[Paper Review] AdS/BCFT correspondence and BTZ black hole within electric field

Fabiano F. Santos|arXiv (Cornell University)|Jun 19, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates the AdS/BCFT correspondence in 3D gravity coupled to a Maxwell field, focusing on a planar BTZ black hole with an external electric field. Using holographic renormalization, it derives the boundary stress-energy tensor and shows that the electric field breaks conformal symmetry, leading to position-dependent fluid-like properties with modified energy density and pressure, while the boundary entropy remains coordinate-independent and proportional to cos(θ).

ABSTRACT

This paper, presents the gravity duals of Conformal Field Theories with boundaries. This theory is known as AdS/BCFT correspondence. In this duality, our system has 3D gravity coupling with the Maxwell field dual to 2D BCFT. On the gravity side, we consider a 3D BTZ black hole. We analyze the effects of the chemical potential on the profile of the extra boundary for the black hole. Performing a holographic renormalization, we calculate the free energy and obtain the total entropy and corresponding area with chemical potential, and the boundary entropy for the black hole. These theories are specified by stress-energy tensors that reside on the extensions of the boundary to the bulk. In this way, the AdS/BCFT appears analogous to the fluid/gravity correspondence with the chemical potential. We discuss the solutions as well as their thermodynamic and fluid properties.

Motivation & Objective

• To extend the AdS/BCFT correspondence to include a 3D BTZ black hole coupled to a Maxwell field with an external electric field.
• To investigate the thermodynamic and fluid-like properties of the boundary conformal field theory (BCFT) in the presence of an electric field.
• To compute the free energy, total entropy, and boundary entropy via holographic renormalization in the AdS/BCFT framework.
• To analyze how the electric field modifies the stress-energy tensor and equation of state on the boundary hypersurface Q.
• To determine the conditions under which the fluid on Q exhibits conformal or non-conformal behavior in low and high-temperature regimes.

Proposed method

• Construct a 3D AdS-BTZ black hole solution coupled to a Maxwell field with Neumann boundary conditions.
• Use the holographic renormalization procedure to compute the Euclidean on-shell action, identifying it as the free energy.
• Derive the boundary stress-energy tensor T_ab on the hypersurface Q extending from the boundary into the bulk, using the induced metric and extrinsic curvature.
• Solve the equations of motion for the profile y(z) of the hypersurface Q, imposing symmetry conditions to achieve p_yy = p_zz.
• Analyze the resulting energy density ρ and pressure p as functions of z and the electric field parameter, using the equation of state Ω = p/ρ.
• Evaluate the entropy density s_Q = (2πL/κ)cos(θ), showing it is independent of spatial coordinates despite inhomogeneous fluid behavior.

Experimental results

Research questions

• RQ1 How does the presence of an external electric field modify the thermodynamic properties of the BTZ black hole in the AdS/BCFT correspondence?
• RQ2 What is the role of the hypersurface Q in encoding the boundary stress-energy tensor and how does it relate to fluid-like behavior in the dual BCFT?
• RQ3 Under what conditions does the fluid on Q remain conformal, and how does the electric field break conformal symmetry?
• RQ4 How does the electric field affect the energy density and pressure profiles on the boundary, and what is the resulting equation of state?
• RQ5 What is the behavior of the boundary entropy in the low- and high-temperature limits, and how does it relate to the chemical potential and electric field?

Key findings

• The boundary entropy s_Q = (2πL/κ)cos(θ) is coordinate-independent, despite the fluid on Q being inhomogeneous due to the electric field.
• In the low-temperature limit (z_h → ∞), the fluid becomes non-conformal with p ≈ −ρ, indicating a breakdown of conformal symmetry due to the electric field.
• In the high-temperature limit (z_h → 0), the fluid exhibits conformal behavior with Ω → −2μ²z_h², and the temperature T_BCFT = 1/(2πz_h) is consistent with standard black hole thermodynamics.
• The energy density ρ and pressure p on Q are modified by the electric field: ρ decreases and p increases, with explicit expressions derived as ρ = (2L cos(θ)/(kz))(1 − √f/2) and p = (L cos(θ)/(2kz√f))(4f − zf′ − 4√f).
• The equation of state Ω = p/ρ = (4f − zf′ − 4√f)/(√f(2 − √f)) captures the effective fluid behavior on Q, showing anisotropy unless f y′² = cot²(θ) is imposed.
• The solution for the Q-profile is numerically obtained and found to be open toward the horizon, satisfying the null (weak) energy condition ensuring non-negative temperature and energy density on Q.