Phys.Rev.D, 113, 12, 124065 (2026)

Kasner singularity of black holes in Einstein-scalar gravity

Ze-Xuan Xiong¹ and Hong Lu^2,1,*

1 The International Joint Institute of Tianjin University, Fuzhou, Tianjin University, Tianjin 300350, China.

2 Center for Joint Quantum Studies, Department of Physics, School of Science, Tianjin University, Tianjin 300350, China.

* xzx_167@tju.edu.cn, mrhonglu@gmail.com

Abstract

We study the spacelike Kasner singularity of spherically-symmetric, static and asymptoti cally 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.