February 18, 1994

This Week's Finds in Mathematical Physics (Week 31)

John Baez

Well, I'm really busy these days trying to finish up a big project, hence the low number of "Weeks" per week, but papers are piling up, and there are some pretty interesting ones, so I thought I'd quickly mention a few. This bunch will mainly concern quantum gravity.

1) Possible implications of the quantum theory of gravity, by Louis Crane, 5 pages in LaTeX format, available as hep-th/9402104.

This is one paper that everyone can read and enjoy, for although one may find it too close to science fiction for comfort, it is far more interesting than most science fiction. Louis Crane has been doing a lot of excellent work on topological quantum field theory for the last few years, strongly advocating the use of category theory as a unifying principle in physics (essentially as an extension of the concept of symmetry embodied in group theory), but this is quite different in flavor.

To begin with, Lee Smolin, one of the originators of the loop representation of quantum gravity, has been spending the last year or so writing a book in a popular style, to be entitled "Life and Light," which tours the cosmos and makes some interesting speculations on "evolutionary cosmology." These speculations are based on 2 hypotheses.

A. The formation of a black hole creates "baby universe," the final singularity of the black hole tunnelling right on through to the initial "big bang" singularity of the new universe thanks to quantum effects. While this must undoubtedly seem outre to anyone unfamiliar with the sort of thing theoretical physicists amuse themselves with these days, in a recent review article by John Preskill on the information loss paradox for black holes, he reluctantly concluded that this was the most conservative solution of that famous problem! Recall the problem: if a black hole evaporates its mass away via Hawking radiation, and that radiation is pure blackbody radiation, hence carries none of the information about the matter that originally formed the black hole, one does not have conservation of information, or more technically speaking, the time evolution is not unitary, since a pure state is evolving into a mixed state. Hawking's original solution to this problem was to bite the bullet and accept the nonunitarity, even though it goes against the basic principles of quantum theory. This appears in:

2) S. W. Hawking, Phys. Rev. D13, 191 (1976).

The "baby universe" solution simply says that the matter seeds a baby universe and the information goes there. Many other solutions have been proposed; two recent review articles are

3) Do Black Holes Destroy Information? by J. Preskill, Caltech report CALT-68-1819, available as hep-th/9209058, Sept. 1992.

4) Black hole information, by Don Page, review lecture to be published in Proceedings of the 5th Canadian Conference on General Relativity and Relativistic Astrophysics, University of Waterloo, 13-15 May, 1993}, edited by R. B. Mann and R. G. McLenaghan (World Scientific, Singapore, 1994), now available in LaTeX form as hep-th/9305040.

Personally, I am a complete agnostic about this problem, since it rests upon so many phenomena that are hypothesized but not yet observed, and since any solution would require a theory of quantum gravity. I am merely reporting the ideas of respected physicists! In any event, the second hypothesis is:

B. Certain parameters of the baby universe are close to but different than those of the parent universe. The notion that certain physical facts that appear as "laws" are actually part of the state of the univese has in fact been rather respectable since the application of spontaneous symmetry breaking to the Weinberg-Salam model of electroweak interactions, part of the standard model. (Again, being my usual cautious self, I must note that a crucial piece of evidence for this model, the Higgs boson, has not yet been seen.) The notion of spontaneous symmetry breaking has become quite popular in particle physics and is a key component of all current theories, such as GUTs or string theory, that attempt to model the messy heap of observed particles and interactions by some pristinely symmetrical Lagrangian. The spontaneous symmetry breaking would be expected to have occured shortly after the big bang, when it got cool enough, much as a hot piece of iron will randomly settle upon some direction of magnetization as its temperature fall below the Curie temperature. One application of this notion to cosmology is already widely popular, namely, inflation. In fact, pursuing the analogy with magnetic domains, i.e. small regions with different directions of magnetization, cosmologists have spend a fair amount of energy thinking about "domain walls," "cosmic strings," monopoles and other defects that might occur as residues of this cooling-down process.

So again, while the idea must seem wild to anyone who has not encountered it before, physicists these days are fairly comfortable with the idea that certain "fundamental constants" could have been other than they were. As for the constants of a baby universe being close to, but different than, those of the parent universe, there is as far as I know no suggested mechanism for this. This is perhaps the weakest link in Smolin's argument (though I haven't seen his book yet). But it is at least conceivable.

