A Paper on the Quantum-Mechanics of De-Sitter Space and the Aspects of Holography

# I. INTRODUCTION/CALCULATION The primary aspect of quantum mechanical de Sitter space that I am talking about there is a holographic theory of the de Sitter space. In original sense, some kind of boundary theory based on conventional quantum mechanics which itself makes no explicit reference of gravity but which encodes the bulk the rest of space within the boundary, in a mysterious way which is not fully understandable But the question we ask, why we work with de Sitter space? I think, de Sitter space is the big elephant in the room! It's a elephant, because it's big! But de Sitter space is not only big but symmetrically big! And also it's important! We may have lived very similar to the de Sitter space! sometimes, we don't see the elephant in the room; perhaps it's just too big. But also sometimes, just because of our fear we pretended the elephant is not there. I think it's a good reason. But the main reason at least from the point of view of peoples, who doing certain kind of physics, last 15 or 20 years, we call it holographic physics. So the main reason is that de Sitter space has no boundary. It has not any natural boundary to anchor the holographic degrees of freedom to the boundary. So that makes the de Sitter space little bit of complicated. That does not mean that no body does not thinking about de Sitter space! When we thinking all the things and read about de Sitter space, another elephant analogy come to our mind. "The Elephant And The Blind Man" -analogy! Five blind man come close to the elephant and touch its different body parts. Just try to guessing what actually it is! And they all guessing wrong. Someone touch it's tail and say it's a rope! Someone touch its pointy horns or teeth and think it's a sword! But de Sitter conjecture with me or with my physics friend, like that it's dS/cFT, or its dS/dS, it's a swampland, or its just dS/dS, or probably just a Matrix‑theory! See, I don't say any of this is just wrong! But I have the feeling that, if we want to say the big picture, we would use different words! So what I am going to present today: Some "admittedly fragmentary circumstantial evidence", that de Sitter space or some aspects of the de Sitter space is described by a kind of matrix-theory. The kind of matrix-theory I'm talking about according to my knowledge: "first proposed by a group of theorists"! They proposed some "fragmentary circumstantial evidence", and I will show some more of it in this paper. And then come up on some other aspects of the de Sitter space! It's began to the idea of "Static Patch"! So the whole frame work, that I am going to discussing is the "Static Patch Holography" or "SPH"! The Static patch of the de Sitter space is the portion we seen from an observer at the center of that Static patch. I would like to call the center of that static patch "the Pole", and this static patch come-up with pair's, at the opposite end of the Static patch called "Anti-Pole"! So I called to Static patches, and pick one which is sort of "gauges choice", and then there is a natural opposing static patch on the other side of de Sitter space which I named "Anti-Pole"! And if we take a mid slice at the de Sitter space, that what you see at the left side a penrose diagram of de Sitter space. But on the right hand side there is an embedding diagram, through at time $t = 0$ , and What you can see is a metric sphere! The pole at one end, the anti‑pole at other end. And the horizon right middle here (fig:a)! If you don't want to slice it at $t = 0$ ; but some later time or before time then you have naturally two horizons! One associated with left static patch, and one associated with right static patch! So I draw them slightly separated (fig: a)! So the basic hypothesis is that all the physics inside of the Static patch can be described by the "Holographic theory". Again the holographic theory, mean the quantum