Einstein Tensor Research Articles

Dynamical evolution of a scalar field coupling to Einstein’s tensor in the Reissner-Nordström black hole spacetime

We study the dynamical evolution of a scalar field coupling to Einstein's tensor in the background of a Reissner-Nordstr\"om black hole. Our results show that the coupling constant $\ensuremath{\eta}$ imprints in the wave dynamics of a scalar perturbation. In the weak coupling, we find that with the increase of the coupling constant $\ensuremath{\eta}$ the real parts of the fundamental quasinormal frequencies decrease and the absolute values of imaginary parts increase for fixed charge $q$ and multipole number $l$. In the strong coupling, we find that for $l\ensuremath{\ne}0$ the instability occurs when $\ensuremath{\eta}$ is larger than a certain threshold value ${\ensuremath{\eta}}_{c}$ which deceases with the multipole number $l$ and charge $q$. However, for the lowest $l=0$, we find that there does not exist such a threshold value and the scalar field always decays for arbitrary coupling constant.

Killing vector fields in three dimensions: a method to solve massive gravity field equations

Killing vector fields in three dimensions play an important role in the construction of the related spacetime geometry. In this work we show that when a three-dimensional geometry admits a Killing vector field then the Ricci tensor of the geometry is determined in terms of the Killing vector field and its scalars. In this way we can generate all products and covariant derivatives at any order of the Ricci tensor. Using this property we give ways to solve the field equations of topologically massive gravity (TMG) and new massive gravity (NMG) introduced recently. In particular when the scalars of the Killing vector field (timelike, spacelike and null cases) are constants then all three-dimensional symmetric tensors of the geometry, the Ricci and Einstein tensors, their covariant derivatives at all orders, and their products of all orders are completely determined by the Killing vector field and the metric. Hence, the corresponding three-dimensional metrics are strong candidates for solving all higher derivative gravitational field equations in three dimensions.

Greybody factor for a scalar field coupling to Einstein's tensor

We study the greybody factor and Hawking radiation for a scalar field coupling to Einstein's tensor in the background of Reissner–Nordström black hole in the low-energy approximation. We find that the presence of the coupling terms modifies the standard results in the greybody factor and Hawking radiation. Our results show that both the absorption probability and Hawking radiation increase with the coupling constant. Moreover, we also find that for the stronger coupling, the charge of black hole enhances the absorption probability and Hawking radiation of the scalar field, which is different from those of ones without coupling to Einstein's tensor in the black hole spacetime.

When scalar field is kinetically coupled to the Einstein tensor

We explore the cosmic evolution of a scalar field with the kinetic term coupled to the Einstein tensor. We find that, in the absence of other matter sources or in the presence of only pressureless matter, the scalar behaves as pressureless matter and the sound speed of the scalar is vanishing. These properties enable the scalar field to be a candidate of cold dark matter. By also considering the scalar potential, we find the scalar field may play the role of both dark matter and dark energy. In this case, the equation of state of the scalar can cross the phantom divide, but this can lead to the sound speed becoming superluminal as it crosses the divide, and so is physically forbidden. Finally, if the kinetic term is coupled to more than one Einstein tensor, we find the equation of state is always approximately equal to -1 whether the potential is flat or not, and so the scalar may also be a candidate for the inflaton.

Cosmological perturbations in the new Higgs inflation

We study the cosmological perturbations created during the New Higgs inflationary phase. In the New Higgs Inflation, the Higgs boson is kinetically coupled to the Einstein tensor and only three perturbative degrees of freedom, a scalar and two tensorial (gravitational waves), propagate during Inflation.Scalar perturbations are found to match the latest WMAP-7yrs data within Standard Model Higgs parameters. Primordial gravitational waves also, although propagating with superluminal speed, are consistent with present data. Finally, we estimate the values of the parameter of the New Higgs Inflation in relation to the Higgs mass, the spectral index and amplitude of the primordial scalar perturbations showing that the unitarity bound of the theory is not violated.

On a Ströbel-Inspired k(t) FLRW Ansatz in a Class of Metric F(R) Models

Assuming a D≥4 dimensional FLRW (Friedmann–Lemaitre–Robertson–Walker) inspired ansatz with spatial curvature being a non-trivial function of time k(t) in a class of metric and Jordan frame F(R) gravity models, non-existence theorems for several types of sources are derived in a simple manner (using specific form of the modified gravity Einstein tensor components).

Remarks on holography with broken Lorentz invariance

Recently a family of solutions of Einstein equations in backgrounds with broken Lorentz invariance was found. We show that the gravitational solution recently obtained by Kachru et al. is a part of the former solution which was derived earlier in the framework of extra-dimensional theories. We show how the energy-momentum and Einstein tensors are related and establish a correspondence between parameters which govern Lorentz invariance violation. Then we demonstrate that scaling behavior of two point correlation functions of local operators in scalar field theory is reproduced correctly for two cases with critical values of scaling parameters. Therefore, we complete the dictionary of 'tree-level' duality for all known solutions of the bulk theory. In the end we speculate on relations between renormalization group flow of a boundary theory and asymptotic behavior of gravitational solutions in the bulk.

