Linearized Einstein Equations Research Articles

Stability off(R)black holes

We investigate the stability of $f(R)$ (Schwarzschild) black hole obtained from the $f(R)$ gravity. It is difficult to carry out the perturbation analysis around the black hole because the linearized Einstein equation is fourth order in $f(R)$ gravity. In order to resolve this difficulty, we transform $f(R)$ gravity into the scalar-tensor theory by introducing two auxiliary scalars. In this case, the linearized curvature scalar becomes a scalaron, showing that all linearized equations are second order, which are the same equations for the massive Brans-Dicke theory. It turns out that the $f(R)$ black hole is stable against the external perturbations if the scalaron does not have a tachyonic mass.

Modes of log gravity

The physical modes of a recently proposed $D$-dimensional ``critical gravity'', linearized about its anti-de Sitter vacuum, are investigated. All ``log mode'' solutions, which we categorize as ``spin-2'' or ``Proca'', arise as limits of the massive spin-2 modes of the noncritical theory. The linearized Einstein tensor of a spin-2 log mode is itself a `nongauge' solution of the linearized Einstein equations whereas the linearized Einstein tensor of a Proca mode takes the form of a linearized general coordinate transformation. Our results suggest the existence of a holographically dual logarithmic conformal field theory.

Scalar contribution to the graviton self-energy during inflation

We use dimensional regularization to evaluate the one loop contribution to the graviton self-energy from a massless, minimally coupled scalar on a locally de Sitter background. For noncoincident points our result agrees with the stress tensor correlators obtained recently by Perez-Nadal, Roura, and Verdaguer. We absorb the ultraviolet divergences using the ${R}^{2}$ and ${C}^{2}$ counterterms first derived by 't Hooft and Veltman, and we take the $D=4$ limit of the finite remainder. The renormalized result is expressed as the sum of two transverse, fourth order differential operators acting on nonlocal, de Sitter invariant structure functions. In this form it can be used to quantum correct the linearized Einstein equations so that one can study how the inflationary production of infrared scalars affects the propagation of dynamical gravitons and the force of gravity.

Non-adiabatic primordial fluctuations

We consider general mixtures of isocurvature and adiabatic cosmological perturbations. With a minimal assumption set consisting of the linearized Einstein equations and a primordial perfect fluid we derive the second-order action and its curvature variables. We also allow for varying equation of state and speed of sound profiles. The derivation is therefore carried out at the same level of generality that has been achieved for adiabatic modes before. As a result we find a new conserved super-horizon quantity and relate it to the adiabatically conserved curvature perturbation. Finally we demonstrate how the formalism can be applied by considering a Chaplygin gas-like primordial matter model, finding two scale-invariant solutions for structure formation.

Solving the effective field equations for the Newtonian potential

Loop corrections to the gravitational potential are usually inferred from scattering amplitudes, which seems quite different from how the linearized Einstein equations are solved with a static, point mass to give the classical potential. In this study we show how the Schwinger–Keldysh effective field equations can be used to compute loop corrections to the potential in a way which parallels the classical treatment. We derive explicit results for the one-loop correction from the graviton self-energy induced by a massless, minimally coupled scalar.

A Possible Reinterpretation of Einstein’s Equations

In this paper, we first review Huei’s formulation in which it is shown that the linearized Einstein equations can be written in the same form as the Maxwell equations. We eliminate some imperfections like the scalar potential which is ill linked to the electric-type field, the Lorentz-type force which is obtained with a time independence restriction and the undesired factor 4 which appears in the magnetic-type part. Second, from these results and in the light of a recent work by C.C. Barros, we propose an extension of the equivalence principle and we suggest a new interpretation for Einstein’s equations by showing that the electromagnetic Maxwell equations can be derived from a new version of Einstein’s ones.

On the energy transported by exact plane gravitational-wave solutions

The energy and momentum transported by exact plane gravitational-wave solutions of Einstein equations are computed using the teleparallel equivalent formulation of Einstein's theory. It is shown that these waves transport neither energy nor momentum. A comparison with the usual linear plane gravitational-waves solution of the linearized Einstein equation is presented.

The initial value problem for linearized gravitational perturbations of the Schwarzschild naked singularity

The coupled equations for the scalar modes of the linearized Einstein equations around Schwarzschild's spacetime were reduced by Zerilli to a (1+1) wave equation , where is the Zerilli ‘Hamiltonian’ and x is the tortoise radial coordinate. From its definition, for smooth metric perturbations the field Ψz is singular at rs = −6M/(ℓ − 1)(ℓ +2), with ℓ being the mode harmonic number. The equation Ψz obeys is also singular, since V has a second-order pole at rs. This is irrelevant to the black hole exterior stability problem, where r > 2M > 0, and rs < 0, but it introduces a non-trivial problem in the naked singular case where M < 0, then rs > 0, and the singularity appears in the relevant range of r (0 < r < ∞). We solve this problem by developing a new approach to the evolution of the even mode, based on a new gauge invariant function, , that is a regular function of the metric perturbation for any value of M. The relation of to Ψz is provided by an intertwiner operator. The spatial pieces of the (1 + 1) wave equations that and Ψz obey are related as a supersymmetric pair of quantum Hamiltonians and . For has a regular potential and a unique self-adjoint extension in a domain defined by a physically motivated boundary condition at r = 0. This allows us to address the issue of evolution of gravitational perturbations in this non-globally hyperbolic background. This formulation is used to complete the proof of the linear instability of the Schwarzschild naked singularity, by showing that a previously found unstable mode belongs to a complete basis of in , and thus is excitable by generic initial data. This is further illustrated by numerically solving the linearized equations for suitably chosen initial data.

