Showing papers on "Introduction to the mathematics of general relativity published in 1966"
TL;DR: In this article, an approach to study the dynamics of thin shells of dust in general relativity is presented. But no mention of admissible or even any space-time co-ordinates is needed.
Abstract: An approach to shock waves, boundary surfaces and thin shells in general relativity is developed in which their histories are characterized in a purely geometrical way by the extrinsic curvatures of their imbeddings in space-time. There is some gain in simplicity and ease of application over previous treatments in that no mention of « admissible » or, indeed, any space-time co-ordinates is needed. The formalism is applied to a study of the dynamics of thin shells of dust.
2,347 citations
TL;DR: In this article, the similarities between the diagram of Kruskal space in general relativity and that of the uniformly accelerated rigid rod in special relativity are shown to be the result of certain physical similarities.
Abstract: The striking formal similarities between the diagram of Kruskal space in general relativity and that of the uniformly accelerated rigid rod in special relativity are shown to be the result of certain physical similarities.
319 citations
TL;DR: In this paper, it was shown that space-time cannot be singularity free if the Einstein equations hold, matter has normal properties, and some reasonable global conditions hold. This would suggest that the Einstein theory probably breaks down in very strong fields.
Abstract: It is shown that space-time cannot be singularity free if the Einstein equations hold, matter has normal properties, and some reasonable global conditions hold. This would suggest that the Einstein theory probably breaks down in very strong fields.
121 citations
TL;DR: Theories of gravitation similar to General Relativity but with an additional R2 term in the Lagrangian are explored in this article. But the experimental consequences of R2 terms are investigated.
Abstract: Theories of gravitation similar to General Relativity but with an additionalR2 term in the Lagrangian are explored. The Schwarzschild metric is not the exterior solution that can be continued to the interior of the body to give a positive definite mass distribution. The experimental consequences ofR2 terms are investigated. Furthermore, it is shown that a theory with anR2 term only possesses an interesting singular dependence on the coupling constant.
102 citations
01 Jan 1966
61 citations
TL;DR: In this paper, it was shown how the ten conserved quantities, recently discovered by E. T. Newman and R. Penrose by essentially geometrical techniques, arise in a direct solution of the Einstein field equations.
Abstract: This paper shows how the ten conserved quantities, recently discovered by E. T. Newman and R. Penrose by essentially geometrical techniques, arise in a direct solution of the Einstein field equations. For static fields it is shown that five of the conserved quantities vanish while the remaining five are expressed in terms of the multipole moments of the source distribution.
53 citations
01 Jan 1966
TL;DR: In this article, a solution of Einstein's vacuum field equations, apparently new, is exhibited, and the metric, which is homogeneous (that is, admits a three-parameter group of motions transitive on space-like hypersurfaces), belongs to Taub Type V.
Abstract: A solution of Einstein's vacuum field equations, apparently new, is exhibited. The metric, which is homogeneous (that is, admits a three-parameter group of motions transitive on space-like hypersurfaces), belongs to Taub Type V. The canonical form of the Riemann tensor, which is of Petrov Type I, is determined.
42 citations
01 Jan 1966
39 citations
30 citations
TL;DR: In this paper, the authors showed that the first-order formula for travel time in the circular-orbit model is strictly independent of the nonlinear term in the theory of general relativity.
Abstract: Using a model of circular coplanar orbits and an analysis accurate to first order in the sun's gravitational radius, Ross and Schiff discussed the recent proposal to test general relativity by measuring round-trip travel times of radar pulses transmitted from the earth towards an inner planet. Their main conclusion, that such measurements would be sensitive to a nonlinear term in Einstein's theory, we find to be invalid. Since first-order differences between Newtonian and Einsteinian orbits are well known to depend on a nonlinear term in the metric, one might expect the round-trip travel times also to depend in first order on such a term. Curiously, this expectation is not realized for circular orbits. When expressed as a function solely of clock readings, the first-order formula for travel time in the circular-orbit model is strictly independent of the nonlinear term. Even were the combined use of radar-pulse travel times and the results of "exact" optical measurements envisioned, their sensitivity to this nonlinear term would be masked almost completely by unavoidable uncertainties in the estimates of other unknown parameters such as the mass of the sun. For noncircular orbits, however, the travel-time measurements will be noticeably sensitive to this nonlinear term through its effect on the advance of the perihelion. In addition to re-examining the circular-orbit model, we describe the operational procedures that we have developed for testing general relativity with data obtained from actual planetary observations. These data cannot be expected in the near future to provide a significant test of more than the first-order influence of solar gravity on radar-pulse travel times and the non-Newtonian advance of Mercury's perihelion, as we previously pointed out.
