Toroidal black holes?
Tuesday 2007 May 22
A well known theorem about black holes (due to Hawking?) states that their boundary must be a sphere, rather than a higher genus surface. For genus greater than 1 the proof seems fine, but the proof for genus 1 (a torus) seems incomplete :it rules them out physically as being unstable, but this seems to allow the possibility that they exist as unstable mathematical objects.
Consider a very thin torus of dust, rotating at just the right speed for the “centrifugal force” to balance the gravitational attraction. If the torus is thin enough this seems to produce a rotating toroidal black hole.
The obvious way to prove these exist is to write down an explicit formula for the metric, but this seems quite hard. There are many known exact solutions with the right symmetry, but they are mostly rather messy and it is hard to see what is going on (and in any case if they gave toroidal black holes the people who found them would probably have noticed).
The complement of a toroidal black hole is not simply connected, so by taking its universal cover one gets a chain of universes which one can travel between by going though the hole in the torus.
Linked and knotted black holes are left as an exercise for the reader.
Wednesday 2007 May 23 at 2:11 am
Wow, this really is a physics blog! Have you come across the higher dimensional toroidal BHs in string theory? I think Horowitz et al call them ‘black rings’. Of course, in classical gravity, it is only relatively recently that people have considered chains of universes (I’m thinking of, eg., Vilenkin). I would be interested to know why you are thinking about toroidal BHs.
Thursday 2007 May 24 at 12:15 am
I don’t see why the metric should be that hard. In the limit as the mass of the ring goes to zero, space is flat except very close to the ring, which is almost non rotating. And near the ring one should be able to ignore everything except a thin line of mass. You may not be able to calculate a metric but I think you can prove that one exists.
I recently set myself the task of reading all arXiv articles written by Chris Doran and Anthony Lasenby. An article that doesn’t quite read on this question came up: Rotating Astrophysical Systems also see Analysing Axisymmetric Gravitational Systems. Their stuff deals with Kerr type metrics.
Thursday 2007 May 24 at 12:59 am
Two questions:
1 – Could the Parker Spiral associated with the sun have a similar counterpart in toroidal black holes?
NASA
http://helios.gsfc.nasa.gov/solarmag.html
alternative depiction
http://en.wikipedia.org/wiki/Parker_spiral
2 – Is the phrase ‘… the “centrifugal force” to balance the gravitational attraction’ equivalent to ‘equlibrium’ or ‘minimax’ of mathematical game theory [von Neumann, Conway, Nash, etc]?
Friday 2007 May 25 at 5:22 pm
Somewhere between 1968 and 1972 I discussed this with Kip Thorne, while I was an undergraduate at Caltech, and reading “Gravitation” by Thorne, Misner, Wheeler, in spiral-bound preprint. I was trying to interest Kip in pubplishing a book with my father, who’d edited and published 11 consecutive best sellers. Kip was dubious, at the time, that any book on General Relativity could be a best seller. This was, of course, before Hawking blew things wide open.
Kip was labriously setting up, with his students, a numerical simulation (which looks so primitive now) of a toroidal dust cloud. He wondered if it could collapse to a naked circle singularity. My intuition, I said, was “no.” The simulation agreed.
I’ve drifted away from Physics, publishing only occasionally, in interdisciplinary conference proceedings, and doing mostly mathematical Economics and Computational Biology. It is long ago that I coauthored with Feynman, albeit that’s been reprinted and adapted.
I am excited to see the issue of toroidal black holes revisted, approximately pi x 10^1 years later.
There is the practical problem of quadrupole and higher order terms radiating away, expecially from symmetry-breaking instabilities. There is the deeper theoretical problems of higher genus manifolds.
Do you like any of John Baez’s n-category theory approaches to the underlying axiomatic mathematical physics, outr of which various QFT should emerge.
Friday 2007 May 25 at 7:01 pm
Another interesting post
. I wonder if the known proofs that rule out black holes of positive genus assume vacuum; one can imagine that if one throws in matter and charge and whatnot then one can get many more solutions to the Einstein equations.
