Spherical shells of positive density - can they be of non-positive mass?

Spherical shells of fluid in general relativity are considered. The density is assumed to be spatially uniform and it is found that there may be three cases of positive, negative and vanishing Schwarzschild mass of the shell although the density and the pressure are both positive throughout. However the negative mass case has to be associated with a singularity representing a negative mass particle and so is unphysical. The zero mass solution has the intriguing feature that the geometry on either side of the shell is Minkowskian and the space is closed. This closure of the space saves the present result from being in contradiction with the positive energy theorems. Earlier investigations claiming zero-mass distributions are also discussed. A short report giving some of the results of the present paper is published inPhys. Lett. A140 285 (1989).     

Vaidya spacetime in the diagonal coordinates

We have analyzed the transformation from initial coordinates (v, r) of the Vaidya metric with light coordinate v to the most physical diagonal coordinates (t, r). An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate v. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates (t, r) is insufficient to cover the entire range of initial coordinates (v, r) outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either g ₀₀ = 0 or g ₀₀ = ∞). The energy–momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false fire-walls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates (η, y), in which the solution is obtained in explicit form, and there is no energy–momentum tensor divergence.     

Killing vector fields and the Einstein-Maxwell field equations with perfect fluid source

This paper is concerned with space-times that satisfy the Einstein-Maxwell field equations in the presence of a perfect fluid, which may be charged. We consider the following question. Suppose that the space-time admits a group of motions (isometries), i.e., that the metric is invariant under a group of transformations. Does it follow that the quantities that describe the source, i.e., the electromagnetic field tensorF _ij, the charge densityε, and the four-velocityu ⁱ, energy densityμ, and pressurep of the fluid, are invariant under the group? It is found that the behavior of these quantities under the group is strongly restricted. In particular in the case of the three-dimensional special orthogonal groupSO(3), which arises in the case of spherically symmetric space-times, it is found that the source quantities are invariant. On the other hand, it is established that there exist groups under whichF _ij is not necessarily invariant. The above question is also considered for the case of homothetic motions.     

Metric gravitation with a two parameter family of static spherically symmetric space-times

Among the variety of all conceivable metric theories of gravitation, Lorentz curvature dynamics is the most geometric extension of Einstein's field equations to fit the solar system data. In this framework two parameters determine the asymptotic form of a static spherically symmetric space-time (without imposing Einstein's conditions); these two parameters are the active gravitational mass of the source and the PPN parameter γ. The Lorentz connection is shown to satisfy covariant evolution equations which preserve either of these two parameters; furthermore, right and left oriented space-times differ in their Lorentz connection. Deviations from the Schwarzschild character find an interpretation in terms of a new object, the Lorentz curvature energy-momentum tensor, which always vanishes identically under the restriction of Einstein's conditions. These deviations contribute strongly to the gravitational force only in the neighbourhood of the Schwarzschild sphere.     

Energy and Momentum of Robinson-Trautman Space-Times

Robinson and Trautman space-times are studied in the context of teleparallel equivalent of general relativity (TEGR). These space-times are the simplest class of asymptotically flat geometries admitting gravitational waves. We calculate the total energy for such space-times using two methods, the gravitational energy-momentum and the translational momentum 2-form. The two methods give equal results of these calculations. We show that the value of energy depends on the gravitational mass M, the Gaussian curvature of the surfaces λ(u,θ) and on the function K(u,θ). The total energy reduces to the energies of Schwarzschild’s and Bondi’s space-times under specific forms of the function K(u,θ).     

