Weyl tensor

The Weyl tensor or Weyl curvature tensor is a 4th order tensor , which in general relativity theory (ART) takes on the role of the Riemann curvature tensor in the field equations for matter - free space (vacuum solutions). It is named after Hermann Weyl . ${\ displaystyle C_ {abcd}}$ ${\ displaystyle R_ {abcd}}$

Like the Riemann curvature tensor, it expresses the gravitational tidal forces that are exerted on a freely falling, extended body as part of the ART. It is formed from the Riemann curvature tensor by subtracting various traces (generated with tensor tapering ) so that the Weyl curvature tensor, in contrast to the full Riemann curvature tensor, is trace-free. In addition, in contrast to the full Riemann curvature tensor, it expresses the shape changes caused by the tidal forces, but does not record the volume change that the Ricci tensor describes, which is created by simple trace formation from the Riemann curvature tensor. The Weyl tensor of the ART agrees in the matter-free space (vacuum solutions of the field equations), where the Ricci tensor vanishes, with the Riemann curvature tensor and thus describes the propagation of gravitational waves .

The Weyl tensor is zero in or dimensions. It is generally non-zero in four and more dimensions. ${\ displaystyle n = 2}$${\ displaystyle n = 3}$

In tensor notation, the Weyl curvature tensor is:

${\ displaystyle C_ {abcd} = R_ {abcd} - {\ frac {1} {n-2}} \ left (R_ {ac} g_ {bd} -R_ {ad} g_ {bc} + R_ {bd} g_ {ac} -R_ {bc} g_ {ad} \ right) + {\ frac {1} {(n-1) (n-2)}} \ left (g_ {ac} g_ {bd} -g_ { ad} g_ {bc} \ right) R}$,

and in ART with : ${\ displaystyle n = 4}$

${\ displaystyle C_ {abcd} = R_ {abcd} - {\ frac {1} {2}} \ left (R_ {ac} g_ {bd} -R_ {ad} g_ {bc} + R_ {bd} g_ { ac} -R_ {bc} g_ {ad} \ right) + {\ frac {1} {6}} \ left (g_ {ac} g_ {bd} -g_ {ad} g_ {bc} \ right) R}$.

Here are the metric tensor , the Ricci tensor and the scalar curvature (it arises from the formation of tracks from the Ricci tensor). ${\ displaystyle g_ {from}}$${\ displaystyle R_ {from}}$${\ displaystyle R}$

The Weyl tensor has the same symmetries as the full Riemann curvature tensor:

${\ displaystyle C_ {abcd} ^ {} = - C_ {bacd} = - C_ {abdc}}$
${\ displaystyle C_ {abcd} + C_ {acdb} + C_ {adbc} ^ {} = 0}$

The disappearance of the trace reads in component notation (with Einstein's sum convention ):

${\ displaystyle {C ^ {a}} _ {bac} = 0.}$

In four spacetime dimensions ( ) it has ten independent components. Generally he has for${\ displaystyle n = 4}$${\ displaystyle n \ geq 3}$

${\ displaystyle N = {\ frac {(n-3) n (n + 1) (n + 2)} {12}}}$

independent components.

Since it is invariant in conformal transformations of the metric , the Weyl tensor is also called conformal tensor . The Weyl tensor vanishes in the Minkowski space and also in every conformal flat space (whose metric is thus connected to that of a Minkowski space via a conformal transformation). ${\ displaystyle g_ {ij} \ to \ lambda g_ {ij}}$

Since the Weyl tensor is included in the vacuum field equations, it also plays a role in the classification of their solutions (Petrow classification). It is used for the geometric analysis of spacetime (singularities of curvature, asymptotically flat spacetime, etc.). From this, invariants such as the Kretschmann scalar can be derived.