It turns out that the most general pp-wave spacetime has only one Killing vector field, the null geodesic congruence k = \ partial _ v.
22.
In the usual coordinates, outside the Killing horizon, the Killing vector field \ partial / \ partial t is timelike, whilst inside it is spacelike.
23.
When the energy-momentum tensor represents an electromagnetic field, a Killing vector field does " not necessarily " preserve the electric and magnetic fields.
24.
While general space-times lack a finite-parameter time-translation symmetry, stationary space-times have such a symmetry, known as a Killing vector.
25.
They have argued that static stars must have " static �ther " solutions, which means that the �ther points in the direction of a timelike Killing vector.
26.
Killing vector field whose integral curves are closed spacelike curves ( circles, in fact ), which moreover degenerate to zero length closed curves on the axis R = 0.
27.
Here, it is useful to know that any Killing vectors which may be present will ( in the case of a vacuum solution ) automatically satisfy the curved spacetime Maxwell equations.
28.
However this is difficult to reconcile with static black holes, as at the event horizon there are no timelike Killing vectors available and so the black hole solutions cannot have static �thers.
29.
Indeed, in the Cartesian chart we can readily find ten linearly independent Killing vector fields, generating respectively one parameter subgroups of time translation, three spatials, three rotations and three boosts.
30.
In connection with this, it may be shown that " every Killing vector field is a matter collineation " ( by the Einstein field equations, with or without cosmological constant ).
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