Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Phase Structure of Scalarized Black Holes in Einstein-Scalar-Gauss-Bonnet Gravity

Carlos Herdeiro, Hyat Huang, Jutta Kunz, Meng-Yun Lai, Eugen Radu, De-Cheng Zou

Abstract

We revisit scalarized black holes in Einstein-scalar-Gauss-Bonnet gravity and analyze the thermodynamic phase transition between the Schwarzschild solution of general relativity and scalarized black holes. Restricting to spherically symmetric configurations, we investigate several classes of scalar-Gauss-Bonnet coupling functions. For the simplest quadratic coupling that triggers spontaneous scalarization, the scalarized solutions are thermodynamically disfavored and no phase transition occurs. For an exponential coupling, the phase structure depends strongly on the coupling parameter, allowing for the absence of a transition, a continuous second-order transition, or a discontinuous first-order transition. For couplings leading to purely nonlinear scalarization, we find either a first-order transition or no transition. These results reveal a rich phase structure of scalarized black holes controlled by the scalar-Gauss-Bonnet coupling.

Phase Structure of Scalarized Black Holes in Einstein-Scalar-Gauss-Bonnet Gravity

Abstract

We revisit scalarized black holes in Einstein-scalar-Gauss-Bonnet gravity and analyze the thermodynamic phase transition between the Schwarzschild solution of general relativity and scalarized black holes. Restricting to spherically symmetric configurations, we investigate several classes of scalar-Gauss-Bonnet coupling functions. For the simplest quadratic coupling that triggers spontaneous scalarization, the scalarized solutions are thermodynamically disfavored and no phase transition occurs. For an exponential coupling, the phase structure depends strongly on the coupling parameter, allowing for the absence of a transition, a continuous second-order transition, or a discontinuous first-order transition. For couplings leading to purely nonlinear scalarization, we find either a first-order transition or no transition. These results reveal a rich phase structure of scalarized black holes controlled by the scalar-Gauss-Bonnet coupling.
Paper Structure (12 sections, 20 equations, 14 figures)

This paper contains 12 sections, 20 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Theoretical Settings
  3. Einstein-scalar-Gauss-Bonnet theory
  4. Radial (in)stability of Schwarzschild
  5. Quantities of interest
  6. Phase transitions
  7. Results
  8. Spontaneous scalarization
  9. A type (i) function: $f(\phi)=\frac{1}{8}\phi^2+\frac{\beta \phi^4}{64}$
  10. A type (ii) function $f(\phi)=\frac{1}{2\beta}[1-\exp(-\frac{\beta\phi^2}{4})]$
  11. Non-linear scalarization with $f(\phi)=\frac{1}{4\beta}[1-\exp(-\frac{\beta\phi^4}{16})]$
  12. Conclusions and Outlook

Figures (14)

  • Figure 1: The scalar charge is shown as a function of the BH mass and as a function of the Hawking temperature. Here and in Figures \ref{['type1-2']}, \ref{['type1-3']} a type (i) scalar field coupling function, $f(\phi)=\frac{1}{8}\phi^2+\frac{\beta \phi^4}{64}$ is considered. In all plots, the quantities are given in units set by the coupling
  • Figure 2: The difference between the free energy of scalarized and Schwarzschild BHs is shown as a function of mass and Hawking temperature.
  • Figure 3: Left: The entropy is shown as a function of scalar charge. Both $S$ and $Q_s$ are given in units of mass $M$, with $S=4\pi M^2$ for the Schwarzschild BH. Right: The entropy is shown as a function of Hawking temperature.
  • Figure 4: Same as Figure \ref{['type1-1']} for a type (ii) coupling function.
  • Figure 5: Same as Figure \ref{['type1-2']} for a type (ii) coupling function.
  • ...and 9 more figures