Black String Thermal Equilibrium

In General Relativity, the spherically symmetric black hole solutions are widely studied due to their relevant role in describing various physical systems of astrophysical and cosmological importance, such as in the description of the final state of a gravitational collapse of stars. The interest of describing black holes with different topology is not considered due the formulation of hoop conjecture, established by Thorne Thorne (1972), which states that event horizons can be created only if a mass is compressed into a region with circumference less than $4\pi GM$ in every direction, which means that only spherically symmetric black holes should exist. However, the hoop conjecture holds only when cosmological constant is assumed to be zero, and therefore, if one admits negative cosmological constant, the spacetime shall become asymptotically Anti-de Sitter. This fact makes possible to investigate cylindrically symmetric black hole solutions, that is, black strings.

The black strings thermodynamic properties are known Chakrian and Santos (2020); Fatima and Saifullah (2012) and in particular the heat capacity positive behavior is interesting for this paper once heat capacity, and as consequence the specific heat, is always positive. This fact can be understood as the property of a system composed by the black string and a thermal reservoir reach a thermal equilibrium state. In this work, it is analyzed the black string behavior when thermal equilibrium is considered. It is assumed that the black string mass per unit of length changes due Hawking radiation since its temperature (Hawking temperature) is different from the temperature of the thermal reservoir $T_{R}$. In this conjecture, the black string temperature can be higher, $T>T_{R}$, or lower $T<T_{R}$ than the reservoir temperature. The temporal evolution of mass per unit of length is investigated in this work.

The papers Lemos (1995); Lemos and Zanchin (1996a) performed by Lemos were a breakthrough in the subject of cylindrical black holes since he has shown that the black strings are predicted when a negative cosmological constant is considered, and moreover, it is possible Lemos and Zanchin (1996b) to relate this (3+1) dimensions solution to the BTZ (2+1) dimension solution. In such initial papers, the black string solution arises from solving the motion equations that result from the Einstein-Hilbert four dimensions action

where $S$ is the Einstein-Hilbert action in four dimensions, g is the determinant of the metric tensor, $\textsl{g}=\text{det}(g_{\mu\nu})$, $R$ is the Ricci scalar, $\Lambda$ is the cosmological constant and $G$ is the Newtonian gravitational constant. The black string solution, considering $G=\hbar=c=1$, is given by Lemos and Zanchin (1996a)

where $-\infty<t<\infty$, the radial coordinate $0\leq r<\infty$, the angular coordinate $0\leq\phi<2\pi$, the axial coordinate $-\infty<z<\infty$, $\alpha^{2}=-\frac{\Lambda}{3}>0$ and $M$ the mass per unit length in the $z$-direction. This solution represents a static straight black string with one event horizon at

The Hamilton-Jacobi method represents an approach to investigate the quantum tunneling from scalar field in black string spacetime. This method is based on the tunneling approach by Parikh and Wilczeck Parikh and Wilczek (2000), the path integrals method by Srinivasan Srinivasan and Padmanabhan (1999) and can be found in the references Gohar and Saifullah (2013a, b). We start from the Klein-Gordon equation on curved spaces for a scalar field $\Phi$

where $\mathrm{g}^{\mu\nu}$ is the metric tensor related to the line element (2) and $m$ is the mass of the field. In order to solve this equation, we use the ansatz

where $S$ represents the trajectory action associated to the particle tunneling. We can recognize this solution as the WKB approximation Birrell et al. (1984). This low wavelength solution is justified since the amount of null geodesics tends to infinity at the event horizon which means that there is a blueshift Parikh and Wilczek (2000). Substituting (5) into (4) and using the first term of the expansion of the action in terms of $\hbar$, we find

The solution to equation (6) can be considered by separation of variables as follows

where $E$ is a constant associated to the energy, $W(r)$ is a function to be determined, $J_{\phi},J_{z}$ are constants associated to angular momentum in $\phi$ and $z$ direction, respectively, and $C$ is a constant. After performing an integration on complex plane, the function $W(r)$ is given by

where it was used the relation (3) and we remark that $W_{+}=-W_{-}$ . The probabilities of crossing the event horizon can be written as

and by the condition that the probability of entering in the event horizon of the black string is $100\%$ Gohar and Saifullah (2013a), we determine that $\operatorname{\mathrm{Im}}(W_{-})=\operatorname{\mathrm{Im}}(C)$ and the replacement of this result into (9) leads to

