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:


1. What is the basis for the claim – does one require general relativity, or can one get it from Newtonian gravity ?
2. What conditions do you require in the model, in order for this to work ?
3. Could someone please refer me to a good paper about this ?



Total energy of the universe
In fact, two very well-known mathematicians, Schoen and Yau, in a much quoted paper, proved the long standing conjecture that the ADM mass is always POSITIVE (except for flat space). Here is the reference and abstract:


Commun. math. Phys. 65, 45--76 (1979)  

On the Proof of the Positive Mass Conjecture in General Relativity  

Richard Schoen and Shing-Tung Yau  

Abstract. Let M be a space-time whose local mass density is non-negative 
everywhere. Then we prove that the total mass of M as viewed from spatial 
infinity (the ADM mass) must be positive unless M is the fiat Minkowski 
space-time. (So far we are making the reasonable assumption of the existence of 
a maximal spacelike hypersurface. We will treat this topic separately.) We can 
generalize our result to admit wormholes in the initial-data set. In fact, we 
show that the total mass associated with each asymptotic regime is non- 
negative with equality only if the space-time :is fiat. 