Gravity from spontaneous Lorentz violation

We investigate a class of theories involving a symmetric two-tensor field in Minkowski spacetime with a potential triggering spontaneous violation of Lorentz symmetry. The resulting massless Nambu-Goldstone modes are shown to obey the linearized Einstein equations in a fixed gauge. Imposing self-consistent coupling to the energy-momentum tensor constrains the potential for the Lorentz violation. The nonlinear theory generated from the self-consistent bootstrap is an alternative theory of gravity, containing kinetic and potential terms along with a matter coupling. At energies small compared to the Planck scale, the theory contains general relativity, with the Riemann-spacetime metric constructed as a combination of the two-tensor field and the Minkowski metric. At high energies, the structure of the theory is qualitatively different from general relativity. Observable effects can arise in suitable gravitational experiments.

Dangers of using the growth equation on large scales in the Newtonian gauge

We examine the accuracy of the growth equation {delta}+2H{delta}-4{pi}G{rho}{delta}=0, which is ubiquitous in the cosmological literature, in the context of the Newtonian gauge. By comparing the growth predicted by this equation to a numerical solution of the linearized Einstein equations in the {lambda}CDM scenario, we show that while this equation is a reliable approximation on small scales (k > or approx. h Mpc{sup -1}), it can be disastrously inaccurate ({approx}10{sup 4}%) on larger scales in this gauge. We propose a modified version of the growth equation for the Newtonian gauge, which while preserving the simplicity of the original equation, provides considerably more accurate results. We examine the implications of the failure of the growth equation on a few recent studies, aimed at discriminating general relativity from modified gravity, which use this equation as a starting point. We show that while the results of these studies are valid on small scales, they are not reliable on large scales or high redshifts, if one works in the Newtonian gauge. Finally, we discuss the growth equation in the synchronous gauge and show that the corrections to the Poisson equation are exactly equivalent to the difference between the overdensities in the synchronous and Newtonianmore » gauges.« less

Perturbations of spacetime around a stationary rotating cosmic string

We consider the metric perturbations around a stationary rotating Nambu-Goto string in Minkowski spacetime. By solving the linearized Einstein equations, we study the effects of azimuthal frame-dragging around the rotation axis and linear frame-dragging along the rotation axis, the Newtonian logarithmic potential, and the angular deficit around the string as the potential mode. We also investigate gravitational waves propagating off the string and propagating along the string, and show that the stationary rotating string emits gravitational waves toward the directions specified by discrete angles from the rotation axis. Waveforms, polarizations, and amplitudes which depend on the direction are shown explicitly.

Explicit solution of the linearized Einstein equations in the transverse-traceless gauge for all multipoles

We write out the explicit form of the metric for a linearized gravitational wave in the transverse-traceless gauge for any multipole, thus generalizing the well-known quadrupole solution of Teukolsky. The solution is derived using the generalized Regge–Wheeler–Zerilli formalism developed by Sarbach and Tiglio.

A rigorous derivation of gravitational self-force

There is general agreement that the MiSaTaQuWa equations should describe the motion of a ‘small body’ in general relativity, taking into account the leading order self-force effects. However, previous derivations of these equations have made a number of ad hoc assumptions and/or contain a number of unsatisfactory features. For example, all previous derivations have invoked, without proper justification, the step of ‘Lorenz gauge relaxation’, wherein the linearized Einstein equation is written in the form appropriate to the Lorenz gauge, but the Lorenz gauge condition is then not imposed—thereby making the resulting equations for the metric perturbation inequivalent to the linearized Einstein equations. (Such a ‘relaxation’ of the linearized Einstein equations is essential in order to avoid the conclusion that ‘point particles’ move on geodesics.) In this paper, we analyze the issue of ‘particle motion’ in general relativity in a systematic and rigorous way by considering a one-parameter family of metrics, gab(λ), corresponding to having a body (or black hole) that is ‘scaled down’ to zero size and mass in an appropriate manner. We prove that the limiting worldline of such a one-parameter family must be a geodesic of the background metric, gab(λ = 0). Gravitational self-force—as well as the force due to coupling of the spin of the body to curvature—then arises as a first-order perturbative correction in λ to this worldline. No assumptions are made in our analysis apart from the smoothness and limit properties of the one-parameter family of metrics, gab(λ). Our approach should provide a framework for systematically calculating higher order corrections to gravitational self-force, including higher multipole effects, although we do not attempt to go beyond first-order calculations here. The status of the MiSaTaQuWa equations is explained.

