Showing papers on "Introduction to the mathematics of general relativity published in 1972"
Abstract: All tensors of contravariant valency two, which are divergence free on one index and which are concomitants of the metric tensor, together with its first two derivatives, are constructed in the four‐dimensional case. The Einstein and metric tensors are the only possibilities.
631 citations
TL;DR: In this paper, a quasi-linear first-order symmetric hyperbolic system of Friedrichs is presented and the existence and uniqueness theorems for the Einstein equations in general relativity are given.
Abstract: A systematic presentation of the quasi-linear first order symmetric hyperbolic systems of Friedrichs is presented A number of sharp regularity and smoothness properties of the solutions are obtained The present paper is devoted to the case ofRn with suitable asymptotic conditions imposed As an example, we apply this theory to give new proofs of the existence and uniqueness theorems for the Einstein equations in general relativity, due to Choquet-Bruhat and Lichnerowicz These new proofs usingfirst order techniques are considerably simplier than the classical proofs based onsecond order techniques Our existence results are as sharp as had been previously known, and our uniqueness results improve by one degree of differentiability those previously existing in the literature
242 citations
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01 Jan 1972
TL;DR: The acceptability of physical theories: Poincare versus Einstein, by G. F. Ellis, G. R. Robinson and J. A. Schild as mentioned in this paper, is discussed.
Abstract: Lanczos, C. Einstein's path from special to general relativity.--Balazs, N. L. The acceptability of physical theories: Poincare versus Einstein.--Ellis, G. F. R. Global and non-global problems in cosmology, by G. F. R. Ellis and D. W. Sciama.--Ehlers, J. The geometry of free fall and light propagation, by J. Ehlers, F. A. E. Pirani and A. Schild.--Trautman, A. Invariance of Lagrangian systems.--Penrose, R. The geometry of impulsive gravitational waves.--Exact solutions of the Einstein-Maxwell equations for an accelerated charge.--Taub, A. H. Plane-symmetric similarity solutions for self-gravitating fluids.--Robinson, I. Equations of motion in the linear approximation, by I. Robinson and J. R. Robinson.--Florides, P. W. Rotating bodies in general relativity.--Chandrasekhar, S. A limiting case of relativistic equilibrium.--Israel, W. The relativistic Boltzmann equation.--Thompson, W. B. The self-consistent test-particle approach to relativistic kinetic theory.
198 citations
TL;DR: In this paper, it was shown that given any static solution of the Einstein vacuum equations, a corresponding family of static vacuum solutions of the Brans-Dicke equations can be written down by inspection.
Abstract: It is shown that, given any static solution of the Einstein vacuum equations, a corresponding family of static vacuum solutions of the Brans-Dicke equations can be written down by inspection. Spherically and axially symmetric fields are considered explicitly. It is demonstrated how some solutions of the Brans-Dicke equations may be obtained without having to solve any field equations explicitly at all.
44 citations
TL;DR: In this article, an exact calculation that leads to the equations of motion (which naturally contain gravitational radiation reaction terms) of a system subject to no external forces is presented, where the system is to be considered as the source of an asymptotically flat space and all the revelant physical quantities such as the velocity νμ, 4 −momentum pμ, angular momentum center of mass tensor Sμν (as well as higher moments) are then defined in terms of surface integrals taken at infinity.
Abstract: We present an exact calculation that leads to the equations of motion (which naturally contain gravitational radiation reaction terms) of a system subject to no external forces. The novelty of our approach lies in the fact that the system is to be considered as the source of an asymptotically flat space and that all the revelant physical quantities such as the velocity νμ, 4‐momentum pμ, angular momentum‐center of mass tensor Sμν (as well as higher moments) are then defined in terms of surface integrals taken at infinity. A subset of the Einstein equations (equivalent to Bondi's supplemantary conditions) then yields the time‐evolution equations for these variables.
39 citations
TL;DR: In this paper, it was shown that if the Hamiltonian constraint of general relativity is imposed as a restriction on the Hamilton principal functional in the classical theory, or on the state functional in quantum theory, then the momentum constraints are automatically satisfied.
Abstract: It is shown that if the Hamiltonian constraint of general relativity is imposed as a restriction on the Hamilton principal functional in the classical theory, or on the state functional in the quantum theory, then the momentum constraints are automatically satisfied. This result holds both for closed and open spaces and it means that the full content of the theory is summarized by a single functional equation of the Tomonaga-Schwinger type.
