Logarithmic corrections to the entropy of near-extremal black holes in New Massive Gravity

Published 11 Jun 2026 in hep-th and gr-qc | (2606.13546v1)

Abstract: We study the one-loop correction to the entropy of near-extremal black holes in three-dimensional massive gravity at the special point where the theory exhibits a unique maximally symmetric vacuum and non-constant curvature hairy black holes can achieve extremality even in the static case. Focusing on the near-horizon AdS$_2\times S^1$ geometry, we evaluate the contribution of boundary graviton modes that become exact zero modes in the extremal limit. We show that the resulting one-loop partition function generates logarithmic corrections to the semiclassical entropy, providing a new extension to higher-curvature gravity of what has been recently obtained for near-extremal black holes in General Relativity.

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Summary

The paper demonstrates that boundary graviton modes yield a universal (3/2) log T quantum correction to the semiclassical entropy of near-extremal black holes in NMG.
It employs a one-loop functional integral method around the extremal geometry to isolate nonanalytic, temperature-dependent corrections arising from zero modes.
The analysis extends the Schwarzian effective description beyond General Relativity, opening new avenues for holography and higher-curvature gravity research.

Logarithmic Entropy Corrections for Near-Extremal Black Holes in 3D New Massive Gravity

Introduction and Motivation

This paper analyzes quantum corrections to the entropy of near-extremal black holes in three-dimensional New Massive Gravity (NMG) at its "special point," extending prior results from General Relativity (GR) and Einstein-Maxwell theory to higher-derivative models with local degrees of freedom. The focus is on the quantum entropy correction in the near-extremal, low-temperature regime, where classical thermodynamics is challenged by a vanishing energy gap and the emission of typical Hawking quanta becomes kinematically suppressed. This work elucidates the role of boundary graviton modes—emergent from large diffeomorphisms in the near-horizon AdS $_2 \times S^1$ geometry—and their contribution to quantum entropy via the 1-loop partition function.

Theoretical Framework: NMG and Hairy Black Holes

NMG supplements the Einstein-Hilbert action with a specific curvature-squared term, yielding fourth-order field equations while remaining ghost-free and parity-even. For generic couplings, it admits two maximally symmetric vacua; at the "special point", $\lambda = m^2$ , the vacua merge and the linearized gauge symmetry is enhanced. This enables the existence of static "hairy" black hole solutions characterized by a gravitational hair parameter $b$ , which distinguishes them from locally AdS $_3$ BTZ solutions and allows extremality in the static case.

Analytically, the metric for these solutions is

$ds^2 = f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\phi^2,\quad f(r) = \frac{r^2}{l^2} + b r + c,$

with $b < 0$ for the existence of inner and outer horizons. The extremal limit ( $r_+ = r_-$ ) yields vanishing entropy and mass, making these the minimal-energy states for fixed charge or hair. The entropy and mass scale linearly and quadratically with $T$ near extremality, and admit a Cardy formula interpretation on the dual CFT side.

One-Loop Functional Integral in Near-Extremal Geometry

The central computation focuses on the one-loop path integral for gravitational fluctuations, where the operator $\mathcal{O}$ acting on perturbations $h_{\mu\nu}$ is expanded around the extremal geometry:

$\lambda = m^2$ 0

The analysis identifies that only the modes with vanishing eigenvalue in the extremal limit ( $\lambda = m^2$ 1) yield nonanalytic, logarithmic corrections in temperature $\lambda = m^2$ 2. These are associated with boundary graviton (Schwarzian) modes, which are normalizable metric perturbations generated by large, non-normalizable diffeomorphisms acting on AdS $\lambda = m^2$ 3 and produce legitimate physical fluctuations in the near-horizon region.

Main Results: Entropy Corrections from Boundary Graviton Modes

The explicit evaluation of the 1-loop determinant, focusing on the boundary graviton tensor modes, leads to a quantum correction to the entropy:

$\lambda = m^2$ 4

where $\lambda = m^2$ 5 is the semiclassical (Wald) entropy. This logarithmic term arises due to the temperature-induced lifting of degeneracy for the boundary (Schwarzian) zero modes, and the prefactor $\lambda = m^2$ 6 is robust across the fluctuation spectrum for NMG in the considered regime. The gravitational hair parameter $\lambda = m^2$ 7 regulates the near-horizon $\lambda = m^2$ 8 size in extremality.

The analysis also considers vector (rotational) modes, which contribute only at subleading order, $\lambda = m^2$ 9. Bulk massive graviton excitations are exponentially suppressed by the graviton mass and do not contribute at leading order, which is consistent with the decoupling scenario validated for BTZ in pure GR.

Geometric Interpretation: Kerr-Schild Construction and Schwarzian Sector

A geometric realization of the relevant boundary modes is achieved via a Kerr-Schild construction based on null geodesics of the hairy black hole background. The boundary graviton perturbations that contribute to the entropy correction correspond to principal null directions, and their structure in the near-horizon extremal limit exactly matches the analytically derived Schwarzian zero modes. This geometric viewpoint enables a potential connection to the classical double copy program, suggesting that such boundary graviton modes may admit gauge-theoretic analogues.

Implications and Future Directions

The results demonstrate that the characteristic logarithmic correction to black hole entropy due to boundary graviton modes persists in a higher-derivative, unitary model with local propagating d.o.f.—not just in GR or 4D/5D Einstein-Maxwell or Einstein-Gauss-Bonnet gravity. This supports the universality of Schwarzian effective descriptions in low-temperature black hole thermodynamics.

Extensions naturally include:

Generalizing to rotating hairy black holes in NMG, where new branches of extremality and corresponding vector/rotational modes may provide further insight into the structure of logarithmic corrections.
Studying the status of quantum corrections in a broader class of higher-curvature (potentially all-unitary) models admitting similar black hole solutions.
Reformulating the effective near-horizon theory in terms of a deformed Schwarzian action, possibly via an auxiliary field formalism, opening new analytic angles for understanding quantum corrections and AdS $b$ 0/CFT $b$ 1 holography.
Exploring double copy frameworks for the full spectrum of near-horizon modes in higher derivative gravity.

Conclusion

The paper systematically evaluates log corrections to the entropy of near-extremal black holes in NMG, confirming that boundary graviton modes—emergent from the near-horizon AdS $b$ 2 geometry—produce a $b$ 3 contribution at 1-loop order. This establishes the persistence of Schwarzian-type universality in quantum black hole thermodynamics beyond GR, highlights the decoupling of bulk modes at low $b$ 4, and sets the stage for further inquiries into holography, higher curvature gravity, and quantum aspects of hairy black hole dynamics.