Now, given these hypotheses a marvelous consequence ensues: Darwinian evolution! Those universes whose parameters are such that many black holes are formed will have many progeny, so the constants of physics can be expected to be "tuned" for the formation of many black holes. As Smolin emphasizes, while the hypotheses A and B may seem impossible to test directly at present, we do at least have a hope of testing this consequence. He has studied the marvelously intricate process of star formation in the galaxy and attempted to see whether altering the constants of physics appear "tuned" for maximizing black hole production, and he argues in his book that they do appear so tuned. Of course, this is an extremely delicate business, since our understanding of galaxy formation, star formation and black hole formation even in this universe is still rather weak - much less for other conceivable universes in which the fundamental constants take different values.

Crane enters the fray at this point, and proposes an additional conjecture:


(The capital letters are his.) He breaks it up into two parts for us:




and argues for each. The result, as any good evolutionist will recognize, is a kind of feedback loop whereby intelligence and baby universe formation both affect each other. Indeed, Crane calls his hypothesis the "meduso-anthropic hypothesis," after certain jellyfish with a two-stage life cycle in which medusids produce polyps and vice versa. This has the charm of completely destroying the usual approach (dare I say "paradigm"?) of physics in which the parameters of the universe are regarded as indifferent to the existence of intelligence. Of course, the anthropic hypothesis is a previous attempt to breach this firewall, but a much less dramatic one, since the only role intelligence plays in that is noticing the laws of the universe.

At this point let me leave off with a quote from Crane's paper:

"It is not hard to see that if these ideas are true, they will be the victims of abuse to dwarf quantum healing and even quantum golf. That is not sufficient reason to ignore them."

and let me gradually turn towards slightly less speculative realms, eventually finishing with some papers containing rigorous mathematics! To begin with, some more on black hole entropy:

5) Some Speculations about Black Hole Entropy in String Theory, Leonard Susskind, 11 pages in AMSTeX, available as hep-th/9309145.

Black hole entropy in canonical quantum gravity and superstring theory, by L. Susskind and J. Uglum, 29 pages, available as hep-th/9401070.

The fact that the entropy of a black hole is (at least under certain circumstances) proportional to the area of its event horizon is a curious relationship between general relativity, quantum field theory and statistical mechanics that many people believe to pointing somewhere, but unfortunately nobody is sure where. Part of the reason is that the standard derivations are somewhat indirect, and the event horizon is not a physical object, so the sense in which it is the locus of entropy is difficult to understand. These authors suggest that in string theory it can be explained in terms of open strings having both ends attached to the horizon.

6) Black hole evaporation without information loss, by C.R. Stephens, G. 't Hooft and B. F. Whiting, 35 pages in TeX format, 3 figures in postscript, available as gr-qc/9310006.

This is an attempt to make black holes radiate away and disappear in a manner that preserves unitarity. I've been too busy to read it. And now for some wormholes:

7) Complementarity in Wormhole Chromodynamics, by Hoi-Kwong Lo, Kai-Ming Lee, and John Preskill, 12 pages and 2 figures, phyzzx macros required, available as hep-th/9308044.

Let me just quote the abstract and note that there is probably some quite interesting topology to be obtained by applying this sort of idea to mathematics:

The electric charge of a wormhole mouth and the magnetic flux "linked" by the wormhole are non-commuting observables, and so cannot be simultaneously diagonalized. We use this observation to resolve some puzzles in wormhole electrodynamics and chromodynamics. Specifically, we analyze the color electric field that results when a colored object traverses a wormhole, and we discuss the measurement of the wormhole charge and flux using Aharonov-Bohm interference effects. We suggest that wormhole mouths may obey conventional quantum statistics, contrary to a recent proposal by Strominger.

Finally, lest the mathematicians think I have abandoned ship, some rigorous results:

8) "No Hair" Theorems - Folklore, Conjectures, Results, by Piotr T. Chrusciel, Garching preprint MPA 792, 30 pages available in LaTeX form as gr-qc/9402032.

The famous "no hair" theorem says that in general relativity static black hole solutions are determined by very few parameters - typically listed as mass, angular momentum and charge in "rest frame" of the black hole. There have been many attempts to extend this result, especially because no actual black hole is likely to be utterly static, since it presumably formed at some time. I have not read this but Chrusciel is a very careful person so I expect it will be up to the standards of his nice review of work on the cosmic censorship hypothesis,

9) On uniqueness in the large of solutions of Einstein's equations ("Strong cosmic censorship"), by Piotr T. Chrusciel, in Mathematical Aspects of Classical Field Theory, Contemp. Math. 132, eds. Gotay, Marsden and Moncrief, AMS, Rhode Island, 1992, pp. 235-274.

Addendum: See "week33" for a paper by Smolin on his evolutionary cosmology theory. His book came out in 1997 under the title "The Life of the Cosmos" - see "week101" for details.

The thing that makes things and the thing that makes things fall apart - they're the same thing. Entropy maximization! - Chris Lee

© 1994 John Baez