# Photon emissions from NHEK and near-NHEK equatorial emitters

Haopeng Yan1 Zezhou Hu2 Minyong Guo3∗ Bin Chen1,2,4

###### Abstract

We consider isotropic and monochromatic photon emissions from equatorial
emitters moving along future-directed timelike geodesics in the near-horizon
extremal Kerr (NHEK) and near-horizon near-extremal Kerr (near-NHEK) regions,
to asymptotic infinity. We obtain numerical results for the photon escaping
probability (PEP) and derive analytical expressions for the maximum observable
blueshift (MOB) of the escaping photons, both depending on the emission radius
and the emitter’s proper motion. In particular, we find that for all anti-
plunging or deflecting emitters that can eventually reach to asymptotic
infinity, the PEP is greater than $50\%$ while for all plunging emitters the
PEP is less than $55\%$, and for the bounded emitters in the (near-)NHEK
region, the PEP is always less than $59\%$. In addition, for the emitters on
unstable circular orbits in the near-NHEK region, the PEP decreases from
$55\%$ to $50\%$ as the orbital radius decreases from the one of the innermost
stable circular orbit to the one of the horizon. Furthermore, we show how the
orientation of the emitter’s motion along the radial or azimuthal direction
affects the PEP and the MOB of the emitted photons.

1Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing
100871, P. R. China

3Department of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871,
P.R. China

3 Department of Physics, Beijing Normal University, Beijing 100875, P. R.
China

4 Collaborative Innovation Center of Quantum Matter, No.5 Yiheyuan Rd, Beijing
100871, P. R. China

Email:haopeng.yan, z.z.hu, minyongguo<EMAIL_ADDRESS>
$~{}~{}~{}~{}~{}~{}\ast$ Corresponding author.

##  1 Introduction

Since the release of the first image of the supermassive black hole M87* by
the Event Horizon Telescope (EHT) collaboration [1] in 2019, there have been
ongoing progresses in the community to explore physical information of the
black hole. This M87* image matches very well with the simulation of a
rotating Kerr black hole based on the Einstein’s theory of General Relativity.
Very recently, an improved image with polarization information was released,
which would help us to explore the structure of the magnetic fields near the
black hole [2]. Understanding these observations relies not only on the study
of photon emissions and propagations in the black hole backgrounds, but also
on the rich physics in the surroundings of the black holes, such as the
magnetohydrodynamics in plasmas, the distribution of magnetic fields and even
the nature of dark matter halo. Consequently the images of the black holes may
allow us to investigate various problems in the strong gravitational field
region.

Besides the silhouette of a black hole shadow, there are other possible
observational signatures with interesting features, for examples, the image of
an hot spot surrounding a black hole [3, 4, 5, 6], light rings (LRs) [7, 8, 9]
and photon rings [10, 11, 12, 13, 14, 15]. Theoretically, as the first step,
one has to understand clearly the photon emissions around the black hole,
especially near the black hole horizon. On the other hand, studying the photon
emissions from the near-horizon area is important to explore various high-
energy processes[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] happened in the
near-horizon region (inside the ergosphere).

Recently, the photon escaping probability (PEP) and maximum observable
blueshift (MOB)111We use the term “MOB” for the reason that the photons with
maximum net blueshift can always reach at infinity and thus are observable in
principle. But in practice these escaping photons may not be observed by a
distant observer since the MOB could be negative and the PEP could also be
vanishingly small. of photon emissions from near-horizon sources around
rotating black holes had been studied by several groups. The PEP of a zero-
angular momentum source (ZAMS) near a Kerr-Newmann black hole was first
studied in [27], and later the study was generalized to the Kerr-Sen spacetime
in [28]. The PEP of a ZAMS approaching the event horizon is zero for a non-
extremal Kerr black hole and tends to $\sim 29\%$ for an extremal Kerr black
hole. Moreover, the PEP and MOB of photon emissions from equatorial (stable)
circular orbiters near a Kerr black hole was studied in [29] and [30, 31] with
different methods clarifying the photon escaping condition. The PEP for a
circular orbiters increases as the orbital radius increases from the innermost
stable circular orbit (ISCO), while it decreases as the black hole spin
parameter increases at a given orbital radius. Remarkably, the value of PEP
for an emitter sitting at the ISCO of an extremal Kerr black hole is still
about $55\%$, and the proper motion of the emitter blueshifts the energy of
the escaped photons. Very recently, photon emissions from the plunging
orbiters222The appearance of a bright hot spot falling into a non-spinning
black hole and its relation with LRs were studied in Ref. [9]. starting from
the ISCO of a black hole with arbitrary spin was studied in [32], and it was
found that the PEP is always more than $50\%$ for the emitter at approximately
halfway between the ISCO and the event horizon. Moreover, the effects of the
proper motion of a plunging emitter on the PEP and the MOB were discussed in
[32]. While these studies applied to the emitter of arbitrary radius and the
black hole of arbitrary spin, special attentions had been paid to the near-
horizon and near-extremal cases in [30] since such cases can be dealt with
analytically. In a previous paper [33], we reinvestigated the photon emissions
from a ZAMS near a Kerr black hole by adapting the method proposed in [30] and
explored the observability of the very deep region in the near-horizon throat
of a high-spin black hole. To that end, we analytically computed the PEP and
the MOB of the ZAMSs both in the near-horizon extremal Kerr (NHEK) and the
near-horizon near-extremal Kerr (near-NHEK) geometries and found that the PEP
tends to $\sim 13\%$ at the radius of the innermost photon shell.

In this paper, we would like to study the effects of the emitter’s proper
motion on the PEP and the MOB of the escaping photons, with the ZAMS as a
“static” reference [27, 33]. We will focus on the equatorial emitters that
follow timelike geodesics in the (near-)NHEK333We use “(near-)NHEK” for both
NHEK and near-NHEK. geometries. We will consider not only the marginally
plunging emitters starting from the ISCO [32] but also other possible plunging
emitters. Moreover, we will consider the emitters with other geodesic motions
as well, including the anti-plunging, the deflecting and the bounded motions.
In [32], only the marginally plunging emitters which has the critical angular
momentum $l_{s}=l_{\text{ISCO}}$ and critical energy
$\omega_{s}=\omega_{\text{ISCO}}$ had been studied, while the black holes were
allowed to have arbitrary spin. Here, instead, we will only focus on the high-
spin black hole but consider the emitters with more possible values of $l_{s}$
and $\omega_{s}$. Note that in the high-spin case the energy $\omega_{s}$ of
the emitter is constrained to be near the superradiant bound
$\omega_{\ast}=l_{s}/2M$ with the corrections appearing at order
$\mathcal{O}(\epsilon^{q})$ [see Eqs. (2.10) and (2.11)]. The timelike
geodesics in the (near-)NHEK regions were well classified in [34, 35, 36, 37]
based on the conserved quantities (2.18) in the (near-)NHEK geometries. As we
restrict the emitters on the equatorial plane, the Carter constant for these
emitters is set to zero, $Q_{s}=0$, thus each orbit of an emitter is
specifically labeled by its (near-)NHEK energy $E_{s}$, angular momentum
$L_{s}$ and radial orientation $s_{r}$. We will consider the behaviors of the
PEP and the MOB of the escaping photons emitted from the orbit labeled by
$(L_{s},E_{s},s_{r})$ and study how they are affected when each of these
parameters varies. Our main results are summarized in Eqs. (4.18), (4.19),
(4.20) and in Fig. 2. In addition, we provide detailed illustrations and
discussions for the results in Figs. 3, 4, 5 and 6.

The remaining part of this paper is organized as follows. In Sec. 2, we
briefly introduce the Kerr geometry and the geodesics in it, as well as its
near-horizon and (near-)extremal limits. In Sec. 3, we define the problem of
photon emissions from an arbitrary near-horizon equatorial emitter, and find
the equations to read the PEP and the MOB. In Sec. 4, we perform the
computations for the (near-)NHEK emitters. In Secs. 5 and 6, we display our
main results with plots and discuss them in detail. In Sec. 7, we summarize
and conclude this work.

##  2 Geometry and geodesics

In this section, we review the timelike and the null geodesics in the Kerr,
NHEK and near-NHEK geometries [38, 34, 35, 36, 37]. In particular, we review
the classifications of the (near-)NHEK geodesics based on their constants of
motions. The timelike geodesics provide the orbits for the photon emitters
with various motions. For the null geodesics, we focus on the unstable
spherical photon orbits, which correspond to the threshold between the
escaping photons and the captured ones.

###  2.1 Kerr geometry and geodesics

The Kerr metric in the Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ is
given by

$ds^{2}=-\frac{\Sigma\Delta}{\Xi}dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}+\frac{\Xi\sin^{2}\theta}{\Sigma}\mathopen{}\mathclose{{}\left(d\phi-\frac{2Mar}{\Xi}dt}\right)^{2},$
(2.1)

where

$\Delta=r^{2}-2Mr+a^{2},\qquad\Sigma=r^{2}+a^{2}\cos^{2}\theta,\qquad\Xi=(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta.$
(2.2)

This metric describes a rotating black hole with a mass $M$ and a spin
parameter $a\equiv J/M$. The event horizons of the black hole are located at
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$.

A free particle possesses four conserved quantities of motion: the mass $\mu$,
the energy $\omega$, the axial angular momentum $l$ and the Carter constant
$Q$. Using the Hamilton-Jacobi method, one can derive the four-momentum
$p^{\mu}$ of this particle [38]

$\displaystyle p^{r}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma}\pm_{r}\sqrt{\mathcal{R}(r)},$ (2.3)
$\displaystyle p^{\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma}\pm_{\theta}\sqrt{\Theta(\theta)},$ (2.4)
$\displaystyle p^{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma}\mathopen{}\mathclose{{}\left[\frac{a}{\Delta}[\omega(r^{2}+a^{2})-al]+\frac{l}{\sin^{2}\theta}-a\omega}\right],$
(2.5) $\displaystyle p^{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma}\mathopen{}\mathclose{{}\left[\frac{r^{2}+a^{2}}{\Delta}[\omega(r^{2}+a^{2})-al]+a(l-a\omega\sin^{2}\theta)}\right],$
(2.6)

where

$\displaystyle\mathcal{R}(r)$ $\displaystyle=$
$\displaystyle[\omega(r^{2}+a^{2})-al]^{2}-\Delta[Q+(l-a\omega)^{2}+\mu^{2}r^{2}],$
(2.7) $\displaystyle\Theta(\theta)$ $\displaystyle=$ $\displaystyle
Q+a^{2}(\omega^{2}-\mu^{2})\cos^{2}\theta-l^{2}\cot^{2}\theta,$ (2.8)

are the radial and angular potentials, respectively, and $\pm_{r}$ and
$\pm_{\theta}$ denote the radial and polar orientations, respectively.

For a massive particle, we have $\mu>0$ and $p^{\mu}=\mu\frac{\partial
x^{\mu}}{\partial\tau}$ with $\tau$ being the proper time. For a massless
particle (photon), we have $\mu=0$ and $p^{\mu}=\frac{\partial
x^{\mu}}{\partial\tau}$ with $\tau$ being an affine parameter. For null
geodesics, the energy $\omega$ may be scaled out from the expressions of
$p^{\mu}$ under a reparameterization and it is convenient to introduce a pair
of impact parameters for the null geodesics:

$\lambda=\frac{l}{\omega},\qquad\eta=\frac{Q}{\omega^{2}}.$ (2.9)

###  2.2 (Near-)NHEK geometry and geodesics

In order to study the near-horizon geometry of a (near-)extremal Kerr black
hole, it is convenient to consider the near-extremal limit [34, 39, 40]

$a=M\sqrt{1-(\epsilon\kappa)^{2}},\qquad 0<\epsilon\ll
1,\qquad\kappa\,\,\,\text{finite},$ (2.10)

and work with the Bardeen-Horowitz coordinates [41, 34, 39, 40]

$T=\epsilon^{q}\frac{t}{2M},\qquad
R=\frac{r-M}{\epsilon^{q}M},\qquad\Phi=\phi-\frac{t}{2M},\qquad 0<q\leq 1.$
(2.11)

Under the transformations (2.11) and taking the limit $\epsilon\rightarrow 0$
for $0<q<1$, the Kerr metric (2.1) yields the NHEK metric [41]

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
2M^{2}\Gamma\mathopen{}\mathclose{{}\left[-R^{2}dT^{2}+\frac{dR^{2}}{R^{2}}+d\theta^{2}+\Lambda^{2}(d\Phi+RdT)^{2}}\right],$
(2.13) $\displaystyle\Gamma(\theta)=\frac{1+\cos^{2}\theta}{2},\hskip
12.91663pt\Lambda(\theta)=\frac{2\sin\theta}{1+\cos^{2}\theta}.$

In the NHEK geometry, the event horizon is mapped to $R=0$ and the ISCO is
mapped to $R=2^{1/3}$. However, since the NHEK metric is invariant under a
dilation $(R,T)\rightarrow(\varepsilon R,T/\varepsilon)$ for an arbitrary
constant $\varepsilon$, the ISCO can actually stay at anywhere in the
intrinsic NHEK geometry and $R=2^{1/3}$ gets meaningful only after the NHEK
region being glued onto the asymptotic far region [35, 40].

