If a particle travels on a geodesic with 4-momentum $P^\mu$ in a spacetime with a Killing vector $K_\mu$ then we have a constant of motion, $K$, given by: $$K=K_\mu P^\mu$$ Using the relationships: $$P^\mu=mU^\mu$$ and $$K_\mu=g_{\mu\nu}K^\nu$$ we obtain: $$K=mg_{\mu\nu}K^\nu U^\mu$$ Let us assume that $K^\nu$ is a timelike Killing vector so that we have: $$K^\nu=(1,0,0,0)$$ Then the constant $K$ is the total energy $e$ of the particle (including gravitational energy) given by: $$e=-mg_{00}\frac{dt}{d\tau}$$ The above argument is my generalisation from @StanLiou's answer to the question Potential Energy in General Relativity where he gives an expression for the total energy of a particle in geodesic motion in Schwarzschild spacetime. I hope I have got the algebra correct!
My question is: could this definition of total particle energy be carried over to cosmology where one has the FRW metric?
The FRW metric does not have a timelike Killing vector so that one cannot expect the total energy $e$ of a co-moving particle to be constant. But the FRW metric does have a timelike conformal Killing vector so that it seems reasonable that the particle energy should scale in some way with conformal time $\eta$.
We can write the flat FRW metric in conformal co-ordinates: $$ds^2=a(\eta)^2(-d\eta^2+dx^2+dy^2+dz^2)$$ Thus we have: $$g_{00}=-a(\eta)^2$$ $$\frac{d\eta}{d\tau}=\frac{1}{a(\eta)}$$ Therefore $$e=m\ a(\eta)$$ Thus it seems that the total energy of a co-moving particle, when expressed in conformal co-ordinates, scales with the Universal scale factor $a(\eta)$.
Is this reasoning valid?
