No-go theorem for false vacuum black holes

Abstract

We study the possibility of non-singular black hole solutions in the theory of general relativity coupled to a non-linear scalar field with a positive potential possessing two minima: a 'false vacuum' with positive energy and a 'true vacuum' with zero energy. Assuming that the scalar field starts at the false vacuum at the origin and comes to the true vacuum at spatial infinity, we prove a no-go theorem by extending a no-hair theorem to the black hole interior: no smooth solutions exist which interpolate between the local de Sitter solution near the origin and the asymptotic Schwarzschild solution through a regular event horizon or several horizons.

Static spherically symmetric black holes with scalar field

“…First, let us consider ϕ and V in the region outside the horizon. From (19), (14) and (16) it follows that…”

“…Metric is given by ( 12) and (13). Scalar field ϕ and its potential V are given by (22) (or (14), see Remark 1) and ( 16).…”

“…Later studies on the Einstein-scalar equations led to a number of no-hair theorems on static spherically symmetric solutions representing black holes or particle like solutions which are asymptotically flat [6,7,8,9,10,11,12,13,14,15,16]. They show that in the case of black holes potential V of the scalar field cannot be positive definite simultaneously at all points outside the event horizon.…”

Correction to: Nonlinear electrodynamics and regular black holes

“…The third solution is the solution with thin shell layer. The thin shell layer can be time-like, space-like or null [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46].…”

An exact black hole spacetime with scalar field and its shadow together with quasinormal modes

“…This solution brings us many interesting properties, for example, the logarithmical divergence problem of total energy for the static field. The subsequent studies on the Einstein-scalar theories lead to the construction of no-hair theorem on black holes or particle-like asymptotically flat spacetimes [2][3][4][5][6][7][8][9][10][11][12]. However, there are other black holes [13][14][15][16][17] which can evade no-hair theorem.…”