Linearized Einstein Field Equations Research Articles

The general set of Noetherian energy-momentum tensors in linearized gravity: mathematical framework

Energy-momentum tensors are foundational objects which are uniquely defined in standard physical field theories such as electrodynamics and Yang-mills theory. In general relativity, and in particular linearized gravity where symmetries required for an energy-momentum tensor derived from Noether’s first theorem are well defined, there exists a long standing non-uniqueness problem; numerous distinct energy-momentum expressions exist in the literature, and there is not consensus which, if any, is the unique expressions for the theory. Recently, the viability of the superpotential ‘improvement’ method was shown to be insufficient for addressing the non-uniqueness problem of energy-momentum tensors in linearized gravity. In the present article, the mathematical framework for the general set of Noetherian energy-momentum tensors in linearized gravity is derived using Noether’s first theorem, which consists of all possible energy-momentum tensors from the Noether current which yield the linearized Einstein field equations in the corresponding Euler-Lagrange equation of the Noether identity without introducing any ‘improvement’ terms. This result has several advantages in addition to not requiring ‘improvements’, such as the ability to impact the Lagrangian proportional piece of the energy-momentum tensor. Numerous common published gravitational energy-momentum expressions are then compared to these general results to assess which can be classified as Noetherian, and which cannot. Standard physical criteria such as symmetry and tracelessness are then used to prove that it is possible to directly obtain an energy-momentum tensor which is simultaneously symmetric and traceless from the Noether current; such an expression is derived from the general results. Consequences of these results and their relation to the aforementioned non-uniqueness problem are discussed.

Gravitational Perturbations of Rotating Black Holes in Lorenz Gauge.

Perturbations of Kerr spacetime are typically studied with the Teukolsky formalism, in which a pair of gauge invariant components of the perturbed Weyl tensor are expressed in terms of separable modes that satisfy ordinary differential equations. However, for certain applications it is desirable to construct the full metric perturbation in the Lorenz gauge, in which the linearized Einstein field equations take a manifestly hyperbolic form. Here we obtain a set of Lorenz-gauge solutions to the linearized vacuum field equations on Kerr-Newman-Unti-Tamburino spacetimes in terms of homogeneous solutions to the spin-2, spin-1, and spin-0 Teukolsky equations. We also derive Lorenz-gauge completion pieces representing mass and angular momentum perturbations of Kerr spacetime.

Gravitational wave cosmology: High frequency approximation

In this paper, we systematically study gravitational waves (GWs) produced by remote compact astrophysical sources. To describe such GWs properly, we introduce three scales, $\lambda, \; L_c$ and $L$, denoting, respectively, the typical wavelength of GWs, the scale of the cosmological perturbations, and the size of the observable universe. For GWs to be detected by the current and foreseeable detectors, the condition $\lambda \ll L_c \ll L$ holds, and such GWs can be well approximated as high-frequency GWs. In order for the backreaction of the GWs to the background to be negligible, we must assume that $\left|h_{\mu\nu}\right| \ll 1$, in addition to the condition $\epsilon \ll 1$, which are also the conditions for the linearized Einstein field equations for $h_{\mu\nu}$ to be valid, where $g_{\mu\nu} = \gamma_{\mu\nu} + \epsilon h_{\mu\nu}$, and $\gamma_{\mu\nu}$ denotes the background. To simplify the field equations, we show that the spatial, traceless, and Lorentz gauge conditions can be imposed simultaneously, even when the background is not vacuum, as long as the high-frequency GW approximation is valid. However, to develop the formulas that can be applicable to as many cases as possible, we first write down explicitly the linearized Einstein field equations by imposing only the spatial gauge. Applying the general formulas together with the geometrical optics approximation to such GWs, we find that they still move along null geodesics and its polarization bi-vector is parallel-transported, even when both the cosmological scalar and tensor perturbations are present. In addition, we also calculate the gravitational integrated Sachs-Wolfe effects, whereby the dependences of the amplitude, phase and luminosity distance of the GWs on these two kinds of perturbations are read out explicitly.

Propagation of quantum gravity-modified gravitational waves on a classical FLRW spacetime

The linearized Einstein field equations provide a low-energy wave equation for the propagation of gravitational fields which may originate from a high energy source. Motivated by loop quantum gravity, we propose the polymer quantization scheme to derive the effective propagation of such waves on a classical Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. To overcome the challenge of polymer quantizing a time-dependent Hamiltonian, we rewrite such a Hamiltonian in a time-independent manner in the extended phase space, polymerize it, and then transform it back to the usual phase space. In this way we obtain a time-dependent polymer Hamiltonian for the gravitational waves. We then derive the effective equations of motion and show that (i) the form of the waves is modified, (ii) the speed of the waves depends on their frequencies, and (iii) quantum effects become more apparent as waves traverse longer distances.

