Aether theory is introduced to implement the violation of the Lorentz invariance in general relativity. For this purpose a unit timelike vector field introduced to theory in addition to the metric tensor. Aether theory contains four free parameters which satisfy some inequalities in order that the theory to be consistent with the observations. We show that the G{\" o}del type of metrics of general relativity are also exact solutions of the Einstein-aether theory. The only field equations are the 3-dimensional Maxwell field equations and the parameters are left free except $c_{1}-c_{3}=1$.
We formulate the nonminimal aether-modified gravity whose action represents itself as a sum of the usual Einstein-Hilbert action and the CPT-even Lorentz-breaking aether-like gravity term proposed by Carroll. For this theory, we show that the Gödel metric solves the modified Einstein equations under a corresponding modification of the matter, while, for the small Lorentz-breaking corrections, the matter turns out to be usual. On the other side, if the matter is suggested to be the same as in the usual general relativity, the Gödel metric is modified by additive aether terms.
It was previously proved that the Gödel-type metrics with flat three-dimensional background metric solve exactly the field equations of the Einstein-Aether theory in four dimensions. We generalize this result by showing that the stationary Gödel-type metrics with nonflat background in $D$ dimensions solve exactly the field equations of the Einstein-Aether theory. The reduced field equations are the $(D-1)$-dimensional Euclidean Ricci-flat and the $(D-1)$-dimensional source-free Maxwell equations, and the parameters of the theory are left free except $c_{1}-c_{3}=1$. We give a method to produce…
In the framework of the Einstein-aether theory we consider a cosmological model, which describes the evolution of the unit dynamic vector field with activated rotational degree of freedom. We discuss exact solutions of the Einstein-aether theory, for which the space-time is of the Gödel-type, the velocity four-vector of the aether motion is characterized by a non-vanishing vorticity, thus the rotational vectorial modes can be associated with the source of the universe rotation. The main goal of our paper is to study the motion of test relativistic particles with vectorial internal degree of fr…
A recent paper studies a modification of Einstein-aether theory in which the aether vector is restricted, at the level of the action, to be the gradient of a scalar. In this comment we note that this scalar version of Einstein-aether theory is equivalent to the projectable version of the IR limit of Hořava gravity when the potential for the scalar is constant. This provides a generally covariant formulation for projectable Hořava gravity.
We present exact solutions in Einstein-aether theory in a static spherically symmetric background space with a spacelike aether field, as a difference with the usual selection of timelike aether field. We assume a coupling between the scalar field and the aether field introduced in the aether coefficients. The exact spacetimes describe hairy black hole solutions for which the limits of the Schwarzschild, de-Sitter Schwarzschild and Reissner-Nordström metrics are recovered.
Energy positivity is established for a class of solutions to Einstein-aether theory and the IR limit of Hořava gravity within a certain range of coupling parameters. The class consists of solutions where the aether 4-vector is divergence free on a spacelike surface to which it is orthogonal (which implies that the surface is maximal). In particular, this result holds for spherically symmetric solutions at a moment of time symmetry.
We show that the Einstein-aether theory of Jacobson and Mattingly (J&M) can be understood in the framework of the metric-affine (gauge theory of) gravity (MAG). We achieve this by relating the aether vector field of J&M to certain post-Riemannian nonmetricity pieces contained in an independent linear connection of spacetime. Then, for the aether, a corresponding geometrical curvature-square Lagrangian with a massive piece can be formulated straightforwardly. We find an exact spherically symmetric solution of our model.
We use a dynamical systems analysis to investigate the future behaviour of Einstein-Aether cosmological models with a scalar field coupling to the expansion of the aether and a non-interacting perfect fluid. The stability of the equilibrium solutions are analysed and the results are compared with the standard inflationary cosmological solutions and previously studied cosmological Einstein-Aether models.
We consider an Einstein-aether type Lorentz-violating theory of gravity in which the aether vector field $V_{μ}$ is represented as the gradient of a scalar field $ϕ$, $V_{μ}=\nabla _{μ}ϕ$. A self interacting potential for the scalar aether field is considered, as well as the possibility of a coupling between the hydrodynamic matter flux and the aether field, with the imposition of the timelike nature of the aether vector. The gravitational field equations and the equation of motion of the scalar field are derived by varying the action with respect to the metric and $ϕ$. In the absence of matte…
The primary purpose of this study is to investigate the constant-roll inflationary scenario with anisotropic conditions concerning the Einstein-aether Scalar-tensor cosmology in noncommutative phase space. We first introduce an Einstein-aether scalar-tensor cosmological model. In this structure, one can introduce an aether field with aether coefficients in the action integral of scalar-tensor. It will be a function of the scalar field, which is, in fact, a kind of extender of the Lorentz-violating theories. Hence, we present the point-like Lagrangian, which represents the field equations of th…
We show how to derive several families of accelerating universe solutions to an Einstein-Aether gravity theory. These solutions provide possible descriptions of inflationary behaviour in the early universe and late-time cosmological acceleration.
