Showing papers on "Riemann curvature tensor published in 1968"

Feynman rules for the gravitational field from the coordinate independent field theoretic formalism

Abstract: By using the method developed in the preceding paper, the Feynman-DeWitt perturbation expansion for the gravitational $S$ matrix is shown to follow from the field-theoretic formalism. Again our methd is to express the path-dependent Green's functions in terms of auxiliary, path-independent Green's functions, in such a way that the path-dependence equation is automatically satisfied. The formula relating the path-dependent to the path-independent Green's functions will be similar to the classical formula relating the path-dependent Riemann tensor to the metric tensor. The equations for the auxiliary Green's functions are found and solved in a perturbation series. If the result is expressed as a sum of Feymann diagrams, one obtains the expected vertices, together with closed loops of fictitious vector particles.

On hypersurfaces satisfying a certain condition on the curvature tensor

Abstract: where the endomorphism R(X, Y) operates on R as a derivation of the tensor algebra at each point of M. Conversely, does this algebraic condition (•*) on the curvature tensor field R imply that M is locally symmetric (i.e. Vi? = 0) ? We conjecture that the answer is affirmative in the case where M is irreducible and complete and d i m M ^ 3 . For partial and related results, see [4], p.ll, [9], Theorem 8, and [6]. The main purpose of the present paper is to give an affirmative answer in the case where M is a complete hypersurface in a Euclidean space. More precisely, we prove

Lorentzian $4$ dimensional manifolds with ``local isotropy''

Abstract: We define “locally isotropic” spaces, as spaces in which there exists, in the tangent space at each pointP, a subgroupA (P) (of dimension at least 1) of the Lorentz groupL + ↑ , leaving the Riemann tensor and its 2 first covariant derivatives invariant; the subgroupsA(P) are assumed to be conjugate inL + ↑ . These spaces admit a group of local isometriesG. IfI P denotes the subgroup ofG leavingP fixed, thendA (P)=I P . All spaces of petrov type D, admitting local isotropy are determined.

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.

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.

Conformal tensor discontinuities in general relativity

Abstract: The postulate is made that across a given hypersurfaceN the metric and its first derivatives are continuous. This postulate is used to derive conditions which must be satisfied by discontinuities in the Riemann tensor acrossN. These conditions imply that the conformal tensor jump is uniquely determined by the stress-energy tensor discontinuity ifN is non-null (and to within an additive term of type Null ifN is lightlike). Alternatively,\([C^{\alpha \beta } _{\gamma \delta } ]\) and [R] determine\(\left[ {R_{\mu v} - \frac{1}{4}Rg_{\mu v} } \right]\) ifN is non-null. These relationships between the conformal tensor and stress-energy tensor jumps are given explicitly in terms of a three-dimensional complex representation of the antisymmetric tensors. Application of these results to perfect-fluid discontinuities is made:\([C^{\alpha \beta } _{\gamma \delta } ]\) is of type D across a fluid-vacuum boundary and across an internal, non-null shock front.\([C^{\alpha \beta } _{\gamma \delta } ]\) is of type I (non-degenerate) in general across fluid interfaces across which no matter flows, except for special cases.

The algebraic structure of the vacuum Riemann tensor

Abstract: A (2j+1)-spinor formalism is used to discuss the Bel-Petrov-Penrose classification of the Weyl conformal tensor A convenient pictorial representation of this classification is presented in the form of a series of intersecting manifolds nested in a four-dimensional projective space The relation to other formalisms is considered briefly