Vector-tensor Theories Research Articles

Generalised Proca theories in teleparallel gravity

Generalised Proca theories of gravity represent an interesting class of vector-tensor theories where only three propagating degrees of freedom are present. In this work, we propose a new teleparallel gravity analog to Proca theories where the generalised Proca framework is extended due to the lower order nature of torsion based gravity. We develop a new action contribution and explore the example of the Friedmann equations in this regime. We find that teleparallel Proca theories offer the possibility of a much larger class of models in which do have an impact on background cosmology.

Disformal transformation of stationary and axisymmetric solutions in modified gravity

The extended scalar-tensor and vector-tensor theories admit black hole solutions with the nontrivial profiles of the scalar and vector fields, respectively. The disformal transformation maps a solution in a class of the scalar-tensor or vector-tensor theories to that in another class, and hence it can be a useful tool to construct a new nontrivial solution from the known one. First, we investigate how the stationary and axisymmetric solutions in the vector-tensor theories without and with the $U(1)$ gauge symmetry are disformally transformed. We start from a stationary and axisymmetric solution satisfying the circularity conditions, and show that in both the cases the metric of the disformed solution in general does not satisfy the circularity conditions. Using the fact that a solution in a class of the vector-tensor theories with the vanishing field strength is mapped to that in a class of the shift-symmetric scalar-tensor theories, we derive the disformed stationary and axisymmetric solutions in a class of these theories, and show that the metric of the disformed solutions does not satisfy the circularity conditions if the scalar field depends on the time or azimuthal coordinate. We also confirm that in the scalar-tensor theories without the shift symmetry, the disformed stationary and axisymmetric solutions satisfy the circularity conditions. Second, we investigate the disformal transformations of the stationary and axisymmetric black hole solutions in the generalized Proca theory with the nonminimal coupling to the Einstein tensor, the shift-symmetric scalar-tensor theory with the nonminimal derivative coupling to the Einstein tensor, the Einstein-Maxwell theory, and the Einstein-conformally coupled scalar field theory. We show that the disformal transformations modify the causal properties of the spacetime.

Suppressed cosmic growth in coupled vector-tensor theories

We study a coupled dark energy scenario in which a massive vector field $A_{\mu}$ with broken $U(1)$ gauge symmetry interacts with the four-velocity $u_c^{\mu}$ of cold dark matter (CDM) through the scalar product $Z=-u_c^{\mu} A_{\mu}$. This new coupling corresponds to the momentum transfer, so that the background vector and CDM continuity equations do not have explicit interacting terms analogous to the energy exchange. Hence the observational preference of uncoupled generalized Proca theories over the $\Lambda$CDM model can be still maintained at the background level. Meanwhile, the same coupling strongly affects the evolution of cosmological perturbations. While the effective sound speed of CDM vanishes, the propagation speed and no-ghost condition of a longitudinal scalar of $A_{\mu}$ and the CDM no-ghost condition are subject to nontrivial modifications by the $Z$ dependence in the Lagrangian. We propose a concrete dark energy model and show that the gravitational interaction on scales relevant to the linear growth of large-scale structures can be smaller than the Newton constant at low redshifts. This leads to the suppression of growth rates of both CDM and total matter density perturbations, so our model allows an interesting possibility for reducing the tension of matter density contrast $\sigma_8$ between high- and low-redshift measurements.

Rotating black holes in valid vector-tensor theories after GW170817

Vector-tensor theories beyond General Relativity have widely been studied in the context of ultraviolet completion of gravity, endowing a mass to the graviton and explaining dark energy phenomena. We here construct rotating black hole solutions in vector-tensor theories valid after the binary neutron star merger event GW170817 that placed very stringent bound on the propagation speed of gravitational waves away from the speed of light. Such valid vector-tensor theories are constructed by performing a generic conformal transformation to Einstein-Maxwell theory, and the new rotating black hole solutions are constructed by applying the same conformal transformation to the Kerr-Newman solution. These theories fall outside of beyond generalized Proca theories but are within an extended class of vector-tensor theories that satisfy a degenerate condition to eliminate instability modes and are thus healthy. We find that such conformal Kerr-Newman solutions preserve the location of the singularities, event horizons and ergoregion boundary from Kerr-Newman, as well as the multipole moments and the Petrov type. On the other hand, the Hamilton-Jacobi equation is no longer separable, suggesting that the Carter-like constant does not exist in this solution. The standard Newman-Janis algorithm also does not work to construct the new solutions. We also compute the epicyclic frequencies, the location of the innermost stable circular orbits, and the Schwarzschild precession and apply the latter to the recent GRAVITY measurement to place bounds on the deviations away from Kerr-Newman for Sgr A$^*$.

