The last post was sparked off by a talk I heard at a conference in Oberwolfach. Here I will write about a topic where another talk at that conference looks like a big step forward. This was by Jared Speck. He was describing work of his with Igor Rodnianski which is not yet fully written up.

These days there is a wide consensus among astrophysicists that there is strong observational evidence to indicate that the expansion of the universe is accelerated. In other words it is not only the case that all distant galaxies are moving away from us (and from each other) but the velocity of recession is actually increasing. In the standard view this is only consistent with general relativity if there is a positive cosmological constant or some exotic matter called dark energy. For convenience I will not distinguish between these two in what follows. Dark energy leads to accelerated expansion and accelerated expansion causes spatial irregularities to be damped. The

geometry of spacetime and the matter distribution are smoothed. This kind of idea can be turned into a precise mathematical statement (maybe not uniquely) called the cosmic no hair theorem. From a mathematical point of view this is rather a conjecture than a theorem – at least it has been that way for most of the time it has existed. The name originates from a phrase of John Wheeler, ‘a black hole has no hair’. The idea of this was that a particular solution of the Einstein equations describing a black hole, the Kerr solution, should be attractor for the evolution of more general solutions containing a black hole. In other words a general class of solutions should evolve so as to look more and more like the Kerr solution. The Kerr solution depends only on two parameters. Thus in this scenario all the details get lost dynamically, leaving a very simple object with no complicated features, no hair. In models for an expanding universe with positive cosmological constant the smoothing process mentioned above also seems to drive all solutions towards an attractor, the de Sitter solution. It is this analogy which gave rise to the name ‘cosmic no hair theorem’.

The mathematical formalization of the cosmic no hair theorem says that a solution of the Einstein-matter equations with positive cosmological constant converges to the de Sitter solution at late times in a suitable sense. A weaker statement is that this should be true for solutions which start close to the de Sitter solution. The latter version can also be thought of as a kind of stability statement for de Sitter space. In the case of the vacuum Einstein equations the stability of de Sitter space was proved by Helmut Friedrich in 1986. Since our universe is certainly not empty the relevance of this result to cosmology is not immediately obvious. It turns out, however, that there are reasons to believe that the cosmological constant can often have a dominant effect on the late-time cosmological expansion which tends to make the effect of the matter into a higher-order correction. It is important to confirm these ideas by a theorem which includes the effect of matter. The most commonly used matter model in cosmology is a perfect fluid with linear equation of state. It contains a parameter which is often restricted by an inequality corresponding to perfect fluids which are less stiff than radiation. The result of Rodnianski and Speck is a form of the cosmic no hair theorem for precisely this class of matter models. The proofs build on previous work of Hans Ringström. Friedrich’s proof uses a technique (the conformal method) which is very powerful but rather rigid. It is difficult to see how to modify the proof to include matter such as a fluid, or indeed to replace the cosmological constant by some other kind of dark energy, such as a nonlinear scalar field. Ringström introduced more flexible methods which allowed him to obtain a version of the cosmic no hair theorem for dark energy modelled by certain types of nonlinear scalar field. His methods open up the perspective of including matter and this is what Rodnianski and Speck have now done. These methods use energy estimates, the workhorse of the theory of nonlinear hyperbolic equations, in a clever way. (I might say more clever than I am, since I once tried very hard to do this, without success.)

The result of Rodnianski and Speck is restricted to the case of irrotational fluids. I see no fundamental reason why this should be necessary. Nevertheless there is a clear technical reason – in the irrotational case the Euler equation of the fluid is equivalent to a nonlinear wave equation. On the level of formal power series the case with rotation works out, as shown in a paper of mine (Ann. H. Poincare 5, 1041). Another question is what happens for large data. In that case there are various restrictions.For sufficiently large data it is to be expected that black holes would be formed (even in the vacuum case). Moreover, the fluid can be expected to form shocks which means that the solution cannot be continued, at least in the realm of smooth solutions. I find it remarkable that the expansion caused by a positive cosmological constant is strong enough to suppress formation of shocks in a small data regime. There is just one result available on this subject for large initial data and inhomogeneous solutions. In this work, due to Blaise Tchapnda and myself (arising from Blaise’s PhD thesis, Class. Quantum Grav. 20, 3037) we treated plane-symmetric solutions of the Einstein-Vlasov system with positive cosmological constant. In this case the symmetry prevents formation of black holes and the choice of matter model allows any analogue of shocks to be avoided.

October 26, 2009 at 11:31 am |

Hello

could you say a word or two about these clever energy estimates discovered by Speck et al?

By the way, I am not entirely sure about this, but wouldn’t Friedrichs techniques work for a perfect fluid with p=1/3\rho, that is the trace of

the energy tensor vanishes.

