# The shadows of accelerating Kerr-Newman black hole and constraints from M87*

Tao-Tao<EMAIL_ADDRESS>corresponding author Qi-Ming
<EMAIL_ADDRESS>Wen-Di<EMAIL_ADDRESS>aCollege of
Physics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106,
China
bInstitute of Physics, Shaanxi University of Technology, Hanzhong 723000,
China
cDepartment of Physics, College of Sciences, Northeastern University, Shenyang
110819, China
dInstitute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, China

###### Abstract

In this paper, we study the influence of the parameters for the accelerating
Kerr-Newman black hole on the shadows and the constraints, extensively. We
find that the rotating parameter $a$, the charge parameter $e$, and the
inclination angle $\theta_{0}$ affect the shadow qualitatively similar to that
of Kerr-Newman black holes. The result shows that the size of the shadow will
scale down with the accelerating factor $A$. Besides, the factor $A$ also can
affect the best viewing angles, which make the observations maximum deviate
from $\theta_{0}=\frac{\pi}{2}$, and the degree of the deviations are less
than $1\%$. Then, we assume the M87* as an accelerating Kerr-Newman black hole
with the mass $M=6.5\times 10^{9}M_{\odot}$ and the distance $r_{0}=16.8Mpc$.
Combining the EHT observations, we find that neither the observations,
circularity deviation $\Delta C$ or axial ratio $D_{x}$ can distinguish the
accelerating black hole or not. However, the characteristic areal-radius of
the shadow curve $R_{a}$ can give corresponding constraints on the parameters
of the accelerating Kerr-Newman black hole. The results shows that the bigger
accelerating factor $A$ is, the stronger constraints on the rotating parameter
$a$ and charged parameter $e$. The maximum range of the accelerating factor is
$Ar_{0}\leq 0.558$ for a accelerating Schwarzschild case with $(a/M=e/M=0)$,
and for an extremely slow accelerating case $(Ar_{0}\leq 0.01)$, the ranges of
rotating parameter $a$ and charged parameter $e$ are $a/M\in(0,1)$ and
$e/M\in(0,0.9)$.

##  I Introduction

When the photons move around a black hole, they will face three kinds of
destinies, i.e., absorbed by the black hole, reflected by the black hole, or
revolving around the black hole. For the third destiny, the photons are
trapped in unstable circular geodesics, defining the photon sphere. The photon
sphere can describe the outline of the black hole, also named with the shadow
of the black hole. Besides, perturbations within the photon sphere give rise
to the familiar excitations known as photon sphere quasinormal modes
Cardoso:2008bp ; Cardoso:2017soq .

In the 1960s, Synge calculated the shadow of a spherically symmetric
Schwarzschild black hole synge1966escape . Then, Luminet considered the case
that the Schwarzschild black hole surrounded by an accretion disk, and
calculated the size of the shadow luminet1979image . In Ref. hawking1973black
, the authors studied the shadow of a rotating black hole, and the result
shows that with the dragging effect of rotation, the shadow will deviate from
the perfect circle. Then, the study on the shadow of different spacetime
geometries has been blooming with the motivation that the non-circular shadow,
including black hole Hioki:2008zw ; Grenzebach:2014fha ; Wang:2017qhh ;
Guo:2018kis ; Li:2020drn ; Yan:2019etp ; Hennigar:2018hza ; Konoplya:2019sns ;
Bambi:2008jg ; Bambi:2010hf ; Konoplya:2019fpy ; Wei:2019pjf ; Wei:2018xks ;
Wang:2018prk ; Liu:2020ola ; Kumar:2019pjp ; Cunha:2018acu , and wormhole
Ohgami:2015nra ; Nedkova:2013msa ; Shaikh:2018kfv ; Amir:2018szm ;
Amir:2018pcu ; Wang:2020emr ; Chang:2020lmg ; Gralla:2020yvo .

More recently, the Event Horizon Telescope (EHT) Collaboration gave the image
of the supermassive black hole M87*, which is the first image of the shadow
for the black hole and makes the theoretical prophecy of the black hole shadow
become physical reality Akiyama:2019cqa ; Akiyama:2019brx ; Akiyama:2019sww ;
Akiyama:2019bqs ; Akiyama:2019fyp ; Akiyama:2019eap . The shadow of M87* gives
the observations with the derivation circularity $\Delta C\lesssim 0.1$, the
axis ratio $1<D_{x}\lesssim 4/3$, and the angular shadow diameter
$\psi_{d}=3\sqrt{3}(1\pm 0.17)\psi_{g}$, with the angular gravitional radius
$\psi_{g}=3.8\pm 0.4\mu as$. Combining with these observations of M87*, the
authors have effectively constrained the characteristics of Kerr-like black
holes Meng ; Kuang:2022ojj ; Xavier:2023exm . Furthermore, various methods
exist for constraining black hole properties. For instance, Kuang and Ali
employ gravitational lensing to constrain Kerr-like black holes Kuang:2022xjp
, while Destounis et al. establish limitations for non-Kerr black holes
Destounis:2020kss ; Destounis:2021 ; Destounis:2021pd ; Destounis:2023  as
well as exotic compact objects Maggio:2021uge ; Destounis:2023khj  through
gravitational-wave observables. Additionally, there are endeavors to constrain
non-vacuum black holes situated within astrophysical contexts Cardoso:2021wlq
; Cardoso:2022whc ; Destounis:2022obl ; Figueiredo:2023gas .

According to previous works, we can find that many rotating black boles share
a similar characteristic, that the inclination angle of the observer equals
$\pi/2$, which makes the observations of the shadows for these black holes
maximum. Besides, all these researches show that the apparent characteristics
of the black hole shadows vary with the parameters of the black holes, which
are closely connected with the essential properties of gravity theory and the
background fields. In other words, the different characteristics of black hole
shadows can reflect the different forms of black hole solutions.

On the other hand, there is a special family of black hole solutions in the
Einstein’s general relativity named as C-metric Berlin ; Newman ; Robinson ;
Griffiths ; Kinnersley:1970zw , which can describe the accelerating black hole
and belongs to the well-known Plebański-Demiański spacetime Plebanski:1976gy ;
Griffiths:2005qp . Many works not only focus on the thermodynamics of the
C-metric Appels:2016uha ; Appels:2017xoe ; Astorino:2016ybm ; Gregory:2017ogk
, but also on its stability, superradiance effect and the validity of strong
cosmic censorship in these spacetimes Destounis1 ; Destounis2 ; Destounis3 ;
Fontana . In Refs. Grenzebach:2015oea ; Zhang:2020xub , the authors just
analyzed the shadows of the accelerating Kerr black holes theoretically. The
result shows that the acceleration of the black hole has an unique influence
on the properties of the shadow.

In Ref. Anabalon:2018qfv , the authors obtained the consistent thermodynamic
description of the accelerating charged and rotating black hole, which also
can be named as accelerating Kerr-Newman (KN) black hole. Inspired by the
previous works Grenzebach:2015oea ; Zhang:2020xub , we can not only
investigate the influence of the parameters, i.e. the rotating parameter, the
charged parameter, and the accelerating parameter on the shadow for the
accelerating KN black hole, and analyze the characterized observables of the
black bole shadow, theoretically. But also, we can consider the supermassive
black hole M87* as the accelerating KN black hole and constrain the
corresponding parameters with the EHT observations.

The main parts of this paper are organized as follows. In Sec. II, we will
investigate the circular orbits of the photons around the accelerating Kerr-
Newman black hole. In Sec. III, we will consider the influence of the
parameters on the shadow for the accelerating Kerr-Newman black hole. We
presuppose the M87* as the accelerating Kerr-Newman black hole, and get the
constraints of the corresponding parameters from the EHT observations in Sec.
IV. The last section contributes to our closing summary.

