Malament–Hogarth spacetime

A Malament–Hogarth (M-H) spacetime, named after David B. Malament and Mark Hogarth, is a relativistic spacetime that possesses the following property: there exists a worldline $\lambda$ and an event p such that all events along $\lambda$ are a finite interval in the past of p, but the proper time along $\lambda$ is infinite. The event p is known as an M-H event. The significance of M-H spacetimes is that they allow for the implementation of certain non-Turing computable tasks (hypercomputation). The idea is for an observer at some event in ps past to set a computer (Turing machine) to work on some task and then have the Turing machine travel on $\lambda$, computing for all eternity. Since $\lambda$ lies in ps past, the Turing machine can signal (a solution) to p at any stage of this never-ending task. Meanwhile, the observer takes a quick trip (finite proper time) through spacetime to p, to pick up the solution. The set-up can be used to decide the halting problem, which is known to be undecidable by an ordinary Turing machine. All the observer needs to do is to prime the Turing machine to signal to p if and only if the Turing machine halts.

The Kerr metric, which describes empty spacetime around a rotating black hole, possesses these features: a computer can orbit the black hole indefinitely, while an observer falling into the black hole experiences an M-H event as they cross the inner event horizon. (This, however, neglects the effects of black hole evaporation.)^[1]

References

1. G. Etesi and I. Nemeti (2002). "Non-Turing computations via Malament-Hogarth space-times". Int. J. Theor. Phys. 41: 341–370. arXiv:gr-qc/0104023.

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• Manchak, John Byron (2009) On the Possibility of Supertasks in General Relativity. [Preprint]