mechanics especially the different version of quantum mechanics, which does not contain gravity! Here is the metric of the de-sitter space: $$ \mathrm {ds^2 = -f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2 = - f (r) d t ^ {2} + (d r^{2}/f(r)) r ^ {2} d \Omega^ {2}} $$ $$ \mathrm {f (r) = (1 - r ^ {2} / R ^ {2})} $$ In this paper, I consider all the physics in 4d; not any higher number of dimension. r: the radial coordinate of a black hole! And R: is the radius of the de-sitter space. De-sitter space has thermodynamics! And it's natural the thermodynamics of the Static patch has a temperature T, which is equal to $1 / 2\pi \mathrm{R}$ . The bigger the space the colder it is! And it has some entropy which i called $\mathbf{S}_0$ : it's the starting basic entropy of the de-sitter space it-self! No perturbation in de-sitter space, just the static de-sitter space! And: $$ \mathrm {S} _ {0} = (\pi / \mathrm {G}) \mathrm {R} ^ {2} $$ R² is the area of the horizon. But here are some hidden assumptions! In order to define the entropy or the temperature we make a huge assumption! Entropy is the part of statistical mechanics, basically quantum statistics! And to defining it we assume to make some assumptions. First of all unitary "quantum mechanics"! There are another kind of entropy that we usually don't use in general! So unitary quantum mechanics, in our case is the static patch! Or equivalently says, "A Hilbert Space Of States". It's possible to count the number of states! There is a Hamiltonian and a notion of energy! And important to distinguishing whether we really have the de-sitter space or rather than some other object! If we do quantum mechanics and look at the space, may be it describe a black hole, may be it describe something else! And i think, the important thing is the Symmetry of the de-sitter space! But in this paper, i don't want to describe about the Symmetry of de-sitter space! But it's very important the hamiltonian and the other generators of symmetry are that they close in a algebra $\mathrm{O}(4,1)$ : which is the Symmetry of de-sitter space! Let's now ask: where the holographic degrees of freedom resides! In the context of AdS they reside at the boundary of AdS! I mean the asymptotic cold boundary of "Anti de-sitter space" (AdS)! But de-sitter space has no boundary! The Static patch does not have any boundary! you most likely know that "the boundary of the static patch is it's horizon"! In the Penrose diagram of de-sitter space you might think the degrees of freedom reside at the boundary of pole and anti-pole! But that's doesn't work very well! Because it has a connection with covariant entropy bound! Means, that the region around the pole, if i draw a surface around the pole here (very close to the pole), then it's area will be very small! So the number of degress of freedom is sufficient to describe what inside the pole! But not the rest of the Static patch! So if you consider the entire static patch (fig: b), then the degrees of freedom or dof reside at horizon or symmetric horizon of de-sitter space! Now let me come to another question very briefly! I want to presume that the holographic degrees of freedom are some Q-bits, or they might be Matrix degrees of freedom, but I'm going to assume they are more simpler than that! So what are the simple degrees of freedom? If the holographic degrees of freedom reside at the stretched horizon, they have nothing to do any boundary, they don't have to do directly degrees of freedom near the pole! They have to do degrees of freedom or dof near the horizon! So thats not like to be the quasi normal modes! I mean the quasi normal modes of oscillation of the horizon! And also not the operators, which create simple disturbances near the pole! In AdS things are far from the holographic dof where they located! Those thing's tends to be complex! But things that are near the dof tends to be simple! So I would expect the dof associated with throwing