 it is called (locally) static if this Killing vector is hypersurface orthogonal. Thus if, in a special coordinate system, we take then from Equation (91) we conclude that stationarity reduces to the condition . If we take to be the tangent vector field to the congruence of curves , i.e., if , then a necessary and sufficient condition for hypersurface-orthogonality is
A generalisation of Killing vectors are conformal Killing vectors for which with an arbitrary smooth function holds. In purely affine spaces, another type of symmetry may be defined: ; they are called affine motions .
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