Counting Cats in Zanzibar Rotating Header Image

What happens when a Turing Machine meets a Blackhole?

A while back I promised to write this. It has taken some time.

Well the short answer is nobody knows. In principle. Let me explain…

But seeing as this is not QI and I am not the curly haired loon Alan Davis I’d best try to explain why…

There are questions that are unanswered and there are the unanswerable.

Magic don’t exist. Science does and it is a kinda magic (or is that Queen?). Robert Oppenheimer certainly proved that. Well, with his pencil he certainly wielded more power than Dumbledore did with a twig. Real science is magic and it is magic beyond anything these sort of numpties could dream of.

In 1995 at Nottingham University a geezer dressed in a manner that would make Arch-Chancellor Ridcully look under-dressed handed me a piece of paper that was the official recognition of my getting the keys to the Universe. It was emotional, I can tell you. It was a BSc in Physics.

So what I am getting at in this preamble is that magic is real and it’s magic because it doesn’t always make “common sense”.

So, to the point!

Black-holes are essentially collapsed stars that have all their mass within their Schwarzchild radius. This is the radius which even classically, light can’t escape from because as you know Neil, Buzz and Mike had to go rather fast to get off this rock but seeing as the speed of light is a cosmic speed limit once the gravity of a collapsed star gets to that having an escape velocity above that of light then you are in it for the duration. By the way I’m sticking with the non-charged, non-rotational solutions here. Hence Schwarzchild will suffice.

If you really want to muck about with Kerr-Newman metric then knock yourselves out. We’ll keep it without angular velocity (or charge). Now I appreciate the revolutions of Beyonce’s twerking her fundamental singularity as much as the next person but for the current porpoise the entertainer on the stage might as well be Noel Coward in a dinner-suit singing some old nonsense that the late Queen Ma would like.

Because the simple truth is black holes have a property which is awesome. It brings information theory (one of the grandest achievements of C20 maths) into a kind of conflict with one of the grandest achievements of C20th physics. And it’s dead simple. By which I mean it lacks complexity. Now, modeling weather is complicated because modeling multiple processes is. This is a different kind of hard. It does not involve the kind of recursive computation that gets a Julia Set on screen. It is conceptually hard rather than computationally so. Hold that thought – it will matter. There are incredible complicated things that are hard and there are simple things that are hard. Things can be hard in qualitatively different ways.

Now, in the 1930s Alan Turing came up with a theoretical model for computers. I’m typing on one now. Now Turing proved (as did Alonzo Church) by a different route (and Kurt Gödel had a look in too) proved this. Now some of this was purely formal such as the disproof of Peano Arithmetic which set out to prove essentially that the mathematics of integers can be based on a single finite and logically consistent axiom-set. That this was a bit of an embuggeration (especially to David Hilbert) is to say the least. Hilbert had proposed a program in which he hoped all mathematics could be reduced to a single axiom set. Peano Arithmetic was a jolly good punt at that. Essentially he’d proposed an idea to develop something much like the characteristica universalis of Leibniz. Essentially an attempt to reduce everything to rule-following. The idea was this…

When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a “natural number”. Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don’t define anything. In 1900, David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twenty-three problems. In 1931, Kurt Gödel proved his second incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself

That’s from wikipedia

So it is entirely a bust flush. Or is it?

Anyway, I realized I’m hundreds of words in and haven’t answered the question! Sorry, but I had to foreground and the essential problem is to do with words like “definable”, “consistent” and especially “infinite”. Now the final one is the impenetrable. Now way back when Galileo realized infinity was a tough nut to crack when he put as an aside the idea (which is true) that the set of Naturals had exactly the same cardinality as the set of Perfect Squares. Now the first set is {1,2,3…} and the second starts {1,4,9…} but both have the same number of members. Huh? You might be thinking there are more of the first because of the gaps in the second? But as Galileo argued and Cantor proved the cardinality (the size of the sets – sort of) can be proven to be the identical for both because both can be put in an exact 1-1 bijection – essentially for each member of the first set there is a one on the other side – kinda like the perfect tea-dance – with integers He (Cantor) called this number Aleph-null. The first of the transfinite cardinals (there are many more cardinals – more than they have in Rome – and some are fucking enormous – a technical term I owe to a Leeds University number theorist). I am getting a bit OT here but I must mention a collection known (and I know this sounds rather “Father Ted” but there are, amongst many others, ineffable cardinals. Please read this because it conveys the total Woo-Woo.

Yes, there are numbers that are infinite in ways that can’t be uttered but must exist, logically. Yes this sounds mystic and it is but it is provable. This is not homeopathy. This is reality.