Under the transformations (2.11) and taking the limit $\epsilon\rightarrow 0$
for $q=1$, we find that the Kerr metric (2.1) yields the near-NHEK metric444We
will use the same coordinates $(T,R,\theta,\Phi)$ [Eq. (2.11)] to describe the
near-NHEK geometry which has already been used for the NHEK metric (2.13). In
the following, we will also use the same notations for the conserved
quantities $(E,L,C,\lambda,\eta)$ both in the NHEK and near-NHEK geometries.
However, these should be distinguishable from the context. [42, 40]

$ds^{2}=2M^{2}\Gamma\mathopen{}\mathclose{{}\left[-(R^{2}+\kappa^{2})dT^{2}+\frac{dR^{2}}{(R^{2}+\kappa^{2})}+d\theta^{2}+\Lambda^{2}(d\Phi+RdT)^{2}}\right].$
(2.14)

In the near-NHEK geometry, the event horizon is mapped to $R=\kappa$ and the
ISCO is mapped to $R=\infty$ [35, 40].

Note that the NHEK metric (2.13) is equivalent to the near-NHEK metric (2.14)
with zero near-horizon temperature, $\kappa=0$ [42, 34]. Therefore, in the
following we will work with Eq. (2.14) for both the NHEK case ($0<q<1$ and
$\kappa=0$) and near-NHEK case ($q=1$ and $\kappa\neq 0$), for simplicity.

The four-momentum $p^{\mu}$ of a free particle in the (near-)NHEK limit
becomes [34, 36]

$\displaystyle p^{R}$ $\displaystyle=$
$\displaystyle\pm_{r}\frac{\sqrt{\mathcal{R}_{n}(R)}}{2M^{2}\Gamma},$ (2.15a)
$\displaystyle p^{\theta}$ $\displaystyle=$
$\displaystyle\pm_{\theta}\frac{\sqrt{\Theta_{n}(\theta)}}{2M^{2}\Gamma},$
(2.15b) $\displaystyle p^{\Phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2M^{2}\Gamma}\mathopen{}\mathclose{{}\left(-\frac{R(E+LR)}{R^{2}-\kappa^{2}}+\frac{L}{\Lambda^{2}}}\right),$
(2.15c) $\displaystyle p^{T}$ $\displaystyle=$
$\displaystyle\frac{1}{2M^{2}\Gamma}\mathopen{}\mathclose{{}\left(\frac{E+LR}{R^{2}-\kappa^{2}}}\right),$
(2.15d)

where

$\displaystyle\mathcal{R}_{n}(R)$ $\displaystyle=$ $\displaystyle-
CR^{2}+2ELR+E^{2}+(C+L^{2})\kappa^{2},$ (2.16)
$\displaystyle\Theta_{n}(\theta)$ $\displaystyle=$ $\displaystyle
C+\mathopen{}\mathclose{{}\left(1-\frac{1}{\Lambda^{2}}}\right)L^{2}-2M^{2}\Gamma\mu^{2},$
(2.17)

are respectively the radial and angular potentials in the NHEK region, with
$C$, $E$ and $L$ being the (near-)NHEK conserved quantities: the Casimir, the
energy and the angular momentum, respectively. The (near-)NHEK conserved
quantities are related to the Kerr conserved quantities by [34, 36]

$l=L,\qquad\omega=\frac{L}{2M}(1+\epsilon^{q}\frac{E}{L}),\qquad
Q=C+\frac{3}{4}L^{2}-\mu^{2}M^{2}.$ (2.18)

For the null geodesics we have $\mu=0$, then we may define a pair of shifted
impact parameters

$\lambda_{0}=\frac{E}{L},\qquad\eta_{0}=\frac{C}{L^{2}},$ (2.19)

which are related to the Kerr impact parameters (2.9) by555This means that the
photon emissions are near the superradiant bound. Similar parameters have also
been introduced in [3].

$\lambda=2M(1-\epsilon^{q}\lambda_{0})\qquad\eta=M^{2}(3+4\eta_{0}).$ (2.20)

It is convenient to introduce a critical angular momentum [35, 36]

$L^{\ast}=l^{\ast}=\frac{2}{\sqrt{3}}\sqrt{M^{2}\mu^{2}+Q},$ (2.21)

which is for the geodesic at the innermost stable spherical orbit (ISSO). In
terms of this critical angular momentum, the Casimir $C$ can be rewritten as

$C=-\frac{3}{4}[L^{2}-(L^{\ast})^{2}].$ (2.22)

We will restrict to the future-directed geodesics which require $p^{T}>0$ [Eq.
(2.15d)], i.e.,

$E+LR>0.$ (2.23)

Next, by analyzing the root structures of the radial potential (2.16), the
equatorial orbits are classified in the phase space $(L,E)$ regarding to their
radial motions [34, 36]. The classifications666Note that there was a forgotten
region for the near-NHEK deflecting orbits in [36], which was corrected in
[37]. Note also that we use “anti-plunging” for the “outward” orbits in [36].
for the NHEK orbits and for the near-NHEK orbits are collected in Table 1 and
Table 2, respectively, where $R_{\pm}$ and $R_{0}$ are the root(s) for the
corresponding cases

$R_{\pm}=\frac{E}{C}L\pm\frac{1}{|C|}\sqrt{(L^{2}+C)(E^{2}+\kappa^{2}C)},\qquad
R_{0}=-\frac{E^{2}+\kappa^{2}L^{2}}{2EL}.$ (2.24)

The meaning of these distinct orbits are as follows [35, 36]. Each ISSO orbits
at a constant radius. A plunging orbit enters the NHEK geometry from the
(near-)NHEK boundary and eventually crosses the horizon in finite affine time.
An anti-plunging orbit emerges out of the horizon and eventually leaves the
(near-)NHEK boundary in finite affine time. A deflecting orbit falls in from
the (near-)NHEK boundary, bounces off at the radial turning point, and returns
to the boundary in finite affine time. Marginal (plunging/anti-
plunging/deflecting) orbits exist only in the NHEK geometry and leave the NHEK
geometry in infinite affine time. A bounded orbit stays in the (near-)NHEK
region between the horizon and the radial turning point. The anti-plunging and
deflecting orbits can escape from the (near-)NHEK to infinity while the
plunging and the bounded ones can not. Note that all near-NHEK spherical
orbits are unstable while all NHEK spherical orbits are ISSO [36].

Table 1: future-directed geodesics in NHEK [34, 36]. Classification | $L$ | $E$ | $\pm_{r}$ | Range
---|---|---|---|---
ISSO | $L=L^{\ast}$ | $E=0$ | $\pm 1$ | $0\leq R\leq\infty$
Marginal | $L>L^{\ast}$ | $E=0$ | $\pm 1$ | $0\leq R\leq\infty$
Plunging | $L\geq L^{\ast}$ | $E>0$ | $-1$ | $0\leq R\leq\infty$
anti-plunging | $L\geq L^{\ast}$ | $E>0$ | $+1$ | $0\leq R\leq\infty$
Deflecting | $L>L^{\ast}$ | $E<0$ | $\pm 1$ | $R_{+}\leq R\leq\infty$
| $-L^{\ast}<L<L^{\ast}$ | $E>0$ | $\pm 1$ | $0\leq R\leq R_{+}$
Bounded | $L=-L^{\ast}$ | $E>0$ | $\pm 1$ | $0\leq R\leq R_{0}$
| $L<-L^{\ast}$ | $E>0$ | $\pm 1$ | $0\leq R\leq R_{-}$
Table 2: future-directed geodesics in near-NHEK [34, 36]. Classification | $L$ | $E$ | $\pm_{r}$ | Range
---|---|---|---|---
Spherical | $L>L^{\ast}$ | $E=-\kappa\sqrt{-C}$ | $\pm 1$ | $R=\frac{\kappa L}{\sqrt{-C}}$
Plunging | $L=L^{\ast}$ | $E\geq 0$ | $-1$ | $\kappa\leq R\leq\infty$
| $L>L^{\ast}$ | $E>-\kappa\sqrt{-C}$ | $-1$ | $\kappa\leq R\leq\infty$
anti-plunging | $L=L^{\ast}$ | $E\geq 0$ | $+1$ | $\kappa\leq R\leq\infty$
| $L>L^{\ast}$ | $E>-\kappa\sqrt{-C}$ | $+1$ | $\kappa\leq R\leq\infty$
Deflecting | $L>L^{\ast}$ | $E<-\kappa\sqrt{-C}$ | $\pm 1$ | $R_{+}\leq R\leq\infty$
| $-L^{\ast}<L<L^{\ast}$ | $E>-\kappa L$ | $\pm 1$ | $\kappa\leq R\leq R_{+}$
Bounded | $L=L^{\ast}$ | $-\kappa L<E<0$ | $\pm 1$ | $\kappa\leq R\leq R_{0}$
| $L=-L^{\ast}$ | $E>-\kappa L$ | $\pm 1$ | $\kappa\leq R\leq R_{0}$
| $L<-L^{\ast}$ | $E>-\kappa L$ | $\pm 1$ | $\kappa\leq R\leq R_{-}$

##  3 General setup

In this section, we consider the photon emissions from an equatorial source
near a Kerr black hole and discuss the near-horizon and near-extremal limits
of this problem. In the following, we use $k^{\mu}$ and $p^{\mu}$ for the
four-momentums of an emitter and a photon, respectively, and we use the
notation that the quantities with a subscript $``s"$ are for an emitter
(source) while the quantities without subscript are for the photons. In
addition, we let $s_{r/\theta}=\pm_{r/\theta}=\pm 1$ denote the orientations
of the emitters and let $\sigma_{r/\theta}=\pm_{r/\theta}=\pm 1$ denote the
orientations of the photons.

We consider photon emissions from the equatorial emitters that moves along
timelike geodesics. For such emitters we have $\theta_{s}=\pi/2$, thus
$Q_{s}=0$ and

$L^{\ast}_{s}=l^{\ast}_{s}=\frac{2}{\sqrt{3}}M\mu.$ (3.1)

The photon emissions are described by null geodesics, which have

$L^{\ast}=l^{\ast}=\frac{2}{\sqrt{3}}\sqrt{Q}.$ (3.2)

We will first introduce a pair of local emission angles of the photons in the
emitter’s local rest frame (LRF), which are expressed in terms of the impact
parameters of the photons. Next, we consider the unstable spherical photon
orbits, which make up the photon shell and correspond to critical emission
angles on the emitter’s sky. The critical emission angles will line up into a
closed critical curve on the emitter’s sky, which allows us to distinguish the
photons captured by the central black hole from those escaping to asymptotic
infinity [30]. We then define the PEP, whose computation is based on
integrating over the interior region of the critical curve, and we find the
expression for the MOB.

###  3.1 Local emission angles on the emitter’s sky

We use $x^{\mu}$ with $\mu=0,1,2,3$ to represent both the Boyer-Lindquist
coordinates $(t,r,\theta,\phi)$ for the Kerr metric (2.1) and the Bardeen-
Horowitz coordinates $(T,R,\theta,\Phi)$ for the NHEK (2.13) and near-NHEK
(2.14) metrics.

