REVIEW
Exact black hole solutions in Palatini Kalb-Ramond gravity
General Relativity and Quantum Cosmology · Submitted 5 Oct 2026 · 74 references
Pith's verdict: Exact Palatini KR black holes are genuinely new and well argued, but the Mercury perihelion bound is 100x too tight: a per-orbit-to-per-century units error that, once fixed, likely demotes perihelion behind light deflection as the tightest test.
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Referee report
The paper
The paper claims
In the Palatini formulation of Kalb-Ramond gravity, exact static black holes exist with both Lorentz-violating couplings $\ell_1$ and $\ell_2$ nonzero, a configuration the purely metric formulation forbids.
The reading
What carries the argument
The load-bearing device is the auxiliary metric $h_{\mu\nu}$, defined through its densitized inverse $\sqrt{-h}\,h^{\mu\nu} = \sqrt{-g}\left[(1+u)g^{\mu\nu} + s^{\mu\nu}\right]$, which converts the connection field equation into a metric-compatibility condition. Because the independent connection is torsion-free, it is the Levi-Civita connection of $h$, and for the purely pseudo-electric VEV ansatz $b_{10}=-b_{01}=\tilde{E}(r)$ with the constant-norm condition $b_{\mu\nu}b^{\mu\nu}=-b^2$ enforced by the Lagrange-multiplier potential $V=\lambda X$, the radial profile $\tilde{E}(r)=|b|\sqrt{FG/2}$ is fixed and the metric (19) solves the complete field equations. The dimensionless parameter $\sigma^2=(1-\ell_1-\ell_2/2)/(1-\ell_1)$ then encodes all residual Lorentz violation in the geometry and in the exterior geodesic sector.
Extended reading notes
Core claim
The central claim is that the Palatini formulation of nonminimally coupled Kalb-Ramond gravity consistently admits static, spherically symmetric black holes with both Lorentz-violating parameters $\ell_1$ and $\ell_2$ nonzero, in contrast to the purely metric theory where Ref. [40] requires $\ell_1\ell_2=0$. For $\Lambda=0$ the auxiliary metric $h_{\mu\nu}$ is Ricci-flat, so the independent connection's Ricci tensor and scalar vanish and every term in the metric and Kalb-Ramond field equations vanishes separately, which is the mechanism that removes the mutual-exclusivity restriction. The resulting metric recovers Schwarzschild-(A)dS when $\ell_1=\ell_2=0$, and the Kretschmann scalar's explicit dependence on the Lorentz-violating parameters shows the deformation is a genuine geometric modification rather than a coordinate effect. After calibrating the residual radial deformation $\sigma^2=(1-\ell_1-\ell_2/2)/(1-\ell_1)$ against the Keplerian mass parameter $\bar{M}=M/\rho$, the Solar System observables depend on the parameters only through $q=\ell_2/(1-\ell_1)$, with Mercury's perihelion precession yielding $-7.4\times10^{-12} \leq q \leq 3.7\times10^{-11}$.
Load-bearing premise
The whole solution family rests on the purely pseudo-electric VEV ansatz $b_{10}=-b_{01}=\tilde{E}(r)$ as the only nonzero Kalb-Ramond background components, combined with the constant-norm condition imposed by the Lagrange-multiplier potential $V=\lambda X$; these choices are what make every curvature term vanish on shell for $\Lambda=0$, and a different VEV orientation or potential could alter or eliminate the solutions.
Editorial extensions
If this is right
- Both Lorentz-violating couplings can be nonzero, so the Palatini KR theory has a wider class of static black holes than the metric formulation, and observational tests should not assume mutual exclusivity of $\ell_1$ and $\ell_2$.
- For $\Lambda=0$ the auxiliary metric is exactly Schwarzschild after a rescaling, so the classical Solar System phenomenology of the exterior geodesic sector reduces to the single parameter $q=\ell_2/(1-\ell_1)$.
- The extended first law $\delta M = T\,\delta S + V\,\delta P$ and Smarr relation $M=2(TS-VP)$ hold despite the Lorentz violation, so the Iyer-Wald conserved mass acts as a thermodynamic enthalpy.
- Of the three Solar System tests, Mercury's perihelion precession gives the most restrictive constraint on $q$, with light deflection and the Shapiro time delay providing progressively weaker bounds.
- All three tests constrain only the combination $q$ and cannot by themselves separate $\ell_1$ from $\ell_2$.
Reading between the lines
- The same Ricci-flat auxiliary-metric mechanism may generalize to other nonminimally coupled antisymmetric or vector backgrounds in the Palatini formalism, where the metric and affine structures decouple curvature terms; the existing Palatini bumblebee solutions hint at this pattern.
- Because the geodesic sector isolates $q$, disentangling $\ell_1$ from $\ell_2$ would require probes beyond geodesics, such as quasinormal modes, shadows, or linear perturbations, as the authors themselves suggest.