##  II Circular photon orbits around the accelerating Kerr-Newman black hole

In this section, we first give a brief review of the accelerating KN black
hole, the line element can be described as Anabalon:2018qfv

$\displaystyle ds^{2}=$
$\displaystyle\frac{1}{H^{2}}\Big{\\{}-\frac{\Delta_{r}}{\Sigma}\big{(}\frac{dt}{\alpha}-a\sin^{2}\theta
d\phi\big{)}^{2}+\frac{\Sigma}{\Delta_{r}}dr^{2}+\frac{\Sigma}{\Delta_{\theta}}d\theta^{2}$
$\displaystyle+\frac{\Delta_{\theta}}{\Sigma}\Big{(}\frac{adt}{\alpha}-(r^{2}+a^{2})d\phi\Big{)}^{2}\Big{\\}},$
(1)

with

$\displaystyle H$
$\displaystyle=1+Ar\cos\theta,~{}\Sigma=r^{2}+a^{2}\cos^{2}\theta,$ (2)
$\displaystyle\Delta_{r}$
$\displaystyle=(1-A^{2}r^{2})(r^{2}-2Mr+a^{2}+e^{2}),$
$\displaystyle\Delta_{\theta}$
$\displaystyle=1+2MA\cos\theta+A^{2}(a^{2}+e^{2})\cos^{2}\theta,$
$\displaystyle\alpha$
$\displaystyle=\frac{\sqrt{(1-a^{2}A^{2})(1+a^{2}A^{2}+e^{2}A^{2})}}{1+a^{2}A^{2}},$

and the corresponding gauge potential can be expressed as

$\displaystyle F=dB,$ (3) $\displaystyle B=-\frac{e}{\Sigma
r}\big{(}\frac{dt}{\alpha}-a\sin^{2}\theta
d\phi\big{)}+\frac{er_{+}}{(a^{2}+r_{+}^{2})\alpha}dt,$

where $r_{+}$ is the radius of the event horizon of the accelerating KN black
hole with $\Delta_{r}(r_{+})=0$.

In the charged accelerating Kerr black hole, the parameters $m$, $a$ and $e$
are the mass, rotation and charge parameters, respectively. The parameter $A$
represents the acceleration factor of the black hole. $H$ is the conformal
factor, which can affect the conformal boundary $r_{B}$ of the accelerating
black hole with $r_{B}=\frac{1}{|A\cos\theta|}$. The parameter $\alpha$ can
rescale the time coordinate and ensure the Killing vector be normalized at
conformal infinity. The term $\Delta_{r}$ can also be expressed as
$\Delta_{r}=(r-r_{-})(r-r_{+})(r^{2}-r^{2}_{A})$, where $r_{\pm}$ are
identical to the event and Cauchy horizons of the non-accelerating KN black
hole, $r_{A}=1/A$ is already familiar in the context of the C-metric as an
acceleration horizon. With the conformal boundary $r_{B}$, the range of $r$
should be constrained as $r_{+}\leq r\leq r_{B}$. Besides, in order to make
sure that the accelerating KN black hole in the bulk, the term $\Delta_{r}=0$
should have roots in the range $r<r_{A}$ Anabalon:2018qfv .

Then, the Lagrange for the photons can be described by

$\mathcal{L}=\frac{1}{2}g^{\mu\nu}\dot{x}_{\mu}\dot{x}_{\nu},$ (4)

with the definition $\dot{x}^{\mu}=dx^{\mu}/d\lambda=u^{\mu}$, where $u^{\mu}$
is the four-velocity of the photon and the parameter $\lambda$ is the affine
parameter. For this stationary black hole, there are two Killing vectors
$\partial_{t}$ and $\partial_{\phi}$ which can result the conserved total
energy $E$ and z-component of the angular momentum $L_{z}$.

Using the Hamilton-Jacobi equation, we can get the null geodesic equation of
the photon on the background of the accelerating KN black hole

$\displaystyle\frac{\Sigma}{H^{2}}\frac{dt}{d\lambda}=$
$\displaystyle\frac{\alpha\left(a^{2}+r^{2}\right)\left(\alpha
E\left(a^{2}+r^{2}\right)-aL_{z}\right)}{\Delta_{r}}$
$\displaystyle+\frac{a\alpha\left(L_{z}-a\alpha
E\sin^{2}\theta\right)}{\Delta_{\theta}},$ (5)
$\displaystyle\frac{\Sigma}{H^{2}}\frac{d\phi}{d\lambda}=$
$\displaystyle\frac{a\left(\alpha
E\left(a^{2}+r^{2}\right)-aL_{z}\right)}{\Delta_{r}}$
$\displaystyle+\frac{\left(L_{z}\csc^{2}\theta-a\alpha
E\right)}{\Delta_{\theta}},$ (6)
$\displaystyle\left(\frac{\Sigma}{H^{2}}\right)^{2}\left(\frac{dr}{d\lambda}\right)^{2}=$
$\displaystyle\big{[}(a^{2}+r^{2})E-aL_{z}\big{]}^{2}-\Delta_{r}Q\equiv R(r),$
(7)
$\displaystyle\left(\frac{\Sigma}{H^{2}}\right)^{2}\left(\frac{d\theta}{d\lambda}\right)^{2}=$
$\displaystyle\Delta_{\theta}Q-\frac{(L_{z}-aE\sin^{2}\theta)^{2}}{\sin^{2}\theta}\equiv\Theta(\theta),$
(8)

where parameter $Q$ is the Carter constant Carter:1968rr  and the functions
$R(r)$ and $\Theta(\theta)$ can be considered as the radial and longitudinal
effective potential, respectively.

Since the photon orbits should independent of the energy, we can introduce the
following dimensionless abbreviations to describe the photon orbits

$\xi=\frac{L_{z}}{E},\quad\eta=\frac{Q}{E^{2}}.$ (9)

Generally, the unstable spherical circular photon orbits should satisfy with
the conditions

$R(r)=0,\quad\frac{dR(r)}{dr}=0.$ (10)

According to the above two conditions, the dimensionless quantities $\xi$, and
$\eta$ can be solved as

$\displaystyle\xi$
$\displaystyle=\frac{a^{2}\partial_{r}\Delta_{r}+r^{2}\partial_{r}\Delta_{r}-4r\Delta_{r}}{a~{}\partial_{r}\Delta_{r}},$
(11) $\displaystyle\eta$
$\displaystyle=\frac{16r^{2}\Delta_{r}}{(\partial_{r}\Delta_{r})^{2}}.$ (12)

By plugging these expressions into Eq.(8), we can find that the non-negativity
of Eq. (8) gives the condition for the photon region as

$(4r\Delta_{r}-\Sigma\partial_{r}\Delta_{r})^{2}\leq
16a^{2}r^{2}\Delta_{r}\Delta_{\theta}\sin^{2}\theta,$ (13)

For the non-rotating case $a=0$, the circular orbit radius can be solved from
the unstable spherical circular photon orbits conditions (10) as

$r_{p}=\frac{-1+A^{2}e^{2}}{3A^{2}M}+2\sqrt{\beta}\cos\Big{[}\frac{1}{3}\cos^{-1}(\alpha\beta^{-\frac{3}{2}})\Big{]},$
(14)

with

$\displaystyle\alpha$
$\displaystyle=\frac{2\left(A^{2}e^{2}-1\right)^{3}-27A^{2}M^{2}\left(A^{2}e^{2}+1\right)}{54A^{6}M^{3}},$
$\displaystyle\beta$
$\displaystyle=\frac{1}{A^{2}}+\frac{\left(A^{2}e^{2}-1\right)^{2}}{9A^{4}M^{2}}.$
(15)