The Primordial Principle of Self-Interaction

The Standard Model of Particle Physics treats four fields—the gravitational, electromagnetic, weak and strong fields. These fields are assumed to converge to a single field at the big bang, but the theory has failed to produce this convergence. Our theory proposes one primordial field and analyzes the evolution of this field. The key assumption is that only the primordial field exists—if any change is to occur, it must be based upon self-interaction, as there is nothing other than the field itself to interact with. This can be formalized as the Principle of Self-interaction and the consequences explored. I show that this leads to the linearized Einstein field equations and discuss the key ontological implications of the theory.

The Wave-Front Equation of Gravitational Signals in Classical General Relativity

In this paper the dynamical equation for propagating wave-fronts of gravitational signals in classical general relativity (GR) is determined. The work relies on the manifestly-covariant Hamilton and Hamilton–Jacobi theories underlying the Einstein field equations recently discovered (Cremaschini and Tessarotto, 2015–2019). The Hamilton–Jacobi equation obtained in this way yields a wave-front description of gravitational field dynamics. It is shown that on a suitable subset of configuration space the latter equation reduces to a Klein–Gordon type equation associated with a 4-scalar field which identifies the wave-front surface of a gravitational signal. Its physical role and mathematical interpretation are discussed. Radiation-field wave-front solutions are pointed out, proving that according to this description, gravitational wave-fronts propagate in a given background space-time as waves characterized by the invariant speed-of-light c. The outcome is independent of the actual shape of the same wave-fronts and includes the case of gravitational waves which are characterized by an eikonal representation and propagate in a generic curved space-time along a null geodetics. The same waves are shown: (a) to correspond to the geometric-optics limit of the same curved space-time solutions; (b) to propagate in a flat space-time as plane waves with constant amplitude; (c) to display also the corresponding form of the wave-front in curved space-time. The result is consistent with the theory of the linearized Einstein field equations and the existence of gravitational waves achieved in such an asymptotic regime. Consistency with the non-linear Trautman boundary-value theory is also displayed.

The gravitational magnetic component and its magnetic effects in linearized theory of gravity

According to the General Relativity (GR), under the approximation of the weak field, the weak gravitational field not only contains the classical Newtonian gravitational field, but also contains a gravitomagnetic (GM) field analogous to the concept of magnetic field, and the name of GM field borrows the basic idea of the magnetic field in electrodynamics. In order to study the physical properties of GM field and its associated effects, firstly, we use a similarity transformation method to decompose the gravitomagnetic component from the second-order tensor field in linearized Einstein field equation and define the concept of the GM field. Then, on this basis, we consider a circular microtube model with uniform velocity fluids (or superfluids), and study the distribution characteristics of the far GM field by this special model. We use a simpler approximation approach to improve the previous calculation method of the GM field in these kind of rings model, the result shows that the distribution characteristic of the far GM field in this model is analogous to the magnetic field produced by a dipole, it is a good correspondence between this microtube model and the dipole model. After that, we have studied the dynamic characteristics of GM field by analogizing the properties of magnetic field, and studied the test particles’ tracks in the linear time-varying GM field and the cosine time-varying GM field in the first time. In addition, in order to improve the previous research methods of ‘gravitational induction’ and ‘frame dragging’ in the GR, we have designed a circular microtube model which has a double-layer structure and with the accelerating flow fluid (or superfluids), we use simpler mathematics than before to explain these two effects by this special model. In conclusion, this work provides some new methods for the study of GM field and its associated effects.

Rainbow scattering of gravitational plane waves by a compact body

We study the time-independent scattering of a planar gravitational wave propagating in the curved spacetime of a compact body with a polytropic equation of state. We begin by considering the geometric-optics limit, in which the gravitational wave propagates along null geodesics of the spacetime; we show that a wavefront passing through a neutron star of tenuity $R/M = 6$ will be focussed at a cusp caustic near the star's surface. Next, using the linearized Einstein Field Equations on a spherically-symmetric spacetime, we construct the metric perturbations in the odd and even parity sectors; and, with partial-wave methods, we numerically compute the gravitational scattering cross section from helicity-conserving and helicity-reversing amplitudes. At long wavelengths, the cross section is insensitive to stellar structure and, in the limit $M \omega \rightarrow 0$, it reduces to the known low-frequency approximation of the black hole case. At higher frequencies $M \omega \gtrsim 1$, the gravitational wave probes the internal structure of the body. In essence, we find that the gravitational wave cross section is similar to that for a massless scalar field, although with subtle effects arising from the non-zero helicity-reversing amplitude, and the coupling in the even-parity sector between the gravitational wave and the fluid of the body. The cross section exhibits \emph{rainbow scattering} with an Airy-type oscillation superposed on a Rutherford cross section. We show that the rainbow angle, which arises from a stationary point in the geodesic deflection function, depends on the polytropic index. In principle, rainbow scattering provides a diagnostic of the equation of state of the compact body; but, in practice, this requires a high-frequency astrophysical source of gravitational waves.