something in from the pole, or the center of de-sitter space are corresponds to complex operators, and the simple operators have more to do with the disturbances of the horizon itself (fig:c)! So now I'm gonna do in this paper: "The Evidence That The Dof Of Holographic Description For De-sitter Matrix Theory!" The dof (degrees of freedom) or Holographic dof of particular kind! Means a particular way to describing What's in space??? So now, what we gonna talk about the boltzmann-fluctuation! Boltzmann fluctuations in thermal equilibrium are large scale fluctuations, crazy things, have normal frequency fluctuations, for example all the gases in the room, suddenly accumulated in one corner! So they are very very improbable! Here is an example: suppose we consider a de-sitter space, there are some degrees of freedom near the horizon! But then a crazy fluctuation happen and all the degrees of freedom near the horizon move and replace near the pole and anti-pole (fig: d)! There are enough degrees of freedom, so that they can form a macroscopic object! That is extremely improbable in the thermal equilibrium of de-sitter space! Imagine our de-sitter space will have in thermal equilibrium long long in future! But then a fluctuation occurs and a macroscopic object form near the pole and anti-pole! This is highly improbable, but this is possible, as any thing can possible in our de-sitter approximation, in thermal equilibrium! So the only question is what is the probability? So, let's talk about such fluctuations in little bit technical! Firstly take the fluctuations, so that they can form macroscopic object such as black hole, near the pole! The "de Sitter Schwarzschild metric" looks same as de-sitter metric, except that for black hole, there is a extra-term! "De-sitter Schwarzschild Metric" is given by: $$ \mathrm {f (r) = 1 - (r ^ {2} / R ^ {2}) - (2 M G / r)} So, now if we do create a black hole near the pole, there will be two horizons! If we solve that equation: One is cosmic horizon, and the other horizon will be the horizon of black hole! From $\mathrm{f}(\mathrm{r}) = 0$ , we find that there are two positive solutions, and one negative which is unphysical! We come up on this thing later! So let's concentrate on small black holes (fig: e)! Small black holes are the black holes, which are much smaller than the de-sitter radius, or equivalently which entropy much smaller than the de-sitter entropy! Here is the penrose diagram of de-sitter Schwarzschild radius (fig: e)! There are two horizons; $(\mathbf{r}_{+})$ the larger horizon, that's the cosmic horizon! Behind the $\mathbf{r}_{+}$ the space is exponentially expand! The other horizon is the black hole horizon $(\mathbf{r}_{-})$ , which collapses to the singularity, and the space is periodic! So that you can identify the left dotted line with the right dotted line! It's actually two black hole; one is on pole side and another one is on Anti-pole side! And the spatial geometry at $t = 0$ , what it would look like??? Here is the (fig: f) global sphere! At the middle, here is the cosmic r+ horizon! But near the pole and Anti-pole there is a second horizon! The pole side horizon is same as the horizon at Anti-pole side! So, it's periodic in nature! So, that's the geometry of Schwarzschild de-sitter (fig: f)! Now i want to think the Schwarzschild de-sitter is just a result of a fluctuation which can happen primary de-sitter space! And what is the probability for??? The probability for it: $$ \begin{array}{l} \mathrm {P} = e^{-\Delta S} \\ \Delta \mathrm {S} = \mathrm {S} _ {0} - \mathrm {S} \\ \end{array} $$ $S_0$ : Is the starting entropy of de-sitter space at thermal equilibrium! S: The Entropy not only for one black hole, but the whole thing subject to the condition that the two black holes are present! On other words, the conditional entropy that the two black holes are present + everything else! And the difference of this two entropy is small if the black holes are