What was that paragraph about? Well it was really about trying to say (and this is relevant) that there. In an arguably similar way General Relativity contains solutions (such as those due to Kurt Gödel which include the possibility of time-travel but whilst mathematically impeccable do not pertain to our Universe, but could in principle, just don’t). Now that is interesting. So interesting I wrote an MSc thesis on it. What was dear old Gödel up to? Fuck knows! I don’t know fuck so I don’t. It is all a piggy-rotten mystery and no mistake. It’s like Windows 8. No bastard groks that one.

Anyway Blackholes are simpler. Once you cross the event horizon you is doomed and I mean proper Frazer doomed. Not only are you not getting out but you can’t really communicate out either. Because nothing can get out of an event horizon. It’s kinda like dropping car keys down a toilet. Now I’m on my uppers here (but promised to write this piece) but my understanding is that in the vicinity of an event horizon time slows as seen by an external observer but seems the same for the subject falling in. They also get gravitationally red-shifted and dim into the IR. So if you chuck your mate into a blackhole and they wave back at you it seems ever slower and ever redder until you can see nothing of them. Now this time dilation is kinda like working a 36hr day and then a 48hr day and then a 72hr day so you get more done. Essentially any computable problem accelerates (i.e. breaking a code and not whether Miley Cyrus ought to wear pants) because the effective time the Turing machine’s rate of knots has becomes asymptotic because time has slowed for it – though not for the observer at a safe distance. So it can actually solve or get round the likes of the Entsheidungsproblem

Except. As the Turing machine (and if you are reading this – you have one – just not a local blackhole – I hope) hits max and goes infinite it is going beyond the event horizon so you never get to know the results. I know. It sucks. Of course you could leap in after it but you’d never be able to get the data out so what is the point?

Now I have no idea whatsoever where my trash goes. But if it were to be chucked in a blackhole then nothing of it would remain to the external observer other than mass, electric charge and angular momentum. This applies to anything. This applies to pork bellies, gold ingots, the works of Shakespeare and your Aunty Gladys. All meaningful information is lost to the Universe. In a very real sense that is why blackholes are something else. And that is why a Turing machine can achieve infinite speed (even that dreadful Acer you bought five years ago) but anyone outside the hole can do nothing with it. And if you are past the event horizon neither can you.

And for my next trick I have this cat and this box ;-)

25 Comments

  1. Mark says:

    So is it alive or dead or don’t you know yet?

  2. NickM says:

    Well technically, from our perspective we don’t know so by the Copoenhagen Interpretation it is meaningless as a question. Until you open the box and the wavefunction collapses one way or the other and you either have to fetch a spade or a pouch of Felix.

  3. Flaxen Saxon says:

    Thanks for the post. I didn’t know what ‘twerking’ meant. But I googled it and got Miley Cyrus (ugh).

  4. Flaxen Saxon says:

    Mr Cats: As a physicist, what is your opinion on Kalum’s cosmological argument?

  5. NickM says:

    (a) I’m NickM, not Cats and (b) who is Kalum?

  6. NickM says:

    And (c) Kalum is toss but I am interested in Antony Garrett Lisi’s E8 theory.

  7. Flaxen Saxon says:

    My apologies NickM. I should pay more attention. Kalum’s cosmological argument is a revamping of Aristotle’s cosmological argument for the existence of a deity. Basically the first premise is that the universe has a cause. That being so then Kalum (?Aristotle) posits,that this is caused by an uncaused ‘god’. The argument has appeal as the notion of infinite regress is implausible. That said, the concept is so drenched in Aristotelian metaphysics as to be totally repugnant to the modern mind. The refutation of the argument is highly technical. Suffice to say the argument can be refuted on the basis of Occam’s razor (the law of parsimony). Why posit complexity? If you accept this argument you have to come up with reasons for the existence of god(s). Christian theologians use this argument for the existence of god, although even if it were true (which it is not) it would no means support the classical concept of the Christian view of god. Theologians are quick to offer god as the default explanation when scientists can not offer an explanation. My view is that the cause of the universe awaits a naturalistic explanation and to rely on a supernatural explanation is tantamount to surrendering to magic and wand waving.

  8. NickM says:

    Well, I am an agnostic. As I get older I get less interested in philosophy. Yes I know the “God of the gaps” argument and that it is silly. Anyhoo,infinite regress is not implausible.

  9. Flaxen Saxon says:

    I’m an atheist and a scientist. I am genuinely interested in your view on the possibility of infinite regress. It seems counter intuitive but that doesn’t mean that it’s not possible. Much of quantum mechanics is just weird.

