Dynamic noncommutative BTZ black holes

In the first-order formulation of gravity in $(2+1)$ dimensions, the Einstein-Hilbert action $\mathcal{I}_{\text{EH}}$ with nonzero vacuum energy becomes equivalent [44, 45] to the action functional of the CS theory

$$\mathcal{S}_{\text{CS}}=\mathcal{I}_{\text{CS}}[\mathcal{A^{(+)}}]-\mathcal{I}_{\text{CS}}[\mathcal{A^{(-)}}]\equiv\mathcal{I}_{\text{EH}},$$

(2.1)

with two $SU(1,1)\simeq SO(1,2)$ connection $1$-forms

$$\mathcal{A}^{(\pm)a}=\omega^{a}\pm\frac{e^{a}}{\ell},$$

(2.2)

where the CS term, $\mathcal{I}_{\text{CS}}=k/4\pi\int\text{tr}[\mathcal{A}\mathop{}\!\mathrm{\,d}\mathcal{A}+2\mathcal{A}\mathcal{A}\mathcal{A}/3]$, is given in terms of the CS 3-form, Here, $k=-\ell/4G$ represents the CS level, and $e^{a}$ and $\omega^{a}$ denote the vielbeins and spin connections, respectively, for $a=0,1,2$. Here, $G$ is the universal gravitational constant and $\ell$ is the length scale associated with the cosmological constant $\Lambda=-1/\ell^{2}$. The metric of a charged rotating BTZ black hole is given by [46]

$$ds^{2}=-f(r)\mathop{}\!\mathrm{\,d}t^{2}+\frac{\mathop{}\!\mathrm{\,d}r^{2}}{g(r)}+r^{2}\mathop{}\!\mathrm{\,d}\varphi^{2}-\frac{2\gamma}{\ell}\mathop{}\!\mathrm{\,d}t\mathop{}\!\mathrm{\,d}\varphi,$$

(2.3)

with

$$g(r)=f(r)+\frac{\gamma^{2}}{\ell^{2}r^{2}},~{}\quad f(r)=\frac{1}{\ell^{2}}\left(r^{2}-\zeta-\epsilon\ln r\right),~{}\zeta=8GM\ell^{2},~{}\epsilon=8\pi G\ell^{2}Q^{2},~{}\gamma=4GJ\ell,$$

(2.4)

where, $M,Q,J$ denote the total mass, charge and angular momentum of the black hole, respectively. One can choose a set of non-unique vielbeins

$$e^{0}=\sqrt{f(r)}\mathop{}\!\mathrm{\,d}t+\frac{\gamma}{\ell\sqrt{f(r)}}\mathop{}\!\mathrm{\,d}\varphi,\quad e^{1}=\frac{\mathop{}\!\mathrm{\,d}r}{\sqrt{g(r)}},\quad e^{2}=r\sqrt{\frac{g(r)}{f(r)}}\mathop{}\!\mathrm{\,d}\varphi,$$

(2.5)

to rewrite the metric (2.3) in terms of the vielbeins as $ds^{2}=-(e^{0})^{2}+(e^{1})^{2}+(e^{2})^{2}$. The corresponding spin connections are given by

$$\omega^{0}=-\frac{\gamma f^{\prime}(r)}{2\ell r\sqrt{f(r)}}\mathop{}\!\mathrm{\,d}t-\sqrt{f(r)}\mathop{}\!\mathrm{\,d}\varphi,\quad\omega^{1}=\frac{\gamma f^{\prime}(r)}{2\ell rf(r)\sqrt{g(r)}}\mathop{}\!\mathrm{\,d}r,\quad\omega^{2}=-\frac{f^{\prime}(r)}{2}\sqrt{\frac{g(r)}{f(r)}}\mathop{}\!\mathrm{\,d}t.$$

(2.6)

