Regular Compact Objects with Scalar Hair

TL;DR

This paper presents exact, regular higher-dimensional black hole solutions with scalar hair sourced by a phantom scalar field, analyzing their thermodynamics, regularity, and dependence on scalar charge and mass.

Contribution

It introduces new regular black hole solutions with scalar hair in higher dimensions, deriving scalar potentials and analyzing their thermodynamic and regularity properties.

Findings

Solutions are regular at the center for all dimensions.

Scalar potential depends on scalar charge and mass in lower dimensions.

Black holes with secondary scalar hair are asymptotically flat.

Abstract

We discuss exact regular compact object solutions in higher dimensional extensions of General Relativity sourced by a phantom scalar field in arbitrary $D$ spacetime dimensions ($D>2$), for which a central singularity is absent. We follow a bottom-up approach, by means of which, by imposing the desired form of the solution to the metric function, we derive the form of the self-interaction scalar potential, which in general appears to depend on both the scalar-hair charge and the black-hole mass. We discuss in this context the validity of the first law of thermodynamics in such systems. Consistency requires the independence of the potential of the mass, imposing in this way the dependence of the mass on the scalar charge of a type that varies with the value of $D$, and according to the no-hair theorem dressing the regular black hole solution with secondary hair. In $D=3,4$ we demonstrate that the potential depends on the ratio of the scalar charge over the mass, and thus considered as a parameter of the theory. This feature, however, does not characterise higher-dimensional cases. Calculating the $D-$dimensional Kretschmann scalar we show that it is finite at the centre point $r=0$ for arbitrary $D$, rendering the solutions regular. The phantom matter content of the theory is also regular at $r=0$, hence the radial coordinate of our manifold is defined for $r \ge 0$. We explicitly discuss the cases of $D=3,4,5,6,10$, and demonstrate that we can have regular, asymptotically flat, black holes with secondary scalar hair.