Photon escape in the extremal Kerr black hole spacetime

Let us consider the behavior of $b_{i}(r;q)$ to determine the range of $b$ in which photons with $q\geq 0$ satisfy the necessary condition to escape from $r=r_{*}$ to infinity. We can see the typical shape of $b_{i}$ as gray curves in Figs. 2(a) and 2(b) for $0\leq q<3$, in Figs. 2(c)–2(e) for $3\leq q<27$, and in Fig. 2(f) for $q\geq 27$. Gray regions denote forbidden regions of photon motion. Orange and blue regions represent the parameter range of $b$ where photons satisfy a necessary condition for escape with $\sigma_{r}=+$ and $\sigma_{r}=-$, respectively.

In the case $r_{1}<r_{\mathrm{H}}<r_{*}\leq r_{2}$, photons initially emitted outward (i.e., $\sigma_{r}=+$) with $b^{\mathrm{s}}_{2}<b\leq b_{1}^{*}$ can escape [see orange region in Fig. 2(a)], and photons initially emitted inward (i.e., $\sigma_{r}=-$) with $2<b<b_{1}^{*}$ also can escape [see blue region in Fig. 2(a)], where

In the case $r_{1}<r_{\mathrm{H}}<r_{2}<r_{*}$, photons initially emitted outward (i.e., $\sigma_{r}=+$) with $b_{2}^{*}\leq b\leq b_{1}^{*}$ can escape [see orange region in Fig. 2(b)], and photons initially emitted inward (i.e., $\sigma_{r}=-$) with $b_{2}^{*}<b<b^{\mathrm{s}}_{2}$ or $2<b<b_{1}^{*}$ also can escape [see blue region in Fig. 2(b)].

In the case $r_{\mathrm{H}}<r_{*}\leq r_{1}$, only photons initially emitted outward (i.e., $\sigma_{r}=+$) with $b^{\mathrm{s}}_{2}<b<b^{\mathrm{s}}_{1}$ can escape [see orange region in Fig. 2(c)].

In the case $r_{\mathrm{H}}\leq r_{1}<r_{*}\leq r_{2}$, photons initially emitted outward (i.e., $\sigma_{r}=+$) with $b^{\mathrm{s}}_{2}<b\leq b_{1}^{*}$ can escape [see orange region in Fig. 2(d)], and photons initially emitted inward (i.e., $\sigma_{r}=-$) with $b^{\mathrm{s}}_{1}<b<b_{1}^{*}$ also can escape [see blue region in Fig. 2(d)].

In the case $r_{\mathrm{H}}\leq r_{1}<r_{2}<r_{*}$, photons initially emitted outward (i.e., $\sigma_{r}=+$) with $b_{2}^{*}\leq b\leq b_{1}^{*}$ can escape [see orange region in Fig. 2(e)], and photons initially emitted inward (i.e., $\sigma_{r}=-$) with $b_{2}^{*}<b<b^{\mathrm{s}}_{2}$ or $b^{\mathrm{s}}_{1}<b<b_{1}^{*}$ also can escape [see blue region in Fig. 2(e)].

For $q\geq 27$, the allowed region is disconnected. Therefore, if $r_{*}$ is in the outer allowed region, photons must have $b_{2}^{*}\leq b\leq b_{1}^{*}$ and always can escape [see orange and blue region in Fig. 2(f)].

Let us summarize the necessary condition for photon escape in the ($b,q$) parameter region for a fixed $r_{*}$. To perform it, we introduce four ranges of $r_{*}$:

where $\tilde{r}\equiv r_{2}(3)=3.95137...$ is the largest solution of $q_{\mathrm{SPO}}(r)=3$. In addition, we introduce two specific values of $q$,

When $q=q_{\mathrm{max}}$, the functions $b_{i}^{*}$ coincide with each other, and their value is

When $q>q_{\mathrm{max}}$, the position $r_{*}$ enters the forbidden region. Therefore, the maximum value of $q$ is limited by $q_{\mathrm{max}}$. The necessary conditions for photon escape are summarized in Table 1.¹¹1Note that in Refs. Ogasawara:2019mir ; Ogasawara:2020frt , the photon escape regions were considered only for the case (i), i.e., $r_{\mathrm{H}}<r_{*}<3$. Here, we identify them for the entire range of $r_{*}$. It is useful to visualize the necessary condition for photon escape in the $b$-$q$ plane; see Fig. 3. The blue, red, purple, and brown curves denote $b=b_{1}^{*}(r_{*};q)$, $b_{2}^{*}(r_{*};q)$, $b^{\mathrm{s}}_{1}(q)$, and $b^{\mathrm{s}}_{2}(q)$, respectively. The black segment denotes $b=b_{1}(r_{\mathrm{H}};q)=2$ and $q\in[0,3]$. The orange regions show the parameter regions where photons initially emitted outward (i.e., $\sigma_{r}=+$) satisfy the necessary condition for escape. The cyan regions show the parameter regions where photons initially emitted both outward and inward (i.e., $\sigma_{r}=\pm$) satisfy the necessary condition for escape. Figures 3(i)–3(iv) correspond to Tables 1(i)–1(iv), respectively.