Gravitational waves — A review on the theoretical foundations of gravitational radiation

In this paper, we review the theoretical foundations of gravitational waves in the framework of Albert Einstein’s theory of general relativity. Following Einstein’s early efforts, we first derive the linearized Einstein field equations and work out the corresponding gravitational wave equation. Moreover, we present the gravitational potentials in the far away wave zone field point approximation obtained from the relaxed Einstein field equations. We close this review by taking a closer look on the radiative losses of gravitating [Formula: see text]-body systems and present some aspects of the current interferometric gravitational waves detectors. Each section has a separate appendix contribution where further computational details are displayed. To conclude, we summarize the main results and present a brief outlook in terms of current ongoing efforts to build a spaced-based gravitational wave observatory.

Gravity and the Spin-2 Planar Schrödinger Equation

A Schrödinger equation proposed for the Girvin-MacDonald-Platzman gapped spin-2 mode of fractional quantum Hall states is found from a novel nonrelativistic limit, applicable only in 2+1 dimensions, of the massive spin-2 Fierz-Pauli field equations. It is also found from a novel null reduction of the linearized Einstein field equations in 3+1 dimensions, and in this context a uniform distribution of spin-2 particles implies, via a Brinkmann-wave solution of the nonlinear Einstein equations, a confining harmonic oscillator potential for the individual particles.

SCHWARZSCHILD SOLUTION OF THE MODIFIED EINSTEIN FIELD EQUATIONS

A reformulation of the Schwarzschild solution of the linearized Einstein field equations in post-Riemannian Finsler spacetime is derived. The solution is constructed in three stages: the exterior solution, the event-horizon solution and the interior solution. It is shown that the exterior solution is asymptotically similar to Newtonian gravity at large distances implying that Newtonian gravity is a low energy approximation of the solution. Application of Eddington-Finklestein coordinates is shown to reproduce the results obtained from standard general relativity at the event horizon. Further application of Kruskal-Szekeres coordinates reveals that the interior solution contains maximally extensible geodesics.

Simplified derivation of the gravitational wave stress tensor from the linearized Einstein field equations

A conserved stress energy tensor for weak field gravitational waves propagating in vacuum is derived directly from the linearized general relativistic wave equation alone, for an arbitrary gauge. In any harmonic gauge, the form of the tensor leads directly to the classical expression for the outgoing wave energy. The method described here, however, is a much simpler, shorter, and more physically motivated approach than is the customary procedure, which involves a lengthy and cumbersome second-order (in wave-amplitude) calculation starting with the Einstein tensor. Our method has the added advantage of exhibiting the direct coupling between the outgoing wave energy flux and the work done by the gravitational field on the sources. For nonharmonic gauges, the directly derived wave stress tensor has an apparent index asymmetry. This coordinate artifact may be straightforwardly removed, and the symmetrized (still gauge-invariant) tensor then takes on its widely used form. Angular momentum conservation follows immediately. For any harmonic gauge, however, the stress tensor found is manifestly symmetric from the start, and its derivation depends, in its entirety, on the structure of the linearized wave equation.

Creation of quantized particles, gravitons, and scalar perturbations by the expanding universe

Quantum creation processes during the very rapid early expansion of the universe are believed to give rise to temperature anisotropies and polarization patterns in the CMB radiation. These have been observed by satellites such as COBE, WMAP, and PLANCK, and by bolometric instruments placed near the South Pole by the BICEP collaborations. The expected temperature anisotropies are well-confirmed. The B-mode polarization patterns in the CMB are currently under measurement jointly by the PLANCK and BICEP groups to determine the extent to which the B-modes can be attributed to gravitational waves from the creation of gravitons in the earliest universe.As the original discoverer of the quantum phenomenon of particle creation from vacuum by the expansion of the universe, I will explain how the discovery came about and how it relates to the current observations. The first system that I considered when I started my Ph.D. thesis in 1962 was the quantized minimally-coupled scalar field in an expanding FLRW (Friedmann, Lemaitré, Robertson, Walker) universe having a general continuous scale factor a(t) with continuous time derivatives. I also considered quantized fermion fields of spin-1/2 and the spin-1 massless photon field, as well as the quantized conformally-invariant field equations of arbitrary integer and half-integer spins that had been written down in the classical context for general gravitational metrics by Penrose.It was during 1962 that I proved that quanta of the minimally-coupled scalar field were created by the general expanding FLRW universe. This was relevant also to the creation of quantized perturbations of the gravitational field, since these perturbations satisfied linear field equations that could be quantized in the same way as the minimally-coupled scalar field equation. In fact, in 1946, E.M. Lifshitz had considered the classical Einstein gravitational field in FLRW expanding universes and had shown that the classical linearized Einstein field equations reduced, in what is now known as the Lifshitz gauge, to two separate classical minimally-coupled massless scalar field equations. These field equations of Lifshitz, when quantized, correspond to the field equations for massless gravitons, one equation for each of the two independent polarization components of the spin-2 massless graviton. I will discuss this further in this article.

