Symmetric black-to-white hole solutions with a cosmological constant

Abstract

For a system with a Hamiltonian constraint, we demonstrate that its dynamics is invariant under different choices of the lapse function, regardless of whether the Hamiltonian incorporates quantum corrections. Applying this observation to the interior of black-to-white holes, we analyze its dynamics with different choices of the lapse function. The results explicitly show that the leading-order expansion of both metrics proposed by Rovelli et al. (Class. Quant. Grav. 35, 225003 (2018); Class. Quant. Grav. 35, 215010 (2018)) and Ashtekar et al. (Phys. Rev. Lett. 121, 241301 (2018); Phys. Rev. D 98, 126003 (2018)) exhibit identical behavior near the transition surface. Therefore, in this sense the black-to-white hole model proposed by Rovelli et al., (Class. Quant. Grav. 35, 225003 (2018); Class. Quant. Grav. 35, 215010 (2018)) may be interpreted as a coarse-grained version of the solution within the framework of loop quantum gravity. The black-to-white hole solutions with exact symmetry between the black hole and white hole regions are constructed by appropriately fixing the quantum parameters in the effective theory of loop quantum gravity. This approach circumvents the issue of amplification of mass, which could arise from a mass difference between the black hole and white hole, and provides a way to link the solutions obtained by minisuperspace quantization to those in the covariant approach. Finally, the black-to-white hole solutions with a cosmological constant are constructed. The numerical solutions for the interior of the black-to-white hole with a cosmological constant are obtained, and their symmetric behavior is also discussed.