Abstract
We discuss vacuum static, spherically symmetric asymptotically flat solutions of the generalized hybrid metric-Palatini theory of gravity (generalized HMPG) suggested by Böhmer and Tamanini, involving both a metric \(g_{\mu\nu}\) and an independent connection \(\hat{\Gamma}_{\mu\nu}^{\alpha}\); the gravitational field Lagrangian is an arbitrary function \(f(R,P)\) of two Ricci scalars, \(R\) obtained from \(g_{\mu\nu}\) and \(P\) obtained from \({\hat{\Gamma}}_{\mu\nu}^{\alpha}\). The theory admits a scalar-tensor representation with two scalars \(\phi\) and \(\xi\) and a potential \(V(\phi,\xi)\) whose form depends on \(f(R,P)\). Solutions are obtained in the Einstein frame and transferred back to the original Jordan frame for a proper interpretation. In the completely studied case \(V\equiv 0\), generic solutions contain naked singularities or describe traversable wormholes, and only some special cases represent black holes with extremal horizons. For \(V(\phi,\xi)\neq 0\), some examples of analytical solutions are obtained and shown to possess naked singularities. Even in the cases where the Einstein-frame metric \(g^{E}_{\mu\nu}\) is found analytically, the scalar field equations need a numerical study, and if \(g^{E}_{\mu\nu}\) contains a horizon, in the Jordan frame it turns to a singularity due to the corresponding conformal factor.
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Notes
We safely omit the factor \(1/(2\varkappa^{2})\) at the gravitational part of the action since only vacuum configurations, where \(S_{m}=0\), will be considered.
Unlike [4, 8, 17] etc., we are using the metric signature \((+--\,-)\), hence the plus sign before \((\partial\phi)^{2}=g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\) corresponds to a canonical field and a minus to a phantom field. The Ricci tensor is defined as \(R_{\mu\nu}=\partial_{\nu}\Gamma^{\alpha}_{\mu\alpha}-\ldots\), so that, for example, the scalar curvature is positive in de Sitter space-time. We also use the units in which \(c=G=1\), \(c\) being the speed of light and \(G\) the Newtonian gravitational constant.
If we write the general static, spherically symmetric metric in the form (17) with an arbitrary radial coordinate \(u\), this metric is asymptotically flat at some \(u=u_{0}\) if [29]
$$\textrm{e}^{\beta(u)}\equiv r(u)\mathop{\to}\limits_{u\to u_{0}}\infty,\quad|\gamma(u_{0})|<\infty,\quad\textrm{e}^{\beta-\alpha}|\beta^{\prime}|\mathop{\to}\limits_{u\to u_{0}}1.$$Comparing (17) with the Schwarzschild metric, it is easy to obtain a general expression for the Schwarzschild mass at \(u=u_{0}\) :
$$m=\lim\limits_{u\to u_{0}}\textrm{e}^{\beta}\gamma^{\prime}/\beta^{\prime}=\lim\limits_{u\to u_{0}}r^{2}\gamma^{\prime}/r^{\prime}.$$In particular, for the (anti-)Fisher metric (20) we have \(m=h\) at \(u_{0}=0\).
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Funding
This publication was supported by the RUDN University Strategic Academic Leadership Program and by RFBR Project 19-02-00346. K.B. was also funded by the Ministry of Science and Higher Education of the Russian Federation, Project “Fundamental properties of elementary particles and cosmology” no. 0723-2020-0041.
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Bronnikov, K.A., Bolokhov, S.V. & Skvortsova, M.V. Spherically Symmetric Space-Times in Generalized Hybrid Metric-Palatini Gravity. Gravit. Cosmol. 27, 358–374 (2021). https://doi.org/10.1134/S0202289321040046
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DOI: https://doi.org/10.1134/S0202289321040046