Einstein Solution Research Articles

The seed-to-solution method for the Einstein constraints and the asymptotic localization problem

We establish the existence of a broad class of asymptotically Euclidean solutions to Einstein's constraint equations, whose asymptotic behavior at infinity is prescribed arbitrarily. The proposed seed-to-solution method encompasses vacuum as well as matter spaces, and relies on iterations based on the linearized Einstein operator and its dual. It generates a Riemannian manifold (with finitely many asymptotically Euclidean ends) from any seed data set consisting of(1) a Riemannian metric and a symmetric two-tensor and(2) a (density) field and a (momentum) vector field representing the matter content. We distinguish between tame or strongly tame seed data sets, depending whether the data provides a rough or an accurate asymptotic Ansatz at infinity. We encompass classes of metrics and matter fields with low decay (possibly with infinite ADM mass) or strong decay (with Schwarzschild behavior). Our analysis is motivated by Carlotto and Schoen's pioneering work on the localization problem for Einstein's vacuum equations. Dealing with metrics with low decay and, simultaneously, establishing estimates that include (and go beyond) harmonic decay require significantly new arguments which are developed in the present paper. We work in a weighted Lebesgue-Hölder framework adapted to the given seed data, and we analyze the nonlinear coupling between the Hamiltonian and momentum constraints. By establishing elliptic regularity estimates for the linearized Einstein operator and its dual, we uncover the novel notion of mass-momentum correctors which is related to the ADM mass-momentum of the manifold. We derive precise estimates for the difference between the seed data and the associate Einstein solution —a result that should be of interest for future numerical investigations. Furthermore, we introduce here and study the asymptotic localization problem in which we replace Carlotto-Schoen's exact localization requirement by an asymptotic condition at a super-harmonic rate. By applying our seed-to-solution method with a suitably constructed, parametrized family of seed data, we solve this problem by exhibiting mass-momentum correctors with harmonic decay.

Aspects of three-dimensional higher curvature gravities

We present new results involving general higher-curvature gravities in three dimensions. The most general Lagrangian of that kind can be written as a function of , where R is the Ricci scalar, , , and is the traceless part of the Ricci tensor. First, we provide a general formula for the exact number of independent order-n densities, #(n). This satisfies the identity #(n − 6) = #(n) − n. Then, we show that, linearized around a general Einstein solution, a generic order-n ⩾ 2 density can be written as a linear combination of R n , which by itself would not propagate the generic massive graviton, plus a density which by itself would not propagate the generic scalar mode, , plus #(n) − 2 densities which contribute trivially to the linearized equations. Next, we obtain an analytic formula for the quasinormal modes and frequencies of the BTZ black hole as a function of the masses of the graviton and scalar modes for a general theory. Then, we provide a recursive formula as well as a general closed expression for order-n densities which non-trivially satisfy an holographic c-theorem, clarify their relation with Born–Infeld gravities and prove that the scalar mode is always absent from their spectrum. We show that, at each order n ⩾ 6, there exist #(n − 6) densities which satisfy the holographic c-theorem in a trivial way and that all of them are proportional to a single sextic density . Next, we show that there are also #(n − 6) order-n generalized quasi-topological densities in three dimensions, all of which are ‘trivial’ in the sense of making no contribution to the metric function equation. Remarkably, the set of such densities turns out to coincide exactly with the one of theories trivially satisfying the holographic c-theorem. We comment on the meaning of Ω(6) and its relation to the Segre classification of three-dimensional metrics.

Self-dual gravitational instantons in conformal gravity: Conserved charges and thermodynamics

We study non-Einstein Bach-flat gravitational instanton solutions that can be regarded as the generalization of the Taub-NUT/Bolt and Eguchi-Hanson solutions of Einstein gravity to conformal gravity. These solutions include non-Einstein spaces which are either asymptotically locally flat spacetimes (ALF) or asymptotically locally Anti-de Sitter (AlAdS). Nevertheless, solutions with different asymptotic conditions exist: we find geometries that present a weakened AlAdS asymptotia, exhibiting the typical low decaying mode of conformal gravity. This permits to identify the simple Neumann boundary condition that, as it happens in the asymptotically AdS sector, selects the Einstein solution out of the solutions of conformal gravity. All the geometries present non-vanishing Hirzebruch signature and Euler characteristic, being single-centered instantons. We compute the topological charges as well as the Noether charges of the Taub-NUT/Bolt and Eguchi-Hanson spacetimes, which happen to be finite. This enables us to study the thermodynamic properties of these geometries.

Algebraic properties of Einstein solutions in ghost-free bimetric theory

A fact is that an Einstein solution in one sector in ghost-free bimetric theory implies an Einstein solution in the other sector. Earlier studies have also shown that some classes of bimetric models necessitate proportional solutions between the sectors. Here, we consider a general setup of the parameters in the theory as well as the general algebraic form of the potential. We show that, if one sector has an Einstein solution, the solutions are either proportional or block proportional with at most two different eigenvalues in the square root governing metric interactions.

