Complete asymptotically flat spin initial data obeying the Einstein constraints and the dominant energy condition have future-causal ADM four-momentum, under the appropriate decay and boundary hypotheses. A spinorial proof solves the Witten spinor equation and integrates the Nester two-form. Its divergence is a sum of a nonnegative spinor-gradient term and a matter-energy contraction with a future-causal Dirac current. Positivity for every asymptotic constant spinor yields the full inequality, rather than merely nonnegative energy in one frame.
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The Positive Energy Theorem is a significant result in the field of general relativity, primarily related to the properties of spacetime and gravitational fields. It states that, under certain conditions, the total energy (or total mass) of an asymptotically flat spacetime, which describes isolated physical systems, is non-negative. In more technical terms, the theorem asserts that the energy associated with a spacetime configuration cannot be negative, which has implications for the stability of gravitational systems.