small! But what ever the entropy difference is; the probability for formation of this kind of fluctuation is $e^{-\Delta S}$ So: $$ \Delta S = (\pi / G) \times \left[ R ^ {2} - \left(r ^ {2} _ {+} + r ^ {2} _ {-}\right) \right] $$ $\mathbf{r}^{2-}$ : That's the entropy associated with the cosmic horizon, and $\mathbf{r}^{2+}$ : is the entropy associated with the black hole! So the difference $\mathrm{S}_0 - \mathrm{S}$ is exactly the expression: $\Delta \mathrm{S}$ . If there is no black hole, then $\mathbf{r}_{+}$ is just capital "R"! And the entropy difference would be $\Delta S = 0$ ! And then the probability is just 1, but logically there would be no probability such as 1 or close to 1! Now, if you plug-in, what the value of $\mathbf{r}_{+}$ and $\mathbf{r}_{-}$ you get $\Delta S = 2\pi R M!$ You know it's not hard to do; and very straight forward! R: The size of de-sitter space, and M: is the mass of the black hole! So: $$ \Delta S = 2 \pi R M $2\pi \mathrm{RM}$ : This quantity here also happens to be boltzmann weight for the production of black hole! And $2\pi \mathrm{R} \sim 1 / \mathrm{T}$ (one over the temperature)! Or, $2\pi \mathrm{R} = \beta = \mathrm{T}^{-1}$ ; M is also the energy of the black hole; $2\pi \mathrm{R}$ : is the boltzmann weight and it goes to the exponent and gives you the probability for such a fluctuation to be happen! There is another form which i find very interesting! It has higher dimensional generalization! Big "R" is the square root of entropy in de-sitter space! Big "M" is proportional to the entropy of black hole! So if you put this together and go through the calculation you may find: $$\Delta S = \sqrt {S _ {0} \cdot s}$$ Keep this formula in mind, because there is a hint (or, a clue) about the structure of degrees of freedom! That clue first discussed in 2006, and later by L. Susskind (2011)! So the clue or the result of the clue inside the matrix-theory! I mean, the matrix structure for the degrees of freedom in de-sitter space (fig: g)! Let's start with matrix with big square arrays, $\mathbf{N} \times \mathbf{N}$ of them (fig: g)! Roughly speaking matrices show-up in the theory of "D‑brane"! You can think of this D‑brane; but I think you don't have to! The diagonal elements represent the position of D‑brane! But for the moment they are matrices, and I'm going to assume in thermal equilibrium; all of this matrix elements are the degrees of freedom! So there are $\mathbf{N}^2 =$ degrees of freedom! And in thermal equilibrium each of one carry some entropy! And the total entropy is proportional to the number of Degrees of freedom! Means, $\mathrm{S}_0 = \sigma N^{2}$ : represent entropy per little degree of freedom! If you ask me How you calculate? Actually i don't know, how to calculate! So just called it 0! Now how can we describe a system in matrix theory? I am coming-up on the original "BFSS"-matrix theory (fig: h)! But in this paper i don't describe the "BFSS"-matrix theory more detail, it's just an inspiration as you might think! How we would think about some system which divided into two subsystem! In this case, the two divided sub-system are the "Horizon degrees of freedom", and "The horizon that separated the two black holes"(see fig: h)! One who studied matrix theory give you a quick answer! One block of the matrix associated with one-degrees of freedom, and the other block associated with the other degrees of freedom! The off-diagonal elements are unexcited! Classically Say's, this off-diagonal matrix elements are set to be zero! The small subsystem in "BFSS"-matrix have $\mathfrak{m} \times \mathfrak{m}$ degrees of freedom, and the big subsystem in "BFSS"-matrix have $(\mathrm{N} - \mathrm{m})^2$ degrees of freedom! So: $\mathrm{S} = \sigma \cdot (\mathrm{N} - \mathrm{m})^{2}$ :Cosmic Horizon Entropy! S = \sigma m^{2} : Black Hole Horizon Entropy! After thermal fluctuations, the entropy of cosmic horizon decreases, that means there are few degrees of freedom in thermal equilibrium! So, the condition entropy $\mathrm{S} < \mathrm{S}_0!