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In this paper we pursue the study of linear perturbations around a Schwarzschild black hole in a generalized Einstein-Cartan theory of gravity, called torsion bigravity. This theory contains both massless and massive spin-2 excitations. Here we consider non spherically-symmetric perturbations with generic multipolarity $L \geq 1$. We extend the conclusion of linear stability, previously obtained for $L=0$ [Phys. Rev. D 104, 024032], to the generic $L \geq 1$ case. We prove that the mass $\kappa$ of the massive spin-2 excitation must be large enough, namely $\kappa r_h > \sqrt{1+\eta}$, to avoid the presence of singularities in the perturbation equations. The perturbation equations are shown to have a triangular structure, where massive spin-2 excitations satisfy decoupled equations, while the Einstein-like massless spin-2 ones satisfy inhomogeneous equations sourced by the massive spin-2 sector. We study quasi-bound states, and exhibit some explicit complex quasi-bound frequencies. We briefly discuss the issue of superradiance instabilities.

Time-dependent spherically-symmetric perturbations of Schwarzschild black holes are studied within torsion bigravity, i.e., within generalized Einstein-Cartan theories where the dynamical torsion carries massive spin-2 excitation. We reduce linearized perturbations to a Zerilli-like equation. The structure of the potential entering the latter Zerilli-like equation has two important consequences. First, in order to avoid the presence of singularities in generic perturbations, one must restrict the range (or inverse mass) of the spin-2 excitation to be (essentially) smaller than the radius of the considered black hole. Second, we then show that the Schwarzschild black hole is linearly stable against spherically-symmetric perturbations., Comment: 15 pages, 2 figures, 1 Mathematica notebook as ancillary file

International audience; We continue the study of spherically symmetric black hole solutions in torsion bigravity, a class of Einstein-Cartan-type gravity theories involving, besides a metric, a massive propagating torsion field. In the infinite-range limit, these theories admit asymptotically flat black hole solutions related to the presence of attractive fixed points in the asymptotic radial evolution of the metric and the torsion. We discuss these fixed points and the way they are approached at large radii. Several phenomenological aspects of asymptotically flat torsion-hairy black holes are discussed: (i) location of the light ring and of the shadow; (ii) correction to the redshift of orbiting stars; and (iii) modification of the periastron precession of orbiting stars. By comparing the observable properties of torsion-hairy black holes to existing observational data on supermassive black holes obtained by the Event Horizon Telescope Collaboration and by the GRAVITY Collaboration, we derive constraints on the theory parameters of torsion bigravity. The strongest constraint is found to come from the recent measurement of the periastron precession of the star S2 orbiting the Galactic-center massive black hole [R. Abuter et al., Astron. Astrophys. 636, L5 (2020)AAEJAF0004-636110.1051/0004-6361/202037813 and to be a thousand times more stringent than Solar-System gravitational tests.

We study spherically symmetric black hole solutions in a four-parameter Einstein-Cartan-type class of theories, called "torsion bigravity". These theories offer a geometric framework (with a metric and an independent torsionfull connection) for a modification of Einstein's theory that has the same spectrum as bimetric gravity models. In addition to an Einsteinlike massless spin-2 excitation, there is a massive spin-2 one (of range $\kappa^{-1}$) coming from the torsion sector, rather than from a second metric. We prove the existence of three broad classes of spherically-symmetric black hole solutions in torsion bigravity. First, the Schwarzschild solution defines an asymptotically-flat torsionless black hole for all values of the parameters. [And we prove that one cannot deform a Schwarzschild solution, at the linearized level, by adding an infinitesimal torsion hair.] Second, when considering finite values of the range, we find that there exist non-asymptotically-flat torsion-hairy black holes in a large domain of parameter space. Third, we find that, in the limit of infinite range, there exists a two-parameter family of asymptotically flat torsion-hairy black holes. The latter black hole solutions give an interesting example of non-Einsteinian (but still purely geometric) black hole structures which might be astrophysically relevant when considering a range of cosmological size., Comment: 19 pages, 3 figures

It was pointed out long ago by Vainshtein [Phys. Lett. 39B, 393 (1972)] that the weak-field perturbation expansion of generic theories (of the nonlinear Fierz-Pauli type) involving massive spin-2 excitations breaks down below a certain distance around a material source ("Vainshtein radius"), scaling as some inverse power of the spin-2 mass $m_2$, i.e., some positive power of the range $m_2^{-1}$. Here we prove that this conclusion does not apply in a generalized Einstein-Cartan theory (called "torsion bigravity") whose spectrum is made (like that of bimetric gravity) of a massless spin-2 excitation and a massive spin-2 one. Working within a static spherically symmetric ansatz, we prove, by reformulating the field equations in terms of new variables, that one can construct an all-order weak-field perturbative expansion where no denominators involving $m_2$ ever appear in the region $r \ll m_2^{-1}$. In particular, we show how the formal large-range limit, $m_2 \to 0$, leads to a well-defined, finite perturbation expansion, whose all-order structure is discussed in some detail., 18 pages, no figures

