I am reading Kennefick’s “Traveling at the Speed of Thought”. It is quite interesting and it is a perspective on the whole of the history of the development of general relativity. The perspective is from a difficult problem—the question of gravitational waves. Einstein changed his mind at least twice on the question of whether his equations demanded them.
Chapter 7 relates reasons for the 20 dark years for GR development where there was little progress. In 1955 there was a renaissance for several reasons. Kennefick describes several new calculations leading to almost as many distinct and contradictory results. The equations are non-linear and thus the analogies with electro-magnetic waves is imperfect.
There are two problems; one methodological and the other mathematical. It seems clear that the central question is not whether there are gravitational waves but whether GR demands them. GR is the only theory we have that works on a large scale so if GR predicts one thing and observation reveals another, we have discovered new physics.
The other problem is lack of a precise mathematical question of the form “Does general relativity predict gravitational waves?”. I strongly suspect that the various derivations alluded to above are due to various vague questions, answers to which may indeed vary.
I took a Newtonian mechanics course from “Owen Chamberlain” at Berkeley in 1954. It was foundational in nature. We started with Newton’s laws, put them into modern terms, and began to derive results. We derived all of the common conservation laws and then some. In particular we derived conservation of energy. I have not seen the same done for GR. It occurred to me at the time as we derived conservation of angular momentum that it was required to add momentum vectors manifest at different locations. I already knew that this was impossible in GR. In a one minute conversation with Chamberlain, he agreed.
There are known integrals of the stress-energy tensor taken over the boundary of a space-time region that can be taken as energy conservation. If we take such a region to include orbiting neutron stars, then the stress energy tensor is zero on the portion of the boundary thru which the gravitational waves are presumed to pass. Thus the only energy conservation law I know for GR does not serve. I suspect that:
Starting on page 126 he discuses the problem of localizing energy; you can’t. But you can’t even in Newton’s realm. If a large wheel rolls by with a physicist sitting at the hub in a non-rotating chair and we ask how the kinetic energy is distributed around the wheel, the moving physicist will see the energy spread uniformly around the rim while an observer on the ground will ascribe most of it to the high part of the wheel which is moving fastest. If they each compute the total kinetic energy the answers will differ by just mv2/2 where m is the mass of the wheel and v is the velocity of the hub. Leibnitz and Newton knew this. What’s new in GR is energy where there is no stuff; where Tij = 0 but Rijkl does not vanish. The empty space-time manifold has energy that is not here, not there, yet somewhere. It is bent like a spring and wants to unbend. It seems we need to ascribe energy to curvature, but in some non-localized fashion. This is very apropos. An intro to the Stress-energy-momentum pseudotensor.
The comments by Jürgen Ehlers quoted on page 240 are similar to my misgivings.
I wish I understood the strength of Einstein’s equations better.
The story of the debate on waves is not a pretty one. Eminent physicists populated both sides. The question itself is very subtle.