Let us further restrict the above necessary condition for photon escape by the non-negativity of $\Theta$, i.e., Eq. (19). The common region of these conditions provides the necessary and sufficient parameter region in which a photon can escape from $(r,\theta)=(r_{*},\theta_{*})$ to infinity. We call it the escapable region. An example of the escapable region is seen in Fig. 4. The green curve denotes $\Theta=0$, and the other curves and colored regions are the same as in Fig. 3. We can find that the escapable region corresponds to the regions in Fig. 3 further restricted by the condition (19).

We identify the escapable region for $q<0$. The negative $q$ together with the non-negativity of $\Theta$ at $\theta=\theta_{*}$ leads to

This implies that $|b|<1$ for $q<0$, and the minimum value of $q$ is given at $b=0$ as

For $q<0$, the SPOs are not relevant to photon escape because they do not exist. Therefore, we only focus on $R\geq 0$, or equivalently,

The right-hand side is positive for all $|b|<1$ and $r>1$. Hence, the allowed region (35) contains the entire parameter region (33). This means that any radial turning point no longer appears for $q<0$. Finally, we conclude that photons with $q<0$ can escape to infinity if they are initially emitted outward (i.e., $\sigma_{r}=+$) and take the range (33). Figure 4 shows an example of the escapable region (see the orange region of $q<0$).

We introduce four critical polar angles $\theta_{1}$, $\theta_{2}$, $\theta_{3}$, and $\theta_{\mathrm{m}}$, at which the classification of the escapable region varies qualitatively. Solving $\Theta=0$ for $\theta$, we obtain a solution

see Appendix B.

The first special point is $(b,q)=(-2,27)$, where $b^{\mathrm{s}}_{1}$ and $b^{\mathrm{s}}_{2}$ coincide with each other. We define $\theta_{1}$ as $\theta_{*}$ at which $-B(\theta_{*};q)$ passes through $(-2,27)$, i.e., $B(\theta_{1};27)=2$ [see the black dot in Fig. 5(i)]. Then, $\theta_{1}$ is given by

When $\theta_{*}<\theta_{1}$, $b^{\mathrm{s}}_{2}<-B$ holds in the range $q\leq 27$. This implies that the minimum value of $b$ in the escapable region for $q\in[q_{\mathrm{min}},27]$ is always $-B$.

The second special point is $(b,q)=(2,3)$, where $r_{1}=r_{\mathrm{H}}=1$ and $b^{\mathrm{s}}_{1}=b_{1}(r_{\mathrm{H}};q)=2$. We define $\theta_{2}$ as $\theta_{*}$ at which $B(\theta_{*};q)$ passes through $(2,3)$, i.e., $B(\theta_{2};3)=2$ [see the black dot in Fig. 5(ii)]. Then, $\theta_{2}$ is given by

When $\theta_{*}<\theta_{2}$, $B<2$ holds in the range $q\leq 3$. This implies that the maximum value of $b$ in the escapable region for $q\in[q_{\mathrm{min}},3]$ is always $B$.

The third special point is $(b,q)=(b^{\mathrm{s}}_{2}(3),3)$, where $b^{\mathrm{s}}_{2}(3)\simeq-6.71$. We define $\theta_{3}$ as $\theta_{*}$ at which $-B(\theta_{*};q)$ passes through $(b^{\mathrm{s}}_{2}(3),3)$, i.e., $-B(\theta_{3};3)=b^{\mathrm{s}}_{2}(3)$ [see the black dot in Fig. 5(iii)]. Then, $\theta_{3}$ is given by

When $\theta_{*}<\theta_{3}$, $b^{\mathrm{s}}_{2}<-B$ holds in the range $q\leq 3$. This implies that the minimum value of $b$ in the escapable region for $q\in[q_{\mathrm{min}},3]$ is always $-B$.