In terms of the components of the spacetime metric $g_{\mu\nu}$, the locally
nonrotating frame (LNRF) is given by [43]

$e_{(0)}=\zeta(\partial_{0}+\varsigma\partial_{3}),\qquad
e_{(1)}=\xi_{1}\partial_{1},\qquad e_{(2)}=\xi_{2}\partial_{2},\qquad
e_{(3)}=\xi_{3}\partial_{3},$ (3.3)

where

$\zeta=\sqrt{\frac{g_{33}}{g_{03}^{2}-g_{00}g_{33}}},\qquad\varsigma=-\frac{g_{03}}{g_{33}},\qquad\xi_{1}=\frac{1}{\sqrt{g_{11}}},\qquad\xi_{2}=\frac{1}{\sqrt{g_{22}}},\qquad\xi_{3}=\frac{1}{\sqrt{g_{33}}}.$
(3.4)

The LRF of an equatorial emitter can be obtained by performing a Lorentz
transformation on the LNRF, which is given by

$\displaystyle\sigma_{[0]}$ $\displaystyle=$
$\displaystyle\gamma[e_{(0)}+v^{(1)}e_{(1)}+v^{(3)}e_{(3)}]|_{x^{i}=x^{i}_{s}},$
(3.5a) $\displaystyle\sigma_{[1]}$ $\displaystyle=$
$\displaystyle\frac{1}{v}[v^{(3)}e_{(1)}-v^{(1)}e_{(3)}]|_{x^{i}=x^{i}_{s}}$
(3.5b) $\displaystyle\sigma_{[2]}$ $\displaystyle=$ $\displaystyle
e_{(2)}|_{x^{i}=x^{i}_{s}},$ (3.5c) $\displaystyle\sigma_{[3]}$
$\displaystyle=$
$\displaystyle\gamma[ve_{(0)}+\frac{1}{v}(v^{(1)}e_{(1)}+v^{(3)}e_{(3)})]|_{x^{i}=x^{i}_{s}},$
(3.5d)

where $v$ and $v^{(i)}$ are the 3-velocity and its components of the emitter
relative to the LNRF, and $\gamma$ is the boost factor. For an emitter at
$x^{\mu}_{s}$ with momentum $k^{\mu}$, these are given by

$v^{(i)}=\frac{k^{\mu}e_{\mu}^{(i)}}{k^{\mu}e_{\mu}^{(0)}}\Bigg{|}_{x^{i}=x^{i}_{s}},\quad(i=1,2,3),\qquad
v=\sqrt{(v^{(1)})^{2}+(v^{(3)})^{2}},\qquad\gamma=\frac{1}{\sqrt{1-v^{2}}}.$
(3.6)

In order to study photon emissions from an equatorial emitter, we can define a
pair of local emission angles $(\alpha,\beta)$ on the emitter’s sky (see Fig.
1)[30, 27],

$\alpha\equiv\arccos\mathopen{}\mathclose{{}\left[\frac{p_{s}^{[3]}}{p_{s}^{[0]}}}\right]\in[0,\pi],\qquad\beta\equiv\arcsin\mathopen{}\mathclose{{}\left[\frac{p_{s}^{[1]}}{\sqrt{(p_{s}^{[1]})^{2}+(p_{s}^{[2]})^{2}}}}\right]\in[-\frac{\pi}{2},\frac{\pi}{2}],$
(3.7)

where $p^{[a]}_{s}=p^{\mu}\sigma_{\mu}^{[a]}|_{x^{i}=x_{s}^{i}}$ are the
components of the photon’s four-momentum in its LRF. Note that the setup is
symmetric about the equatorial plane and only $\sigma_{\theta}^{2}$ appears in
the local angles, therefore we may only focus on a half sphere of the
emitter’s sky by choosing either $\sigma_{\theta}=1$ or $\sigma_{\theta}=-1$
in $p_{s}^{[2]}$. Then the half sphere is uniquely parameterized by
$(\alpha,\beta)$.

Figure 1: Emitter’s sky parameterized by local emission angles $\alpha$ and
$\beta$.

###  3.2 Photon shell and critical emission angles

The photon shell consists of unstable spherical photon orbits in the Kerr
spacetime, which satisfy

$\mathcal{R}(\tilde{r})=\mathcal{R}^{\prime}(\tilde{r})=0.$ (3.8)

By solving these equations, we can obtain the critical photon impact
parameters [38]

$\displaystyle\tilde{\lambda}(\tilde{r})$ $\displaystyle=$ $\displaystyle
a+\frac{\tilde{r}}{a}\mathopen{}\mathclose{{}\left[\tilde{r}-\frac{2(\tilde{r}^{2}-2M\tilde{r}+a^{2})(\tilde{r})}{\tilde{r}-M}}\right],$
(3.9a) $\displaystyle\tilde{\eta}(\tilde{r})$ $\displaystyle=$
$\displaystyle\frac{\tilde{r}^{3}}{a^{2}}\mathopen{}\mathclose{{}\left[\frac{4M(\tilde{r}^{2}-2M\tilde{r}+a^{2})(\tilde{r})}{(\tilde{r}-M)^{2}}-\tilde{r}}\right].$
(3.9b)

We have $\tilde{\eta}(\tilde{r})\geq 0$ for the photons crossing the
equatorial plane, then these orbits lie in the range
$\tilde{r}_{1}\leq\tilde{r}\leq\tilde{r}_{2}$, where

$\displaystyle\tilde{r}_{1}$ $\displaystyle=$ $\displaystyle
2M\mathopen{}\mathclose{{}\left[1+\cos\mathopen{}\mathclose{{}\left(\frac{2}{3}\arccos\mathopen{}\mathclose{{}\left[-\frac{a}{M}}\right]}\right)}\right],$
(3.10) $\displaystyle\tilde{r}_{2}$ $\displaystyle=$ $\displaystyle
2M\mathopen{}\mathclose{{}\left[1+\cos\mathopen{}\mathclose{{}\left(\frac{2}{3}\arccos\mathopen{}\mathclose{{}\left[\frac{a}{M}}\right]}\right)}\right].$
(3.11)

Hereafter, the quantities adorned with a tilde are evaluated for the critical
photon emissions.

Plugging (3.9) into (3.7) gives the critical emission angles which line up
into a closed critical curve on an emitter’s sky [30]

$\mathcal{C}=\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\tilde{\alpha}(\tilde{r})\,,\tilde{\beta}(\tilde{r})}\right)\bigg{|}\tilde{r}_{1}<\tilde{r}<\tilde{r}_{2}}\right\\}.$
(3.12)

This critical curve separates the emitter’s sky into two distinct regions. To
distinguish these regions, it is useful to introduce the “direction to the
black hole center” on the emitter sky [Eq. (3.7)] which corresponds to [30]

$\lambda=\eta=0,\qquad\sigma_{r}=-1.$ (3.13)

Then we call the region containing the “direction to the black hole center”
the captured region, and call the other complementary one the escaping region.

In the extremal limits (2.10), the intrinsic (near-)NHEK calculations for
unstable spherical photon orbits by solving
$\mathcal{R}_{n}(\tilde{R})=\mathcal{R}^{\prime}_{n}(\tilde{R})=0$ are not
sufficient to resolve the whole photon shell described by (3.9). The reason is
that the whole photon shell stretches over various near-horizon limits for a
near-extremal black hole [30]. In the extremal limit $\epsilon\to 0$, the
radii of the innermost photon shell (3.10) and the outermost photon shell
(3.11) become

$\displaystyle\tilde{r}_{1}$ $\displaystyle=$ $\displaystyle
M\mathopen{}\mathclose{{}\left[1+\epsilon\tilde{R}_{1}+\mathcal{O}(\epsilon^{2})}\right],$
(3.14) $\displaystyle\tilde{r}_{2}$ $\displaystyle=$ $\displaystyle
4M\mathopen{}\mathclose{{}\left[1+\mathcal{O}(\epsilon^{2})}\right],$ (3.15)

where $\tilde{R}_{1}=2\kappa/\sqrt{3}$. We can see that the radius
$\tilde{r}_{1}$ is in the near-NHEK $p=1$ region while the radius
$\tilde{r}_{2}$ is in the Extreme Kerr $``p=0"$ region [see Eq. (2.11)] [34,
30]. Therefore, one must rely on the calculations in the Kerr geometry to
obtain the whole photon shell. By expanding the critical impact parameters of
the Kerr photon shell [Eqs. (3.9)] in $\epsilon$ under the transformations
[30]

$a=M\sqrt{1-(\epsilon\kappa)^{2}},\qquad
r_{s}=M(1+\epsilon^{q}R_{s}),\qquad\tilde{r}=M(1+\epsilon^{p}\tilde{R}),$
(3.16)

we can obtain the shifted critical impact parameters
$(\tilde{\lambda}_{0},\tilde{\eta}_{0})$. For the NHEK case we choose $q=2/3$
and for the near-NHEK case we have $q=1$. We will discuss the expansions for
the (near-)NHEK cases in Sec. 4.

###  3.3 Photon escaping probabilities and net blueshift

We assume that the emitter emits monochromatic photons isotropically in its
LRF (3.5). The redshift factor $g$ and net blueshift $z$ of a photon that
reaches to asymptotic infinity are defined by

$g\equiv\frac{\omega}{p_{s}^{[0]}},\qquad z\equiv\frac{g-1}{g}.$ (3.17)

Let $\mathcal{A}_{e}/\mathcal{A}_{c}$ be the area of the photon
escaping/captured region in the emitter’s sky of unit radius, respectively.
The photon escaping probability is defined by [27, 30]

$\mathcal{P}_{e}\equiv\frac{\mathcal{A}_{e}}{4\pi}=\frac{1-\mathcal{A}_{c}}{4\pi}.$
(3.18)

In order to compute the PEP, we introduce a pair of planar polar coordinates
[31]

$\rho=\sqrt{2(-\cos\alpha+1)},\qquad\varphi=\frac{\pi}{2}+\beta,$ (3.19)

such that the area element $d\mathcal{A}$ on the $(\rho,\varphi)$-plane is
equal to the area element on the sphere of the emitter’s sky $d\Omega$,

$d\mathcal{A}=\rho d\rho\wedge d\varphi=\sin\alpha d\alpha\wedge
d\beta=d\Omega.$ (3.20)

Regarding to the position of the direction to the black hole center on the
emitter’s sky, the area of the interior region of the closed critical curve
(3.12) may correspond to either $\mathcal{A}_{e}$ or $\mathcal{A}_{c}$, which
can be computed by [30]

$\mathcal{A}_{in}=\int_{in}\tilde{\rho}d\tilde{\rho}d\tilde{\varphi}=\int_{in}\frac{1}{2}\tilde{\rho}^{2}d\tilde{\varphi}.$
(3.21)

##  4 Photon emissions from (near-)NHEK emitters

In Sec. 2.2, we have reviewed the (near-)NHEK geometries and the geodesics for
the massive particles and for the photons. In this section, we compute the PEP
and the MOB for the photon emissions from the emitters at (near-)NHEK radius
$R_{s}$, whose motions are characterized by $(L_{s},E_{s},s_{r})$. The
relevant formulae are just defined in Sec. 3. We have $q=2/3$ in the Bardeen-
Horowitz coordinates [Eq. (2.11)] for the NHEK geometry, and we have $q=1$ in
the Bardeen-Horowitz coordinates for the near-NHEK geometry.

###  4.1 Critical emission angles and critical curve

In the last section, we obtained the LRF [Eq. (3.5)] and the local emission
angles [Eq. (3.7)] of an equatorial emitter at arbitrary radius around a black
hole of arbitrary spin. Now we compute the LRF and the local emission angles
for a (near-)NHEK emitter. At the emitter’s position, the LNRF (3.3) in the
(near-)NHEK geometry is given by

$e_{(T)}=\frac{\partial_{T}-R_{s}\partial_{\Phi}}{M\sqrt{R_{s}^{2}-\kappa^{2}}},\qquad
e_{(R)}=\frac{\sqrt{R_{s}^{2}-\kappa^{2}}}{M}\partial_{R},\qquad
e_{(\theta)}=\frac{1}{M}\partial_{\theta},\qquad
e_{(\Phi)}=\frac{1}{2}\partial_{\Phi}.$ (4.1)

The components of 3-velocity [Eq. (3.6)] of an emitter relative to the LNRF
are

$v^{(R)}=\frac{s_{r}\sqrt{\mathcal{R}_{n}(R_{s})}}{E_{s}+L_{s}R_{s}},\qquad
v^{(\Phi)}=\frac{L_{s}\sqrt{R_{s}^{2}-\kappa^{2}}}{2(E_{s}+L_{s}R_{s})},$
(4.2)

where $\mathcal{R}_{n}$ is defined in (2.16). Then the local emission angles
[Eq. (3.7)] of an emitter are given by

$\displaystyle\alpha(\lambda_{0},\eta_{0})$ $\displaystyle=$
$\displaystyle\arccos\mathopen{}\mathclose{{}\left[\frac{-2v^{2}\mathopen{}\mathclose{{}\left(\lambda_{0}+R_{s}}\right)+2\sigma_{r}\sqrt{\mathcal{R}_{n0}(R_{s})}v^{(R)}+v^{(\Phi)}\sqrt{R_{s}^{2}-\kappa^{2}}}{v\mathopen{}\mathclose{{}\left[2\mathopen{}\mathclose{{}\left(\lambda_{0}+R_{s}}\right)-2\sigma_{r}v^{(R)}\sqrt{\mathcal{R}_{n0}(R_{s})}-v^{(\Phi)}\sqrt{R_{s}^{2}-\kappa^{2}}}\right]}}\right],$
(4.3a) $\displaystyle\beta(\lambda_{0},\eta_{0})$ $\displaystyle=$
$\displaystyle\arcsin\mathopen{}\mathclose{{}\left[\frac{2\sigma_{r}v^{(\Phi)}\sqrt{\mathcal{R}_{n0}(R_{s})}-v^{(R)}\sqrt{R_{s}^{2}-\kappa^{2}}}{\sqrt{4v^{2}\mathopen{}\mathclose{{}\left(R_{s}^{2}-\kappa^{2}}\right)\Theta_{n0}(\frac{\pi}{2})+\mathopen{}\mathclose{{}\left[v^{(R)}\sqrt{R_{s}^{2}-\kappa^{2}}-2\sigma_{r}v^{(\Phi)}\sqrt{\mathcal{R}_{n0}(R_{s})}}\right]{}^{2}}}}\right],\,\,\,$
(4.3b)

where

$\displaystyle\mathcal{R}_{n0}(R)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{R}_{n}(R)}{L^{2}}=-\eta_{0}(R^{2}-\kappa^{2})+2\lambda_{0}R+\lambda_{0}^{2}+\kappa^{2},$
(4.4) $\displaystyle\Theta_{n0}(\frac{\pi}{2})$ $\displaystyle=$
$\displaystyle\frac{\Theta_{n}(\theta)}{L^{2}}\Big{|}_{\theta=\frac{\pi}{2}}=\eta_{0}+\frac{3}{4}.$
(4.5)

