# Future Prospects for Constraining Black-Hole Spacetime: Horizon-scale
Variability of Astrophysical Jet

Kotaro Moriyama  Institut für Theoretische Physik, Goethe-Universität
Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-12
Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan  Alejandro Cruz-Osorio
Instituto de Astronomía, Universidad Nacional Autónoma de México, AP 70-264,
Ciudad de México 04510, México  Institut für Theoretische Physik, Goethe-
Universität Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main,
Germany  Yosuke Mizuno  Tsung-Dao Lee Institute, Shanghai Jiao Tong
University, Shengrong Road 520, Shanghai, 201210, People’s Republic of China
School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan
Road, Shanghai, 200240, People’s Republic of China  Institut für Theoretische
Physik, Goethe-Universität Frankfurt, Max-von-Laue-Strasse 1, D-60438
Frankfurt am Main, Germany  Christian M. Fromm  Institut für Theoretische
Physik und Astrophysik, Universität Würzburg, Emil-Fischer-Str. 31, D-97074
Würzburg, Germany  Institut für Theoretische Physik, Goethe-Universität
Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany  Max-
Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
Antonios Nathanail  Research Center for Astronomy, Academy of Athens, Soranou
Efessiou 4, GR-115 27, Athens, Greece  Institut für Theoretische Physik,
Goethe-Universität Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am
Main, Germany  Luciano Rezzolla  Institut für Theoretische Physik, Goethe-
Universität Frankfurt, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main,
Germany  Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1,
60438 Frankfurt, Germany  School of Mathematics, Trinity College, Dublin 2,
Ireland

###### Abstract

The Event Horizon Telescope (EHT) Collaboration has recently published the
first horizon-scale images of the supermassive black holes M87∗ and Sgr A∗ and
provided some first information on the physical conditions in their vicinity.
The comparison between the observations and the three-dimensional general-
relativistic magnetohydrodynamic (GRMHD) simulations has enabled the EHT to
set initial constraints on the properties of these black-hole spacetimes.
However, accurately distinguishing the properties of the accretion flow from
those of the spacetime, most notably, the black-hole mass and spin, remains
challenging because of the degeneracies the emitted radiation suffers when
varying the properties of the plasma and those of the spacetime. The next-
generation EHT (ngEHT) observations are expected to remove some of these
degeneracies by exploring the complex interplay between the disk-jet dynamics,
which represents one of the most promising tools for extracting information on
the black-hole spin. By using GRMHD simulations of magnetically arrested disks
(MADs) and general-relativistic radiative-transfer (GRRT) calculations of the
emitted radiation, we have studied the properties of the jet and the
accretion-disk dynamics on spatial scales that are comparable with the
horizon. In this way, we are able to highlight that the radial and azimuthal
dynamics of the jet are well correlated with the black-hole spin. Based on the
resolution and image reconstruction capabilities of the ngEHT observations of
M87∗, we can assess the detectability and associated uncertainty of this
correlation. Overall, our results serve to assess what are the prospects for
constraining the black-hole spin with future EHT observations.

black hole physics – galaxies: individual (M87) – galaxies: jets – techniques:
high angular resolution – techniques: image processing – magnetohydrodynamics
(MHD) – radiative transfer

##  1 Introduction

The observational exploration of the physical conditions in the vicinity of a
black hole is one of the most promising avenues for an accurate mapping of the
spacetime and understanding of the origin of the relativistic jets. Through
groundbreaking efforts, the Event Horizon Telescope (EHT) Collaboration has
recently provided images of the horizon-scale emission of the accretion flow
in the vicinity of supermassive black holes in the nearby radio galaxy M87
(M87∗) and in the Galactic center (Sgr A∗). More specifically, in 2017, the
EHT array consisted of eight stations at six geographical sites that
successfully observed M87∗ (Event Horizon Telescope Collaboration et al.
2019a, b, c, d, e, f) and Sgr A∗ (Event Horizon Telescope Collaboration et al.
2022a, b, c, d, e, f) with the unprecedented horizon-scale resolution of $\sim
20~{}\mathrm{\mu as}$. The imaging surveys of M87∗ and Sgr A∗ revealed bright
emission rings that are consistent with the theoretical predictions of general
relativity.

