Question

Does the total particle energy increase in FRW Universe?

Answer 1

If a particle travels on a geodesic with 4-momentum PμPμ in a spacetime with a Killing vector KμKμ then we have a constant of motion, KK, given by:

K=KμPμK=KμPμ

Using the relationships:

Pμ=mUμPμ=mUμ

and

Kμ=gμνKνKμ=gμνKν

we obtain:

K=mgμνKνUμK=mgμνKνUμ

Let us assume that KνKν is a timelike Killing vector so that we have:

Kν=(1,0,0,0)Kν=(1,0,0,0)

Then the constant KK is the total energy ee of the particle (including gravitational energy) given by:

e=−mg00dtdτe=−mg00dtdτ

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 eeof 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 ηη.

We can write the flat FRW metric in conformal co-ordinates:

ds2=a(η)2(−dη2+dx2+dy2+dz2)ds2=a(η)2(−dη2+dx2+dy2+dz2)

Thus we have:

g00=−a(η)2g00=−a(η)2

dηdτ=1a(η)dηdτ=1a(η)

Therefore

e=m a(η)e=m a(η)

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(η)a(η).

Is this reasoning valid

Answer 2

the metric is now locally that of Minkowski: g(ea,eb)=ηabg(ea,eb)=ηab. But the key point is as follows: these basis vectors satisfy [ea,eb]=γcabec[ea,eb]=γabcec , where the gamma functions are the commutation functions of the basis and the Riemann tensor is a function of these functions. So this is GR