As discussed in Sec. 3.2, the (near-)NHEK critical parameters
$(\tilde{\lambda}_{0},\tilde{\eta}_{0})$ [Eq. (2.20)] can be obtained by
expanding the Kerr critical impact parameters $(\tilde{\lambda},\tilde{\eta})$
[Eq. (3.9)]. To be specific, let us consider the expansions (3.16) for the
emitter’s radius $r_{s}$ with $q=2/3$ or $1$, and for the photon shell radii
$\tilde{r}$ with $0<p\leq 1$. Then, to the leading order as
$\epsilon\rightarrow 0$, we obtain

$\displaystyle\tilde{\lambda}_{0}^{+}=\frac{1}{2}\tilde{R}^{2},\qquad$
$\displaystyle\tilde{\eta}_{0}^{+}=0,\qquad$ $\displaystyle\text{for}\quad
p=p^{+}=\frac{q}{2},$ (4.6a)
$\displaystyle\tilde{\lambda}_{0}^{-}=-\frac{\kappa^{2}}{\tilde{R}},\qquad$
$\displaystyle\tilde{\eta}_{0}^{-}=-\frac{\kappa^{2}}{\tilde{R}^{2}},\qquad$
$\displaystyle\text{for}\quad p=p^{-}=1,$ (4.6b)
$\displaystyle\tilde{\lambda}_{0}^{m+}=\infty,\qquad$
$\displaystyle\tilde{\eta}_{0}^{m+}=0,\qquad$ $\displaystyle\text{for}\quad
p=p^{m+}\in(0,\frac{q}{2}),$ (4.6c)
$\displaystyle\tilde{\lambda}_{0}^{m-}=0,\qquad$
$\displaystyle\tilde{\eta}_{0}^{m-}=0,\qquad$ $\displaystyle\text{for}\quad
p=p^{m-}\in(\frac{q}{2},1),$ (4.6d)

where $\tilde{R}(+)\in(0,\infty)$ and $\tilde{R}(-)\in[\tilde{R}_{1},\infty)$.
Here, the superscripts “$\pm$” and “$m$” represent the outer/inner and middle
limits of the photon shell, respectively. The critical emission angles and the
critical curve (3.12) can be obtained by plugging the critical impact
parameters (4.6) into the local emissions angles (4.3) with
$\sigma_{r}=\text{sign}(\tilde{r}-r_{s})$. Note that the limits of $p=p^{\pm}$
could complete the critical curve, while those of $p=p^{m\pm}$ only
contributes the points that connect the $p^{\pm}$ curves [30, 33].

Note also that, if we take the transformation $\tilde{R}^{2}\rightarrow
R_{s}\tilde{R}^{2}$ for $(\tilde{\alpha}^{+},\tilde{\beta}^{+})$, then a NHEK
critical curve is independent of $R_{s}$ once the 3-velocity components (4.2)
are fixed, and so does the PEP. This is due to the dilational symmetry of the
NHEK geometry. In contrast, a nonzero near-horizon temperature $\kappa$ in the
near-NHEK geometry breaks the dilational symmetry. Therefore, in contrast with
the NHEK case, a near-NHEK critical curve does depend on $R_{s}$ even when the
3-velocity components (4.2) are fixed, and so does the PEP.

###  4.2 Photon escaping probability

Let us now compute the PEP (3.18) for the photon emissions from a free
(near-)NHEK emitter labeled by $(L_{s},E_{s},s_{r})$ [Eq. (2.18)] at radius
$R_{s}$. To do so, we need to compute the area inside the critical curve
(3.12) in the $(\rho,\varphi)$-plane (3.19). From (2.20) and (3.13), the
“direction to the black hole center” is expressed in terms of the shifted
critical impact parameters as

$\lambda_{0}=\infty,\qquad\eta_{0}=-\frac{3}{4}.$ (4.7)

These parameters correspond to a point in the plane which is contained in the
photon captured region. Then the interior area $\mathcal{A}_{in}$ [Eq. (3.21)]
can be computed numerically

$\mathcal{A}_{in}=\int_{in}\frac{1}{2}\tilde{\rho}^{2}d\tilde{\varphi}=\mathopen{}\mathclose{{}\left|\int_{\tilde{R}_{1}}^{\infty}\tilde{\rho}^{2}(\tilde{\alpha}^{-})\frac{d\tilde{\varphi}(\tilde{\beta}^{-})}{d\tilde{R}}d\tilde{R}+\int_{0}^{\infty}\tilde{\rho}^{2}(\tilde{\alpha}^{+})\frac{d\tilde{\varphi}(\tilde{\beta}^{+})}{d\tilde{R}}d\tilde{R}}\right|.$
(4.8)

Note that we need to integrate over both of the half spheres with
$\sigma_{\theta}=\pm 1$ on the emitter’s sky, thus the factor $1/2$ in front
of $\tilde{\rho}^{2}$ has been canceled out. We will present the results in
Secs. 5 and 6, and discuss them in detail there.

###  4.3 Maximum observable blueshift

As $\epsilon\rightarrow 0$, the redshift factor (3.17) for the photon
emissions from a (near-)NHEK emitter becomes

$g(\lambda_{0},\eta_{0})=\frac{\sqrt{R_{s}^{2}-\kappa^{2}}}{\gamma\mathopen{}\mathclose{{}\left[2(R_{s}+\lambda_{0})-v^{(\Phi)}\sqrt{R_{s}^{2}-\kappa^{2}}-2\sigma_{r}v^{(R)}\sqrt{\mathcal{R}_{\kappa
0}(R_{s})}}\right]}.$ (4.9)

The MOB $z_{\text{mob}}$ is obtained when the redshift factor reaching its
maximum value $g_{\text{max}}$. In order to find the maximum value for the
redshift, we analyze the ranges of the impact parameters
$(\lambda_{0},\eta_{0})$ for the photons that can reach to infinity777The MOB
of the marginally plunging emitters from the ISCO was discussed in [32] by
analyzing the corresponding parameters of the photon emissions.. We have
reviewed the null geodesics in the (near-)NHEK region in Sec. 2.2 and
summarized the classifications in Table 1 and Table 2 for the NHEK and near-
NHEK cases, respectively. For the future-directed photon emissions we have
$L^{\ast}=\frac{2}{\sqrt{3}}\sqrt{Q}\geq 0$ and $\lambda_{0}>-R$, and for the
photons arrived at infinity we have888Note that $\sigma_{r}$ will flip from
$-1$ to $+1$ at the turning point. $\sigma_{r}=1$.

In the NHEK region we always have $r_{s}>\tilde{r}_{1}$. In the near-NHEK
region we may either have $r_{s}>\tilde{r}_{1}$ or have
$r_{s}\leq\tilde{r}_{1}$. For $r_{s}>\tilde{r}_{1}$, both the reflecting and
the anti-plunging photon trajectories can reach to infinity, while for
$r_{s}\leq\tilde{r}_{1}$, only the anti-plunging photon trajectories can reach
to infinity. Then from Tables 1 and 2 we have $-1\leq\eta_{0}\leq 0$ and
$\lambda_{0}>\lambda_{0-}(\eta_{0})$ for the anti-plunging orbits, and
$\lambda_{0}<\lambda_{0-}(\eta_{0})$ for the deflecting orbits, where

$\lambda_{0-}(\eta_{0})=-\kappa\sqrt{-\eta_{0}}.$ (4.10)

Moreover, the positivity of the potentials (4.4) and (4.5)requires

$\lambda_{0}\geq\lambda_{0t}(\eta_{0})=-R_{s}+\sqrt{(1+\eta_{0})(R_{s}^{2}-\kappa^{2})},\qquad\eta_{0}\geq-\frac{3}{4}.$
(4.11)

For the photon emissions from the infalling emitters with $s_{r}=-1$ in Eq.
(4.2), we have $\partial_{\eta_{0}}g>0$ and $\partial_{\lambda_{0}}g<0$. Note
also that we have $\lambda_{0}>\lambda_{0t}$ for $r_{s}>\tilde{r}_{1}$ and
$\lambda_{0}>\lambda_{0-}$ for $r_{s}\leq\tilde{r}_{1}$. Then we obtain

$\displaystyle
g_{\text{max},i}=\begin{cases}g_{i}[\lambda_{0t}(-3/4),-3/4],\quad&\text{if
}r_{s}>\tilde{r}_{1},\\\ g_{i}[\lambda_{0-}(-3/4),-3/4],\quad&\text{if
}r_{s}\leq\tilde{r}_{1},\end{cases}$ (4.12)

where

$\displaystyle g_{i}[\lambda_{0t}(-3/4),-3/4]$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma[1-v^{(\Phi)}]}=\frac{\sqrt{3}L_{s}^{\ast}\sqrt{R^{2}_{s}-\kappa^{2}}}{(2R_{s}-\sqrt{R_{s}^{2}-\kappa^{2}})L_{s}+2E_{s}},$
(4.13) $\displaystyle g_{i}[\lambda_{0-}(-3/4),-3/4]$ $\displaystyle=$
$\displaystyle\frac{\sqrt{R_{s}^{2}-\kappa^{2}}}{\gamma\mathopen{}\mathclose{{}\left[2R_{s}-\sqrt{3}\kappa-v^{(\Phi)}\sqrt{R_{s}^{2}-\kappa^{2}}-v^{(R)}\sqrt{3R_{s}^{2}-4\sqrt{3}R_{s}\kappa+4\kappa^{2}}}\right]}.\,\,\,\,\,\,\,\,\,\,$
(4.14)

Hereafter, we use subscript $``i/o"$ to represent $s_{r}=\mp 1$ in Eq. (4.2),
respectively.

For the outgoing emitter with $s_{r}=-1$ in Eq. (4.2), we also have
$\partial_{\eta_{0}}g<0$, and we have $\partial_{\lambda_{0}}g\gtrless 0$ for
$\lambda_{0}\lessgtr\lambda_{0c}$, where

$\lambda_{0c}(\eta_{0})=-R_{s}+\gamma_{R}\sqrt{1+\eta_{0}}\sqrt{R_{s}^{2}-\kappa^{2}}$
(4.15)

with $\gamma_{R}=1/\sqrt{1-(v^{(R)})^{2}}$. Then we obtain

$\displaystyle
g_{\text{max},o}=\begin{cases}g_{o}[\lambda_{0c}(-3/4),-3/4],\quad&\text{if
}\lambda_{0-}<\lambda_{0c},\\\ g_{o}[\lambda_{0-}(-3/4),-3/4],\quad&\text{if
}\lambda_{0-}>\lambda_{0c},\,\,(r_{s}\leq\tilde{r}_{1}\text{
only}),\end{cases}$ (4.16)

where $g_{o}[\lambda_{0-}(-3/4),-3/4]$ has the same form as
$g_{i}[\lambda_{0-}(-3/4),-3/4]$ [Eq. (4.14)] but with $s_{r}=1$ in $v^{(R)}$,
and

$g_{o}[\lambda_{0c}(-3/4),-3/4]=\frac{1}{\gamma\mathopen{}\mathclose{{}\left[1/\gamma_{R}-v^{(\Phi)}}\right]}=\frac{\sqrt{3}L_{s}^{\ast}}{-L_{s}+\sqrt{3(L_{s}^{\ast})^{2}+L_{s}^{2}}}.$
(4.17)

From (4.13) we can see that, for an infalling emitter, an inward radial motion
and a retrograde angular motion will redshift the energy of an escaping photon
while a prograde angular motion will blueshift the energy of an escaping
photon. From (4.17) we can see that, for an outgoing emitter, the energy of an
escaping photon is affected by its velocity in a complex way but is determined
only by its angular momentum with a simple relation. We also find that
$g_{\text{max},o}$ is always greater than $g_{\text{max},i}$ with the same
parameters $(L_{s},E_{s})$, which means that an outward radial motion will
enhance the energy of escaping photons.