It is worth mentioning that throughout the manuscript, we denote $(\cdot)^{\prime}$ as the first-order derivative with respect to $r$. A noncommutative counterpart of the charged rotating BTZ black hole can be obtained by using the SW map [3] for gauge fields, where the noncommutativity between the radial and angular variables is imposed through the commutation relation [29]

$$[\mathcal{R},\varphi]=2i\theta,\quad\mathcal{R}=r^{2},$$

(2.7)

where, $\theta^{R\varphi}=-\theta^{\varphi R}=2\theta$ for a constant $\theta$, with all other components being zero. The SW map imposes the compatibility condition for gauge transformation between $\mathcal{A}$ and $\tilde{\mathcal{A}}$

$$\tilde{\mathcal{A}}(\mathcal{A})+\tilde{\delta}_{\tilde{\xi}}\tilde{\mathcal{A}}(\mathcal{A})=\tilde{\mathcal{A}}(\mathcal{A}+\delta_{\xi}\mathcal{A}),$$

(2.8)

which when solved using the perturbation theory for infinitesimal $\xi$ and $\tilde{\xi}$, we obtain

	$\displaystyle\tilde{\mathcal{A}}_{\mu}(\mathcal{A})=\mathcal{A}_{\mu}-\frac{i}{4}\theta^{v\rho}\left\{\mathcal{A}_{v},\partial_{\rho}\mathcal{A}_{\mu}+\mathcal{F}_{\rho\mu}\right\}+\mathcal{O}\left(\theta^{2}\right),$		(2.9)
	$\displaystyle\tilde{\xi}(\xi,\mathcal{A})=\xi+\frac{i}{4}\theta^{\mu\nu}\{\partial_{\mu}\xi,\mathcal{A}_{\nu}\}+\mathcal{O}(\theta^{2}).$		(2.10)

Here, $\{\cdot,\cdot\}$ and $\mathcal{F}_{\mu\nu}$ represent the anticommutator with respect to the conventional matrix product and the field strength for the gauge fields $\mathcal{A}_{\mu}$, respectively. For the noncommutativity of the form (2.7), the correction terms to the gauge fields are given by[37]

	$\displaystyle\mathbb{A}_{\mu}(\mathcal{A})=$	$\displaystyle-\frac{i\theta}{2}\left[\frac{1}{2}\eta_{ab}\mathcal{A}_{R}^{a}\left(\partial_{\varphi}\mathcal{A}_{\mu}^{b}+\mathcal{F}_{\varphi\mu}^{b}\right)\mathbb{I}-\frac{1}{2}\eta_{ab}\mathcal{A}_{\varphi}^{a}\left(\partial_{R}\mathcal{A}_{\mu}^{b}+\mathcal{F}_{R\mu}^{b}\right)\mathbb{I}\right.$		(2.11)
		$\displaystyle+i\left(\mathcal{A}_{R}^{a}\tau_{a}+\mathcal{B}_{R}\tau_{3}\right)\left(\partial_{\varphi}\mathcal{B}_{\mu}+\mathcal{F}_{\varphi\mu}^{(\mathcal{B})}\right)-i\left(\mathcal{A}_{\varphi}^{a}\tau_{a}+\mathcal{B}_{\varphi}\tau_{3}\right)\left(\partial_{R}\mathcal{B}_{\mu}+\mathcal{F}_{R\mu}^{(\mathcal{B})}\right)$
		$\displaystyle\left.+i\mathcal{B}_{R}\left(\partial_{\varphi}\mathcal{A}_{\mu}^{b}+\mathcal{F}_{\varphi\mu}^{b}\right)\tau_{b}-i\mathcal{B}_{\varphi}\left(\partial_{R}\mathcal{A}_{\mu}^{b}+\mathcal{F}_{R\mu}^{b}\right)\tau_{b}\right],$

where $a,b=0,1,2$, and $\eta_{AB}=\operatorname{diag}(-1,1,1,-1)$ and $\tau$s are the generators of $U(1,1)$. Notice that the noncommutative extension produces two extra gauge fields $\mathcal{B}_{\mu}^{\pm}$, and in order to match the result of the noncommutative version with the commutative limit, we must have the condition

$$\mathop{}\!\mathrm{\,d}\mathcal{B}_{\mu}^{(\pm)}=0,$$

(2.12)

so that the corresponding field strengths to the gauge fields $\mathcal{B}_{\mu}$ vanish. Therefore, in [37], the extra gauge fields $\mathcal{B}_{\mu}^{\pm}$ were chosen as $\mathcal{B}_{\mu}^{(\pm)}=B\mathop{}\!\mathrm{\,d}\varphi$ ($B$ is a constant), which simplifies (2.11) further, and leads to

