While the focus here has been on energy, analogue definitions for global momentum exist; given a field of angular Killing vectors and following the Komar technique, one can also define global angular momentum.
32.
In connection with this, it may be shown that " every Killing vector field is a matter collineation " ( by the Einstein field equations ( EFE ), with or without cosmological constant ).
33.
Equivalently, a plane-wave is a pp-wave with at least a five-dimensional Lie algebra of Killing vector fields X, including X = \ partial _ v and four more which have the form
34.
A Killing vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative ( equivalently, specifying X and F ) at any point ( for every connected component ) of M.
35.
It can be shown that the G�del solution is, up to local isometry, the " only " perfect fluid solution of the Einstein field equation admitting a five-dimensional Lie algebra of Killing vectors.
36.
It was customary to represent black hole horizons via stationary solutions of field equations, i . e ., solutions which admit a time-translational Killing vector field everywhere, not just in a small neighborhood of the black hole.
37.
For example, Killing vector fields may be used to classify spacetimes, as there is a limit to the number of global, smooth Killing vector fields that a spacetime may possess ( the maximum being 10 for 4-dimensional spacetimes ).
38.
For example, Killing vector fields may be used to classify spacetimes, as there is a limit to the number of global, smooth Killing vector fields that a spacetime may possess ( the maximum being 10 for 4-dimensional spacetimes ).
39.
We can restate this by saying that our symmetry group includes as three-dimensional subgroups examples of Bianchi types I, III and VIII . Four of the five Killing vectors, as well as the curvature tensor, do not depend upon the coordinate y.
40.
In particular, the three spatial Killing vector fields have exactly the same form as the three nontranslational Killing vector fields in a spherically symmetric chart on E 3; that is, they exhibit the notion of arbitrary Euclidean rotation about the origin or spherical symmetry.
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