There are spacetime models in general relativity (i.e. solutions to the Einstein equations) that permit hypercomputation. In a Malament-Hogarth spacetime, there are wordlines such that an object traveling along the worldline will experience infinite time, but for an observer at some point p outside the wordline, the time it takes for the object to traverse the worldline is finite, and the wordline is entirely within p's causal past. So if you had a Turing machine traveling along this wordline, it could send a signal to p if and onliy if it halted, and the observer at p is guaranteed to receive the signal if the machine ever halts.

Don't you have the problem of the signal getting blue-shifted out of the range of your detector?