	$\displaystyle\mathbb{A}_{\mu}^{(\pm)a}$	$\displaystyle=-\frac{\theta B}{2}\left[\partial_{R}\mathcal{A}_{\mu}^{(\pm)a}+\mathcal{F}_{R\mu}^{a}\right],$		(2.13)
	$\displaystyle\mathbb{B}_{\mu}^{(\pm)}$	$\displaystyle=-\frac{\theta}{2}\eta_{ab}\left[\mathcal{A}_{R}^{(\pm)a}\mathcal{F}_{\varphi\mu}^{b}-\mathcal{A}_{\varphi}^{(\pm)a}\mathcal{F}_{R\mu}^{b}-\mathcal{A}_{\varphi}^{(\pm)a}\partial_{R}\mathcal{A}_{\mu}^{(\pm)b}\right].$

By replacing the vielbeins (2.5) and spin connections (2.6) in (2.2) followed by a noncommutative correction (2.13), one can obtain the final forms of the CS gauge fields for noncommutative space, $\tilde{A}^{\pm a}$. The metric of the corresponding black hole can be obtained by reconstructing the vielbeins and spin connections as

$$\tilde{e}^{a}=\frac{\ell}{2}\left(\tilde{A}^{(+)a}-\tilde{A}^{(-)a}\right),\quad\tilde{\omega}^{a}=\frac{1}{2}\left(\tilde{A}^{(+)a}+\tilde{A}^{(-)a}\right),$$

(2.14)

such that the metric of a charged rotating noncommutative BTZ black hole turns out to be [37]

	$\displaystyle ds^{2}=$	$\displaystyle-\left(f(r)-\theta B\frac{2r^{2}-\epsilon}{4\ell^{2}r^{2}}\right)\mathop{}\!\mathrm{\,d}t^{2}+\left(\frac{1}{g(r)}+\theta B\frac{2f(r)\ell^{2}+2r^{2}-\epsilon}{4\ell^{2}r^{2}g^{2}(r)}\right)\mathop{}\!\mathrm{\,d}r^{2}$		(2.15)
		$\displaystyle\quad\quad+\left(r^{2}-\frac{\theta B}{2}\right)\mathop{}\!\mathrm{\,d}\varphi^{2}-\frac{2\gamma}{\ell}\mathop{}\!\mathrm{\,d}t\mathop{}\!\mathrm{\,d}\varphi+\mathcal{O}\left(\theta^{2}\right).$

It has been shown that the space-time noncommutativity eliminates the point-like sources by the smearing of an object over the space [3, 47], which; in turn, affects the propagation of energy and momentum of the object [48]. The fuzziness induced by the noncommutativity, thus, clearly results in the modifications of the distributions of mass and charge of the particle. Therefore, an alternative and straightforward way to obtain the metric of a noncommutative BTZ black hole is by replacing the Dirac’s point-like mass and charge distributions with a distribution function whose width is the same as the minimal length of the corresponding noncommutative background [23, 24, 27, 34, 35]. As for example, the minimal length corresponding to the SW noncommutativity $[x^{\mu},x^{\nu}]=i\theta^{\mu\nu}$ being $\sqrt{\theta}$, one can incorporate such a noncommutativity in mass and charge distributions by considering the corresponding densities to be a Gaussian of width $\sqrt{\theta}$

$$\rho_{c}^{(n+1)D}(r)=\frac{q}{(4\pi\theta)^{n/2}}e^{-r^{2}/4\theta},\quad\rho_{m}^{(n+1)D}(r)=\frac{M}{(4\pi\theta)^{n/2}}e^{-r^{2}/4\theta}.$$

(2.16)

