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where is the Ricci curvature tensor and is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor). The Ricci tensor vanishes in vacuum spacetimes (such as the Schwarzschild solution mentioned above), and hence there the Riemann tensor and the Weyl tensor coincide, as do their invariants.
Here, saying that = is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that our spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime .
The vorticity tensor represents any tendency of the initial sphere to rotate; the vorticity vanishes if and only if the world lines in the congruence are everywhere orthogonal to the spatial hypersurfaces in some foliation of the spacetime, in which case, for a suitable coordinate chart, each hyperslice can be considered as a surface of ...
This statement is equivalent to the more usable condition that the Lie derivative of the tensor under the vector field vanishes: = on M. This has the consequence that, given any two points p and q on M , the coordinates of T in a coordinate system around p are equal to the coordinates of T in a coordinate system around q .
Its chief disadvantage is that in those coordinates the metric depends on both the time and space coordinates. In Eddington–Finkelstein, as in Schwarzschild coordinates, the metric is independent of the "time" (either t in Schwarzschild, or u or v in the various Eddington–Finkelstein coordinates), but none of these cover the complete spacetime.
Killing tensor fields are symmetric tensor fields T such that the trace-free part of the symmetrization of vanishes. Examples of manifolds with Killing tensors include the rotating black hole and the FRW cosmology. [7]
Frame fields of a Lorentzian manifold always correspond to a family of ideal observers immersed in the given spacetime; the integral curves of the timelike unit vector field are the worldlines of these observers, and at each event along a given worldline, the three spacelike unit vector fields specify the spatial triad carried by the observer.
In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static. The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary ; the Schwarzschild solution thus follows.