The fourth special point is $(b,q)=(b_{\mathrm{m}},q_{\mathrm{max}})$, where $b_{1}^{*}$ and $b_{2}^{*}$ coincide with each other. We define $\theta_{\mathrm{m}}$ as $\theta_{*}$ at which $-B(\theta_{*};q)$ passes through $(b_{\mathrm{m}},q_{\mathrm{max}})$, i.e., $-B(\theta_{\mathrm{m}};q_{\mathrm{max}})=b_{\mathrm{m}}(r_{*})$ [see the black dot in Fig. 5(iv)]. Then, $\theta_{\mathrm{m}}$ is given by

Note that we only need to consider $\theta_{\mathrm{m}}$ for $r_{*}\geq 3$ because it does not contribute to specifying the escapable region when $r_{*}<3$. The critical angle $\theta_{\mathrm{m}}$ depends on $r_{*}$ and monotonically decreases with $r_{*}$ in the range

When $\theta_{*}<\theta_{\mathrm{m}}$, $b_{2}^{*}<-B$ holds in the range $q\leq q_{\mathrm{max}}$. This implies that the minimum value of $b$ in the escapable region for $q\in[q_{\mathrm{min}},q_{\mathrm{max}}]$ is always $-B$.

We introduce five critical values $\bar{q}$, $q^{\mathrm{t}}_{\pm}$, and $q^{\mathrm{s}}_{\pm}$, as the values of $q$ at the intersections of $b=2$, $b_{i}^{*}(r_{*};q)$, $b^{\mathrm{s}}_{i}(q)$, and $\pm B(\theta_{*};q)$, at which the classification of the parameter ranges varies qualitatively.

We define $\bar{q}$ as the value of $q$ at the intersection of $b=2$ and $b=B(\theta_{*};q)$ in the range $0\leq q\leq 3$, and we denote the intersection as P$(\bar{q})$ (see the black dot in Fig. 6). Then, $\bar{q}$ is given by²²2This $\bar{q}(\theta_{*})$ is expressed as $q_{1}(\theta_{*})$ in Ref. Ogasawara:2020frt

which only appears for $\theta_{*}\in[\theta_{2},\pi/2)$ and monotonically decreases with $\theta_{*}$ in the range

When $q<\bar{q}$, $B<b_{1}^{*}$ holds. This implies that the maximum value of $b$ in the escapable region for $q<\bar{q}$ is always $B$.

We define $q^{\mathrm{t}}_{+}$ as the value of $q$ at the intersection of $b=b_{1}^{*}(r_{*};q)$ and $b=B(\theta_{*};q)$, and we denote the intersection as P$(q^{\mathrm{t}}_{+})$ (see the blue dot in Fig. 6). On the other hand, we define $q^{\mathrm{t}}_{-}$ as the value of $q$ at the intersection of $b=b_{2}^{*}(r_{*};q)$ and $b=-B(\theta_{*};q)$ for $\theta_{*}\geq\theta_{\mathrm{m}}$ and $b=b_{1}^{*}(r_{*};q)$ and $b=-B(\theta_{*};q)$ for $\theta_{*}\leq\theta_{\mathrm{m}}$, and we denote the intersection as P$(q^{\mathrm{t}}_{-})$ (see the red dot in Fig. 6). Then, $q^{\mathrm{t}}_{\pm}$ are given by³³3These $q^{\mathrm{t}}_{+}(r_{*},\theta_{*})$ and $q^{\mathrm{t}}_{-}(r_{*},\theta_{*})$ are expressed as $q_{2}(r_{*},\theta_{*})$ and $q_{6}(r_{*},\theta_{*})$, respectively, in Ref. Ogasawara:2020frt .

where $q^{\mathrm{t}}_{-}$ only appears for $r_{*}>2$. For $\theta_{*}=\theta_{\mathrm{m}}$, $b=b_{i}^{*}$ and $b=-B$ coincide with each other at $q=q_{\mathrm{max}}$. Note that $q^{\mathrm{t}}_{+}<q^{\mathrm{t}}_{-}$ holds for all $r_{*}$ and $\theta_{*}$. Figures 7(i) and 7(ii) show the values of $q^{\mathrm{t}}_{+}$ and $q^{\mathrm{t}}_{-}$ in the $r_{*}$-$\theta_{*}$ parameter space, respectively.