Comparing the EHT observations with the theoretical simulations enables us to
constrain the black-hole spacetime within the mass and spin with general
relativity as well as alternative theories. The measured shadow diameters
($d\sim 42~{}\mathrm{\mu as}$ for M87∗ and $d\sim 52~{}\mathrm{\mu as}$ for
Sgr A∗) are consistent with the prediction of general relativity (e.g., Event
Horizon Telescope Collaboration et al. 2019e, 2022e; Younsi et al. 2023) and
can be also used to discriminate from general relativity’s possible deviations
of the black-hole spacetime (e.g., Kocherlakota & Rezzolla 2020; Event Horizon
Telescope Collaboration et al. 2019f, 2022f; Cruz-Osorio et al. 2021; Vagnozzi
et al. 2022). The comparison of the observations with a large library of
three-dimensional general-relativistic magnetohydrodynamic (GRMHD) simulations
provides information on the plausible ranges of model parameters for the
spacetime, the magnetic field, and the accretion flow properties (e.g., Event
Horizon Telescope Collaboration et al. 2019e, 2022e).

Furthermore, the horizon-scale detection of the polarized profile around M87∗
has provided additional constraints on the magnetic fields (Event Horizon
Telescope Collaboration et al. 2021a, b). In particular, the polarization
properties of M87∗ can be explained with magnetic fields around the emission
region that are predominantly poloidal, and this has allowed us to obtain the
first measurements of the magnetic field strength, electron density, and
temperature with horizon-scale resolution. Moreover, by comparing the imaging
results with the library of GRMHD polarized images, it was possible to confirm
their consistency with magnetically arrested accretion disks (MADs).

Despite this considerable progress, with the current angular resolution of the
EHT observations, it remains challenging to constrain the detailed structure
of the black-hole spacetime solely from the horizon-scale images. Despite the
significant progress achieved, the current angular resolution of the EHT
observations poses challenges in accurately determining the detailed structure
of black-hole spacetime based solely on horizon-scale images. For instance,
according to general relativity, the diameter of the black-hole shadow is
predicted to depend weakly on the observer’s inclination angle and the black-
hole spin, approximately $5.0\pm 0.2r_{\rm g}$, where $r_{\rm g}$ represents
the gravitational radius (e.g., see Takahashi 2004; Event Horizon Telescope
Collaboration et al. 2022f). Although these parameters are crucial in
characterizing the spacetime of a black hole, achieving a more precise
constraint on the black-hole parameters would require additional criteria
and/or improvements in observations. The essential role of non-thermal
emission has been demonstrated with observations of M87∗ at the regime between
low (43 and 86 GHz) and high (near-IR) frequencies (Pandya et al. 2016;
Davelaar et al. 2019; Cruz-Osorio et al. 2022; Fromm et al. 2022). Overall,
GRMHD simulations have clearly demonstrated the intricate dependence of the
total intensity and polarization map on the properties of both the plasma and
spacetime in a complex manner. Differentiating between them is not a
straightforward endeavor and necessitates the application of innovative
methodologies.

Future developments of the EHT capabilities as those explored with the next
generation EHT (ngEHT) project, represent one of the most promising to address
the observational and theoretical challenges discussed above. The aim of the
development is to achieve improved angular resolution across a wide range of
frequencies, including 86, 230, and 345 GHz. The possibility of improved
spacetime measurements utilizing ngEHT observations is currently being
intensively explored (e.g., Chael et al. 2021; Ricarte et al. 2023), while
some research has investigated the non-thermal emission effects and
constrained the observational results. In particular, Fromm et al. (2022) have
parameterized the non-thermal effect and investigated the characteristics of
model images and broadband spectra spanning from the radio to the near-
infrared range for M87∗. The obtained best-bet models for each spin showed a
strong agreement with the spectral features observed at multiple wavelengths
(Davelaar et al. 2019).

In addition, the horizon-scale imaging of the M87∗ jet represents a promising
tool for extracting information on the spacetime properties and measuring the
non-thermal effects simultaneously. The observation of the M87∗ jet has a long
history and has been conducted across a broad frequency range, from radio to
$\gamma$-ray wavelengths (e.g., Curtis 1918; Reid et al. 1982; Hada et al.
2013; Kim et al. 2018; Snios et al. 2019; MAGIC Collaboration et al. 2020).
The time-averaged jet morphology was further compared with 86 GHz GMVA
observations of M87∗ to constrain the plausible parameter spaces of GRMHD and
general-relativistic radiative transfer (GRRT) simulations (Cruz-Osorio et al.
2022; Fromm et al. 2022). Especially, recent 86 GHz imaging results of M87 (Lu
et al. 2023) have revealed an edge-brightened jet that is connected to the
accretion flow of the black hole. This observational milestone provides strong
motivation for theoretical and observational studies aimed at investigating
the relationship between the jet and photon ring. The recent observational
improvement and forthcoming ngEHT observations hold great potential to detect
the horizon-scale structure of the jet and to capture the disk dynamics of the
M87∗ through long-time monitoring (Roelofs et al. 2023).

