A poor man’s positive energy theorem

We show that positivity of energy for stationary, asymptotically flat, non-singular domains of outer communications is a simple corollary of the Lorentzian splitting theorem. A milestone in mathematical general relativity is the positive energy theorem [8, 10]. It is somewhat surprising that the standard proofs of this result do not have anything to do with Lorentzian geometry. In this note we prove energy positivity using purely Lorentzian techniques, albeit for a rather restricted class of geometries; it seems that in practice our proof only applies to stationary (with or without black holes) space-times. This is a much weaker statement than the theorems in [8, 10] and their various extensions, but the proof here seems of interest because the techniques involved are completely different and of a quite elementary nature. Using arguments rather similar in spirit to those of the classical singularity theorems [5], our proof is a very simple reduction of the problem to the Lorentzian splitting theorem [4]. (In lieu of the Lorentzian splitting theorem, one can impose Partially supported by a Polish Research Committee grant 2 P03B 073 24; email [email protected], URL www.phys.univ-tours.fr/∼piotr Partially supported by NSF grant # DMS-0104042; email [email protected]