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In the last chapter, we saw that in the weak-field approximation, Einstein’s equations for a perturbation of the Minkowski metric can be reduced to $$ \square w_{ab} = - 16\pi T_{ab} ,$$ in the de Donder gauge [see (10.2)]. This is an inhomogeneous wave equation, with the energy-momentum tensor as source. It is strongly reminiscent of Maxwell’s equations for the four-potential in the Lorenz gauge and it has the same implication. Maxwell’s equations imply that moving charges generate electromagnetic waves. Einstein’s equations imply that moving masses generate gravitational waves.

When one observer sends light signals to another, the frequency of the light measured at emission by the first observer is generally not the same as that measured at reception by the second. Even in special relativity, the light is redshifted if the second is moving away from the first. This is the Doppler effect. In general relativity, there is a gravitational redshift when both are at rest in the gravitational field of a static spherically symmetric body, and the first is below the second.