Hairy Black Holes in Nonlocal Quadratic Gravity

• The paper demonstrates that nonlocal quadratic gravity produces hairy black holes via Yukawa-screened deformations that shift the event horizon and alter thermodynamic properties.
• It employs a perturbative expansion in α to solve modified Einstein equations, revealing corrections in gravitational potential and ensuring ghost freedom through positive graviton residues.
• The study highlights enhanced thermodynamic stability and smooth free energy profiles, linking quantum-inspired nonlocal effects with classical black hole physics.

Hairy black holes in nonlocal quadratic gravity are solutions to a class of quantum-inspired modified gravity theories characterized by the presence of nonlocal, quadratic curvature terms in the gravitational action. These theories support static, charged black hole configurations with “hair”—in this context, Yukawa-screened deformations to the standard mass profile—arising from the interplay between nonlocal interactions and the Maxwell field. Nonlocal effects induce significant modifications to the horizon structure, thermodynamics, stability, and spectrum of gravitational excitations, with direct implications for the physical viability and consistency of the theory at both the classical and semiclassical levels (Rocha, 29 Jan 2026).

1. Action, Field Equations, and Metric Ansatz

The foundational action for nonlocal quadratic gravity, minimally coupled to an electromagnetic field, is

$$S = \int d^4x\,\sqrt{-g}\,\left[ \tfrac12\,R + \alpha\Bigl(R_{\mu\nu}\,\mathcal{E}\,R^{\mu\nu} - \tfrac14\,R\,\mathcal{E}\,R\Bigr) - \tfrac14\,F_{\mu\nu}F^{\mu\nu} \right]$$

where $\alpha \ll 1$ is a dimensionless parameter controlling the strength of nonlocal corrections, $\mu$ is a mass scale governing the range of nonlocality, and $\mathcal{E} \equiv (\Box - \mu^2)^{-1}$ encodes nonlocality via a Yukawa operator.

Variation with respect to $g^{\mu\nu}$ yields modified Einstein equations: $$G_{\mu\nu} + \alpha\,H_{\mu\nu}^{\rm NL} = T^{(EM)}_{\mu\nu}$$ where $G_{\mu\nu}$ is the Einstein tensor, $T^{(EM)}_{\mu\nu}$ is the electromagnetic energy-momentum tensor, and $H_{\mu\nu}^{\rm NL}$ records the nonlocal quadratic corrections. The explicit form of $H_{\mu\nu}^{\rm NL}$ entails covariant derivatives acting on nonlocally-smeared Ricci tensors and scalars.

Imposing static spherical symmetry, the ansatz is

$\alpha \ll 1$0

with the metric function expanded perturbatively: $\alpha \ll 1$1 The nonlocal field equations are then solved order by order in $\alpha \ll 1$2 using the Green’s function $\alpha \ll 1$3 associated with the Yukawa kernel.

2. Nonlocal Corrections: Yukawa Screening and Horizon Shift

At leading nontrivial order in $\alpha \ll 1$4, the black hole metric receives short-range corrections from nonlocality: $\alpha \ll 1$5 The primary effect,

$\alpha \ll 1$6

reflects Yukawa-type screening of the central mass, exponentiating away at large $\alpha \ll 1$7.

The event horizon radius $\alpha \ll 1$8 is shifted inward compared to the Reissner-Nordström case: $\alpha \ll 1$9 with $\mu$0 the exponential integral. The horizon shift $\mu$1 is governed by the sign and scale of the nonlocal correction.

3. Black Hole Thermodynamics: Temperature, Entropy, and Potentials

The Hawking temperature is found by

$\mu$2 and acquires nonlocal corrections: $\mu$3

The Bekenstein–Hawking entropy, $\mu$4, is similarly depressed: $\mu$5 The chemical potential (electrostatic potential at the horizon), extracted from the first law $\mu$6, is renormalized by nonlocality: $\mu$7

4. Thermodynamic Stability and Phase Behavior

Evaluation of thermodynamic response functions reveals enhanced stability properties due to nonlocal corrections. The specific heat at constant volume,

$\mu$8

is rendered less negative by the positive $\mu$9-corrections, indicating improved (but not fully positive) thermodynamic stability for small black holes.

The Helmholtz and Gibbs free energies are

$\mathcal{E} \equiv (\Box - \mu^2)^{-1}$0

with explicit expressions remaining analytic for all (physical) values of $\mathcal{E} \equiv (\Box - \mu^2)^{-1}$1, $\mathcal{E} \equiv (\Box - \mu^2)^{-1}$2, and $\mathcal{E} \equiv (\Box - \mu^2)^{-1}$3. The absence of cusps or discontinuities in $\mathcal{E} \equiv (\Box - \mu^2)^{-1}$4 and $\mathcal{E} \equiv (\Box - \mu^2)^{-1}$5 demonstrates that nonlocal quadratic gravity lacks first-order phase transitions in the black hole sector for these solutions.

5. Linearized Spectrum: Propagator Structure and Ghost Freedom

The quadratic expansion of the action about Minkowski space ($\mathcal{E} \equiv (\Box - \mu^2)^{-1}$6) leads to the following form for the graviton propagator in the spin-2 sector (momentum space, $\mathcal{E} \equiv (\Box - \mu^2)^{-1}$7): $\mathcal{E} \equiv (\Box - \mu^2)^{-1}$8 There are two physical poles in the spin-2 channel: a massless graviton at $\mathcal{E} \equiv (\Box - \mu^2)^{-1}$9 and a massive spin-2 resonance at $g^{\mu\nu}$0. The residue at the massive pole,

$g^{\mu\nu}$1

is positive, matching the healthy kinetic normalization of the graviton mode and excluding the presence of ghostlike instabilities. Both classical and quadratic-level perturbation theory are, therefore, free of nonunitary excitations within the effective field theory description.

6. Physical Implications and Outlook

Nonlocal quadratic gravity supports black hole solutions with nontrivial “hair” in the form of Yukawa-screened corrections to the Reissner–Nordström profile. Nonlocality pulls the horizon inward, raises the Hawking temperature, and reduces the entropy, indicating the presence of short-range gravitational modifications. Specific heat corrections lead to a parametric improvement in the thermodynamic stability regime for small black holes. The absence of phase transitions in the analytic form of free energies suggests a smooth thermodynamic landscape across the relevant parameter space. The spectrum of gravitational excitations is free of ghosts at the classical and quadratic quantum levels, with both spin-2 poles exhibiting positive norm and residue. The structure of these solutions demonstrates that quantum-inspired nonlocality can regularize certain pathologies of quadratic gravity without sacrificing consistency, pointing to a fertile connection between infrared-modified gravity and black hole microphysics (Rocha, 29 Jan 2026).