  10. Peter says:

    I thought Hawking radiation escapes from black holes, though that may not be relevant to your point.

  11. NickM says:

    Peter,
    Yes |Hawking radiation does but it carries no info.

  12. Mark says:

    Please excuse my density. I’m sitting with my quad core Turing machine and we’re drifting towards this black hole. I’m sending you whatever results I get from the machine as we solve more and more NP-complete problems that take exponentially-increasing time as they get more complex.

    From my perspective the machine is going at the same speed and taking longer and longer to come up with answers to the increasingly complex problems. From your perspective I am acquiring a reddish sun tan and my relaying of results to you is getting even more slower and slower as your perception of my time changes.

    So where is our accelerating T-machine?

  13. NickM says:

    Mark as time slows more can be done. Imagine you have two days (as seen externally) to paint a house well what if it dilates to 7 days? You can put the kettle on and watch telly for a while. That is how it works. The folks commanding the mission see the short version. You get the long version to work in.

    But… Once your quad-core crosses the event horizon (i.e.) when it speeds to infinity from your ref point you can’t get any data anymore. So it can be done but you won’t know it unless you leap in and whilst you’ll know it nobody outside will or can. It’s like having Marilyn Monroe eternally flirting with you but you can never, in principle, touch.

    And BTW. forget quad core Intel Inside stuff. This would happen with a ZX-81.

    As to the general point of slowing and the red-shift. This is why Soviet physicists called these things “frozen stars”. A good name. They are very odd. Inside the event horizon of a Schwarzchild blackhole something odd happens. time ceases to be the directed co-ordinate and radial distance from the singularity replaces it. Blackholes with angular momentum are different. They have a circular singularity.

    The Universe is very weird. But if you have a quad core Intel please ask me about market rates before you chuck it into the universal garbology truck? I might just make an offer that is better than it just disappearing!

  14. Mark says:

    OK thanks. (I get the ZX-81 thing, I just happen to have a quad core in my hand. It’s not for sale :-)

  15. NickM says:

    Not that it matters Mark, this week I’ll a Motorola RAZR i for 150 quid (less if it is a return) and that has a 2GHz Intel Atom. I remember when that was supercomputer territory with Bond girls in very short lab-coats carrying clipboards and disks the size of family size pizzas in drives the size of tumble driers. So I shall remain Nick and live in a house and wear jeans and a T-shirt and not the Mao suit and plan Global domination with a just above entry level smartphone from an extinct volcano. And the monorail can go hang too. And our little tuxedo cat won’t be traded for a Persian. Modern life is rubbish.

  16. Talwin says:

    NickM. I haven’t understood a fucking word (GCE Gen. Science I, 1961, only). However, I genuinely find this stuff strangely compelling reading. Maybe that’s part of the magic too.

  17. Mark says:

    The first computer I used stored the programmes on paper tape, had 16k of core and 16Meg pizza disks.

  18. Mark says:

    (And we lived in a hole in the road)

  19. NickM says:

    And I once lived in Leeds…

  20. Mr Ecks says:

    Relativistic time dilation doesn’t really confer any practical advantage on the traveller. His/her time expands but only from the view point of the observer who is not travelling. A stationary observer might note that the travellers 2 days had actually lasted 7 of the observers days but from the travellers POV his/her 2 days still only lasted 2 days.

  21. RogerC says:

    Regarding Flaxen Saxon’s point about deities:

    However you look at it, the origin of the universe presents a mystery. As I see it, you have very few alternatives. Feel free to suggest more if you think I’ve missed anything.

    1) It has always been here and will always be here.

    2) It is in some manner circular or cyclic. This is basically a riff off 1).

    3) It sprang into being spontaneously, an effect without a cause. Presumably it might therefore end with a cause having no effect. Or it might carry on forever…

    None of the above make sense from the point of view of our common sense experience of the world!

    However, positing a supreme being, or arguing that the visible universe is just a component part or an emergent property of something larger merely pushes the problem back by one remove, it doesn’t actually do anything to explain it. Where did the supreme being come from, or the larger “multiverse” if you’re less supernaturally inclined? If those things were always here / sprang into being spontaneously, then why not the universe? As you note, Occam’s razor applies.

    In all three cases, the universe is essentially causeless. Either it was always here (either going back infinitely or going round and round in circles) or it just showed up one day. Philosophical argument about the origin of the universe has always seemed to me to be much more about the need of philosophers to have some sort of narrative that makes instinctive sense, than about actually understanding the phenomenon itself.

    Disclaimer: not a professional cosmologist. Feel free to critique.

