Riemann-Christoffel tensor

Riemann-Christoffel tensor

[′rē‚män ′kris·tə·fəl ‚ten·sər]
(mathematics)
The basic tensor used for the study of curvature of a Riemann space; it is a fourth-rank tensor, formed from Christoffel symbols and their derivatives, and its vanishing is a necessary condition for the space to be flat. Also known as curvature tensor.
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References in periodicals archive ?
Lanczos, "A remarkable property of the Riemann-Christoffel tensor in four dimensions," Annals of Mathematics, vol.
Field Equations for Lovelock Gravity: An Alternative Route
Calculating the physically observable components of the Riemann-Christoffel tensor [X.sub.ik] (28) for the de Sitter vacuum bubble, we find
A telemetric multispace formulation of Riemannian geometry, general relativity, and cosmology: implications for relativistic cosmology and the true reality of time
If we use these vectors, the Riemann-Christoffel tensor can be written (see also Section 4.1, Eqs (A1)-(A5)) as
In the difference appearing in the Riemann-Christoffel tensor only four terms remain therefore:
Deformed surfaces in holographic interferometry. Similar aspects in general gravitational fields/Deformeeritud pinnad holograafilises interferomeetrias. Sarnased aspektid uldistes gravitatsioonivaljades
For example: [G.sub.[Mu][Nu]] is obtained from the Riemann-Christoffel tensor, that is, from the quantity which largely determines the structure of space-time, by the most straightforward method, namely by the contraction of two indices.
Eddington's Search for a Fundamental Theory: A Key to the Universe
Moreover, Zelmanov proven that any non-holonomic space has nonzero Riemannian curvature (nonzero Riemann-Christoffel tensor) due to [g.sub.0i] [not equal to] 0.
PLANCK, the satellite: a new experimental test of general relativity
He followed that procedure by which the Riemann-Christoffel tensor was built: proceeding from the non-commutativity of the second derivatives of an arbitrary vector
A theory of the Podkletnov effect based on general relativity: anti-gravity force due to the perturbed non-holonomic background of space
Briefly, as one calculates the Ricci tensor [R.sub.[alpha][beta]] = [g.sup.[mu]v][R.sub.[alpha][mu]v[beta]] by the contraction of the Riemann-Christoffel tensor
Correct linearization of Einstein's equations
Our solution of the deviation equations depends on a specific formula for the space metric whereby we calculate the Riemann-Christoffel tensor. Because the sources of gravitational waves (double stars, pulsars, etc.) are far away from us, we expect received gravitational waves to be weak and plane.
Exact theory of a gravitational wave detector. New experiments proposed
He followed that procedure by which the Riemann-Christoffel tensor was built: proceeding from the noncommutativity of the second derivatives of an arbitrary vector
Zelmanov's Anthropic Principle and the Infinite Relativity Principle
As is well known, the components of the Riemann-Christoffel tensor satisfy the identities
Gravitational waves and gravitational inertial waves in the general theory of relativity: a theory and experiments
To answer the question let us recall that Zelmanov, following the same procedure by which the Riemann-Christoffel tensor was introduced, after considering non-commutativity of the chr.inv.-second derivatives of a vector *[[nabla].sub.i] *[[nabla].sub.k] [Q.sub.l] - +[[nabla].sub.k]*[[nabla].sub.i][Q.sub.l] = 2[A.sub.ik]/[c.sup.2] *[partial derivative][Q.sub.l]/[partial derivative]t + [H.sup....j.sub.lki][Q.sub.j], had obtained the chr.
A new method to measure the speed of gravitation

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