Why Does the amount of gravitational potential energy in the universe increase as it expands?

Question

Ok so this is my thought process... please bear with me

I understand that as the universe expands, there is supposed formation of matter due to dark energy,and with an increase in mass comes and increase in overall gravitational potential energy in the universe as a whole.

That being said isn't the rate of gravitational potential increase equal to the rate of universal expansion where this energy might be converted to the kinetic energy for this expansion? With these two cancelling each other out, what would remain is the distance between universal bodies of great mass increasing, which should reduce the gravitational potential energies between them, reducing the total GPE of the universe as it expands.

Is this confusion due to the input of energy into the closed system that is our universe or am i just thinking about it the wrong way?

Answer 1

I am not particularly familiar with Cosmology. I'm sure others can answer your questions much better, but here are a few of my thoughts that don't take into account dark matter and dark energy.

I understand that as the universe expands, there is supposed formation of matter due to dark energy,and with an increase in mass comes and increase in overall gravitational potential energy in the universe as a whole.

Gravitational potential energy becomes more negative with increasing mass, and more positive (increases) with increasing separation. So to the extent that the separation of normal mass (I think they call it baryonic mass) appears to be increasing due to expansion of the universe, that alone will cause an increase in gravitational potential energy.

That being said isn't the rate of gravitational potential increase equal to the rate of universal expansion where this energy might be converted to the kinetic energy for this expansion?

For an isolated system one normally associates a decrease in potential energy with an increase in kinetic energy, not the other way around. Maybe its different in cosmology.

With these two cancelling each other out, what would remain is the distance between universal bodies of great mass increasing, which should reduce the gravitational potential energies between them, reducing the total GPE of the universe as it expands.

Like I said, we normally associate an decrease in potential energy cancelling out an increase in kinetic energy, for conservation of energy.

Is this confusion due to the input of energy into the closed system that is our universe or am i just thinking about it the wrong way?

Can't say. But maybe the confusion stems from thinking gravitational potential energy decreases with increasing separation of mass.

Hope this helps.

Answer 2

The Hamiltonian constraint of ADM general relativity is $N{\cal H} = 0$, for $N$ a lapse function and $\cal H$ the Hamitlonian. This equation tells us that in general spacetime manifold do not have any way of defining a Gaussian surface from which one can compute mass-energy. I wrote a physics stack exchange

How did the universe shift from "dark matter dominated" to "dark energy dominated"?

how to derive in a purely Newtonian fashion this Hamiltonian, without the $k/a^2$ term for $a$ the scale factor. This Hamiltonian is \begin{equation} {\cal H} = \left(\frac{\mathrm da}{\mathrm dt}\right)^2 - \frac{8\pi G\rho a^2}{3}= 0. \end{equation} This defines the Hubble parameter $H = (\dot a/a)$ that depends on the density of mass-energy $$ \left(\frac{\dot a}{a}\right)^2 = H^2 = \frac{8\pi G\rho}{3}. $$ This is a quick and easy way of deriving the Friedman-Lemaitre-Robertson-Walker equation. The constant vacuum density $\rho$ leads to the conclusion that things in a funny way gravitationally repel, or “fall up.” This is one way to think of the accelerated expansion of the universe. There is then a constant and positive vacuum energy $\rho$ which defines the cosmological constant $\Lambda = 8\pi G\rho/c^2$, where the $c^2$ comes when we consider $\rho$ energy density instead of mass density.

We first see that $\left(\frac{\dot a}{a}\right)~=~\frac{d~ln(a)}{dt}$ which easily leads to the the fact \begin{equation} a(t)~=~a_0~exp\left(t~\sqrt{\frac{8\pi G\rho}{3c^2}}\right). \end{equation} Now consider this with equation 1. This means the potential energy for a mass accelerating away from some coordinate position is decreasing and becoming more negative. However, the potential energy for any two masses separated by the same distance is the same. Two masses connected by a rod that keeps them from comoving with the expansion of space will then have the same gravitational potential. This is of course made up for by the equal amount of kinetic energy that is generated.