# Black holes within scalar fields: is the problem self-consistent?

Many times I see papers devoted to "scalar black holes" or related to black holes with "scalar hairs". All of which confuses me. As an example, consider this 2004 paper from Phys. Rev. Lett. From the Introduction:

If one expects that the global ubiquitous scalar field does exist such that everything, including black holes, is “floating” inside it,...

But then how on Earth one obtains black hole solutions at all, if a "global ubiquitous" scalar field exists? By definition, BH is a vacuum solution to Einstein's equations, but here we are told there is a ubiquitous scalar field, i.e. no vacuum. At best, I expect in such case Janis-Newman-Winicour solutions, but not vacuum Schwarzschild (or Kerr) solution. I must be missing something fundamental, but the topic of these papers seems just self-inconsistent to me from the outset.

• If a black hole is by definition a vacuum solution, doesn't it stop being a black hole as soon as the tiniest piece of dust falls inside? A black hole is a region of spacetime not casually connected to null infinity, i.e., a place from where not even light can escape. This is a more general and more useful definition. – Javier Sep 6 '18 at 12:54
• Well, this is exactly what I don't understand. If there is a background scalar field, then the spherically-symmetric solution is the Janis-Newman-Winicour metric which has no horizons (journals.aps.org/prd/abstract/10.1103/PhysRevD.31.1280). It is not the same as a piece of dust falling into already formed Schwarzschild BH. – Maximko Sep 6 '18 at 13:08

That is wrong. The defining feature of a black hole is an event horizon. The space time does not have to be a vacuum solution, there could be matter. For example Reissner–Nordström solution is a black hole if it has $Q<M$. And it is a solution of Einstein–Maxwell field equations, with electric field. Black hole spacetimes do not even have to be solutions of Einstein field equations with or without whatever matter content, since they could, for example, appear in theories other than classical general relativity.
I expect that OP's confusion stems from the (wrong) belief that there is only one possible theory of scalar field. There are, in fact, many possible theories of scalar field, differing in how this scalar field interacts with itself (specified by its potential $V(\phi)$), and how it interacts with other fields (including gravitational). In the paper referred to in question (hep-th/0408163) black holes were found in the theory with specific form of scalar field potential (equation (1) in the paper), chosen for the integrability of the model. While the JNW solution is for the massless scalar field with trivial potential $V(\phi)\equiv 0$.
• @Maximko: then objects described by Schwarzschild/Kerr metric cannot form? Not at all, just that black hole that formed would not have a 'scalar hair' (having nontrivial static/stationary scalar field around them). Instead, once the solution stabilizes (stops changing considerably with time), we would have a stable stationary (in some ref. frame) black hole (described in a region around it by Schw./Kerr metric), with the scalar field in its vicinity having constant value (and thus its stress-energy ${}=0$). Plus, there may be some scalar radiation escaping to infinity. – A.V.S. Sep 7 '18 at 15:06