Entropy, energy and temperature–length inequality for Friedmann universes

In this paper, we continue the study of the physical consequences of our modified black hole entropy formula in expanding spacetimes. In particular, we apply the new formula to apparent horizons of Friedmann expanding universes with zero, negative and positive spatial curvature. As a first result, we found that apart from the static Einstein solution, the only Friedmann spacetimes with constant (zero) internal energy are the ones with zero spatial curvature. This happens because, in the computation of the internal energy [Formula: see text], the contribution due to the nonvanishing Hubble flow must be added to the usual Misner–Sharp energy giving, for zero curvature spacetimes, a zero value for [Formula: see text]. This fact does not hold when curvature is present. After analyzing the free energy [Formula: see text], we obtain the correct result that [Formula: see text] is stationary only for physical systems in isothermal equilibrium, i.e. a de Sitter expanding universe. This result permits us to trace back a physically reasonable hypothesis concerning the origin of the early and late times de Sitter phase of our universe. Finally, we deduce an interesting temperature–length inequality similar to the time–energy uncertainty of ordinary quantum mechanics but with temperature instead of time coordinate. Remarkably, this relation is independent of the gravitational constant [Formula: see text] and can thus be explored also in nongravitational contexts.

Exact barotropic distributions in Einstein-Gauss-Bonnet gravity

New exact solutions to the field equations in the Einstein--Gauss--Bonnet modified theory of gravity for a 5--dimensional spherically symmetric static distribution of a perfect fluid is obtained. The Frobenius method is used to obtain this solution in terms of an infinite series. Exact solutions are generated in terms of polynomials from the infinite series. The 5--dimensional Einstein solution is also found by setting the coupling constant to be zero. All models admit a barotropic equation of state. Linear equations of state are admitted in particular models with the energy density profile of isothermal distributions. We examine the physicality of the solution by studying graphically the isotropic pressure and the energy density. The model is well behaved in the interior and the weak, strong and dominant energy conditions are satisfied.

Particular solutions in bimetric theory and their implications

Ghost-free bimetric theory can describe gravity in the presence of an extra spin-2 field. In this paper, we study certain aspects of dynamics in this theory: (i) It is shown that if either of the metrics is an Einstein solution, then the other is always forced to be Einstein, too. For a class of bimetric models, this constraint is stronger and as soon as one metric is Einstein, the other metric is forced to be proportional to it. As a consequence, the models in this class avoid a branch of pathological solutions that exhibit determinant singularities or nonlinear ghosts. These constraints persists in a generalized form when sources are included, but are destroyed in the massive gravity limit of the theory. (ii) For another class of bimetric models, we show the existence of solutions that do not admit a massive gravity limit. A bimetric model that could exhibit a nonlinear version of "partially massless" symmetry belongs to both these classes. It is argued that if such a model exist, its symmetry will not survive in the massive gravity limit.

Holographic entanglement entropy for 4D conformal gravity

Using the proposal for holographic entanglement entropy in higher derivative gravities, we compute holographic entanglement entropy for the conformal gravity in four dimensions which turns out to be finite. However, if one subtracts the contribution of the four dimensional Gauss-Bonnet term, the corresponding entanglement entropy has a divergent term and indeed restricted to an Einstein solution of the conformal gravity, the resultant entanglement entropy is exactly the same as that in the Einstein gravity. We will also make a comment on the first law of the entanglement thermodynamics for the conformal gravity in four dimensions.

Higher derivative effects for 4d AdS gravity

Motivated by holography, we explore higher derivative corrections to four-dimensional Anti-de Sitter (AdS) gravity. We point out that in such a theory the variational problem is generically not well-posed given only a boundary condition for the metric. However, when one evaluates the higher derivative terms perturbatively on a leading order Einstein solution, the equations of motion are always second order and therefore the variational problem indeed requires only a boundary condition for the metric. The equations of motion required to compute the spectrum around the corrected background are still generically higher order, with the additional boundary conditions being associated with new operators in the dual conformal field theory. We discuss which higher derivative curvature invariants are expected to arise in the four-dimensional action from a top-down perspective and compute the corrections to planar AdS black holes and to the spectrum around AdS in various cases. Requiring that the dual theory is unitary strongly constrains the higher derivative terms in the action, as the operators associated with the extra boundary conditions generically have complex conformal dimensions and non-positive norms.

