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It is not the case that A Thornian time machine cannot operate in a spatially open universe while consistently satisfying Einstein's field equations and the weak energy condition.
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1.
A partial Cauchy surface must be compact if its future Cauchy horizon is compact (independent of field equations and energy conditions).
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2.
A necessary condition for a Thornian time machine to operate is that the future Cauchy horizon of the initial partial Cauchy surface is compactly generated.
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3.
Einstein's field equations combined with the weak energy condition rule out a non-compact partial Cauchy surface having a compactly generated future Cauchy horizon.
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