Showing papers on "Introduction to the mathematics of general relativity published in 1974"
768 citations
TL;DR: In this article, all one-loop divergences of the quantized Einstein-Maxwell system are calculated, using dimensional regularization and the background-field method, and the resulting counterterms including photon and graviton ghost contributions, are quadratic in curvatures when evaluated on the mass shell, and cannot be renormalized away by rescaling.
Abstract: All one-loop divergences of the quantized Einstein-Maxwell system are calculated, using dimensional regularization and the background-field method. The resulting counterterms, including photon and graviton ghost contributions, are quadratic in curvatures when evaluated on the mass shell, and cannot be renormalized away by rescaling. Brans-Dicke theory is also nonrenormalizable; source-free Einstein theory with cosmological term is (formally) renormalizable, as are Weyl models.
344 citations
TL;DR: The field equations of general relativity with spin and torsion are considered to describe correctly the gravitational properties of matter on a microphysical level in this paper, and it is shown how the singularity theorems of Penrose and Hawking must be modified to apply in this field equation.
Abstract: The field equations of general relativity with spin and torsion (${U}_{4}$ theory) are considered to describe correctly the gravitational properties of matter on a microphysical level. By an averaging procedure one arrives at a macroscopic field equation, which under normal matter densities coincides with Einstein's equation of conventional general relativity. For very high matter densities, even if the spin are randomly distributed, Einstein's equation breaks down and ${U}_{4}$ theory must be applied. It is shown how the singularity theorems of Penrose and Hawking must be modified to apply in ${U}_{4}$ theory. All known cosmological models in ${U}_{4}$ theory which prevent singularities are shown to violate an energy condition of a singularity theorem.
204 citations
TL;DR: The initial value equations of general relativity are formulated as a system of four coupled quasilinear elliptic equations as discussed by the authors, which result from a covariant orthogonal decomposition of symmetric tensors and a generalized technique of conformal deformation.
Abstract: The initial-value equations of Einstein's theory of general relativity are formulated as a system of four coupled quasilinear elliptic equations. These equations result from a covariant orthogonal decomposition of symmetric tensors and a generalized technique of conformal deformation of initial data. Mathematical properties and global integrability conditions of the equations are discussed. Physical interpretation of the independent and dependent data is given for both spatially closed and asymptotically flat initial-data sets. In the latter case, the four dependent functions constitute long-range scalar and vector potentials which determine the total mass and total linear and angular momenta of an isolated system. The definitions of linear and angular momenta suggest a unique extension to asymptotically flat three-spaces of the group of translations and rotations of flat three-space. In turn, the "almost symmetries" thus defined lead to Gaussian theorems expressing the equality of certain surface and volume integrals for total linear and angular momenta. An interpretation of the scalar and vector potentials for closed three-spaces is also given. In the Appendix we treat the special case of conformally flat initial data.
157 citations
TL;DR: In this article, exact exterior solutions of a rotating infinite cylinder were obtained, in addition to two different solutions obtained before, and a new solution was found which is finite on the axis.
Abstract: We obtain exact exterior solutions of a rotating infinite cylinder. In addition to two different solutions obtained before, a new solution is found which is finite on the axis. It seems that these three solutions exhaust the possibilities.
151 citations
TL;DR: The interactions between the quantized Einstein and quantized Yang-Mills fields are one-loop non-renormalizable as discussed by the authors, and the interactions between these two fields are non-linear.
53 citations
Journal Article•
53 citations
TL;DR: In this article, an exact solution to the Einstein-Dirac equations for a static, plane-symmetric spacetime generated by neutrinos is presented, which correspond to a neutrino current along the symmetry axis of the space.
Abstract: An exact solution to the Einstein-Dirac equations is presented for a static, plane-symmetric spacetime generated by neutrinos. We find the neutrino field to be nonzero and correspond to a neutrino current along the symmetry axis of the space. The neutrinos yield a zero energy-momentum tensor and therefore the gravitational field is exactly the same as for the vacuum case. A comparison with other solutions is presented along with a discussion of the possible physical significance of this "ghost neutrino" field.