On a physical interpretation of the Schwarzschild metric

A method is devised for giving a physical interpretation to the customary Schwarzschild coordinates in the vicinity of a charged or uncharged isolated mass. The construction is accomplished by introducing systems that are allowed to freely fall in toward the mass from infinity (drift-systems). It is demonstrated that the Schwarzschild spatial coordinates and their increments have a full physical significance in terms of rod and clock measurements performed in the drift-systems. The time coordinate and its increment are not so amenable to treatment and cannot be considered as having been given such physical significance. In the discussion the Schwarzschild metric about an uncharged and charged mass is derived, in part, by heuristic classical arguments employing conservation of energy. The arguments are then shown to be valid by consulting the Field Equations. In the derivation the gravitational singularity (at 2GM/C ²) takes on the significance of being the location at which a drift-system achieves the speed of light relative to a proper system at the same point.     

Curvature collineations and the determination of the metric from the curvature in general relativity

It is shown that for a very general class of space-times, the componentsR _bcd ^a of the curvature tensor determine the metric components up to a constant conformal factor. This general class contains most of those cases which are usually considered to be interesting from the point of view of Einstein's general relativity theory. The connection between the above result and the existence of proper curvature collineations is given.     

Establishment of the Riemannian structure of space-time by classical means

Efforts at providing a physical-axiomatic foundation of the space-time structure of the general theory of relativity have led, when based on simple empirical facts about freely falling particles and light signals, in a satisfying manner only to a Weyl space-time. By adding postulates based on quantum theory, however, the usual pseudo-Riemannian space-time can be reached. We present a newclassical postulate which provides the same results. It is based upon the notion of the radar distance between freely falling particles and demands the approximate equality of the growth of the radar distance for particle pairs of equal, small initial velocities. We show that given this, a property results, as found in earlier work by the author, that distinguishes between Weyl and Lorentz space-times. The property refers to a special metric and decides whether its metric connection has the given free-fall worldlines as geodesics or not. It consists in the vanishing of the mixed spatiotemporal componentsg _i4 of this metric in suitable coordinates along the worldline of the freely falling observer, as the rest system of which the coordinates are constructed.     

Neutrino radiation in spherically-symmetric gravitational fields II. The structure of the radiation field

It is shown that the neutrino radiation field emitted by a star may be described by Vaidya's radiating Schwarzschild metric. The gravitational energy shift of the neutrino field is also considered, both in terms of an exact solution and in the weak field approximation.     

Quantum fluctuations in the conformally flat and the schwarzschild space-times

A general technique is described for dealing with the quantum fluctuations between conformally flat space-times. The second part of the paper deals with the Schwarzschild spacetime. It is shown there that this space-time is stable against fluctuations of mass, but transitions between two space-times of different masses can be obtained via conformai fluctuations. Purely conformal fluctuations of the Schwarzschild metric are, however, damped at the event horizon. Similar conclusions are drawn about the Reissner-Nordstrom space-time.     

The Schwarzschild metric and de Donder condition

The nonordinary mathematical form of the Schwarzschild metric in harmonic coordinates is considered. The ordinary property of energy density for it is demonstrated.     

Quasinormal Modes for Schwarzschild–AdS Black Holes: Exponential Convergence to the Real Axis

We study quasinormal modes for massive scalar fields in Schwarzschild–anti-de Sitter black holes. When the mass-squared is above the Breitenlohner–Freedman bound, we show that for large angular momenta, ?, there exist quasinormal modes with imaginary parts of size exp(??/C). We provide an asymptotic expansion for the real parts of the modes closest to the real axis and identify the vanishing of certain coefficients depending on the dimension.     

Zero-rest-mass fields in an algebraically special curved space-time

Zero-rest-mass higher-spin fields in algebraically special vacuum background space-times are considered. It is shown that the algebraic speciality of the background metric strongly restricts the form of the solutions of these fields. These results are used to study perturbations of the Schwarzschild black hole.     