On the elastic analysis of a circular lined tunnel considering the delayed installation of the support

This paper revisits the problem of excavation of a circular lined tunnel in an infinite elastic medium under plane strain conditions, subjected to non-uniform stresses, solved by Einstein and Schwartz in their article ‘Simplified Analysis for Tunnel Supports’ published in 1979 (ASCE J Geotech Eng Div, 106–7), and in related publications by Schwartz and Einstein. In contrast with the solution by Einstein and Schwartz from 1979 which considers that the support is installed at the very same time the tunnel is excavated, and also in contrast with related publications by Schwartz and Einstein which account for the delayed effect of support installation by multiplying the values of loads and displacements on the support by a reduction factor (in the same expressions from their 1979 article), this paper formulates the problem as two separate problems to account for the two-stage excavation process. In a first stage, the initial tractions on the periphery of the tunnel to be excavated are decreased (or ‘relaxed’) by a factor fR, to account for the presence of the tunnel front, as normally done in analysis of two dimensional sections of lined tunnels using commercial finite element software. For this first stage, the solution of the field quantities (stresses and displacements) in the ground are provided. In a second stage the support is installed, and the previously imposed tractions on the boundary of tunnel and support are removed, resulting in loading and deformation of the support as it interacts with the ground. For this second stage, the field quantities in the ground and the resulting values of load and displacements on the support are also provided. This paper shows that by decreasing the tractions on the tunnel periphery in a ratio fR of the initial ground stresses before the installation of the support (i.e., with 0≤fR<1), the resulting final loads and displacements on the support are reduced in the same ratio fR, with respect to the corresponding loads and displacements computed by Einstein and Schwartz's solution (i.e., when considering the case fR=1). This observation confirms the agreement of the two-stage procedure followed in this study and the procedure followed in publications by Schwartz and Einstein of multiplying the values of load and displacements on the support (for the case fR=1) by a correction factor to account for the delayed installation of the support. The paper also shows that when the stresses and displacements in the ground for the first stage are subtracted from the corresponding ground stresses and displacements for the second stage (after the support is installed), the resulting values of stresses and displacements are reduced in exactly the same ratio fR. All governing equations, boundary conditions and steps needed to arrive to a dimensionless form of the two-stage analytical solution of Einstein and Schwartz problem are presented, including the solutions for stresses and displacements in the ground which were not included in publications by Einstein and Schwartz. Also, particular forms of the two-stage solution are presented, including cases of an infinitely soft support, an infinitely rigid support, and uniform initial far-field stresses, which are shown to be equivalent to the classical expressions known as Kirsch and Lamé solutions, respectively. Although the outstanding solution by Einstein and Schwartz published in 1979 is basically correct, Einstein and Schwartz dropped a term in the solution for bending moment because its effect is small in most of the situations (particularly when the thickness of the support is small compared with the radius of the tunnel). Since the two-stage solution for bending moment presented in this paper is equivalent to the Einstein and Schwartz's solution only when the missing term in their solution is included, this paper presents a revised form of the expressions by Einstein and Schwartz published in 1979 which include the missing term, and it examines what the effect of retaining or dropping the term is. Finally a comparison of results obtained with the two-stage analytical solution and the finite difference code FLAC3D is presented, together with the outline of a computer spreadsheet that implements the two-stage analytical solution of this fundamental problem of tunnel ground–support interaction analysis.

The solution of complete Schwarzschild's planetary equation with the method of successive approximations

The Einstein's solution of the planetary equation of motion from Schwarzschild's line element is well known. In this paper, we solve the complete Schwarzschild's planetary equation with the method of successive approximation for the corresponding precession and compare the result with that from the line element. Keywords: precession angle, successive approximation, line element, planetary equation, perturbation term Nigerian Journal of Physics Vol. 17, 2005: 7-10

Multi-black-hole solutions in five dimensions

Using a recently developed generalized Weyl formalism, we construct an asymptotically flat, static vacuum Einstein solution that describes a superposition of multiple five-dimensional Schwarzschild black holes. The spacetime exhibits a $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ rotational symmetry. It is argued that for certain choices of parameters the black holes are collinear and so may be regarded as a five-dimensional generalization of the Israel-Khan solution. The black holes are kept in equilibrium by membranelike conical singularities along the two rotational axes; however, they still distort one another by their mutual gravitational attraction. We also generalize this solution to one describing multiple charged black holes, with fixed mass-to-charge ratio, in Einstein-Maxwell-dilaton theory.