October 26, 2009 at 2:12 pm |

Hello Uwe

When talking about ‘clever energy estimates’ I was thinking of those used by Ringstrom. As I understood from Speck’s talk the case with a fluid is a lot more difficult but I know very little about the details. As to the orginal estimates I would sum up the situation as follows. The only tool available is energy estimates and so what is necessary (and difficult) is to find the right energy functionals. One procedure which is useful in doing this is to identify what are likely to be the main terms and produce a simplified model system by discarding all other terms Analysing the dynamics of the model system then helps to guess how to treat the original system. Doing these things has a lot to do with intuition and is anything but a deterministic algorithm.

October 30, 2009 at 10:53 am |

Hello Alan,

let me ask a more precise question then. Does the energy in the case of the Einstein equations with a perfect fluid have some geometric or kinetic interpretation? I am especially interested in the interpretation for the part of the energy which corresponds to the fluid.

October 30, 2009 at 10:45 am |

I just realized that I only answered one of the two questions which Uwe asked. Concerning the second, it is plausible that Friedrich’s techniques would work in the case of a radiation fluid but nobody has ever worked it out.

October 30, 2009 at 10:49 am |

Right, since you did not answer I presumed the answer would be no.

I do remember having made some calculations, but, since this is a long time ago, I am not sure whether it was trivial or difficult.

I presume it was quite straightforward, but since it was basically a copy of Friedrichs original work I did not bother to write it up.

November 17, 2009 at 12:30 am |

Alan, quick question about the term “no hair theorem”.

I’ve been under the impression before (similar to what you described) that the “hair” represents higher order terms in the asymptotic expansion at spatial infinity. The higher multipole moments decays faster and thus cannot be resolved as well at large distances. So an analogy is drawn to human hair: one cannot see it at large distances, but one can examine it when one gets closer.

Recently Gilbert Weinstein offered me (at the Boca Raton AMS meeting) an alternate interpretation of the term: he uses “no hair” to refer to the fact that the set of stationary AF solutions to electro-vac (regularity conditions yada yada) embeds in the set of smooth solutions as a 3 dimensional submanifold. No hair is used to described the fact that the set of stationary solutions do not bifurcate.

In his language, no hair is strictly weaker than local/weak rigidity (which is taken to mean that there exists a tubular neighborhood of the Kerr-Newman family [in some suitable topology] which does not intersect any other stationary solutions; no hair allows another sheave of solution to come arbitrary close to the KN family), which is strictly weaker than uniqueness. Whereas, I think, in your parlance no hair is equivalent to uniqueness?

Anyway, I think I arrived at my interpretation after reading Carter’s Les Houches lectures. My question actually is this: do you know where Wheeler’s phrase was first used? I’ve seen that referred to a lot, but mostly as folklore.

November 17, 2009 at 10:03 am |

Hi Willie,

The mental picture I have is of the black hole at late times being like a very round sphere, resembling a bald head. I never heard the bifurcation idea before. As for the meaning of ‘no hair theorem’ for black holes, I see two versions. The first (weak) says that the Kerr-Newman solutions are unique under certain assumptions. The second (strong, rather vague) says that all dynamical solutions containing a black hole converge at late time to Kerr-Newman solutions. Clearly any reasonable formulation of the second implies any reasonable formulation of the first. But what about the other way around?. Here is a possble ‘physics’ argument. The system is dissipative due to emission of gravitational radiation. Therefore (?) it converges to an equilibrium solution. The first version tells us about the equilibrium solutions. Of course what is being ignored in this ‘therefore’ could be described as ‘hair-raising’.

Now for your concrete question. It seems that Wheeler did really write this phrase. I have not seen the paper myself but the reference is given in a paper of Thorne and Zurek, Foundations of Physics, 16, 79. Interestingly this paper also gives some information which tends to support the idea that Wheeler invented the name ‘black hole’, an idea which is sometimes contested.

December 1, 2009 at 1:31 pm |

Rodnianski and Speck have now posted their paper on this subject as http://arxiv.org/abs/0911.5501

October 15, 2010 at 6:19 pm |

Hi all,

is there a no-hair theorem for extensions of General Relativity? For f(R) gravity for example?

October 16, 2010 at 9:37 am |

Hi,

there are some results on f(R) gravity which are obtained by using the alternative formulation via a nonlinear scalar field. They only treat the case of homogeneous cosmologies. See the paper arXiv:0810.3558 by Lucy Macnay. It might be possible to do something similar in the inhomogeneous case, using the results on nonlinear scalar fields due to Hans Ringstrom, but I do not think anybody did it yet.

February 14, 2011 at 9:22 am |

I was pleased to see that Jared Speck has now extended the result discussed in the post to the general case, i.e. the case with rotation. The reference is http://arxiv.org/abs/1102.1501.

July 21, 2011 at 7:23 am |

[…] In the meantime there are more satisfactory results on this question useing other methods (see this post) but this example does show that the Fuchsian method can be applied to problems in general […]