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You seem to be conflating "causally linked" with "in the same reference frame". This is not correct. Two points in the spacetime are "causally linked" if there is a causal (timelike or null) curve that connects them (meaning you can reach one point from the other without moving faster than $c$), so two observers can be ...

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The choice of Riemannian metric $h_{\mu \nu}$ is itself arbitrary, since there are multiple inequivalent rank-2 non-degenerate tensors on an arbitrary space; and different choices of $h_{\mu \nu}$ will lead to different "preferred time directions." For example, consider the following two rank-2 tensors on Minkowski space with coordinates $t$, $x$, ...

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There is a fundamental problem with this scheme: constructing a warp drive spacetime. The papers exploring such spacetimes all work by (1) assuming some negative energy density exotic matter to hold together the bubble, and (2) that the bubble already exists. (1) is problematic since we have never seen such exotic matter, but (2) is more fundamental: there ...

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The principle is that information cannot move from Point 1 to Point 2 faster than the speed of light. In the example you give, information moves from the initial positions of the two participants to Planet B at less than the speed of light, so the principle is not violated.

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Some work along these lines has been done by Bob Wald and his colleagues over the years: Wald, R. Dynamics in nonglobally hyperbolic, static space‐times. J. Math. Phys. 21, 2802 (1980). Ishibashi, A. & R. Wald. Dynamics in non-globally-hyperbolic static spacetimes: II. General analysis of prescriptions for dynamics. Class. Quantum Grav. 20, 3815 (...

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The limit here is about how far the information can travel between two points. If people who originate at those points move while the message is propagating, that can reduce the time it takes for the people to get the message, of course, because it cuts the distance that the message has to travel to reach them. (But it does not reduce the minimum time ...

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