Quantization of fermions on Kerr space-time

We study a quantum fermion field on a background nonextremal Kerr black hole. We discuss the definition of the standard black hole quantum states (Boulware, Unruh, and Hartle-Hawking), focussing particularly on the differences between fermionic and bosonic quantum field theory. Since all fermion modes (both particle and antiparticle) have positive norm, there is much greater flexibility in how quantum states are defined compared with the bosonic case. In particular, we are able to define a candidate Boulware-like state, empty at both past and future null infinity, and a candidate Hartle-Hawking-like equilibrium state, representing a thermal bath of fermions surrounding the black hole. Neither of these states have analogues for bosons on a nonextremal Kerr black hole and both have physically attractive regularity properties. We also define a number of other quantum states, numerically compute differences in expectation values of the fermion current and stress-energy tensor between two states, and discuss their physical properties.

Figure 1

Part of the Carter-Penrose diagram for the complete Kerr geometry, showing the future event horizon ${\mathcal{H}}^{+}$, past event horizon ${\mathcal{H}}^{-}$, future null infinity ${\mathcal{I}}^{+}$, and past null infinity ${\mathcal{I}}^{-}$. Region $I$ corresponds to the space-time exterior to the event horizon and is the region on which we study the quantum fermion field. Region IV will be required in Sec. 3 for defining some of our quantum states. A more complete Carter-Penrose diagram for the Kerr geometry can be found in Ref. 67.

Figure 2

The cross section of the stationary limit (shorter-dashed red curve) and speed-of-light (longer-dashed blue curve) surfaces, for $a<a_0=M\sqrt{2[\sqrt{2}-1]}$ (left), $a=a_0$ (center), and $a>a_0$ (right). In each case we have plotted cross sections on a plane of fixed azimuthal angle $\phi$. The axis of rotation of the black hole is a vertical line through the center of each diagram, and the equatorial plane a horizontal line through the center of each diagram. The black circle denotes the region inside the event horizon.

Figure 3

Examples of typical angular and radial mode functions. Plot (a) shows the spin-half spheroidal harmonics ${}_{1}S_{\Lambda }$ and ${}_{2}S_{\Lambda }$ for $\ell =\frac{5}{2}$, $m=\frac{1}{2}$ for a range of spheroidal couplings $a\omega =-2,-1.5,\dots ,1.5,2$. The symmetries (2.29, 2.30) are apparent. Plot (b) shows the radial functions ${}_{1}R_{\Lambda }$ and ${}_{2}R_{\Lambda }$ for the in modes, defined by boundary conditions (2.33), for $\ell =m=3/2$, $M\omega =0.4$, and two cases: $a/M=0$ (dashed lines) and $a/M=0.5$ (dotted lines). Plot (c) shows the radial functions for the up modes, defined by boundary conditions (2.34), with the same parameters.

Figure 4

Frequency integrals and mode-sum convergence. These plots show a typical integrand, $[1+\exp (\tilde{\omega }/T_H){]}^{-1}{t}_{\theta \theta }^{(\mathrm{in})}$ [where $t_{\theta \theta }$ is defined in Eq. (c22)], as a function of frequency $\omega$, for the in modes with $\frac{1}{2}\le \ell \le \frac{21}{2}$, $m\ge \ell -1$, in the special case $a=a_0\approx 0.910M$. The integrand is evaluated on the equatorial plane ($\theta =\pi /2$) at $r=$ (a) $1.6M$, (b) $1.8M$, and (c) $2.0M$. The mode sum in cases (a) and (b) appears to be convergent, whereas the sum in the case (c) does not seem to converge. We note that the speed-of-light surface intersects the equatorial plane at $r=2M$ in this case, so plot (c) indicates that this particular mode sum will diverge outside the speed-of-light surface. The physical implications of this result are discussed in Sec. 4c.