In this paper, we investigate the dynamics of jets and accretion disks that
can be probed through the ngEHT observations. The previous studies in this
area (e.g., Raymond et al. 2021; Roelofs et al. 2023) have provided evidence
for the detectability of the jet base and photon ring. We report on three-
dimensional GRMHD simulations of MAD disks with the numerical code BHAC to
obtain an accurate modeling of the accretion flow in the vicinity of the black
hole (Porth et al. 2017; Olivares et al. 2019). In a comparative study with
other GRMHD codes utilized in the EHT Collaboration, BHAC demonstrated
maturity, capability, and consistency in simulation results (Porth et al.
2019). Additionally, a comprehensive convergence study examining the Riemann
problem, shocks, wave propagation, and accretion is available in Porth et al.
(2017) and Olivares et al. (2019). The synthetic images of the emission are
calculated with the GRRT scheme implemented in the numerical code BHOSS,
introducing a hybrid thermal-non-thermal particle distribution function
optimized by the observed continuum spectra of M87∗ (Fromm et al. 2022; Cruz-
Osorio et al. 2022). The Radiative Transfer Scheme was rigorously tested to
ensure its consistency and accuracy within the context of the general
relativistic radiative transfer codes used by the EHT Collaboration (Gold et
al. 2020). We primarily focus on the dynamic properties of the jet and
accretion disk and demonstrate that the azimuthal variation and radial jet
propagation provide information on the black-hole spacetime. Furthermore, by
comparing the simulations and reconstructed images, we measure the uncertainty
and detectability of the black-hole characteristics under the planned ngEHT
observations.

The plan of this paper is as follows: In Section 2, we introduce the setup of
the GRMHD and GRRT simulations and the strategy of the synthetic ngEHT
observations. In Section 3, we demonstrate the fundamental properties of the
utilized GRMHD and GRRT models. In Section 4, we investigate the properties of
the jet-disk dynamics and demonstrate the detectability with the expected
ngEHT observations. Section 5 is instead devoted to investigating the physical
origin of the detected jet dynamics discussed in Section 4. Finally, in
Section 6, we present our conclusions and future prospects.

##  2 Methods

###  2.1 General-Relativistic magnetohydrodynamic simulations and radiative
transfer

Our investigation of the dynamics of jet launching and accretion flow around a
black hole has employed three-dimensional GRMHD simulations performed with the
numerical code BHAC (Porth et al. 2017; Olivares et al. 2019), which solves
the GRMHD equations on a Kerr background expressed in the spherical Kerr-
Schild coordinates $(t,r,\theta,\phi)$. The grid spacing is logarithmic in the
radial and linear in the polar and azimuthal directions. Hereafter, we use
geometrized units, in which the gravitational constant, $G$, and the speed of
light, $c$, are set to be unity and define the gravitational radius as $r_{\rm
g}:=GM/c^{2}$, where $M$ is the black-hole mass and the gravitational time
$t_{\rm g}:=r_{\rm g}/c$.

The conditions of GRMHD simulations (the convention, resolution of the grid,
and initial conditions employed in our analysis) are the same as those in
previous simulations, which were used to compare with M87∗ observations (Fromm
et al. 2022; Cruz-Osorio et al. 2022). We take the time window with a quasi-
stationary mass accretion rate between $13000\leq t/t_{\rm g}\leq 15000$, with
a time resolution of $t=10~{}t_{\rm g}\sim 3.6~{}{\rm days}$ for M87∗. The
scaling factors of the accretion rate (summarized in Table 1) and model
parameters are selected to reproduce a consistent total flux at 230 GHz ($\sim
1~{}{\rm Jy}$) and a continuum spectrum density within the frequency range of
$10^{10}~{}{\rm Hz}\leq\nu\leq 10^{16}~{}{\rm Hz}$.

