First law of thermodynamics for dynamical apparent horizons and the entropy of Friedmann universes

Abstract

Recently, we have generalized the Bekenstein–Hawking entropy formula for black holes embedded in expanding Friedmann universes. In this letter, we begin the study of this new formula to obtain the first law of thermodynamics for dynamical apparent horizons. In this regard we obtain a generalized expression for the internal energy U together with a distinction between the dynamical temperature $T_D$ of apparent horizons and the related one due to thermodynamics formulas. Remarkable, when the expression for U is applied to the apparent horizon of the universe, we found that this internal energy is a constant of motion. Our calculations thus show that the total energy of our spatially flat universe including the gravitational contribution, when calculated at the apparent horizon, is an universal constant that can be set to zero from simple dimensional considerations. This strongly support the holographic principle.

Appendices

Appendix 1

In this appendix, we extend the theorem in [21] for the the de Sitter cosmological solution. The results below do not include black holes in the static patch of the expanding de Sitter universe. As a consequence, the static Schwarzschild de Sitter solution is excluded from this tractation.

The inequality (1) is obtained in [21] starting from an initial data set for the Einstein equations given by the four objects: the energy density $\rho$, the unperturbed metric ${\tilde{g}}_{ab}$, the extrinsic curvature ${\tilde{K}}_{ab}$ and the matter current density ${\tilde{J}}_a$ defined on a three-dimensional spherically symmetric manifold $\Sigma$. These quantities cannot be arbitrary, but they must satisfy the Hamiltonian and the momentum constraints. The momentum constraint is automatically satisfied since ${\tilde{K}}_{ab}$ is pure trace and we set ${\tilde{J}}_a=0$. In the de Sitter case we have:

$$\begin{aligned} a(t)=e^{H_{\Lambda }t},\;\;\;H_{\Lambda }=c\;\sqrt{\frac{\Lambda }{3}}, \end{aligned}$$

(23)

We must satisfy only the Hamiltonian constraint that for the background metric is trivially given by:

$$\begin{aligned} H^2=\frac{c^2\Lambda }{3}. \end{aligned}$$

(24)

By following the same reasoning present in [21], in the presence of a spherical perturbation, the Hamiltonian constraint becomes

$$\begin{aligned} R^{(3)}[g]-K_{ab}K^{ab}+Tr(K_{ab})^2=16\pi G\rho , \end{aligned}$$

(25)

where $R^{(3)}$ is the Ricci scalar on $\Sigma$ and:

$$\begin{aligned} \rho ={\rho }_{\Lambda }+\delta {\rho }_S,\;\; {\rho }_{\Lambda }=\frac{\Lambda c^2}{8\pi G}. \end{aligned}$$

(26)

Also in this case, under the condition that $K_{a}^{a}=const$, i.e. the rate of expansion of the volume is not perturbed ($\delta J_b=0$), is supposed to hold. As a result, the Hubble rate of the perturbed metric is left unchanged. In this way, thanks to the form of the spatial part of the de Sitter unperturbed metric (${\tilde{g}}_{ab}=a^2(t){\delta }_{ab}$), the only difference with the calculations present in [21] is that the mass excess is calculated with respect to the ‘unperturbed’ density ${\rho }_{\Lambda }$. Hence, instead of (4) we can write:

$$\begin{aligned} \frac{G}{c^2}\delta M_S < \frac{L_S}{2}+\frac{A}{4\pi }\sqrt{\frac{\Lambda }{3}}. \end{aligned}$$

(27)

By following the reasonings of [15], we obtain again expression (2) with H given by (24).

Appendix 2

As stated at the end of Sect. 4, if we consider quantum effects, the introduction of $\hbar$ can lead to a constant non-zero value for $U_h$ in presence of a cosmological constant. In this case, after introducing an effective energy density ${\rho }_h$ at the Hubble radius of our universe, we can write $U_h=c^2{\rho }_h\frac{4\pi c^3}{3H^3}$: we have

$$\begin{aligned} U=const.=K=c^5{\rho }_h\frac{4\pi }{3H^3}. \end{aligned}$$

(28)

Equation (28) implies that ${\rho }_h\sim H^3$. More precisely, consider an initial time $t_i$ with $H(t_i)=H_i, {\rho }_h(t_i)={\rho }_{hi}$: we have

$$\begin{aligned} {\rho }_h(t)={\rho }_{hi}\frac{H^3}{H_i^3}. \end{aligned}$$

(29)

Since it is expected that the semiclassical approximation leading to (2) is certainly valid after the Planck era, we expect the time $t_i$ be near the inflationary epoch, where quantum effects on the geometry are negligible. Suppose now that the universe is dominated by this effective energy-density ${\rho }_h$. By inserting ${\rho }_h$ into Einstein’s equation $H^2=8\pi G/3({\rho }_h)$ we obtain $H=3/(8\pi G)H_i^3/{\rho }_i$, i.e. a constant value near to the Planck scale. This simple fact suggests that, if the matter content of the universe were dominated at some time by ${\rho }_h$, then a de Sitter phase begins.

We now investigate the possible functional relation between ${\rho }_h$ and the usual energy density content of the universe. Suppose that the universe is filled with usual matter of some species ${\rho }_s(t)$. Friedmann equations dictate that $H^2=8\pi G/3 {\rho }_s$. From (29) we obtain

$$\begin{aligned} {\rho }_h=B_i{\rho }_s(t) H(t),\;\;B_i=\frac{{\rho }_{hi}}{{\rho }_{si} H_i}. \end{aligned}$$

(30)

Equation (30) shows a coupling between ordinary matter and ${\rho }_h$. This coupling is negligible at late times but is huge soon after the Planck era. Suppose that the primordial time $t_i$ is the begin of the primordial inflation $t_I$. The expression (30) is similar to the ones present in the so called bulk cosmology [38] related to the entropic force (in particular, also in [36–38], an ‘entropic pressure’ proportional to $H^2$ in the Friedmann’s equations is analyzed). After introducing the density parameters for the species s as ${\Omega }_s=8\pi {\rho }_s/(3H^2)$ and by denoting with the subscript “0” the actual time, we have:

$$\begin{aligned} \frac{{\Omega }_{h0}}{{\Omega }_{s0}}=\frac{{\Omega }_{hI}}{{\Omega }_{sI}}\frac{H_0}{H_I}. \end{aligned}$$

(31)

If we identify the parameter ${\Omega }_{h0}$ with the actual dark energy component and with the parameter ${\Omega }_{s0}$ the actual dark matter component, we have $\frac{{\Omega }_{h0}}{{\Omega }_{s0}}\simeq 2.3$. Concerning the ratio $H_0/H_I$, from theoretical estimates (see [43]) we have $H_I/H_0\sim (10^{20},10^{61})$. By putting these values in (31), we obtain $\frac{{\Omega }_{hI}}{{\Omega }_{sI}}\sim (10^{20},10^{61})$. As a consequence, if primordial inflation is due to ${\rho }_h$, then the huge ratio $\frac{{\Omega }_{hI}}{{\Omega }_{sI}}$ can account for the actual value of $H_0$ and also for the actual value of the cosmological constant.