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

56 Waves and energyFinally, what is of the most interest is the energy density as a fraction of theclosure or critical cosmological density, given by the Hubble constant H 0 asϱ c = 3H0 2/8π. The resulting ratio is the symbol gw( f ) that we met in theprevious lecture: gw ( f ) = 32π 3f 3 | ¯h( f )| 2 . (5.21)5.3.3 Other approaches3H 2 0We finish this lecture by observing that there is no unique approach to definingenergy for gravitational radiation or indeed for any solution of Einstein’sequations. Historically this has been one of the most difficult areas for physiciststo get to grips with. In the textbooks you will find discussions of pseudotensors, ofenergy measured at null infinity and at spacelike infinity, of Noether theorems andformulae for energy, and so on. None of these are worse than we have presentedhere, and in fact all of them are now known to be consistent with one another, ifone does not ask them to do too much. In particular, if one wants only to localizethe energy of a gravitational wave to a region of the size of a wavelength, andif the waves have short wavelength compared to the background curvature scale,then pseudotensors will give the same energy as the one we have defined here.Similarly, if one takes the energy flux defined here and evaluates it at null infinity,one gets the so-called Bondi flux, which was derived by H Bondi in one of thepioneering steps in the understanding of gravitational radiation. Many of theseissues are discussed in the Schutz–Sorkin paper referred to earlier [23].5.4 Exercises for chapter 5(e)In the notes above we give the general gauge transformationh µν → h µν − ξ µ;ν − ξ ν;µ .Use the formula for the derivation of Einstein’s equations from an actionprinciple,δI = 1 ∫ √ δ(R −g)h µν d 4 x16π δg µνwithδ(R √ −g)=−G µν√ −g,δg µνbut insert a pure gauge h µν . Argue that since this is merely a coordinatetransformation, the action should be invariant. Integrate the variation of theaction to prove the contacted Bianchi identityG µν ,ν = 0.