Approximation of Light‐Ray Deflection Angle and Gravitational Lenses in the Schwarzschild Metric. II. Lensing Magnification in a Binary System

We apply an exact solution of a light deflection angle in the Schwarzschild metric, written with elliptic integrals, to the basic lensing theory. The magnifications for a point and a surface source are calculated and compared to the magnification provided by the usually applied traditional Einstein solution. We propose a simple correction to the Einstein angle 4/b that fixes it closer to the solution written with an elliptic integral. The results are applied to an eclipsing binary system, and the luminosity of a binary system at the secondary eclipse with lensing effects is calculated. Our results show that the radius of the Einstein ring, where the magnification diverges, is slightly underestimated by the Einstein solution. This leads to an error of the overall brightness of a point-mass lens. The effect is pronounced with systems of comparable physical and Einstein radii, especially when the Einstein ring is partially inside the physical radius of the lens object. The difference is not usually large, but in the worst case of a tight double neutron star, the error made in total flux can be on the order of ~10% if the higher order effects are neglected.

Instability of the Randall-Sundrum Model and Exact Bulk Solutions

Five dimensional geodesic equation is used to study the gravitational force acted on a test particle in the bulk of the Randall-Sundrum two-brane model.This force could be interpreted as the gravitational attraction from matters on the two branes and may cause the model to be unstable. By analogy with star models in astrophysics, a fluid RS model is proposed in which the bulk is filled with a fluid and this fluid has an anisotropic pressure to balance the gravity from the two branes. Thus a class of exact bulk solutions is obtained which shows that any 4D Einstein solution with a perfect fluid source can be embedded in $y=$ constant hypersurfaces in the bulk to form an equilibrium state of the brane model. By requiring a 4D effective curvature to have a minimum, the compactification size of the extra dimension is discussed.

Approximation of Light‐Ray Deflection Angle and Gravitational Lenses in the Schwarzschild Metric. I. Derivation and Quasar Lens

We calculate a second-order approximation for the light-ray deflection in a Schwarzschild metric. The approximation is based on Darwin's exact solution for the deflection angle, which is written in terms of elliptic integrals. Even if the derived equation is almost as simple as the usual Einstein deflection angle, it follows the Darwin solution more faithfully. This enables relatively simple calculations of gravitational lensing in axially symmetric conditions with a small value of the impact parameter b. We demonstrate the effects of second-order accuracy by modeling the lensed quasar QSO B0957+561A, B. Our simple toy model leads to a 4% difference in mass estimate for the deflector, when comparing the Einstein solution with the approximation. The naive point mass model is certainly inadequate when modeling a realistic quasar lens, but it demonstrates clearly the effects of the second-order accuracy and could, nevertheless, be used for some supermassive black hole models.

Cosmic strings in a braneworld theory with metastable gravitons

If the graviton possesses an arbitrarily small (but nonvanishing) mass, perturbation theory implies that cosmic strings have a nonzero Newtonian potential. Nevertheless in Einstein gravity, where the graviton is strictly massless, the Newtonian potential of a cosmic string vanishes. This discrepancy is an example of the van Dam--Veltman--Zakharov (VDVZ) discontinuity. We present a solution for the metric around a cosmic string in a braneworld theory with a graviton metastable on the brane. This theory possesses those features that yield a VDVZ discontinuity in massive gravity, but nevertheless is generally covariant and classically self-consistent. Although the cosmic string in this theory supports a nontrivial Newtonian potential far from the source, one can recover the Einstein solution in a region near the cosmic string. That latter region grows as the graviton's effective linewidth vanishes (analogous to a vanishing graviton mass), suggesting the lack of a VDVZ discontinuity in this theory. Moreover, the presence of scale dependent structure in the metric may have consequences for the search for cosmic strings through gravitational lensing techniques.

Classical and quantum Lovelock cosmology.

least for positive Lorentzian energy densities we show that this solution is uniquely determined by the rather natural physical requirement that the Euclidean solution is flat when we add no matter to the system. In these cases, the given classical solution can be interpreted in terms of perturbations to the corresponding Einstein solution. We also discuss the associated Wheeler-DeWitt equation and apply our results to the model with a cosmological constant and a massive scalar field model.

The vanishing likelihood of space-time singularity in quantum conformal cosmology

A general formalism is developed for studying the behavior of quantized conformal fluctuations near the space-time singularity of classical relativistic cosmology. It is shown that if the material contents of space-time are made of massive particles which obey the principle of asymptotic freedom and interact only gravitationally, then it is possible to estimate the quantum mechanical probability that, of the various possible conformal transforms of the classical Einstein solution, the actual model had a singularity in the past. This probability turns out to be vanishingly small, thus indicating that within the regime of quantum conformal cosmology it is extremely unlikely that the universe originated out of a space-time singularity.

On the viscosity of suspensions of solid spheres

A cell theory is used to derive the dependence of the zero-shear-rate viscosity on volume concentration for a suspension of uniform, solid, neutrally buoyant spheres. This result reduces to Einstein's solution at infinite dilution and to Frankel and Acrivos's expression in the limit as the concentration approaches its maximum value. Good agreement is found between the solution and the available data for the entire concentration range, provided that the maximum concentration is determined from the viscosity data themselves.