# Could dark energy be gravitational potential energy?

I was wondering if dark energy might actually be gravitational potential energy.

If one assumes that space is flat and that the Universe is approximately a sphere with mass M and the Hubble radius R then we find that we have the approximate relation:

$\frac{G M }{R} = c^2$

A mass m at the center of the sphere would have approximate gravitational potential energy

$PE = - \frac{G M m}{R} = - m c^2$

This potential energy would have a gravitational repulsion that might counteract the gravitational attraction due to mass m causing the Universal expansion to accelerate.

I would be interested to hear what you think of this line of thought.

bcrowell
Staff Emeritus
Gold Member
No. As explained in our FAQ, general relativity doesn't have a conserved scalar mass-energy that can be defined in all spacetimes: https://www.physicsforums.com/showthread.php?t=506985 Therefore the PE you're discussing can't be meaningfully defined in cosmology.

Also, the negative sign of the PE in Newtonian gravity represents its attractive nature. (If you flip the sign, you get a universe in which gravity is repulsive, like the electrical repulsion between like charges.) So you can't use it to explain a repulsion.

The nonrelativistic limit of GR with $\Lambda=0$ is Newtonian gravity, so you can't use Newtonian gravity to derive $\Lambda\ne0$.

A mass m at the center of the sphere

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Well, it's not entirely far fetched. The standard model of cosmology describes the matter of the universe as a homogeneous and isotropic ideal fluid of dust particles. Now, we know that on the scales of galaxies and galaxy clusters, the universe is far from homogeneous today.

As one coarse grains the description of "dust" to larger and larger scales, effects of the gravity become more and more unclear. For the FRW metric to apply at all scales, one needs to be able to carry out this coarse graining up to scales of galaxy clusters and beyond, at which point the "particles" of ideal fluid can be themselves expanding or contracting.

I was wondering if dark energy might actually be gravitational potential energy.
Nope.

A mass m at the center of the sphere would have approximate gravitational potential energy

$PE = - \frac{G M m}{R} = - m c^2$

This potential energy would have a gravitational repulsion that might counteract the gravitational attraction due to mass m causing the Universal expansion to accelerate.
Aside from the problems with calculating total energy with GR, you run into the problem that you flipped a sign. Things with negative potential energy are *attractive*. As you move two objects closer together, the potential energy goes lower, which means that the negative sign in the potential energy causes objects to *attract*.

Also, the negative sign of the PE in Newtonian gravity represents its attractive nature. (If you flip the sign, you get a universe in which gravity is repulsive, like the electrical repulsion between like charges.) So you can't use it to explain a repulsion.
I am assuming that the negative PE is itself a source of gravitational repulsion on normal matter. It is as though it is a gravitational source with negative mass.

No. As explained in our FAQ, general relativity doesn't have a conserved scalar mass-energy that can be defined in all spacetimes
I admit my argument is only Newtonian with the addition of the equivalence of mass and energy.