Electrovacuum Solution - Test Fields

Test Fields

Sometimes one can assume that the field energy of any electromagnetic field is so small that its gravitational effects can be neglected. Then, to obtain an approximate electrovacuum solution, we need only solve the Maxwell equations on a given vacuum solution. In this case, the electromagnetic field is often called a test field, in analogy with the term test particle (denoting a small object whose mass is too small to contribute appreciably to the ambient gravitational field).

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.

Note that this procedure amounts to assuming that the electromagnetic field, but not the gravitational field, is "weak". Sometimes we can go even further; if the gravitational field is also considered "weak", we can independently solve the linearised Einstein field equations and the (flat spacetime) Maxwell equations on a Minkowksi vacuum background. Then the (weak) metric tensor gives the approximate geometry; the Minkowski background is unobservable by physical means, but mathematically much simpler to work with, whenever we can get away with such a sleight-of-hand.