Spontaneous vectorization in the presence of vector-field coupling to matter

We examine the possibility of spontaneous vectorization in the vector-tensor theories with the vector conformal and disformal couplings to matter. We study the static and spherically symmetric solutions of the relativistic stars with the nontrivial profile of the vector field satisfying the boundary conditions ${A}_{\ensuremath{\mu}}\ensuremath{\rightarrow}0$ at the spatial infinity, where ${A}_{\ensuremath{\mu}}$ represents the vector field. First, we study the linear perturbations about the general relativistic (GR) stellar solutions with the vanishing vector field ${A}_{\ensuremath{\mu}}=0$. We show that the pure vector disformal coupling causes the ghost or gradient instability of the GR stars, indicating the breakdown of the hyperbolicity on the GR stellar backgrounds. On the other hand, the pure conformal coupling causes the tachyonic instability of the GR solutions and would lead to the spontaneous growth of the vector field toward the nontrivial solutions as in the manner of spontaneous scalarization in the scalar-tensor theories. We then construct the static and spherically symmetric solutions of the relativistic stars with 0 and 1 nodes of the vector field in the presence of the pure disformal coupling. We find that the properties of the solutions with the nontrivial vector field are similar to those of the generalized Proca theories. As in the mass-radius diagram the branches of the 0 and 1 node solutions are disconnected to that of the GR solutions in the lower density regimes, they may be formed from the selected choice of the initial conditions. We also construct the 0 node solutions in the case of the pure conformal coupling, and show that the branch of these solutions is connected to that of the GR solutions in both the low and high density regions and hence would arise spontaneously via the continuous evolution from the GR solutions. Finally, we briefly discuss the combined effects of the conformal and disformal couplings on the nontrivial solutions.

Role of matter in gravitation: Going beyond the Einstein-Maxwell theory

For field theories in curved spacetime, defining how matter gravitates is part of the theory building process. In this letter, we adopt Bekenstein's multiple geometries approach to allow part of the matter sector to follow the geodesics on a general pseudo-Riemannian geometry, constructed from a tensor and a $U(1)$ gauge field. This procedure allows us to generate a previously unknown corner of vector-tensor theories. In the Jordan frame, apparent high-derivative terms of the vector field are reduced by integrating out an auxiliary variable, at the cost of introducing new matter interactions. As a simple example, we consider a conformal relation between different geometries and demonstrate the presence of an auxiliary degree. We conclude with a discussion of applications, in particular for the early universe.

NAT black holes

We study some physical properties of black holes in Null Aether Theory (NAT) – a vector-tensor theory of gravity. We first review the black hole solutions in NAT and then derive the first law of black hole thermodynamics. The temperature of the black holes depends on both the mass and the NAT “charge” of the black holes. The extreme cases where the temperature vanishes resemble the extreme Reissner–Nordström black holes. We also discuss the contribution of the NAT charge to the geodesics of massive and massless particles around the NAT black holes.

Kinetic field theory applied to vector-tensor gravity

The formation of cosmic structures is an important diagnostic for both the dynamics of the cosmological model and the underlying theory of gravity. At the linear level of these structures, certain degeneracies remain between different cosmological models and alternative gravity theories. It is thus indispensable to study the non-linear, late-time evolution of cosmic structures to try and disentangle their fundamental properties caused by the cosmological model or gravity theory itself. Conventionally, non-linear cosmic structure formation is studied by means of computationally expensive numerical simulations. Since these inevitably suffer from shot noise and are too time consuming to systematically scrutinize large parameter spaces of cosmological models or fundamental theories, analytical methods are needed to overcome the limitations of numerical simulations.Recently, a new analytic approach to non-linear cosmic structure formation has been proposed based on kinetic field theory for classical particle ensembles. Within this theory, a closed, analytic, non-perturbative and parameter-free equation could be derived for the non-linear power spectrum of cosmic density perturbations which agrees very well with numerically simulated results to wave numbers k≲10hMpc−1 at redshift z=0. In this Letter, we study for the first time the implications of alternative gravity theories for non-linear cosmic structure formation applying this promising new analytic framework. As an illustrative example, we consider vector-tensor theories, which support very interesting isotropic cosmological solutions.

Charged particle dynamics in the vicinity of black hole from vector-tensor theory of gravity immersed in an external magnetic field

Abstract The dynamics of charged particle in the vicinity of event horizon of a rotating charged black hole (BH) with gravitomagnetic charge from a class of vector-tensor theories of modified gravity immersed in an axially symmetric magnetic field (Bfield) has been studied. The presence of Bfield reduces the radii of innermost stable circular orbit (ISCO) of charged particle and the opposite phenomenon occurs due the parameter β which describes the deviation of modified gravity considered here from the usual Einstein-Maxwell gravity also acts to increase the radius of ISCO. Angular momentum also plays the role regarding the motion of particle which implies that larger the angular momentum easier for a particle to escape. We study the stability of orbits using Lyapunov characteristic exponent which implies more stability in the presence of Bfield. Generally, in the presence of Bfield it is easier for a particle to escape than its absence. We find twist in the trajectories of charged particles close to event horizon due to spin of BH in the presence of Bfield and there is no twist in the absence of it. Finally, we comment on possible utilization of our findings for the relativistic jets and magnetohydrodynamical out flows.

