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

54 Waves and energyAs in linearized theory, so also in the general case, the quantity h µν behaveslike a tensor with respect to background coordinate transformations, and so doesT µν (GW) . However, it is not gauge-invariant and so it is not physically observable.Since the integral of the action is independent of coordinate transformations thathave compact support, so too is the integral of the effective stress-energy tensor.In practical terms, this makes it possible to localize the energy of a wave to withina region of about one wavelength in size where the background curvature doesnot change significantly. In fact, if we restrict our gauge transformations to havea length scale of a wavelength, and if we average (integrate) the stress-energytensor of the waves over such a region, then any gauge changes will be smallsurface terms.By evaluating the effective stress-energy tensor on a smooth backgroundmetric in a Lorentz gauge, and performing the averaging (denoted by symbol〈···〉), one arrives at the Isaacson tensor:T (GW)αβ= 132π 〈h µν;αh µν ;β〉. (5.13)This is a convenient and compact form for the gravitational stress-energytensor. It localizes energy in short-wavelength gravitational waves to regions ofthe order of a wavelength. It is interesting to remind ourselves that our onlyexperimental evidence of gravitational waves today is the observation of the effecton a binary orbit of the loss of energy to the gravitational waves emitted by thesystem. So this energy formula, or equivalent ones, is central to our understandingof gravitational waves.5.3 Practical applications of the Isaacson energyIf we are far from a source of gravitational waves, we can treat the waves bylinearized theory. Then if we adopt the TT gauge and specialize the stress-energytensor of the radiation to a flat background, we getT (GW)αβ= 132π 〈hTT ij,α hTTij ,β〉. (5.14)Since there are only two components, a wave travelling with frequency f(wavenumber k = 2π f ) and with a typical amplitude h in both polarizationscarries an energy F gw equal to (see exercise (f) at the end of this lecture)F gw = π 4 f 2 h 2 . (5.15)Putting in the factors of c and G and scaling to reasonable values gives[ ]F gw = 3mWm −2 h 2 [ ]f 21 × 10 −22 , (5.16)1 kHz