Energy Density of the gravitational field

[Geometrick]

Why is there no satisfactory way to define the energy density of a gravitational field? Why must we use global energy concepts? I'm currently reading proofs of the Positive Energy Theorem, but none of the references explains why this is so. Can anyone explain it or point me to a good reference?

 

[haushofer]

Because such an expression would depend on your coordinates due to the equivalence principle. If I measure a graviational field and say it has energy density in point x, then you can Lorentz-transform yourself to an accelerating observer and say that there is no gravitational field, so the energy density is 0.

This is completely different from, let's say, the electromagnetic field.