##  III The shadows of the charged accelerating Kerr black holes

For calculating the character of the black hole shadow, which can be seen by
an observer, we should assume a normalized and orthogonal frame where the
observer is located at, and this frame can be expressed as following

$\displaystyle\hat{e}_{(t)}=$
$\displaystyle\sqrt{\frac{g_{\phi\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}}\left(\partial_{t}-\frac{g_{t\phi}}{g_{\phi\phi}}\partial_{\phi}\right),$
(16) $\displaystyle\hat{e}_{(r)}=$
$\displaystyle\frac{1}{\sqrt{g_{rr}}}\partial_{r},~{}\hat{e}_{(\theta)}=\frac{1}{\sqrt{g_{\theta\theta}}}\partial_{\theta},~{}\hat{e}_{(\phi)}=\frac{1}{\sqrt{g_{\phi\phi}}}\partial_{\phi}.$
(17)

In this frame, we can see that the observer is locally static and zero angular
momentum with respect to infinity for $\hat{e}_{(t)}\cdot\partial_{\phi}=0$.
Hence, this frame also can be named as zero-angular-momentum-observer (ZAMO)
reference frame.

For describing the shadow of the accelerating KN black hole, we should project
the 4-momentum $p^{\mu}$ of photon onto the ZAMO reference frame where the
observer located at, and the corresponding quantities measured by the observer
can be expressed as

$p^{(t)}=-p_{\mu}\hat{e}_{(t)}^{\mu},\quad
p^{(i)}=p_{\mu}\hat{e}_{(i)}^{\mu},~{}~{}i=r,\theta,\phi.$ (18)

In the ZAMO frame, the spatial component of momentum $|\vec{P}|$ should
satisfy $|\vec{P}|=p^{(t)}$ for the massless photon. In order to express the
specific 3-momentum $p^{i}$, we can introduce the observation angles
$(\alpha,\beta)$ as Cunha:2016bpi

$\displaystyle p^{(r)}$ $\displaystyle=$
$\displaystyle|\vec{P}|\cos\alpha\cos\beta,~{}p^{(\theta)}=|\vec{P}|\sin\alpha,$
$\displaystyle p^{(\phi)}$ $\displaystyle=$
$\displaystyle|\vec{P}|\cos\alpha\sin\beta.$ (19)

Combining the geodesic equations (II)-(8) together, we can obtain

$\displaystyle\sin\alpha$ $\displaystyle=$
$\displaystyle\frac{p^{(\theta)}}{p^{(t)}}=\pm\frac{H}{\zeta-\gamma\xi}$ (20)
$\displaystyle\times\sqrt{\frac{\eta}{\Sigma}-\frac{(a\sin^{2}\theta-\xi)^{2}}{\Sigma\Delta_{\theta}\sin^{2}\theta}}|_{(r_{O},\theta_{O})},$
$\displaystyle\tan\beta$ $\displaystyle=$
$\displaystyle\frac{p^{(\phi)}}{p^{(r)}}=\frac{\xi\sqrt{\Sigma\Delta_{r}}}{H\sqrt{g_{\phi\phi}}}$
(21)
$\displaystyle\times\Big{[}(r^{2}+a^{2}-a\xi)^{2}-\Delta_{r}\eta\Big{]}^{-\frac{1}{2}}|_{(r_{O},\theta_{O})}.$

Here, to simplify the above expression, we set the abbreviations
$\zeta\equiv\hat{e}_{(t)}^{t}$, and $\gamma\equiv\hat{e}_{(t)}^{\phi}$. The
parameters $r_{O}$ and $\theta_{O}$ are the radial position and the
inclination angle between the observer and the direction of the rotation axis
for the black hole.

###  III.1 The Shadows

Furthermore, in order to obtain the apparent position on the plane of the sky
for the observer, we need to introduce the Cartesian coordinate $(x,y)$ as

$\displaystyle x\equiv-r_{O}\beta,\quad y\equiv r_{O}\alpha.$ (22)

We can see that both the coordinates $x$ and $y$ are the functions of
$(r_{0},\theta_{O},a,e,A)$. We show the shadows of the accelerating KN black
hole for the variables $a$, $e$, $A$ and $\theta_{O}$ in Figs. 1 and 2, where
we set $M=1$ and $r_{O}=100$. From Fig. 1(a), we can see that with the
increasing of the angular momentum $a$, the shape of the rotating black hole
shadow becomes non-circular as the result due to the dragging effect. Besides,
Fig. 1(b) shows that the observable dragging effect which can result in the
shadows of rotating black hole deviating from the standard circle also depends
on the observation angle $\theta_{O}$, even though the charged accelerating
Kerr black hole with large rotating parameter. From Fig 2, we should note that
the size of the black hole shadow decreases with the charge parameter $e$ and
the acceleration factor $A$. At the same time, as the charge parameter $e$
increases, the non-circular behavior of black hole shadows becomes more
significant. Summing up, we can draw a conclusion that the phenomena for the
shadow of Kerr black hole deviating from the standard circle are mainly caused
by the rotating of the black hole, and the presence of charge of the black
hole will assist in the generation of the phenomena, which is similar to that
of the Kerr-Newman cases Perlick ; Grenzebach ; Tsukamoto ; Xavier .
Furthermore, the acceleration factor $A$ can have an effect on the size of the
shadow for the black hole, and we will further investigate other influences of
the factor $A$ on the shadow in the next part.

(a) $\ a$

(b) $\ \theta_{O}$

Figure 1: The examples of shadows for the accelerating KN black holes with
different parameters. The other parameters are set as
$\theta_{O}=\frac{\pi}{2},e=0.4,A=0.002$ for (a) and $e=0.4,a=0.9,A=0.002$ for
(b).

(a) $~{}e$

(b) $~{}A$

Figure 2: The patterns of shadows of the accelerating KN black holes with
different parameters. The other parameters are set as
$\theta_{O}=\frac{\pi}{2},a=0.7,A=0.002$ for (a) and
$\theta_{O}=\frac{\pi}{2},a=0.7,e=0.7$ for (b).

###  III.2 The Observables

In the previous section, we analyzed the influence of the parameters for the
accelerating KN black holes on their shadows. However, we only can get the
shadows of the black hole by astronomical observation. So, in order to know
the characters of the accelerating KN hole by the astronomical observed data,
we should construct some easily measured and reliable astronomical
observables. Here, we will consider the following observables with the radius
$R_{s}$ and the distortion parameter $\delta_{s}$, which are defined by Hioki
and Maeda Hioki:2009na . The parameter $R_{s}$ can be considered as a radius
of the reference circle which should pass the top point $(x_{t},y_{t})$, the
bottom point $(x_{b},y_{b})$, and the right point $(x_{r},0)$ of the shadow,
and the distortion parameter $\delta_{s}$ can measure the distortion of the
black hole shadow compared with the reference circle, which can be defined as

$\displaystyle
R_{s}=\frac{(x_{t}-x_{r})^{2}+y_{t}^{2}}{2(x_{r}-x_{t})},~{}~{}\delta_{s}=\frac{x_{l}-\tilde{x}_{l}}{R_{s}}.$
(23)

where $(x_{l},0)$ and $(\tilde{x}_{l},0)$ are the left points of the shadow
and reference circle, respectively.

