Are static black holes solitons?

If we start with the Einstein-Hilbert action with no matter, and consider time independent finite energy field configurations, then any static solution (e.g Schwarzchild metric) seems to be a soliton-like solution of the equations of motion corresponding to the Lagrangian.

Now topological solitons have the property that perturbative quantum fluctuations don't decay solitons to trivial ground state configurations, i.e they are topologically protected (I don't know whether something like this which prevents decay from solitonic configuration to the ground state configuration is true of non-topological solitons). However in the case of black holes, Hawking radiation processes decay the black hole metric to flat space metric.

Considering the two viewpoints, can static black hole configurations be considered as solitonic configurations? If yes, how is the doubt in the second paragraph resolved?

• What's your definition of "soliton"? – ACuriousMind Sep 27 '17 at 8:17

3 Answers

Being a static, localized and finite energy density solution to a non-linear field equation is not enough to be a soliton. It also has to be stable. Topological solitons such as $\phi^4$ or sine-Gordon kink, vortices and monopoles have they stability guaranteed by topological conditions in the form of conserved topological charges. These topological charges are normally associated to the topology of the vacuum manifold. On the other hand, non-topological solitons such as the KdV soliton have their stability secured by an infinite number of Noether conserved charges whose origin relies on symmetries.

You are mixing two frameworks, classical general relativity and its black holes, and quantum mechanics ( hawking radiation) .

Effective quantization of gravity seems to work in cases like hawking radiation, but note, effective . Black holes as solutions of classical general relativity cannot be understood quantum mechanically until or when a definitive quantization of gravity is attained.

The effective quantization solution for the beginning of the Big Bang with its indeterminate quantum mechanical region instead of a singularity is not a soliton solution of anything classical; so my guess is that , like all singularities, the soliton solution to the classical frame will be a mathematical formula, not a model for observations.

• Let me rephrase the question. The idea here is that topological solitons are topologically protected against going from their non-trivial configurations to ground state configurations by quantum fluctuations. But classical static black holes aren't, they evaporate off by Hawking radiation. So my question is, can they still be considered as some special class of non-topological solitons, since they satisfy the condition of static finite energy field configurations? – Bruce Lee Sep 27 '17 at 5:57
• And my answer is that until gravity is definitevely quantized, from singularities up ( and black holes depend on singularities in the classical solution) the question cannot be answered, because there is no proof that the classical soliton solution is valid/relevant in a quantized general relativity theory. – anna v Sep 27 '17 at 6:08

It depends how you define a "soliton". For example, the "inverse spectral method/transformation" is a method to exactly solve inital value problems of non-linear partial differential equations (e.g. the Korteweg-de-Vries equation). A characteristic of solutions obtained by the inverse scattering method is the existence of solitons. Furthermore, the stationary axisymmetric Einstein equations in vacuum are equivalent to the complex Ernst equation. It can be solved by writing the inverse scattering problem as a Riemann–Hilbert factorization problem and the solutions are indeed black holes. So in a sense one could consider black holes to be some kind of soliton.

• I see. This makes the situation clear classically. However there is another aspect which is the stability of the soliton. Asymptotically Flat space black holes decay. However in the case of large AdS black holes, reflecting boundary conditions make the black hole eternal, and it is a soliton by both your as well as this criteria. – Bruce Lee Jan 31 '20 at 0:24