# Is there energy density arising from expansion velocity?

I'm taking a first course in cosmology, and I'm having a conflict of intuitions when it comes to the relative motion of objects arising from the growth of the scale factor.

Intuition 1: The growth of the scale factor can't exert a force on anything, so this movement shouldn't have any energy associated with it - otherwise surely the energy of the universe would be increasing.

Intuition 2: The growth of the scale factor increases the relative distances of things over time, which is the same as giving them a relative velocity. Relative velocity is the only kind of velocity, so this motion ought to have an associated kinetic energy. This should in turn contribute to the universe's energy density, and thus the growth of the scale factor.

Intuition 2 seems stronger to me, but I don't see how it's compatible with conservation of energy. Where does this 'kinetic energy' go when we transform into conformal co-ordinates? Also, I've not seen any contribution from this supposed kinetic energy in Friedmann equations - I realise this could just be because it's negligibly small for most expansion rates.

Energy isn't conserved in general relativity, particularly in an expanding universe. So if you try to reason than x must be true becasue otherwise energy wouldn't be conserved then you could go down a road of thinking many incorrect things. So don't do it.

Intuition 1: The growth of the scale factor can't exert a force on anything

Forces are not exerted by gravity, gravity isn't a force. Forces are exerted by pressure and stress. But we can't conclude there is no energy. Dust has non stress and no forces, but it has a nonzero energy density. And it has different amounts of energy density in different frames. Just assuming something doesn't have an energy would be wrong.

Intuition 2: The growth of the scale factor increases the relative distances of things over time, which is the same as giving them a relative velocity.

You do realize that velocity is relative to a frame? If you frame is to comoving frame of the cosmological fluid then everything is at rest in that frame if the peculiar velocity is zero. Being at rest relative to your frame right there is a real thing. Having your distance compared to something else somewhere else isn't even physics. Physics is local.

Relative velocity is the only kind of velocity, so this motion ought to have an associated kinetic energy.

It's not motion. Motion is relative to a frame. It's just breaking a tangent of a worldline into frame dependent parts. There was a worldline, it had a tangent. That was real. And then if you pick a frame you can brake that one tangent vector into parts, if you wanted to for some reason. Like if you hate energy-momentum and want energy and momentum instead.

This should in turn contribute to the universe's energy density, and thus the growth of the scale factor.

The energy density does change. But not because there is zero velocity in the comoving frame.

Intuition 2 seems stronger to me, but I don't see how it's compatible with conservation of energy.

Both intuitions are wrong. But there isn't conservation of energy.

I've not seen any contribution from this supposed kinetic energy in Friedmann equations

Because there isn't kinetic energy in the comoving frame from a zero velocity in the comoving frame. Physics is local.

• Ah OK - so the motion doesn't have an associated energy unless there is local motion - makes sense. I didn't get how evaluation of velocity in the comoving frame could have some kind of priority if the physics was supposed to be frame-invariant, makes more sense now. And somehow my entire general relativity course managed to skip over the fact that energy isn't conserved in general relativity...
– Ben
Mar 7, 2016 at 13:32
• @Ben I'm not sure what you mean by a priority. In an expanding universe peculiar velocities get smaller over time under geodesic motion. The point is energy doesn't even exist unless you pick a global frame, and energy density depends on frame (though it always exists). And your GR course should have said that zero covariant divergence of the Stress-Energy tensor is the GR equivalent of conservation of energy and momentum, and that it's different than zero coordinate divergence which is conservation of energy and momentum in a (possibly local) coordinate system. Mar 7, 2016 at 16:07