We formulate the concept of time machine structure for spacetimes exhibiting a compactely constructed region with closed timelike curves. After reviewing essential properties of the pseudo Schwarzschild spacetime introduced by A. Ori, we present an analysis of its geodesics analogous to the one conducted in the case of the Schwarzschild spacetime. We conclude that the pseudo Schwarzschild spacetime is geodesically incomplete and not extendible to a complete spacetime. We then introduce a rotating generalization of the pseudo Schwarzschild metric, which we call the the pseudo Kerr spacetime. We establish its time machine structure and analyze its global properties.
We investigate sufficient conditions for real-valued functions on product spaces to be bounded from above by sums or products of functions which depend only on points in the respective factors.We suppose that the result in the Theorem below is known to hold for sufficiently regular spaces, but we were not able to find it anywhere in the standard literature. Also note that the statement of the Theorem can essentially be seen as a statement about the poset of real valued continuous functions on a space M × N . The theorem states, that any finite subset of C 0 (M × N, R) has an upper bound in the sub-poset of functions (x, y) → F (x) + G(y).Theorem. Suppose M and N are two locally compact Hausdorff spaces that are countable at infinity. Then for any continuous function f : M × N → R there are continuous functionsProof: We can assume that f (t, x) ≥ 0 for all t ∈ M and x ∈ N . If this is not the case, we just replace f (t, x) by max{f (t, x), 0} ≥ f (t, x).The statement is obvious if M or N is compact: If M is compact, we have f (t, x) ≤ max t∈M f (t, x) =: G(x). The same argument applies if N is compact. Therefore, we assume that neither M nor N is compact.By assumption (locally compact and countable at infinity), there are exhaustionssuch that the K i 's are compact subsets of M for all i ∈ N 0 and ∅ = L 0 , L i ⊂ Li+1 , such that the L i 's are compact subsets of N for all i ∈ N 0 .
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