Corrections to the thermodynamic quantities of Bose system by the generalized uncertainty principle

In 1973, inspired by the black hole’s area theorem[1], Bekenstein[2] noticed the striking similarities between black hole mechanics and thermodynamics and pointed out that a black hole is a thermodynamic system. Hawking[3], who was initially opposed to the idea, proved that black holes have thermal radiation by means of the method of quantum field theory in curved spacetime, showing that the four laws of black hole mechanics are essentially the four laws of conventional thermodynamics. Since the black hole radiation is purely thermal, it is natural that there will be a concomitant loss of information when a black hole radiates all its mass, known as the black hole information puzzle. At the same time, the existence of the singularity in black holes is so incredible and incomprehensible that people begin to search for a quantum gravity theory that could accommodate both general relativity and quantum theory. As strong candidates for quantum gravity, studies[4, 5, 6, 7, 8, 9, 10, 11, 12, 13] such as string theory and loop quantum gravity suggested that the Heisenberg uncertainty principle (HUP) should be modified when the gravitational effect is non-negligible. Then the counterpart is the generalized uncertainty principle (or the gravitational uncertainty principle)

where $\alpha\sim G$. All these studies have shown that the GUP is a general conclusion which is independent of the specific gravitational quantization scheme. However, the deficiency of HUP is that it does not take the gravitational interaction into account. As previous studies of black hole thermodynamics have been using the HUP, some of the earlier results may need to be modified if the GUP replaces the HUP. Through a heuristic approach where the HUP prevents the total collapse of hydrogen atoms, R.J.Adler et al.[14] utilised the GUP to prove that the complete evaporation of a black hole can be prevented by dynamics rather than by symmetry or quantum numbers. Many works [14, 15, 16, 17, 18] have shown that considering the GUP would allow a black hole to evaporate to the point where there would be a remnant of what is considered to be one of the candidates for cold dark matter left at the end. Furthermore, A.J.M.Medved and E.C.Vagenas [19] obtained an unfolded quantum modified Bekenstein–Hawking entropy through the GUP, proving that the prefactor of the logarithmic term of the entropy can be represented by a parameter derived from a more fundamental theory. Numerous calculations [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] have used the GUP method to study black hole thermodynamics to derive corrections to the Hawking radiation spectrum and Bekenstein-Hawking entropy.

In 2014, Ref.[29] introduced the GUP into a shell with Bose gas to obtain a modified Stefan-Boltzmann law, which showed a $T^{6}$ correction term. Applying the modified law to the black hole radiation yields a black hole remnant with the Planck scale and a maximum radiation temperature at that time. In addition, the lifespan of a black hole is also modified. We believe some corrections will appear in the Bose system by considering the GUP. Ref.[29] did not further research the corrections to other thermodynamic quantities of the Bose system, so our work builds on this.

The remaining part of the paper proceeds as follows: The second part briefly describes the method for obtaining the energy-momentum relation for Bose gas in the Schwarzschild spacetime. In the third part, we derive the corrections to the thermodynamic quantities by means of the grand partition function. The fourth part contains a brief discussion and conclusion. The unit $G=c=1$ is adopted throughout this paper.

In order to study the thermodynamic properties of a black hole, ’t Hooft [31] proposed the famous ”brick wall” model in 1985. In this paper, we adopt this model and move the distant wall to the vicinity of the black hole horizon, so that a thin shell adjacent to the horizon is formed between the two walls. The thermal equilibrium quantum gas(Hawking radiation) within this shell near the Schwarzschild black hole horizon is selected as the research object. The Hawking radiation in the shell is

and this spectrum is the particle number distribution law of the Bose system[32]. Since the shell is infinitely close to the horizon, a coordinate system which does not diverge at the horizon should be adopted. Introduce the following coordinate transformation into the Schwarzschild metric,

where $t_{s}$ is the Schwarzschild time. Then the Schwarzschild spacetime line element can be written as

where $f=1-\frac{2M}{r}$, M for the black hole mass, $r_{H}=2M$ for the event horizon. It is a stationary coordinate system without divergence behaviour at the event horizon $r_{H}$, first proposed by Painlevé [33]. We take a thin shell layer for discussion which is based on the brick-wall model at the horizon, whose inner surface is infinitely close to the horizon. The coordinate distance between the inner surface and the horizon is $\epsilon$ and the thickness of the shell is $\delta$.

We consider that the shell is filled with Bose gas. The Klein-Gordon equation satisfied by Bose particles is

Use the Wentzel-Kramers-Brillouin (WKB) approximation and set the wave function as

Combining Eq.(5) and Eq.(6) yields

where $\omega=p_{t},p_{r}=\frac{\partial S}{\partial r},p_{\theta}=\frac{\partial S}{\partial\theta},p_{\varphi}=\frac{\partial S}{\partial\varphi}$. For a spherically symmetric system, only the radial motion is taken into consideration and then the energy-momentum relation for the boson can be found as

Since the modulus of momentum cannot be negative, only the positive root is taken. Considering the shell is close to the horizon, $f\to 0$, therefore we have

Then we obtained the energy-momentum relation for Bose gas in the vicinity of the event horizon.

To understand the behaviour and thermodynamic properties of the Bose system in a shell, we will discuss it in a 6-dimensional phase space. By the GUP there is

where $\Delta q_{i}$ and $\Delta p_{i}$ represent the position and momentum uncertainty, respectively. In a 6-dimensional phase space, the degeneracy $\omega_{l}$ is equal to the volume $\upd x\upd y\upd z\upd p_{x}\upd p_{y}\upd p_{z}$ divided by the uncertainty, i.e.

The above equation is the expression in a Cartesian coordinate system, it can be rewritten in the spherically symmetric coordinate as

For a Bose system in a shell, thermodynamic quantities can be derived from the partition function in statistical theory. Introduce a grand partition function as follow

Now, substituting the energy-momentum relation (9) into the above equation gives

In the vicinity of the event horizon, expanding $f(r)$ with the Taylor series and taking the first approximation lead to

Finally we obtained the concrete expression of the grand partition function

where $V=4\pi r_{H}^{2}\delta$. The internal energy $U$ and pressure $p$ of the Bose gas can be derived from the partition function as

Comparing the two equations above, the relationship between $p$ and the internal energy density $u$ can be obtained as follows

where

Also we can obtain the entropy of the Bose gas in the shell layer as

Obviously, the second term of Eq.(21) is a correction, which is related to the GUP. Since $V=4\pi r_{H}^{2}\delta=A\delta$, by taking a proper cutoff factor $\frac{\delta}{\epsilon^{3}}=\frac{45\hbar^{3}}{8\pi^{2}k_{B}}$, the first term of Eq.(21) can be expressed as $A/4$, and the second term becomes $-\frac{9\alpha A}{28\hbar^{2}\epsilon^{2}}$. In addition, if we let the corrected entropy in Eq.(21) equal $A/4$, then the cutoff factor $\epsilon$ will be a function of $\delta$, satisfying the equation $315\hbar^{5}\epsilon^{5}-56\pi^{2}k_{B}\delta\epsilon^{2}+72\pi^{2}k_{B}\delta=0$.