Yes, that formula applies only the limit of linearized gravity and small strain. The electromagnetic flux ExH is expressed in terms of the fields, and there is no explicit frequency dependence. If instead, it is expressed in terms of the vector potential A, then there is an f^2. The gravitational metric, which includes the strain, is a potential, and there is an f^2. But the "gravitational acceleration", the 9.8 m/s^2 we feel on Earth, is a Christoffel symbol. Expressed thusly, I would expect the flux to have no frequency dependence. Note that the Christoffel symbol is not a tensor, and there is always a choice of coordinates in which it vanishes on a given world line. Thus, while the electromagnetic energy-momentum tensor is localized, a function of spacetime point, the gravitational energy-momentum pseudotensor is not, and is meaningful only when integrated over a few wavelengths. A true tensor, if it vanishes at a point in one coordinate system, vanishes at that point in all coordinate systems. -- Gene From: Warren D Smith <warren.wds@gmail.com> To: math-fun@mailman.xmission.com Sent: Tuesday, February 23, 2016 11:47 AM Subject: Re: [math-fun] energy flux of gravitational and other fields (Trying again, emailer crashed.) E.Salamin:
From this reference http://www.aei.mpg.de/~schutz/download/lectures/AzoresCosmology/Schutz.Azore... the flux (power per unit area) carried by a gravitational wave of frequency f and strain s is
F = (pi/4) (c^3/G) f^2 s^2 WDS: 1. this formula cannot be right for large strains, at best it can be right only in the limit of small strains so can use linearized GR. Should be derivable from "stress-energy pseudotensor" https://en.wikipedia.org/wiki/Stress%E2%80%93energy%E2%80%93momentum_pseudot... 2. The contrast with Maxwell is interesting. Maxwell energy flux = ExB where E=electric & B=magnetic field, x=vector cross product note NO frequency dependence. 3. For a scalar field https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor https://en.wikipedia.org/wiki/Scalar_field_solution again I see an f^2 frequency dependence. 4. So what is going on? Why for even-spin fields (gravity spin=2, scalar spin=0) do we have f^2 dependence while for odd-spin (Maxwell, photon spin=1) we have no frequency dependence? We get frequency f^n dependence if the stress energy tensor involves DERIVATIVES of the field, which in turn happens if the lagrangian density involves n field derivatives. For spin=0 & 2 we have n=2 (arising as squaring a first derivative). For spin=1 (Maxwell, Proca) we have n=0. The Dirac, Majorana, and Weyl (spin=1/2) fields all have n=1 according to http://arxiv.org/pdf/1006.1718.pdf so for them I would expect f^1 dependence. Rarita-Schwinger, which is one proposal for spin=3/2 fields, has n=1 according to https://en.wikipedia.org/wiki/Rarita%E2%80%93Schwinger_equation so for it also I would expect f^1 dependence. Is there some obvious unified reason these n had to happen in all these cases? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)