Cosmological Black Holes and the Direction of Time

Abstract

Macroscopic irreversible processes emerge from fundamental physical laws of reversible character. The source of the local irreversibility seems to be not in the laws themselves but in the initial and boundary conditions of the equations that represent the laws. In this work we propose that the screening of currents by black hole event horizons determines, locally, a preferred direction for the flux of electromagnetic energy. We study the growth of black hole event horizons due to the cosmological expansion and accretion of cosmic microwave background radiation, for different cosmological models. We propose generalized McVittie co-moving metrics and integrate the rate of accretion of cosmic microwave background radiation onto a supermassive black hole over cosmic time. We find that for flat, open, and closed Friedmann cosmological models, the ratio of the total area of the black hole event horizons with respect to the area of a radial co-moving space-like hypersurface always increases. Since accretion of cosmic radiation sets an absolute lower limit to the total matter accreted by black holes, this implies that the causal past and future are not mirror symmetric for any spacetime event. The asymmetry causes a net Poynting flux in the global future direction; the latter is in turn related to the ever increasing thermodynamic entropy. Thus, we expose a connection between four different “time arrows”: cosmological, electromagnetic, gravitational, and thermodynamic.

Appendix: Technical Details of the Cosmological Growth of Black Holes

1.1 Cosmological Expansion

Black holes are co-moving with the cosmological expansion of the universe. In order to describe the increase of size of the black hole due to this expansion, we adopt the Hawking–Hayward quasilocal mass (Hawking 1968; Hayward 1994), which measures the mass of a bound source of gravitation in an asymptotically FRWL universe (Nolan 1998). Specifically, we adopt:

$$\begin{aligned} M_{\small \mathrm{BH}}^{\small \mathrm{HH}}(t) = M_\mathrm{0}\, a(t), \end{aligned}$$

(22)

where $M_\mathrm{{0}}$ is the black hole present mass and a(t) stands for the normalized scale factor. To implement this proposal, we replace Hawking–Hayward quasi-local mass in McVittie metrics (McVittie 1933) for a mass-particle embedded in a Friedmann cosmological model and, via Einstein field equations, we study the corresponding energy-momentum tensor.

For a cosmological black hole embedded in the flat Friedmann model, the generalized McVittie metric in isotropic coordinates $(t,r,\theta ,\phi )$ is (Gao et al. 2008):

$$\begin{aligned} ds^2 = -\frac{\left\{ 1-\frac{M(t)}{2\,r a(t)}\right\} ^2}{\left\{ 1+\frac{M(t)}{2\,ra\,(t)}\right\} ^2} dt^2\nonumber + a(t)^2 \left\{ 1+\frac{M(t)}{2\,r\,a(t)}\right\} ^4 \left( dr^2+r^2d\varOmega ^2\right) , \end{aligned}$$

(23)

where M(t) is an arbitrary function of the cosmic time t. For the open and closed Friedmann models, we propose the following metrics:

$$\begin{aligned} ds^2=- & {} \frac{\left\{ 1-\frac{M(t)}{2\,r\,a(t)} \left( 1\pm \frac{r^2}{4R_0^2}\right) ^{1/2}\right\} ^2}{\left\{ 1+\frac{M(t)}{2\,r\,a(t)} \left( 1 \pm \frac{r^2}{4R_0^2}\right) ^{1/2}\right\} ^2} dt^2+ \frac{\left\{ 1+\frac{M(t)}{2\,r\,a(t)} \left( 1 \pm \frac{r^2}{4R_0^2}\right) ^{1/2}\right\} ^4}{\left( 1 \pm \frac{r^2}{4R_0^2}\right) ^2} a(t)^2 dr^2\nonumber \\&+ \frac{\left\{ 1+\frac{M(t)}{2\,r\,a(t)} \left( 1 \pm \frac{r^2}{4R_0^2}\right) ^{1/2}\right\} ^4 a(t)^2 r^2 d\varOmega ^2}{\left( 1 \pm \frac{r^2}{4R_0^2}\right) ^2}, \nonumber \end{aligned}$$

(24)

where ± is the sign of the spatial curvature. If we adopt for the function M(t) the Hawking–Hayward quasi-local mass, given by Eq. 22, the metrics above take the form:

$$\begin{aligned} ds^2 = -\frac{\left\{ 1-\frac{M_0}{2\,r}\right\} ^2}{\left\{ 1+\frac{M_0}{2\,r}\right\} ^2} dt^2\nonumber + a(t)^2 \left\{ 1+\frac{M_0}{2\,r}\right\} ^4 \left( dr^2+r^2d\varOmega ^2\right) ,\end{aligned}$$

(25)

$$\begin{aligned} ds^2 = - \frac{\left\{ 1-\frac{M_0}{2\,r} \left( 1\pm \frac{r^2}{4R_0^2}\right) ^{1/2}\right\} ^2}{\left\{ 1+\frac{M_0}{2\,r} \left( 1 \pm \frac{r^2}{4R_0^2}\right) ^{1/2}\right\} ^2} dt^2 \nonumber + \frac{\left\{ 1+\frac{M_0}{2\,r} \left( 1 \pm \frac{r^2}{4R_0^2}\right) ^{1/2}\right\} ^4}{\left( 1 \pm \frac{r^2}{4R_0^2}\right) ^2} a(t)^2 dr^2 \nonumber + \frac{\left\{ 1+\frac{M_0}{2\,r} \left( 1 \pm \frac{r^2}{4R_0^2}\right) ^{1/2}\right\} ^4}{\left( 1 \pm \frac{r^2}{4R_0^2}\right) ^2} a(t)^2 r^2 d\varOmega ^2, \end{aligned}$$

(26)

Notice that if $a(t)= 1$ and $r<<R_0$, the metrics (23) and (24) reduce to the Schwarzschild metric in physical isotropic coordinates $(t,a(t)r,\theta ,\phi )$ with central mass $M_0 a(t)$. Conversely, if $M_0=0$, we recover the Friedmann geometries.

