New Rotating Black Hole Solutions With Imperfect Fluid Energy-Momentum Tensor In $f(R)$ Gravity

Technical Summary: New Rotating Black Hole Solutions With Imperfect Fluid Energy-Momentum Tensor in f(R) Gravity

Problem Statement
The holographic correspondence between rotating black holes and dual Conformal Field Theories (CFTs) has been extensively studied within Einstein gravity, particularly for extremal and non-extremal black holes. While the Kerr/CFT correspondence and the concept of "hidden conformal symmetry" have been established for General Relativity (GR), the extension of these dualities to Modified Theories of Gravity (MTG), specifically $f(R)$ gravity, remains an open challenge. The field equations in $f(R)$ gravity are non-trivial functions of the Ricci scalar, which may hinder the existence of a dual CFT description. This paper addresses the question of whether rotating black hole solutions in $f(R)$ gravity, specifically those involving inverse Ricci scalar terms, admit a holographic dual description. Furthermore, it investigates whether two physically distinct types of rotating black holes—one with inherent rotation in the presence of matter and one generated via Lorentz boost in a vacuum—can be distinguished through their dual CFT properties.

Methodology
The authors employ two distinct methods to construct rotating black hole solutions within the framework of $f(R)$ gravity, defined by the action $I = \frac{1}{16\pi G} \int d^4x \sqrt{-g} (f(R) + L_{matter})$ with $f(R) = R - \frac{16\Lambda^2}{3R}$.

1. Type I Solution (Inherent Rotation with Matter):

   • The authors utilize a modified Newman-Janis algorithm (NJA) proposed by Azreg-Aïnou to generate a rotating line element from a static seed metric.
   • This solution inherently possesses a rotational parameter $a$ and exists in the background of a non-trivial energy-momentum tensor ($T_{\mu\nu}^{matter}$), which is identified as an imperfect fluid.
   • The authors numerically verify that this energy-momentum tensor satisfies the weak, strong, and dominant energy conditions.
   • To establish the holographic dual, a massless scalar probe field is introduced. The radial part of the Klein-Gordon equation is analyzed in the near-horizon region. By expanding the metric function near the outer horizon and comparing the resulting wave equation to the quadratic Casimir operator of an $SL(2, \mathbb{R})_L \times SL(2, \mathbb{R})_R$ algebra, the authors derive the hidden conformal symmetry.

2. Type II Solution (Boosted Rotation in Vacuum):

   • Starting from the static, spherically symmetric vacuum solution of the same $f(R)$ theory, the authors apply a Lorentz boost transformation mixing the time and angular coordinates ($\hat{\phi} = \Xi\phi + at$, $\hat{t} = \Xi t + a\phi$).
   • This generates a "boosted" rotating metric where the rotation is a coordinate artifact rather than an inherent property of the spacetime geometry; the energy-momentum tensor remains zero ($T_{\mu\nu}^{matter} = 0$).
   • Similar to Type I, the authors analyze the scalar probe field in this background. They expand the metric function near the horizon and compare the radial wave equation to the Casimir operator to identify the hidden conformal symmetry and extract CFT parameters.

Key Contributions and Results

• Existence of Exact Solutions: The paper presents two new classes of exact rotating black hole solutions in $f(R)$ gravity. Type I represents a black hole with inherent rotation supported by an imperfect fluid, while Type II represents a vacuum black hole where rotation is induced via a Lorentz boost.
• Verification of Energy Conditions: For the Type I solution, the authors demonstrate numerically that the effective energy-momentum tensor satisfies all standard energy conditions (weak, strong, and dominant) across a range of parameters (mass, cosmological constant, and rotation parameter).
• Establishment of Hidden Conformal Symmetry: For both solution types, the authors successfully demonstrate the existence of a local hidden conformal symmetry. The radial wave equations for the scalar probes in the near-horizon limit are shown to be eigenfunctions of the quadratic Casimir operator of an $SL(2, \mathbb{R})_L \times SL(2, \mathbb{R})_R$ algebra.
• Derivation of CFT Parameters: The authors explicitly calculate the left and right CFT temperatures ($T_L, T_R$), mode numbers ($n_L, n_R$), and central charges ($c_L, c_R$) for both black hole types. They verify that the microscopic Cardy entropy of the dual CFT matches the macroscopic Bekenstein-Hawking entropy of the black holes (calculated using the Wald entropy formula for $f(R)$ gravity).
• Distinction via Holography: A comparative analysis of Type I and Type II black holes with identical mass and horizon radii reveals a distinct difference in their dual CFT properties. The Type I black hole (inherent rotation with matter) exhibits higher CFT temperatures and higher CFT mode numbers compared to the Type II black hole (boosted rotation in vacuum).

Significance and Claims
The paper claims to extend the black hole holography paradigm from General Relativity to the realm of $f(R)$ gravity, demonstrating that the duality holds even when the field equations are non-trivial and when matter fields (imperfect fluids) are present.

The primary significance of the work lies in the proposed conjecture regarding the distinguishability of rotating black holes. The authors suggest that while two rotating black holes may appear identical to a distant observer (sharing the same mass and horizon geometry), their nature can be distinguished by examining their dual CFT properties. Specifically:

"Two identical rotating black holes (the same mass and the horizon) in f(R) theory could be distinguished by looking at their dual CFT temperatures, and the dual CFT mode numbers. The rotating black hole in the presence of matter has higher CFT temperatures and higher CFT mode numbers, compared to the vacuum boosted rotating black hole."

This conjecture implies that the "hidden" nature of the rotation (inherent vs. coordinate-induced) and the presence of matter are encoded in the thermodynamic and spectral properties of the dual CFT. The authors frame this as a novel method to probe the internal structure and matter content of black holes in modified gravity theories through holographic duality.

The paper concludes by noting that the $SL(2, \mathbb{R})_L \times SL(2, \mathbb{R})_R$ symmetry is spontaneously broken to $U(1)_L \times U(1)_R$ due to the angular identification $\phi \sim \phi + 2\pi$, resulting in a dual description involving finite temperatures. Future work is suggested to explore charged rotating black holes and the potential existence of super-conformal field theories in this context.