In terms of $z_{\text{mob}}$, the MOB for a (near-)NHEK emitter with
$r_{s}>\tilde{r}_{1}$ can be summarized as

$\displaystyle z_{\text{mob},o}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{\sqrt{3}L_{s}^{\ast}}\mathopen{}\mathclose{{}\left(\sqrt{L_{s}^{2}+3(L_{s}^{\ast})^{2}}-L_{s}}\right),$
(4.18) $\displaystyle z_{\text{mob},i}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{\sqrt{3}L_{s}^{\ast}}\mathopen{}\mathclose{{}\left(\frac{2(L_{s}R_{s}+E_{s})}{\sqrt{R_{s}^{2}-\kappa^{2}}}-L_{s}}\right),$
(4.19)

while the MOB for a (near-)NHEK emitter with $r_{s}\leq\tilde{r}_{1}$ can be
summarized as

$\displaystyle
z_{\text{mob}}=\begin{cases}1-\frac{1}{g[\lambda_{0-}(-3/4),-3/4]},\quad&\text{if
}s_{r}=-1,\text{ or }s_{r}=1\,\,\&\,\,\lambda_{0-}>\lambda_{0c},\\\
1-\frac{1}{g[\lambda_{0c}(-3/4),-3/4]},\quad&\text{if
}s_{r}=1\,\,\&\,\,\lambda_{0-}<\lambda_{0c}.\end{cases}$ (4.20)

This is one of our main results and we will analyze the expressions of the MOB
for specific emitters in Sec. 5 and plot them in Figs. 2, 4, 5 and 6.

##  5 MOB and PEP for different emitters

Figure 2: The MOB $z_{\text{mob}}(R_{s})$ and the PEP $\mathcal{P}_{e}(R_{s})$
for photon emissions from several representative emitters moving along
(near-)NHEK equatorial geodesics. The conserved quantities $(L_{s},E_{s})$
label the emitters. We show these with the rescaled radial coordinates,
representing the proper radial distance [34]. The dotted curves are used as
the reference emitters: the red and blue dotted curves are for ZAMS (which
does not follow a geodesic) and the emitters at circular geodesics,
respectively, the black and gray dotted curves are for the emitters on the
marginal (plunging/anti-plunging/deflecting) orbits with $s_{r}=\pm 1$. The
dashed and solid curves are for the emitters on the outgoing ($s_{r}=+1$) and
ingoing ($s_{r}=-1$) orbits, respectively. We have set $M=1$, $\mu=1$ and
$\kappa=1$ (for the near-NHEK case). Figure 3: Top to bottom: examples for the
PEPs for the emitters along the plunging, the anti-plunging, the bounded and
the deflecting orbits, respectively. The conserved quantities $(L_{s},E_{s})$
[Eq. (2.18)] label the emitters. The dashed and solid curves are for the
sources on the outgoing ($s_{r}=+1$) and infalling ($s_{r}=-1$) orbits,
respectively. The dotted curves are displayed as the references.

In Fig. 2, we display the results of $z_{\text{mob}}(R_{s})$ and
$\mathcal{P}_{e}(R_{s})$ of the escaping photons emitted from different orbits
labeled by $(L_{s},E_{s},s_{r})$. We can see that the behaviors of PEP are
closely related to that of MOB. In addition, we also present more examples of
the PEPs in Fig. 3 for the emitters along the plunging, the anti-plunging, the
bounded and the deflecting orbits, respectively. We take an ZAMS as the
reference emitter which has $v^{(R)}=0$ and $v^{(\Phi)}=0$. Note that the ZAMS
does not follow a geodesic and we will use the term “orbits” only for the
geodesics. We take the emitters moving on circular orbits as the references as
well. We use red and blue dotted curves to show the results for ZAMSs and
circular orbits, respectively. The dotted curves with other colors are used as
additional reference emitters whose motions are described in the plots. The
dashed and solid curves are, respectively, for the infalling ($s_{r}=-1$) and
the outgoing ($s_{r}=1$) emitters along several orbits labeled by
$(L_{s},E_{s})$. We show these curves with the rescaled radial coordinates

$s_{\text{NK}}=\log(R+1),\qquad
s_{\text{nNK}}=\log\mathopen{}\mathclose{{}\left[R+\sqrt{R^{2}-\kappa^{2}}}\right],$
(5.1)

in terms of which, the proper radial distances in the NHEK and near-NHEK
regions are [34]

$ds_{\text{NK}}(\bar{R}_{1},\bar{R}_{2})=s_{\text{NK2}}-s_{\text{NK1}},\qquad
ds_{\text{nNK}}(R_{1},R_{2})=s_{\text{nNK2}}-s_{\text{nNK1}},\qquad\bar{R}=R+1,$
(5.2)

while the proper radial distance between the NHEK and near-NHEK regions scales
as $\log|\epsilon|$ [43, 34]. Note that from the perspective of near-NHEK
geometry, the $R_{\text{nNK}}\rightarrow\infty$ limit glues onto the NHEK
region. Here the subscripts “NK” and “nNK” represent the quantities in the
NHEK and near-NHEK geometries, respectively. In practice, we choose $M=1$,
$\mu=1$ for the emitter’s mass, and $\kappa=1$ for the near-NHEK case.

Before describing the main results of this paper, we recall the relevant
results for photon emissions from ZAMS and from the emitters on circular
geodesics. For photon emissions from a ZAMS, we have

$\displaystyle
z_{\text{mob}}(R_{s})|_{\text{ZAMS}}=\begin{cases}0,\quad&\text{if
}r_{s}>\tilde{r}_{1},\\\
1-\frac{2R_{s}-\sqrt{3}\kappa}{\sqrt{R_{s}^{2}-\kappa^{2}}},\quad&\text{if
}r_{s}\leq\tilde{r}_{1}.\end{cases}$ (5.3)

It was found in [27, 33] that the PEP for a NHEK ZAMS is about $29\%$ and the
PEP for a near-NHEK ZAMS decreases as the source radius decreases from near-
NHEK infinity towards horizon radius and reaches approximately $13\%$ at
$\tilde{r}_{1}$. For photon emissions from the emitters on circular orbits in
the range $r_{s}>\tilde{r}_{1}$ ($\tilde{r}_{1}$ is also the last allowed
radius for a circular orbit [43], at which $L_{s}\rightarrow\infty$), we have

$\displaystyle
z_{\text{mob}}(R_{s})|_{\text{cir}}=\begin{cases}1-\frac{1}{\sqrt{3}},\quad&\text{if
}r_{s}=M(1+\epsilon^{\frac{2}{3}}R_{s}),\quad(\text{ISCO}),\\\
1-\frac{1}{\sqrt{3}L_{s}^{\ast}}\mathopen{}\mathclose{{}\left(\frac{2(L_{s}R_{s}-\kappa\sqrt{-C_{s}})}{\sqrt{R_{s}^{2}-\kappa^{2}}}-L_{s}}\right),\quad&\text{if
}r_{s}=M(1+\epsilon R_{s}),\quad R_{s}>\tilde{R}_{1}.\end{cases}$ (5.4)

All circular orbits in the NHEK region are the ISCO while the ones in the
near-NHEK region are unstable. For a given unstable circular orbit in the
near-NHEK region, the orbital radius is uniquely determined by $R_{c}=\kappa
L_{s}/\sqrt{-C_{s}}$, then we obtain
$z_{\text{cir},u}=z_{\text{mob}}(R_{c})|_{\text{cir}}=z_{\text{mob},o}$. Here,
the subscript $``u"$ represents “unstable”. For photon emissions from the
ISCO999For photon emissions from stable circular orbits outside the ISCO of an
extremal Kerr black hole, the PEP decreases monotonously as the orbital radius
decreases and reaches approximately $55\%$ at the ISCO [29, 30]., the PEP is
about $55\%$ [29, 30]. We find that, for photon emission from unstable
circular orbits, the PEP decreases from $\sim 55\%$ to $\sim 50\%$ as the
orbital radius decreases from the one of ISCO towards $\tilde{r}_{1}$.

Let us first look at the MOB (see Fig. 2) for a (near-)NHEK emitter labeled by
$(L_{s},E_{s},s_{r})$. For photon emissions from an outgoing emitter
$(L_{s},E_{s},+1)$, the MOB is a constant along a given orbit, i.e., the MOB
does not depend on $R_{s}$. For photon emissions from a marginal infalling
emitter which exists only in the NHEK region, the MOB does not depend on
$R_{s}$ as well and we have

$z_{\text{mar},i}=z_{\text{mob},i}|_{\text{marginal}}=1-\frac{L_{s}}{\sqrt{3}L_{s}^{\ast}}.$
(5.5)

For photon emissions from a bound/deflecting emitter along the infalling part
of its orbit, $z_{\text{mob},i}(R_{s})$ decreases/increases monotonously as
the emitter moves from (near-)NHEK infinity towards horizon. For photon
emissions from the plunging emitters, the behavior of
$z_{\text{mob},i}(R_{s})$ between the NHEK and near-NHEK cases are different:
for photon emissions from a NHEK plunging emitter or from a near-NHEK plunging
emitter with $E_{s}\geq 0$, $z_{\text{mob},i}(R_{s})$ decreases monotonously
as the emitter moves from (near-)NHEK infinity towards horizon; while for
photon emissions from a near-NHEK plunging emitter with
$-\kappa\sqrt{-C_{s}}<E_{s}<0$, $z_{\text{mob},i}(R_{s})$ increases at the
beginning as the emitter leaves from the near-NHEK infinity until reaching an
extreme value at

$R_{e}=-\frac{\kappa^{2}L_{s}}{E_{s}},$ (5.6)

then $z_{\text{mob},i}(R_{s})$ begins to decrease as the emitter falls towards
the horizon, and at $R_{e}$ we find
$z_{\text{mob},i}=z_{\text{mob},o}=z_{\text{cir},u}$ as
$E_{s}\rightarrow-\kappa\sqrt{-C_{s}}$. Moreover, at the (near-)NHEK infinity,
we have $z_{\text{mob},i}(R_{\infty})=z_{\text{mar},i}$ for a plunging or a
deflecting emitter, and at the the turning point of a bounded or a deflecting
orbit, we have $z_{\text{mob},i}=z_{\text{mob},o}$. These are summarized in
Tables 4 and 6.

Next, we turn to the PEP (see Figs. 2 and 3) for a (near-)NHEK emitter labeled
by $(L_{s},E_{s},s_{r})$. For photon emissions from a plunging emitter,
$\mathcal{P}_{e}(R_{s})$ decreases monotonously along its orbit as the emitter
moves from (near-)NHEK infinity towards the horizon. For photon emissions from
a bounded emitter, $\mathcal{P}_{e}(R_{s})$ decreases monotonously along its
orbit as the emitter moves outward from the horizon towards the turning point
and then bounces off back to the horizon. For photon emissions from a
deflecting emitter, $\mathcal{P}_{e}(R_{s})$ increases monotonously along its
orbit as the emitter moves inward from (near-)NHEK infinity towards the
turning point and then bounces off back towards (near-)NHEK infinity. For
photon emissions from the anti-plunging emitters, there are differences
between the NHEK and near-NHEK cases: for a NHEK anti-plunging emitter and for
a near-NHEK anti-plunging emitter with $E_{s}>\kappa\sqrt{-C_{s}}$,
$\mathcal{P}_{e}(R_{s})$ decreases monotonously along its orbit from the
horizon towards infinity; while for a near-NHEK anti-plunging emitter with
$E_{s}<\mathopen{}\mathclose{{}\left|\kappa\sqrt{-C_{s}}}\right|$,
$\mathcal{P}_{e}(R_{s})$ increases monotonously along its orbit from the
horizon towards infinity. Furthermore, we find that for all (anti-plunging and
deflecting) emitters that can eventually leave from the NHEK infinity, we
always have $\mathcal{P}_{e}>50\%$ while for all plunging emitters that fall
into the black hole we always have $\mathcal{P}_{e}<55\%$. In addition, for
the bounded emitters in the (near-)NHEK region, we always have
$\mathcal{P}_{e}<59\%$. These are also summarized in Tables 4 and 6.

##  6 Influence of the emitter’s proper motion

From Fig. 2, we can also see that the PEP and MOB are clearly affected by the
proper motion of an emitter. For an emitter with proper motion labeled by
$(L_{s},E_{s},s_{r})$, the behaviors of PEP and MOB along its orbit are
clarified in the previous section. Next, we consider the behaviors of PEP and
MOB as the orbital constants $(L_{s},E_{s},s_{r})$ vary. To do so, we study
the functions $z_{\text{mob}}(L_{s},E_{s},s_{r})$ and
$\mathcal{P}_{e}(L_{s},E_{s},s_{r})$ at several representative points along
each emitter’s orbit.

In the last section, we find that $z_{\text{mob},o}$ [Eq. (4.18)] depends only
on $L_{s}$ and tends to $1$ (infinite blueshift) as $L_{s}\rightarrow\infty$
while tends to $-\infty$ (infinite redshift) as $L_{s}\rightarrow-\infty$.
Moreover, $z_{\text{mob},i}$ [Eq. (4.19)] of an infalling emitter is always
less than that of an outgoing one with the same parameters $(L_{s},E_{s})$,
and $z_{\text{mob},i}$ depends on both $L_{s}$ and $E_{s}$. We now consider
only photon emissions with $z_{\text{mob}}\geq 0$ and make a cutoff along a
given orbit (if relevant) at $R_{z}$ where $z_{\text{mob}}(R_{z})=0$. In other
words, we focus on the prograde orbits ($L_{s}>0$) for the emitters since no
photon emissions from the retrograde emitters ($L_{s}<0$) are blueshifted.

