What,s the energy of the gravitational field?

[eljose]

What,s the "energy" of the gravitational field?..

If we can define for the Electro-Magnetic field an "energy"...

$$Energy= \alpha \int_{V} (E^{2}+B^{2})dv$$

where E and B are the electric and magnetic field..but my question is...¿why can not define an "energy" for the gravitational field so H=Energy where H is the Hamiltonian?..

 

[pervect]

The very short answer is "Noether's theorem".

http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html
gives some of the history

Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as 'the failure of the energy theorem '. In a correspondence with Klein [3], he asserted that this 'failure' is a characteristic feature of the general theory, and that instead of 'proper energy theorems' one had 'improper energy theorems' in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether. In the note to Klein he reports that had requested that Emmy Noether help clarify the matter. In the next section this problem will be described in more detail and an explanation given of how Noether clarified, quantified, and proved Hilbert's assertion. One might say it is a lemma of her Theorem II.

"Local" here is a rather ambibuous word - basically, when the author says that GR doesn't conserve energy "locally", he really means that we can't write the intergal you write above.

Basically, Noether's theorem says that the symmetry group of GR is too large (it's infinte, as the theory is diffemorphism invariant) to have a conserved energy in the sense above.

 

[Garth]

Remember that energy is a frame dependent concept whereas energy-momentum is not.

Thus in GR it is energy-momentum that is conserved (in all frames) not generally energy.

There is one theory that also includes the local conservation of energy, but that is not "mainstream" (although published and about to be tested).

Garth