# Conservation of Energy in Friedman Equation

1. Apr 25, 2012

### paultsui

The first law of thermodynamics states that $$dU = -PdV + dQ$$
We can apply this to the expansion of the universe by assuming the expansion of space is a adiabatic process, i.e. $dQ = 0$. Together with the FW metric, we end up with one of the Friedman Equations: $$\dot{\rho} = -\frac{3\dot{a}}{a}(p + P/c^{2})$$

Now let's focus on a cubic region of space. In the case P=0, this means that $\dot{\rho} = -\frac{3\dot{a}}{a}(p)$. This makes sense because this accounts for the fact that when the universe expands, the energy density within get diluted. However, when P != 0, we also have the term $-\frac{3\dot{a}}{a}(P/c^{2})$. This terms correspond to the $-pdV$ term in the thermodynamics equation. In other words, the energy in the region decreases not just because of dilution, but also because part of the energy is used to "push" the matter outside the region.

My question is, where has the energy that used to "push" the matter outside the region gone? Obviously energy has to be conserved. So the lost energy must have gone to somewhere.

Thanks!

2. Apr 25, 2012

### Mentz114

Can this be fixed by including a cosmological constant ?

If not, it looks like a case of lack of global energy conservation in GR.

3. Apr 25, 2012

### paultsui

What do you mean by including a cosmological constant?
This equation can be obtained by eliminating the cosmological constant from the other two Friedman Equations. So I guess cosmological constant is irrelavant here.

I think this is a local energy conservation problem - we are considering a "small" region in space, where energy flow out of the region to its neighbour.

Please correct me if I am wrong.

4. Apr 25, 2012

### Mentz114

OK.

Locally the energy has gone to some neighbours. I guess the divergence of the flow is zero so it must be escaping at the boundary.

5. Apr 25, 2012

### tom.stoer

Unfortunately energy as an integral over a 3-volume cannot even be defined in an expanding universe.