$ is the original entropy in de-sitter space! And the black hole entropy $\mathrm{s} = \sigma \cdot \mathrm{m}^2$ So, now what about $\Delta S???$ $$ \Delta S = S_0 - S - s $$ $$ \Delta \mathrm {S} = \sigma \cdot \mathrm {N} ^ {2} - \sigma \cdot (\mathrm {N} - \mathrm {m}) ^ {2} - \sigma \cdot \mathrm {m} ^ {2} $$ $$ \Delta \mathrm {S} = 2 \sigma (\mathrm {m} \cdot \mathrm {N}) $$ $\Delta S$ : is simply the number of constraints $\times \sigma$ ! If we looked at the "BFSS"-matrix structure; we may find out that (fig: h) there are two off-diagonal matrix structures, where off-diagonal elements are set to zero! So that the entropy difference by constraining the system in that way! $$\Delta S = 2\sqrt{S \times s}$$ The big "S" proportional to $\sim (\mathrm{N - m})^2$ or $(\mathrm{N - m})\sim \sqrt{\mathrm{S}})$ And small "s" is proportional to $\sim \mathrm{m}^2$ , or $\mathrm{m}\sim \sqrt{\mathrm{s}}!$ And the whole result is $2\sqrt{(\mathrm{S}\bullet\mathrm{s})}$ . $\Delta S = 2\sqrt{(S\bullet s)}$ is exactly the same formula that i derived-out starting with the de-sitter Schwarzschild black hole! And calculating the entropy difference! It's lil, bit of surprise that they two match together despite the factor of 2! The entropy is the product of two subsystem entropy is not a normal thing! It's not the thing that we see everyday! Because the general relativistic calculation of $\Delta S$ , and Matrix calculation of $\Delta S$ is just equal to each other! Let me give an example which does not give that type of formula! Here is the "Toy-Model" ( see fig: i)! (fig: i) You have a potential which has a "peak" at the center, a "up side down" harmonic oscillator, and all of the particles or most of the particles sit at the bottom of (minimum) the potential! And give the system a large entropy! In thermal equilibrium the particles typically sit at the bottom of the well! But there is a tiny probability that the particles or some particle sit top of the potential, due to thermal fluctuations! I won't go through the calculations, it's elementary just elementary statistical mechanics! $$ \Delta S = m \cdot \log\left(\frac{V}{v}\right) = 2 m \cdot \log\!\left(\frac{s}{S_0}\right) $$ \Delta \mathrm {S} \sim \mathrm {m} \cdot \log (\mathrm {m / N}) \Delta \mathrm {S} \sim \mathrm {s} \cdot \log (\mathrm {s} / \mathrm {S}) $$ So, the $\Delta S$ or differential entropy, proportional to the entropy of small subsystem (where few degrees are sit on) times logarithm of small subsystem entropy "s": $-$ : logarithm of large subsystem entropy "S"! And it does not like anything like the product: \Delta S = 2 \sigma N \cdot m \sim \sqrt {(s \cdot S)} $$ There are many of the mode you can construct and none of them looked like $\sim \sqrt{(\mathrm{s} \cdot \mathrm{S})} \rightarrow$ this! Except the matrix construction! So this type of formula $\Delta \mathrm{S} \sim \sqrt{(\mathrm{s} \cdot \mathrm{S})}$ intimately connected with the constraints $\Delta \mathrm{S}$ formula connected with little "m" and "(N-m) So, $\Delta \mathrm{S} \approx \sqrt{(\mathrm{s} \cdot \mathrm{S})}$ show's a very interesting correspondence between "Matrix-theory"; "Statistical Mechanics"; and "Schwarzschild de Sitter!! Now i come-up on a very difficult problem; not too much difficult! But giving up the assumptions, of small black holes, we study various kind of black holes of entire mass range; from smaller possible black holes, to large mass black holes in de Sitter space!. # II. FOR LARGE BLACK HOLES $$ (\mathrm {s} / \mathrm {S}) {\sim} 1 $$ Means that the black hole at the same order as cosmic horizon! So that's why, $s = S$ , equal to each other! $$ \mathrm {f (r) = 1 - (r ^ {2} / R ^ {2}) - (2 M G / r)} $$ $$ \mathrm {f(r) = r - (r ^ {3} / R ^ {2}) - 2 M G = 0} $$ Suppose I want to find out where the horizon is? What is the value of r, R???? Then i have to set $\mathrm{f(r) = 