Figure 3 displays the influence of the rotation parameter $a$ and charge
parameter $e$ on the observable parameters $R_{s}$ and $\delta_{s}$. From Fig.
3, we can see that the radius parameter $R_{s}$ decreases rapidly with the
increasing charge parameter $e$. While, the rotating parameter $a$ slightly
affect the reference circle $R_{s}$. For the distortion parameter
$\delta_{s}$, Fig. 3 shows that the distortion of the shadow for the
accelerating KN black hole increases with the rotation parameter $a$, and the
effect of charged parameter $e$ is slight. The effects of the rotating
parameter $a$ and the charged parameter $e$ on the astronomical observables
$R_{s}$ and $\delta_{s}$, are similar to that of Kerr-Newman black hole
Kumar:2019pjp .

In previous chapter, the result shows that the acceleration factor $A$ can
affect the size of the shadow of the black hole, which can also be verified in
Fig. 4. From Fig. 4, we can see that the observation angle $\theta_{O}$
results a slighter influence on the radius $R_{s}$, but a larger effect on the
distortion $\delta_{s}$. Furthermore, when the inclination angle
$\theta_{0}\approx\frac{\pi}{2}$, both $R_{s}$ and $\delta_{s}$ reach the
maximum value, with other parameters are fixed.

In order to ascertain the influence of the acceleration factor $A$ on the best
viewing angle $\theta_{0}$ which makes the radius $R_{s}$ or $\delta s$
maximum, we can introduce the deviation angle $\theta_{R}$ and $\theta_{s}$
with the definition as Zhang:2020xub

$\displaystyle\theta_{R}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}-\theta_{b}|_{(R_{s}=R_{max})},$ (24)
$\displaystyle\theta{s}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}-\theta_{b}|_{(\delta_{s}=\delta_{s_{max}})}.$ (25)

Figure 5 shows the variation of the degree of deviation angles $\theta_{R}$
and $\theta_{s}$ as the result of the accelerating factor $A$. For the
deviation angle $\theta_{R}$, Fig. 5 shows that the deviation angle
$\theta_{R}$ decreases with the charged parameter $e$, while the rotating
parameter $a$ has no effect on $\theta_{R}$. From Fig. 5, we can see that the
deviation angle $\theta_{s}$ decreases with both the rotating parameter $a$
and charged parameter $e$. Besides, both the deviation angles $\theta_{R}$ and
$\theta_{s}$ increase with the accelerating factor $A$. While, the the
accelerating factor $A$ has a greater influence on the deviation angle
$\theta_{R}$ than on $\theta_{s}$, and the degree of the deviations are less
than $1\%$.

Figure 3: The density plots of the observable parameters $R_{s}$ and $\delta
s$ with the variables ($a$,$e$). The other parameters are set as
$\theta_{O}=\frac{\pi}{2}$, $r_{0}=100$, and $A=0.005$.

Figure 4: The density plots of the observable parameters $R_{s}$ and $\delta
s$ with the variables ($A$,$\theta_{0}$). Here, the other parameters are set
as $r_{0}=100$, $a=0.7$, and $e=0.7$. The dashed black lines denote $R_{s}$
and $\delta s$ with $\theta_{0}=\frac{\pi}{2}$.

Figure 5: The diagrams for the deviation of the inclination angle which makes
the shadows radius $R_{s}$ and the distortion $\delta_{s}$ maximum. Here, we
set $M=1$ and $r_{0}=100$. The dashed lines are black ($e=0.2$), green
($e=0.4$), and red ($e=0.59$) with $a=0.8$. The solid lines are black
($a=0.2$), green ($a=0.4$), and red ($a=0.7$) with $e=0.7$. Besides, the solid
blue line is ($a=0.05,e=0$).

##  IV Constraints of the parameters from M87*

The shadow image of the supermassive black hole M87* as crescent shaped was
photographed by the EHT collaboration with the mass $M=6.5\times
10^{9}M_{\odot}$, the distance $r_{0}=16.8Mpc$, and the inclination angle
$\theta_{0}=17^{o}$ Akiyama:2019cqa ; Akiyama:2019brx ; Akiyama:2019sww ;
Akiyama:2019bqs ; Akiyama:2019fyp ; Akiyama:2019eap . The preliminary analysis
of the EHT observations constrain the root-mean-square distance from the
average radius of the black hole shadow as $\Delta C\lesssim 0.1$ which also
named as circularity deviation, the axis ratio as $1<D_{x}\lesssim 4/3$. Then,
in Refs. Psaltis ; EventHorizonTelescope:2021dqv , the authors addressed that
the shadow size of M87* should lie in the range as $R_{a}\approx
3\sqrt{3}(1\pm 0.17)M$. Many works have used these shadow observables $\Delta
C$, $D_{x}$ and $R_{a}$ to constrain the parameters of corresponding black
holes Afrin ; Bambi:2019tjh ; Afrin:2021wlj . Inspired by these works, we
consider the accelerating charged rotating black hole as the supermassive
black hole in M87*, and use the EHT observations to give the constraints on
the corresponding parameters.

To obtain the constraints, we should introduce the explicit definitions of the
observables $\Delta C$, $D_{x}$ and $\psi_{d}$, respectively. The center of
the shadow can be set as ($x_{c}=\frac{x_{r}+x_{l}}{2},y_{c}=0$). The boundary
of the shadow can be expressed with the polar corrdinates ($\phi,R(\phi)$) as

$\displaystyle\phi=\tan^{-1}\Big{(}\frac{y}{x-x_{c}}\Big{)},~{}~{}R(\phi)=\sqrt{(x-x_{c})^{2}+y^{2}},$
(26)

and the circularity deviation $\Delta C$ of the black hole shadow can be
defined as Bambi:2019tjh

$\displaystyle\Delta
C=\frac{1}{\bar{R}}\sqrt{\int^{2\pi}_{0}(R(\phi)-\bar{R})^{2}d\phi},$ (27)

with the average radius

$\displaystyle\bar{R}=\frac{1}{2\pi}\int^{2\pi}_{0}R(\phi)d\phi.$ (28)

The axis ratio is expressed Kumar:2018ple ; Banerjee:2019nnj

$\displaystyle D_{x}=\frac{y_{t}-y_{b}}{x_{r}-x_{l}},$ (29)

and the definition of the characteristic areal radius of the shadow curve can
be expressed as Banerjee:2019nnj ; Abdujabbarov:2015xqa

$\displaystyle
R_{a}=\sqrt{\frac{2}{\pi}\int^{r_{max}}_{r_{min}}\Big{(}y(r)\frac{dx(r)}{dr}\Big{)}}.$
(30)

From the definitions of the obsverations, we can see that both the circularity
deviation $\Delta C$ and the axis ratio $D_{x}$ are the relative obserations,
which means the expansion or reduction of the outline for the shadow will not
affect $\Delta C$ or $D_{x}$. Roughly speaking, Fig. 2(b) shows that with the
accelerating factor $A$ increasing, the diagrams of the shadow for the
accelerating KN black hole will scale down, which implies that the EHT
observables $\Delta C,D_{x}$ might not impose limitations on the range of
accelerating factor $A$. This conclusion gains further support from Fig. 6,
which confirms that the EHT observations $\Delta C\lesssim 0.1$ and
$D_{x}\lesssim 4/3$ are insufficient to effectively constrain the accelerating
factor $A$. Additionally, Fig. 6 reveals a diminishing trend in circularity
deviation $\Delta C$ with increasing accelerating factor $A$, while the impact
of accelerating factor $A$ on the axis ratio $D_{x}$ remains negligible.
Furthermore, for a KN black hole with near-extreme behavior or large rotation
parameter $a$, both $\Delta C$ and $D_{x}$ are more pronounced. Consequently,
to establish constraints on the parameters of the accelerating KN black hole
using the EHT observables $\Delta C$ and $D_{x}$, it is advisable to present
the outcomes in the $(a,e)$ plane. From Fig. 7, we can see that the whole
parameters space of ($a,e$) satisfy the EHT observations $\Delta C\lesssim
0.1$ and $D_{x}\lesssim 4/3$, which indicates that it is impossible to make
any constraint on the parameters of the accelerating KN black hole.