We present, for the first time, the quantization process for the Einstein-aether scalar field cosmology. We consider a cosmological theory proposed as a Lorentz violating inflationary model, where the aether and scalar fields interact through the assumption that the aether action constants are ultra-local functions of the scalar field. For this specific theory there is a valid minisuperspace description which we use to quantize. For a particular relation between the two free functions entering the reduced Lagrangian the solution to the Wheeler-DeWitt equation as also the generic classical solu…
The exact static spherically symmetric solutions for pure-aether theory and Einstein-aether theory are presented. It is shown that both theories can deliver the Schwarzschild metric, but only the Einstein-aether theory contains solutions with "almost-Schwarzschild" metrics that satisfy Einstein's experiments. Two specific solutions are of special interest: one in pure-aether theory that derives the attractive nature of gravitation as a result of Minskowski signature of the metric, and one - the Jacobson solution- of Einstein-aether theory with "almost-Schwarzschild" metric and non-zero Ricci t…
The Noether charge method for defining the Hamiltonian of a diffeomorphism-invariant field theory is applied to "Einstein-aether" theory, in which gravity couples to a dynamical, timelike, unit-norm vector field. Using the method, expressions are obtained for the total energy, momentum, and angular momentum of an Einstein-aether space-time. The method is also used to discuss the mechanics of Einstein-aether black holes. The derivation of Wald, and Iyer and Wald, of the first law of black hole thermodynamics fails for this theory, because the unit vector is necessarily singular at the bifurcati…
We propose accurate calculations of quasinormal modes of black holes in the Einstein-aether theory, which were previously considered in the literature, partially, with insufficient accuracy. We also show that the arbitrarily long-lived modes, quasiresonances, are allowed in the Einstein-aether theory as well and demonstrate that the asymptotic tails, unlike quasinormal frequencies, are indistinguishable from those in the Einstein theory.
Exact and analytic solutions in Einstein-Aether scalar field theory with Kantowski-Sachs background space are determined. The theory of point symmetries is applied to determine the functional form of the unknown functions which defines the gravitational model. Conservation laws are applied to reduce the order of the field equations and write the analytic solution. Moreover, in order to understand the physical behaviour of the cosmological model a detailed analysis of the asymptotic behaviour for the gravitational field equations is performed.
We study how the dynamical vector degree of freedom in modified gravity affects the CMB B-mode polarization in terms of the Einstein-aether theory. In this theory, vector perturbations can be generated from inflation, which can grow on superhorizon scales in the subsequent epochs and thereby leaves imprints on the CMB B-mode polarization. We derive the linear perturbation equations in a covariant formalism, and compute the CMB B-mode polarization using the CAMB code modified so as to incorporate the effect of the aether vector field. We find that the amplitude of the B-mode signal from the aet…
We study the impact of modifying the vector sector of gravity on the CMB polarization. We employ the Einstein-aether theory as a concrete example. The Einstein-aether theory admits dynamical vector perturbations generated during inflation, leaving imprints on the CMB polarization. We derive the perturbation equations of the aether vector field in covariant formalism and compute the CMB B-mode polarization using the modified CAMB code. It is found that the amplitude of the B-mode signal from the aether field can surpass the one from the inflationary gravitational waves. The shape of the spectru…
The local Lorentz violation (LV) in gravity sector should show itself in derivation of the characteristic quasinormal modes (QNMs) of black hole mergers from their general relativity case. In this paper, I study QNMs of the gravitational field perturbations to Einstein-aether black holes and, at first compare them to those in Schwarzschild black hole, and then some other known LV gravity theories. By comparing to Schwarzschild black hole, the first kind aether black holes have larger damping rate and the second ones have lower damping rate. And they all have smaller real oscillation frequency …
We discuss the set of constraints for Einstein-aether theories, comparing the flat background case with what is expected when the gravitational fields are dynamic. We note potential pathologies occurring in the weak gravitational field limit for some of the Einstein-aether theories.
Einstein-aether theory is general relativity coupled to a dynamical, unit timelike vector. If this vector is restricted in the action to be hypersurface orthogonal, the theory is identical to the IR limit of the extension of Horava gravity proposed by Blas, Pujolàs and Sibiryakov. Hypersurface orthogonal solutions of Einstein-aether theory are solutions to the IR limit of this theory, hence numerous results already obtained for Einstein-aether theory carry over.
