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Google has not been helpful because so many derivations of gravitational potential energy discuss $r$ at infinity. My understanding of this

https://www.youtube.com/watch?v=IcxptIJS7kQ

is: Dark energy increases with volume The volume of the universe is increasing. Therefore the total amount of dark energy is increasing. If the universe is flat , it can expand forever at an increasing rate. So the amount of dark energy $\to\infty$. The conservation of energy is not violated because gravitational energy is infinite.

We know that $$U=-G\frac{mM}{r}$$

so when $r\to0$ $U \to\infty$.

But no particle has zero radius so $r$ cannot be 0 so $U$ must be finite. So how can dark energy $\to\infty$?

Kyle Kanos
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monzie
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You're mixing up two different things.

Firstly, for the universe as a whole energy is (probably) not conserved. I say probably because although most people think that energy is not conserved I have heard the opposing view argued with some conviction. I didn't watch the (hour and 20 minute!!) video you linked to, but as far as I know the problems regarding conservation of energy that dark energy raises are not simply linked to gravitational potential energy. Energy is not conserved because the universe as a whole violates time translation invariance, so Noether's theorem doesn't apply and energy is not necessarily conserved.

Your second point is that for a point mass the potential energy becomes infinite as $r$ tends to zero. We simply don't know if elementary particles are pointlike or not, though string theory suggests not. In any case we expect that some theory of quantum gravity will become important at very short distances, and this will prevent the gravitational potential from becoming infinite. In any case, as far as we know this is unrelated to dark energy.

I seem to have used the phrase as far as we know quite a lot. The problem is that these areas aren't fully understood, or in the case of quantum gravity hardly understood at all!

John Rennie
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