###  6.1 NHEK emitters

Figure 4: Values of $z_{\text{mob}}(L_{s})$ and $\mathcal{P}_{e}(L_{s})$
between the endpoints of the part of each NHEK equatorial orbit with
$z_{\text{mob}}\geq 0$. The radial ranges are illustrated explicitly in Table
4. The blue solid line in the left is for $z_{\text{mob},o}$ and the red solid
line has $z_{\text{mob}}=0$. The lightpurple, lightgreen and lightorange
ranges are for photon emissions from the infalling ($s_{r}=-1$) bounded,
plunging and deflecting emitters, respectively, while the purple, green and
orange ranges are for photon emissions from the outgoing ($s_{r}=+1$) bounded,
anti-plunging and deflecting emitters, respectively.

Table 3 shows the components of the 3-velocities of the NHEK emitters relative
to the ZAMS at the endpoints of these emitters’ orbits or at the cutoff point

$R_{z}=\frac{2E_{s}}{\sqrt{3}L_{s}^{\ast}-L_{s}}.$ (6.1)

We find that the velocities are independent of $E_{s}$ at these points. And a
change of $E_{s}$ will only extend or shrink the range with
$z_{\text{mob}}\geq 0$ by shifting the points $R_{z}$ or $R_{+}$. As
$\mathopen{}\mathclose{{}\left|E_{s}}\right|$ increases, $R_{+}$ and $R_{z}$
increase as well. For a bounded orbit, the radial range $R\in(0,R_{+})$ is
extended while the entire region $R\in(R_{z},R_{+})$ appears farther away from
the horizon. For a deflecting orbit, the radial range $R\in(R_{+},\infty)$ is
shrunk while the entire region $R\in(R_{+},R_{z})$ appears closer to the
horizon. For a plunging emitter, the radial range $R\in(R_{z},\infty)$ is
shrunk. We find that the PEP and MOB are also independent of $E_{s}$ at the
endpoints or at the cutoff point.

Table 3: The 3-velocity components at the endpoints ($0$, $R_{+}$ or $R_{\infty}$) or the cutoff point ($R_{z}$) of a NHEK equatorial orbit. $R_{s}$ | $0$ | $R_{z}$ | $R_{+}$ | $R_{\infty}$
---|---|---|---|---
$v^{(R)}$ | $\pm 1$ | $\pm\sqrt{2\sqrt{3}L_{s}^{\ast}L_{s}}/(\sqrt{3}L_{s}^{\ast}+L_{s})$ | $0$ | $\pm\sqrt{-C_{s}}/L_{s}$
$v^{(\Phi)}$ | $0$ | $L_{s}/(\sqrt{3}L_{s}^{\ast}+L_{s})$ | $L_{s}/\sqrt{3(L_{s}^{\ast})^{2}+L_{s}^{2}}$ | $1/2$
Table 4: Behaviors of $z_{\text{mob}}(L_{s},E_{s},s_{r})$ and $\mathcal{P}_{e}(L_{s},E_{s},s_{r})$ between the endpoints of the part of each NHEK equatorial orbit with $z_{\text{mob}}\geq 0$, where $z_{\text{mar},i}$ and $z_{\text{mob},o}$ are given in Eqs. (5.5) and (4.18), respectively. $L_{s}$ | $E_{s}$ | $s_{r}$ | $z_{\text{mob}}$ | $\mathcal{P}_{e}$ | motion
---|---|---|---|---|---
$0<L_{s}<L_{s}^{\ast}$ | $E_{s}>0$ | $+$ | $z_{\text{mob},o}$ | $\mathcal{P}_{e}(R_{+})<\mathcal{P}_{e}<\mathcal{P}_{e}(0)$ | bounded
$0<L_{s}<L_{s}^{\ast}$ | $E_{s}>0$ | $-$ | $(0,\,\,z_{\text{mob},o})$ | $\mathcal{P}_{e}(R_{z})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{+})$ | bounded
$L_{s}=L_{s}^{\ast}$ | $E_{s}=0$ | $\shortmid$ | $1-1/\sqrt{3}$ | $\sim 55\%$ | ISCO
$L_{s}\geq L_{s}^{\ast}$ | $E_{s}>0$ | $+$ | $z_{\text{mob},o}$ | $\mathcal{P}_{e}(R_{\infty})<\mathcal{P}_{e}<\mathcal{P}_{e}(0)$ | anti-plunging
$L_{s}\geq L_{s}^{\ast}$ | $E_{s}>0$ | $-$ | $(0,\,\,z_{\text{mar},i})$ | $\mathcal{P}_{e}(R_{z})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{\infty})$ | plunging
$L_{s}>L_{s}^{\ast}$ | $E_{s}=0$ | $\pm$ | $z_{\text{mob},o}/z_{\text{mar},i}$ | $\mathcal{P}_{\text{mar},o}/\mathcal{P}_{\text{mar},i}$ | marginal
$L_{s}>L_{s}^{\ast}$ | $E_{s}<0$ | $+$ | $z_{\text{mob},o}$ | $\mathcal{P}_{e}(R_{+})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{\infty})$ | deflecting
$L_{s}^{\ast}<L_{s}<2$ | $E_{s}<0$ | $-$ | $(0,\,\,z_{\text{mob},o})$ | $\mathcal{P}_{e}(R_{z})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{+})$ | deflecting
$L_{s}>2$ | $E_{s}<0$ | $-$ | $(z_{\text{mar},i},\,\,z_{\text{mob},o})$ | $\mathcal{P}_{e}(R_{\infty})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{+})$ | deflecting

In Table 4, we display distinct behaviors of
$z_{\text{mob}}(L_{s},E_{s},s_{r})$ and $\mathcal{P}_{e}(L_{s},E_{s},s_{r})$
between the endpoints of the part of an emitter’s orbit with
$z_{\text{mob}}\geq 0$. Meanwhile, we display in Fig. 4 the corresponding
values of $z_{\text{mob}}$ and $\mathcal{P}_{e}$. From Fig. 4, we can see that
the maximum PEP is increased monotonously as $L_{s}$ is increased from $0$.
For $L_{s}=0$, the maximum PEP is about $33\%$ and the PEP at $R_{z}$ is about
$29\%$. For $L_{s}=L_{s}^{\ast}$, the maximum PEP for the anti-plunging
emitters is about $59\%$ and the minimum/maximum PEP for the anti-
plunging/plunging emitters is about $55\%$, and the PEP for the plunging
emitters at $R_{z}$ is about $39\%$. For $L_{s}=\sqrt{3}L_{s}^{\ast}$, the PEP
for the anti-plunging emitters is in a narrow range around $73\%$ and the PEP
for the plunging emitters in $(R_{z},R_{\infty})$ is about $50\%$. However, as
$L_{s}\rightarrow\sqrt{3}L_{s}^{\ast}$, the cutoff point $R_{z}$ tends to
infinity and thus the radial region with $z_{\text{mob}}\geq 0$ is actually
shrunk to zero. For $L_{s}>\sqrt{3}L_{s}^{\ast}$, photon emissions from the
anti-plunging and deflecting orbits always have a PEP greater than $50\%$
while photon emissions from the plunging orbits never have a PEP greater than
$50\%$.

As expected, an anti-plunging or deflecting emitter can be observed ( i.e,
with considerable PEP and carrying enough energy at infinity) by a distant
observer, since the emitter itself can also reach to asymptotic far region.
However, for a plunging emitter that eventually drops into the central black
hole, it is interesting to ask to what depth in the near-horizon throat can
the emitter still be observed. We find that, among all plunging emitters, the
one with critical parameters $(L_{s}^{\ast},0)$ has a maximum PEP and the
range of $(R_{z},R_{\infty})$ extends to the entire NHEK region.

###  6.2 Near-NHEK emitters

Unlike those for the NHEK emitters, the 3-velocity components relative to the
ZAMS for an near-NHEK plunging emitter with $-\kappa\sqrt{-C_{s}}<E_{s}<0$ are
no longer monotonous along its orbits, and neither does the MOB of such an
emitter. Instead, there is an extreme velocity along each of these orbits at
the critical point $R_{e}$ [Eq. (5.6)]. Table 5 shows the components of an
emitter’s 3-velocity at the endpoints, the critical point of the emitter’s
orbit or the cutoff point

$R_{z}=\frac{2E_{s}L_{s}\pm(\sqrt{3}L_{s}^{\ast}+L_{s})\sqrt{E_{s}^{2}+\kappa^{2}(C_{s}+\frac{\sqrt{3}}{2}L_{s}^{\ast}L_{s})}}{2C_{s}+\sqrt{3}L_{s}^{\ast}L_{s}},$
(6.2)

where the plus/minus sign is taken as $L_{s}\lessgtr\sqrt{3}L_{s}^{\ast}$,
respectively. In the table, the 3-velocity components at the critical points
are

$v^{(R)}(R_{e})=-\sqrt{(E_{s}^{2}+\kappa^{2}C_{s})/(E_{s}^{2}-\kappa^{2}L_{s}^{2})},\hskip
12.91663ptv^{(\Phi)}(R_{e})=\kappa
L_{s}/\sqrt{4(E_{s}^{2}-\kappa^{2}L_{s}^{2})}.$ (6.3)

Note that, as $E_{s}\rightarrow-\kappa\sqrt{-C_{s}}$, the 3-velocity
components obtained at $R_{e}$ have the same forms as those obtained at the
turning point $R_{+}$ of a bounded or deflecting orbit. We find that the
velocities do not depend on $E_{s}$ at the endpoints ($\kappa$, $R_{+}$ or
$R_{\infty}$) and at the cutoff point $R_{z}$, but do depend on $E_{s}$ at the
critical point $R_{e}$. As in the NHEK cases, a change of $E_{s}$ will extend
or shrink the range with $z_{\text{mob}}\geq 0$ in a similar way. However, for
the near-NHEK case, the PEP and MOB depend on $E_{s}$ at the endpoints, the
cutoff point and the critical point. And in the limit
$\mathopen{}\mathclose{{}\left|E_{s}}\right|\rightarrow\infty$, the PEP and
MOB tend to the values of the corresponding NHEK cases.