Figure 8 shows the constraints of the parameters for the black hole by the EHT
observation with the shadow size $R_{a}$. All the real curves represent the
lower bound $R_{a}=4.31M$, and the reasonable parameters with the constraints
below these curves. The results illustrate that the bigger accelerating factor
$A$, the stronger restriction on the rotating parameter $a$ and charge
parameter $e$. From the extremely slow accelerating case, i.e., $Ar_{0}\leq
0.01$, the charge parameter $e$ has maximum range as $e/M\in(0,0.9)$, which is
consistent with the conclusion in EventHorizonTelescope:2021dqv . While,
according to the range of shadow size $R_{a}$, we can not make any constraints
on the rotating parameter $a$. Furthermore, we consider the maximum value
range of the accelerating factor $A$ by setting the parameters as $a/M=e/M=0$,
and the results shows that $Ar_{0}\in(0,0.558)$, which is shown in Fig. 9.

Figure 6: The density diagrams of the circularity $\Delta C$ and axial ratio
$D_{x}$ with the ETH data $M=6.5\times 10^{9}M_{\odot}$, $r_{0}=16.8Mpc$, and
$\theta_{0}=17^{o}$. The solid lines are red ($a=0.6$), green ($a=0.7$), and
blue ($a=0.86$) with $e=0.5$. The dashed lines are red ($e=0.3$), green
($e=0.5$) and green ($e=0.76$) with $a=0.8$.

Figure 7: The density diagrams of the circularity $\Delta C$ and axial ratio
$D_{x}$ in the $(a,e)$ plane with the ETH data $M=6.5\times 10^{9}M_{\odot}$,
$r_{0}=16.8Mpc$, and $\theta_{0}=17^{o}$. The parameter $A$ is set as
$A=0.01\frac{1}{r_{0}}$. Figure 8: The constraints of the ETH observation
$4.31M\leq R_{a}\leq 6.08M$ on the parameters ($a,e$). The real curves
represent $R_{a}=4.31M$ with different accelerating factor $A$. Figure 9: The
constraint of the ETH observation $4.31M\leq R_{a}\leq 6.08M$ on the
accelerating factor $A$ with the parameters $a/M=e/M=0$. Here, we set
$A=\rho\frac{r}{r_{0}}$.

##  V Summary

In this paper, we concentrated on the shadow of the accelerating KN black
hole, and analyzed the influence of the parameters on the shadow and its
observables. The rotating parameter $a$, the charged parameter $e$, and the
inclination angle $\theta_{0}$ affect the shadow with the qualitatively
similar to that of Kerr-Newman black holes Perlick ; Grenzebach ; Tsukamoto ;
Xavier ; Meng . Besides, the size of the shadow for the accelerating KN black
hole case will scale down with the accelerating factor $A$.

Then, we analyzed the influence of the parameters on the shadow observables,
i.e., the reference radius $R_{s}$ and the distortion $\delta s$. The rotating
parameter $a$ dominates the change of the distortion $\delta s$, and the
reference radius $R_{s}$ decreases with the charge parameter $e$, rapidly. The
result shows that although the accelerating factor $A$ can affect the best
viewing angle, which makes $R_{s}$ or $\delta s$ maximum, the deviations from
$\theta_{0}=\frac{\pi}{2}$ are limited, i.e., the deviations are less than
$1\%$.

Finally, we assumed the accelerating KN black hole as the supermassive M87*
black hole, and used the EHT observables to constrain the black hole
parameters. We found that the whole parameters space ($a,e$) satisfy the EHT
observables $\Delta C$ and $D_{x}$ constraints, which means that it can not
rule out the supermassive M87* black hole is an accelerating KN black hole
through the EHT observables $\Delta C$ and $D_{x}$. On the other hand,
according to the observable shadow size $R_{a}$, the result shows that the
bigger accelerating factor $A$ is, the smaller ranges of parameter $a$ and $e$
are. For an extremely slow accelerating case $(Ar_{0}\leq 0.01)$, the ranges
of rotating parameter $a$ and charge parameter $e$ are $a/M\in(0,1)$ and
$e/M\in(0,0.9)$. The maximum range of the accelerating factor is $Ar_{0}\leq
0.558$ for a accelerating Schwarzschild case with $(a/M=e/M=0)$.

## Acknowledgements

This work is supported in part by the National Natural Science Foundation of
China (Grants No. 12147175, No. 12205129, No. 12175105, No. 11575083, No.
11565017), and the China Postdoctoral Science Foundation (Grant No.
2021M701529, No. 2021M700729).

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M}_{\phi}$ is a closed subspace of $C^{k,\beta}(M)$ by [37], Theorem 1.5.4,
which is a version of the Schauder estimates.

We claim that the index of $D{\cal M}_{\phi}$ is zero. That would imply that

$codim\,Im(D{\cal M}_{\phi})=1.$

Since $Im(D{\cal M}_{\phi})\subset\tilde{C}^{k,\beta}(M)$, and since $\dim
C^{k,\beta}(M)/\tilde{C}^{k,\beta}(M)=1$, it follows that $Im(D{\cal
M}_{\phi})=\tilde{C}^{k,\beta}(M)$.

It remains to prove that the index of $D{\cal M}_{\phi}$ is zero. For linear
elliptic operators with infinitely smooth coefficients on real valued
functions this is always true since the adjoint operator has the same symbol
as the original operator. The operators with Hölder coefficients require more
care. By [37], Theorem 1.5.4, the image of a linear elliptic operator of
second order with $C^{k,\alpha}$-coefficients, $k\geq 2$,

$P\colon C^{k+2,\alpha}(M)\longrightarrow C^{k,\alpha}(M)$

is equal to the orthogonal complement of the kernel of its adjoint $P^{*}$.
Let us show that in our case the adjoint to $D{\cal M}_{\phi}$ is equal to
itself using Corollary 10.3(b):

$\displaystyle(D{\cal
M}_{\phi}(\psi),\xi)=2\int_{M}\xi\det(G_{0}+Hess_{\mathbb{O}}\phi,Hess_{\mathbb{O}}\psi)\Theta=$
$\displaystyle
2\int_{M}\psi\det(G_{0}+Hess_{\mathbb{O}}\phi,Hess_{\mathbb{O}}\xi)\Theta=(D{\cal
M}_{\phi}(\xi),\psi).$

Thus $D{\cal M}_{\phi}$ is self-adjoint and the kernel of its adjoint is also
1-dimensional. Q.E.D.

  3. 3

######  22.2 Proposition.

Let $M^{16}$ be a compact $Spin(9)$-manifold. Let $k\geq 2$ be an integer,
$\beta\in(0,1)$. Then the map

${\cal M}\colon{\cal U}^{k+2,\beta}\longrightarrow{\cal V}^{k,\beta}$

is a diffeomorphism of Banach manifolds, in particular it is onto.

Proof. ${\cal M}$ is one-to-one by the uniqueness of the solution (Theorem
12.1).

By Proposition 22.1 it suffices to show that ${\cal M}$ is onto. Since ${\cal
V}^{k,\beta}$ is obviously connected (and even convex), and since the image of
${\cal M}$ is open by Proposition 22.1, it suffices to show that the image of
${\cal M}$ is a closed subset of ${\cal V}^{k,\beta}$.

