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It is shown that if
AB is an exact solution of the Einstein vacuum field equations in 4 + 1 dimensions, R^
AB = 0, and l
A is a null vector field, then
AB + l
A
l
B is also an exact solution of the Einstein equations R^
AB = 0 if and only if the perturbation l
A
l
B satisfies the Einstein equations linearized about
AB. Then, making use of the Kaluza–Klein approach, it is shown that this result allows us to obtain exact solutions of the Einstein–Maxwell equations (possibly coupled to a scalar field) by solving a system of linear equations. 相似文献
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