 Okay, thank you mercy. Okay. That is not a black hole evaporating That's the Kizumun volcano in Kamchatka that took a picture out of the out of the window an airplane when I was flying from Vancouver to I Can't remember whether it was about a ten years ago. Either to Korea or to or to Japan I Forgot to leave and look up the year, but it's on it's in the Wikipedia article about Kizumun volcano, so Okay, so Well, we've we've heard a lot from Jose and that and he's covered a lot that some of the things that I'm trying to cover too So there's some overlap in some sense classically black holes Maybe the simplest objects in the university and put what once they settle down They're just characterized by mass and angular momentum in asymptotically flat space, but quantum mechanically they may be the most complex they seem to have an enormous number of states and Enormous number of different behaviors, so I was asked to talk a little bit about some of the timescale So I'll talk a bit about that. There are a number of timescales associated with a black hole in this talk, yeah, I'm Going to whoops, where's the point to the pointers the top part? Oh There we go. Anyway for simplicity, I'm going to just talk about four-dimensional space time Maybe most of formulas generalized other dimensions, but let's do that. So if you if you imagine well We the biggest black hole that we've seen now. I mean it's some question about what the mass is but of order 40 Billion solar masses at least as some example Wikipedia the number is very questionable But anyway of that order, there's others of order 30 billion solar masses black hole So let's suppose we take the biggest one to get some of the biggest numbers of this Then of course if you say that we have that much energy Then what's the quantum oscillation period for the for the phase to change by 2 pi then that that period is is about 10 to the 91 seconds and then the Hawking decay time is around 10 to the 106 seconds, so there's a ratio of those times of that and then there's also Larger timescales of black hole if it's if it's stable in the micro canonical and in an anti-decider space time Well as the box can't be too big as Jose said in this case for this black hole The box can't be bigger than about 10 to the 31 light years Then there's a classical recurrence time or I think what was it called this Hilbert time e to the the s It's roughly that it's roughly the 170 30 37th root of a Google plex And then there's a quantum recurrence time for the quantum state to get back to within some small number of that Which is doubly exponential, so it's roughly exponential of 10 to the 10 to the 10 to the 100 and then times alpha so Google plex times alpha raise it to the power of 10 Exponentiated anyway, so there's enormous range of times for for some of these black holes and Okay, I'll probably can skip over the details of this but to avoid Historical accidents of units like the foot and and so on all use plank units and here You know the plank mass and the and the plank length and the plank time and the plank and the plank energy Which is roughly I think the most kinetic energy of an airplane that I've asked the pilots about Afterward what their mass it was about 10 plank units for the kinetic energy of a jumbo jet smaller Just tend to be less than one plank unit of kinetic energy Plank power is that plank energy density while it's it's 85% of 10 to the 200 electron volts for megaparsec cube If you want to use crazy units for this thing Okay, and observe physical parameters of plank units. Well the elementary charge I'm setting H bar and C and G is is roughly that proton mass About 10 to the minus 20. It's very close to the inverse square root of the largest prime ever found by human without the use of computers But off by a little bit electron mass 10 to the minus 23 solar mass is Roughly 10 to 37. It's it's it's it's close to the inverse square of this by Stellar evolution stellar model things supermassive black hole of the biggest one I'll Take as this which is roughly that a year well a Julian year that astronomers use is that age of universe is about 5 trillion days Cosmological cause the interesting is very these numbers very close to Unity that fit in there so you can describe it by 10 by three English words 10 square out of Hertz is the cosmological constant And the given Hawking energy is also very close to within the uncertainties of lamba. It's it's it's very close to this this number Okay, that's just the background Well as we heard before after suggestions from David Beckenstein that black holes have an entropy proportional to their horizon area Which Stephen Hawking originally noted would be inconsistent with the then-believed fact that black holes can only absorb radiation but then after independent predictions by Jacob Zeldovich and Alexi Starbinski and then others of us Larry Ford and I sort of independently realized later That the black rotating black holes would emit then Hawking heard about this and he talked to Zeldovich and Starbinski he liked her idea didn't like their derivation So he went back to do it to do the derivation and found his way that even non rotating black holes He met radiation and have a temperature And so here I'll give it for a non rotating uncharted black hole in other words short shield