Let us have a sequence of points in the image ${\cal
M}\phi_{i}\overset{C^{k,\beta}}{\longrightarrow}e^{f}\in{\cal V}^{k,\beta}$
where $\phi_{i}\in{\cal U}^{k+2,\beta}$. By Theorems 13.5, 17.4, 21.1, there
exist $\alpha\in(0,1)$ and a constant $C$ both depending on $||f||_{C^{2}}$,
$M$, $G_{0}$, $\Theta$ such that $||\phi_{i}||_{C^{2,\alpha}}<C$ for $i\gg 1$.
By the Arzelà-Ascoli theorem choosing a subsequence we may assume that
$\phi_{i}\longrightarrow\phi$ in $C^{2}(M)$. Clearly $\phi\in C^{2,\alpha}(M)$
and one has ${\cal M}(\phi)=e^{f}$. Explicitly one has

$\displaystyle\det(G_{0}+Hess_{\mathbb{O}}\phi)=e^{f}\det G_{0}.$ (22.1)

Moreover, $G_{0}+Hess_{\mathbb{O}}\phi\geq 0$. This, the Sylvester criterion
(Proposition 5.4), and (22.1) imply that the inequality is strict, i.e.
$G_{0}+Hess_{\mathbb{O}}\phi>0$, and hence the equation (22.1) is elliptic
with $C^{\infty}$-smooth coefficients on the left hand side and with
$C^{k,\beta}$ on the right hand side. Hence Lemma 17.16 from [27] implies that
$\phi\in C^{k+2,\beta}(M)$. Thus $\phi\in{\cal V}^{k+2,\beta}$ and
$e^{f}={\cal M}(\phi)\in Im({\cal M})$. Q.E.D.

  4. 4

Finally let us state the main result of the paper which is an immediate
consequence of Proposition 22.2.

######  22.3 Theorem.

Let $M^{16}$ be a compact connected $Spin(9)$-manifold. Let $G_{0}$ be
$C^{\infty}$-smooth positive definite section of the bundle $\underline{{\cal
H}_{2}}(M)$. Let $k\geq 2$ be an integer, $\beta\in(0,1)$, and $f\in
C^{k,\beta}(M)$.

Then there is a unique constant $A>0$ such that the octoionic Monge-Ampère
equation

$\displaystyle\det(G_{0}+Hess_{\mathbb{O}}\phi)=Ae^{f}\det G_{0}$ (22.2)

has a unique, up to a constant, $C^{2}$-smooth solution $\phi$ which
necessarily belongs to $C^{k+2,\beta}(M)$. If $f$ is $C^{\infty}$-smooth, then
the solution $\phi$ is also $C^{\infty}$-smooth.

######  22.4 Remark.

Recall that the constant $A$ in the theorem is defined by the following
condition

$\int_{M}(Ae^{f}-1)\cdot\det G_{0}\cdot\Theta=0,$

where $\Theta$ is a parallel positive $3/4$-density on $M$.

##  23 Appendix

In this appendix we describe the commutative algebra computer computation
necessary to prove the existence of a solution for system (9.5) for arbitrary
closed smooth current $T$ satisfying conditions (9.1). Ehrenpreis’ Theorem 9.3
combined with this computation implies existence of a solution.

Setting $q_{1}:=x$ and $q_{2}:=y$ we have

$\displaystyle x=\sum_{k=0}^{7}x_{k}e_{k},\quad y=\sum_{k=0}^{7}y_{k}e_{k}.$
(23.1)

In terms of these variables system (9.5) reduces to the following system of
second order differential equations

$\displaystyle\left\\{\begin{array}[]{l}\frac{\partial^{2}u}{\partial\bar{x}\partial
x}=\sum_{i=0}^{7}\frac{\partial^{2}u}{\partial x_{i}^{2}}=T_{\bar{1}1},\\\ \\\
\frac{\partial^{2}u}{\partial\bar{y}\partial
y}=\sum_{i=0}^{7}\frac{\partial^{2}u}{\partial y_{i}^{2}}=T_{\bar{2}2},\\\ \\\
\frac{\partial^{2}u}{\partial\bar{x}\partial
y}=\sum_{i,j=0}^{7}e_{i}\bar{e}_{j}\frac{\partial^{2}u}{\partial x_{i}\partial
y_{j}}=T_{\bar{1}2}.\end{array}\right.$ (23.7)

This system is equivalent to a following system of $10$ scalar equations

$\displaystyle P_{1}u=T_{\bar{1}1},\quad P_{2}u=T_{\bar{2}2},$ $\displaystyle
P_{k+3}u=T_{\bar{1}2}^{k},\quad k=0,\cdots,7,$ (23.8)

where an upper index $k$ in $T_{\bar{1}2}$ stands for a $k$-th component of
octonion $T_{\bar{1}2}$ and $P_{i}$ are the second order differential
operators. These operators, after substitution $\partial_{x_{i}}\longmapsto
x_{i}$, $\partial_{y_{i}}\longmapsto y_{i},$ become quadratic polynomials that
read

$\displaystyle P_{1}=x\bar{x}=\sum_{i=0}^{7}x_{i}^{2},\quad
P_{2}=y\bar{y}=\sum_{i=0}^{7}y_{i}^{2},$ (23.9)
$\displaystyle\left\\{\begin{array}[]{r}P_{3}=x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+x_{4}y_{4}+x_{5}y_{5}+x_{6}y_{6}+x_{7}y_{7},\\\
P_{4}=x_{1}y_{0}-x_{0}y_{1}+x_{3}y_{2}-x_{2}y_{3}+x_{5}y_{4}-x_{4}y_{5}-x_{7}y_{6}+x_{6}y_{7},\\\
P_{5}=x_{2}y_{0}-x_{3}y_{1}-x_{0}y_{2}+x_{1}y_{3}+x_{6}y_{4}+x_{7}y_{5}-x_{4}y_{6}-x_{5}y_{7},\\\
P_{6}=x_{3}y_{0}+x_{2}y_{1}-x_{1}y_{2}-x_{0}y_{3}+x_{7}y_{4}-x_{6}y_{5}+x_{5}y_{6}-x_{4}y_{7},\\\
P_{7}=x_{4}y_{0}-x_{5}y_{1}-x_{6}y_{2}-x_{7}y_{3}-x_{0}y_{4}+x_{1}y_{5}+x_{2}y_{6}+x_{3}y_{7},\\\
P_{8}=x_{5}y_{0}+x_{4}y_{1}-x_{7}y_{2}+x_{6}y_{3}-x_{1}y_{4}-x_{0}y_{5}-x_{3}y_{6}+x_{2}y_{7},\\\
P_{9}=x_{6}y_{0}+x_{7}y_{1}+x_{4}y_{2}-x_{5}y_{3}-x_{2}y_{4}+x_{3}y_{5}-x_{0}y_{6}-x_{1}y_{7},\\\
P_{10}=x_{7}y_{0}-x_{6}y_{1}+x_{5}y_{2}+x_{4}y_{3}-x_{3}y_{4}-x_{2}y_{5}+x_{1}y_{6}-x_{0}y_{7}.\end{array}\right.$
(23.18)

We note that polynomials $P_{k+3}$ for $k=0,\cdots,7$ are coefficient in front
of $e_{k}$ in the expression $x\bar{y}.$

Setting $f_{1}:=T_{\bar{1}1},$ $f_{2}:=T_{\bar{2}2}$ and
$f_{k+3}:=T_{\bar{1}2}^{k},\quad k=0,\cdots,7,$ we now will apply Theorem 9.3.
To verify that the assumptions of this theorem are met we implement the
strategy described in Section 9 part 6.

Setting polynomial ring $A:=\mathbb{Q}[x_{0},\cdots,x_{7},y_{0},\cdots,y_{7}]$
(by Lemma 9.5 it suffices to consider rational rather than real coefficients)
we now look for generators of the submodule $N_{\mathbb{Q}}$ of the free
$A$-module $A^{10}$

$N_{\mathbb{Q}}=\\{(Q_{1},\dots,Q_{10})\in
A^{10}|\,\sum_{i=1}^{10}Q_{i}P_{i}=0\\}.$

To obtain such generators we use symbolic computations which we implement in a
well known program Macaulay2. We give a printout of an executed program for
Macaulay2 for computing generators of the submodule $N$ at the end of this
appendix. The program was executed at Macaulay2Web interface located at
https:/ /www.unimelb-macaulay2.cloud.edu.au/. Interested reader can download
our program from
http://www.math.kent.edu/$\sim$gordon/MA/generator_of_submoduleXYb.txt
The output of the program produces $16$ first order polynomials as generators.
These generators result in $16$ first order polynomials which after back-
substitution $x_{i}\longmapsto\partial_{x_{i}}$,
$y_{i}\longmapsto\partial_{y_{i}}$ give a system of $16$ first order
differential equations that read:

$\displaystyle\left(\begin{array}[]{l}\partial_{x_{0}}\\\ \partial_{x_{1}}\\\
\partial_{x_{2}}\\\ \partial_{x_{3}}\\\ \partial_{x_{4}}\\\
\partial_{x_{5}}\\\ \partial_{x_{6}}\\\
\partial_{x_{7}}\end{array}\right)T_{\bar{2}2}=\left(\begin{array}[]{rrrrrrrr}\partial_{y_{0}}&-\partial_{y_{1}}&-\partial_{y_{2}}&-\partial_{y_{3}}&-\partial_{y_{4}}&-\partial_{y_{5}}&-\partial_{y_{6}}&-\partial_{y_{7}}\\\
\partial_{y_{1}}&\partial_{y_{0}}&\partial_{y_{3}}&-\partial_{y_{2}}&\partial_{y_{5}}&-\partial_{y_{4}}&-\partial_{y_{7}}&\partial_{y_{6}}\\\
\partial_{y_{2}}&-\partial_{y_{3}}&\partial_{y_{0}}&\partial_{y_{1}}&\partial_{y_{6}}&\partial_{y_{7}}&-\partial_{y_{4}}&-\partial_{y_{5}}\\\
\partial_{y_{3}}&\partial_{y_{2}}&-\partial_{y_{1}}&\partial_{y_{0}}&\partial_{y_{7}}&-\partial_{y_{6}}&\partial_{y_{5}}&-\partial_{y_{4}}\\\
\partial_{y_{4}}&-\partial_{y_{5}}&-\partial_{y_{6}}&-\partial_{y_{7}}&\partial_{y_{0}}&\partial_{y_{1}}&\partial_{y_{2}}&\partial_{y_{3}}\\\
\partial_{y_{5}}&\partial_{y_{4}}&-\partial_{y_{7}}&\partial_{y_{6}}&-\partial_{y_{1}}&\partial_{y_{0}}&-\partial_{y_{3}}&\partial_{y_{2}}\\\
\partial_{y_{6}}&\partial_{y_{7}}&\partial_{y_{4}}&-\partial_{y_{5}}&-\partial_{y_{2}}&\partial_{y_{3}}&\partial_{y_{0}}&-\partial_{y_{1}}\\\
\partial_{y_{7}}&-\partial_{y_{6}}&\partial_{y_{5}}&\partial_{y_{4}}&-\partial_{y_{3}}&-\partial_{y_{2}}&\partial_{y_{1}}&\partial_{y_{0}}\end{array}\right)\left(\begin{array}[]{c}T_{\bar{1}2}^{0}\\\
T_{\bar{1}2}^{1}\\\ T_{\bar{1}2}^{2}\\\ T_{\bar{1}2}^{3}\\\
T_{\bar{1}2}^{4}\\\ T_{\bar{1}2}^{5}\\\ T_{\bar{1}2}^{6}\\\
T_{\bar{1}2}^{7}\end{array}\right)$ (23.42)

$\displaystyle\left(\begin{array}[]{l}\partial_{y_{0}}\\\ \partial_{y_{1}}\\\
\partial_{y_{2}}\\\ \partial_{y_{3}}\\\ \partial_{y_{4}}\\\
\partial_{y_{5}}\\\ \partial_{y_{6}}\\\
\partial_{y_{7}}\end{array}\right)T_{\bar{1}1}=\left(\begin{array}[]{rrrrrrrr}\partial_{x_{0}}&\partial_{x_{1}}&\partial_{x_{2}}&\partial_{x_{3}}&\partial_{x_{4}}&\partial_{x_{5}}&\partial_{x_{6}}&\partial_{x_{7}}\\\
\partial_{x_{1}}&-\partial_{x_{0}}&-\partial_{x_{3}}&\partial_{x_{2}}&-\partial_{x_{5}}&\partial_{x_{4}}&\partial_{x_{7}}&-\partial_{x_{6}}\\\
\partial_{x_{2}}&\partial_{x_{3}}&-\partial_{x_{0}}&-\partial_{x_{1}}&-\partial_{x_{6}}&-\partial_{x_{7}}&\partial_{x_{4}}&\partial_{x_{5}}\\\
\partial_{x_{3}}&-\partial_{x_{2}}&\partial_{x_{1}}&-\partial_{x_{0}}&-\partial_{x_{7}}&\partial_{x_{6}}&-\partial_{x_{5}}&\partial_{x_{4}}\\\
\partial_{x_{4}}&\partial_{x_{5}}&\partial_{x_{6}}&\partial_{x_{7}}&-\partial_{x_{0}}&-\partial_{x_{1}}&-\partial_{x_{2}}&-\partial_{x_{3}}\\\
\partial_{x_{5}}&-\partial_{x_{4}}&\partial_{x_{7}}&-\partial_{x_{6}}&\partial_{x_{1}}&-\partial_{x_{0}}&\partial_{x_{3}}&-\partial_{x_{2}}\\\
\partial_{x_{6}}&-\partial_{x_{7}}&-\partial_{x_{4}}&\partial_{x_{5}}&\partial_{x_{2}}&-\partial_{x_{3}}&-\partial_{x_{0}}&\partial_{x_{1}}\\\
\partial_{x_{7}}&\partial_{x_{6}}&-\partial_{x_{5}}&-\partial_{x_{4}}&\partial_{x_{3}}&\partial_{x_{2}}&-\partial_{x_{1}}&-\partial_{x_{0}}\end{array}\right)\left(\begin{array}[]{c}T_{\bar{1}2}^{0}\\\
T_{\bar{1}2}^{1}\\\ T_{\bar{1}2}^{2}\\\ T_{\bar{1}2}^{3}\\\
T_{\bar{1}2}^{4}\\\ T_{\bar{1}2}^{5}\\\ T_{\bar{1}2}^{6}\\\
T_{\bar{1}2}^{7}\end{array}\right)$ (23.66)

Noting that $T_{\bar{2}1}=\bar{T}_{\bar{1}2}$ one can directly verify that the
systems of equations above is a scalar form of the closed current condition
(9.1).

