In the context of Einstein-aether scalar field cosmology we solve the field equations and determine exact and analytic solutions. In particular, we consider a model proposed by Kanno and Soda where the aether and the scalar fields interact through the aether coefficient parameters, which are promoted to be functions of the scalar field. For this model, we write the field equations by using the minisuperspace approach and we determine the scalar field potentials which leads to Liouville--integrable systems. We solve the field equations for five families of scalar field potentials and, whether it is feasible, we write down the analytic solutions by using closed-form functions.
We study the Einstein-aether theory in Weyl integrable geometry. The scalar field which defines the Weyl affine connection is introduced in the gravitational field equation. We end up with an Einstein-aether scalar field model where the interaction between the scalar field and the aether field has a geometric origin. The scalar field plays a significant role in the evolution of the gravitational field equations. We focus our study on the case of homogeneous and isotropic background spacetimes and study their dynamical evolution for various cosmological models.
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.
Most early twentieth century relativists --- Lorentz, Einstein, Eddington, for examples --- claimed that general relativity was merely a theory of the aether. We shall confirm this claim by deriving the Einstein equations using aether theory. We shall use a combination of Lorentz's and Kelvin's conception of the aether. Our derivation of the Einstein equations will not use the vanishing of the covariant divergence of the stress-energy tensor, but instead equate the Ricci tensor to the sum of the usual stress-energy tensor and a stress-energy tensor for the aether, a tensor based on Kelvin's ae…
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…
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 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 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…
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.
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 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 determine exact and analytic solutions of the gravitational field equations in Einstein-aether scalar model field with a Bianchi I background space. In particular, we consider nonlinear interactions of the scalar field with the aether field. For the model under consideration we can write the field equations by using the minisuperspace description. The point-like Lagrangian of the field equations depends on three unknown functions. We derive conservation laws for the field equations for specific forms of the unknown functions such that the field equations are Liouville integrable. Furthermor…
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…
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…
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 $F(K)$…
We study the cosmological evolution of the field equations in the context of Einstein-Aether cosmology by including a scalar field in a spatially flat Friedmann--Lema\^ıtre--Robertson--Walker spacetime. Our analysis is separated into two separate where a pressureless fluid source is included or absent. In particular, we determine the critical points of the field equations and we study the stability of the specific solutions. The limit of general relativity is fully recovered, while the dynamical system admits de Sitter solutions which can describe the past inflationary era and the future late-…
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.
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…
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.
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 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.
Here we briefly discuss the Einstein-Aether gravity theory by modification of Einstein-Hilbert action. We find the modified Friedmann equations and then from the equations we find the effective density and pressure for Einstein-Aether gravity sector. These can be treated as dark energy provided some restrictions on the free function $F(K)$, where $K$ is proportional to $H^{2}$. Assuming two types of the power law solutions of the scale factor, we have reconstructed the unknown function $F(K)$ from HDE and NADE and their entropy-corrected versions (ECHDE and ECNADE). We also obtain the EoS para…
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…
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 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.
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.
We present the solution space of the field equations in the Einstein-aether theory for the case of a vacuum Bianchi Type V space-time. We also find that there are portions of the initial parameters space for which no solution is admitted by the reduced equations. Whenever solutions do exist, their physical interpretation is examined through the behavior of Ricci and/or Kretsmann scalar, as well as with the identification of the effective energy momentum tensor in terms of a perfect fluid. There are cases in which no singularities appear and others where the effective fluid is isotropic.
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 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 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…
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.
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.
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 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…
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 $c_{1}-c_{3}=1$.
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 …
Einstein-aether theory is extended by allowing for spinning degrees of freedom of the aether. In addition to the acceleration, shear, expansion, and vorticity of the aether velocity field, a spin rotation describing the dynamics of a classical intrinsic angular momentum of the aether is introduced as a kinematic quantity. The action of Einstein-aether theory is augmented by a term quadratic in the spin rotation and by coupling terms with the vorticity and the acceleration. Besides breaking the Lorentz boost invariance, the theory breaks the invariance under spatial rotations in the direction o…
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…
The so-called Einstein-Aether theory is General Relativity coupled (at second derivative order) to a dynamical unit time-like vector field (the aether). It is a Lorentz-violating theory, and gained much attention in the recent years. In the present work, we study the cosmological evolution of Einstein-Aether models with power-law-like potential, by using the method of dynamical system. In the case without matter, there are two attractors which correspond to an inflationary universe in the early epoch, or a de Sitter universe in the late time. In the case with matter but there is no interaction…
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 develop a new method for calculation of quasi-normal modes of black holes, when the effective potential, which governs black hole perturbations, is known only numerically in some region near the black hole. This method can be applied to perturbations of a wide class of numerical black hole solutions. We apply it to the black holes in the Einstein-Aether theory, a theory where general relativity is coupled to a unit time-like vector field, in order to observe local Lorentz symmetry violation. We found that in the non-reduced Einstein-Aether theory, real oscillation frequency and damping rate…
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.
We consider an Einstein-aether scalar field cosmological model where the aether and the scalar field are interacting. The model of our consideration consists the two different interacting models proposed in the literature by Kanno et al. and by Donnelly et al. We perform an extended analysis for the cosmological evolution as it is provided by the field equations by using methods from dynamical systems; specifically, we determine the stationary points and we perform the stability analysis of those exact solutions.
Inflationary spatially homogeneous cosmological models within an Einstein-Aether gravitational framework are investigated. The matter source is assumed to be a scalar field which is coupled to the aether field expansion and shear scalars through the generalized harmonic scalar field potential. The evolution equations are expressed in terms of expansion-normalized variables to produce an autonomous system of ordinary differential equations suitable for numerical and local stability analysis. An analysis of the local stability of the equilibrium points indicate that there exists a range of value…
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 this paper, second post-Newtonian approximation of Einstein-aether theory is obtained by Chandrasekhar's approach. Five parameterized post-Newtonian parameters in first post-Newtonian approximation are presented after a time transformation and they are identical with previous works, in which $γ=1$, $β=1$ and two preferred-frame parameters remain. Meanwhile, in second post-Newtonian approximation, a parameter, which represents third order nonlinearity for gravity, is zero the same as in general relativity. For an application for future deep space laser ranging missions, we reduce the metric …
`Einstein-Aether' theory, in which gravity couples to a dynamical, time-like, unit-norm vector field, provides a means for studying Lorentz violation in a generally covariant setting. Demonstrated here is the effect of a redefinition of the metric and vector field in terms of the original fields and two free parameters. The net effect is a change of the coupling constants appearing in the action. Using such a redefinition, one of the coupling constants can be set to zero, simplifying studies of solutions of the theory.
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…
A Covariant Phase Space Approach to Einstein-AEther Gravity
SISSA–International School for Advanced Studies（国际高等研究院）；INFN, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；IFPU–Institute for Fundamental Physics of the Universe（宇宙基础物理研究所）
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…
