Exact Taub-NUT-like Black Holes in Einstein-bumblebee gravity: their thermodynamics and thermodynamic topology

TL;DR

The paper constructs an exact three-parameter, non-rotating Taub-NUT-like black hole in Einstein-bumblebee gravity, where Lorentz-symmetry breaking is encoded by a bumblebee field with vacuum expectation value. The authors derive the metric and bumblebee configuration, show how the Lorentz-violating parameter $\ell$ modifies the mass and temperature while preserving the horizon structure, and establish a Smarr-type relation with a Misner charge $N$ and its conjugate potential $\psi$. They compute the thermodynamic quantities: $M = m/\sqrt{1+\ell}$, $T = 1/(4\pi \sqrt{1+\ell} \ r_+)$, $S = \pi (r_+^2 + n^2)$, $N = -4\pi n^3/r_+$, and $\psi = 1/(8\pi n \sqrt{1+\ell})$, with $M(S,N) = \frac{\sqrt{S - \pi n^2}}{2\sqrt{\pi}\sqrt{1+\ell}} + \frac{N}{8\pi n \sqrt{1+\ell}}$. Using a Poincaré-Hopf-based thermodynamic topology analysis on a cylinder, they show the topological class is unchanged from the Lorentz-invariant Taub-NUT BH (Class I). The work thus demonstrates the robustness of thermodynamic topology under Lorentz-violating gravity and connects Schwarzschild-like, Taub-NUT, and their Lorentz-violating deformations.

Abstract

We re-derive an exact analytic three-parameter expressions for the non-rotating metric, describing a Taub-NUT-like black hole (BH), and its associated bumblebee field that are solutions to the Einstein-bumblebee gravity. We construct a consistence thermodynamics for the Taub-NUT-like BH and determine its thermodynamic topological class. The Lorentz symmetry breaking affects the mass and temperature of the BH but does not affect its thermodynamic topological classification.

Figures (2)

• Figure 1: Vector field plots of the unit vector field $\phi/||\phi||$ for the Taub-NUT-like solution \ref{['sol']} taking $\tau=6\pi r_0\sqrt{1+\ell}$, where $r_0$ represents an arbitrary length scale defined by the size of a cavity surrounding the black hole. The blue dots represent CPs located at $(r/r_0,\Theta)=(3/2,\pi/4),\,(3/2,3\pi/4)$. For the upper CP (corresponding to $\Theta =3\pi /4$), as one moves counterclockwise once around the red closed curve, the vector field rotates clockwise $2\pi$ radians, so that the index or the winding number is $-1$.
• Figure 2: $F(r_h)$ versus $r_h$ where $F(r_h)$ has three extreme values. The figure depicts the case where the two arrows, as those shown in \ref{['class1']}, are both downward. As one moves on the line $\Theta =3\pi /4$ from $r=r_m$ to $\infty$, the first extreme value of $F(r_h)$ that one meets is a minimum (occurring at $r_h=0.4 r_0$), so the first arrow is downward. Now, as one moves on the line $\Theta =3\pi /4$ from $\infty$ to $r=r_m$, the first extreme value of $F(r_h)$ that one meets is again a minimum (occurring at $r_h=2.53 r_0$), so the second arrow is also downward. We do not care about the nature of intermediate extreme value, as the maximum value occurring at $r_h=1.38 r_0$.