Practical tools for third order cosmological perturbations

We discuss cosmological perturbation theory at third order, derivingthe gauge transformation rules for metric and matter perturbations,and constructing third order gauge invariant quantities. We presentthe Einstein tensor components, the evolution equations for a perfectfluid, and the Klein-Gordon equation at third order, including scalar,vector and tensor perturbations. In doing so, we also give all secondorder tensor components and evolution equations in full generality.

Quantum behaviors on an excreting black hole

Often, geometries with horizons offer insights into the intricate relationships between general relativity and quantum physics. However, some subtle aspects of gravitating quantum systems might be difficult to ascertain using static backgrounds, since quantum mechanics incorporates dynamic measurability constraints (such as the uncertainty principle, etc). For this reason, the behaviors of quantum systems on a dynamic black hole background are explored in this paper. The velocities and trajectories of representative outgoing, ingoing and stationary classical particles are calculated and contrasted, and the dynamics of simple quantum fields (both massless and massive) on the spacetime are examined. Invariant densities associated with the quantum fields are exhibited on the Penrose diagram that represents the excreting black hole. Furthermore, a generic approach for the consistent mutual gravitation of quanta in a manner that reproduces the given geometry is developed. The dynamics of the mutually gravitating quantum fields are expressed in terms of the affine parameter that describes local motions of a given quantum type on the spacetime. Algebraic equations that relate the energy–momentum densities of the quantum fields to Einstein's tensor can then be developed. An example mutually gravitating system of macroscopically coherent quanta along with a core gravitating field is demonstrated. Since the approach is generic and algebraic, it can be used to represent a variety of systems with specified boundary conditions.

Thermodynamic structure of Lanczos-Lovelock field equations from near-horizon symmetries

It is well known that, for a wide class of spacetimes with horizons, Einstein equations near the horizon can be written as a thermodynamic identity. It is also known that the Einstein tensor acquires a highly symmetric form near static, as well as stationary, horizons. We show that, for generic static spacetimes, this highly symmetric form of the Einstein tensor leads quite naturally and generically to the interpretation of the near-horizon field equations as a thermodynamic identity. We further extend this result to generic static spacetimes in Lanczos-Lovelock gravity, and show that the near-horizon field equations again represent a thermodynamic identity in all these models. These results confirm the conjecture that this thermodynamic perspective of gravity extends far beyond Einstein's theory.

Decomposition of geometric perturbations

For an infinitesimal deformation of a Riemannian manifold, we prove that the scalar, vector, and tensor modes in decompositions of perturbations of the metric tensor, the scalar curvature, the Ricci tensor, and the Einstein tensor decouple if and only if the manifold is Einstein. As an example, a four-dimensional space–time whose three-dimensional spatial foliations satisfy the condition of the theorem is homogeneous and isotropic. Cosmological applications are discussed.

Kerr–Schild ansatz in Einstein–Gauss–Bonnet gravity: an exact vacuum solution in five dimensions

As is well known, Kerr–Schild metrics linearize the Einstein tensor. We shall see here that they also simplify the Gauss–Bonnet tensor, which turns out to be only quadratic in the arbitrary Kerr–Schild function f when the seed metric is maximally symmetric. This property allows us to give a simple analytical expression for its trace, when the seed metric is a five-dimensional maximally symmetric spacetime in spheroidal coordinates with arbitrary parameters a and b. We also write in a (fairly) simple form the full Einstein–Gauss–Bonnet tensor (with a cosmological term) when the seed metric is flat and the oblateness parameters are equal, a = b. Armed with these results we give in a compact form the solution of the trace of the Einstein–Gauss–Bonnet field equations with a cosmological term and a ≠ b. We then examine whether this solution for the trace does solve the remaining field equations. We find that it does not in general, unless the Gauss–Bonnet coupling is such that the field equations have a unique maximally symmetric solution.

Explicit form of the Mann-Marolf surface term in (3+1) dimensions

The Mann-Marolf surface term is a specific candidate for the 'reference background term' that is to be subtracted from the Gibbons-Hawking surface term in order make the total gravitational action of asymptotically flat spacetimes finite. That is, the total gravitational action is taken to be: (Einstein-Hilbert bulk term)+(Gibbons-Hawking surface term)-(Mann-Marolf surface term). As presented by Mann and Marolf, their surface term is specified implicitly in terms of the Ricci tensor of the boundary. Herein, I demonstrate that, for the physically interesting case of a (3+1)-dimensional bulk spacetime, the Mann-Marolf surface term can be specified explicitly in terms of the Einstein tensor of the (2+1)-dimensional boundary.