Then there is of course the issue of how exactly one defines a black hole (for instance, many definitions require a notion of asymptotic spatial infinity, which may become a little problematic (though not too much) when dealing with the universal cover of the complement of a torus. But perhaps the topology of a black hole is largely invariant under what the definition of a black hole is.
Friday 2007 May 25 at 8:20 pm
In response to some of the questions above, I dont know enough about the Parker spiral or game theory or n-categories to say anything useful.
I found Hawking’s proof that toroidal black holes cannot exist; it’s given on pages 335-337 of Hawking and Ellis, “The large scale structure of spacetime”. It does not require a vacuum spacetime, but just needs a “dominant energy condition”. It has several details missing and is not easy to follow, and in particular in the middle of page 337 I cannot figure out what is going on: he seems to assume (incorrectly) that if a function θ of w vanishes at w=0 and has non-negative derivative at w=0 then it is non-positive for small negative values of w.
Friday 2007 May 25 at 9:45 pm
Instabilities of a torus were found empirically in plasma confinement experiments. These include, I think, the flute instability, as in “Solitary vortex in a flute instability” Pavlenko, V. P.; Petviashvili, V. I., (Fizika Plazmy, vol. 9, Sept.-Oct. 1983, p. 1034-1037) Soviet Journal of Plasma Physics (ISSN 0360-0343), vol. 9, Sept.-Oct. 1983, p. 603, 604. Translation.
Crudely, I think that there are: (1) instabilities which change the topology, such as pinching off the torus into a bent cylinder (more accurately: a cylider capped at both ends by a cone, bent so that the the two end-cones share an apex), by shrinking the radius of a circle to a point; and (2) instabilities which don’t change the topology, such as those that smoothly deform the torus, for which one can often do a Fourier analysis.
In the latter case, one is dealing with oscillatory perturbations on the lateral circles, or transverse circles of the torus, or both, or twisted combinations (i.e. helices wrapped around circles, such as we see in torus knots and the usual drawings in introductory twistor theory). Even if we restrict oscillations to standing waves, we get modes that can blow up in amplitude.
As to “Linked and knotted black holes are left as an exercise for the reader” we have (with my molecular biology hat on) “catenane” = A supramolecular species consisting of mechanically interlocked macrocyclic rings. Also, as suggested earlier, knot theory in all its tangles glory.
A general theory which shows the evolution of a toroidal black hole should include all of knot theory. Which, bringing us full circle, Thomson’s vortex-atom theory which led Tait to invent knot theory. Of course, our knots are not embedded in R^3, but something that asymptotically approaches Minkowski’s R^4 in the limit of gravitation G approaching 0.
Monday 2007 May 28 at 7:24 am
It may be worth noting that an infinite-dimensional family of stationary, axially symmetric solutions to Einstein’s equation in vacuo may be obtained by using a species of Backlund transformation, as described, for example, here:
http://eom.springer.de/e/e120160.htm
The reference at the conclusion of this entry to Geroch’s 1971 paper, “A method for generating solutions of Einstein’s equations,” may well be the most practically useful of the lot. Geroch writes well, and his paper provides a very clear, straightforward exposition of the method.
But the trouble with the solutions this method provides is that it’s hard to tell what their asymptotic behavior is at null infinity — and of course it’s only solutions that have the right behavior at null infinity that could reasonably be said to represent a toroidal black hole.
Oh, and finally, I suppose someone ought to note that it’s been generally recognized for decades that Hawking-Ellis contains a number of faulty proofs. I haven’t yet gone through the one presented on pp. 335-7, but I certainly won’t be surprised to find that it does indeed suffer from serious gaps.
Most of these gaps, though, have been filled in over the years by determined and sophisticated readers of the book, and my impression is that actually rather few of the theorems stated in Hawking-Ellis are actually known to be false.
The trouble is that the vast majority of these gap-filling results amassed by generations of determined and sophisticated readers of Hawking-Ellis have never been published. They constitute a sort of folklore, with which most students of global structure and the Cauchy problem in general relativity become acquainted at their doctoral advisors’ knees.
