# 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
• @Diracology Thanks for the clarification, you could have put it down as an answer. – Bruce Lee Sep 30 '17 at 22:57
• @BruceLee I did it! – Diracology Oct 1 '17 at 0:58

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.