We define $q^{\mathrm{s}}_{+}$ as the value of $q$ at the intersection of $b=b^{\mathrm{s}}_{1}(q)$ and $b=B(\theta_{*};q)$, and we denote the intersection as P$(q^{\mathrm{s}}_{+})$ (see the purple dot in Fig. 6). On the other hand, we define $q^{\mathrm{s}}_{-}$ as the value of $q$ at the intersection of $b=b^{\mathrm{s}}_{2}(q)$ and $b=-B(\theta_{*};q)$ for $\theta_{*}\geq\theta_{1}$ and $b=b^{\mathrm{s}}_{1}(q)$ and $b=-B(\theta_{*};q)$ for $\theta_{*}\leq\theta_{1}$, and we denote the intersection as P$(q^{\mathrm{s}}_{-})$ (see the brown dot in Fig. 6). Then, $q^{\mathrm{s}}_{\pm}$ are given by⁴⁴4This $q^{\mathrm{s}}_{+}(\theta_{*})$ is expressed as $q_{3}(\theta_{*})$, and $q^{\mathrm{s}}_{-}(\theta_{*})$ is expressed as $q_{4}(\theta_{*})$ for $\theta_{*}\geq\theta_{1}$ and as $q_{5}(\theta_{*})$ for $\theta_{*}<\theta_{1}$, respectively, in Ref. Ogasawara:2020frt .

where $q^{\mathrm{s}}_{+}$ only appears for $\theta_{*}\in(0,\theta_{2})$ and monotonically decreases with $\theta_{*}$ in the range

As $\theta_{*}$ increases from $0$ to $\pi/2$, $q^{\mathrm{s}}_{-}$ begins with $q^{\mathrm{s}}_{-}(0)=11+8\sqrt{2}$, monotonically increases to the maximum value $27$ at $\theta_{*}=\theta_{1}$, and monotonically decreases to $q^{\mathrm{s}}_{-}(\pi/2)=0$. For $\theta_{*}=\theta_{1}$, $b=b^{\mathrm{s}}_{i}$ and $b=-B$ coincide with each other at $q=27$.

In addition, we define $\mathrm{P}(q_{*})$ as a point in the $b$-$q$ plane that represents $(b_{1}^{*}(r_{*};q_{*}),q_{*})$ for $r_{*}<3$ and $(b_{2}^{*}(r_{*};q_{*}),q_{*})$ for $3\leq r_{*}\leq 4$ (see the gray dot in Fig. 6).

These are summarized in Table 2.

It is worth noting that $q^{\mathrm{t}}_{\pm}$ and $q_{*}$ are not always involved in classification; i.e., the intersections P($q^{\mathrm{t}}_{+}$), P($q^{\mathrm{t}}_{-}$), and P$(q_{*})$ are not always involved in classification.

When $q^{\mathrm{t}}_{+}>q_{*}$ for $r_{*}<3$, the intersection P($q^{\mathrm{t}}_{+}$) does not contribute to specifying the escapable region [see the gray region in Fig. 7(i)]. On the other hand, when $q^{\mathrm{t}}_{+}\leq q_{*}$ for $r_{*}<3$, the intersection P($q^{\mathrm{t}}_{+}$) is a special point, which contributes to specifying the escapable region. For $r_{*}\geq 3$, the intersection P($q^{\mathrm{t}}_{+}$) always contributes to it. The colored region in Fig. 7(i) represents the parameter region where $q^{\mathrm{t}}_{+}$ contributes to specifying the escapable region.

For the same reason as for $q^{\mathrm{t}}_{+}$, the relative values of $q^{\mathrm{t}}_{-}$ and $q_{*}$ determine whether $q^{\mathrm{t}}_{-}$ contributes to specifying the escapable region. In the case of $r_{*}<3$, only when $q^{\mathrm{t}}_{-}\leq q_{*}$ for $\theta_{*}<\theta_{1}$, the intersection P($q^{\mathrm{t}}_{-}$) is included in the escapable region. In the case of $r_{*}\geq 3$, when $q^{\mathrm{t}}_{-}\leq q_{*}$ for $\theta_{*}<\theta_{1}$ and when $q^{\mathrm{t}}_{-}\geq q_{*}$, the intersection P$(q^{\mathrm{t}}_{-})$ is included. The colored region in Fig. 7(ii) represents the parameter region where $q^{\mathrm{t}}_{-}$ contributes to specifying the escapable region.

In the case of $r_{*}<3$, when $q^{\mathrm{t}}_{+}\leq q_{*}\leq q^{\mathrm{t}}_{-}$ and when $q^{\mathrm{t}}_{-}\leq q_{*}$ for $\theta_{*}\geq\theta_{1}$, the point P($q_{*}$) contributes to specifying the escapable region. In the case of $r_{*}\geq 3$, the point P($q_{*}$) contributes to it only when $q^{\mathrm{t}}_{-}\leq q_{*}$ for $\theta_{*}\geq\theta_{1}$. The blue region in Fig. 8 represents the parameter region where $q_{*}$ contributes to specifying the escapable region.