On the uniqueness of the space-time energy in General Relativity: the illuminating case of the Schwarzschild metric

The case of asymptotic Minkowskian space-times is considered. A special class of asymptotic rectilinear coordinates at the spatial infinity, related to a specific system of free falling observers, is chosen. This choice is applied in particular to the Schwarzschild metric, obtaining a vanishing energy for this space-time. This result is compared with the result of some known theorems on the uniqueness of the energy of any asymptotic Minkowskian space, showing that there is no contradiction between both results, the differences becoming from the use of coordinates with different operational meanings. The suitability of Gauss coordinates when defining an intrinsic energy is considered and it is finally concluded that a Schwarzschild metric is a particular case of space-times with vanishing intrinsic 4-momenta.     

The structure of the b-completion of space-time

Structured space, as a natural generalization of the manifold concept, is defined to be a topological space with a sheaf of real function algebras which are suitably localized and closed with respect to composition with smooth Euclidean functions. Vector fields, differential forms, linear connection and curvature are introduced on structured spaces. It is shown that structured spaces correctly model space-times with singularities. Schmidt's b-boundary of space-time is constructed in the category of structured spaces, and well known difficulties with the b-boundaries of the closed Friedman and Schwarzschild space-times are disentangled. It is argued that the b-boundary of space-time, when considered in the category of structured spaces, can serve as a good definition of classical singularities.     

Hawking radiation, Unruh radiation, and the equivalence principle

We compare the response function of an Unruh-DeWitt detector for different space-times and different vacua and show that there is a detailed violation of the equivalence principle. In particular comparing the response of an accelerating detector to a detector at rest in a Schwarzschild space-time we find that both detectors register thermal radiation, but for a given, equivalent acceleration the fixed detector in the Schwarzschild space-time measures a higher temperature. This allows one to locally distinguish the two cases. As one approaches the horizon the two temperatures have the same limit so that the equivalence principle is restored at the horizon.     

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.     

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.     

Spherically symmetric space-times transparent to scalar multipole waves

Using nonscattering potentials of Chang and Janis, a large class of spherically symmetric space-times is constructed on which all multipole solutions to the minimally coupled scalar wave equation are expressible in terms of characteristic data functions in essentially as simple a fashion as for flat space-time. The space-times are transparent to multipole waves in the same sense that flat space-time is. Both conformally flat and not conformally flat space-times are obtained. Some examples are discussed which show that the variety of transparent space-times is large even within the class of Robertson-Walker spaces.     

Ricci recurrent space-times with torsion

A Ricci recurrent space-time with covariantly constant stress tensor is an Einstein space-time. We extend this result to Ricci recurrent space-times with torsion. The result is applied to the case of Riemann-Cartan space-times with spin density.     

Regge calculus and observations. I. Formalism and applications to radial motion and circular orbits

A method is developed for tracing geodesics of particles and light rays through Regge calculus space-times. In the flat interiors of the blocks, the geodesics are straight lines, and at the boundaries between blocks they are refracted according to an extremal distance law. The method is then tested for the Regge calculus version of the Schwarzschild space-time. For radial motion, it gives good agreement with the analytic solution, provided the blocks are sufficiently small. Exactly circular orbits are impossible in Regge calculus, but it is shown how to construct approximately circular ones. These examples are in preparation for an investigation of more general orbits and more general space-times, using Regge calculus.     

The Lynden-Bell and Katz definition of gravitational energy: Applications to singular solutions

We apply the Lynden-Bell and Katz (LK) definition of gravitational energy to static and spherically symmetric space-times which admit a curvature singularity. These are the Tolman V, Tolman VI and the interior Schwarzschild solutions, the latter with the boundary limit of 9/8th of the gravitational radius. We show that the LK definition can still be applied to these solutions despite the presence of a singularity which nonetheless appears to carry no energy in the LK sense. While in the solutions that we mentioned the KL gravitational energy is positive definite everywhere in space time, this is not the case for the overcharged Reissner-Nordström space-time. In the latter case in fact the LK energy density becomes negative sufficiently close to the singularity hence we use the positivity criterion to impose a more stringent limit of validity to the Reissner-Nordström solution.     

Scalar torsion and a new symmetry of general relativity

We reformulate the general theory of relativity in the language of Riemann–Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to Riemann–Cartan space-times with scalar torsion and obtain the conservation laws for auto-parallel motions in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.     

