On Monday, October 19, 2020 at 7:06:54 PM UTC-5 Bruce wrote:
> On Mon, Oct 19, 2020 at 11:08 PM Jason Resch <[email protected]> wrote: > >> On Sun, Oct 18, 2020, 11:55 PM Bruce Kellett <[email protected]> wrote: >> >>> On Mon, Oct 19, 2020 at 3:31 PM Jason Resch <[email protected]> wrote: >>> >>>> On Sun, Oct 18, 2020, 10:33 PM Bruce Kellett <[email protected]> >>>> wrote: >>>> >>>>> On Mon, Oct 19, 2020 at 1:09 PM Jason Resch <[email protected]> >>>>> wrote: >>>>> >>>>>> On Sun, Oct 18, 2020, 8:47 PM Bruce Kellett <[email protected]> >>>>>> wrote: >>>>>> >>>>>>> >>>>>>> Remember that entropy is basically related to the volume of phase >>>>>>> space, not of ordinary space. And phase space relates to the number of >>>>>>> particles (hence mass-energy). Spatial volume is essentially irrelevant >>>>>>> for >>>>>>> volumes greater than that of the corresponding black hole. >>>>>>> >>>>>> >>>>>> No. Consider an infinite length. With a single atom you can encode >>>>>> infinite information through placement of the atom along that length. >>>>>> This >>>>>> is with finite mass energy, but unrestricted spatial volume. >>>>>> >>>>> >>>>> That does not encode infinite information. There is, after all, only >>>>> one particle, and it can have only one position. If you want to encode >>>>> more >>>>> information, you need more particles. You might need an infinite number >>>>> of >>>>> bits to encode the position of one particle as a real number, but the >>>>> single particle cannot encode this. >>>>> >>>> >>>> This is plainly false. Every 1 mile distance that particle is placed >>>> along the line encodes a unique number. Travel up to 2^N miles and you can >>>> encode N bits. With infinite range there's no upper bound. >>>> >>> >>> A single particle can be in only one place, and encode on ly one bit. >>> >> >> If I were building a hard drive and could only write a fixed number of >> crosses, let's say 5 crossed, on a LxL grid, the number of possible >> combinations would be (L^2 choose 5). >> >> So for L = 3 that is (9 choose 5) = 126. >> Which means I can encode Log2(126) = 6.97 bits >> >> For L = 4 that is (16 choose 5) = 4368. >> Which means I can encode Log2(4368) = 12.09 bits >> >> My total number of crosses (let's say I use a single atom to represent >> each, is 5 in both cases). It is constant. But if I have more space, I have >> more ways of arranging them, and can use them to encode more bits, or >> conversely it takes more information to describe the system. >> >> Why does this analogy not extend to a quantum system of particles in >> larger or smaller regions of space? >> > > > I think you are forgetting the physical nature of your atoms and your > grid. Because information is physical, it requires mass-energy to encode. > Look again at the Bekenstein bound you have used: > > S <= 2pi RE > > That does imply that if you increase the volume, you can fit in a greater > entropy. But it does not mean that increasing the volume for fixed > mass-energy automatically increases the entropy. In order to increase the > entropy to that allowed in the larger volume, you have to also increase the > mass-energy. > > Or think of a grid of naughts and crosses, with a larger grid but fixed >>>> number of crosses, the number of possible combinations for drawing a fixed >>>> number of crosses still increases with more spaces to place them. >>>> >>> >>> Each combination encodes only one combination. >>> >> >> >> More unique combinations mean more states a system can possibly be in, >> meaning it takes more information to uniquely define the state the system >> can be in, or alternatively the more information the system may encode. >> >> >>> An arbitrary volume can only hold a limited amount of energy, or >>>>> entropy, as given by the Bekenstein bound. >>>>> >>>> >>>> Energy isn't the same thing as entropy. >>>> >>> >>> Bekenstein relates them. >>> >>>> But the maximum entropy for a particular mass is given when that mass >>>>> forms a black hole -- which saturates the Bekenstein bound. >>>>> >>>> >>>> The bound is always satisfied. Black holes just reach the maximum of >>>> the bound at a given VOLUME. >>>> >>> >>> I said saturated, not 'satisfied'. The bound gives the maximum possible >>> enclosed mass for a given volume, or the volume is that for which entropy >>> is maximum for a given mass which saturates the bound. >>> >> >> I think you're thinking of the black hole entropy equation, which is >> related to, but distinct from, the Bekenstein bound. >> > > > The Bekenstein bound as you have used it merely means that the amount of > entropy in a given volume is limited. Increasing the volume will allow for > greater entropy, but the entropy at the bound increases only if the mass is > also increased. Entropy (information) is a physical thing, and coding or > storing information requires energy. > > I realize that it is difficult to say this clearly and precisely, because > in general statistical physics, the entropy is so far below the maximum > possible in the considered volume, that the Bekenstein bound is largely > irrelevant. It becomes an issue only if you look at situations, such as > black holes, where the bound is in fact saturated, and you consider > increasing either the mass or the volume. It is then that the fact that > the bound depends on their interdependence becomes important. > This thread is getting a bit labyrinthine, so I will comment just here. The event horizon represents the smallest area that can bound a given amount of entropy. This entropy is just a measure of information that is not accessible to observers "at infinity." The interior of an ideal Schwarzschild black hole [image: penrose diagram for Schwarzschild BH.jpg] marked by region III is increasing in volume. The region IV is the dual while hole, and is analogous to a^† while the black hole is a. The white hole generates quantum particles while the black hole absorbs them. The more physical truncated diagram [image: Penrose diagram for Schwarzschild truncated.jpg] has the black hole horizon at r = 2m and the interior has a nonstationary volume that increases. The first diagram describes two entangled black holes, where the quantum states associated with the event horizon are entangled with each other. The interior region III is then the Einstein-Rosen bridge connecting these two. Two observers entering either of these black holes may meet in the middle, but can never return or escape to regions I or II. Unless the two observers are entangled their presence