TITLE: Total energy of the universe
QUESTION [11 upvotes]: In popular science books and articles, I keep running into the claim that the total energy of the Universe is zero, "because the positive energy of matter is cancelled out by the negative energy of the gravitational field".
But I can't find anything concrete to substantiate this claim.
As a first check, I did a calculation to compute the gravitational potential energy of a sphere of uniform density of radius R using Newton's Laws and threw in $E=m{c}^2$ for energy of the sphere, and it was by no means obvious that the answer is zero !
So, my questions:

What is the basis for the claim – does one require general relativity, or can one get it from Newtonian gravity ?

What conditions do you require in the model, in order for this to work ?

Could someone please refer me to a good paper about this ?

REPLY [6 votes]: When people claim that total energy is zero in a gravitational theory, they usually mean something kind of trivial. In an ordinary field theory, we can define the stress-energy tensor by varying the Lagrangian with respect to the metric. But in a gravitational theory, the metric is a dynamical field, so this variation just gives an equation of motion for the metric that should be zero on a classical solution.
What I've just said sounds fairly classical; the Wheeler-DeWitt equation is a version of this idea in the quantum context.
At low energies, where gravity is weak, you can of course just talk about the energy of a configuration of matter, independent of gravity, but this will not be conserved in general relativity. (A cosmological constant is a standard example where you can see that it's hard to make sense of energy conservation in the presence of gravity.)
As the others have said, the ADM energy is sometimes a more useful definition of energy in GR.