  A theorem due to Krasnikov (2002, 2003 [Other Internet Resources], 2014a) might seem to demonstrate that no relativistic spacetime can count as embodying a Thornian time machine so understood. Following Krasnikov, let us say that a spacetime condition \(C\) is local just in case, for any open covering \(\{V_{\alpha}\}\) of an arbitrary spacetime \((\mathcal{M}, g_{ab}), C\) holds in \((\mathcal{M}, g_{ab})\) iff it holds in \((V_{\alpha}, g_{ab}|_{V_{\alpha}})\) for all \(\alpha\). Examples of local conditions one might want to impose on physically reasonable spacetimes are Einstein’s gravitational field equations and so-called energy conditions that restrict the form of the stress-energy tensor \(T_{ab}\). An example of the latter that will come into play below is the weak energy condition that says that the energy density is non-negative.[18] Einstein’s field equations (sans cosmological constant) require that \(T_{ab}\) is proportional to the Einstein tensor which is a functional of the metric and its derivatives. Call a \(C\)-spacetime \((\mathcal{M}', g'_{ab})\) a \(C\)-extension of a \(C\)-spacetime \((\mathcal{M}, g_{ab})\) spacetime if the latter is isometric to an open proper subset of the former; and call \((\mathcal{M}, g_{ab}) C\)-extensible if it admits a \(C\)-extension and \(C\)-maximal otherwise. (Of course, \(C\) might be the empty condition.) Krasnikov’s theorem shows that every \(C\)-spacetime \((\mathcal{M}, g_{ab})\) admits a \(C\)-maximal extension \((\mathcal{M}^{max}, g^{max}_{ab})\) such that all CTCs in \((\mathcal{M}^{max}, g^{max}_{ab})\) are to the chronological past of the image of \(\mathcal{M}\) in \((\mathcal{M}^{max}, g^{max}_{ab})\). So start with some candidate spacetime \((\mathcal{M}, g_{ab})\) for a Thornian time machine, and apply the theorem to \((D^+ (\Sigma_0), g_{ab}|_{D^+ (\Sigma_0)})\).  Conclude that no matter what local conditions the candidate spacetime is required to satisfy, \(D^+ (\Sigma_0)\) has extensions that also satisfies said local conditions but does not contain CTCs to the future of \(\Sigma_0\). Thus, the candidate spacetime fails to exhibit the crucial feature identified above necessary for underwriting the contention that the conditions on \(\Sigma_0\) are responsible for the development of CTCs.  Hence, it appears as if Krasnikov’s theorem effectively prohibits time machines. 