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. > 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. 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. > > 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. > 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. 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. Bruce Jason > > In terms of the cosmological problem, the initial state has a particular >> total mass, and that does not increase with the expansion of the universe. >> Consequently, the maximum possible entropy does not increase either. The >> point of the Past Hypothesis is that the initial state of this mass was of >> low entropy since the gravitational degrees of freedom were not saturated >> (it did not form a black hole), so there is a large amount of room >> available for the entropy to increase. >> >> 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/CAFxXSLTohsYxw1TObr0MurptwzKcp%3D9pG3j_2oWOQC2SsZ8eRg%40mail.gmail.com.