New Weyl-invariant vector-tensor theory for the cosmological constant

We introduce a new Weyl-invariant and generally-covariant vector-tensor theory with higher derivatives. This theory can be induced by extending the mimetic construction to vector fields of conformal weight four. We demonstrate that in gauge-invariant variables this novel theory reduces to the Henneaux-Teitelboim description of the unimodular gravity. Hence, compared with the standard general relativity, our new higher derivative vector-tensor theory has only one new global degree of freedom—the cosmological constant. Finally we discuss potential extensions of this vector-tensor theory.

Hawking radiation from cubic and quartic black holes via tunneling of GUP corrected scalar and fermion particles

We analyze the effect of the generalized uncertainty principle (GUP) on the Hawking radiation from the hairy black hole in U(1) gauge-invariant scalar–vector–tensor theory by utilizing the semiclassical Hamilton–Jacobi method. To do so, we evaluate the tunneling probabilities and Hawking temperature for scalar and fermion particles for the given spacetime of the black holes with cubic and quartic interactions. For this purpose, we utilize the modified Klein–Gordon equation for the Boson particles and then Dirac equations for the fermion particles, respectively. Next, we examine that the Hawking temperature of the black holes do not depend on the properties of tunneling particles. Moreover, we present the corrected Hawking temperature of scalar and fermion particles which look similar in both interactions, but there are different mass and momentum relationships for scalar and fermion particles in cubic and quartic interactions.

A Modified Gravity Theory: Null Aether**Supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK)

General quantum gravity arguments predict that Lorentz symmetry might not hold exactly in nature. This has motivated much interest in Lorentz breaking gravity theories recently. Among such models are vector-tensor theories with preferred direction established at every point of spacetime by a fixed-norm vector field. The dynamical vector field defined in this way is referred to as the “aether”. In this paper, we put forward the idea of a null aether field and introduce, for the first time, the Null Aether Theory (NAT)—a vector-tensor theory. We first study the Newtonian limit of this theory and then construct exact spherically symmetric black hole solutions in the theory in four dimensions, which contain Vaidya-type non-static solutions and static Schwarzschild-(A)dS type solutions, Reissner-Nordström-(A)dS type solutions and solutions of conformal gravity as special cases. Afterwards, we study the cosmological solutions in NAT: We find some exact solutions with perfect fluid distribution for spatially flat FLRW metric and null aether propagating along the x direction. We observe that there are solutions in which the universe has big-bang singularity and null field diminishes asymptotically. We also study exact gravitational wave solutions—AdS-plane waves and pp-waves—in this theory in any dimension . Assuming the Kerr-Schild-Kundt class of metrics for such solutions, we show that the full field equations of the theory are reduced to two, in general coupled, differential equations when the background metric assumes the maximally symmetric form. The main conclusion of these computations is that the spin-0 aether field acquires a “mass” determined by the cosmological constant of the background spacetime and the Lagrange multiplier given in the theory.

Jordan frame beyond scalar-tensor theories

We study the Jordan frame formulation of generalizations of scalar-tensor theories conceived by replacing the scalar with other fields such as vectors. The generic theory in this family contains higher order time derivative terms in the Jordan frame action which is indicative of ill-posedness. However, we show that equations of motion can always be reduced to a second-order-in-time form as long as the original Einstein frame formulation is well posed. The inverse transformation from the Jordan frame back to the Einstein frame is not possible for all field values in all theories, but we obtain a fully invertible transformation for vector-tensor theories by a redefinition of the vector field. Our main motivation is a better understanding of spontaneous scalarization and its generalizations, however our conclusions are applicable to a wide class of theories. Jordan frame has been traditionally used for certain calculations in scalar-tensor theories of gravitation, and our results will help researchers generalize these results, enabling comparison to observational data.

