Energy of comoving particle in FRW spacetime goes like scale factor?
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As far as I know ηαηα is unique in FRW space time as it is the only timelike conformal Killing vector field. The isometry of this field defines a global notion of energy proportional to scale factor a. I want this as I believe that the Einstein field equations in natural units in FRW space time should read Gμν=1/(m2pa2)TμνGμν=1/(mp2a2)Tμν.
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The line element for the flat FRW metric using conformal time , Cartesian spatial coordinates and scale factor , is given by
A particle of unit mass moving on a geodesic, parametrized by from to
, maximizes the proper time given by
where the Lagrangian is given by
Consider the time-component Euler-Lagrange equation
Now we have
where and .
Now we also have
Let us assume that the particle is at rest in the comoving frame so that we have
.
Therefore we have
The time-component Euler-Lagrange equation becomes
Integrating and using we obtain
The left-hand side can be identified with the energy, , of the unit mass particle. Therefore
Is this right? Is the energy of a unit mass comoving particle, in an expanding FRW spacetime, proportional to the scale factor
A particle of unit mass moving on a geodesic, parametrized by from to
, maximizes the proper time given by
where the Lagrangian is given by
Consider the time-component Euler-Lagrange equation
Now we have
where and .
Now we also have
Let us assume that the particle is at rest in the comoving frame so that we have
.
Therefore we have
The time-component Euler-Lagrange equation becomes
Integrating and using we obtain
The left-hand side can be identified with the energy, , of the unit mass particle. Therefore
Is this right? Is the energy of a unit mass comoving particle, in an expanding FRW spacetime, proportional to the scale factor
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