black hole With event horizon radius that so well for general Black hole it's in four dimensions. It's it's related to the the surface gravity And then okay, if you put in the numbers, it's what in plank unit is one over eight pie in So this thermal Hawking radiation made Beckenstein's idea of entropy consistent It wouldn't just be infinite entropy if the black hole only absorbed and it fixed the unknown constant of proportionality So that the dimensionless entropy entropy divided by Boltzmann's constant Is is then given by in plank units a quarter of their horizon area or for a short shield black hole four pi m squared So we heard in the last talk the entropy goes proportional to the square of the of the of the mass of the energy So then Okay, so then Vaporation rates I did numerical calculations from Hawking's formulas of various rates for a black hole and Roughly if you have a big black hole of stellar masses and assuming that all three neutrinos Have masses that are within an order of magnitude or took our couple of them Then then these massive black holes basically can only emit photons and gravitons and then the the mass rate loss is given by this This is not the fine structure constant, but just a numerical calculation constant that I calculated Then the entropy of the Beckenstein Hawking entropy of the black hole Decreases by that The radiation because it goes through this barrier. There's there's some extra entropy Denoted it's bigger by a factor that for photons and gravitons is about this The last digits are probably uncertain, but I put them here is just if anybody wants to repeat the calculation I'd be curious to know how close I got it, but Anyway, if it's way off, I don't know do I lose my PhD if If somebody gets finally different numbers Okay, so semi classical evolution of the black hole it's it's decaying and four dimensions and because well we heard from the last talk the lifetime goes is in cubed and So therefore the the the masses a function of time goes like this the decay time is that it's it put in the numbers It's it's almost nine thousand times the in plank units times the cube of the mass or in solar masses This is the mass of the Sun 10 to the 67 years for a solar mass and a 10 solar mass would be 10 to the 70 years and so on and Then there's semi classical beccasite Hawking black hole entropy then has this time dependence and Whereas the radiation entropy because there's this extra factor beta because of the scattering through the barrier outside has this sort of form then Okay, if Hawking well as we heard also from the people talking assuming local quantum field theory and a fixed dynamical space-time background And particularly assuming the horizons Smooth well, I guess this is really what's outside right? I suppose it doesn't really matter. What's inside the horizon? but if what's outside the rise of the smooth assuming local quantum field theory then It would evaporate in black hole with its absolute event horizon That's the boundary of what can get to the outside then leads the loss of information From the exterior Hawking therefore predicted when the black hole evaporates away And information would be lost from the universe and a pure initial state would become a mixed state so he did this breakdown of I Think he originally Tyler breakdown of physics and a referee required him to change that so he'd break or break down our predictability And this argument did depend on a semi classical analysis the first paper to dispute it was the following one Which gave a number of options and but didn't really prove which one was right But suggested that maybe there was no loss of information Well the issue laid in the doldrum for years But now most opinion but not all is switched to mine and even Hawking has as conceded that But there are still you know holdouts like bill undrew and Bob wall that so on think that maybe information really is lost Now interest in this in this black hole information thing Surge recently with a paper by on Hyrey Meroff Plochinsky and sully black hole complimentary are firewalls 2013 so they gave a provocative argument that suggests that an in falling Observer burns up at the horizon or he gets destroyed at the horizon of a sufficiently old black hole So the horizon becomes what they call the firewall So the argument basically was that if unitary evolution suggests at late times The Hawking radiation is maximally entangled with the remaining black hole and neighborhood including the modes just outside the horizon But then this further suggests that what's just outside cannot be insufficiently entangled with what's inside There's a monogamy theorem that's that something can't be maximally entangled with two different things So if the Hawking radiation that's about to come out is maximally entangled with the earlier Thing what you'd expect to be at least after it came out You'd expect it to be maximally entangled with the the the earlier Hawking radiation if it is also maximally entangled when it's just about To come out it can't be maximally entangled with just inside and then without this entanglement you'd have fields basically oscillating into or Fluctuating independently in both sides of the horizon which would lead to huge gradients in the fields and therefore large energy densities So without this