Spherically symmetric perturbations of spatially flat Friedmann models with imperfect fluid

A solution to the linearized Einstein equations is obtained for spherically symmetric perturbations of an imperfect fluid in a spatially flat Friedman model. The fluid density and pressure are assumed to be related by a linear equation of state. We consider perturbations with some spatial configuration which is nonsingular at the point r = 0. The instability problem for gravitational perturbations is studied, and solution are obtained depending on the functional form of the bulk and shear viscosity.

Can rigidly rotating polytropes be sources of the Kerr metric?

We use a recent result by Cabezas et al (2007 Gen. Rel. Grav. 39 707) to build up an approximate solution to the gravitational field created by a rigidly rotating polytrope. We solve the linearized Einstein equations inside and outside the surface of zero pressure including second-order corrections due to rotational motion to get an asymptotically flat metric in a global harmonic coordinate system. We prove that if the metric and their first derivatives are continuous on the matching surface up to this order of approximation, the multipole moments of this metric cannot be fitted to those of the Kerr metric.

Toward the AdS/CFT gravity dual for high energy collisions. II. The stress tensor on the boundary

In this second paper of the series, we calculate the stress tensor of excited matter, created by debris of high energy collisions at the boundary. The falling open strings, connected to receding charges, produce a nonzero stress tensor which we found analytically from time-dependent linearized Einstein equations in the bulk. It corresponds to exploding nonequilibrium matter: we discuss its behavior in some detail, including its internal energy density in a comoving frame and the 'freeze-out surfaces'. We then discuss what happens for the ensemble of strings.

Vacuum stress–energy density and its gravitational implications**This paper reports work in progress on a project combining researchers in Texas, Louisiana and Oklahoma. It is supported by NSF Grants PHY-0554849 and PHY-0554926.

In nongravitational physics the local density of energy is often regarded as merely a bookkeeping device; only total energy has an experimental meaning—and it is only modulo a constant term. But in general relativity the local stress–energy tensor is the source term in Einstein's equation. In closed universes, and those with Kaluza–Klein dimensions, theoretical consistency demands that quantum vacuum energy should exist and have gravitational effects, although there are no boundary materials giving rise to that energy by van der Waals interactions. In the lab there are boundaries, and in general the energy density has a nonintegrable singularity as a boundary is approached (for idealized boundary conditions). As pointed out long ago by Candelas and Deutsch, in this situation there is doubt about the viability of the semiclassical Einstein equation. Our goal is to show that the divergences in the linearized Einstein equation can be renormalized to yield a plausible approximation to the finite theory that presumably exists for realistic boundary conditions. For a scalar field with Dirichlet or Neumann boundary conditions inside a rectangular parallelepiped, we have calculated by the method of images all components of the stress tensor, for all values of the conformal coupling parameter and an exponential ultraviolet cutoff parameter. The qualitative features of contributions from various classes of closed classical paths are noted. Then the Estrada–Kanwal distributional theory of asymptotics, particularly the moment expansion, is used to show that the linearized Einstein equation with the stress–energy near a plane boundary as source converges to a consistent theory when the cutoff is removed.

Stress tensor of static dipoles in strongly coupledN=4gauge theory

In the context of the AdS/CFT correspondence we calculate the induced stress tensor of static dipoles (electric-electric and electric-magnetic) in a strongly coupled $\mathcal{N}=4$ SYM gauge theory, by solving the linearized Einstein equation with Maldecena string as a source. Analytic expressions are given for the far-field and a near-field close to one charge, and compared to what one has in weak coupling. The result can be compared to lattice results for QCD-like theories in a deconfined but strongly coupled regime.

Gravitational bending of light by planetary multipoles and its measurement with microarcsecond astronomical interferometers

AbstractGeneral relativistic deflection of light by mass, dipole, and quadrupole moments of gravitational field of a moving massive planet in the Solar system is derived in the approximation of the linearized Einstein equations. All terms of the order of 1 μas and larger are taken into account, parameterized, and classified in accordance with their physical origin. We discuss the observational capabilities of the near-future optical and radio interferometers for detecting the Doppler modulation of the radial deflection, and the dipolar and quadrupolar light-ray bending by Jupiter and Saturn.

Graviton localization and Newton’s law for brane models with a nonminimally coupled bulk scalar field

Brane world models with a nonminimally coupled bulk scalar field have been studied recently. In this paper we consider metric fluctuations around an arbitrary gravity-scalar background solution, and we show that the corresponding spectrum includes a localized zero mode which strongly depends on the profile of the background scalar field. For a special class of solutions, with a warp factor of the RS form, we solve the linearized Einstein equations, for a pointlike mass source on the brane, by using the brane bending formalism. We see that general relativity on the brane is recovered only if we impose restrictions on the parameter space of the models under consideration.