The vacuum conservation theorem

Abstract

A version of the vacuum conservation theorem is proved which does not assume the existence of a time function nor demands stronger properties than the dominant energy condition. However, it is shown that a stronger stable version plays a role in the study of compact Cauchy horizons.Comment: 9 pages. v2,v3: fixed some typo

Area Theorem and Smoothness of Compact Cauchy Horizons

“…2.4 and Theor. 14, was posted on the Archive (arXiv: 1406.5919) as the last of a series of three papers (the others being [58,59]). The very next day a related work by E. Larsson (arXiv:1406.6194 recently published in [54]), reaching similar conclusions on smoothness of Cauchy horizons and topology change, was also posted ( [54] mentions that it also appeared some days before on a public web repository of theses of the KTH Institute, Stockholm).…”

“…Proof. By Theorem 25 and the energy condition the stress energy tensor vanishes on S, thus by Hawking's conservation theorem [44] as improved and clarified in [59] the stress energy tensor vanishes on D(S). The last statement follows from Theorem 18.…”

“…This observation leads us to the weakened stable dominant energy condition which states that the stable dominant energy condition holds wherever T = 0. It requires that the above endomorphism sends the futuredirected causal cone into the future-directed timelike cone plus the zero vector [59]. In the physical four dimensional case it allows diagonal stress energy tensors in which the energy density is larger than the absolute value of the principal pressures.…”

“…Suppose that H − (S) has a compactly generated component, then by Theorem 18 H − (S) has just one compact C 2 component and S is compact. By the weakened stable dominant energy condition and by the conservation theorem as clarified and improved in [59,Prop. 3.5] T (n, n) = 0 somewhere on H − (S) where n is the semitangent to the horizon, which is impossible because T (n, n) = R(n, n) and by Theorem 18, R(n, n) = 0 on the horizon.…”