ladder entanglement observer falling into the black hole should be burned up by high energy radiation Okay, I'm not gonna solve this this this problem here I think I mean it does make assumptions in particular local quantum field theory outside the horizon whereas in quantum gravity I don't think we have locality. So it's there's at least one assumption. That's almost certainly not true but it is a little bit surprising how Locality is a pretty good approximation now now you cannot Externally observe entanglement across the horizon because at least if it is an absolute horizon because you can't see across it But it's but this should be eventually transferred to the radiation So one thing we'd like to know is the retarded time dependence of the von Neumann entropy the Hawking radiation if you look at Scribe plus at the future in all infinity and ask What's the entropy of up to what some point? How does it depend on time and the Strominger? Raise this question gave five back can it answer? Maybe it's a bad question. Maybe the information is destroyed Maybe there's a long-ring rev net Maybe there's a non-local remnet and then max or maxible information return Well, I'm going to assume unitarity and maxible information return that the black hole does go away It doesn't leave a remnet and that this this comes out. So let me just give a little bit things about the The time scales so I'll assume unitary evolution no loss of information I'll assume that there's an initial approximately pure state So for example if you form it from a star the entropy of a star is of order that which is a lot bigger than the Beckenstein Hawking entropy of the black hole for a solar mass is of order 20 of orders of magnitude bigger And you remember this gets Exponentiated so the number of states is just enormously bigger For a black hole then for the star So but we assume that since it's for maybe from a low entropy state that initially the black hole really The number of states that are composed of is much smaller than this. It's actually a tiny subset of that Okay, I'll assume that there's nearly maxible entanglement between the hole and the radiation that and also People asked me to and ask how do I measure entropy? I'm going to measure entanglement across a fuzzy boundary if you put a sharp boundary You tend to get depending on what your cutoff is if your cutoffs have ordered the plank energy you tend to get Entropies that have ordered the at quarter the area and I want to avoid that So well one thing is you could take as the if it's a pure if the whole thing's a pure state Then half the mutual information would be the entropy of each part So instead of taking with a sharp boundary you could say one of the systems inside oracles 3m Where's the circular photon orbits? I mean this is arbitrary, but and the other ones outside are equal 6m Which is the isco the innermost stable circular orbits for massive things So if you admit that region in between then the the mutual information becomes finite You're sort of missing that region But you'd expect just Hawking radiation without it without including the high energy modes to have you know Over one particle crossing this region at each time. Okay. I'm assuming complete evaporation in the just final Hawking radiation I'm assuming a non rotating uncharged in other words short shield black hole in four dimensions initial Mac black hole is large massless photons and gravitons if this is true then Then I'm assuming that there's no other particles that have less mass than this and this is quite a lot less than the differences Is in the masses for neutrinos? We don't know what the lightest neutrino, but I'm just assuming the lightest ones bigger than that So I'm going to soon just essentially photons and gravitons emitted. That's basically just to get the time scales I mean you can modify it to that. Okay, I just made up another slide this morning Just explain a little bit more or another idea of how to regularize this thing So suppose I said that there is a von Neumann entropy in a region Let's suppose we take spherical shells and Suppose we say that is a region between our time a times a m That's why I can make a to be just a pure number and our b times a m So for example, I'll use a equals 3 and b equals 6 for example And suppose you use the Planck scale cutoff with areas of that and Let's suppose that I want the I really want the entropy not counting these Contributions that have ordered the area or if you do the cutoff just right you can get it a over 4 of the boundaries So I'm I'm really interested in the entropy of taking off these high values that you get at the cutoffs So suppose I did that Well, let's suppose I have a Cauchy surface that maybe cuts across and goes into the black hole But it's it's it's regulars to Cauchy surface and I want to and so on the whole Cauchy surface I have some state, but I don't want to divide it into subsystems. So say one that goes all away from the middle I mean, I have to have the surface go back and before the singularity. So it's it's a regular Cauchy surface So from zero radius to 3m and then between 3m 6m and then from 6m on to infinity and Suppose that except for the high frequency mode, there's negligible entanglement between this region I mean after all it's sort of it's a small region So there