39 citations
TL;DR: In this paper, it was shown that for test fields of spin-2 in vacuum spaces, solutions of the propagation equation are restricted to constant multiples of the Weyl spinor.
Abstract: The propagation of a zero rest-mass test field of arbitrary spins>1 through curved space-time is found to be subject to strong constraints. A null test field is shown to be possible only in a restricted class of spaces previously introduced by Kundt and Thompson. This result is in fact a simultaneous generalization of the theorems of Robinson and of Goldberg and Sachs. For test fields of spin-2 in vacuum spaces, solutions of the propagation equation are restricted, save in a few exceptional cases, to constant multiples of the Weyl spinor. The exceptional cases are discussed, and appear to be physically uninteresting.
28 citations
TL;DR: The scalar-tensor formalism is implicitly embodied in the theory of general relativity, thus illustrating the considerable freedom available in specifying the nature and physical content of the "matter tensor" in the Einstein equation.
Abstract: The various versions of the scalar-tensor theory (e.g., the theories of Jordan, Hoyle, and Brans-Dicke) are derived from a general variational principle. It is shown that scalar-conformal transformations not only interconvert the various current versions of the scalar-tensor theory (i.e., Brans-Dicke theory \ensuremath{\rightleftarrows} Hoyle steady-state theory), but also convert the scalar-tensor variational principle into the variational principle of general relativity. The scalar-tensor formalism is therefore implicitly embodied in the theory of general relativity, thus illustrating the considerable freedom available in specifying the nature and physical content of the "matter tensor" in the Einstein equation.
26 citations
TL;DR: In this paper, all solutions of Einstein's field equations which represent a static perfect fluid are considered and a number of results for vacuum fields obtained by Ehlers and Kundt are generalized.
Abstract: All solutions of Einstein's field equations which represent a static perfect fluid are considered and a number of results for vacuum fields obtained by Ehlers and Kundt are generalized. All such solutions with a degenerate Weyl tensor are found explicitly. As is the case with vacuum fields of Ehlers and Kundt all the solutions obtained are found to admit at least a two-parameter group of local isometries.
01 Jan 1972
TL;DR: In this paper, the Heisenberg equations of motion for the spin-1/2 wave equation in general relativity were obtained by a covariant procedure, and they were found to be similar to the equations for a classical pole-dipole test-particle.
Abstract: The Heisenberg equations of motion for the spin-1/2 wave equation in general relativity are obtained by a covariant procedure. They are found to be similar to the equations of motion for a classical pole-dipole test-particle in general relativity. The identification is complete when the Heisenberg equations are taken to be satisfied by the respective expectation values.
TL;DR: In this article, it is shown that the Riemann curvature tensor has twenty independent components, ten of which appear in the Weyl tensor, and nine of these components appearing in the Einstein tensor.
Abstract: In a four-dimensional curved space-time it is well-known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley-Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.
TL;DR: In this paper, the authors consider the static, spherically symmetric, perfect fluid solution for a given massm in general relativity, and show that its average density is bounded.
Abstract: We consider the static, spherically symmetric, perfect fluid solution for a given massm in general relativity, and show that its average density is bounded. We then show by an example that this need not be so for non-spherical bodies.
TL;DR: In this article, all space-times admitting a neutrino radiation field are obtained, characterised by the Weyl tensor being of type D, N (or O) or III.
Abstract: All space-times admitting a neutrino radiation field are obtained. Three classes of such space-times exist, characterised by the Weyl tensor being of type D, N (or O) or III.
TL;DR: In this article, an attempt is made to clarify the physical and the mathematical reasonings that underlie Einstein's laws of gravitation, and it appears that only by a mixture of physical reasonableness, mathematical simplicity, and aesthetic sensibility can one arrive at Einstein's field equations.
Abstract: An attempt is made to clarify the physical and the mathematical reasonings that underlie Einstein's laws of gravitation. It appears that only by a mixture of physical reasonableness, mathematical simplicity, and aesthetic sensibility can one arrive at Einstein's field equations. The general theory of relativity is in fact an example of “the power of speculative thought.” The topics considered include a discussion of the principle of equivalence and the view of space-time as a geometric manifold. Two “derivations” of Einstein's equations are given: one based on physical reasonableness and the other based on a variational principle and mathematical simplicity.
TL;DR: In this article, exact static solutions to the field equations of general relativity that represent axially symmetric distributions of isolated bodies have been developed; the solutions obtained do not involve negative mass; the gravitational attraction between the bodies is counterbalanced by the Coulomb repulsion of electrical charges placed on the bodies.
