# Information, Entropy and Holographic Screen

### Problem 1:

Show that information density on holographic screen is limited by the value $\sim 10^{69} \: \mbox{bit/m}^2.$

### Problem 2:

Choosing the Hubble sphere as a holographic screen, find its area in the de Sitter model (recall that in this model the Universe dynamics is determined by the cosmological constant $\Lambda >0$).

### Problem 3:

Find the entropy of a quantum system composed of $N$ spin-$1/2$ particles.

### Problem 4:

Consider a 3D lattice of spin-$1/2$ particles and prove that entropy of such a system defined in a standard way is proportional to its volume.

### Problem 5:

Find the change of the holographic screen area when it is crossed by a spin-$1/2$ particle.

### Problem 6:

What entropy change corresponds to the process described in the previous problem?

### Problem 7:

According to Hawking, black holes emit photons (the black hole evaporation) with thermal spectrum and effective temperature $$T_{BH}=\frac{\hbar c}{4\pi k_{_B}}r_{g},$$ where $r_{g}={2GM}/{c^2}$ is the black hole Schwarzschild radius. Neglecting accretion and CMB absorption, determine the life time of a black hole with initial mass $M_{0}.$ Calculate the lifetime for black holes with the mass equal to Planck mass, mass of the Earth and Solar mass.

### Problem 8:

Show that black holes have negative thermal capacity.

### Problem 9:

Show that the black hole evaporation process is accompanied by increase of its temperature. Find the relative variation of the black hole temperature if its mass gets twice smaller due to evaporation.

### Problem 10:

Show that in order to produce an entropy force when a particle approaches the holographic screen, its temperature must be finite.

### Problem 11:

Determine the entropic acceleration of a particle crossing the holographic screen with temperature ${{T}_{b}}$.

### Problem 12:

Show that the Unruh temperature is the Hawking radiation temperature $$T_{BH}=\frac{\hbar c^3}{8\pi k_{_B}GM}$$ with substitution $a\to g,$ where $g$ is the surface gravity of the black hole.

### Problem 13:

In order to verify experimentally the Unruh effect, it is planned to accelerate particles with acceleration of the order of $10^{26} m/sec^2$. What vacuum temperature does this acceleration correspond to?

### Problem 14:

Derive the second Newton's law from the holographic principle.

### Problem 15:

Show that the inertia law can be obtained from the holographic principle.

### Problem 16:

Obtain the Newton's law of universal gravitation using the holographic principle.

### Problem 17:

Compare how close to a black hole are Earth, Sun and observable Universe.

### Problem 18:

Estimate the temperature of the Hubble sphere, considering it as a holographic screen.

### Problem 19:

Show that equilibrium between the relic radiation and holographic screen is possible only at Planck temperature.

### Problem 20:

According to the holographic ideology, all physical phenomena can be described by the boundary layer theory. Therefore a conclusion comes that one should account for contribution of such surface terms while deriving the equations of General Relativity. Show that consideration of the boundary terms in the Einstein-Hilbert action is equivalent to introduction of non-zero energy-momentum tensor into the standard Einstein equations.

### Problem 21:

Derive the Friedman equations from the holographic principle.

### Problem 22:

Derive the Friedman equations from the holographic principle for a non-flat Universe.

### Problem 23:

Consider an effective field theory with the ultraviolet cutoff parameter equal to $\Lambda$ and entropy satisfying the inequality $S\equiv L^3\Lambda^3\le S_{BH}\simeq L^2 M_{Pl}^2,$ where $S_{BH}$ is the Beckenstein-Hawking entropy, and show that such a theory necessarily contains the states with Schwarzschild radius $R_S$ much greater than the linear dimensions $L$ of the system. (see Zimdahl, Pavon, 0606555)

### Problem 24:

Estimate the dark energy density assuming that the total energy in a region with linear size $L$ cannot exceed the black hole mass of the same size.

### Problem 25:

Find the correspondence between the ultraviolet and infrared cutoff scales.

### Problem 26:

Show that the entropy force produces negative pressure.

### Problem 27:

Taking the Hubble sphere for the holographic screen, and using the SCM parameters, find the entropy, the force acting on the screen and the corresponding pressure.

### Problem 28:

The most popular approach to explain the observed accelerated expansion of the Universe assumes introduction of dark energy in the form of cosmological constant into the Friedman equations. As we have seen in the corresponding Chapter, this approach is successfully realized in SCM. Unfortunately, it leaves aside the question of the nature of the dark energy. An alternative approach can be developed in the frame of holographic dynamics. In this case it is possible to explain the observations without the dark energy. It is replaced by the entropy force, acting on the cosmological horizon (Hubble sphere) and leading to the accelerated Universe's expansion.
Show that the Hubble sphere acceleration obtained this way agrees with the result obtained in SCM.