Equivalence of the Regge and Einstein equations for almost flat simplicial space-times

The theory of distributions is applied to almost flat simplicial space-times. Explicit expressions are given for the first-order defects. It is shown explicitly that the Riemann tensor for an almost flat simplicial space-time contains delta-functions on the bones and derivatives of delta-functions on the 3-dimensional faces of the boundary of the space-time. The latter terms have not previously been seen in the Regge calculus. It is shown that the Regge and Hilbert actions have equal values on almost fiat simplicial space-times and that the Einstein equations lead directly to the Regge field equations.     

A Fermat principle for stationary space-times and applications to light rays

We present an extension of the classical Fermat principle in optics to stationary space-times. This principle is applied to study the light rays joining an event with a timelike curve. Existence and multiplicity results of light rays are proved. Moreover, Morse Relations relating the set of rays to the topology of the space-time are obtained, by using the number of conjugate points of the ray. The results hold also for stationary space-times with boundary, in particular the Kerr space-time outside the stationary limit surface.     

The Embedding of Schwarzschild in Braneworld

The braneworlds models were inspired partly by Kaluza-Klein’s theory, where both the gravitational and the gauge fields are obtained from the geometry of a higher dimensional space. The positive aspects of these models consist in perspectives of modifications it could bring in to particle physics, such as: unification in a TeV scale, quantum gravity in this scale and deviation of Newton’s law for small distances. One of the principles of these models is to suppose that all space-times can be embedded in a bulk of higher dimension. The main result in these notes is a theorem showing a mathematical inconsistency of the Randall-Sundrum braneworld model, namely that the Schwarzschild space-time cannot be embedded locally and isometrically in a five dimensional bulk with constant curvature (for example AdS-5). From the point of view of semi-Riemannian geometry this last result represents a serious restriction to the Randall-Sundrum’s braneworld model.     

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.     

Generalized Klein–Gordon field equations with octonion space-time (OST) algebra

The concept of space-time representation is redefined using the octonion space-time (OST) algebra. In this study, describing the properties of octonions and their possible connection with Euclidean space-times, the internal and external space-time events are represented within the OST algebra. Keeping in mind the octonionic dual-Euclidean space-times, we express the homogeneous field equations which leads to the symmetrical nature of internal and external space-times. We derive the generalized Proca–Maxwell equations for massive-dyons in the case of the OST algebra. Accordingly, we have obtained a new set of octonionic Klein–Gordon potential (KGP) and Klein–Gordon field (KGF) equations for massive dyons from the generalized Proca–Maxwell equations. This formalism demonstrates that the octonionic KGP and KGF equations can be expressed in a single equation and it is equivalent to energy-momentum relation for dyons. As such, we have made an attempt to write the conservation of Noetherian current from the octonionic Klein–Gordon equations.     

Mass Defect in the Noncommutative Schwarzschild Space-Time

In this paper we investigate the mass defect and other gravitational effects in noncommutative Schwarzschild space-time obtained by considering particles as smeared objects. The effects of space-time noncommutativity on mass defect of a test particle and a homogeneous spherical shell are calculated. The NC corrections to gravitational redshift, and light-speed in Schwarzschild field are briefly discussed. The results show that the NC corrections have weakening action on these gravitational effects comparing with those in commutative cases.     

Mappings of empty space-times leaving the curvature tensor invariant

It is shown that if in some local coordinate system the componentsR ⁱ _jkl of the curvature tensor of an empty space-time are known, then, provided the space-time is not of Petrov typeN with hypersurface orthogonal geodesic rays, the components of the metric tensor are uniquely determined up to a trivial constant scaling factor. The Petrov type-N empty space-times with hypersurface orthogonal geodesic rays are investigated. The most general mappings leaving the curvature tensorR ⁱ _jkl invariant are found for each class of these space-times.     

Quantum Detectors in Generic Non Flat FLRW Space-Times

We discuss a quantum field theoretical approach, in which a quantum probe is used to investigate the properties of generic non-flat FLRW space-times. The probe is identified with a conformally coupled massless scalar field defined on a space-time with horizon and the procedure to investigate the local properties is realized by the use of Unruh-DeWitt detector and by the evaluation of the regularized quantum fluctuations. In the case of de Sitter space, the coordinate independence of our results is checked, and the Gibbons-Hawking temperature is recovered. A possible generalization to the electromagnetic probe is also briefly indicated.     

Regge calculus and observations. II. Further applications

The method, developed in an earlier paper, for tracing geodesies of particles and light rays through Regge calculus space-times, is applied to a number of problems in the Schwarzschild geometry. It is possible to obtain accurate predictions of light bending by taking sufficiently small Regge blocks. Calculations of perihelion precession, Thomas precession, and the distortion of a ball of fluid moving on a geodesic can also show good agreement with the analytic solution. However difficulties arise in obtaining accurate predictions for general orbits in these space-times. Applications to other problems in general relativity are discussed briefly.