The asymptotically flat metric (23) and its corresponding energy-momentum tensor has been analyzed by Gao et al. (2008). The metric turns to be consistent with the energy-momentum of an imperfect pressureless accreting fluid. We emphasize that, asymptotically, the energy-momentum tensor coincides with the one of the cosmological model:

$$\begin{aligned} T_{\mu \nu }=\rho (r,t) u_\mu u_\nu + q_\mu u_\nu + q_\nu u_\mu \rightarrow \; \rho _\infty (t) u_\mu u_\nu . \end{aligned}$$

(27)

Here $u^\mu \rightarrow (|g_{00}(r,t)|^{-1/2},0,0,0)$, $q^\mu \rightarrow (0,q(r,t),0,0)$, and $\rho _\infty \propto H^2(t)$, in agreement with Friedmann equations.⁶

We maintain that the same result can be extend to the asymptotically curved metrics (24). According to Einstein field equations, the matter content of these geometries is an imperfect accreting fluid that, asymptotically, coincides with the one of curved Friedmann models:

$$\begin{aligned} T_{\mu \nu } &= [P(r,t)+\rho (r,t)] u_\mu u_\nu \nonumber \\&+ P(r,t) g_{\mu \nu } + q_\mu u_\nu + q_\nu u_\mu \nonumber \\&\rightarrow \; \rho _\infty (t) u_\mu u_\nu , \nonumber \end{aligned}$$

where $\rho _\infty (t) \propto H^2(t) \pm C a^{-2}(t)$, with C a positive constant, in accordance with Friedmann equations.

We have shown that the spacetime metrics given by (23) and (24) are consistent with Hawking–Hayward quasi-local mass proposal for a black hole embedded in Friedmann cosmological models. A detailed analysis of the existence of trapped surfaces for those metrics will be given in a future work. In what follows, we will assume Hawking–Hayward quasi-local mass as the effective mass of a black hole embedded in a Friedmann cosmological model.

1.2 Black Hole Accretion of Cosmic Microwave Background Radiation

The rate of accretion of CMB photons onto a black hole is given by:

$$\begin{aligned} \dot{M}_{\small \mathrm{BH}}^{\small \mathrm{CMB}} = \pi b^{2} n_{\small \mathrm{CMB}}(z) <{\epsilon }_{\mathrm{ph}}> c, \end{aligned}$$

(29)

where $b = \left( 3 \sqrt{3}r_\mathrm{S}\right) /2$ is the critical impact parameter of a Schwarzschild black hole ($r_\mathrm{S}$ stands for one Schwarzschild radius); $n_{\small \mathrm{CMB}}(z)$ and $<{\epsilon }_{\mathrm{ph}}>$ are the density and energy of CMB photons as a function of redshift z, respectively:

$$\begin{aligned} n_{\small \mathrm{CMB}}(z) &= 5\,10^{-2} \mathrm{cm}^{-3} \left( 1+z\right) ^{3}, \end{aligned}$$

(30)

$$\begin{aligned} <{\epsilon }_{\mathrm{ph}}> &= 6\,10^{-6} \mathrm{erg} \left( 1+z\right) . \end{aligned}$$

(31)

For simplicity, we consider only supermassive black holes⁷ of $M_{0} = 10^{8}\,M_{\odot }$. Equation (29) and (22) then yields:

$$\begin{aligned} \dot{M}_{\small \mathrm{BH}}^{\small \mathrm{CMB}} (t)= 5.5\,10^{25}\,\pi \left( \frac{a_\mathrm{0}}{a(t)}\right) ^{2}\, \frac{\mathrm{erg}}{\mathrm{s}}. \end{aligned}$$

(32)

The variation of black hole mass by accretion of CMB photons during the cosmic time interval $\varDelta t = t-t_{0}$ ($t_{0}$ is the value of the present cosmic time) can be obtained by integration of the latter equation:

$$\begin{aligned} \varDelta M_{\small \mathrm{BH}}^{\small \mathrm{CMB}}(t) = {\int _{t_\mathrm{0}}}^{t} \frac{\dot{M}_{\small \mathrm{BH}}^{\small \mathrm{CMB}} (t^\prime )}{c^{2}} dt^\prime . \end{aligned}$$

(33)

As mentioned above, it is expected that accretion of matter and currents also occurs, as it is actually observed in active galactic nuclei (Romero and Vila 2014). In this sense, our estimation for the growth of the black hole mass due to CMB accretion gives the absolute minimum.