Assume that we have a particle [itex]i[/itex] at rest according to co-moving coordinates.(adsbygoogle = window.adsbygoogle || []).push({});

Then the total energy of the particle at rest, [itex]E_i[/itex], is its rest mass energy plus the sum of the (negative) gravitational potential energies between it and every other particle in the Universe. Thus we have

[itex] \large E_i = m_i c^2 - \sum^{\infty}_{j=1} \frac{G m_i m_j}{r_{ij}} [/itex] where [itex] i \ne j [/itex].

Do we expect this total energy to stay constant as the Universe expands?

If the gravitational potential terms become less negative as the Universe expands then the rate at which time passes at particle [itex]i[/itex] will increase causing its energy/frequency to increase. A clock at position [itex]i[/itex] would not tick at a constant rate and therefore could not register the passage of cosmological time properly which seems strange as it is in an inertial frame and therefore should be a "good" timekeeper.

In order that the total energy of each particle in the Universe should stay constant we would require that the number of particles in the Universe and hence its total mass should be proportional to the Universal scale factor. This puts a strong constraint on the scale factor forcing it to be linear in time.

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# Is the total energy of each particle constant in an expanding universe?

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