Monday 2007 May 28 at 7:59 am
Ah, yes. I see, upon consulting Wald’s textbook, that all asymptotically flat axisymmetric stationary solutions to Einstein’s equation in vacuo are known to arise from Minkowski spacetime by the application of a suitable sequence of those Backlund-type transformations described by (for example) Geroch.
But it seems that, at least as of the early 80s, the algebraic computations needed to obtain these solutions explicitly were, to use Wald’s word, “formidable.” Evidently the obstacles were great enough back then to keep this method from yielding much of anything in the way of practical results.
But I wonder whether, with the advent of sophisticated computer algebra systems, some of these computations may by now have become quite manageable.
Naively, it would seem at least possible that they have — and in that case, one might actually be in a position to undertake an exhaustive direct search for exact solutions to Einstein’s equation capable of being construed as a toroidal black hole.
But since it’s likely that the Hawking-Ellis theorem is essentially correct, it’s probably wiser just to keep fiddling with the proof in the book, until the gaps in it disappear.
Monday 2007 May 28 at 1:23 pm
A more up to date account of exact solutions is given in
Exact Solutions of Einstein’s Field Equations
by Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt
ISBN-13: 978-0521461368
It has some solutions with the right symmetry, though
as far as I can figure out none of them are toroidal black holes, but it also refers to more solutions in the literature that I have not checked.
I think the proof in Hawking and Ellis is essentially correct for holes of genus greater than 1, and puts severe restrictions on holes of genus 1, but I’m not sure whether it is complete for genus 1.
Monday 2007 May 28 at 3:41 pm
Ah, yes, I see. I had not noticed the new edition of the EXACT SOLUTIONS book. Its original edition appeared a bit too early to include a number of significant discoveries concerning these stationary axisymmetric solutions arising from Minkowski spacetime by the action of the (infinite-dimensional) Geroch group. (In particular, Xanthopoulos’s work, showing how to generate solutions with prescribed multipole moments at null infinity, came too late for inclusion.) But no doubt the 2002 edition covers everything.
Thanks for pointing that out — I look forward to studying it! — and forgive my inadvertance in my previous post.
Saturday 2007 June 2 at 2:42 am
Black holes with horizons of nonzero genus exist for negative cosmological constant. See the paper:
Thermodynamics of (3+1)-dimensional black holes with toroidal or higher genus horizons
Authors: Dieter R. Brill, Jorma Louko, Peter Peldan
http://arxiv.org/abs/gr-qc/9705012
These black holes are not interesting for astrophysics but may be useful for the Anti-DeSitter/Conformal Field theory correspondence. In this setting the following paper is a good starting point:
Toroidal Black Holes and T-duality
Massimiliano Rinaldi
http://arxiv.org/abs/hep-th/0208026
Tuesday 2007 July 17 at 7:49 am
Asymptotically flat toroidal black holes have been shown not to exist about 10 years ago — see gr-qc/9410004.
Friday 2007 July 20 at 9:05 pm
Jose,
Your new paper
Gravitational Collapse to Toroidal and Higher Genus asymptotically AdS Black Holes
Authors: Filipe C. Mena, Jose Natario, Paul Tod
http://arxiv.org/abs/0707.2519
is very interesting.
The formation of ADS4 blackholes should have a CFT3 dual description of Choptuik scaling for example.
Friday 2007 July 20 at 10:04 pm
That is a cool paper indeed. I had not known the earlier references you cited, including Brill & Louko [1997] and Vanzo [1997]. That means that I failed to understand where this thread came from at all.
I knew less differential gemotry than anyone else learning from Kip Thorne back in the late 60s/early 70s, and have forgotten most of that and only relearned some. Still, I can stupidly ask the question: can there be two toroidal black holes linked together in Anti deSitter space?? A knotted black hole?
Monday 2007 July 23 at 5:22 pm
Interesting question. Usually when two black holes collide
the bigger one swallows the smaller one and classically one ends up with a black hole with the same horizon topology as the bigger one.
This is speculative but for two small black holes in Anti-DeSitter space quantum mechanics may allow the possiblity for the horizon topology to change. For example two toroidal black holes could combine into a genus two black hole. This isn’t a knotted black hole but the horizon topology is the direct sum of the two tori which are joined.