- A direct test of the mechanism's robustness would be to substitute the same ansatz with a polynomial potential (e.g., $V=\lambda X + \mu X^2$) or a pseudo-magnetic VEV orientation and check whether any both-branches solution survives.
- If the Ricci-flat mechanism holds, the $\Lambda<0$ and $\Lambda>0$ branches should also admit both-parameters-nonzero solutions, and the thermodynamics established for $\Lambda\leq 0$ could be probed for stability against linear perturbations.
Original abstract
We investigate static and spherically symmetric black holes in Palatini Kalb-Ramond gravity, where a nonzero vacuum expectation value of the Kalb-Ramond field spontaneously breaks local Lorentz symmetry. Treating the metric and the torsion-free affine connection as independent variables, we derive the modified field equations and obtain an exact family of solutions encompassing positive, vanishing, and negative values of the cosmological constant $\Lambda$. Unlike the corresponding purely metric formulation, the Palatini theory consistently allows both Lorentz-violating parameters $\ell_1$ and $\ell_2$ to be nonzero. Using the Iyer-Wald formalism, we determine the conserved mass of the black hole solutions. For the $\Lambda\leq0$ sector, we further verify the first law and the Smarr relation, identifying the conserved mass as the thermodynamic enthalpy. We then confront the $\Lambda=0$ solution with Mercury's perihelion precession, solar light deflection, and the Shapiro time delay. Once the central mass parameter is calibrated by planetary dynamics, these observables constrain the Lorentz-violating parameters only through $q\equiv\ell_2/(1-\ell_1)$. Among these tests, Mercury's perihelion precession yields the most stringent constraint, $-7.4\times10^{-12} \leq q \leq 3.7\times10^{-11}$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We investigate static and spherically symmetric black holes in Palatini Kalb-Ramond gravity, where a nonzero vacuum expectation value of the Kalb-Ramond field spontaneously breaks local Lorentz symmetry. Treating the metric and the torsion-free affine connection as independent variables, we derive the modified field equations and obtain an exact family of solutions encompassing positive, vanishing, and negative values of the cosmological constant $\Lambda$. Unlike the corresponding purely metric formulation, the Palatini theory consistently allows both Lorentz-violating parameters $\ell_1$ and $\ell_2$ to be nonzero. Using the Iyer-Wald formalism, we determine the conserved mass of the black hole solutions. For the $\Lambda\leq0$ sector, we further verify the first law and the Smarr relation, identifying the conserved mass as the thermodynamic enthalpy. We then confront the $\Lambda=0$ solution with Mercury's perihelion precession, solar light deflection, and the Shapiro time delay. Once the central mass parameter is calibrated by planetary dynamics, these observables constrain the Lorentz-violating parameters only through $q\equiv\ell_2/(1-\ell_1)$. Among these tests, Mercury's perihelion precession yields the most stringent constraint, $-7.4\times10^{-12} \leq q \leq 3.7\times10^{-11}$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate static and spherically symmetric black holes in Palatini Kalb-Ramond gravity, where a nonzero vacuum expectation value of the Kalb-Ramond field spontaneously breaks local Lorentz symmetry. | contribution 0.82 | 0.50 | 0.08 | asserted 0.97 |
| s1 | Treating the metric and the torsion-free affine connection as independent variables, we derive the modified field equations and obtain an exact family of solutions encompassing positive, vanishing, and negative values of the cosmological constant $\Lambda$. | contribution 0.51 | 0.97 | 0.24 | asserted 1.00 |
| s2 | Unlike the corresponding purely metric formulation, the Palatini theory consistently allows both Lorentz-violating parameters $\ell_1$ and $\ell_2$ to be nonzero. | result 0.94 | 0.70 | 0.24 | asserted 1.00 |
| s3 | Using the Iyer-Wald formalism, we determine the conserved mass of the black hole solutions. | method 0.86 | 0.96 | 0.05 | asserted 1.00 |
| s4 | For the $\Lambda\leq0$ sector, we further verify the first law and the Smarr relation, identifying the conserved mass as the thermodynamic enthalpy. | result 0.70 | 0.95 | 0.12 | asserted 1.00 |
| s5 | We then confront the $\Lambda=0$ solution with Mercury's perihelion precession, solar light deflection, and the Shapiro time delay. | method 0.63 | 0.92 | 0.05 | asserted 0.98 |
| s6 | Once the central mass parameter is calibrated by planetary dynamics, these observables constrain the Lorentz-violating parameters only through $q\equiv\ell_2/(1-\ell_1)$. | result 0.85 | 0.88 | 0.15 | asserted 0.99 |
| s7 | Among these tests, Mercury's perihelion precession yields the most stringent constraint, $-7.4\times10^{-12} \leq q \leq 3.7\times10^{-11}$. | result 1.00 | 0.91 | 0.39 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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