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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.


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TL;DR: In this article, a method which uses a generalized tensorial $\zeta$-function to compute the renormalized stress tensor of a quantum field propagating in a (static) curved background is presented.
Abstract: A method which uses a generalized tensorial $\zeta$-function to compute the renormalized stress tensor of a quantum field propagating in a (static) curved background is presented. The starting point of the method is the direct computation of the functional derivatives of the Euclidean one-loop effective action with respect to the background metric. This method, when available, gives rise to a conserved stress tensor and produces the conformal anomaly formula directly. It is proven that the obtained stress tensor agrees with statistical mechanics in the case of a finite temperature theory. The renormalization procedure is controlled by the structure of the poles of the stress-tensor $\zeta$ function. The infinite renormalization is automatic and is due to a ``magic'' cancellation of two poles. The remaining finite renormalization involves conserved geometrical terms arising by a certain residue. Such terms renormalize coupling constants of the geometric part of Einstein's equations (customary generalized through high-order curvature terms). The method is checked on particular cases (closed and open Einstein`s universe) finding agreement with other approaches. The method is also checked considering a massless scalar field in the presence of a conical singularity in the Euclidean manifold (i.e. Rindler spacetimes/large mass black hole manifold/cosmic string manifold). There, the method gives rise to the stress tensor already got by the point-splitting approach for every coupling with the curvature regardless of the presence of the singular curvature. Comments on the measure employed in the path integral, the use of the optical manifold and different approaches to renormalize the Hamiltonian are made.

42 citations

Journal ArticleDOI
TL;DR: In this article, Wong's canonical coordinate form of torsion-free connections with skew-symmetric Ricci tensors on surfaces is extended to connections with torsions.
Abstract: Some known results on torsionfree connections with skew-symmetric Ricci tensor on surfaces are extended to connections with torsion, and Wong’s canonical coordinate form of such connections is simplified

42 citations

Journal ArticleDOI
TL;DR: In this article, a recursive algorithm for the computation of the complete asymptotic series, for small time, of the amount of heat inside a domain with smooth boundary in a Riemannian manifold was given.
Abstract: We give a recursive algorithm for the computation of the complete asymptotic series, for small time, of the amount of heat inside a domain with smooth boundary in a Riemannian manifold; we consider arbitrary smooth initial data, and we impose Dirichlet condition on the boundary. When the Ricci curvature of the domain and the mean curvature of its boundary are both nonnegative, we also give sharp upper and lower bounds of the heat content which hold for all values of time. These estimates extend to convex sets of the Euclidean space having arbitrary boundary.

42 citations

Journal ArticleDOI
TL;DR: In this paper, a scalar curvature invariant which is quartic in the second derivatives of the Riemann tensor has been shown to describe radiation fields outside bounded sources.
Abstract: Scalar curvature invariants are studied in type N solutions of vacuum Einstein's equations with in general non-vanishing cosmological constant Lambda. Zero-order invariants which include only the metric and Weyl (Riemann) tensor either vanish, or are constants depending on Lambda. Even all higher-order invariants containing covariant derivatives of the Weyl (Riemann) tensor are shown to be trivial if a type N spacetime admits a non-expanding and non-twisting null geodesic congruence. However, in the case of expanding type N spacetimes we discover a non-vanishing scalar invariant which is quartic in the second derivatives of the Riemann tensor. We use this invariant to demonstrate that both linearized and the third order type N twisting solutions recently discussed in literature contain singularities at large distances and thus cannot describe radiation fields outside bounded sources.

42 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202364
2022152
2021169
2020163
2019174
2018180