New exact solutions of quadratic curvature gravity

It is a known fact that the Kerr-Schild type solutions in general relativity satisfy both exact and linearized Einstein field equations. We show that this property remains valid also for a special class of the Kerr-Schild metrics in arbitrary dimensions in generic quadratic curvature theory. In addition to the anti-de Sitter (AdS) wave (or Siklos) metric which represents plane waves in an AdS background, we present here a new exact solution, in this class, to the quadratic gravity in $D$ dimensions which represents a spherical wave in an AdS background. The solution is a special case of the Kundt metrics belonging to spacetimes with constant curvature invariants.

Gravitational spectral shift caused by Casimir stresses

The linearized Einstein field equations with the renormalized stress tensor of a massless quantum scalar field as source are solved in the 4-dimensional spacetime near an infinite plane boundary. The motion of particles and light is investigated disclosing a variety of effects. For instance, it is shown that light rays initially parallel to the boundary do not bend. On the other hand, under the same initial conditions, particles will be affected by the presence of the quantum field, being accelerated toward the plane boundary or departing from it depending on the value of the curvature coupling parameter $\ensuremath{\xi}$. It is suggested that $\ensuremath{\xi}$ can be set by measuring the spectral deviation of a light ray travelling perpendicularly to the boundary.

Localization of Gravity in Brane World with Arbitrary Extra Dimensions

We study the induced 4-dimensional linearized Einstein field equations in an m-dimensional bulk space by means of a confining potential. We used the confining potential in this model to localized gravitons on the brane. It is shown that in this approach the mass of graviton is quantized. The cosmological constant problem is also addressed within the context of this approach. We show that the difference between the values of the cosmological constant in particle physics and cosmology stems from our measurements in two different scales, small and large.

Gravitomagnetic effects of a massive and slowly rotating sphere with an equatorial mass current on orbiting test particles

AbstractWithin the framework of linearized Einstein field equations we compute the gravitomagnetic effects on a test particle orbiting a slowly rotating, spherical body with a rotating matter ring fixed to the equatorial plane. Our results show that the effect on the precession of particle orbits is increased by the presence of the ring.

Stress-energy tensor of a quark moving through a strongly-coupledN=4supersymmetric Yang-Mills plasma: Comparing hydrodynamics and AdS/CFT duality

The stress-energy tensor of a quark moving through a strongly-coupled $\mathcal{N}=4$ supersymmetric Yang-Mills plasma, at large ${N}_{\mathrm{c}}$, is evaluated using gauge/string duality. The accuracy with which the resulting wake, in position space, is reproduced by hydrodynamics is examined. Remarkable agreement is found between hydrodynamics and the complete result down to distances less than $2/T$ away from the quark. In performing the gravitational analysis, we use a relatively simple formulation of the bulk to boundary problem in which the linearized Einstein field equations are fully decoupled. Our analysis easily generalizes to other sources in the bulk.

A solution of linearized Einstein field equations in vacuum used for the detection of the stochastic background of gravitational waves

A solution of linearized Einstein field equations in vacuum is given and discussed. First it is shown that, computing from our particular metric the linearized connections, the linearized Riemann tensor and the linearized Ricci tensor, the linearized Ricci tensor results equal to zero. Then the effect on test masses of our solution, which is a gravitational wave, is discussed. In our solution test masses have an apparent motion in the direction of propagation of the wave, while in the transverse direction they appear at rest. In this way it is possible to think that gravitational waves would be longitudinal waves, but, from careful investigation of this solution, it is shown that the tidal forces associated with gravitational waves act along the directions orthogonal to the direction of propagation of waves. The computation is first made in the long wavelengths approximation (wavelength much larger than the linear distances between test masses), then the analysis is generalized to all gravitational waves. In the last sections of this paper it is shown that the frequency dependent angular pattern of interferometers can be obtained from our solution and the total signal seen from an interferometer for the stochastic background of gravitational waves is computed.

Gravitational modification of the Berry phase and some other effects induced by high-power lasers

We study some effects when a probe light pulse propagates in the spacetime curved by the gravitational field of high-power laser pulses. The gravitational field produced by a high-power laser is calculated by using the linearized Einstein field equation under the weak field approximation. It is found that the gravitational modification of the photon Berry phase occurs when the probe light pulse transports in a uniform winding fiber in this gravitational background. We also find the gravitational redshifting and defocusing for the probe pulse caused by the gravitational field of the high-power lasers.