Here, $q,M$ and $n$ are the charge, mass and the number of spatial dimensions of the object, respectively. A black hole solution in $AdS_{3}$ space follows from the solution of the Einstein-Maxwell equations (in $8G=c=1$ unit)

	$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\pi\left(T_{\mu\nu}^{\text{matter}}+T_{\mu\nu}^{\text{em}}\right)+\frac{1}{\ell^{2}}g_{\mu\nu},$		(2.17)
	$\displaystyle\qquad\qquad\quad\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}F^{\mu\nu}\right)=J^{\nu},$		(2.18)

where the notations are taken to be standard. Since the charge distribution is assumed to be static, the nonzero component of the four current, $J^{\nu}(r)=\rho_{c}(r)\delta^{\nu}_{0}$, is the charge density only, and the non-vanishing terms in the field strength are $F^{r0}=-F^{0r}=E(r)$. Now, the continuity equation $T^{\mu\nu}{}_{;\nu}=0$ suggests that the matter distribution be anisotropic so that the most general form of the matter energy-momentum tensor becomes [49]

$$T_{\mu\nu}^{\text{matter}}=\left(\rho+p_{t}\right)u_{\mu}u_{\nu}+p_{t}g_{\mu\nu}+\left(p_{r}-p_{t}\right)\chi_{\mu}\chi_{\nu},$$

(2.19)

where $\rho$ is the energy density and $p_{t}$ and $p_{r}$ are the tangential and radial pressure, respectively. Here, $u^{i}$ and $\chi^{i}$ represent the $(2+1)D$ velocity and the unit vector in the radial direction, respectively. The electromagnetic stress-energy tensor is taken to be of the standard form

$$T_{\mu\nu}^{\text{em}}=-\frac{1}{4\pi}\left(F_{\mu\alpha}g^{\alpha\beta}F_{\beta\nu}-\frac{1}{4}g_{\mu\nu}F_{\sigma\alpha}g^{\alpha\beta}F_{\beta\rho}g^{\rho\sigma}\right).$$

(2.20)

Now, to solve the set of Einstein-Maxwell equations, let us assume a spherically symmetric solution of the form

$$ds^{2}=-f(r)\mathop{}\!\mathrm{\,d}t^{2}+[g(r)]^{-1}\mathop{}\!\mathrm{\,d}r^{2}+r^{2}\mathop{}\!\mathrm{\,d}\phi^{2},$$

(2.21)

with $f(r)=e^{2\alpha(r)}$ and $[g(r)]^{-1}=e^{2\beta(r)}$. By using the energy-momentum tensors given by (2.19) and (2.20) as well as the metric in (2.21), the field equations given by (2.17) can be rewritten as

	$\displaystyle\frac{\beta^{\prime}(r)e^{-2\beta(r)}}{r}=\pi\rho-\frac{1}{\ell^{2}}+\frac{E^{2}(r)}{8}e^{2[\alpha(r)+\beta(r)]},\quad\frac{\alpha^{\prime}(r)e^{-2\beta(r)}}{r}=\pi p_{r}+\frac{1}{\ell^{2}}-\frac{E^{2}(r)}{8}e^{2[\alpha(r)+\beta(r)]}$
	$\displaystyle e^{-2\beta(r)}\left\{[\alpha^{\prime}(r)]^{2}+\alpha^{\prime\prime}(r)-\alpha^{\prime}(r)\beta^{\prime}(r)\right\}=\pi p_{t}+\frac{1}{\ell^{2}}+\frac{E^{2}(r)}{8}e^{2[\alpha(r)+\beta(r)]}.$		(2.22)

The equations in (2.2) suggest that $\alpha(r)=-\beta(r)$ and, therefore, one can use (2.21) to solve the associated smeared electric field from (2.18) as

$$E(r)=\frac{1}{r}\int_{0}^{r}\tilde{r}\rho_{c}^{3D}\left(\tilde{r}\right)\mathop{}\!\mathrm{\,d}\tilde{r}=\frac{q}{2\pi r}\left(1-e^{-r^{2}/4\theta}\right),$$

