Exact black hole solutions in Palatini Kalb-Ramond gravity

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 $\Lambda \le 0$ 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, $-7.4\times 10^{-12}\le q\le 3.7\times 10^{-11}$.

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