Table 5: The 3-velocity components at the endpoints ($\kappa$, $R_{+}$ or $R_{\infty}$), the cutoff point ($R_{z}$), or the critical point ($R_{e}$) of a near-NHEK equatorial orbit. $R_{s}$ | $\kappa$ | $R_{z}$ | $R_{e}$ | $R_{+}$ | $R_{\infty}$
---|---|---|---|---|---
$v^{(R)}$ | $\pm 1$ | $\pm\sqrt{2\sqrt{3}L_{s}^{\ast}L_{s}}/(\sqrt{3}L_{s}^{\ast}+L_{s})$ | $v^{(R)}(R_{e})$ | $0$ | $\pm\sqrt{-C_{s}}/L_{s}$
$v^{(\Phi)}$ | $0$ | $L_{s}/(\sqrt{3}L_{s}^{\ast}+L_{s})$ | $v^{(\Phi)}(R_{e})$ | $L_{s}/\sqrt{3(L_{s}^{\ast})^{2}+L_{s}^{2}}$ | $1/2$
Table 6: Behaviors of $z_{\text{mob}}(L_{s},E_{s},s_{r})$ and $\mathcal{P}_{e}(L_{s},E_{s},s_{r})$ between the endpoints of the part of each near-NHEK equatorial orbit with $z_{\text{mob}}\geq 0$. $L_{s}$ | $E_{s}$ | $s_{r}$ | $z_{\text{mob}}$ | $\mathcal{P}_{e}$ | motion
---|---|---|---|---|---
$L_{0}<L_{s}<L_{s}^{\ast}$ | $E_{s}>-\kappa L_{s}$ | $+$ | $z_{\text{mob},o}$ | $\mathcal{P}_{e}(R_{+})<\mathcal{P}_{e}<\mathcal{P}_{e}(\kappa)$ | bounded
$L_{0}<L_{s}<L_{s}^{\ast}$ | $E_{s}>-\kappa L_{s}$ | $-$ | $(0,\,\,z_{\text{mob},o})$ | $\mathcal{P}_{e}(R_{z})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{+})$ | bounded
$L_{s}=L_{s}^{\ast}$ | $E_{s}\geq 0$ | $+$ | $z_{\text{mob},o}$ | $\mathcal{P}_{e}(R_{\infty})<\mathcal{P}_{e}<\mathcal{P}_{e}(\kappa)$ | anti-plunging
$L_{s}>L_{s}^{\ast}$ | $E_{s}>\kappa\sqrt{-C_{s}}$ |  |  |  |
$L_{s}>L_{s}^{\ast}$ | $E_{s}<\mathopen{}\mathclose{{}\left|\kappa\sqrt{-C_{s}}}\right|$ | $+$ | $z_{\text{mob},o}$ | $\mathcal{P}_{e}(\kappa)<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{\infty})$ | anti-plunging
$L_{s}>L_{s}^{\ast}$ | $-\kappa\sqrt{-C_{s}}<E_{s}<0$ | $-$ | $(0,\,\,z_{\text{mob},i}(R_{e}))$ | $\mathcal{P}_{e}(R_{z})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{\infty})$ | plunging
$L_{s}\geq L_{s}^{\ast}$ | $E_{s}\geq 0$ | $-$ | $(0,\,\,z_{\text{mar},i})$ | $\mathcal{P}_{e}(R_{z})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{\infty})$ | plunging
$L_{s}>L_{s}^{\ast}$ | $E_{s}=-\kappa\sqrt{-C_{s}}$ | $\shortmid$ | $z_{\text{mob},o}$ | $\mathcal{P}_{\text{cir},u}$ | circular
$L_{s}>L_{s}^{\ast}$ | $E_{s}<-\kappa\sqrt{-C_{s}}$ | $+$ | $z_{\text{mob},o}$ | $\mathcal{P}_{e}(R_{+})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{\infty})$ | deflecting
$L_{s}^{\ast}<L_{s}<2$ | $E_{s}<-\kappa\sqrt{-C_{s}}$ | $-$ | $(0,\,\,z_{\text{mob},o})$ | $\mathcal{P}_{e}(R_{z})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{+})$ | deflecting
$L_{s}>2$ | $E_{s}<-\kappa\sqrt{-C_{s}}$ | $-$ | $(z_{\text{mar},i},\,\,z_{\text{mob},o})$ | $\mathcal{P}_{e}(R_{\infty})<\mathcal{P}_{e}<\mathcal{P}_{e}(R_{+})$ | deflecting
Figure 5: Values of $z_{\text{mob}}(L_{s})$ and $\mathcal{P}_{e}(L_{s})$
between the endpoints of the part of each near-NHEK equatorial orbit with
$z_{\text{mob}}\geq 0$. The ranges are illustrated explicitly in Table 6. For
the emitters with $L_{0}<L_{s}<\frac{L_{s}^{\ast}}{\sqrt{5}}$ and $E_{s}=0$,
the MOB $z_{\text{mob}}$ is obtained for photon emissions from
$\tilde{r}<\tilde{r}_{1}$, otherwise, it is given in Eqs. (4.18) and (4.19).
We have chosen $E_{s}=0$ for the bounded, the plunging and the anti-plunging
emitters, and have chosen $E_{s}=-\kappa L_{s}$ for the deflecting emitters.
The solid blue line in the left is for $z_{\text{mob},o}$ and the solid red
line has $z_{\text{mob}}=0$, while the dotted red line in the right has
$z_{\text{mob}}\rightarrow-\infty$. The lightpink, lightblue and lightmagenta
ranges are for photon emissions from the infalling ($s_{r}=-1$) bounded,
plunging and deflecting emitters, respectively, while the pink and magenta
ranges are for photon emissions from the outgoing ($s_{r}=+1$) bounded and
deflecting emitters, respectively, and the blue range are the overlap range
for photon emissions from the anti-plunging and outgoing deflecting emitters.
Figure 6: $z_{\text{mob}}(R_{s})$ and $\mathcal{P}_{e}(R_{s})$ for photon
emissions from several near-NHEK emitters. The red dotted curve is for ZMAS,
and the blue dotted curve is for the emitters on circular geodesic orbits. The
dashed curves are for $s_{r}=+1$ and the solid curves are for $s_{r}=-1$. The
darkcyan, cyan and lightcyan have $L_{s}=\sqrt{3}L_{s}^{\ast}$ and
$E_{s}=-\kappa\sqrt{-C_{s}}$, $0$ and $2\sqrt{3}L_{s}^{\ast}$, respectively.
The magenta curve has $L_{s}=\sqrt{3}L_{s}^{\ast}$ and $E_{s}=-\kappa L_{s}$.
The darkpink, pink and lightpink curves have $L_{s}=\frac{4}{5}L_{s}^{\ast}$
and $E_{s}=-\frac{(8-\sqrt{3})}{10}L_{s}^{\ast}$, $0$ and
$\frac{\sqrt{3}}{2}L_{s}^{\ast}$, respectively.

Table 6 displays distinct behaviors of $z_{\text{mob}}$ and $\mathcal{P}_{e}$
between the endpoints of the part of an emitter’s orbit with
$z_{\text{mob}}\geq 0$. $z_{\text{mob},o}$ is given in Eq. (4.18) if
$r_{s}\geq\tilde{r}_{1}$ or in Eq. (4.20) if $r_{s}<\tilde{r}_{1}$,
$z_{\text{mar},i}$ is given in Eq. (5.5) and
$z_{\text{mob},i}(R_{e})=1+\frac{L_{s}}{\sqrt{3}L_{s}^{\ast}}-\frac{2\sqrt{\kappa^{2}L^{2}_{s}-E_{s}^{2}}}{\sqrt{3}\kappa
L_{s}^{\ast}}$. The value of $L_{0}$ is obtained for $z_{\text{mob},o}=0$.
Meanwhile, we display in Fig. 5 the corresponding values of $z_{\text{mob}}$
and $\mathcal{P}_{e}$, in which $E_{s}$ is chosen to be $0$ for the bounded,
the plunging and the anti-plunging emitters, and $E_{s}$ is chosen to be
$-\kappa L_{s}$ for the deflecting emitters. In order to see the effects of
$E_{s}$, we display $z_{\text{mob}}(R_{s})$ and $\mathcal{P}_{e}(R_{s})$ for
the emitters with $L_{s}=\sqrt{3}L_{s}^{\ast}$ and $\frac{4}{5}L_{s}^{\ast}$,
and each with several values of $E_{s}$ in Fig. 6. The effects of $E_{s}$ can
also be seen from Fig. 3. Fig. 5 shows that, if $E_{s}$ is fixed,
$z_{\text{mob}}(L_{s})$ and $\mathcal{P}_{e}(L_{s})$ for the near-NHEK
emitters have similar behaviors as those for the NHEK emitters. However, the
values of PEP and MOB do depend on $E_{s}$ and tend to the NHEK values as
$\mathopen{}\mathclose{{}\left|E_{s}}\right|\rightarrow\infty$. For
$L_{s}\rightarrow L_{0}=\frac{L_{s}^{\ast}}{\sqrt{5}}$ and $E_{s}=0$, the
turning point $R_{+}$ approaches $\tilde{R}_{1}$ and the maximum PEP is about
$31\%$. For $L_{s}\geq L_{s}^{\ast}$ and $E_{s}\rightarrow
E_{c}=-\kappa\sqrt{-C_{s}}$, the PEP for photon emissions from unstable
circular orbits at $R_{c}=\kappa L_{s}/\sqrt{-C_{s}}$ can be formally written
as $\mathcal{P}_{\text{cir},u}=\mathcal{P}_{e}(L_{s})$. We find that
$\mathcal{P}_{\text{cir},u}$ is the same as the PEP of the marginal infalling
emitters in the NHEK region, $\mathcal{P}_{\text{mar},i}$, that is
$\mathcal{P}_{\text{cir},u}=\mathcal{P}_{\text{mar},i}$ (see the dotted blue
curve on the right in Fig. 5 and the lightblue curve above the lightgreen
region on the right in Fig. 4). As $E_{s}$ is perturbed from $E_{c}$ for
$L_{s}\in[L_{s}^{\ast},\sqrt{3}L_{s}^{\ast})$, an unstable circular orbits
would become a deflecting/anti-plunging/plunging orbit and
$\mathcal{P}_{e}(L_{s})$ starts to depart from $\mathcal{P}_{\text{cir},u}$.
Among all the plunging emitters with a given
$L_{s}\in[L_{s}^{\ast},\sqrt{3}L_{s}^{\ast})$, the PEP takes a maximum value
for $E_{s}\rightarrow E_{c}^{+}$, which corresponds to the marginal plunging
emitter starting from $R_{c}$. For the marginal plunging emitter with
$L_{s}=L_{s}^{\ast}$, the orbital radius $R_{c}$ tends to the radius of the
ISCO, and we have $E_{c}=0$ and

$\mathcal{P}_{e}(R_{\text{ISCO}})|_{(L_{s}^{\ast},0,-1)}\approx
55\%,\qquad\mathcal{P}_{e}(R_{z})|_{(L_{s}^{\ast},0,-1)}\approx 12\%,$ (6.4)

where

$R_{z}=\sqrt{1+\frac{2}{\sqrt{3}}}\kappa.$ (6.5)

Here we only give a few selected illustrations for several representative
features for photon emissions from the near-horizon emitters and a complete
picture can be built up from the figures in this paper.

##  7 Summary and conclusion

In this paper, we studied isotropic and monochromatic photon emissions from
equatorial emitters moving on timelike geodesics in the (near-)NHEK regions
[see Eqs. (2.13) and (2.14)]. The orbits for these geodesics were classified
in Tables 1 and 2 whose orbital radii were written as
$r_{s}=M(1+\epsilon^{q}R_{s})$. Along each emitter’s orbit, we had the
conserved quantities for the emitter: the angular momentum $l_{s}$ and the
energy $\omega_{s}$. For a (near-)NHEK emitter, we introduced $l_{s}=L_{s}$
and $\omega_{s}=\frac{l_{s}}{2M}+\frac{E_{s}}{2M}\epsilon^{q}$. We had
$q=\frac{2}{3}$ for the NHEK case in the ISCO scale and $q=1$ for the near-
NHEK case. In Sec. 4 we computed the PEP and MOB for photon emissions from
these emitters. In general, the results depend on the emitter radius $R_{s}$
and the constants of the emitter’s motion: $L_{s}$, $E_{s}$ and $s_{r}$, with
$s_{r}$ being the radial orientation. Then, in Sec. 5 we analyzed the
behaviors of PEP and MOB along each given orbit to see their dependence on
$R_{s}$, and we summarized the results in Tables 4 and 6. After that, we
analyzed the effects of an emitter’s proper motion on the MOB and PEP in Sec.
6. The main analytical result for MOB was summarized in Eqs. (4.18), (4.19)
and (4.20). Moreover, the numerical results for MOB and PEP were displayed in
Figs. 2, 3, 4, 5 and 6, from which we found that the MOB and PEP had similar
behaviors as the parameters of the emitters vary.

In Fig. 2, we showed that the PEP and MOB for the emitters belonging to
distinct classes of motions were different. In addition, in Fig. 3 we
displayed more examples on the PEP for the emitters belonging to each class of
motion. Furthermore, from Figs. 4, 5 and 6 we could see that the overall
observability of an emitter, that said the magnitudes of PEP and MOB, were
determined by the angular momentum $l_{s}$ of the emitter. This is reasonable
since the energy $\omega_{s}$ of a (near-)NHEK emitter is constrained to be
near the superradiant bound $\omega_{\ast}=\frac{l_{s}}{2M}$. And, as
expected, the photons from the outgoing emitters are brighter than those from
the infalling ones that have the same parameters $(l_{s},\omega_{s})$. As the
angular momentum $l_{s}$ of an emitter increases, the MOB and the maximum of
PEP for the emitters with all allowed motions increase accordingly. On the
other hand, for the emitters with the same angular momentum $l_{s}$, the
corrections of their energies to the critical energy,101010Note that, since in
the NHEK region we have $\Delta\omega$ at $\mathcal{O}(\epsilon^{2/3})$ while
in the near-NHEK region we have $\Delta\omega$ at $\mathcal{O}(\epsilon)$, the
near-NHEK emitters correspond to the NHEK emitters with $\Delta\omega=0$.
$\Delta\omega=\omega_{s}-\omega_{\ast}=\frac{E_{s}}{2M}\epsilon^{q}$,
distinguish their motions and affect the MOB and PEP for these emitters as
well. Simply speaking, while $l_{s}$ plays a dominant role on the overall
feature of MOB and PEP, $\Delta\omega$ starts to play a role after $l_{s}$
being fixed.