We show that Hořava gravity can be obtained from Einstein-aether theory in the limit that the twist coupling constant goes to infinity, while holding fixed the expansion, shear and acceleration couplings. This limit helps to clarify the relation between the two theories, and allows Hořava results to be obtained from Einstein-aether ones. The limit is illustrated with several examples, including rotating black hole equations, PPN parameters, and radiation rates from binary systems.
We perform numerical simulations of gravitational collapse in Einstein-aether theory. We find that under certain conditions, the collapse results in the temporary formation of a white hole horizon.
We show that one-parameter family of Godel-type metrics are exact solutions in a wide class of Einstein-Maxwell-scalar field theories. In these solutions, the gauge field has a 1-form expectation value parallel to the timelike contact form that represents the rotation of the Godel universe.
Einstein-aether theory is general relativity coupled to a dynamical unit timelike vector field. A brief review of current theoretical understanding and observational constraints on the four coupling parameters of the theory is given.
In the framework of axionic extension of the Einstein-aether theory we establish the model, which describes a stiff regulation of the behavior of axionic dark matter by the dynamic aether. The aether realizes this procedure via the modified Higgs potential, designed for modeling of nonlinear self-interaction of pseudoscalar (axion) field; the modification of this potential is that its minima are not fixed, and their positions and depths depend now on the square of the covariant derivative of the aether velocity four-vector. Exact solutions to the master equations, modified correspondingly, are…
We study gravitational collapse of a spherically symmetric scalar field in Einstein-aether theory (general relativity coupled to a dynamical unit timelike vector field). The initial value formulation is developed, and numerical simulations are performed. The collapse produces regular, stationary black holes, as long as the aether coupling constants are not too large. For larger couplings a finite area singularity occurs. These results are shown to be consistent with the stationary solutions found previously.
In this paper, we will analyse the cosmological models in Einstein-aether gravity, which is a modified theory of gravity in which a time-like vector field breaks the Lorentz symmetry. We will use this formalism to analyse different cosmological models with different behavior of the scale factor. In this analysis, we will use a certain functional dependence of the dark energy on the Hubble parameter. It will be demonstrated that the aether vector field has a non-trivial effect on these cosmological models. We will also perform the \emph{Om} diagnostic in Einstein-aether gravity. Thus, we will f…
We investigate the field equations in the Einstein-aether theory for static spherically symmetric spacetimes and a perfect fluid source and subsequently with the addition of a scalar field (with an exponential self-interacting potential). We introduce more appropriate dynamical variables that facilitate the study of the equilibrium points of the resulting dynamical system and, in addition, we discuss the dynamics at infinity. We study the qualitative properties of the models with a particular interest in their asymptotic behaviour and whether they admit singularities. We also present a number …
We briefly describe the modified Friedmann equations for Einstein-Aether gravity theory and we find the effective density and pressure. The purpose of our present work is to reconstruction of Einstein-Aether Gravity from other modified gravities like $f(T)$, $f(R)$, $f(G)$, $f(R,T)$ and $f(R,G)$ and check its viability. The scale factor is chosen in power law form. The free function $F(K)$ for Einstein-Aether gravity (where $K$ is proportional to $H^{2}$) have been found in terms for $K$ by the correspondence between Einstein-Aether gravity and other modified gravities and the nature of $F(K)$…
The space of the solutions of the differential equations resulting from considering matter fluids of scalar field type or perfect fluid in Einstein-aether theory is analyzed. The Einstein-aether theory of gravity consists of General Relativity coupled to a vector field of unit time type, called the aether. In this effective theory, Lorentz invariance is violated, but locality and covariance are preserved in the presence of the vector field. For the mathematical formulation of the models, the 1 + 3 formalism is used that allows writing field equations for spherically symmetric inhomogeneous met…
We compare the initial value formulation of the low-energy limit of (non-projectable) Horava gravity to that of Einstein-aether theory when the aether is assumed to be hypersurface orthogonal at the level of the field equations. This comparison clearly highlights a crucial difference in the causal structure of the two theories at the non-perturbative level: in Horava gravity evolution equations include an elliptic equation that is not a constraint relating initial data but needs to be imposed on each slice of the foliation. This feature is absent in Einstein-aether theory. We discuss its physi…
We establish a nonlinear version of the SU(N) symmetric theory, which describes self-consistently the interaction between the gravitational, gauge, vector and pseudoscalar (axion) fields. In the context of this theory the SU(N) symmetric multiplet of vector fields is associated with the color aether, the decay of which in the early Universe produced the canonic dynamic aether and the axionic dark matter. The SU(N) symmetric Yang-Mills field, associated with the color aether, forms the source, which transfers the energy of the decaying color aether to the axion field. The nonlinear modification…
In the framework of the Einstein-Maxwell-aether theory, we present two new classes of exact charged black hole solutions, which are asymptotically flat and possess the universal as well as Killing horizons. We also construct the Smarr formulas, and calculate the temperatures of the horizons, using the Smarr mass-area relation. We find that, in contrast to the neutral case, such obtained temperature is not proportional to its surface gravity at any of the two kinds of the horizons. Einstein-Maxwell-aether black holes with the cosmological constant and their topological cousins are also given.