    ~R~

  22. Flaxen Saxon says:

    Hi Roger C. I’m not a cosmologist either. I think the evidence for the universe having a beginning is very strong and the notion of an eternal ‘steady state’ universe is not in favour with cosmologists, these days; everything seems to point to a big bang. At that instant, space and time came into existence. A consideration of existence or anything prior to this, becomes hopelessly mired in contradiction and paradox. Of course, the question why the big bang happened in the first place is something that philosophers and scientists have been pondering for a while. Does our concept of causality have any meaning when attempting to explain the big bang? How can you have a cause when nothing, neither time nor space exists – my head hurts. Sophisticated theologians argue that ‘god’ exists outside time and space ie outside the naturalistic world and consequently not subject to natural law. How very convenient. God is an external, supernatural being.

    We poor mortals exist in the natural world subject to natural law. The problem of positing supernatural beings (of any type- devils and angels, take note), is that by definition they are unknowable. There is no empirical evidence for their existence. Even if they did actually exist we would never know. This of course leads us to agnosticism. However, my point is: because supernatural beings and realms are by definition unknowable they are tantamount to non existence. The onus is on the ‘believer’ of supernatural beings and phenomena to provide evidence for their existence. Of course this is never possible. If they did then this would constitute natural knowledge and therefore the ‘being’ or concept would leave the supernatural realm and become knowledge.

  23. NickM says:

    Good points but I assume RogerC and Flaxen Saxon are aware of the weak anthopomorphic principle? Basically it goes like this. If this universe didn’t exist in the way it does (settings for fundamental constants and such) there would be no bugger to debate the existence of God, Allah, Shiva or the Flying Spaghetti Monster. It seems a cop-out but the more you think on it… Perhaps there are/have been/will be other universes where no life form with the capacity to even conceive of such questions ever existed/exists/shall exist. Perhaps there are other sentient forms who know the answer (or an answer that suits them) but they’re in Andromeda. We are simultaneously cursed and blessed as H sapiens sapiens. Maybe that is the only real truth we can really know. OK, I know there are others like the law of the excluded middle but I was getting whimsical. Perhaps TS Eliot had the right idea with…

    “We shall not cease from exploration, and the end of all our exploring will be to arrive where we started and know the place for the first time.”

    And perhaps my cat, Timmy, got it by snoozing on the sofa, lying in front of the fire, chasing squirrels, eating Felix and Dreamies (Feline Crack) and generally having a hell of a time. Maybe cats are smart enough to manage in Timmy’s case to get a valet and house-keeper who love him dearly and cater for his every need and whim and not give a damn about the sort of issues that have had the learned scratching their heads for thousands of years.

    My wife and I really on the NHS. Timmy has private medical insurance. So who is smarter? Lizzy and I go out and buy and each other him food. He doesn’t. He doesn’t even catch the squirrels though I once had the delightful task of retrieving from a water-butt the corpse of a squirrel. It was white, hairless and bloated. It was rank I can tell you with a Dame Judy beyond the Pale. I then had to bury it. Maybe Timmy chased it into the water-butt (I have seen him perform manoeuvre kills before of sheer genius – I once saw him getting beaten-up by another cat and shooed the critter – Timmy did a Superman leap onto the fleeing beast’s back with fangs and claws bared – there was a horrendous noise and we never did see that cat again). That was a Guderian move.

    I’m writing this tripe and Timmy is probably asleep on the bed dreaming of mice. So we may be more intelligent but are we smarter?

    He does bring us presents mind. Small dead birds mainly. Not actually useful but I guess it is the thought that counts.

  24. Flaxen Saxon says:

    I suppose what you are saying is that it’s all about ‘subjective perspective’. Is real objectivity ever possible? Perhaps not, but we can only work with what we have. We must be ‘seeing’ something from our perspective which is meaningful or science (the scientific method) wouldn’t work. I’m not a philosopher, I’m a professional biologist, and get to work with tangible entities. I get to stick pins in things, but only if I want to. There is a strain in cosmology that leads to mysticism, and I can understand why without condoning. I’ve been accused of being a crusty, old empiricist (Kant and Popper can bugger off). I haven’t always been this way. Anyway, ask me a question about dandelions or evolution- I feel I’m on firmer ground.

  25. NickM says:

    Flaxen, yes and no as to “subjective perspective”. “Yes” as in you can sometimes know things but can’t communicate them – hardly a new idea really. It just seems odd after the invention of the telegraph etc. “No” as in you can know that in principle a solution to certain mathematical problems has been found but never know what it is. It’s a bit like knowing your car keys exist somewhere despite being buggered as to where they exist. Again less radical than the black-hole thing sounds at first blush. Sort of.

Leave a Reply

%d bloggers like this: