I have been reading through the paper hep-th/9902121 and have a few questions about the first five lines of the introduction:
1) "In a generally covariant theory, it is unnatural to assign a local energy-momentum density to the gravitational field."
I'm not sure I understand this. Take for example the Einstein equation, the LHS would have local quantities e.g. metric and so surely the RHS (energy-momentum tensor) should also be local? Furthermore, aren't the ADM and Komar descriptions of energy generally covariant - don't these involve the energy-momentum tensor?
What do the authors mean here?
2) Any "candidate expressions depending only on the metric and its first derivatives will always vanish at a given point in locally flat coordinates"
I understand that if we use normal coordinates, we can locally make the first derivatives of the metric vanish and the metric take the standard Minkowski form (with subleading 2nd derivative terms) - why does this mean the energy-momentum needs to vanish?
And why can't I build the energy-momentum tensor out of higher derivative objects e.g. Ricci/Riemann tensor?
3) What is wrong with the ADM and Komar methods of defining energy and that I thought (until reading this and getting confused) depended on a local energy-momentum. Why do we need this quasilocal energy-momentum?