Interior volume of Banados–Teitelboim–Zanelli black hole

Abstract

We calculate the interior volume for the three-dimensional Banados-Teitelboim-Zanelli black hole in this paper. To that end, we obtain the volume functional and numerically find a path to extremize it, before choosing a spacelike hypersurface to get an asymptotic analytical expression for the interior volume based on the characteristics of the path.

Effects of charge on the interior volume of BTZ black holes

“…The volume of this hypersurface defining the black hole volume increases monotonically with the advanced time v. As discussed by many [4,6,17,19,24], such a behaviour can be appropriately used in defining and understanding the black hole entropy. We will touch on this in latter sections.…”

Information paradox and corrected thermodynamics for black holes

“…The concept of simultaneity surface does not work in curved spacetime any more, so we are left with finding the extreme value of V Σ to obtain the interior volume of the black hole. Using Lagrangian method, it was found that the functional (13) can be extremal when the coordinate r of the two-dimensional surface S satisfies [8,13] dr(λ) dλ = 0.…”

“…Here in (23), we have considered the retarded time v the black hole mass m as variations. The retarded time v is in fact v(λ f ) which denotes the end of the collapse, contrasting with v(λ i ), which corresponds to the beginning of the collapse [8,13]. v(λ i ) is the time that the horizon forms (which has been set to be zero in above).…”

“…Along this way, the Christodoulou-Rovelli (CR) volume of a spherically symmetric black hole was defined [8][9][10][11][12][13]. The volume of the spinning Kerr black hole was also raised [14].…”

Information evolution in the interior of an axially symmetric BTZ black hole

“…In Ref. [14], M. Zhang followed the CR work and found the interior volume of BTZ black hole increases with advance time. This work showed that the above stated discussion could be extended to (2 + 1)− dimensional BTZ black holes.…”