Astrophysical signatures of black holes in generalized Proca theories

Explaining the late time acceleration is one of the most challenging tasks for theoretical physicists today. Infra-red modification of Einstein's general theory of relativity (GR) is a possible route to model late time acceleration. In this regard, vector-tensor theory as a part of gravitational interactions on large cosmological scales, has been proposed recently. This involves generalization of massive Proca lagrangian in curved space time. Black hole solutions in such theories have also been constructed. In this paper, we study different astrophysical signatures of such black holes. We first study the strong lensing and time delay effect of such static spherically symmetric black hole solutions, in particular for the case of gravitational lensing of the star S2 by Sagittarius A* at the centre of Milky Way. We also construct the rotating black hole solution from this static spherically symmetric solution in Proca theories using the Newman-Janis algorithm and subsequently study lensing, time delay and black hole shadow effect in this rotating black hole space time. We discuss the possibility of detecting Proca hair in future observations.

Interpreting deviations between AR-VTG and GR

The cosmic microwave background (CMB) anisotropies predicted by two cosmological models are compared, one of them is the standard model of general relativity with cold dark matter and cosmological constant, whereas the second model is based on a consistent vector-tensor theory of gravitation explaining solar system and cosmological observations. It is proved that the resulting differences — between the anisotropies of both models — are due to the so-called late integrated Sachs–Wolfe effect and, consequently, cross-correlations between maps of CMB temperatures and tracers of the dark matter distribution could be used in future to select one of the above models. The role of reionization is analyzed in detail.

Stable cosmological solutions in degenerate theory of gravity

We study cosmological applications of extended vector–tensor theories, whose Lagrangians contain up to two derivatives with respect to metric and vector field. We derive background equations under the assumption of homogeneous and isotropic universe and study the nature of cosmological perturbations on top of the background. We found an example of healthy cosmological solutions in non-trivial degenerate theory of gravity for the first time, where those perturbations do not suffer from any instabilities, that is, ghost and gradient instabilities.

Black hole and cosmos with multiple horizons and multiple singularities in vector-tensor theories

A stationary and spherically symmetric black hole (For example, Reissner-Nordstrom black hole or Kerr-Newman black hole) has at most one singularity and two horizons. One horizon is the outer event horizon and the other is the inner Cauchy horizon. Can we construct static and spherically symmetric black hole solutions with N horizons and M singularities? De Sitter cosmos has only one apparent horizon. Can we construct cosmos solutions with N horizons? In this article, we present the static and spherically symmetric black hole and cosmos solutions with N horizons and M singularities in the vector-tensor theories. Following these motivations, we also construct the black hole solutions with a firewall. The deviation of these black hole solutions from the usual ones can be potentially tested by future measurements of gravitational waves.

Relativistic stars in vector-tensor theories

We study relativistic star solutions in second-order generalized Proca theories characterized by a $U(1)$-breaking vector field with derivative couplings. In the models with cubic and quartic derivative coupling, the mass and radius of stars become larger than those in general relativity for negative derivative coupling constants. This phenomenon is mostly attributed to the increase of star radius induced by a slower decrease of the matter pressure compared to general relativity. There is a tendency that the relativistic star with a smaller mass is not gravitationally bound for a low central density and hence dynamically unstable, but that with a larger mass is gravitationally bound. On the other hand, we show that the intrinsic vector-mode couplings give rise to general relativistic solutions with a trivial field profile, so the mass and radius are not modified from those in general relativity.

Speed of gravitational waves and black hole hair

The recent detection of GRB 170817A and GW170817 constrains the speed of gravity waves $c_T$ to be that of light, which severely restricts the landscape of modified gravity theories that impact the cosmological evolution of the universe. In this work, we investigate the presence of black hole hair in the remaining viable cosmological theories of modified gravity that respect the constraint $c_T=1$. We focus mainly on scalar-tensor theories of gravity, analyzing static, asymptotically flat black holes in Horndeski, Beyond Horndeski, Einstein-Scalar-Gauss-Bonnet, and Chern-Simons theories. We find that in all of the cases considered here, theories that respect $c_T=1$ do not allow for hair, or have negligible hair. We further comment on vector-tensor theories including Einstein Yang-Mills, Einstein-Aether, and Generalised Proca theories, as well as bimetric theories.

Fate of Large-Scale Structure in Modified Gravity After GW170817 and GRB170817A.

The coincident detection of gravitational waves (GW) and a gamma-ray burst from a merger of neutron stars has placed an extremely stringent bound on the speed of GWs. We showed previously that the presence of gravitational slip (η) in cosmology is intimately tied to modifications of GW propagation. This new constraint implies that the only remaining viable source of gravitational slip is a conformal coupling to gravity in scalar-tensor theories, while viable vector-tensor theories cannot now generate gravitational slip at all. We discuss structure formation in the remaining viable models, demonstrating that (i)the dark-matter growth rate must now be at least as fast as in general relativity (GR), with the possible exception of that beyond the Horndeski model, and (ii)if there is any scale dependence at all in the slip parameter, it is such that it takes the GR value at large scales. We show a consistency relation that must be violated if gravity is modified.