43 citations
35 citations
TL;DR: In this article, a class of algebraically special space-times with twisting rays is studied, where the Weyl tensor satisfies the peeling-off property along the repeated principal null congruence, and the Ricci tensor exhibits an equally simple asymptotic behavior.
Abstract: A recently characterized class of (in general nonvacuum) algebraically special space‐times with twisting rays is studied. The Weyl tensor satisfies the peeling‐off property along the repeated principal null congruence, and the Ricci tensor exhibits an equally simple asymptotic behavior, which is in fact compatible (via the Einstein field equations) with the presence of a suitably restricted electromagnetic or neutrino field. If the gravitational and source fields are nonradiative, the above asymptotic behavior is restricted. In this case we explicitly solve the Einstein vacuum field equations, the Einstein‐Maxwell equations and the Einstein‐Weyl (combined gravitational‐neutrino) equations. The solutions obtained are related to the known algebraically special solutions of these equations.
26 citations
TL;DR: In this article, an analogy between the Lienard-Wiechert solutions of the Maxwell equations and the Robinson-Trautman solution of the Einstein equations was established by virtue of the fact that a principal null vector field of either the Maxwell or Weyl tensor in each case satisfies the following four conditions: (1) the field is a geodesic field, (2) it has non-vanishing divergence, (3) it is shear free, and (4) twist (or curl) free.
Abstract: An analogy is established between the Lienard‐Wiechert solutions of the Maxwell equations and the Robinson‐Trautman solutions of the Einstein equations by virtue of the fact that a principal null vector field of either the Maxwell or Weyl tensor in each case satisfies the following four conditions: (1) The field is a geodesic field, (2) it has nonvanishing divergence, (3) it is shear free, and (4) it is twist (or curl) free.
TL;DR: In this paper, it was shown that Teukolsky's equation can be derived from a second-order wave equation for the Riemann tensor in a way that emphasizes the modern tensor analysis content of the Newman-Penrose formalism.
Abstract: It is shown that Teukolsky's equation can be derived from a second-order wave equation for the Riemann tensor. The derivation is done in a way that emphasizes the modern tensor-analysis content of the Newman-Penrose formalism.
TL;DR: In this paper, all spherically symmetric static space times for which a nonredundant stationary Killing tensor of rank two exists are found and discussed, including explicit forms for the line elements, the Killing tensors, and corresponding quadratic constants of geodesic motion.
Abstract: All spherically symmetric static space‐times for which a nonredundant stationary Killing tensor of rank two exists are found and discussed. The results include explicit forms for the line elements, the Killing tensors, and the corresponding quadratic constants of geodesic motion. Included as special cases are relativistic analogs of the Lenz vector and of the quadratic constant of the motion peculiar to the spatial harmonic oscillator.
TL;DR: In this paper, the general solution of Einstein's equations for a stationary cylindrically symmetric distribution of pressure-free matter is obtained. But it contains a function which may be freely prescribed.
Abstract: The general solution of Einstein's equations for a stationary cylindrically symmetric distribution of pressure-free matter is obtained. It contains a function which may be freely prescribed. Using this freedom examples are given of new types of singularity in General Relativity.
TL;DR: In this paper, an exact static solution of Einstein's field equations of general relativity in the presence of zero-rest-mass scalar fields has been obtained when both the metric tensor gij and the zero-position scalar field φ exhibit plane symmetry in the sense of Taub [9].
Abstract: An exact static solution of Einstein's field equations of general relativity in the presence of zero-rest-mass scalar fields has been obtained when both the metric tensor gijand the zero-rest-mass scalar field φexhibit plane symmetry in the sense of Taub [9]. Our solution generalizes the empty space-time solution with plane symmetry previously obtained by Taub to the situation when static zero-rest-mass scalar fields are present. The static plane symmetric solutoins of Einstein's field equations in the presence of massive scalar fields, and the difference between the massless and non-massless scalar fields are being investigated, and will be published separately later on. We also hope to discuss non-static plane symmetric solutions of Einstein's field equations in the presence of scalar fields in future.