(2.23)

which when is utilized for solving the Einstein-Maxwell equations given by (2.2) and (2.18), we obtain the line element (2.21) with

$$e^{-2\beta(r)}=g(r)=f(r)=-M\left(1-e^{-r^{2}/4\theta}\right)+\frac{r^{2}}{\ell^{2}}-\frac{q^{2}}{16\pi^{2}}\left[\ln|r|+\Gamma\left(0,\frac{r^{2}}{4\theta}\right)-\frac{1}{2}\Gamma\left(0,\frac{r^{2}}{2\theta}\right)\right].$$

(2.24)

Here, $\Gamma\left(0,x\right)$ denotes an upper incomplete gamma function.

In this section, we shall study the thermodynamic properties of the metric given by (2.15), which corresponds to a charged rotating noncommutative BTZ black hole. Although, the same for the uncharged case has been studied earlier in [38], however; to our knowledge, the thermodynamics of the charged case has not been explored yet. Recall that the Hawking temperature in the tunneling formalism, as described in Sec. 5.0.1, applies to the metric having a spherically symmetric form only. However, our metric (2.15) is not in the required form and, thus, it claims a coordinate transformation

$$\mathop{}\!\mathrm{\,d}\varphi\rightarrow\mathop{}\!\mathrm{\,d}\varphi^{\prime}+\frac{J}{2r^{2}-\theta B}\mathop{}\!\mathrm{\,d}t,$$

(5.10)

so that the transformed spherically symmetric metric in natural units ($8G=c=1$) takes the form

$$ds^{2}=-\left(\mathcal{G}(r)-M+\frac{\theta BJ^{2}}{8r^{4}}\right)\mathop{}\!\mathrm{\,d}t^{2}+\left(\mathcal{G}(r)-M+\frac{\theta BM}{2r^{2}}-\frac{\theta B}{2\ell^{2}}\right)^{-1}\mathop{}\!\mathrm{\,d}r^{2}+\left(1-\frac{\theta B}{2r^{2}}\right)r^{2}\mathop{}\!\mathrm{\,d}\varphi^{\prime 2},$$

(5.11)

where

$$\mathcal{G}(r)=\frac{r^{2}}{\ell^{2}}+\frac{J^{2}}{4r^{2}}-\mathcal{\pi}q^{2}\ln r-\frac{\theta B}{2\ell^{2}}+\frac{\theta B\pi q^{2}}{4r^{2}}.$$

(5.12)

Therefore, by following (5.6), the Hawking temperature acquires the form

$$T_{H}=\frac{\lambda(r_{+})}{2\pi\ell^{2}}\left[r_{+}^{2}-\theta B\lambda^{-1}(r_{+})\left(1-\lambda(r_{+})+\frac{\ell^{2}M}{2r_{+}^{2}}\right)+\mathcal{O}\left(\theta^{2}\right)\right]^{1/2},$$

(5.13)

with

$$\lambda(r_{+})=1-\frac{\ell^{2}J^{2}}{4r_{+}^{4}}-\frac{\pi q^{2}\ell^{2}}{2r_{+}^{2}}.$$

(5.14)

Thus, the total mass of the black hole (5.11) is computed from the condition that at the outer horizon $r_{+}$, the quantity, $\mathcal{G}(r)-M+\frac{\theta BM}{2r^{2}}-\frac{\theta B}{2\ell^{2}}$, vanishes, which implies that

$$M=\left(1-\frac{\theta B}{2r_{+}^{2}}\right)^{-1}\left(\mathcal{G}(r_{+})-\frac{\theta B}{2\ell^{2}}\right)=\mathcal{G}(r_{+})+\frac{\theta B}{2}\left(\frac{\mathcal{G}(r_{+})}{r_{+}^{2}}-\frac{1}{\ell^{2}}\right)+\mathcal{O}\left(\theta^{2}\right).$$

(5.15)