wouldn't be expected to be many Hawking quadricles crossing it So except for the high energy modes, you wouldn't expect that much entanglement between this region and the other two regions. So therefore the the this this Finite entropy or this entropy is not too big, but it would be Between there and there and between there and there in other words There's not too much entanglement. So it's basically a sum and also from r equals 3m out to infinity is roughly this Okay, then you can just by putting this in you I can write these things in terms of these Unregularized well, you have to get a cut off. I guess to make them finite But these ones that you cut that you'd look include the high energy modes If you did that you could you could you could get this and then that's an approximation for that And then this is approximation for that so if I use those approximations I get finite numbers because basically There's one boundary here and one boundary for there and but this has two boundaries So therefore the minus sign that the contribution from the boundaries that are that are like a over four Cancel between here and here and similarly down here. So these are then finite things So that's what I'm going to be sort of talking about when I talk about these these things Now the arguments for nearly maxible entanglement is that if you take a if you take a sub a Pure state a system in a pure state and it's a the Hilbert space is a tensor product of two parts Then if you trace over one part you get the the von Neumann entropy the other part and Similarly if you trace over this part you get the von Neumann entropy that if the whole things appear state those two are the same But then you can ask that of course that depends I mean once you set the decomposition of the Hilbert space it depends on what the total pure state is But you can take the average You if you integrate over this over this this complex projective space that Gary talked about with the The the standard measure on it the fubini of the volume element of the fubini study metric on that you get what's called the Har measure on these pure states so I can do a hard averaging over all pure states to calculate what the average was and that's what I What I calculated in here. I had an approximate formula And I had a conjecture for an exact formula that other people later proved And then if you apply it that that comes out that there's a less than the half unit of information on average in the smaller sub system In other words, it's entropy. It's von Neumann entropy The average is within a half a unit of the and actually if the if the two systems are unequal on sides It's much closer, but that's sort of the maximum difference Then I applied this to black holes if all the information going into gravitation collapse Escapes gradually from the black hole it would likely come out in such a slow rate or be so spread out It can never be found or excluded by perturbative analysis. This is like Jose is saying it's do you've got e to the minus s effects? So it's non perturbative. You can't you can't exclude it by perturbative analysis Well, then also so kind of on Susskin's Conjecture that black holes are fat fast scramblers and that they take a time of logarithmic in the number of degrees of freedom and That black holes are the fastest scramblers in nature So that they rapidly give a state that gives something like this They don't well it doesn't get hard averaging over this time But it gives it so that almost all subsystems the average the average entropy would be very nearly the same as that So they support like results on using an average overall peer states of the total system of black hole plus radiation so Okay, so let's let's just do some calculation suppose we take this these semi-class These are now really going to be that the ones have taken out the the the the a things at the boundary So I'm going back to capital letters here Suppose they be a alright suppose these are approximate upper bounds on the von Neumann entropy See, I'm thinking the Beckenstein Hawking entropy is it's not always the actual von Neumann entropy of a black hole In fact the black hole the form from the star with the actual von Neumann entropy Would be much much less But that would be the maximum for a black hole of that size and then the radiation will assume that you know You have some volumes and you have this some this amount of energy to raise. What's the maximum amount of that? So that's what these two numbers are the von Neumann entropy The Hawking radiation which is assumed to initially pure initial state of uniterity is the same as the von Neumann entropy The black hole since the co-amidations that appear state and it shouldn't be greater than either this or these these are upper bounds So then okay, I can make this well This is partly coming up on an acronym. That's the first letters of my wife's name Kathy But anyway, conjectured annex or anorexic triangle hypothesis that the triangular inequalities for the joint entropy Are nearly saturated. So it's a nearly and nearly anorexic Well, actually was my wife. I told her it was a very skinny triangle and she asked me if it was anorexic So anyway, she gave me the idea. It's an anorexic triangle that the inequalities are very nearly saturated So this leads the assumption of nearly