0}$ , but that's nasty things nasty because there is a $(-2\mathrm{MG / r})$ in $\mathrm{f(r)}$ expression! $$ \operatorname {f} (\mathrm {r}) = 0 $$ Now, you have a polynomial cubic equation! This cubic equation has three roots! The cubic that i drawn in deep-black (see fig: j)! So (according to fig: j) one root is -ve, and it has no physical meaning! But other two roots $(\mathbf{r} - )$ and $(\mathbf{r}_{+})$ are positive. The same $(\mathbf{r} - )$ and $(\mathbf{r}_{+})$ that i talk about, before! And right in the middle, there is a radius: $\mathbf{r}_{+}$ is "Nariai"! If MG is very large enough, then there is just one root at the point, means the two roots $(\mathbf{r} - )$ and $(\mathrm{r}_{+})$ collapse into one root, and that can happen at the point $\mathbf{r}_{\mathrm{n}}:$ "Nariai" black hole (see fig: j)! Instead of working with the mass of black hole, or the Schwarzschild radius of black hole (as the independent parameter here), We work with a coordinate called X. $$ \mathrm {X} = \left[ (\mathrm {r} _ {-}) - (\mathrm {r} _ {+}) \right] / \mathrm {R} $$ X: is the difference between $(\mathbf{r}_{-})$ and $(\mathbf{r}_{+})$ horizon, here $[(r_{-}) - (r_{+})]$ that entity is normalized by the size or the radius of de-sitter space "R"! And X varies between $-1$ to $+1$ . $$ - 1 < X < 1 $$ When $\mathrm{X} = -1$ ; then $(\mathrm{r}_{+}) = \mathrm{R}$ (No black hole formation)! But for $\mathrm{X} = +1$ , when you interchange $(\mathrm{r}_{+})$ and $(\mathrm{r}_{-})$ ; $(\mathrm{r}_{-})$ is always smaller than $(\mathrm{r}_{+})$ ! But I'm going to extend and continue the definition! So, if you interchange them $(\mathrm{r}_{+}) > (\mathrm{r}_{-})!$ If you study this analytically as a function of $\mathrm{X}(\mathrm{r}_{+};\mathrm{r}_{-})$ , then $[(r_{+}) - (r_{-})]$ going to $-1$ to $+1$ . So what's going on in $+1???$ For simplification you can imagine the cosmic horizon has shrunken down to be very small and the black hole horizon has grown to be very large! So, they interchange! The black hole horizon become cosmic horizon, and the cosmic horizon become the black hole horizon! At least the system has that kind of Symmetry! So what you may find that $\Delta S$ : The entropy difference is given by the surprisingly simple formula: $$ \Delta S = \Delta S _ {n} \cdot [ 1 - X ^ {2} ] $$ N: stands for "Nariai" if $\mathrm{X} = 0$ ; then $$ \Delta \mathrm {S} = \Delta \mathrm {S} _ {\mathrm {n}} $$ And if you calculate the entropy difference at N, Before turning into the matrix theory, lets imagine a transition! The transition starts at $X = -1$ . That's mean the Tiny-Tiny black hole in the big de-sitter space! Now let's imagine the Tiny-Tiny black hole starts absorbing energy from the cosmic horizon! Now, you might be say that violate the 2nd law of Thermodynamics! you get $\Delta S = (1 / 3)\bullet S_{\mathrm{n}}$ ! Here $(1 / 3)$ just a numerical constant! $S_0$ is the entropy of de-sitter space at thermal equilibrium! That's calculation is very easy to do, as there is a lot of symmetry! But if you go far from the Symmetry point the answer becomes very hard to defining! But it's not hard at the point N. If $\mathrm{X} = 0$ , then according to the formula black hole shrinks into zero size, and hence there is no $\Delta S$ ! Just: Our horizon is much colder than the black hole! So it violate the 2nd law of thermodynamics! But the 2nd law of thermodynamics not a law! It's just a statistical tendency! It's simply improbable for energy to flow from the cosmic horizon to black hole horizon! So, don't worry about! I will show you in a moment!!! But let's take that there is a boltzmann fluctuation, so energy can flow from cosmic horizon to black hole! And in that case the black hole begin to grow! Eventually the black hole horizon and the cosmic horizon in same size at the point "N"! But if it keeps going, then cosmic horizon shrunk back and become Black Hole Horizon! Or maybe no black hole at all. So, Now we