Maxwell-Jüttner distribution for rigidly rotating flows in spherically symmetric spacetimes using the tetrad formalism

We consider rigidly rotating states in thermal equilibrium on static spherically symmetric spacetimes. Using the Maxwell-Jüttner equilibrium distribution function, constructed as a solution of the relativistic Boltzmann equation, the equilibrium particle flow four-vector, stress-energy tensor and the transport coefficients in the Marle model are computed. Their properties are discussed in view of the topology of the speed-of-light surface induced by the rotation for two classes of spacetimes: maximally symmetric (Minkowski, de Sitter and anti-de Sitter) and nonrotating black hole (Schwarzschild and Reissner-Nordström) spacetimes. To facilitate our analysis, we employ a nonholonomic comoving tetrad field, obtained unambiguously by applying a Lorentz boost on a fixed background tetrad.

Figure 1

Comparison between the mean of the Möller velocity $⟨g_{\varphi }⟩$ (2.35a) and of the modulus of the velocity $⟨v⟩$ (2.35b) as functions of the relativistic coldness $\zeta$. The limits (2.8) at small and large $\zeta$ are confirmed: $⟨g_{\varphi }⟩$ goes to $\frac{4}{5}$ at small $\zeta$ and is well approximated by $\sqrt{16/\pi \zeta }$ at large $\zeta$; while $⟨v⟩$ goes to 1 (the speed of light) as $\zeta \to 0$, while at large $\zeta$, it behaves like $\sqrt{8/\pi \zeta }$.

Figure 2

Plots of (a) the effective bulk viscosity $\tilde{\eta }\equiv a^2\eta /m$; (b) the thermal conductivity $\tilde{\lambda }\equiv a^2\lambda$; and (c) the shear viscosity $\tilde{\mu }\equiv a^2\mu /m$. Each plot shows two curves, corresponding to the cases when the mean velocity in Eq. (2.34) is taken to be the average of the Möller velocity $⟨g_{\varphi }⟩$ and the average of the particle velocity $⟨v⟩$, respectively. It can be seen that, in both cases, the coefficient of shear viscosity has a maximum located at ${\zeta }_{⟨g_{\varphi }⟩}=1.342$ and ${\zeta }_{⟨v⟩}=1.535$, having the values $\tilde{\eta }({\zeta }_{⟨g_{\varphi }⟩})=3.292\times {10}^{-4}$ and $\tilde{\eta }({\zeta }_{⟨v⟩})=3.554\times {10}^{-4}$, respectively.

Figure 3

The dependence of the energy density $E$ (top) and equation of state $w=P/E$ (bottom) on the inverse temperature ${\beta }^{-1}$, for various values of the mass $m$. The expressions for $E$ and $P$ can be found in Eqs. (3.21a) and (3.21b), respectively.

Figure 4

The SOL structure of (a) dS and (b) AdS. The vertical axis represents the coordinate $\tilde{z}\equiv \omega r\cos \theta$ along the rotation axis, while the horizontal axis represents the distance $\tilde{\rho }\equiv \omega r\sin \theta$ from the rotation axis. In the dS case, the rightmost line corresponding to $\tilde{\mathrm{\Omega }}=0$ represents the cosmological horizon, located at $\tilde{r}=\pi /2$. On AdS, no SOL forms when $\tilde{\mathrm{\Omega }}<1$. The Minkowski case, when $\rho ={\tilde{\mathrm{\Omega }}}^{-1}$, is not shown here.

Figure 5

The proper radial distance ${\tilde{s}}_{\mathrm{SOL}}$ (4.15) between the origin and the SOL in the equatorial plane ($\sin \theta =1$) with respect to $\tilde{\mathrm{\Omega }}=\mathrm{\Omega }/\omega$. The bottom, middle and top lines correspond to the cases $\epsilon =1$ (dS), $\epsilon =0$ (Minkowski) and $\epsilon =-1$ (AdS). In the case of Minkowski spacetime, we adopt the convention $\tilde{s}=s={\mathrm{\Omega }}^{-1}$ and $\tilde{\mathrm{\Omega }}=\mathrm{\Omega }$ (i.e. the parameter $\omega$ is immaterial in this case).