Figure 5

Ratio test to examine the divergence/regularity of expectation values of quantum states, for the stress-energy tensor component $⟨{\widehat{T}}_{\theta \theta }⟩$. The differences in expectation values for the six states defined in Eqs. (4.18, 4.19, 4.20, 4.21, 4.22, 4.23) are considered. In particular, these are (a) $⟨{\widehat{T}}_{\theta \theta }{⟩}^{U^{-}-B^{-}}$, (b) $⟨{\widehat{T}}_{\theta \theta }{⟩}^{CC{H}^{-}-B^{-}}$, (c) $⟨{\widehat{T}}_{\theta \theta }{⟩}^{B-B^{-}}$, (d) $⟨{\widehat{T}}_{\theta \theta }{⟩}^{H-B^{-}}$, (e) $⟨{\widehat{T}}_{\theta \theta }{⟩}^{H-B}$, and (f) $⟨{\widehat{T}}_{\theta \theta }{⟩}^{H-\tilde{B}}$. In each case, the ratio ${\rho }_{\ell }$ (4.30) is plotted for $\ell \sim 20$ as a function of $r$, $\theta$, where $z=r\cos \theta$ and $x=r\sin \theta$. The axis of rotation of the black hole is a vertical line through the center of each diagram, and the equatorial plane a horizontal line through the center of each diagram. The green dotted line is the speed-of-light surface; the purple dotted line the stationary limit surface (we use the value $a=a_0=M\sqrt{2[\sqrt{2}-1]}$ for which these two surfaces touch in the equatorial plane). The black circle is the region inside the event horizon. Divergent regions (where ${\rho }_{\ell }>1$) are blue (darker) and convergent regions (where ${\rho }_{\ell }<1$) are yellow (lighter).

Figure 6

Ratio test to examine the divergence/regularity of expectation values of quantum states, for the stress-energy tensor component $⟨{\widehat{T}}_{\theta \theta }⟩$. The differences in expectation values for the states defined in Eqs. (4.33, 4.34) are considered. In particular, these are (a) $⟨{\widehat{T}}_{\theta \theta }{⟩}^{H-U^{-}}$, (b) $⟨{\widehat{T}}_{\theta \theta }{⟩}^{\tilde{B}-B^{-}}$. The structure of the plots follows that in Fig. 5, and the same parameters are used.

Figure 7

The expectation values $⟨{\widehat{J}}^{\mu }{⟩}^{CC{H}^{-}-B^{-}}$ for components of the fermion current, multiplied by $\Delta$ (2.2). The expectation values have been computed using (4.14, 4.15, 4.16, 4.17) with $L=+1$ (for $L=-1$ the components have the same magnitude but the opposite sign). The expectation values are plotted on the vertical axis as functions of $(r,\theta )$, with $x=r\sin \theta$ and $z=r\cos \theta$. In the horizontal plane, positive values are shaded in red, while blue denotes negative values. We use the value $a=a_0=M\sqrt{2[\sqrt{2}-1]}$ for the rotation parameter of the Kerr black hole.

Figure 8

The expectation values $⟨{\widehat{T}}_{\mu \nu }{⟩}^{CC{H}^{-}-B^{-}}$ for components of the stress-energy tensor [multiplied by various powers of $\Delta$ (2.2); note that we do not claim that the power of $\Delta$ used necessarily corresponds to the rate of divergence of the components near the horizon], using the expressions (c15, c16, c17, c18, c19, c20, c21, c22) with $L=+1$ (for $L=-1$, all components have the same values). The parameters used and format of the plots are the same as in Fig. 7.

Figure 9

The rate of rotation ${\Omega }_{\mathrm{ZEFO}}$ of the zero energy flux observer (ZEFO) (4.42) for the expectation values $⟨{\widehat{T}}_{\mu \nu }{⟩}^{CC{H}^{-}-B^{-}}$ (left) and $⟨{\widehat{T}}_{\mu \nu }{⟩}^{H-\tilde{B}}$ (right). In both plots, we also show the angular velocity of the horizon ${\Omega }_H$ (2.8) (denoted “rigid rotation”), and on the left-hand plot we also show ${\Omega }_{\mathrm{ZAMO}}$ (4.38) and ${\Omega }_{\mathrm{Carter}}$ (4.39). All quantities are plotted as functions of the coordinates $(r,\theta )$. The expectation value in the right-hand plot diverges on the speed-of-light surface, which can be seen in the numerical noise in the red (non-flat) surface. As in previous figures, we use the value $a=a_0=M\sqrt{2[\sqrt{2}-1]}$ for the rotation parameter of the Kerr black hole.