The LONGEST time - Numberphile

Key Takeaways

you got another number for us I have yeah yeah it's the um I'll just write it down shall I that's probably the the best thing to do so it is 10 to the 10 to the 10 to the 10 to the 10 and then this is the strange bit to the 1.1 okay this has been claimed to be the largest finite time that has ever been calculated by a physicist in a in a published paper this is the paper it's a bit of a weird paper it's not had a huge impact or anything it's about black hole information loss and conscious beings we won't we won't go down that route he actually calculates something called the uh panker recurrence time for um a certain type of universe within a certain cosmological model and this is the the number that he gets out so this is this is the one I'm I'm on about let me just check I got the number of tens right yeah I did okay here it's equation 16 so he's put plank times Millennial or whatever because basically it don't really make any difference whether you use seconds plank times Millennia years when the number is this big but there's other interesting numbers in here which is this one here you know you got 310 to the 2.08 that's the point career recurrence time for for our universe what's all this about dude okay so Point career recurrence time this is this is something that's Arisen from statistical mechanics very simply you know we if we had um a pack of cards and we you know we' only got a finite number of cards in that pack and let's say we keep dealing each other you know hands of five cards eventually if we do it for long enough you know you're going to get a royal flush Brady that's guaranteed to happen and then if you wait long enough again you'll get another one and so on and so on and that's true because there's a finite number of cards now what panker realized is if you take a gas of particles you can put a gas in a box now you can put all the particles in one corner of the box and then they'll dispair and then they'll move around and but what pranker recurrence tells us that after a very very very very long time those particles will eventually return to the corner of the Box you always get a repetition and it's basically because the thing that controls the evolution of that system only has a finite region of what we call phase space of of sort of solution space that's accessible to it and So eventually you always be come back to it arbitrary close to where you started the the time scales are truly enormous before you start expecting it to happen so you can apply this Quantum mechanically as well so when you're doing in quantum mechanics what you're really talking about is is sort of the evolution of micro States within the quantum system so you know the sort of building M Quantum building blocks of your system and they will eventually return they will evolve but eventually return to their initial State and what Don paage and this paper has tried to do is he's actually applied that to various types of universe you models of various types of universe it's a bit of a cheat because a universe is what would call a macroscopic object it's a large object it's not really a mic Quantum micro state in some sense it what what it is is is it's it's a an ensemble an average of all the micro States so what he's tried to do is he's tried to say okay I'm going to um treat the universe as this as this average of all these states but then I'm going to count up all the possible averages and treat them as a as a sort of micro state in itself so it's a bit of a cheat but he gets an extra exponential out from that are we are we talking about Jupiter's over there the androma Galax over there I'm in this room filming you yeah yeah yeah yeah yeah all that okay now it's a very there's a very large number of possibilities that you can have but it's finite and so as the system evolves it's only got it and it can only evolve the system can only evolve through a finite number of possibilities and eventually it evolves back to where it started so actually it's um I've often he heard it said that the Universe for example will um you know will evolve will expand eventually everything will be very far part you know spread out very far because of the expansion of the universe that all objects will have collapsed the forms black holes that then those black holes will have evaporated from Hawking radiation and all you will have is this very sort of you know sort of Bleak landscape of um of just radiation and that's come out of these evaporated black holes that's uniformly distributed and it' be very boring okay but that isn't the end state of the universe the end state of the universe is is that after these truly epic time scales you will eventually have a panker recurrence and you'll get back you'll wind up back where you started from and it's quite easy to see how it might happen imagine you have this sort of Bleak Universe right just have a little fluctuation that little fluctuation sort of you know gathers together builds up other things eventually it sort of forms a a a a sort of Galaxy even you know from that Galaxy you get planets stars and you keep going you keep going eventually you'll get back to a situation where it looks like it is today now what I think is is fair I think it's a fair point is that there's no way of ever being aware of these rep repetitions over these large times and the reason is you can never build a device you could never be an observer that could measure this and the and that's that's because over these huge time scales such a device or such an observer would definitely thermalize would definitely become part of this recurrence itself so there's never anybody or anything that could measure it there's this sort of idea you know this sort of notion within physics is that if you can't measure it it's irrelevant in some sense so but according to this in this in this number