Now, if one compares the emission factor (III) to the canonical ensemble Boltzmann factor $\exp\left\{E/T_{H}\right\}$, the Hawking temperature is written as, (we set $\hbar=1$)

This result can also be found in the references Fatima and Saifullah (2012); Cai and Zhang (1996) by a different approach and in the references Gohar and Saifullah (2013a, b) by using the Hamilton-Jacobi method. It is interesting to see that the temperature behavior is linear in terms of the event horizon radius $r_{+}$. If one uses the relation (3) to write this temperature in terms of mass per unit of length associated to the event horizon radius, one obtains

which makes it clear that $T_{H}\rightarrow 0$ only if $M_{+}\rightarrow 0$. From this point, unless stated otherwise we shall consider $T_{H}=T$.

The Stefan-Boltzmann equation can be used to investigate the energy flux due the difference of temperature of the black string and the empty space as presented in the reference de Santi and Santarelli (2019). We adapt this procedure in order to study the temporal evolution of the black string mass per unit length. The Stefan-Boltzmann relation can be written as

where $T$ is identified, in this work, as the Hawking temperature associated to the black string, $T_{R}$ is the temperature associated to the reservoir, $L$ represents the energy flux through the system components, $\lambda$ is the Stefan-Boltzmann constant: $\lambda=\pi^{2}k_{B}^{4}/60\hbar^{3}c^{2}$ in SI units and $A$ is the area per unit of length of the black string $A=2\alpha\pi r_{+}^{2}$ Cai and Zhang (1996). The system proposed is defined as the black string, in vacuum, with temperature $T_{H}$, we use $T_{R}=0$ and, for this case, the relation (14) reduces to

It is important to point out that it shall be used arbitrary units because, in the point of view of this work, the general behavior of the results are interesting and their values could be investigated considering any choice of units. Then, rewriting the relation (15) using the equation (3) to write $r_{+}$ in terms of $M_{+}$ and considering $k_{B}=G=c=\hbar=1$, we have

Now, for the investigated system we can consider $J$ as the temporal variation of energy of the black string and, as consequence, its temporal variation of the mass per unit length since the energy is modified by emission (or absorption) of radiation. In this perspective, it is possible to write $J=-dM_{+}/dt$ and using the relation (16) one finds

Integrating this relation, considering the initial conditions $t_{0}=0$ and $M_{+}(t_{0})=M_{0}$, as

we obtain the temporal evolution of the black string mass per unit length considering the empty space as the thermal reservoir:

Here, $a=\frac{27\alpha^{3}}{160\pi}$. This relation establishes that the “complete evaporation” ($M\rightarrow 0$) would demand an infinite time. This fact is physically plausible because if $M\rightarrow 0$, then $T\rightarrow 0$ from relation (13) and $T\rightarrow 0$ should not happen due the third law of thermodynamics. In other words, it is not possible to the black string completely evaporate. We remark that this prediction suggests a different behavior in comparison to the Schwarzchild black hole since for the last one it is predicted the existence of a lifetime which depends on the initial mass of the black hole.

The figure 1 presents a plot of the relation (19) which represents the analytic solution to the equation (17) and also presents a numerical¹¹1It was applied the Classical Runge-Kutta method using the open source software of mathematics SageMath. All numerical calculations presented in this work were performed using the mentioned software. solution to such equation. It is possible to observe the decrease in the value of the mass per unit length. This fact was expected since the black string temperature is higher than the empty space temperature $T>T_{R}$ and therefore the black string radiates and evaporates continuously until the system reach the equilibrium temperature.

