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.

Almost-Schur lemma

Abstract: Schur’s lemma states that every Einstein manifold of dimension n ≥ 3 has constant scalar curvature. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci tensor is assumed to be small rather than identically zero. In particular, we provide an optimal L2 estimate under suitable assumptions and show that these assumptions cannot be removed.

The effective gravitational Lagrangian and the energy-momentum tensor in the inflationary Universe

Abstract: The inflationary model proposed by Guth (1981), is based on non-classical behaviour of the energy-momentum tensor. The authors try to evaluate the corrections to the usual linear relation between the scalar curvature and the trace of the energy-momentum tensor, R approximately T, replacing the usual Einstein-Hilbert Lagrangian R by an unknown function f(R) and imposing the inflationary solution upon the scale factor a(t) of the Robertson-Walker metric. Solving for f(R) enables one to evaluate the corrections to the relation R approximately T, which may be developed in powers of R or T at will.

Complexity factor for charged spherical system

Abstract: In this paper, we study the complexity factor for a charged anisotropic self-gravitating object. We formulate the Einstein–Maxwell field equations, Tolman–Opphenheimer–Volkoff equation, and the mass function. We form the structure scalars by the orthogonal splitting of the Riemann tensor and then find the complexity factor with the help of these scalars. Finally, we investigate some astrophysical objects for the vanishing of complexity condition. It is found that the presence of the electromagnetic field decreases the complexity of the system.

Primordial tensor modes of the early Universe

Abstract: We study cosmological tensor perturbations on a quantized background within the hybrid quantization approach. In particular, we consider a flat, homogeneous and isotropic spacetime and small tensor inhomogeneities on it. We truncate the action to second order in the perturbations. The dynamics is ruled by a homogeneous scalar constraint. We carry out a canonical transformation in the system where the Hamiltonian for the tensor perturbations takes a canonical form. The new tensor modes now admit a standard Fock quantization with a unitary dynamics. We then combine this representation with a generic quantum scheme for the homogeneous sector. We adopt a Born-Oppenheimer ansatz for the solutions to the constraint operator, previously employed to study the dynamics of scalar inhomogeneities. We analyze the approximations that allow us to recover, on the one hand, a Schr\"odinger equation similar to the one emerging in the dressed metric approach and, on the other hand, the ones necessary for the effective evolution equations of these primordial tensor modes within the hybrid approach to be valid. Finally, we consider loop quantum cosmology as an example where these quantization techniques can be applied and compare with other approaches.