of in this time in this number of years you and I are going to make this video again you and I are going to make this video again I know this is the it's less than that this this is for a special type of of universe that's particularly large okay the number for a for for us is at least less than this other number that he's written down here which is 10 of 10 to of 10 to the 10 to the 2.08 as I said I'll just say years okay so this one is the one that applies to us in this unit in our visible Universe what we call our causal patch this one applies to uh seeing what is the panker recurrence time for a truly vast domain of of universe that you can get out of certain models of um of cosmology so uh they all they're all based of course on the on the same idea and we can work through where this where these numbers come from so the pranker recurrence time of of any system is roughly proportional to the uh to the number of sort of states in that system because we're applying this to the universe this is really the number of macro States the number of these averages of micro states that you can talk about so let's call this n macro this then we'd expect it to be about the exponential of the number of micro States now why is that well this doesn't really have to be an e this could be a two or whatever basically any micro state is either in or out of the averaging with some weighting and so this is the the number that you get out the number of micro States when you relate to the entropy is e to the entropy let's let's look at some volume of the universe okay of some radius R then the entropy we've done this before and it's the same argue as we have before the entropy that you could possibly have in this this region of space basically it's proportional to I'll just be a bit sloppy with factors R 2 over the plank length squared so the next step is e to the e to the so now let's apply so let's apply it to the the really big number that he does okay so the question is what's R well the radius of the universe is about 10^ the 26 M yeah our visible Universe okay so as far as we can see what we call our causal patch the plank length is about 10- 34 M though R over ml plank squared is about 10^ 120 which is a number you often see in h in in physics actually this is sort of the number that's associated with the cosmological constant problem but anyway well 120 is about 10^ 2.08 that's where that is coming from I don't know why he's so precise about this to 2.08 because what he's going to do next is so the the number we have is e to the e to the 10 to the 10 to the 2.08 but what he does here is he just approximates these e as 10 which is fine really in the broader scheme saying E 10 for the sake of cosmology they're more or less the same thing all e become 10 and you get 10 to the 10 to the 10 to the 2.08 which hopefully is what he's got there and it is okay so that's where that number comes from so so this is basically the point career recurence time for our visible Universe our patch why is the other number bigger well because the other number is is is um there he's looking at he's trying to get a big number is why he get he gets he's trying to get a bigger number what what he's looking at there is is um model of inflation now inflation is some a model of the very early Universe where the universe grew really quickly out of a very small patch of a universe the amount by which it blows up depends on various parameters in the model but basically this the thing that you get is you get that the size of the universe so R for that case is of order e to the 4 pi over m^2 okay so this is actually R over L plank it really he's done everything in plank units so that's for M here is the mass of some of this infl what we call the inflaton field it's just some field in the model that causes the expansion 1 m^2 is 10 12 I think this might be 13 actually 10 12 so this is about 10^ the 13 right overall because 13 is about 10 okay bear with me so this is e to the 10^ 13 roughly the 13 well is about 10 to the 1.1 that's where the 1.1 comes from he's very precise about this 1.1 yeah he's very sluy with some of the other factors get e to the E to another e to the 10 to the 10 to the 1.1 and then we do the same thing we turn all the e into tens help me understand cuz you know I'm just how big numbers are how long a time is this okay so this this is truly vast like I said there's no device there's no Observer that anything that could survive this kind of length of time scale in fact you would probably say that the universe is more likely to Tunnel out of the current state before this could happen in some sense this is such a long time scale that one might say actually the probability of tunneling to a new phase of the universe completely different is actually going to dominate over this that would occur first so maybe this is kind of irrelevant point but um yeah I mean it's truly vast I mean it's it's bigger than um than a Google clearly way bigger is it bigger than a Google Plex I think so so uh you know let's just check that so yeah clearly it is it's enormous it's it's prob as big as Graham's number but you know grahames number is the daddy right so well I didn't actually know about this paper but yeah I just I was um I just thought oh big numbers let's see what's interesting about big numbers and uh and then I stumbled across this paper it's the lot it well he claims in fact he says it he claims it's the so so far as I know these are the longest finite times that have been explicitly calculated by any physicist so whether somebody has calculated a longer one and I'm sure some of the viewers will try to calculate a long one and then claim that theyve but this is in a published paper right so you got to get the paper published but um the challeng is on I guess to find a long one it's probably they probably has been since but he was the one that pointed it out