What is Malament-Hogarth spacetime?

Continuing the discussion from Halting Oracles And Law of information Non Growth:

Perhaps one of the physicists wants to help us out with this one? @pevaquark @PdotdQ @dga471

Beats me. I should probably add despite the gracious notion of what a doctorate in physics could mean, I’m a biophysics type of guy where I solve things like equations for fluid or ion flow in constricted geometries (similar to ion channels/voltage gates/bacterial pores).

Looking at maybe 150 or so citations on the original paper most seem to be in computing research of which I know nothing. Perhaps we can ask the person who wrote this webpage:
https://ncatlab.org/nlab/show/Malament‚ÄďHogarth+spacetime

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Ooooh, this is something that I know quite a bit about. I have to go to work now, but I’ll respond later today.

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The point with the Malament-Hogarth spacetime is this:

There are some spacetimes in which the clock carried by a person in a spaceship moving in a particular path through the spacetime ticks an infinite amount, while:

  1. The spaceship is always in the past of a particular point in spacetime, called the Malament-Hogarth Event.
  2. And more strongly, the spaceship is within the past lightcone of the Malament-Hogarth Event, i.e. in principle the person in the spaceship can send signals (which travels at the speed of light) to a detector located at the Malament-Hogarth Event.

Since an infinite amount of time elapses for the person in the spaceship, this person can conduct supertasks. The canonical example is: a mean mathematics professor sent their graduate student in such a spaceship, while they themself enjoy tenure. In this spaceship, the graduate student performs the task he was given by the professor: to check that Goldbach’s Conjecture is true by explicitly computing it for every even integers. If the student found a counter-example to Goldbach’s conjecture, they will beam a message to the Malament-Hogarth event. Because the student has an infinite amount of time, this is a doable task.

It turns out that there are examples of Malament-Hogarth spacetimes in which this is possible, including anti-de sitter (a universe with a negative cosmological constant) and inside a rotating black hole.

Some problems:

  1. In many spacetimes (e.g. anti-de Sitter), the path of the spaceship requires the use of an infinite amount of fuel, which requires the spaceship to be able to constantly distill energy from its environment throughout its travel.
  2. The Malament-Hogarth event necessarily predicts a breakdown in determinism (it necessitates what in general relativity is called a naked singularity). This is not good philosophically.
  3. Some spacetimes are only Malament-Hogarth in extremely specific conditions. For example, for a rotating black hole, a little perturbation blows up the location known as the ‚Äúinner horizon‚ÄĚ, which is where the Malament-Hogarth event is located.

Note that supertasks can be conducted in non-Malament-Hogarth spacetimes if we allow the detector (in addition to the spaceship) to also be moving. These spacetimes are called Pitowsky Spacetimes.

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Very interesting. Thanks @PdotdQ.

Easy Peasy!

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So this is interesting.

What sounds like a proof that it is possible, actually looks much more like a proof that it is impossible. This is so thoroughly beyond even conceivable technology and the world as we find it, that I‚Äôm not sure why we would call it ‚Äúpossible.‚ÄĚ In what sense is it possible? @BruceS

I was not using possible in the sense of physically possible. I was using it in the sense of logically possible. as defined in the modality models in philosophy.

At the conclusion of our previous exchange, I understood that you agreed that questions like the halting problem could be answered by God because they were in the logical region of the diagram.

I also formed the opinion that when you used the phrase ‚Äúlogically possible‚ÄĚ earlier in the example of compression algorithms, you meant that it was not logically possible for humans to determine the optimal compression because any approach to doing so that was available to humans would involve running an algorithm on something that was equivalent to a Turing Machine and that if such an algorithm existed, it would mean that the halting problem could be solved on a TM, which was a logical contradiction.

However, I must admit I did not post that opinion because I was satisfied with the discussion’s ending there.

If my opinion of your use of ‚Äúlogically impossible‚ÄĚ is correct, then one could get into issues of whether the Church-Turing Thesis is in fact true:

But I am not sure if that issue is germane to the discussions PS is meant for. Plus I really don’t know much about it!

I always wonder when people say things like that.

CT is not the kind of thing that could be ‚Äúin fact true.‚ÄĚ It could only be a matter of human convention (our meaning conventions for ‚Äúcomputation‚ÄĚ).

Fair point. It might have better for me to ask whether ‚Äúintuitively computable‚ÄĚ could have any other useful meaning than Turing-computable. At least that would replace issues of what ‚Äėtruth‚Äô could mean in this context (if anything) with what ‚Äėuseful‚Äô could mean, which might be a more fruitful topic for discussion.

Philosophers seem to have found something to talk about on this issue: The Church-Turing Thesis . Although I guess that almost goes without saying.

Edit to add: The linked SEP article on the CTT also discusses the CTT and the philosophy of Mind. Piccinini has an article which discusses this issue in more detail for any that are interested: Computationalism, The Church‚ÄďTuring Thesis, And The Church-Turing Fallacy. Here he is using ‚Äėcomputationalism‚Äô to refer to the computational theory of mind.

Yes, ‚Äúintuitively computable‚ÄĚ seems appropriate here. Roughly speaking, C-T is a set of conventions that attempt to make our intuitions precise.

I have not come across any good objections to using C-T to define ‚Äúcomputation‚ÄĚ.

Well, Malament-Hogarth spacetimes are physically possible, just very improbable :rofl:

Note that the Pitowski spacetimes are much less demanding than Malament-Hogarth spacetimes (flat spacetime is Pitowskian), but most require crazy (or even non-physically reasonable) paths for either the spaceship or the detector.

I think a cool sci-fi premise would be a hyper-advanced civilization that could engineer these spacetimes to do their computations in. A lot of Malament-Hogarth spacetimes require materials that disobey the energy conditions to built. These would be materials like dark energy or some sort of vacuum energy, and a hyper-advanced civilization might be able to produce and do engineering with them.

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Sean Carroll in recent mindscape podcast

(Click on transcript link to see quote)