Now, by using (5.13) and (5.15), it is straightforward to obtain the entropy by using (5.9). However, the expression of the entropy being slightly lengthy, we do not present it explicitly. More importantly, our analysis in the present scenario does not require the explicit form of entropy. Therefore, we avoid such an expression, which, anyway does not carry significant importance in our context. A more important entity is the Hawking temperature provided by (5.13). The analysis of temperature is shown in Fig. 2 for two sets of values of the charge, angular momentum and the noncommutative parameter. The orange lines correspond to the noncommutative cases associated with the fixed value of the noncommutative parameter, whereas the purple lines show the same in the commutative limit $\theta\rightarrow 0$. As it turns out that the behavior of the black hole temperature in the noncommutative case is similar to the commutative one, we only obtain a correction of the temperature due to the noncommutativity. We also do not notice any restriction on the noncommutative parameter in the sense that the black hole is physical and behaves normally for all values of the noncommutative parameter.

Thermodynamics of (3.6) can be studied in the tunneling formalism but, now in this case, the temperature will be time-dependent. The line element (3.6) is not yet in a spherically symmetric form and, therefore, in order to apply the tunneling formalism, we apply the following coordinate transformation

$$\mathop{}\!\mathrm{\,d}\varphi\rightarrow\mathop{}\!\mathrm{\,d}\varphi^{\prime}+\frac{J}{2r^{2}-\theta B}\mathop{}\!\mathrm{\,d}t,\quad\mathop{}\!\mathrm{\,d}{r}\rightarrow\mathop{}\!\mathrm{\,d}{r}^{\prime}-\frac{\theta C}{2}\mathop{}\!\mathrm{\,d}{t}.$$

(5.16)

The line element up to the first-order correction in $\theta$ is, thus, modified as

$$ds^{2}=-\mathcal{F}(r^{\prime})\mathop{}\!\mathrm{\,d}t^{2}+\mathcal{H}^{-1}(r^{\prime})\mathop{}\!\mathrm{\,d}r^{\prime 2}+\left[r^{\prime 2}-\frac{\theta Ct}{2}\left(1+2r^{\prime}\right)\right]\mathop{}\!\mathrm{\,d}\varphi^{\prime 2},$$

(5.17)

where

	$\displaystyle\mathcal{F}(r^{\prime})=\frac{r^{\prime 2}}{\ell^{2}}-M+\frac{J^{2}}{4r^{\prime 2}}-\pi q^{2}\ln r^{\prime}-\frac{\theta Ct\lambda(r^{\prime}_{+})}{2\ell^{2}}(1+2r^{\prime})$		(5.18)
	$\displaystyle\mathcal{H}(r^{\prime})=\frac{r^{\prime 2}}{\ell^{2}}-M+\frac{J^{2}}{4r^{\prime 2}}-\pi q^{2}\ln r^{\prime}-\frac{\theta Ct}{2}\left[\frac{2(1+r^{\prime})}{\ell^{2}}-\frac{M}{r^{\prime 2}}-\frac{\pi q^{2}(1+2r^{\prime})}{2r^{\prime 2}}-\frac{J^{2}}{2r^{\prime 3}}\right].$		(5.19)

Since the radius of the horizon $r^{\prime}$ is obtained as a solution of $\mathcal{H}(r^{\prime})=0$, it is worthwhile to mention that, interestingly, it becomes time-dependent. Clearly, the Hawking temperature also turns out to be time-dependent

$$T_{H}=\frac{\lambda(r^{\prime}_{+})}{2\pi}\left[\frac{r_{+}^{{}^{\prime}2}}{\ell^{4}}-\frac{\theta Ct\lambda^{-1}(r^{\prime}_{+})}{2\ell^{2}}\left\{\frac{M}{r_{+}^{{}^{\prime}2}}+\frac{3J^{2}}{2r_{+}^{{}^{\prime}3}}+\frac{\pi q^{2}}{r^{\prime}_{+}}+\frac{2-2\lambda(r^{\prime}_{+})+2r^{\prime}_{+}}{\ell^{2}}\right\}+\mathcal{O}\left(\theta^{2}\right]\right]^{1/2}.$$

(5.20)