TL;DR: The exterior solution for a radiating sphere in general relativity is known as mentioned in this paper, and the complete exterior solution when the radiating spheres is charged is presented in the paper "Exterior Solution for a Radial Sphere in General Relativity".
Abstract: The exterior solution for a radiating sphere in general relativity is known. The complete exterior solution when the radiating sphere is charged is presented.
TL;DR: In this article, a new approach to the problem of motion in General Relativity, based upon the systematic approximation procedure of Synge, is presented, where equations of transnational motion for a system of spherical bodies moving under their mutual gravitational attractions are derived.
Abstract: A new approach to the problem of motion in General Relativity, based upon the systematic approximation procedure of Synge, is presented. The equations of transnational motion for a system of spherical bodies moving under their mutual gravitational attractions are derived. Approximations are based upon the weakness of the field and on the distance between any two of the bodies being considered large by comparision with their radii. The most general stress distribution consistent with maintaining the symmetry of the bodies throughout the motion is chosen. The use of controlled errors enables us to derive equations of motion applicable to a wider class of physical systems than the original equations of Einstein, Infeld and Hoffmann and Fock-Papapetrou.
TL;DR: In this paper, the constraints equations of General Relativity are reduced on an initial maximal submanifold, by the use of conformai techniques, to a non-linear elliptic equation for the conformal factor φ.
Abstract: The constraints equations of General Relativity are reduced on an initial maximal submanifold, by the use of conformai techniques, to a non-linear elliptic equation for the conformal factor φ. Some existence, uniqueness, and nonexistence theorems are proved for this equation, in the case of closed manifolds, and also for open manifolds (in particular for manifolds homeomorphic to ℝ3).
TL;DR: Some similarities between P-P wave space-times and complex recurrent and conformally symmetric space times in general relativity are discussed in this article, where the similarities between the two types of wave spaces are discussed.
Abstract: Some similarities between 'P-P' wave space-times and complex recurrent and conformally symmetric space-times in general relativity are discussed.
TL;DR: In this paper, it was shown how General Relativity may be derived from a conformally invariant theory of gravitation in interaction with matter after recognising the existence of a family of canonical gauges.
TL;DR: The most general Lagrange density for which the associated Euler-Lagrange equations are precisely Maxwell's equations is obtained in this paper, which is more general than the Lagrangian which is commonly used, it still has essentially the same energy momentum tensor.
Abstract: The most general Lagrange density (which is a concomitant of the metric tensor together with a vector field and its first derivatives) for which the associated Euler-Lagrange equations are precisely Maxwell's equations is obtained Although it is more general than the Lagrangian which is commonly used, it still has essentially the same energy momentum tensor
TL;DR: In this paper, the Einstein field equations are solved subject to the assumptions that the source of the gravitational field is a non-rotating perfect fluid, the Weyl tensor is algebraically special, and the repeated principal null direction is tangent to a geodesic, shearfree and twistfree congruence.
Abstract: The Einstein field equations are solved subject to the assumptions that (1) the source of the gravitational field is a non-rotating perfect fluid, (2) the Weyl tensor is algebraically special, (3) the repeated principal null direction is tangent to a geodesic, shearfree and twistfree congruence, and is parallely transferred along the fluid congruence. The solutions in which the line-element admits a multiply transitive group of motions have been studied by Stewart and Ellis; the remaining ones are new, and appear to represent inhomogeneous anisotropic cosmological models.
TL;DR: In this paper, a criterion for stability of a charged sphere is obtained by use of two different methods, i.e., stability criterion for the stability of the charged sphere and stability of its dynamics.
Abstract: A criterion for stability of a charged sphere is obtained by use of two different methods. The result is applied to a charged dust in order to investigate its stability.
TL;DR: In this paper, general-relativistic analogues of some classical theorems and equations of magneto-hydrodynamics are found and their relevance to astrophysics is briefly discussed.
Abstract: General-relativistic analogues of some classical theorems and equations of magneto-hydrodynamics are found and their relevance to astrophysics is briefly discussed.