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Would a non-hyperbolic model of our universe have fundamentally different laws than our universe?

For example, in Gödel's metric, its laws would be described by general relativity. But if the Gödel spacetime admitted any boundaryless temporal hyperslices (e.g. a Cauchy surface), any CTC (Closed Timelike Curve: https://en.wikipedia.org/wiki/Closed_timelike_curve) would have to intersect it an odd number of times, contradicting the fact that the spacetime is simply connected. Therefore, this spacetime is not globally hyperbolic. Would then laws be different?

Since it is theorised that with CTC hypercomputation would be possible, would then laws be based on hypercomputational-like processes?

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Short of having actual experimental evidence, it is hard to say what physical laws may differ in the presence of non-globally hyperbolic spacetimes, so for the most part, in the analysis of non-globally hyperbolic spacetimes, it is assumed that the same laws apply. Of course, this doesn't mean that the behaviours are similar to globally hyperbolic spacetimes, as there are a lot of assumptions we usually make that do not hold for general spacetimes.

Not much changes for the basics of classical physics : you solve the equations of motion given by the Lagrangian given some initial conditions. Things get a bit tricky there, though : for a start, defining boundary conditions is not trivial as there may be no spacelike hypersurface. Given some initial conditions, it may not be possible to get a unique solution. There may be several solutions satisfying it or even none (which isn't a good sign). If the spacetime lacks time orientability, you may not be able to define separate retarded and advanced waves which is generally important for EM.

Some non-globally hyperbolic spacetimes do allow hypercomputations (assuming fairly unphysical things of course, like a machine that can run for an arbitrarily long time and infinite memory), and more generally hypertasks. They are called Malament-Hogarth spacetimes. A Malament-Hogarth spacetime is one such that there exists an event $p$ such that a curve $\gamma$ of infinite proper time exists entirely in the past of $p$, that is,

$$\exists p, \exists \gamma, \gamma \subset I^-(p), l_\gamma = \infty$$

It can be shown[1] that every M-H spacetime is non-globally hyperbolic. On the other hand, it's not a terribly useful construct as information can't actually be communicated to $p$ : any signal from $\gamma$ to $p$ will be infinitely blueshifted, making it unlikely to be a well-defined example of a spacetime with the backreaction.

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The laws of physics in GR are the Einstein field equations (EFE). The only reason we talk about the Gödel spacetime as an interesting example is that it's a solution of the EFE. It has the property of not being hyperbolic, which means that we lose the ability to make meaningful predictions. You could hypothesize that there are some other laws of physics that would allow us to make predictions in a non-hyperbolic spacetime, but that would be pure speculation not grounded in any facts.

Since it is theorised that with CTC hypercomputation would be possible, would then laws be based on hypercomputational-like processes?

This seems like ungrounded speculation, and also seems to assume that physics is computational, whereas the standard picture is that computation is physical.

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