maximal entangle what we need hole in radiation So the von Neumann entropy be near the minimum of that It can't be bigger than either one and as time changes then these well first This is big and then the radiation is small and there's a ratio come out to cross. So That's what then leads to this the von Neumann entropy going up and down So it's the semi classical which is spontaneously creasing with time of the radiation and the semi classical entropy the black hole Decreasing as the black hole evaporates the maximum occurs at that this crossover point at this time Which you see it's about half the decay time, but it's not exactly and the epsilon depends on this ratio of the of the How much entropy is produced by the thing and the mass of the black hole? It's about seventy seven percent initial black hole and the Becking Hawking's the Beckenstein Hawking entropy is about 60 percent of the of the initial value It's a little bit off off from the half, which is you know But this is yeah, this is the point at which the thing Turns over so at this time The von Neumann entropy the radiation and that the entropy then has a value of that about 60 percent of the initial one Or in terms of solar masses, it's about six times ten to the seventy six times the square of the mass and solar units so it's about twenty percent greater than half the original semi classical one and This time it's about eighty three percent of the of the half of the decay Sorry, it's about It's it's it's it's it's roughly it's that times this time Which is the time for the black hole to lose half its area and half its semi classical Beckenstein Hawking entropy so Okay, then you can put in if you want to you can then get an explicit formula for the von Neumann entropy Because you've got this at the early times before this t star You you've got this this theta functions once you've got this term And then after that time you have this term and then putting in numbers in turn in plank units You get this this sort of thing so here's a here's a graph of that thing You can see it goes up and it curves up a little bit because of course as the black hole loses mass It evaporates faster and then here it goes down and then it curves it bends down because basically as you get to the end The black hole evaporates faster. So that's it's the this is even more more curve for for this this thing So, okay, that's what you that's what we sort of expect. I mean, it's not proved that this is exactly right But that's that's what I would sort of expect it to be Well, I'd also expect some correct correct some corrections to those formulas From fluctuations of the mass the black hole could spit out a little more particles or a little less So there may be more changes in each for the mass Fluctions of the position There's going to be brownie in motion to the black hole and in fact by time a solar mass black hole evaporates the uncertainty in the position is Going to be I forget 20 orders of magnitude in the than the size of the observable universe today or something It's enormous how much uncertainty there is in the position, but that doesn't add that much to the entropy Entropy of the motion, okay, there's a log m and Maybe non-maximal entangle, but the fact that it's it's not going to be exactly to peak There's this half units of order one and entropy near the black hole I'm not counting the high frequency modes, but just caulking particles near the border one Fuzziness in the boundary of order one. So there's you know, there's there's corrections of our but that whole entropy's of order in Square, so these are all relatively small correction. So Okay, so on the assumption that a short shield black hole of initial mass Much bigger than the solar mass basically to make it too massive to mid anything but photons and gravitons If it starts at a nearly pure quantum state at least relatively speaking and decays away completely by unitary process While being nearly maximal scrambled at all time the von Neumann entropy increases up to this maximum entropy At a time given by this and then decreases back to near zero Okay, I don't have time to give it here, but if you did start with a black hole on a maximum mixed state This is from some other people where I talked about the time dependence the von Neumann entropy the Hawking radiation increases from zero Up to a maximum of a bit of about 20% more than the initial Beccasite Hawking entropy at this time and then the increases back this so you can do these curves for different Vies of an initial entropy, but here I don't have only time I made just that just to summarize what I've calculated in some other paper now so far I've been talking about black holes that decay in Asymptotically flat space time, but an anti-decentre Space time with a negative cosmological constant if you impose either thermal or reflecting boundary conditions that it's time like conformal boundary at radial infinity The thing it acts like a finite box So either you keep it perfectly reflecting and fix the energy or you keep it thermal at the box Then sufficiently large black holes can be stable in either the canonical or micro canonical ensemble as Hawking and I found So if we use this notation for the word so B is what was it called L? it's the scale length of the black hole and Canonical ensemble gives