general relativity - Is the total energy of the universe zero?

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 $E=mc^2$ 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?

   
   
   

Answer

On my blog, I published a popular text why energy conservation becomes trivial (or is violated) in general relativity (GR).

To summarize four of the points:

1. 
   

   In GR, spacetime is dynamical, so in general, it is not time-translation invariant. One therefore can't apply Noether's theorem to argue that energy is conserved.

   
   


2. 
   
   

   One can see this in detail in cosmology: the energy carried by radiation decreases as the universe expands since every photon's wavelength increases. The cosmological constant has a constant energy density while the volume increases, so the total energy carried by the cosmological constant (dark energy), on the contrary, grows. The latter increase is the reason why the mass of the universe is large - during inflation, the total energy grew exponentially for 60+ $e$-foldings, before it was converted to matter that gave rise to early galaxies.

   
   


3. 
   

   If one defines the stress-energy tensor as the variation of the Lagrangian with respect to the metric tensor, which is okay for non-gravitating field theories, one gets zero in GR because the metric tensor is dynamical and the variation — like all variations — has to vanish because this is what defines the equations of motion.

   
   


4. 
   

   In translationally invariant spaces such as Minkowski space, the total energy is conserved again because Noether's theorem may be revived; however, one can't "canonically" write this energy as the integral of energy density over the space; more precisely, any choice to distribute the total energy "locally" will depend on the chosen coordinate system.