### Problem 29:

Plot the dependence of the deceleration parameter on the red shift in the Universe composed of non-relativistic matter. Take into account the negative pressure generated by the entropy force (see Problem \ref{HU26}). Compare the result with the SCM.

### Problem 30:

Show that the coincidence problem does not arise in the models with the holographic dark energy.

### Problem 31:

Show that the holographic screen in the form of Hubble sphere cannot explain the accelerated expansion of Universe.

### Problem 32:

Find the dependence of holographic dark energy density on the scale factor taking the cosmological particle horizon $R_p$ as the holographic screen. Show that such choice cannot explain the accelerated expansion of the Universe.

### Problem 33:

Obtain the equation of motion for the relative density of the holographic dark energy in the case when the particle horizon serves as the holographic screen.

### Problem 34:

Solve the equation of motion obtained in the previous problem.

### Problem 35:

Find dependence the of holographic dark energy density on the scale factor taking the cosmological event horizon $R_h$ as the holographic screen. Find the equation of state parameter for such a dark energy.

### Problem 36:

Obtain the equation of motion for the relative density of the holographic dark energy in the case when the event horizon serves as a holographic screen.

### Problem 37:

Solve the equation of motion obtained in the previous problem.

### Problem 38:

Find the dependence of cosmological parameters (Hubble parameter $H$, state equation parameter $w$ and deceleration parameter $q$) on red shift in the model of the Universe composed of radiation, non-relativistic matter and holographic dark energy, taking as the holographic screen the Ricci scalar's $R$ characteristic length $$\rho_{_{RDE}}=-\frac{\alpha}{16\pi} R =\frac{3\alpha}{8\pi}\left(\dot{H}+2H^2+\frac{ k}{a^2}\right),$$ where $\alpha$ is a positive constant, $k$ is the sign of space curvature.

### Problem 39:

Find the dependence of holographic dark energy density on the scale factor for the model of the previous problem.

### Problem 40:

Find the pressure of holographic dark energy for the model of Problem.

### Problem 41:

In the frame of the holographic dark energy model considered in Problem , find the dependence $w(z)$ using the SCM parameters: $\Omega_{k0}=0$, $\Omega_{m0}=0.27$, $\Omega_{r0}=8.1 \times 10^{-5}$,$\Omega_{X0}=0.73$, $w_0=-1$.

### Problem 42:

Find the state equation parameter for the holographic dark energy in the agegraphic dark energy model, in which the holographic screen is taken in the form of the surface situated at the distance traversed by light during the Universe's lifetime $T$.

### Problem 43:

Find the equation of motion for the relative density of agegraphic dark energy in the model considered in the previous problem.

### Problem 44:

Using the equation of motion for the relative density of agegraphic dark energy, obtained in the previous problem, find its dependence on scale factor in the Universe dominated by non-relativistic matter.

### Problem 45:

Using the equation of motion for the relative density of agegraphic dark energy, obtained in Problem, find its dependence on scale factor in the case of the dark energy domination.

### Problem 46:

Derive the exact solution of the equation of motion obtained in Problem.

### Problem 47:

Find the state equation parameter for the agegraphic dark energy in the case when this form of dark energy interacts with dark matter: $\dot{\rho}_m+3H\rho_m=Q,$ $\dot{\rho}_q+3H(1+w_q)\rho_q=-Q.$

### Problem 48:

In the model of interacting agegraphic dark energy and dark matter find the equation of motion for the relative density of agegraphic dark energy.

Isolated systems are known to evolve towards the equilibrium state in such a way that entropy $S(x)$ never decreases $\frac{dS(x)}{dx}\ge0,$ while $\frac{d^2S(x)}{dx^2}<0.$ In the context of eternally expanding FRW-cosmology the above mentioned conditions imply that the entropy available to comoving observer plus the entropy on the cosmological horizon satisfy the conditions (generalized second law of thermodynamics) $S'(a)\ge0,\ S''(a)\le0$ where the prime stands for differentiation with respect to scale factor $d/da$ and $a\to\infty.$ The latter condition means that we deal below with late stages of the Universe evolution.

### Problem 49:

Show that in the Friedmannian Universe with energy density $\rho$ the cosmological horizon area equals $A=\frac{3}{2G}\frac 1 \rho.$

### Problem 50:

Show that in the Friedmannian Universe filled with a substance with state equation $p=w\rho$ the cosmological horizon area grows with the Universe's expansion under the condition $1+w>0$.

### Problem 51:

Using the results of Problem \ref{HU49}, find the entropy of the Universe.

### Problem 52:

Find the state equation for the substance, filling the Universe with $A''\le0$.

### Problem 53:

Obtain the following expression for $A''$ in the case $w\ne\,const$: $A''=\frac{9}{2Ga\rho}\left[w'+\frac{(1+w)(2+3w)}{a}\right].$