Riemann curvature tensor

About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.

Spectra of Compact Locally Symmetric Manifolds of Negative Curvature.

Abstract: Let S be a Riemannian symmetric space of noncompact type, and let G be the group of motions of S. Then the algebra L-~ of G-invariant differential operators on S is commutative, and its spectrum A(S) can be canonically identified with ~/w where ~ is a complex vector space with dimension equal to the rank of S, and to is a finite subgroup of G L ( ~ ) generated by reflexions. Let P be a discrete subgroup of G that acts freely on S and let X = E \\ S . Then the members of 5~ may be regarded as differential operators on X. Let us now assume that X is compact and define the spectrum A of X as the set of those elements of A(S) for which one can find a nonzero eigenfunction defined on X. In this paper we study the relationship of A to the geometry of X and determine the asymptotic growth of A as a subset of A(S). In subsequent papers we plan to study the asymptotic behaviour of the eigenfunctions and to examine the problem of obtaining improvements on the error estimates. It is well-known that G, which is transitive on S, is a connected real semisimple Lie group with trivial center, and that the stabilizers in G of the points of S are the maximal compact subgroups of G. So we can take S = G/K, X =F\\G/K, where K is a fixed maximal compact subgroup of G, and F is a discrete subgroup of G containing no elliptic elements (= elements conjugate to an element of K) other than e, such that F\\G is compact. Let G = K A N be an Iwasawa decomposit ion of G; let o be the Lie algebra of A; and let to be the Weyl group of (G, A). If we take ,~to be the dual of the complexification a c of a, then A ( S ) ~ / w canonically. In what follows we shall commit an abuse of notation and identify A(S) with ,~, but with the proviso that points of ~ in the same w-orbit represent the same element of A(S).

Bianchi identities in higher dimensions

Abstract: A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension n. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in n − 4 spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers–Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.

Holography of gravitational action functionals

Abstract: Einstein-Hilbert (EH) action can be separated into a bulk and a surface term, with a specific (``holographic'') relationship between the two, so that either can be used to extract information about the other. The surface term can also be interpreted as the entropy of the horizon in a wide class of spacetimes. Since EH action is likely to just the first term in the derivative expansion of an effective theory, it is interesting to ask whether these features continue to hold for more general gravitational actions. We provide a comprehensive analysis of Lagrangians of the form $\sqrt{\ensuremath{-}g}L=\sqrt{\ensuremath{-}g}Q_{a}{}^{bcd}R^{a}{}_{bcd}$, in which $Q_{a}{}^{bcd}$ is a tensor with the symmetries of the curvature tensor, made from metric and curvature tensor and satisfies the condition ${\ensuremath{ abla}}_{c}Q_{a}{}^{bcd}=0$, and show that they share these features. The Lanczos-Lovelock Lagrangians are a subset of these in which $Q_{a}{}^{bcd}$ is a homogeneous function of the curvature tensor. They are all holographic, in a specific sense of the term, and---in all these cases---the surface term can be interpreted as the horizon entropy. The thermodynamics route to gravity, in which the field equations are interpreted as $TdS=dE+pdV$, seems to have a greater degree of validity than the field equations of Einstein gravity itself. The results suggest that the holographic feature of EH action could also serve as a new symmetry principle in constraining the semiclassical corrections to Einstein gravity. The implications are discussed.

Republication of: On the physical significance of the Riemann tensor

Abstract: This Golden Oldie is a reprinting of a paper by F. A. E. Pirani first published in 1956. It is accompanied by a reprinting of a paper by J. L. Synge first published in 1934. Together these papers pointed the way to the interpretation of geodesic deviation and its relation to the curvature tensor. These two Golden Oldies are accompanied by an Golden Oldie Editorial containing an editorial note written by A. Trautman, and by the biography of F. Pirani written by himself and commented by A. Trautman.