We study spherically symmetric solutions in a four-parameter Einstein-Cartan-type class of theories. These theories include torsion, as well as the metric, as dynamical fields, and contain only two physical excitations (around flat spacetime): a massless spin-2 excitation and a massive spin-2 one (of mass $ m_2 \equiv \kappa$). They offer a geometric framework (which we propose to call "torsion bigravity") for a modification of Einstein's theory that has the same spectrum as bimetric gravity models. We find that the spherically symmetric solutions of torsion bigravity theories exhibit several remarkable features: (i) they have the same number of degrees of freedom as their analogs in ghost-free bimetric gravity theories ( i.e. one less than in ghost-full bimetric gravity theories); (ii) in the limit of small mass for the spin-2 field ($ \kappa \to 0$), no inverse powers of $\kappa$ arise at the first two orders of perturbation theory (contrary to what happens in bimetric gravity where $1/\kappa^2$ factors arise at linear order, and $1/\kappa^4$ ones at quadratic order). We numerically construct a high-compactness (asymptotically flat) star model in torsion bigravity and show that its geometrical and physical properties are significantly different from those of a general relativistic star having the same observable Keplerian mass., Comment: 28 pages, 3 figures

We continue the exploration of the consistency of a modified-gravity theory that generalizes General Relativity by including a dynamical torsion in addition to the dynamical metric. The six-parameter theory we consider was found to be consistent around arbitrary torsionless Einstein backgrounds, in spite of its containing a (notoriously delicate) massive spin-2 excitation. At zero bare cosmological constant, this theory was found to admit a self-accelerating solution whose exponential expansion is sustained by a non-zero torsion background. The scalar-type perturbations of the latter torsionfull self-accelerating solution were recently studied and were found to preserve the number of propagating scalar degrees of freedom, but to exhibit, for some values of the torsion background some exponential instabilities (of a rather mild type). Here, we study the tensor-type and vector-type perturbations of the torsionfull self-accelerating solution, and of its deformation by a non-zero bare cosmological constant. We find strong, "gradient" instabilities in the vector sector. No tuning of the parameters of the theory can kill these instabilities without creating instabilities in the other sectors. Further work is needed to see whether generic torsionfull backgrounds are prone to containing gradient instabilities, or if the instabilities we found are mainly due to the (generalized) self-accelerating nature of the special de Sitter backgrounds we considered., 32 pages, 2 figures; four references added in the Introduction; more explanations on the structure of the action of torsion gravity; additional discussion of the crossing between the two branches of de Sitter-like solutions; to be published in Phys. Rev. D

We consider the model of modified gravity with dynamical torsion. This model was found to have promising stability properties about various backgrounds. The model admits a self-accelerating solution. We have shown previously that if the parameters are adjusted in such a way that the torsion is much greater than the effective cosmological constant, the self-accelerating solution is unstable: there are exponentially growing modes. Here we study the scalar perturbations in the case when the torsion is of the order of the effective cosmological constant. We find that there are no exponential instabilities., Comment: arXiv admin note: substantial text overlap with arXiv:1705.00856

We consider the model of modified gravity with dynamical torsion. This model was previously found to have promising stability properties about various backgrounds. Here, we study the stability of linear perturbations about the self-accelerating solution. We apply the (3[Formula: see text]+[Formula: see text]1)-decomposition and consider the scalar sector of perturbations. We find that the number of degrees of freedom is equal to 2, which is the same as in Minkowski background. However, there is at least one instability in the scalar sector, if the value of background torsion is large enough. This does not rule out the possibility of stable self-acceleration with torsion of the order of the effective cosmological constant.

We consider a model belonging to the class of Poincarè gauge gravities. The model is free of ghosts, tachyons and gradient instabilities about Minkowski and torsionless Einstein backgrounds of sufficiently small curvature. At zero cosmological constant, the model admits a self-accelerating solution with non-Riemannian connection. We study scalar perturbations about the self-accelerating solution and find that the number of scalar modes is the same as in Minkow ski background; moreover, in the limit of small effective cosmological constant and below the UV cutoff of the low energy effective theory, the scalar sector does not have pathologies like ghosts or rapid gradient instabilities.