has broken the exact entanglement of these black holes. The entanglement of black holes is a sort of idealization representing how the entanglement of states entering or exiting (by Hawking radiation) can have complicated entanglements. This interior region is not constant or stationary and the dynamical evolution of this volume is dV/dt = Aℓ_ads T, for A = 4πr^2 (r = 2m) the horizon area, ℓ_ads the AdS scale and T = 1/8πm temperature. Now I have done a sleight of hand here by putting this in AdS spacetime, which is because this business becomes easier to work there. The physics is not substantially different. This interior volume increases linearly with time, here time is Rindler time. This is analogous to the growth in complexity of a quantum computing circuit, and this suggests that the growth in this area is a measure of the amount of Hilbert space the interior region evolves through. The relationship between the volume and this area is of particular interest when the area does change. This happens with the absorption or emission of a quantum of radiation or particle. One way to see this is that a quantum state may be very near the horizon and an observer near I^∞ will not be able to observe then or any quantum emitted by this state, say a photon emitted by an atom, for it will be arbitrarily red shifted. However, if this observer elects to either put themselves on an accelerated frame close to the horizon or to fall into the black hole this state becomes observable very close to the horizon. This is though a sort of Heisenberg microscope issue, where the energy this observer imposes on the black hole results in an uncertainty in localizing the state not only on the horizon, but as Hawking radiation removed from the horizon. This entanglement of two black holes is a form of entanglement of states of the black hole with interior states and Hawking radiation. At this juncture in physics there are subtle matters of how this changes the quantum complexity of the interior and the horizon entropy or equivalently the Bekenstein bound. Remember, the Bekenstein bound is a classical approximation and in fact this should be S = A/4ℓ_p^2 + quantum corrections where those quantum corrections are involved with adjusting the interior volume or the amount of quantum complexity therein. LC > > >>>> Increasing the volume does not increase the actual entropy unless you >>>>> simultaneously increase the mass. >>>>> >>>> >>>> You keep saying this but don't provide any justification or sources. I >>>> implore you to read the wikipedia article and if it is wrong, please point >>>> me to a source with the right/corrected equation. >>>> >>> >>> The justification is that it is impossible to increase the mass of a >>> black hole without at the same time increasing its radius (volume). For a >>> black hole, the radius is 2M, in natural units. So the mass and radius are >>> directly related. Any greater volume for the same mass does not saturate >>> the bound. >>> >> >> Forget about saturating the bound, that's not the point. Saturating the >> bound requires maximizing entropy for a given volume. On that we agree. >> >> My point is that the bound implies that a larger amount of volume, for >> fixed energy, allows for higher entropy. >> > > That is correct, provided you realize that increasing the entropy beyond a > saturated bound requires the input of more mass-energy. > > Put your black hole in a larger volume and now the black hole has a very >> well defined position in that volume, which is more information than you >> had before. >> >> It's a generally accepted in computer science that a turing machine >> allowed to use infinite space could store infinite information, even with >> fixed total mass/energy. >> > > > You do not have massless tapes on which to store your infinite > information. So this would appear to be nonsensical. A Turing machine in a > physical object, and it is subject to the laws of physics. > > As explained on that page, the bound is not limited to black holes, it >>>> says something more general which relates entropy bounds to the product of >>>> spherical radius and mass. >>>> >>> >>> The entropy bound you are talking about is >>> >>> S <= 2pi RE. >>> >>> This is saturated when the radius and energy are related as for a black >>> hole: >>> >>> R = 2M, for which S = 4pi M^2. >>> >>> Nothing mysterious here. I was talking about maximum possible entropy, >>> which occurs when the bound is saturated, as for a black hole. >>> >>> That is really all that the Bekenstein bound says. It is a bound, after >>> all, and has information about the entropy only when that bound is >>> saturated. >>> >> >> If the bound strictly depends on energy, why is R included in the >> formulation? >> > > > That specifies the volume within which the energy is enclosed. But > increasing the volume does not, of itself, increase the entropy. The > maximum entropy for a fixed mass-energy is fixed by the surface area of a > black hole of radius R = 2M. > > > > >> So for a fixed amount of mass, the entropy is maximized when that mass is >>> in the form of a black hole. Increasing the volume surrounding the BH makes >>> no difference to the entropy maximum for that mass. >>> >> >> For the system as a whole it does. Now the black hole has coordinates in >> a larger volume which did not exist before, and must be included in any >> description of that system. >> >> A non-collapsed relativistic gas sits right on the edge of becoming a >> black hole and satisfying the bound. Consider such a relativistic gas >> confined to a 1 meter volume. Now considering that gas is given more space >> to occupy, it is placed in a sphere of 1 light-year. >> >> Are there not now many more degrees of freedom possible for that same >> mass energy in a 1 light-year space than when it was confined to 1 meter? >> Are not more bits and precision required to specify the coordinates of each >> particle? >> > > > Putting a black hole in a bigger volume does not increase the entropy of > that black hole. Specifying coordinates for the constituents of the BH is > either irrelevant, or requires additional mass. > > The upshot of all of this is that the expansion of space in a > cosmology does not increase the maximum possible entropy. The maximum > entropy is set by the amount of mass-energy in the cosmology, and that does > not increase with the expansion. > > Bruce > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/e183766c-26a0-42e3-b99e-5932295d127bn%40googlegroups.com.