Monday 2007 August 6 at 8:01 pm
Thank you, McGuigan. The topological problems appear to me (as a naive observer) to become more acute for dimension greater than 4.
“…the discovery of a black hole with a ring shaped horizon that can only exist when there are more than four dimensions…” [Emparan & Reall, Phys Rev Lett 2002]
arXiv:0705.0117 (replaced)
Title: Particle Motion in the Rotating Black Ring Metric
Authors: James Hoskisson
Comments: 33 pages, 19 figures, updated references
Subjects: High Energy Physics – Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
[3 August 2007]
Tuesday 2007 August 7 at 7:05 pm
And, by the way, are you this McGuigan, of this very lovely paper?
Riemann Hypothesis, Matrix/Gravity Correspondence and FZZT Brane Partition Functions
Authors: Michael McGuigan
Comments: 17 pages, 2 figures, 1 table
Subjects: Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th)
We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large N matrix model. Using a related function $\Xi(z)$ we develop an analogy between this function and the Airy function Ai(z) of the Gaussian matrix model. The analogy gives an intuitive physical reason why the zeros lie on a critical line. Using a Fourier transform of the $\Xi(z)$ function we identify a Kontsevich integrand. Generalizing this integrand to $n \times n$ matrices we develop a Kontsevich matrix model which describes n FZZT branes. The Kontsevich model associated with the $\Xi(z)$ function is given by a superposition of Liouville type matrix models that have been used to describe matrix model instantons.
Friday 2007 November 16 at 1:25 pm
I have write , with Remy Mosseri, in a book published by Cambridge University Press: “Geometrical Frustration”
something on disclinations. These are topological defects found in condensed matter (liquid crystals for instance) which are curvature concentration.
Page 94 paragraph 4.5 we describe -4/pi disclinations which are very close to toroidal worm hole.
Far away from space-time physics !
Tuesday 2008 November 18 at 4:05 am
A torus knot of p=1290 q=235 should yeild a stable structure
1290 is right out of The Book of Daniel
235 is a Lunar Cycle (Metonic Cycle)
I’ve rendered this on Knot Plot and you can actually see through the center which would imply that it could be used as a “wormhole” without dying during the attempt.
I have Vista 64 and 4 gig of ram so I was able to allocate lots of RAM to the progam. This is not a rendering error.
Monday 2009 February 2 at 9:57 pm
I figure out the situation in a very intuitive way: the torus of dust collapsing into a toroidal black hole, will behave like a torus (or cylinder) of water in the void, under the action of capillarity (except that the capillarity “force” acts here more as a speed of deformation than as an acceleration): its instability will make it split into two or or more parts that become separate black holes. These black holes will orbit around each other, and eventually merge into only two. Orbiting more, they emit gravitational waves that take the cinetic momentum of the system with them, until all merges into a single black hole.
Monday 2009 February 2 at 11:10 pm
After more thoughts, well, sorry I did not pay attention to everything in my previous message:
First, the torus of dust won’t make any horizon (nothing special will happen) even if it collapses onto a circle (the escape velocity converges to something below c). So, if the sound speed in dust increases with density then the collapse will end to a stable value of the thickness. And the closer to the circle it is focused, the faster the instability will take place (as its characteristic time is just proportional to the thickness) to divide the dust torus, and then only collapse it into a series of black holes.
Thursday 2009 March 26 at 2:53 am
@ carlbrannen
The main reason finding a metric of this sort is relatively difficult is because we’re (gravitationally) lensing at the braneworld level. Black holes aren’t the simple configurations that sci-fi films make them out to be. In theory – and this is a rather blunt hypothesis – black holes energistically are the intitial fields for stellar evolution; not the other way around. In stringlish (lay application of English to string theory, or vice versa) there actually is no (braneworld) model that generates inflation or any type of inflaton field. Therein lies the problem of defining a functional metric that suits Einstein’s equations.
Friday 2009 April 24 at 11:19 am
Hey, cool tips. Perhaps I’ll buy a glass of beer to the man from that chat who told me to visit your blog