FTheory: Quasi-Local Charges for Energy, Angular Momentum and Centre of Mass and Possible Applications in LQG
Witten’s proof of the positivity of the ADM mass provides a definition of energy in terms of three-surface spinors. I give a generalisation in terms of twistors that returns the remaining six Poincaré charges at spacelike infinity, which are the angular momentum and centre of mass (provided certain fall-off and parity conditions are satisfied). The construction can be brought back to finite distance, where it defines an infinite tower of quasi-local charges, only ten of which survive at spatial infinity. I point out that the underlying three-surface equations are well-defined on surfaces of arbitrary signature, and I study them on the entire boundary of a compact four-dimensional spacetime region. This gives us certain spinors that are inserted into the quasi-local expressions for energy and angular momentum, which are now integrals over an arbitrary two-dimensional cross-section of the boundary. For any two such cross-sections, conservation laws are derived that determine the influx (outflow) of matter and gravitational radiation. If the four-dimensional region is bounded by an outgoing null surface, it is shown that the quasi-local energy can only increase in time. I close with some comments on possible applications in loop quantum gravity.