Showing papers on "Mathematics of general relativity published in 1968"

An Interior Solution in General Relativity

Abstract: Formulae are given for the field of a sphere of constant gravitational mass density.

A symmetric-Tensor-antisymmetric-tensor theory of gravitation and electromagnetism from a quaternion representation of general relativity

Abstract: The factorization of Einstein’s formalism into a pair of simultaneous quaternion field equations in general relativity entails the enlargement of the original formalism from 10 to 16 independent relations. By iteration, the quaternion field equations are shown to be physically equivalent to a symmetric-tensor-antisymmetric-tensor formalism. The symmetric-tensor part is in one-to-one correspondence with the 10 relations of Einstein’s original theory of gravitation. The remaining 6 (antisymmetric tensor) relations have no counterpart in the earlier gravitational theory. In addition to the generalization of the metrical field (and therefore the description of gravitational forces) that follows from the incorporation of the antisymmetric-tensor contribution, the covariant divergence of the latter formalism automatically leads to a system of field equations whose structure is in one-to-one correspondence with the Maxwell theory for electromagnetism. It is shown that Einstein’s original formalism entails 10, rather than 16 relations, because, in addition to its covariance under the group of general relativity (aconnected topological group) it is also covariant under time and space reflection transformations. The latter is an undue restriction since it is not required by the principle of general relativity alone. When these discrete symmetry elements are dropped, the (more general) quaternion formalism results. With the expression of the latter (which is not sensitive to the «handedness» of space or the direction of time) as thesum of two formalisms—one that is even and the other odd under space or time reflections—the original 10 relations of Einstein’s equationsplus the 6 relations that lead to the Maxwell field equations follow. Finally, with the application of the Schwarzschild conditions in the anti-symmetric-tensor part of the field equations, they reduce (in orderv/c) to a scalar-field formalism. With this approximation then, the full exploitation of the principle of general relativity—in terms of a symmetric-tensor-antisymmetric-tensor formalism—reduces to a scalar-tensor formalism, such as the one that has been studied by Brans and Dicke. It follows that to within the approximations that have been considered in this comparison (which would be applicable to a derivation of planetary motion) the present theory and a trulyscalar-tensor theory would be experimentally indistinguishable. In the derivation presented in this paper, however, the generalization of the metrical field to incorporate with the usual formalism an antisymmetric-tensor part (and therefore a scalar part in the application to planetary problems) follows in a natural way from the lowest-dimensional irreducible representations of the group of general relativity and no new fundamental constants need be introduced.

Vierbein Field Theory of Gravitation

Abstract: The sixteen components of the vierbein field which factorize the metric tensor are used to construct a simple nonlinear field theory of gravitation which, although it is shown to be equivalent to Einstein's theory physically, is based on a scalar action function of first order, replacing the Riemann scalar which serves as a second-order action function in the conventional approach.

New foundation of general relativity.

Abstract: The essential point of this note is to establish the uniqueness of general relativity by postulating the derivability of the equations of motion from the (gravitational) field equations. It is also emphasized that even with regard to agreement with experiment (including cosmology) there exist, in my opinion, no compelling arguments against general relativity.

Nonstatic electromagnetic fields in general relativity. ii.

Abstract: The Rainich equations of the `already unified field theory' are studied in the case of non-static electromagnetic fields, and a solution is obtained for a space-time metric which admits a group G4 of automorphisms. There exists a divergence-free electromagnetic field for x4 > 0, except for x4 -> infinity. It is shown that the electromagnetic field vanishes for large values of time, and the solution for a completely empty flat space is then obtained.

A Variational Formulation of Standard General Relativity which Includes the Full Bianchi Identities in Consequence of an Extended General Invariance Property

Abstract: The central properties of the usual variational formulations of EINSTEIN's general theory of relativity are sketched and a more general variational formulation is introduced, which is more appropriate to the geometric foundations of the theory. In particular, this formulation leads to the BIANCHI identities in their non-contracted form following in consequence of a general invariance property. This invariance property of the general variational principle relates to transformations (not, in general, coordinate transformations) that contain arbitrary fifth (or third) order tensor “generators”. These results can be interpreted as implying that the variational principle introduced here admits an “extended principle of general covariance” (i.e., a covariance principle more general than the usual principle relating to general coordinate covariance). Some of the formal implications of these results as well as their connection with general coordinate covariance are discussed briefly. In particular these results point to the existence of a fundamental transformation theory connecting all of the RIEMANNian spacetimes of general relativity.

Spherically symmetric space-time of class one and electromagnetism

Abstract: It is well known that a spherically symmetric space-time is, in general, of class two. A necessary and sufficient condition for a spherically symmetric space-time to be of class one has been obtained in terms of the Riemann curvature tensor. By means of a transformation property of s.s. space-time, three distinct cases are shown to exist. The incompatibility of class one spherically symmetric space-times with Rainich algebraic conditions is established in these three cases. It is concluded that spherically symmetric electromagnetic fields cannot be embedded in a flat space of 5-dimensions.

A class of generalized singular electromagnetic fields

Abstract: The Bel-Lapiedra-Montserrat equations are solved in a conformally flat space-time to give a class of generalized singular electromagnetic fields In the Minkowski space-time of special relativity it is shown that these solutions consist of three arbitrary holomorphic functions

Changes of the Petrov Type of a Space-Time

Abstract: A theoretical study is made of the "evolution" of a space-time of one Petrov type into another. The word "evolution" is used to mean that change of type which a point observer would consider his local space-time had experienced as a discontinuous front swept over him. The study is motivated by a desire to better understand the Petrov classification and thus improve its use as a tool to deal with solutions in general relativity. An algorithm is developed that may in principle be used to test the evolution of any closed-form solution known. A flow diagram is found which shows the routes that each Petrov type may use and the Petrov type of the discontinuity causing each change. During this development some results of the study of discontinuous hypersurfaces are found essentially equivalent to Trautman's information approach to waves, thus yielding further motivation for the use of the latter. The Weyl technique is understood as an important method for at least partially separating the coordinate and physical behaviors.

Topological aspects of the bel-petrov classification

Abstract: The topological aspects of the Bel-Petrov classification of the curvature tensor are examined for compact orientable space-times in which the Einstein equations for the exterior case are satisfied. It is shown that for such space-times of Bel Case III the metric tensor is singularity-free and that the Pontrjagin number identically vanishes. Bel Cases I and II are examined and conditions are given for which the metric is singularity-free and the Pontrjagin number vanishes. Applications to gravitional radiation in general relativity are discussed.

Use of delta functions in general relativity for determination of the integration constants

Abstract: During the integration of the Einstein—Maxwell equations, integration constants appear and their interpretation is often very difficult. There is therefore, a requirement for a calculus of delta functions which will automatically relate integration constants to sources. In this paper a calculus of this kind is developed and applied to the spherically symmetric problem. In this way we get a method of distinction between pure mathematical and physical singularities.

Type‐Null Vacuum Solutions in General Relativity

Abstract: It is shown that the necessary and sufficient conditions for the vacuum solutions of a certain metric to be of type null according to the Pirani‐Petrov classification are that a certain other metric be conformally flat. The general solution is obtained.