Figure 6

Dependence of the effective coefficient of bulk viscosity $\tilde{\eta }$ (2.33a) divided by its maximum value ${\tilde{\eta }}_{\max }$ on $\tilde{r}$ on (a) dS and (b) AdS spacetimes in the absence of rotation ($\tilde{\mathrm{\Omega }}=0$). Each curve corresponds to a different value of ${\zeta }_0=m{\beta }_0$, where ${\beta }_0\equiv \beta (r=0)$ is the inverse temperature at the coordinate origin. The local maxima exhibited by $\tilde{\eta }$ (highlighted by circular points) appear only when ${\zeta }_0$ is large (dS) or small (adS), its locations being given by Eqs. (4.16) and (4.17) for the dS and adS spaces, respectively.

Figure 7

Dependence of the effective coefficient of bulk viscosity $\tilde{\eta }$ (2.33a) in the equatorial plane ($\theta =\pi /2$) divided by its maximum value ${\tilde{\eta }}_{\max }$ on (a) dS, (b) Minkowski and (c), (d) AdS spacetimes. Each curve corresponds to a different value of $\tilde{\mathrm{\Omega }}$, while the parameter ${\zeta }_0=4$ for (a)–(c) and ${\zeta }_0=1$ for (d). It should be noted that when $\tilde{\mathrm{\Omega }}=1$, $\tilde{\eta }$ is constant in the equatorial plane of AdS space, cf. (4.14).

Figure 8

The dependence of ${\beta }^2/{\beta }_0^2$ on $\tilde{r}=r/2M$ in the equatorial plane $\sin \theta =1$ for (a)-(c) $\tilde{Q}$ fixed at (a) 0 (Schwarzschild space), (b) 0.5 and (c) 1.0 (extremal Reissner-Nordström space), for various values of $\tilde{\mathrm{\Omega }}=2M\mathrm{\Omega }$. In (d), $\tilde{\mathrm{\Omega }}$ is fixed at 0.4 and the charge is varied from $\tilde{Q}=0$ to $\tilde{Q}=1.0$. The regions where ${\beta }^2>0$ represent “allowed” regions (where the temperature is finite and well defined), while the points where $\beta =0$ represent horizons.

Figure 9

The SOL structure of the Reissner-Nordström spacetime for four values of $\tilde{Q}=Q/M$. The vertical axis represents the coordinate $\tilde{z}\equiv z/2M$ along the rotation axis, while the horizontal axis represents the distance $\tilde{\rho }=r\sin \theta /2M$ from the rotation axis. The contours represent the surfaces where $\beta =0$, i.e. either the black hole horizon (only the outer horizons are shown) or the rotation horizon (i.e. where the SOL induced by the rotation forms). The case (a) shows the horizon structure for the Schwarzschild space ($Q=0$), while the case (d) represents an extremal Reissner-Nordström black hole ($Q=M$).

Figure 10

Dependence of the ratio $\tilde{\eta }/{\tilde{\eta }}_{\max }$ (2.33a) of the effective coefficient of bulk viscosity divided by its maximum value, evaluated in the equatorial plane ($\theta =\pi /2$), with respect to the distance from the rotation axis for the Schwarzschild space-time ($\tilde{Q}=0$) in the following cases: (a) $\tilde{\mathrm{\Omega }}=0$ for various values of ${\zeta }_0$; (b) ${\zeta }_0=7$ for various values of $\tilde{\mathrm{\Omega }}$; (c) $\tilde{\mathrm{\Omega }}=0.36$ for various values of ${\zeta }_0$.

Figure 11

Same as Fig. 10 for the Reissner-Nordström black hole. (a) $(\tilde{\mathrm{\Omega }},{\zeta }_0)=(0,1.1)$; (b) $(\tilde{\mathrm{\Omega }},{\zeta }_0)=(0,4)$; (c) $(\tilde{\mathrm{\Omega }},{\zeta }_0)=(0.36,10)$, for various values of $\tilde{Q}=Q/M$.