine review paper
http://xxx.lanl.gov/abs/gr-qc/0004016
Wrt the preprints of Brill, note that the RT spacetimes have the very interesting property that their horizons are crinkled such that one cannot obtain smooth extensions! There are several recent preprints on RT vacuums, including
http://xxx.lanl.gov/abs/gr-qc/0412016
Those who recall my attempt to exposit some of the beautiful mathematics of soliton theory will be intrigued by the conjecture that RT equation, the striking fourth order PDE which governs the RT solutions, is exactly solvable, like the famous KdV equation! Such equations have an unusual degree of "symmetry", which enforces some beautiful structure. This is why pure mathematicians are interested in the RT equation, independently of its origin. Interestingly enough, the mathematical physicist Plebanski once wrote down a "heavenly equation" by analogy, which I think has recently been proven to be completely integrable. These are all nonlinear wave equations of a kind.
There are some other noteworthy points here. You probably know that introducing rotation drastically changes the causal structure (compare maximal Kerr and maximal Schwarzschild in good textbooks). Clearly at most one of these pictures can be qualitatively correct in the generic situation! This raises the question studied by Poisson and Israel and many others since: what is the generic picture of a realistic black hole interior? This is a topic of current research and I think it is fair to say that is far from settled, but you can find some interesting educated guesses in arXiv preprints.
One of the most interesting sources of such educated guesses is due to a beautiful discovery of Chandrasekhar: the theory of black hole interiors is closely tied to the theory of colliding plane waves!
You may have heard that the ultra-relavistic boost of a Schwarzschild hole gives an -axisymmetric- impulsive pp wave (not a plane wave), the Aichelburg-Sexl ultraboost. Similarly for Kerr and indeed any asymptotically flat Ernst vacuum, except that the axisymmetric pp waves get more complicated. (Caveat: the limiting procedure is tricky and it seems safest to say that there need not be a unique choice which is clearly preferred.) So physicists like t'Hooft have long suggested taking two of these guys and making them collide head on, to study the scattering of two black holes (perhaps the tiny kind which might be produced in a collider). This amounts to studying highly idealized black hole collision and has the appearance of a kind of classical "wave-particle duality", although I suggest this suggestion be taken with a massive dose of salt, or better yet, not swallowed at all :-/
But I am talking about colliding -plane- waves, in which all points on the planar wavefronts are equivalent. That is, initially we have a flat vacuum region, then along come two waves which collide head on. In the classic example of Penrose and Khan, two impulsive plane waves collide. Regions I, II, III are locally flat, but the fourth region is -curved- and ends in a singularity. Other models include weak nonscalar singularities which might be survivable physically, but which kill gtr's ability to predict beyond a certain point.
88
8 8
/\ /\
/ \/ \
/ /\ \
/ II/ \III\
/ / I \ \The surprising fact is that when two such waves collide, the region left behind when they scatter each other can be locally isometric to (part of) the -interior- of a Schwarzschild or Kerr hole! Because the topology is different, this allows a classical peek into the local geometry of the outer portions -inside- the horizon of a black hole! Hence civilian interest in colliding plane waves in "analog gravity".
Returning to the construction of collapse models by matching a fluid region to a vacuum or null dust region: taking a different tack, you can try matching to something more interesting than an FRW dust, such as a fluid with radial heat flow (as above), or an LTB dust, one of the most popular choices of exact dust solution for studying simple gravitational collapse models. For the latter, you can look for arXiv preprints giving a useful form of the LTB (Lemaitre-Tolman-Bondi) dust.
By the way, Hermann Bondi, a leading Humanist who died on 10 September after a long and very distinguished career, was not only a coauthor of the steady-state theory, but a pioneer of the theoretical study of gravitational radiation in gtr, as well as the theoretical study of gravitational collapse.
"T. Essel" (hiding somewhere in cyberspace)
.
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