(ed.). Gravitational waves (IOP, 2001)(422s).

Chapter 5Waves and energyHere we discuss wave-like perturbations h µν of a general background metric g µν .The mathematics is similar to that of linearized theory: h µν is a tensor with respectto background coordinate transformations (as it was for Lorentz transformationsin linearized theory) and it undergoes a gauge transformation when one makes aninfinitesimal coordinate transformation. As in linearized theory, we will assumethat the amplitude of the waves is small. Moreover, the waves must have awavelength that is short compared to the radius of curvature of the backgroundmetric. These two assumptions allow us to visualize the waves as small ripplesrunning through a curved and slowly changing spacetime.5.1 Variational principle for general relativityWe start our analysis of the small perturbation h µν by introducing the standardHilbert variational principle for Einstein’s equations. The field equations ofgeneral relativity can be derived from an action principle using the Ricci scalarcurvature as the Lagrangian density. The Ricci scalar (second contraction of theRiemann tensor) is an invariant quantity which contains in addition to g µν and itsfirst derivatives also the second derivatives of g µν , so our action can be writtensymbolically as:I[g µν ] = 1 ∫R(g µν , g µν,α , g µν,αβ ) √ −g d 4 x (5.1)16πwhere √ −g is the square root of the determinant of the metric tensor. As usual invariational principles, the metric tensor components are varied g µν → g µν +h µν ,and one demands that the resulting change in the action should vanish to firstorder in any small variation h µν of compact support:50δI = I[g µµ + h µν ] − I[g µν ]= 116π∫ δ(R√ −g)δg µνh µν d 4 x + O(2) (5.2)