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Here is a purely geometrical way to think about this Edward says it is possible to cut a wedge out of a flat spacetime and glue the edges together. So in my mind this looks like a paper cone. A cone is flat precisely because it can be created by rolling up a flat sheet. Rolling up preserves the metric on the interior of the sheet (not on the boundaries ...

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The most intuitive way to express this, as far as I know, is to start by taking the limit of the ratio of the circumference of a circle about the singularity to its radius, with the radius tending to zero. This ratio must be $2\pi$ for a manifold, and the conical deficit indicates that the space-time is singular at the centre of the circle. You may find more ...

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What motivates the assumption that a closed timelike curve must cross a spacelike slice an odd number of times? Its not an assumption. And it isn't true of all manifolds. Consider $\mathbb S^2\times\mathbb R^2$ as a subset of $\mathbb R^6,$ or just $$\{(a,b,A,B,y,z)\in\mathbb R^6:a^2+b^2=A^2+B^2=1\}$$ with the metric ...

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