a stable black hole if the temperature is bigger than the square of three over 2 pi B This is in four-dimensional space time is generalized you can generalize our other dimensions and the mass is bigger than that so if if the scale is is B divided by giga years then then the Then the mass would would have to be bigger than than this if that were a giga year of core For for the canonical ensemble and B is related to the negative cosmological constant by this formula the micro canonical ensemble allows the box to be much bigger than the black hole and Then basically the the energy of the hole plus radiation has to be bigger than this Well, this comes out from numbers and then the root of some cubic So it's about this this mass where B is in giga years So if we had a link scale of giga years, then you'd have Then even a a black hole smaller than a solar mass could be stable with radiation so you could have You know so solar mass or less it's stable with its radiation in this in this anti-decentre space If its size is at least that are bigger Okay, then we have various recurrence times if you have it in anti-decentre space to keep the thing from evaporate Then the the black hole can can can change and you can if the if RH is the radius of the rise in an anti-decentre space Then the mass and the entropy are this that the mass and entropy are non-linear functions And you can see that basically at large mass then the mass goes is the Cube and so the entropy goes is the two-thirds power Which is what which is a D what you'd expect for conformal field theory with two plus one dimensions one time and two space dimensions so the number of quantum states of a black hole of roughly this size is approximately e to the s and This multiplied by the time for light to travel a distance comparable to the hole say that is a classical recurrence time It's the expected time to return to near the initial state at the black hole Say if you imagine this is not really full-quantary if we're undergoing a sequence of classical transitions So for represented by roughly s classical bits ignoring factors like log 2 that each could take vise of 0 1 This would be expected time until the black hole returned to the same sequence of Cubans Okay, I'm not I don't fully understand the arguments and they're not sure I can repeat them But Lenny Susskin and others have argued that black holes formed by collapse from smooth initial conditions Remain free of firewalls for a time at least of the order of the classical recurrence time Assuming they're kept from evaporating and if they did evaporate then big black holes They wouldn't last anywhere near this time so that they would never become unsmooth But even if they're in the center There are arguments that they that it won't get Again, I'm not completely competent to discuss the details and I don't have time to discuss the details Now if you want just not just the microscopic classical configuration to return to near some previous value, but also the quantum state Then you need all the amplitudes for all well all to run to return to very near the same values So that you don't get a orthogonal state So the sum of the absolute squares of the differences of these amplitudes are small So this is a much stronger constraint and it takes a quantum recurrence time Which for black holes is estimated to be doubly exponential So X exponentiate the number of that's the number of states But you have to have an exponential of that because you're you're basically having this number of of states with with phases You want all basically nearly all the phases to come back to be close to zero So doubly exponential the entropy or the number of cubits representing the black hole So okay, Jose talked a little bit about complexity at the end Suskin and others like Adam Brown have argued that the complexity of a black hole quantum state Which is defined as the minimal number of a certain set of quantum gates needed to produce the state Well within some epsilon. That's again not specified front To produce a safe on a simple fiducial state goes linear with the time for order this this is I think what? Oh, they called it the the Heisenberg time that the the classical recurrence time And then it reaches the maximum possible value and then the complexity stays there with small fluctuations But then if you want it to have a fluctuation that goes down very near to zero that takes this quantum recurrence time. So Anyway, so he's saying that well, maybe the black hole stays smooth up until at least this this or This time and then it might have firewalls. I mean, okay Nobody's really sure it might have firewalls, but then occasionally maybe the firewalls would go away if it went down to that Okay, this is very speculative. I don't know. I mean, I guess I tend to favor the fact that is the view that they hardly ever have firewalls But it's that's a subject of Okay, let me go back to the the time scales and let's collect a number of them and do them for For two masses for a solar mass even though that's too small to actually astrophysically form a black hole But it's a convenient mass unit and then also for these ones It's roughly the highest estimate of any supermassive black hole to subserve so far So the shortest one I can think of is the quantum period in black units 2 pi over