Researchers Derive Three-Dimensional BTZ Black Hole Horizons Via Novel Action Integrals

Scientists are tackling the complex problem of constructing and studying black holes that accurately incorporate the effects of matter. Yuanfan Cao and Andrew Svesko, both from the Department of Mathematics at King’s College London, present a novel approach to deriving the horizon of three-dimensional charged and rotating black holes using tree-level gravitational partition functions. Their research focuses on holographic BTZ black holes, which are dual to accelerating black holes in anti-de Sitter space, and crucially provides a first-principles derivation of the generalized entropy for these systems. By regulating the bulk Euclidean geometry with Karch-Randall end-of-the-world branes, they compute the on-shell action and demonstrate that thermal entropy aligns with generalized entropy, offering significant insight into black hole thermodynamics and gravity itself.

Scientists are tackling the complex problem of constructing and studying black holes that accurately incorporate the effects of matter.

BTZ Black hole thermodynamics from Holography

This breakthrough circumvents the difficulties traditionally encountered when studying black hole horizons by leveraging braneworld holography and a carefully constructed geometric setup. This approach provides a means to consistently realize the semi-classical approximation, a regime where quantum matter fields influence the classical geometry, and offers a pathway to explore black hole dynamics beyond the Planck scale where current approximations typically fail. By strategically positioning ETW branes on umbilic surfaces, codimension-1 hypersurfaces with specific curvature properties, the scientists automatically satisfy Israel junction conditions, ensuring a well-posed variational problem and simplifying the analysis. Experiments show that this braneworld construction allows for the exact construction of quantum black holes in (2+1)-dimensions, a feat previously hindered by the difficulty of solving the coupled system of geometry and renormalized quantum correlators. The study leverages the AdS/CFT correspondence, where the bulk theory is governed by Einstein-Hilbert gravity in the large central charge limit, and the brane gravity theory represents an asymptotic expansion valid when metric fluctuations are suppressed relative to matter loops. This work opens avenues for exploring quantum gravity effects in black hole systems and provides a foundation for future investigations into the fundamental nature of spacetime and quantum information.

BTZ Horizon Derivation via Holographic Partition Function yields

The study pioneered a method extending Kudoh and Kurita’s work on braneworld black hole thermodynamics, specifically deriving the thermodynamics of “quantum BTZ” black holes related to classical Bañados-Teitelboim-Zanelli (BTZ) black holes modified by perturbative backreaction from a conformally coupled scalar field. A key innovation involved placing an ETW brane at the second umbilic surface, functioning as an infrared regulator in the Euclideanized spacetime and ensuring a finite on-shell action without ambiguous background subtraction or local counterterms. Furthermore, the four-dimensional classical Bekenstein-Hawking entropy matched the three-dimensional generalized entropy, both derived via the bulk canonical partition function. This research harnessed a semi-classical saddle-point approximation to derive the generalized entropy from an on-shell Euclidean action, expressed as (β∂β −1)Ion-shell E (β) = S(3) gen = S(3) BH + νS(3) CFT + ν2S(3) Iyer-Wald +., where β represents the inverse temperature.

The team engineered a “double holography” framework, establishing equivalence between the classical bulk action and the on-shell Euclidean action of the induced brane theory, encompassing contributions from gravitational and CFT matter sectors. This equality holds for any backreaction strength, even non-perturbatively, while the perturbative expansion is controlled by the dimensionless parameter ν ≡G3ħc/l3 < 1, defining the AdS3 length scale. The generalized entropy decomposes into Bekenstein-Hawking entropy, CFT fine-grained entropy, and Iyer-Wald entropy from higher-derivative corrections, with ellipsis indicating contributions mixing gravitational and matter corrections at higher orders in ν.

BTZ Horizon Derived via Holographic Backreaction

Experiments revealed that the induced geometry on the brane at x = 0 is ds2|x=0 = −H(r)dt2 + H−1(r)dr2 + r2 dφ −a r dt 2, where H(r) is given by equation (2.6). The gauge field at x = 0 is Aμdxμ|x=0 = −2 l⋆ lq r dt, though this does not imply the Maxwell gauge field localizes on the brane in the same manner as gravity. To achieve canonically normalized coordinates (t, r, φ), the following transformations were applied: t = η( t − al3 φ), r = s r2 −r2s (1 − a2)η2, and φ = η φ − a t l3, where η ≡∆φ 2π, a ≡ax2 1 l3, and rs ≡l3 aη x1 q 2 −κx2 1. Measurements confirm that the resulting canonically normalized metric on the brane is ds2|x=0, a complex expression detailed in equation (2.19), with points identified along orbits of the Killing vector ∂ φ.

The gauge potential in these coordinates, as shown in equation (2.20), reduces to Abd xb|x=0 = −2qlη l⋆r( r)d t + 2 aqll3η l⋆r( r) d φ on the brane at x = 0. The ETW brane at x = 0 intersects the bulk black hole horizon, preserving the inner and outer horizons of the bulk black hole, with roots r± of H(r) corresponding to null generators of the (bulk) Killing horizon. Tests prove that the surface gravity at the horizon, κ±, is given by η(1 − a2) 1 + a2x2 1 r2 ± H′(r±) 2, consistent with equation (2.40) of the referenced work. In the tensionless limit, where l→∞, the charged and rotating AdS4 C-metric reduces to the Kerr-Newman-AdS4 black hole after appropriate coordinate and parameter rescalings. Analysis of the induced brane couplings reveals that G3 = G4 2l4, 1 L2 3 = 2 l2 4 (1 −2πG4l4τ), and l2 ⋆= 5 4l2 ⋆, with the central charge of the CFT3 being c3 = l2 4 G4. Expanding for small (l/l3) shows that 2c3G3 = L4 and G3c3 ∼l, demonstrating that in the limit of an effectively massless 3D graviton, the semi-classical theory is an asymptotic expansion.

BTZ Horizon Derivation via Holographic Entropy formulas reveal

The authors acknowledge a limitation in that their calculations are performed within a specific regime of acceleration, and the acceleration horizon imprints on the brane when the length scale is outside the slow acceleration regime. Future research could explore the behaviour of these black holes in different acceleration regimes or investigate modifications to the model to account for more complex gravitational effects.