On the Backward Stability of the Schwarzschild Black Hole Singularity

We study the backwards-in-time stability of the Schwarzschild singularity from a dynamical PDE point of view. More precisely, considering a spacelike hypersurface \({\Sigma_0}\) in the interior of the black hole region, tangent to the singular hypersurface \({\{r = 0\}}\) at a single sphere, we study the problem of perturbing the Schwarzschild data on \({\Sigma_0}\) and solving the Einstein vacuum equations backwards in time. We obtain a local backwards well-posedness result for small perturbations lying in certain weighted Sobolev spaces. No symmetry assumptions are imposed. The perturbed spacetimes all have a singularity at a “collapsed” sphere on \({\Sigma_0}\), where the leading asymptotics of the curvature and the metric match those of their Schwarzschild counterparts to a suitably high order. As in the Schwarzschild backward evolution, the pinched initial hypersurface \({\Sigma_0}\) ‘opens up’ instantly, becoming a regular spacelike (cylindrical) hypersurface. This result thus yields classes of examples of non-symmetric vacuum spacetimes, evolving forward-in-time from regular initial data, which form a Schwarzschild type singularity at a collapsed sphere. We rely on a precise asymptotic analysis of the Schwarzschild geometry near the singularity which turns out to be at the threshold that our energy methods can handle.     

A class of static charged dust spheres in general relativity

Using static spherically symmetric space-times with associated 3-spaces obtained as hypersurfacest= const as 3-spheroidal, a class of physically viable relativistic models for spherical distributions of uniformly charged dust in equilibrium is obtained. The charged analog of Schwarzschild interior solution given by Cooperstock and de la Cruz follows as a particular case of this class.     

ADM pseudotensors,conserved quantities and covariant conservation laws in general relativity

The ADM formalism is reviewed and techniques for decomposing generic components of metric, connection and curvature are obtained. These techniques will turn out to be enough to decompose not only Einstein equations but also covariant conservation laws. Then a number of independent sets of hypotheses that are sufficient (though not necessary) to obtain standard ADM quantities (and Hamiltonian) from covariant conservation laws are considered. This determines explicitly the range in which standard techniques are equivalent to covariant conserved quantities.The Schwarzschild metric in different coordinates is then considered, showing how the standard ADM quantities fail dramatically in non-Cartesian coordinates or even worse when asymptotically flatness is not manifest; while, in view of their covariance, covariant conservation laws give the correct result in all cases.     

Velocity and velocity bounds in static spherically symmetric metrics

We find simple expressions for velocity of massless particles with dependence on the distance, r, in Schwarzschild coordinates. For massive particles these expressions give an upper bound for the velocity. Our results apply to static spherically symmetric metrics. We use these results to calculate the velocity for different cases: Schwarzschild, Schwarzschild-de Sitter and Reissner-Nordström with and without the cosmological constant. We emphasize the differences between the behavior of the velocity in the different metrics and find that in cases with naked singularity there always exists a region where the massless particle moves with a velocity greater than the velocity of light in vacuum. In the case of Reissner-Nordström-de Sitter we completely characterize the velocity and the metric in an algebraic way. We contrast the case of classical naked singularities with naked singularities emerging from metric inspired by noncommutative geometry where the radial velocity never exceeds one. Furthermore, we solve the Einstein equations for a constant and polytropic density profile and calculate the radial velocity of a photon moving in spaces with interior metric. The polytropic case of radial velocity displays an unexpected variation bounded by a local minimum and maximum.     

Lightlike simultaneity,comoving observers and distances in general relativity

We state a condition for an observer to be comoving with another observer in general relativity, based on the concept of lightlike simultaneity. Taking into account this condition, we study relative velocities, Doppler effect and light aberration. We obtain that comoving observers observe the same light ray with the same frequency and direction, and so gravitational redshift effect is a particular case of Doppler effect. We also define a distance between an observer and the events that it observes, called lightlike distance, obtaining geometrical properties. We show that lightlike distance is a particular case of radar distance in the Minkowski space-time and generalizes the proper radial distance in the Schwarzschild space-time. Finally, we show that lightlike distance gives us a new concept of distance in Robertson–Walker space-times, according to Hubble law.