It should be noted that the time-dependence of temperature and horizon radius is not a surprising result, because the corresponding metric is non-static and non-stationary. These time-dependent thermodynamic quantities may give rise to dynamic characteristics of the Hawking radiation and evaporation of the black hole. To realize this, we can simplify (5.20) for the chargeless and nonrotating scenario by taking the limits $J\rightarrow 0$ and $q\rightarrow 0$. Thereafter, if we choose $\ell=10,\theta C=1$ and the Hawking temperature $T_{H}$ to be zero, the equation (5.20) will be left with two free parameters, $r_{+}$ and $t$. Therefore, we can easily obtain a set of possible values of the horizon radius after the evaporation is complete and the corresponding time taken to evaporate, which are, however, not possible to determine exactly unless we know how the radius of the horizon changes with time. Explicit expressions of time-dependent horizon radius and total mass of a dynamic noncommutative black hole are yet unexplored and out of our scope in the present study. This problem itself is a new and separate project that may require a longer time to explore. One of the possible ways to address the time-dependent mass is to follow the method prescribed in [55]; however, such a study in a noncommutative background may demand more sophistication and, thus, we keep it as a separate problem to be addressed in future.

The aim of this section is to study the thermodynamic properties of the metric obtained in (4.3) using the Lorentzian distribution function. However, before that, in order to introduce an interesting discussion, let us perform the same analysis for the metric (2.24) associated with the Gaussian distribution function. The semi-classical Hawking temperature as per (5.6) turns out to be

$$T_{H}=\frac{r_{+}}{2\pi\ell^{2}}\left[1-\frac{M\ell^{2}}{4\theta}e^{-r_{+}^{2}/4\theta}-\frac{q^{2}\ell^{2}}{32\pi^{2}r_{+}^{2}}\left(1+e^{-r_{+}^{2}/2\theta}-2e^{-r_{+}^{2}/4\theta}\right)\right],$$

(5.21)

whereas the total mass acquires the form

$$M=\frac{1}{\left(1-e^{-r_{+}^{2}/4\theta}\right)}\left[\frac{r_{+}^{2}}{\ell^{2}}-\frac{q^{2}}{16\pi^{2}}\left[\ln|r_{+}|+\Gamma\left(0,\frac{r_{+}^{2}}{4\theta}\right)-\frac{1}{2}\Gamma\left(0,\frac{r_{+}^{2}}{2\theta}\right)\right]\right].$$

(5.22)

Therefore, the entropy can be calculated easily from (5.9) as follows

$$S=4\pi\int_{r_{0}}^{r_{+}}\frac{1}{\left(1-e^{-\psi^{2}/4\theta}\right)}\mathop{}\!\mathrm{\,d}\psi.$$

(5.23)

The Hawking temperature for the noncommutative charged BTZ black hole with Lorentzian distribution given by (4.3) is computed from (5.6) as

$$T_{H}=\frac{r_{+}}{4\pi}\left[\frac{2}{\ell^{2}}-\frac{M\sqrt{\theta}}{(r_{+}^{2}+\theta)^{\frac{3}{2}}}-\frac{q^{2}}{16\pi^{2}r_{+}^{2}}\left(1-\frac{\sqrt{\theta}}{\sqrt{r_{+}^{2}+\theta}}\right)^{2}\right].$$

(5.24)

The total mass of the same being

$$M=\left(1-\frac{\sqrt{\theta}}{\sqrt{r_{+}^{2}+\theta}}\right)^{-1}\left[\frac{r_{+}^{2}}{\ell^{2}}-\frac{q^{2}}{8\pi^{2}}\left\{\ln(\sqrt{r_{+}^{2}+\theta}+\sqrt{\theta})-\frac{\ln\left(\left|r_{+}^{2}+\theta\right|\right)}{4}\right\}\right],$$

(5.25)

the entropy from (5.9) becomes

$$S=4\pi\int_{r_{0}}^{r_{+}}\frac{\sqrt{\psi^{2}+\theta}}{\sqrt{\psi^{2}+\theta}-\sqrt{\theta}}\mathop{}\!\mathrm{\,d}\psi.$$

(5.26)