Using the solution phase space method, we investigate the thermodynamics of black holes in Einstein-aether-Maxwell theory, for which the traditional Wald method (covariant phase space method) fails. We show the first laws of thermodynamics and definitive entropy expressions at both Killing and universal horizons for some examples of exact black hole solutions, including 3-dimensional static charged quasi-BTZ black hole, two 4-dimensional static charged black holes and 3-dimensional rotating solution. At Killing horizons the entropies are exactly one quarter of the horizon area, but at universa…
We review the status of "Einstein-Aether theory", a generally covariant theory of gravity coupled to a dynamical, unit timelike vector field that breaks local Lorentz symmetry. Aspects of waves, stars, black holes, and cosmology are discussed, together with theoretical and observational constraints. Open questions are stressed.
The extended Einstein-Maxwell-aether-axion model describes internal interactions inside the system, which contains gravitational, electromagnetic fields, the dynamic unit vector field describing the velocity of an aether, and the pseudoscalar field associated with the axionic dark matter. The specific feature of this model is that the axion field controls the dynamics of the aether through the guiding functions incorporated into the Jacobson's constitutive tensor. Depending on the state of the axion field these guiding functions can control and switch on or switch off the influence of accelera…
We consider a pp-wave symmetric model in the framework of the Einstein-Maxwell-aether-axion theory. Exact solutions to the equations of axion electrodynamics are obtained for the model, in which pseudoscalar, electric and magnetic fields were constant before the arrival of a gravitational pp-wave. We show that dynamo-optical interactions, i.e. couplings of the electromagnetic field to a dynamic unit vector field, attributed to the velocity of a cosmic substratum (aether, vacuum, dark fluid...), provide the response of axionically active electrodynamic system to display anomalous behavior.
Local Lorentz invariance violation can be realized by introducing extra tensor fields in the action that couple to matter. If the Lorentz violation is rotationally invariant in some frame, then it is characterized by an ``aether'', i.e. a unit timelike vector field. General covariance requires that the aether field be dynamical. In this paper we study the linearized theory of such an aether coupled to gravity and find the speeds and polarizations of all the wave modes in terms of the four constants appearing in the most general action at second order in derivatives. We find that in addition to…
The gravitational consequence of local Lorentz violation (LV) should show itself in derivation of the characteristic quasinormal ringing of black hole mergers from their general relativity case. In this paper, we study quasinormal modes (QNMs) of the scalar and electromagnetic field perturbations to Einstein aether black holes. We find that quasinormal ringing of the first kind aether black hole is similar to that of another Lorentz violation model---the QED-extension limit of standard model extension. These similarities between completely different backgrounds may imply that LV in gravity sec…
We analyze the observational and theoretical constraints on ``Einstein-aether theory", a generally covariant theory of gravity coupled to a dynamical, unit, timelike vector field that breaks local Lorentz symmetry. The results of a computation of the remaining post-Newtonian parameters are reported. These are combined with other results to determine the joint post-Newtonian, vacuum-Cerenkov, nucleosynthesis, stability, and positive-energy constraints. All of these constraints are satisfied by parameters in a large two-dimensional region in the four-dimensional parameter space defining the theo…
We extend the Einstein-aether theory to take into account the interaction between a pseudoscalar field, which describes the axionic dark matter, and a time-like dynamic unit vector field, which characterizes the velocity of the aether motion. The Lagrangian of the Einstein-aether-axion theory includes cross-terms based on the axion field and its gradient four-vector, on the covariant derivative of the aether velocity four-vector, and on the Riemann tensor and its convolutions. We follow the principles of the Effective Field theory, and include into the Lagrangian of interactions all possible t…
We extend the Einstein-aether theory to include the Maxwell field in a nontrivial manner by taking into account its interaction with the time-like unit vector field characterizing the aether. We also include a generic matter term. We present a model with a Lagrangian that includes cross-terms linear and quadratic in the Maxwell tensor, linear and quadratic in the covariant derivative of the aether velocity four-vector, linear in its second covariant derivative and in the Riemann tensor. We decompose these terms with respect to the irreducible parts of the covariant derivative of the aether vel…
