Black Hole Singularity, Generalized (Holographic) $c$-Theorem and Entanglement Negativity

In this paper we revisit the question that in what sense empty $AdS_{5}$ black brane geometry can be thought of as RG-flow. We do this by first constructing a holographic $c$-function using causal horizon in the black brane geometry. The UV value of the $c$-function is $a_{UV}$ and then it decreases monotonically to zero at the curvature singularity. Intuitively, the behavior of the $c$-function can be understood if we recognize that the dual CFT is in a thermal state and thermal states are effectively massive with a gap set by the temperature. In field theory, logarithmic entanglement negativity is an entanglement measure for mixed states. For example, in two dimensional CFTs on infinite line at finite temperature, the renormalized entanglement negativity of an interval has UV (Low- T) value $c_{UV}$ and IR (High-T) value zero. So this is a potential candidate for our $c$-function. In four dimensions we expect the same thing to hold on physical grounds. Now since the causal horizon goes behind the black brane horizon the holographic $c$-function is sensitive to the physics of the interior. Correspondingly the field theory $c$-function should also contain information about the interior. So our results suggest that high temperature (IR) expansion of the negativity (or any candidate $c$-function) may be a way to probe part of the physics near the singularity. Negativity at finite temperature depends on the full operator content of the theory and so perhaps this can be be done in specific cases only. The existence of this $c$-function in the bulk is an extreme example of the paradigm that space-time is built out of entanglement. In particular the fact that the $c$-function reaches zero at the curvature singularity correlates the two facts : loss of quantum entanglement in the IR field theory and the end of geometry in the bulk which in this case is the formation of curvature singularity.