After the temperature and density decrease due Universe expansion, the interactions among photons and hydrogen atoms also decreased. The photons, earlier with sufficient energy to remove electrons from atoms, became less energetic and could no longer remove electrons, but moved freely without interacting with matter. Hence, the Big Bang model predicts the cosmic radiation background and its energy is associated to the hydrogen atom ionization energy. Such radiation was detected as black body radiation in microwave frequency by Arno Penzias and Robert Wilson in 1965 Penzias and Wilson (1965); Dicke et al. (1965). The current approximated value associated to the Cosmic Microwave Background - CMB is $T_{\mbox{\scriptsize CMB}}\approx 2.7255\mbox{K}$. We consider the CMB as the thermal reservoir and we study the energy exchange in the system. Therefore, it is possible to associate a thermal current (energy flux) by considering the Stefan-Boltzmann relation, writing $T_{R}=T_{\mbox{\scriptsize CMB}}$, as

Performing the procedure adopted in the previous section, the temporal evolution of the black string mass per unit of length can be determined by the following differential equation

where

The relation (21), which represents the temporal evolution of the mass per unit of length considering the CMB as thermal reservoir, can be investigated for the purpose of obtaining a clue with respect to the mass behavior of the system. It is possible to calculate an equilibrium mass relation: after some modifications, we integrate the equation (21) as

where $x={M^{\prime}_{+}}^{\frac{1}{3}}$, $a$ and $b$ are given in equation (22). Performing the integration, we obtain

where $K$ is an integration constant. We notice a change in behavior when the logarithms arguments are zero, that is,

The second relation above results negative mass and for this reason such relation is not important. However, the first relation provides the mass associated to the change in system behavior as

The mass per unit of length shown in equation (27) represents the mass value for which the thermal equilibrium of the system is reached. This interpretation can be confirmed if we calculate what should be the mass associated with the CMB temperature. Considering the relation (12) applied for CMB one obtains

and now, considering the relation (3), one obtains the following result which confirms the interpretation concerning the equation (27)

The figure 2 shows a plot of the solution to the equation (21). It can be seen that the predictions provided by the previous discussion in the section I have been confirmed. One can see that the mass per unit of length assumes a constant value, $M=1$ and such fact corroborates two predictions: first, the mass per unit of length reaches a constant value and it indicates that the temperature gradient ceases and there is no energy exchange between black string and CMB anymore; second, the equilibrium mass per unit of length $M_{e}=1$ confirms relation (27) since, for this plot, it was set $a=b=1$ and the relation (27) predicts that, for such choice of parameters, $M_{e}=1$.

The CMB evolves according to the temporal evolution of the universe. We consider this time evolution divided into three eras: the radiation era, the matter era, and the dark energy era. Each one represents the era when there is a predominance of each one of the cosmic fluids (radiation, matter and dark energy) considering the big bang as the beginning of time. The energy exchange between the black string and the CMB is investigated by considering that the temperature $T_{\mbox{\scriptsize CMB}}$ assumes different behaviors for each era of the Universe. The temperature for each era can be written, in SI units, as²²2These relations, presented in SI units are known and their demonstration can be found in de Santi and Santarelli (2019). In this work, we apply such relations in arbitrary units to understand the black string temporal evolution.

where $\rho_{R_{0}}$,$\rho_{M_{0}}$, $\rho_{\Lambda_{0}}$ represent the current densities of radiation, matter and dark energy, respectively. Photons, neutrinos, cosmic microwave background and gravitons are associated to the radiation era; hadronic and non-hadronic matter are associated to the matter era; and the dark energy era is associated to dark energy.

This paper represents a conjecture regarding the presence of the black string in the universe as we know it considering its eras. This conjecture is interesting to situate the black string model in the temporal evolution of the universe. In principle, the black string subjected to the universe as we currently understand it would be an object in thermal equilibrium with the universe, with respect to current measurements of cosmic microwave background radiation temperature. This interpretation refers to figure 2 in which it is clearly seen that as time passes, the mass per unit length assumes a constant value. This means that the rate of energy exchange between the black string and the reservoir (CMB) has ceased and thermal equilibrium has been reached. The study considering the change in reservoir temperature as time passes performed in sections VI.1, VI.2 and VI.3 shows that the mass values decrease rapidly indicating a rapid evaporation of the black string as can be seen by investigating figures 3, 4 and 5. These conjectures are important for an understanding of the thermal behavior of objects representing singularities and for using a method for such studies.