Abstract: We develop exact static solutions to the field equations of general relativity that represent axially symmetric distributions of isolated bodies. The solutions obtained do not involve negative mass; the gravitational attraction between the bodies is counterbalanced by the Coulomb repulsion of electrical charges placed on the bodies. Since the assignment of physical interpretations to axially symmetric solutions is not straightforward in general relativity, we also present additional arguments showing that our multiparticle interpretation is indeed correct. Thus we analyse, for the case of two isolated particles, the behavior of the differential invariants and the geodesic motion of test particles, and for all the examples considered we find agreement with our multiparticle interpretation. Our solutions also contain nonsingular event horizons, which we relate to an electrovac theorem recently proved by Israel.
TL;DR: In this paper, the authors consider a metric which admits a four-parameter isometry group and obtain exact solutions to the field equations for a tachyon dust particle with spacelike 4-velocities.
Abstract: Special relativity allows the possibility of a class of particles, called tachyons, which travel with speeds greater than the speed of light in vacuum. These particles have spacelike 4‐velocities. Since tachyons have energy and momentum, they will contribute to the gravitational field through the energy‐momentum tensor. One question then is what types of solutions to the Einstein field equations will tachyons yield. We consider a metric which admits a four‐parameter isometry group. When this metric is used in the field equations using a dust energy‐momentum tensor, solutions exist only for spacelike 4‐velocity of the dust. We interpret these as solutions for a tachyon dust. Exact solutions to the field equations are obtained.
TL;DR: In this article, it was shown that the Newtonian potential that generates the static Schwarschild and Reissner-Nordstrom metrics also generates the NUT space metric from the class of Papapetrou stationary fields.
TL;DR: In this paper, the Plebanski and Stachel and Goenner-Stachel lists of metrics which are solutions of Einstein's field equations, have two double eigenvalues and admit 3-parameter groups of isometries with 2-dimensional spacelike orbits are completed by the addition of metrics that result from the use of a more general metric form.
Abstract: The Plebanski and Stachel and Goenner and Stachel lists of metrics which are solutions of Einstein's field equations, have two double eigenvalues and admit 3-parameter groups of isometries with 2-dimensional spacelike orbits are completed by the addition of metrics which result from the use of a more general metric form.
TL;DR: In this article, the one-dimensional general relativistic motion of a falling body in a uniform gravitational field is obtained, where the inertial mass of the particle is a function of its velocity and position.
Abstract: The one-dimensional general relativistic motion of a falling body in a uniform gravitational field is obtained. The inertial mass of the particle is a function of its velocity and position. The results are compared with the special relativity and Newtonian motions.
TL;DR: In this paper, a method to obtain a class of exact solutions for uniform fluid spheres surrounded by empty space is presented, where it is shown that there is a reversal in the motion of contraction or expansion of the sphere, while in other cases there is no bouncing at all.
Abstract: The author presents a method to obtain a class of exact solutions for uniform fluid spheres surrounded by empty space. It is shown that in some cases there is a reversal in the motion of contraction or expansion of the sphere, while in other cases there is no bouncing at all.
TL;DR: In this paper, it was shown that Einstein equations cannot hold as operator equations if written in terms of a potential hµν(D) which is a weakly local field.
Abstract: The problem of formulating a local quantum theory of Einstein equations is examined. It is proved that Einstein equations cannot hold as operator equations if written in terms of a potentialhµν(D) which is a weakly local field. This result is independent of the kind of metric chosen in the Hilbert space and it doesn't require covariance ofhµν.
TL;DR: In this paper, a 3+1-dimensional decomposition of the Einstein field equations is obtained for a general spacetime, and the resulting equations treat kab as the metric in the space-like hypersurfacest=constant.
Abstract: A new (3+1)-dimensional decomposition of the Einstein gravitational field equations is obtained for a general spacetime. The metric is taken in the form
$$ds^2 = e^{ - 2u} k_{ab} (dx^a + \xi ^a dt)(dx^b + \xi ^b dt) - c^2 e^{2u} dt^2 $$
and the resulting equations treatkab as the metric in the space-like hypersurfacest=constant. It is shown that this decompostion forms a more convenient starting point for slow motion approximations than does their usual 4-dimensional formulation. This is illustrated by a derivation of the first post-Newtonian approximation to the field equations, the simplicity there resulting fromkab being still flat to this order.