Black holes with a nonconstant kinetic term in degenerate higher-order scalar tensor theories

TL;DR

This paper finds new black hole solutions in shift-symmetric DHOST theories allowing a nonconstant scalar kinetic term, demonstrating their stability and extending the understanding of scalar-tensor gravity models beyond traditional solutions.

Contribution

It provides the first analytic vacuum black hole solutions in Class Ia DHOST theories with nonconstant kinetic terms, including models with linear time dependence of the scalar field.

Findings

Some solutions are free of ghost and Laplacian instabilities.

Solutions differ from Schwarzschild and Schwarzschild-(A)dS black holes.

Analytic and numerical examples of nontrivial black holes with scalar hair.

Abstract

We investigate static and spherically symmetric black hole (BH) solutions in shift-symmetric quadratic-order degenerate higher-order scalar-tensor (DHOST) theories. We allow a nonconstant kinetic term $X=g^{\mu\nu} \partial_\mu\phi\partial_\nu \phi$ for the scalar field $\phi$ and assume that $\phi$ is, like the spacetime, a pure function of the radial coordinate $r$, namely $\phi=\phi(r)$. First, we find analytic static and spherically symmetric vacuum solutions in the so-called {\it Class Ia} DHOST theories, which include the quartic Horndeski theories as a subclass. We consider several explicit models in this class and apply our scheme to find the exact vacuum BH solutions. BH solutions obtained in our analysis are neither Schwarzschild or Schwarzschild (anti-) de Sitter. We show that a part of the BH solutions obtained in our analysis are free of ghost and Laplacian instabilities and are also mode stable against the odd-parity perturbations. Finally, we argue the case that the scalar field has a linear time dependence $\phi=qt+\psi (r)$ and show several simple examples of nontrivial BH solutions with a nonconstant kinetic term obtained analytically and numerically.