Kasner Singularity of Black Holes in Einstein-scalar Gravity

Abstract

We study the spacelike Kasner singularity of spherically-symmetric, static and asymptotically flat black holes in Einstein gravity minimally coupled to a massless scalar with a suitable self-interacting scalar potential. We focus on how the asymptotic information such as the mass and scalar charge affect the properties of the Kasner singularity, including the Kasner exponents. We show how a nontrivial integration constant can be extracted from the near-singularity geometry and find a general pattern that this integration constant asymptotes to a linear combination of the mass and scalar charge at large mass limit. We also find that there may be a black hole upper bound on the maximum surviving time of a massive particle inside such a black hole before it falls into the Kasner singularity, and the Schwarzschild black hole saturate this bound.

Figures (12)

• Figure 1: In all three graphs, we have chosen $r_0=1$. The top panel describes a black hole with $\phi_0= 0.44802$. The left and right panels describe solutions that do not have asymptotic Minkowsk region. The left panel has $\phi_0=0.50000$ and the function $f$ diverges positively at some finite $r$, whilst the right panel has $\phi_0=0.44686$ and the function $f$ diverges negatively at some finite $r$.
• Figure 2: We present how the horizon scalar hair $\phi_0$, the mass $M$ and scalar charge $\Sigma$ depend on the horizon radius $r_0$. The solid line in the left panel describes the Schwarzschild black hole, which indicates that scalar hair's contribution to the spacetime geometry in the exterior region is small.
• Figure 3: The dots are the numerical data of the mass/entropy relation. The red and blue solid lines are the mass/entropy approximate formulae \ref{['massentropy']} for large and small black holes respectively. They fit the data accurately in their respective parameter regions.
• Figure 4: The left and right panels give the temperature as a function of entropy for both small and larger black holes respectively. The solid lines reflect the temperature based on the first law \ref{['Ttheory']}, while the dots are black hole temperatures \ref{['Tdata']} read off directly from the black hole numerical solutions.
• Figure 5: The Kasner exponents of Branch-1 solutions are functions of the black hole radius. They satisfy the two identities \ref{['Psconstraints']} with extremely high accuracy ($\le 10^{-7}$.) Furthermore, $P_T$ is positive and bounded above as in \ref{['PTbound']}.