TL;DR: In this article, a method of obtaining Komar's covariant formulation of the conservation laws of general relativity directly from the variation of the scalar curvature density is presented, free of the addition of arbitrary elements and only tensorial terms and operations are employed in the development.
Abstract: A method of obtaining Komar's covariant formulation of the conservation laws of general relativity directly from the variation of the scalar curvature density is presented. The procedure of obtaining this expression is free of the addition of arbitrary elements and only tensorial terms and operations are employed in the development.
TL;DR: The Rayski method was used to calculate the energy of a static and spherically symmetric gravitational system as mentioned in this paper, which was used for the first time in the 1990s.
Abstract: The Rayski method is used to calculate the energy of a static and spherically symmetrical gravitational system.
TL;DR: In this article, it is shown that if the special theory of relativity is valid in the infinitesimal neighborhood of any point of a radially symmetric and stationary gravitational field, then a single new assumption will suffice to yield equations that predict all the three Einstein effects.
Abstract: The first part of this paper is concerned with test particles in a radially symmetric and stationary gravitational field. It is shown that if the special theory of relativity is valid in the infinitesimal neighborhood of any point of such a field, then a single new assumption will suffice to yield equations that predict all the three Einstein effects. Notably, the equations suggest also an energy process associated with the motion of a test particle in a gravitational field that may give an explanation to such phenomena as novae and radio stars. The second part of this paper shows that without further assumptions the equations for the radially symmetric and stationary field may be extended to a general scalar field theory. A simple solution of the latter theory reveals effects that may prove to be of interest to astronomy. The paper contains also a brief discussion of a necessary modification of the corpuscular theory of light in order to bring it in line with the present theory.
01 Jan 1966
TL;DR: In this article, the relationships between Witten and Penrose and Bergmann's spinor approach to the theory of general relativity were analyzed. But the relationships were not discussed in detail.
01 Jan 1966
TL;DR: In bimetric relativity, involving a Riemannian metric tensor gμν and a flat-space metric tensors γμν, one can write down a variational principle which gives the field equations and also leads to a gravitational energy-momentum density tensor as mentioned in this paper.
TL;DR: In this article, a general relativistic extension of some simple notions, e.g., total angular momentum, is proposed to construct an integral object, conserved in time and covariant.
Abstract: In this paper we attempt a general relativistic extension of some simple notions, e.g., total angular momentum, so far correctly defined only in the frame of special relativity. The mathematical aspect of the problem leads us to ask, given a tensor of vanishing divergence, how to construct an integral object, conserved in time and covariant.A formalism based upon bitensors ensures the covariance, while the assumption of an exploding (or imploding) schema of matter seems to be the only available means to preserve conservation when the space‐time is curved. This can be formulated in a general theorem then applied to different physical situations. The total linear momentum occurs as a vector fixed at the point of explosion (or implosion). Its length turns out (at least when a pure matter schema is concerned) to be superior to the total mass of the fluid. The excess appears as the energy carried by the explosion. The case of a so called ``uncompressible'' holonomic fluid gives a quite analogous result. In ord...
TL;DR: In this article, a spherically symmetric mass of material in radial motion which may either collapse on to the central point (gravitational collapse) or be exploding from that point (spherical blast) was determined from three general conditions: (1) from a certain pre-assigned form for the coefficients of the metric; (2) from the existence of co-moving co-ordinates; (3) from assuming that the stress is spatially isotropic.
Abstract: I HAVE recently obtained certain exact solutions of Einstein's equations of general relativity which, so far as I can trace, have not been noticed before They refer to a spherically symmetric mass of material in radial motion which may either be collapsing on to the central point (gravitational collapse) or be exploding from that point (spherical blast) The solutions are determined from three general conditions: (1) from a certain pre-assigned form for the coefficients of the metric; (2) from the existence of co-moving co-ordinates; (3) from the assumption that the stress is spatially isotropic The results have analogies with the spherical blast waves of classical gas-dynamics1, but with the additional feature that the gravitational self-attraction of the material is a controlling element of the motion A by-product of the investigation is the discovery of the class of solutions of Einstein's equations to which belongs a particular case2 obtained in 1933 by a different method