    Macaulay2, version 1.20.0.1
    with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, Isomorphism,
    LLLBases, MinimalPrimes, OnlineLookup, PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone

    i1 : A=QQ[x_0..y_7];

    i2 : P_1 = x_0*x_0 + x_1*x_1 + x_2*x_2 + x_3*x_3 + x_4*x_4 + x_5*x_5 + x_6*x_6 + x_7*x_7;

    i3 : P_2 = y_0*y_0 + y_1*y_1 + y_2*y_2 + y_3*y_3 + y_4*y_4 + y_5*y_5 + y_6*y_6 + y_7*y_7;

    i4 : P_3 = x_0*y_0 + x_1*y_1 + x_2*y_2 + x_3*y_3 + x_4*y_4 + x_5*y_5 + x_6*y_6 + x_7*y_7;

    i5 : P_4 = x_1*y_0 - x_0*y_1 + x_3*y_2 - x_2*y_3 + x_5*y_4 - x_4*y_5 - x_7*y_6 + x_6*y_7;

    i6 : P_5 = x_2*y_0 - x_3*y_1 - x_0*y_2 + x_1*y_3 + x_6*y_4 + x_7*y_5 - x_4*y_6 - x_5*y_7;

    i7 : P_6 = x_3*y_0 + x_2*y_1 - x_1*y_2 - x_0*y_3 + x_7*y_4 - x_6*y_5 + x_5*y_6 - x_4*y_7;

    i8 : P_7 = x_4*y_0 - x_5*y_1 - x_6*y_2 - x_7*y_3 - x_0*y_4 + x_1*y_5 + x_2*y_6 + x_3*y_7;

    i9 : P_8 = x_5*y_0 + x_4*y_1 - x_7*y_2 + x_6*y_3 - x_1*y_4 - x_0*y_5 - x_3*y_6 + x_2*y_7;

    i10 : P_9 = x_6*y_0 + x_7*y_1 + x_4*y_2 - x_5*y_3 - x_2*y_4 + x_3*y_5 - x_0*y_6 - x_1*y_7;

    i11 : P_10= x_7*y_0 - x_6*y_1 + x_5*y_2 + x_4*y_3 - x_3*y_4 - x_2*y_5 + x_1*y_6 - x_0*y_7;

    i12 : N=matrix{{P_1,P_2,P_3,P_4,P_5,P_6,P_7,P_8,P_9,P_10}};

o12 : Matrix $A^{1}\longleftarrow A^{10}$

    i13 : kernel N

$\texttt{o13~{}=~{}image}\
\begin{array}[]{l}\left\\{2\right\\}\vphantom{00000000-y_{0}y_{1}y_{2}y_{3}y_{4}y_{5}y_{6}y_{7}}\\\
\left\\{2\right\\}\vphantom{x_{0}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}00000000}\\\
\left\\{2\right\\}\vphantom{-y_{0}-y_{1}-y_{2}-y_{3}-y_{4}-y_{5}-y_{6}-y_{7}x_{0}-x_{1}-x_{2}-x_{3}-x_{4}-x_{5}-x_{6}-x_{7}}\\\
\left\\{2\right\\}\vphantom{y_{1}-y_{0}y_{3}-y_{2}y_{5}-y_{4}-y_{7}y_{6}x_{1}x_{0}-x_{3}x_{2}-x_{5}x_{4}x_{7}-x_{6}}\\\
\left\\{2\right\\}\vphantom{y_{2}-y_{3}-y_{0}y_{1}y_{6}y_{7}-y_{4}-y_{5}x_{2}x_{3}x_{0}-x_{1}-x_{6}-x_{7}x_{4}x_{5}}\\\
\left\\{2\right\\}\vphantom{y_{3}y_{2}-y_{1}-y_{0}y_{7}-y_{6}y_{5}-y_{4}x_{3}-x_{2}x_{1}x_{0}-x_{7}x_{6}-x_{5}x_{4}}\\\
\left\\{2\right\\}\vphantom{y_{4}-y_{5}-y_{6}-y_{7}-y_{0}y_{1}y_{2}y_{3}x_{4}x_{5}x_{6}x_{7}x_{0}-x_{1}-x_{2}-x_{3}}\\\
\left\\{2\right\\}\vphantom{y_{5}y_{4}-y_{7}y_{6}-y_{1}-y_{0}-y_{3}y_{2}x_{5}-x_{4}x_{7}-x_{6}x_{1}x_{0}x_{3}-x_{2}}\\\
\left\\{2\right\\}\vphantom{y_{6}y_{7}y_{4}-y_{5}-y_{2}y_{3}-y_{0}-y_{1}x_{6}-x_{7}-x_{4}x_{5}x_{2}-x_{3}x_{0}x_{1}}\\\
\left\\{2\right\\}\vphantom{y_{7}-y_{6}y_{5}y_{4}-y_{3}-y_{2}y_{1}-y_{0}x_{7}x_{6}-x_{5}-x_{4}x_{3}x_{2}-x_{1}x_{0}}\end{array}\left(\\!\begin{array}[]{cccccccccccccccc}\vphantom{\left\\{2\right\\}}0&0&0&0&0&0&0&0&-y_{0}&y_{1}&y_{2}&y_{3}&y_{4}&y_{5}&y_{6}&y_{7}\\\
\vphantom{\left\\{2\right\\}}x_{0}&x_{1}&x_{2}&x_{3}&x_{4}&x_{5}&x_{6}&x_{7}&0&0&0&0&0&0&0&0\\\
\vphantom{\left\\{2\right\\}}-y_{0}&-y_{1}&-y_{2}&-y_{3}&-y_{4}&-y_{5}&-y_{6}&-y_{7}&x_{0}&-x_{1}&-x_{2}&-x_{3}&-x_{4}&-x_{5}&-x_{6}&-x_{7}\\\
\vphantom{\left\\{2\right\\}}y_{1}&-y_{0}&y_{3}&-y_{2}&y_{5}&-y_{4}&-y_{7}&y_{6}&x_{1}&x_{0}&-x_{3}&x_{2}&-x_{5}&x_{4}&x_{7}&-x_{6}\\\
\vphantom{\left\\{2\right\\}}y_{2}&-y_{3}&-y_{0}&y_{1}&y_{6}&y_{7}&-y_{4}&-y_{5}&x_{2}&x_{3}&x_{0}&-x_{1}&-x_{6}&-x_{7}&x_{4}&x_{5}\\\
\vphantom{\left\\{2\right\\}}y_{3}&y_{2}&-y_{1}&-y_{0}&y_{7}&-y_{6}&y_{5}&-y_{4}&x_{3}&-x_{2}&x_{1}&x_{0}&-x_{7}&x_{6}&-x_{5}&x_{4}\\\
\vphantom{\left\\{2\right\\}}y_{4}&-y_{5}&-y_{6}&-y_{7}&-y_{0}&y_{1}&y_{2}&y_{3}&x_{4}&x_{5}&x_{6}&x_{7}&x_{0}&-x_{1}&-x_{2}&-x_{3}\\\
\vphantom{\left\\{2\right\\}}y_{5}&y_{4}&-y_{7}&y_{6}&-y_{1}&-y_{0}&-y_{3}&y_{2}&x_{5}&-x_{4}&x_{7}&-x_{6}&x_{1}&x_{0}&x_{3}&-x_{2}\\\
\vphantom{\left\\{2\right\\}}y_{6}&y_{7}&y_{4}&-y_{5}&-y_{2}&y_{3}&-y_{0}&-y_{1}&x_{6}&-x_{7}&-x_{4}&x_{5}&x_{2}&-x_{3}&x_{0}&x_{1}\\\
\vphantom{\left\\{2\right\\}}y_{7}&-y_{6}&y_{5}&y_{4}&-y_{3}&-y_{2}&y_{1}&-y_{0}&x_{7}&x_{6}&-x_{5}&-x_{4}&x_{3}&x_{2}&-x_{1}&x_{0}\end{array}\\!\right)$

o13 : $A$-module, submodule of $A^{10}$

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