Reconstructing $f(R)$ gravity from generalized entropies: exact Lagrangians

Technical Summary: Reconstructing $f(R)$ Gravity from Generalized Entropies

Problem Statement
The Bekenstein-Hawking entropy law ($S_{BH} = A/4G$) establishes a fundamental link between gravitation, thermodynamics, and quantum theory. However, various theoretical frameworks (including non-extensive statistics, relativistic generalizations, and quantum-gravitational effects) suggest that horizon entropy may deviate from the standard area law, leading to generalized entropy functionals (e.g., Tsallis, R'enyi, Kaniadakis, Barrow). A central inverse problem arises: if the entropy of a gravitational horizon is modified, what are the corresponding modifications to the underlying gravitational dynamics? Specifically, can one reconstruct the metric $f(R)$ gravitational Lagrangian from a prescribed generalized entropy-area relation?

Methodology
The authors employ the Wald entropy construction, which relates the entropy of a stationary horizon to the derivative of the gravitational Lagrangian with respect to the Ricci scalar ($R$). The master relation is given by:
$$\partial_R f = \frac{4G S(A)}{A}$$
where $S(A)$ is the prescribed generalized entropy.

A critical methodological challenge identified is the area-curvature ambiguity. The entropy is a function of the horizon area $A$, while the Lagrangian is a function of the curvature $R$. To reconstruct $f(R)$, a specific relation $A(R)$ must be supplied. The paper demonstrates that for Schwarzschild-de Sitter geometries, $A$ is not uniquely determined by $R$ alone due to the presence of the mass parameter $M$. Consequently, the reconstruction is intrinsically branch-dependent. The authors analyze two distinct physical branches:

1. The Massive Branch: Corresponds to a black hole of fixed mass $M$. The area-curvature relation is derived via an expansion around the Schwarzschild solution ($R=0$), yielding $A(R) \approx A_0(1 + cR)$. This approach retains dependence on $M$ and is accurate only to leading order in curvature.
2. The Maximally Symmetric Branch: Corresponds to the cosmological horizon with vanishing mass ($M=0$). Here, the relation is exact and mass-independent: $A = 48\pi/R$. This allows for a non-perturbative reconstruction without expanding in the entropy deformation parameter.

The authors derive a general mapping rule for the maximally symmetric branch: an entropy term proportional to $S_{BH}^q$ generates a curvature term proportional to $R^{2-q}$. They also derive branch-independent stability criteria relating the Dolgov-Kawasaki condition ($f_{RR} > 0$) and the scalaron mass ($m_{sc}^2$) directly to the monotonicity of the entropy functional $S(s)/s$ and its derivative $S'(s)$.

Key Contributions and Results

• Exact Lagrangians: Using the exact quadrature formula derived for the maximally symmetric branch, the authors obtain closed-form $f(R)$ Lagrangians for several generalized entropies without perturbative expansion:

  • Tsallis-Cirto & Barrow: Generate power-law corrections $R^{2-\delta}$ and $R^{1-\Delta/2}$, respectively.
  • R'enyi: Generates an elementary logarithmic structure involving $\ln(1 + \lambda\sigma/R)$.
  • Kaniadakis: Generates a hyperbolic sine integral structure, leading to a leading-order $1/R$ correction.
  • Logarithmic Entropy: Generates an $R^2 \ln R$ term.
  • Power-law Entanglement: Generates $R^{1-\beta}$ corrections.

• Exponent Reflection: On the maximally symmetric branch, the exponent mapping $q \to 2-q$ implies that entropy contributions with $q > 1$ (super-extensive) map to infrared curvature corrections (powers $< 1$). For instance, the Kaniadakis entropy ($q \approx 1 + \kappa^2 s^2/6$) produces the $1/R$ modification proposed by Carroll et al. as a geometric origin for late-time acceleration. Conversely, logarithmic entropy corrections map to ultraviolet $R^2 \ln R$ structures.

• Stability Analysis:

  • Maximally Symmetric Branch: Stability is determined by the sub-extensivity of the entropy ($d(S/s)/ds < 0$). The R'enyi reconstruction is found to be Dolgov-Kawasaki stable and tachyon-free. In contrast, Barrow and Kaniadakis reconstructions are found to be unstable (Dolgov-Kawasaki instability and tachyonic scalarons) on this branch.
  • Massive Branch: The stability condition is reversed ($d(S/s)/ds > 0$ required). Here, Barrow and Kaniadakis reconstructions are stable (or perturbatively stable), while the R'enyi reconstruction fails to satisfy both Dolgov-Kawasaki and non-tachyonic conditions simultaneously in any positive parameter window.

• Horizon Charges: The authors verify the consistency of the reconstructed theories by expressing them in first-order scalar-tensor form and calculating the conserved boost charge on weak isolated horizons. In all cases, the conserved charge reproduces the original generalized entropy, providing an independent quasi-local consistency check.

Significance and Claims
The paper establishes a direct, non-perturbative link between generalized horizon thermodynamics and modified gravitational dynamics. Its primary claim is that the correspondence between entropy and the gravitational Lagrangian is not unique but depends fundamentally on the choice of the vacuum branch (fixed-mass vs. maximally symmetric).

The authors emphasize that these two branches are not merely different approximations of the same theory but represent distinct reconstructions associated with different horizon families (black holes vs. cosmological horizons). Consequently, physical conclusions regarding stability and viability (e.g., whether a specific entropy model yields a viable $f(R)$ theory) are branch-dependent. For cosmological applications, the exact maximally symmetric reconstruction is presented as the natural choice, while the massive branch is appropriate for individual black hole thermodynamics. The work provides a rigorous framework to translate thermodynamic modifications directly into gravitational Lagrangians and their associated stability properties.