![]() It sports a really nice graphical interface with plenty of tools. Tokyo Press, pp.VTrace Portable is a very nice application that comes with plenty of features including ping options, traceroute, IANA information (WhoIs, ASN for BGP systems), DNS records (like nslookup or DIG), geographical placement, open TCP ports (simple port scan). (1969), "The curvature of 4-dimensional Einstein spaces", Global Analysis (Papers in Honor of K. ![]() (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 2-9. Petersen, Peter (2006), Riemannian geometry, Graduate Texts in Mathematics, vol. 171 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 0387292462, MR 2243772.(1973), The Large Scale Structure of Space-Time, Cambridge University Press, ISBN 6-4 "On the Significance of the Weyl Curvature in a Relativistic Cosmological Model". ![]() Christoffel symbols provides a coordinate expression for the Weyl tensor.The valence (0,3) tensor on the right-hand side is the Cotton tensor, apart from the initial factor. The (0,4) valence Weyl tensor is then ( Petersen 2006, p. 92)Ĭ = R − 1 n − 2 ( R i c − s n g ) ∧ ◯ g − s 2 n ( n − 1 ) g ∧ ◯ g This is most easily done by writing the Riemann tensor as a (0,4) valence tensor (by contracting with the metric). The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This fact was a key component of Nordström's theory of gravitation, which was a precursor of general relativity. If the Weyl tensor vanishes in dimension ≥ 4, then the metric is locally conformally flat: there exists a local coordinate system in which the metric tensor is proportional to a constant tensor. In dimensions ≥ 4, the Weyl curvature is generally nonzero. In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. More generally, the Weyl curvature is the only component of curvature for Ricci-flat manifolds and always governs the characteristics of the field equations of an Einstein manifold. ![]() In general relativity, the Weyl curvature is the only part of the curvature that exists in free space-a solution of the vacuum Einstein equation-and it governs the propagation of gravitational waves through regions of space devoid of matter. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Measure of the curvature of a pseudo-Riemannian manifold ![]()
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