Quantum BTZ black hole

The non-holographic stress-tensor may be expressed as an infinite sum over images (4.68), with coefficients complicated functions of M and J; an analogous structure holds for the remaining components.

“The non-holographic stress-tensor may be expressed as an infinite sum over images, e.g.,”

The A→∞ limit of the rotating C-metric with k=+1 is a double-Wick rotation of Kerr-AdS4 metric.

“Notably, the A → ∞ limit of the rotating C-metric (see below), with k = +1, is a double-Wick rotation of Kerr-AdS 4 metric considered in [9] (see Appendix B of [25]).”

G3 in mass (4.29) is interpreted as an all-order resummation of the higher-derivative corrections to the mass, whilst G3 is the 'bare' Newton constant, since it is the physical constant G3 M which is being corrected.

“As such, G 3 in mass (4.29) is interpreted as an all-order resummation of the higher-derivative corrections to the mass [25], whilst G 3 is the 'bare' Newton constant, since it is the physical constant G 3 M which is being corrected.”

Setting μ = 0 and performing the coordinate transformation (4.4) brings the C-metric line element to empty AdS4, foliated by AdS3 slices with radius L4 cosh(σ) at constant σ.

“This brings the C-metric line element (4.1) to the form”

The qBTZ solution is characterized by the two branches (4.38), however, a finer subdivision of the two branches is Branch 1a: κ = +1, 0 < x1 < 1, Branch 1b: κ = -1, 0 < x1 < √3, Branch 2: κ = -1, √3 < x1 < ∞.

“Succinctly, the qBTZ solution is characterized by the two branches (4.38), however, a finer subdivision of the two branches is [25] Branch 1a:”

The non-zero stress-energy of the M > 0 states among the branch 2 black holes is due to Hawking radiation in thermal equilibrium with the finite temperature black hole, with the dominant Casimir energy subtracted from the quantum state appearing in ⟨Tij⟩.

“This suggests the non-zero stress-energy of the M > 0 states among the branch 2 black holes is due to Hawking radiation in thermal equilibrium with the finite temperature black hole (where the dominant Casimir energy has been subtracted from the quantum state appearing in ⟨T ij ⟩ [25]).”

For l ≠ 0, the non-vanishing components of the renormalized quantum-stress tensor (4.31) are (4.64).

“For ℓ ̸ = 0, the non-vanishing components of the renormalized quantum-stress tensor (4.31) are found to be [25]”

Pictorially, the branches of quantum black holes have a similar representation as Figure 6 (see Figure 6 of [25]).

“Pictorially, the branches of quantum black holes have a similar representation as Figure 6 (see Figure 6 of [25]).”

With the coordinate change (4.48), the brane metric does not quite have the canonical asymptotic form of a rotating AdS black hole; introducing a normalized radial coordinate r (4.51) achieves it.

“With the coordinate change (4.48), the brane metric does not quite have the canonical asymptotic form of a rotating AdS black hole.To do so, one introduces a normalized radial coordinate r such that [25].”

In the case of the semi-classical induced theory on the brane (4.19), the Iyer-Wald entropy is (5.27).

“In the case of the semi-classical induced theory on the brane (4.19), the Iyer-Wald entropy is [25] 35”