find Black hole horizon become the cosmic horizon! I will call it "Inside-Out" transition (See the fig: l)*. So for an observer (see fig: m)* first the tiny black hole absorb energy from de-sitter space and grow! At some point or "N" point the both cosmic and black hole horizon at same size! And then the black hole horizon grow and become the cosmic horizon and the cosmic horizon turn into a tiny-black hole horizon hole horizon! And the probability for this to happen: $$ \begin{array}{l} P_{\text{prob}} = e^{-\Delta S} = e^{-\left(S_{0}/3\right)} \\ \end{array} $$ For that case: $\Delta S = \Delta S_{\mathrm{n}}$ $S_0$ for big de-sitter space is a large number, so $e^{-(S_0/3)}$ is a tiny-tiny number! That is a "Boltzmann-Fluctuation"! But it still an interesting Probability for the quantum mechanics of de Sitter space! Now the quest is: "Can we construct $\Delta S = \Delta S_{\mathrm{n}} \cdot [1 - X^{2}]$ for a holographic model??? The answer is yes, we can! Again it's the good-old "Matrix-Theory" (see the fig: n)*! Dividing the system into two subsystems, means two blocks! And off-diagonal boxes are just as constraints! Consider the case of "Nariai" black hole! The "Nariai" black hole has some kind of symmetry between two blocks! It has many degrees of freedom in Big-block as in the small block! I mean two equal blocks (see fig: o)*! That is the matrix analogous for "Nariai" black hole! $$ \Delta S = \sigma m \cdot (N - m) $$ In gravity theory S: Schwarzschild radius! $$ \begin{array}{l} \mathrm {S} \sim \mathrm {r} ^ {2} \\ \text {And} (\mathrm {r} _ {-}) = \sqrt {\mathrm {S} _ {-}}; (\mathrm {r} _ {+}) = \sqrt {\mathrm {S} _ {+}} \\ \mathrm {So: X = [ (r _ {-}) - (r _ {+}) ] / R} \\ = [ (\sqrt {S _ {-}}) - (\sqrt {S _ {+}}) ] / \sqrt {S _ {0}} \\ = [ \mathrm {m} - (\mathrm {N} - \mathrm {m}) ] / \mathrm {N} \\ = (2 \mathrm {m} / \mathrm {N}) - 1 \\ \end{array} $$ So, we have now two quantity! One is X, another is $\Delta S$ . By squaring X And rearranging things little bit: $$ \Delta S = \Delta S _ {n} \cdot [ 1 - X ^ {2} ] $$ $$ \Delta \mathrm {S} _ {\mathrm {n}} \sim (\mathrm {S} _ {0} / 2) $$ Now, let's talk about the dynamics of the constraints! You already know that, only "Nariai" black hole $(\mathrm{m - N}) = \mathrm{N / 2}$ and $\mathrm{m} = \mathrm{N} / 2$ But in general according to the "BFSS"-matrix theory there are different orders of degrees of freedom! And In BFSS-there is 11-Bosonic Matrices. But now assume there is a finite number of degrees of freedom! But the all of them represented in block diagonal form! Now it would take some interesting dynamics to assure that if one matrix takes the block-diagonal form, all of the others also wind up [for energetic reason] in block diagonal form! I will give you an example a model, in which you can see how it can come about just by virtue of one matrix taking a block diagonal form. And also all the other takes the block diagonal form! So, that depends on the dynamics! That's way I'm going to introduce one more matrix degree of freedom! I denoted it as "R" (The script_R, see the fig: p)*! This matrix "R" as same in "BFSS"-matrix theory is intended to represent the coordinate distance from the horizon! In a sense that it's diagonal elements roughly speaking, it's eigen values: represent the distances (if you like, you can think in terms of D0-branes) from the center of some set of degrees of freedom or "D0-brane degrees of freedom". I'm going to write down a formula for Lagrangian $$ \mathrm {L} = \sum \mathrm {Tr} (\mathrm {c} ^ {2} \mathrm {R} ^ {4} \mathrm {A} ^ {2} - [ \Re , \mathrm {A} ] [ \mathrm {A}, \Re ]) + \dots \dots \dots . $$ Lagrangian is the function of some degrees of freedom, which i called A. The summation Here, over the finite number of different matrices (11-Bosonic Matrices and some number of fermionic matrices)! Here Tr represent the ordinary $\mathrm{N}\times \mathrm{N}$ trace! c: is just a number. R: is the size or radius of