Generalized Vaidya spacetime in Lovelock gravity and thermodynamics on the apparent horizon

We present a kind of generalized Vaidya solutions in a generic Lovelock gravity. This solution generalizes the simple case in Gauss-Bonnet gravity reported recently by some authors. We study the thermodynamics of apparent horizon in this generalized Vaidya spacetime. Treating those terms except for the Einstein tensor as an effective energy-momentum tensor in the gravitational field equations, and using the unified first law in Einstein gravity theory, we obtain an entropy expression for the apparent horizon. We also obtain an energy expression of this spacetime, which coincides with the generalized Misner-Sharp energy proposed by Maeda and Nozawa in Lovelock gravity.

Sharp bounds on 2m/r for static spherical objects

Sharp bounds are obtained, under a variety of assumptions on the eigenvalues of the Einstein tensor, for the ratio of the Hawking mass to the areal radius in static, spherically symmetric spacetimes.

Metrics with vanishing quantum corrections

We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor Tμν(gαβ, ∂τgαβ, ∂τ∂σgαβ, …,) constructed from sums of terms, the involving contractions of the metric and powers of arbitrary covariant derivatives of the curvature tensor. A classical solution, such as an Einstein metric, is called universal if, when evaluated on that Einstein metric, Tμν is a multiple of the metric. A Ricci flat classical solution is called strongly universal if, when evaluated on that Ricci flat metric, Tμν vanishes. It is well known that pp-waves in four spacetime dimensions are strongly universal. We focus attention on a natural generalization; Einstein metrics with holonomy Sim(n − 2) in which all scalar invariants are zero or constant. In four dimensions we demonstrate that the generalized Ghanam–Thompson metric is weakly universal and that the Goldberg–Kerr metric is strongly universal; indeed, we show that universality extends to all four-dimensional Sim(2) Einstein metrics. We also discuss generalizations to higher dimensions.

Lovelock tensor as generalized Einstein tensor

We show that the splitting feature of the Einstein tensor, as the first term of the Lovelock tensor, into two parts, namely the Ricci tensor and the term proportional to the curvature scalar, with the trace relation between them is a common feature of any other homogeneous terms in the Lovelock tensor. Motivated by the principle of general invariance, we find that this property can be generalized, with the aid of a generalized trace operator which we define, for any inhomogeneous Euler-Lagrange expression that can be spanned linearly in terms of homogeneous tensors. Then, through an application of this generalized trace operator, we demonstrate that the Lovelock tensor analogizes the mathematical form of the Einstein tensor, hence, it represents a generalized Einstein tensor. Finally, we apply this technique to the scalar Gauss-Bonnet gravity as an another version of string-inspired gravity.

Black hole evaporation in a spherically symmetric non-commutative spacetime

Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat spacetime and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in non-commutative geometry have shown that, in general relativity, the effects of non-commutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy–momentum tensor as a source on the right-hand side. Relying on the recently obtained non-commutativity effect on a static, spherically symmetric metric, we have considered from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes has been shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F have been derived which are compatible with the adiabatic approximation.

On the naturalness of Einstein’s equation

We compute all 2-covariant tensors naturally constructed from a semi-Riemannian metric which are divergence-free and have weight greater than −2. As a consequence, it follows a characterization of the Einstein tensor as the only, up to a constant factor, 2-covariant tensor naturally constructed from a semi-Riemannian metric which is divergence-free and has weight 0 (i.e., is independent of the unit of scale). Since these two conditions are also satisfied by the energy–momentum tensor of a relativistic space–time, we discuss in detail how these theorems lead to the field equation of General Relativity.

Two 3-branes in Randall–Sundrum setup and current acceleration of the universe

Five-dimensional spacetimes of two orbifold 3-branes are studied, by assuming that the two 3-branes are spatially homogeneous, isotropic, and independent of time, following the so-called “bulk-based” approach. The most general form of the metric is obtained, and the corresponding field equations are divided into three groups, one is valid on each of the two 3-branes, and the third is valid in the bulk. The Einstein tensor on the 3-branes is expressed in terms of the discontinuities of the first-order derivatives of the metric coefficients. Thus, once the metric is known in the bulk, the distribution of the Einstein tensor on the two 3-branes is uniquely determined. As applications, we consider two different cases, one is in which the bulk is locally AdS 5 , and the other is where it is vacuum. In some cases, it is shown that the universe is first decelerating and then accelerating. The global structure of the bulk as well as the 3-branes is also studied, and found that in some cases the solutions may represent the collision of two orbifold 3-branes. The applications of the formulas to the studies of the cyclic universe and the cosmological constant problem are also pointed out.