Quantum charged black holes

The lack of logarithmic behavior arises from the fact that the bulk four-dimensional gauge field does not localize on the brane in the same way as gravity.

“This lack of logarithmic behavior arises from the fact the bulk four-dimensional gauge field does not localize on the brane in the same way as a gravity [99].”

The temporal component vanishes in the limit l → 0 while the azimuthal component is independent of backreaction; this is consistent with a three-dimensional dyonic defect in conical AdS3 or a BTZ black hole.

“Interestingly, the temporal component vanishes in the limit ℓ → 0 while the azimuthal component is independent of backreaction.This is consistent with three-dimensional dyonic defect in conical AdS 3 or a BTZ black hole [99].”

To determine the range, two conditions are imposed: (i) the bulk has a horizon r0 > 0, and (ii) x1 is finite, having a maximum xmax1, such that constant (t, r)-surfaces are compact.

“To determine this range, two conditions are imposed [99]: (i) the bulk has a horizon r 0 > 0, and (ii) x 1 is finite, having a maximum x max 1 , such that constant (t, r)-surfaces are compact.”

Upon imposing bulk regularity conditions, the induced geometry on the brane at x = 0 in canonically normalized coordinates is (4.85).

“Upon imposing bulk regularity conditions (equivalent to those for the neutral, static geometry), the induced geometry on the brane at x = 0, in terms of canonically normalized coordinates (t, r, ϕ) = (η t, η -1 r, η φ), is [99] (see also [100])”

Computing charge Q from the brane perspective would require a resummation of the infinite tower of higher-derivative terms in the induced theory; alternatively, the bulk theory performs this resummation, and the charge of the brane black hole is identified with the electric charge of the bulk black hole.

“As with mass or angular momentum, computing charge Q from the brane perspective would require a resummation of the infinite tower of higher-derivative terms appearing in the induced theory.Alternatively, the bulk theory performs this resummation, and the charge of the brane black hole is identified with the electric charge of the bulk black hole [99] (see also [102])”

For arbitrary κ, the extremal mass is μ_ext^(κ) = ... (4.96).

“Explicitly, for arbitrary κ, the extremal mass is [99]”

When the non-backreacted solution has q ≥ μ/2, backreaction results in a spacetime possessing a naked (timelike) singularity.

“Specifically, when the non-backreacted solution has q ≥ µ/2, then backreaction will result in a spacetime possessing a naked (timelike) singularity [99].”

The only essential difference between entropies of charged and neutral black holes is that, rather than reaching zero, the charged black hole entropy ends at a finite, non-zero value-the entropy of the extremal black hole.

“The only essential difference between entropies of the charged and neutral black holes is that, rather than reaching zero, the charged black hole entropy ends at a finite, non-zero value -the entropy of the extremal black hole [99].”