We picked out several characteristic values of $l_{s}$ to illustrated our
results. If $l_{s}=0$, then $z_{\text{mob},o}=0$ [see Eqs. (4.18) and (4.20)]
which means that no photon emissions from retrograde emitters ($l_{s}<0$) were
blueshifted. We focused on the case $l_{s}>0$, which has blueshifted photon
emissions. Around $l_{s}=l_{s}^{\ast}$ [see Eq. (3.1)], the emitters start to
appear with distinct motions and different observational features. For
$0<l_{s}<l_{s}^{\ast}$, the emitters are bounded and require
$\Delta\omega<-\epsilon^{q}\kappa\omega_{\ast}$, and the PEP for them are
always less than $59\%$. In this case, the maximum value is reached as
$l_{s}\rightarrow l_{s}^{\ast}$ and the emitter is bounced outward at the
inner boundary of its orbit. For $l_{s}=l_{s}^{\ast}$ and $\Delta\omega=0$,
the emitters are on the ISCO in the NHEK region and have the marginal
(anti-)plunging motion in the near-NHEK region, while for $l_{s}=l_{s}^{\ast}$
and $\Delta\omega\gtrless 0$, the emitters have the (anti-)plunging/deflecting
motions, respectively. And, for the emitter at the ISCO, the PEP is about
$55\%$ and the MOB is $1-1/\sqrt{3}\approx 0.42$. Similarly, for
$l_{s}>l_{s}^{\ast}$, the emitters with
$\Delta\omega=-\epsilon\kappa\omega_{\ast}\sqrt{-C_{s}/l_{s}^{2}}$ are on
unstable circular orbits while the emitters with
$\Delta\omega\gtrless-\epsilon^{q}\kappa\omega_{\ast}\sqrt{-C_{s}/l_{s}^{2}}$
take the plunging (or anti-plunging)/deflecting motions, respectively. The PEP
for the emitter on unstable circular orbits, $\mathcal{P}_{\text{cir},u}$,
decreases from $55\%$ to $50\%$ as the orbital radius decreases from the ISCO
radius towards the horizon one. The PEP for the deflecting and anti-plunging
emitters are always greater than $50\%$ and the MOB for the outgoing parts of
the orbits are always greater than $0.42$, which implies that the emitters
with $l_{s}>l_{s}^{\ast}$ and with outward radial motions could be well
observed by distant observers. On the other hand, the PEPs for the plunging
emitters decreases as the emitters move from the NHEK boundary towards the
horizon and are always less than $55\%$ (the value for the emitters at the
ISCO). We found that $\mathcal{P}_{\text{mar},i}=\mathcal{P}_{\text{cir},u}$
with $\mathcal{P}_{\text{mar},i}$ the PEP for the plunging emitters at the
NHEK boundary. Moreover, as a plunging emitter moves from NHEK infinity
towards the horizon, the inward radial velocity for the emitter with
$\Delta\omega>0$ increases monotonously while that for the emitter with
$\Delta\omega<0$ decreases at the beginning and then increases to the end. The
maximum value of MOB for a plunging emitter with $\Delta\omega>0$ is obtained
at the NHEK infinity, and this maximum value ($z_{\text{mar},i}$) decreases as
$l_{s}$ is increased and $z_{\text{mar},i}=0$ for
$l_{s}=\sqrt{3}l_{s}^{\ast}$. In addition, as $l_{s}$ increases for
$\Delta\omega\rightarrow-\epsilon\kappa\omega_{\ast}\sqrt{-C_{s}/l_{s}^{2}}$,
the maximum value of MOB for a plunging emitter increases, and this maximum
value ($z_{\text{cir},u}$) is obtained at the corresponding orbital radius of
an unstable circular orbit and actually $z_{\text{cir},u}=z_{\text{mob},o}$.
Furthermore, we found that the PEP and MOB decreased faster for the plunging
emitters with larger $\Delta\omega$, which means that for the plunging
emitters, a smaller $\Delta\omega$ would extend the observable range towards
the deeper region in the near-horizon throat of a high-spin black hole.

## Acknowledgments

We thank Peng-Cheng Li for his helpful discussions and comments on the
manuscript. The work is in part supported by NSFC Grant No. 11735001.

## References

  * [1] Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. Lett. 875 (2019) L1, arXiv:1906.11238 [astro-ph.GA].
  * [2] Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon,” Astrophys. J. Lett. 910 no. 1, (2021) L13, arXiv:2105.01173 [astro-ph.HE].
  * [3] S. E. Gralla, A. Lupsasca, and A. Strominger, “Observational Signature of High Spin at the Event Horizon Telescope,” Mon. Not. Roy. Astron. Soc. 475 no. 3, (2018) 3829–3853, arXiv:1710.11112 [astro-ph.HE].
  * [4] M. Guo, N. A. Obers, and H. Yan, “Observational signatures of near-extremal Kerr-like black holes in a modified gravity theory at the Event Horizon Telescope,” Phys. Rev. D 98 no. 8, (2018) 084063, arXiv:1806.05249 [gr-qc].
  * [5] H. Yan, “Influence of a plasma on the observational signature of a high-spin Kerr black hole,” Phys. Rev. D 99 no. 8, (2019) 084050, arXiv:1903.04382 [gr-qc].
  * [6] M. Guo, S. Song, and H. Yan, “Observational signature of a near-extremal Kerr-Sen black hole in the heterotic string theory,” Phys. Rev. D 101 no. 2, (2020) 024055, arXiv:1911.04796 [gr-qc].
  * [7] P. V. P. Cunha and C. A. R. Herdeiro, “Stationary black holes and light rings,” Phys. Rev. Lett. 124 no. 18, (2020) 181101, arXiv:2003.06445 [gr-qc].
  * [8] M. Guo and S. Gao, “Universal Properties of Light Rings for Stationary Axisymmetric Spacetimes,” Phys. Rev. D 103 no. 10, (2021) 104031, arXiv:2011.02211 [gr-qc].
  * [9] V. Cardoso, F. Duque, and A. Foschi, “The light ring and the appearance of matter accreted by black holes,” arXiv:2102.07784 [gr-qc].
  * [10] S. E. Gralla, D. E. Holz, and R. M. Wald, “Black Hole Shadows, Photon Rings, and Lensing Rings,” Phys. Rev. D 100 no. 2, (2019) 024018, arXiv:1906.00873 [astro-ph.HE].
  * [11] E. Himwich, M. D. Johnson, A. Lupsasca, and A. Strominger, “Universal polarimetric signatures of the black hole photon ring,” Phys. Rev. D 101 no. 8, (2020) 084020, arXiv:2001.08750 [gr-qc].
  * [12] M. D. Johnson et al., “Universal interferometric signatures of a black hole’s photon ring,” Sci. Adv. 6 no. 12, (2020) eaaz1310, arXiv:1907.04329 [astro-ph.IM].
  * [13] S. E. Gralla, A. Lupsasca, and D. P. Marrone, “The shape of the black hole photon ring: A precise test of strong-field general relativity,” Phys. Rev. D 102 no. 12, (2020) 124004, arXiv:2008.03879 [gr-qc].
  * [14] J. Peng, M. Guo, and X.-H. Feng, “Influence of quantum correction on black hole shadows, photon rings, and lensing rings,” Chin. Phys. C 45 no. 8, (2021) 085103, arXiv:2008.00657 [gr-qc].
  * [15] J. Peng, M. Guo, and X.-H. Feng, “Observational Signature and Additional Photon Rings of Asymmetric Thin-shell Wormhole,” arXiv:2102.05488 [gr-qc].
  * [16] R. Penrose, “Gravitational collapse: The role of general relativity,” Riv. Nuovo Cim. 1 (1969) 252–276.
  * [17] J. K. T. Piran, J. Shaham, “High efficiency of the penrose mechanism for particle collisions,” Astrophys.J. 196 (1975) L107.
  * [18] E. Berti, R. Brito, and V. Cardoso, “Ultrahigh-energy debris from the collisional Penrose process,” Phys. Rev. Lett. 114 no. 25, (2015) 251103, arXiv:1410.8534 [gr-qc].
  * [19] J. D. Schnittman, “Revised upper limit to energy extraction from a Kerr black hole,” Phys. Rev. Lett. 113 (2014) 261102, arXiv:1410.6446 [astro-ph.HE].
  * [20] M. Zhang and J. Jiang, “Escape probability of particle from Kerr-Sen black hole,” Nucl. Phys. B 964 (2021) 115313, arXiv:2004.03367 [gr-qc].
  * [21] M. Banados, J. Silk, and S. M. West, “Kerr Black Holes as Particle Accelerators to Arbitrarily High Energy,” Phys. Rev. Lett. 103 (2009) 111102, arXiv:0909.0169 [hep-ph].
  * [22] M. Guo and S. Gao, “Kerr black holes as accelerators of spinning test particles,” Phys. Rev. D 93 no. 8, (2016) 084025, arXiv:1602.08679 [gr-qc].
  * [23] Y.-P. Zhang, B.-M. Gu, S.-W. Wei, J. Yang, and Y.-X. Liu, “Charged spinning black holes as accelerators of spinning particles,” Phys. Rev. D 94 no. 12, (2016) 124017, arXiv:1608.08705 [gr-qc].
  * [24] R. D. Blandford and R. L. Znajek, “Electromagnetic extractions of energy from Kerr black holes,” Mon. Not. Roy. Astron. Soc. 179 (1977) 433–456.
  * [25] F. Camilloni, G. Grignani, T. Harmark, R. Oliveri, and M. Orselli, “Force-free magnetosphere attractors for near-horizon extreme and near-extreme limits of Kerr black hole,” Class. Quant. Grav. 38 no. 7, (2021) 075022, arXiv:2007.15662 [gr-qc].
  * [26] N. Callebaut, M. J. Rodriguez, and H. Verlinde, “Electro-Magnetic Energy Extraction from Rotating Black Holes in AdS,” JHEP 12 (2020) 018, arXiv:2006.09901 [hep-th].
  * [27] K. Ogasawara, T. Igata, T. Harada, and U. Miyamoto, “Escape probability of a photon emitted near the black hole horizon,” Phys. Rev. D 101 no. 4, (2020) 044023, arXiv:1910.01528 [gr-qc].
  * [28] M. Zhang and J. Jiang, “Emissions of photons near the horizons of Kerr-Sen black holes,” Phys. Rev. D 102 no. 12, (2020) 124012, arXiv:2004.11087 [gr-qc].
  * [29] T. Igata, K. Nakashi, and K. Ogasawara, “Observability of the innermost stable circular orbit in a near-extremal Kerr black hole,” Phys. Rev. D 101 no. 4, (2020) 044044, arXiv:1910.12682 [astro-ph.HE].
  * [30] D. E. A. Gates, S. Hadar, and A. Lupsasca, “Photon emission from circular equatorial Kerr orbiters,” Phys. Rev. D 103 no. 4, (2021) 044050, arXiv:2010.07330 [gr-qc].
  * [31] D. E. A. Gates, S. Hadar, and A. Lupsasca, “Maximum observable blueshift from circular equatorial Kerr orbiters,” Phys. Rev. D 102 no. 10, (2020) 104041, arXiv:2009.03310 [gr-qc].
  * [32] T. Igata, K. Kohri, and K. Ogasawara, “Photon emission from inside the innermost stable circular orbit,” Phys. Rev. D 103 no. 10, (2021) 104028, arXiv:2102.13427 [gr-qc].
  * [33] H. Yan, M. Guo, and B. Chen, “Observability of Zero-angular-momentum Sources Near Kerr Black Holes,” arXiv:2104.07889 [gr-qc].
  * [34] D. Kapec and A. Lupsasca, “Particle motion near high-spin black holes,” Class. Quant. Grav. 37 no. 1, (2020) 015006, arXiv:1905.11406 [hep-th].
  * [35] G. Compère, K. Fransen, T. Hertog, and J. Long, “Gravitational waves from plunges into Gargantua,” Class. Quant. Grav. 35 no. 10, (2018) 104002, arXiv:1712.07130 [gr-qc].
  * [36] G. Compère and A. Druart, “Near-horizon geodesics of high-spin black holes,” Phys. Rev. D 101 no. 8, (2020) 084042, arXiv:2001.03478 [gr-qc]. [Erratum: Phys.Rev.D 102, 029901 (2020)].
  * [37] G. Compère, Y. Liu, and J. Long, “Classification of radial Kerr geodesic motion,” arXiv:2106.03141 [gr-qc].
  * [38] J. M. Bardeen, “Timelike and null geodesics in the Kerr metric,” in Les Houches Summer School of Theoretical Physics: Black Holes.  1973\.
  * [39] S. Hadar, A. P. Porfyriadis, and A. Strominger, “Gravity Waves from Extreme-Mass-Ratio Plunges into Kerr Black Holes,” Phys. Rev. D 90 no. 6, (2014) 064045, arXiv:1403.2797 [hep-th].
  * [40] S. E. Gralla, A. P. Porfyriadis, and N. Warburton, “Particle on the Innermost Stable Circular Orbit of a Rapidly Spinning Black Hole,” Phys. Rev. D 92 no. 6, (2015) 064029, arXiv:1506.08496 [gr-qc].
  * [41] J. M. Bardeen and G. T. Horowitz, “The Extreme Kerr throat geometry: A Vacuum analog of AdS(2) x S**2,” Phys. Rev. D 60 (1999) 104030, arXiv:hep-th/9905099.
  * [42] I. Bredberg, T. Hartman, W. Song, and A. Strominger, “Black Hole Superradiance From Kerr/CFT,” JHEP 04 (2010) 019, arXiv:0907.3477 [hep-th].
  * [43] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, “Rotating black holes: Locally nonrotating frames, energy extraction, and scalar synchrotron radiation,” Astrophys. J. 178 (1972) 347.