m 2 pi over the energy So a plank units. It's that so that many seconds for a solar mass for a 40 billion solar mass. It's really short Next shortest time I can think it might be the time just travel one short shield radius You might call it the light crossing time. So for a solar mass, that's about 10 to the minus 5 seconds For a supermassive black holes. It's a border four and a half days for light to go go that distance Okay, then all right, then there's a scrambling time and this is a time for the information to essentially get scrambled over the black hole or Well other things that that Preskel and Hayden show or even Preskel showed that If you knew all the quantum information that came out of the black hole up to some time and now you've somebody fit in some New information how long would it take for you to decode that new information and that's also board of the scrambling time? so it's the light time crossing time times a log of ass so then for that the Solar mass black hole. It's it's it's about three Milliseconds and for this one. It's a border 5.6 years. So it's bigger than the four and a half days by this log factor All right then another scale again This is a little arbitrary But if you if you ask what sort of anti-decider space could have this black hole in it as a micro conical ensemble the time scale for Oscillation of a particle in this anti-decider space that would give another scale that you you could get and Basically because it goes the e to the five thirds Then that's about 10 to the 65 plank units or about 10 to the 14 years for a solar mass black hole That's the if you if you'd have to have an anti-decider space With it with a length scale or a time scale less than this to hold it It was bigger than that the block would be more favorable for the black hole evaporate away for this 40 billion mass one You get 8 times 10 to the 31 years for this Then this time to reach the maximum von Neumann entropy That's what some people called the the the page time. Well, maybe they haven't defined it Well, so I don't know maybe I'm taking liberties with it to say it's the time when the von Neumann It is the maximum, but I assume that's what people mean. Anyway, I didn't make up the name So I'm I guess I should leave it to who how they define it But having the peak of the von Neumann entropy at least if you start as reasonable thing So that time scale for a solar mass black hole is it's six times in the 66 years for the supermassive It's about four times in the 98 years the Hawking evaporation time is you know of order two times this but not quite so that's 10 to the 67 years for the solar mass and almost 10 to the 99 years for the for the the supermassive black hole Then these recurrence times if we have reflecting boundary conditions to prevent the evaporation the classical recurrence time Because it's a huge exponential. It doesn't really matter whether I use plank units or a short-shield radius So here I could basically just give it in plank Well, whatever units you multiply this by the plank or by the short-shield radius That's going to make some tiny change in the very far number of digits of these far more digits than I've given So it's 10 to that or 10 to this for this So this second one is roughly a Google plex raised to the power of the fine structure constant if you want a mnemonic for it And the quantum recurrence time is get exponential of that So that's of order this for the solar mass one and and very this and so it's very nearly So we have a Google and then this is a Google plex so Google plex to the alpha power Factorial then take a factorial of that It's not quite as big as when my children were younger. They wanted to know big number. Well, I told them a Google plex They want something bigger. So the shortest two-word thing I can say is Google plex factorial So they kept going around saying they want a Google plex cactorial of cake or Google plex factorial I can't of course. I can't say it's a Google plex factorial I have to take a Google plex to the alpha power and then take a factorial to get this to get this number for forty billion mass black hole Okay, so to conclude Classically a vacuum spherical black hole is very simple described by the short shield metric when the cosmological constant zero or by the short shield Anti-decider metric that I gave back earlier when that when the cosmological constant is negative and there's analogous one When the cosmological constant is positive But a quantum black hole has a complicated behavior of many different time scales The quantum oscillation time that the light crossing time the scrambling time the largest ads period for micro Canonical ensemble that's that the so-called page time the Hawking decay time the classical recurrence time and the quantum recurrence time so For a supermassive black hole of 40 billion solar masses these times raised from less than 10 to 91 seconds to very roughly the factorial of 137th root of a Google plex, so Anyway, okay, so anyway, so black holes are extremely fascinating and when you go to the Quantum they become very complex and there's lots of mysteries about them We don't yet understand exactly. How does the information come out? Are there firewalls that if so when? But it's a very fascinating subject to to consider so thanks for your attention