We examine the consequences of Lorentz violation during slow-roll inflation. We consider a canonical scalar inflaton coupled, through its potential, to the divergence of a fixed-norm timelike vector field, or "aether." The vector is described by Einstein-aether theory, a vector-tensor model of gravitational Lorentz violation. We derive and analyze the cosmological perturbation equations for the metric, inflaton, and aether. If the scale of Lorentz violation is sufficiently small compared to the Planck mass, and the strength of the scalar-aether coupling is suitably large, then the spin-0 and s…
In this paper, we systematically study spacetimes of gravitational plane waves in Einstein-aether theory. Due to the presence of the timelike aether vector field, now the problem in general becomes overdetermined. In particular, for the linearly polarized plane waves, there are five independent vacuum Einstein-aether field equations for three unknown functions. Therefore, solutions exist only for particular choices of the four free parameters $c_{i}$'s of the theory. We find that there exist eight cases, in two of which any form of gravitational plane waves can exist, similar to that in genera…
The time independent spherically symmetric solutions of General Relativity (GR) coupled to a dynamical unit timelike vector are studied. We find there is a three-parameter family of solutions with this symmetry. Imposing asymptotic flatness restricts to two parameters, and requiring that the aether be aligned with the timelike Killing field further restricts to one parameter, the total mass. These "static aether" solutions are given analytically up to solution of a transcendental equation. The positive mass solutions have spatial geometry with a minimal area 2-sphere, inside of which the area …
We consider a Lorentz violating scalar field cosmological model given by the modified Einstein-aether theory defined in Weyl integrable geometry. The existence of exact and analytic solutions is investigated for the case of a spatially flat Friedmann--Lema\^ıtre--Robertson--Walker background space. We show that the theory admits cosmological solutions of special interests. In addition, we prove that the cosmological field equations admit the Lewis invariant as a second conservation law, which indicates the integrability of the field equations.
We study the quasi-normal modes of the charged scalar perturbations in the background of the Einstein-Maxwell-aether black hole through three methods (WKB method, continued fraction method, generalized eigenvalue method). Then we propose the specific treatment for the generalized effective potential with $ω$-dependence and the complete procedure of transforming calculation continued fraction method into finding the zero point of the corresponding complex function numerically. These methods are valid because the results from different methods are consistent. We also investigate the allowed regi…
This paper reviews the theory, phenomenology, and observational constraints on the coupling parameters of Einstein-aether gravity, i.e. General Relativity coupled to a dynamical unit timelike vector field. A discussion of open questions concerning both phenomenology and fundamental issues is included.
We reconsider spherically symmetric black hole solutions in Einstein-Aether theory with the condition that this theory has identical PPN parameters as those for general relativity, which is the main difference from the previous research. In contrast with previous study, we allow superluminal propagation of a spin-0 Aether-gravity wave mode. As a result, we obtain black holes having a spin-0 "horizon" inside an event horizon. We allow a singularity at a spin-0 "horizon" since it is concealed by the event horizon. If we allow such a configuration, the kinetic term of the Aether field can be larg…
An effective Lagrangian approach based on an extended Einstein-Aether (EA) model is presented and formulated in a generic non flat Fridman-Lemaitre-Robertson-Walker (FLRW) space-time. For a flat FLRW space-time, a Friedmann equation similar to the one obtained in Quantum Loop Cosmology (QLC) is reproduced, with related non singular bounce solution. Finally, the Static Spherically Symmetric (SSS) solutions are investigated.
We study gravitational waves emitted by a binary system of non-spinning bodies in a quasi-circular inspiral within the framework of Einstein-aether theory. In particular, we compute explicitly and analytically the expressions for the time-domain and frequency-domain waveforms, gravitational wave polarizations, and response functions for both ground- and space-based detectors in the post-Newtonian approximation. We find that, when going beyond leading-order in the post-Newtonian approximation, the non-Einsteinian polarization modes contain terms that depend on both the first and the second harm…
Einstein's theory of general relativity describes gravity as the interaction of particles with space-time geometry, as opposed to interacting with a physical fluid, as in the old gravitational aether theories. Moreover, any theoretical physicist would tell you that, despite its counter-intuitive structure, general relativity is one of the simplest, most beautiful, and successful theories in physics, that has withstood a diverse battery of precision tests over the past century. So, is there any motivation to relax its fundamental principle, and re-introduce a gravitational aether? Here, I give …