de-sitter space! And $[\Re ,\mathrm{A}][\mathrm{A},\Re ]$ : that represent the commutator of $\Re$ with all the degrees of freedom $\mathrm{A^2}$ . I'm not going to discuss why it is in that way! I just represent it as a Model for how this kind of division into blocks can be dynamically explain (see the fig: q)\*! The small block represent the degree of freedoms near the node! And the big block represent the degree of freedoms near at the Horizon (*Cosmic or Black hole, whatever. See the fig: r)! If you take this form of $\Re$ and commuted with A's, then take its trace; you will find that the Lagrangian: $$ \mathrm {L} = \mathrm {c} ^ {2} \mathrm {R} ^ {4} \mathrm {a} _ {\mathrm {i j}} \mathrm {a} _ {\mathrm {j i}} - \mathrm {R} ^ {2} \mathrm {a} _ {\mathrm {i j}} \mathrm {a} _ {\mathrm {j i}} $$ $a_{ij}a_{ji}$ are the quadratic terms in time derivative, and $a_{ij}a_{ji}$ are quadratic terms in a's. # III. CONCLUSION So, what you get by this result that I finally derived? You get a system of harmonic oscillator. If you work out the frequency of the harmonic oscillator you may get: $$ \omega = 1 / \mathrm{cR} $$ $$ \mathrm{T} = 1 / 2 \pi \mathrm{R} $$ Suppose the frequency of the oscillator is very large, $\omega >> T$ , in other words the energy scale of the oscillator is very large! Larger than anything in this problem! Then you can expect this oscillations will in order to keep the energy low, you have to freeze the oscillators into the ground state! That's the kind of constraints that we can imagine earlier (The off-diagonal elements are set to be zero)! The other scale of this problem is the temperature! If we want to know the degrees of freedom (which frozen), then what or which Temperature they frozen-out??? That's the temperature of remaining degrees of freedom! Which roughly $\sim 1 / 2\pi R$ . If $\omega >> T$ , the $a_{ij}$ are frozen into ground state and carry no entropy! If $\omega \sim T$ the $a_{ij}$ carry entropy: $\sigma -\sigma^{\prime}$ Means the off-diagonal elements $\mathbf{a}_{ij}$ 's are not completely frozen! And instead of carrying zero entropy they will carry a entropy $\sigma - \sigma' = \Delta \sigma$ . $\sigma$ : Entropy of unconstrained degrees of freedom. $\Delta \sigma$ : the entropy of "off-diagonal degrees of freedom". So in Case of Small Black Hole: $$ \Delta S = 2 \left(\sigma^ {\prime} / \sigma\right) \times \sqrt {(s \cdot S)} \approx \sqrt {(s \cdot S)} $$ $$ (\sigma^ {\prime} / \sigma) _ {\text {small}} \approx 1 / 2 $$ For Large Black Hole $$ \Delta S = (1 / 2) \cdot (\sigma^ {\prime} / \sigma) S _ {0} (1 - X ^ {2}) $$ $$ \approx (1 / 3) \times S _ {0} \cdot (1 - X ^ {2}) $$ $$ (\sigma^ {\prime} / \sigma) _ {-} \text {big} \approx 2 / 3 $$  Fig:(a)   HOLOGRAPHIC DOF AT STRETCHED HORIZON  STRETCHED HORIZON (Fig:b)   Not Simple. Fig. (c)  Fig. (d) SMALL BLACK HOLES   Fig. (f) $$ DOF = \mathrm{Finite} $$ MATRICES! $$ A = \left[ \begin{array}{l l l l} a_{11} & a_{12} & a_{13} & \dots \\ a_{21} & \cdot & \cdot & \cdot \\ \vdots & \cdot & \cdot & \cdot \\ \vdots & \cdot & \cdot & \cdot \\ \vdots & \cdot & \cdot & \cdot \\ \vdots & \cdot & \cdot & \cdot \\ a_{n1} & & \cdot & \cdot \end{array} \right]; S _ {0} = \sqrt {N ^ {2}} $$ $$ fig: (g) $$          DYNAMICS OF MATRIX THEORY A_{t}\cos \tau = R But at the point. $\left( 0\right) : r = 0$ $$ \mathbf {R} = \left[ \begin{array}{l l l l l} r_{11} & r_{12} & r_{13} & \dots & r_{1n} \\ r_{21} & \bullet & & & \\ \bullet & & \bullet & & \\ \bullet & & \bullet & & \\ \bullet & & & \bullet & \\ r_{n1} & & & & \end{array} \right] $$ $$ \mathrm{fig}: (P) $$ $$ R = $$ $$ \begin{array}{r l} & {\left[ \begin{array}{l l l l l} R & 0 & 0 & 0 & 0 \\ 0 & R & 0 & 0 & 0 \\ 0 & 0 & R & 0 & 0 \\ 0 & 0 & 0 & R & 0 \\ 0 & 0 & 0 & 0 \end{array} \right],} \\ & {\left[ \begin{array}{l l l l l} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array} \right]} \end{array} $$ $$ \mathrm{fig}: (g) $$ 

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