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.
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Notes
That the former characterization implies the latter is clear. For the converse, if at the given point \(T= 0\) or if no causal vector is sent to zero we have finished. Thus assume \(T\ne 0\) and there is a f.d. causal vector \(w\) which has zero image. Since \(T\ne 0\) there is a f.d. causal vector \(v\) such that \(u^b:=-T^b_{\ a} v^a\ne 0\), necessarily f.d. timelike by assumption, then \(0=T(w,v)=-g(w,u)\) which is a contradiction since it must be positive.
Hawking [6, p. 293] claims a similar result but his proof and claim seem incorrect. He assumes a weaker form of energy condition, which does not exclude the possibility \(T^{ab}=\alpha n^a n^b\), where \(n\) is a lightlike vector, and then he applies a version of the conservation theorem which he has not really proved. Probably he used Eq. (8) missing the difference between \(\hat{S}\) and \(S\), and that between \(V\) and \(n\).
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Work partially supported by GNFM of INDAM.
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Minguzzi, E. The vacuum conservation theorem. Gen Relativ Gravit 47, 32 (2015). https://doi.org/10.1007/s10714-015-1878-4
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DOI: https://doi.org/10.1007/s10714-015-1878-4