Question

Does rest mass increase in the FRW metric?

Answer 1

The flat FRW metric can be written in conformal co-ordinates:

ds2=a2(η)(dη2−dx2−dy2−dz2)ds2=a2(η)(dη2−dx2−dy2−dz2)

where ηη is conformal time. Let us assume that a(η0)=1a(η0)=1 when η0η0 is the present conformal time.

Now the energy of a massive particle EE is given by:

E=PμVμ=mgμνUμVνE=PμVμ=mgμνUμVν

where the 4-momentum of the particle is Pμ=mUμPμ=mUμ and UμUμ,VνVν are the 4-velocities of the particle and observer respectively.

Let us assume that both the particle and the observer are co-moving at the same conformal time ηη. Therefore the spatial components of their 4-velocities are zero. As the 4-velocities must also be normalised we have:

g00U0U0=g00V0V0=1g00U0U0=g00V0V0=1

Therefore the 4-velocities of the co-moving particle and observer are given by:

Uμ=Vμ=(1a(η),0,0,0)Uμ=Vμ=(1a(η),0,0,0)

Thus the energy EE of a co-moving particle at time ηη, as measured by a co-moving observer at time ηη, is given by:

E=m g00 U0 V0=m a2(η)1a(η)1a(η)=mE=m g00 U0 V0=m a2(η)1a(η)1a(η)=m

Thus, using this definition of energy, the energy of individual co-moving massive particles is constant. Therefore, for example, we can say that the mass density of cosmological "dust", used in the Friedmann equations, simply goes like ρm∝1/a3ρm∝1/a3. This is the conventional viewpoint.

But we can define an energy E0E0 which is the energy of a comoving particle at time ηη with respect to a comoving observer at the present time η0η0 when a(η0)=1a(η0)=1:

E0=m g00 U0 V0=m a2(η)1a(η)11=m a(η)E0=m g00 U0 V0=m a2(η)1a(η)11=m a(η)

Answer 2

Explanation:

The concept of relativistic mass, which increases with velocity, is not compatible with the ... The same difficulty occurs with the term rest mass. To get ... Metric-first & entropy-first surprises.