Utilizing the GRMHD results, we calculated synthetic images using the general-
relativistic ray-tracing code employed in the BHOSS (Younsi et al. 2012,
2020), which solves the co-variant radiative-transfer equation. The
calculation of the synthetic images is performed with the fast light
approximation, in which the fluid is assumed not to change during the photon
propagation so that an image corresponds to a single time slice. Synchrotron
radiation is estimated to be the main (only) source of emission at 230 GHz and
the electron energy distribution is characterized by the so-called ”kappa”
model (Xiao 2006) with a non-thermal energy contribution from the jet
(Davelaar et al. 2019). We focus specifically on the best-bet models of MAD
with five different spins ($a_{*}=-0.9375,-0.5,0.0,0.5,$ and $0.9375$, where
$a_{*}=J/M^{2}$ represents the normalized spin and $J$ denotes the angular
momentum) at an inclination angle of $i=160^{\circ}$. These models exhibit a
continuum spectrum consistent with the observations of M87∗ (Cruz-Osorio et
al. 2022; Event Horizon Telescope Collaboration et al. 2019e). The black-hole
mass is set to be $6.5\times 10^{9}M_{\odot}$, and the distance is set to be
$16.8~{}{\rm Mpc}$. Additionally, we set the azimuthal angle of the spin axis
as $\phi_{\rm obs}(=\arctan(y_{\rm obs}/x_{\rm obs}))=18^{\circ}$ (Fromm et
al. 2022).

Table 1: The mass accretion rate of the GRMHD simulations used in this study for each black-hole spin. The model names are referenced from Fromm et al. (2022).  Model | Spin $a_{*}$ | Mass accretion rate $\langle\dot{M}\rangle~{}{\rm[M_{\odot}~{}\mathrm{yr}^{-1}]}$
---|---|---
MT.M.1 | $-0.9375$ | $3.99\times 10^{-4}$
MT.M.2 | $-0.5$ | $3.42\times 10^{-4}$
MT.M.3 | $0$ | $2.50\times 10^{-4}$
MT.M.4 | $0.5$ | $2.32\times 10^{-4}$
MT.M.5 | $0.9375$ | $1.06\times 10^{-4}$

###  2.2 Synthetic ngEHT observations and Imaging strategy

Figure 1: Aggregate baseline coverage for the ngEHT observation with the
expected telescopes (Roelofs et al. 2023). The blue and red data points
represent the $(u,v)$ coverage obtained from the telescopes integrated in 2017
and the ngEHT, respectively, with corresponding observational time durations
of 7 hours and 24 hours. The axes of $(u,v)$ plane are in units of giga-
wavelengths at the spatial frequency of 230 GHz.  Figure 2: Comparison between
the top panels displaying snapshot images obtained from the GRRT simulation
and the middle panels demonstrating reconstructed images using synthetic data
based on the expected ngEHT observations at a frequency of $230~{}{\rm GHz}$
(Section 4), specifically focusing on the case characterized by
$a_{*}=0.9375$. The images in the top right and middle right panels are
restored using a circular Gaussian beam with a full width at half maximum
(FWHM) of $20~{}\mathrm{\mu as}$, as indicated in the lower right corners. The
unit represented in each image is the logarithm of the brightness temperature,
denoted as $T_{\rm B}$. The bottom panels depict the ratio between the GRRT
and reconstructed images, with the bottom left panel displaying the ratio
without restoration, and the bottom right panel representing the ratio after
restoration with the Gaussian beam.

As already anticipated, the primary aim of this paper is to explore the
dynamics of the jet and of the disk, which contain information about the
properties of spacetime, and to assess their detectability with the
anticipated capabilities of the ngEHT. In order to comprehensively analyze the
detectable properties, it is essential to assess the impact of observational
uncertainties with the expected imaging results. The observed visibility
$V(t,u,v)$ is defined as the Fourier transform of the source intensity
$I(t,x_{\rm obs},y_{\rm obs})$,

$V(t,u,v):=\int I(t,x_{\rm obs},y_{\rm obs})e^{2\pi i(ux_{\rm obs}+vy_{\rm
obs})}dx_{\rm obs}dy_{\rm obs},$

where $(x_{\rm obs},y_{\rm obs})$ are the Cartesian coordinates on the image
plane, while $(u,v)$ are the components of the baseline vector (commonly
referred to as spatial frequencies) between two antennas, which is normalized
by the observing wavelength and projected on the image plane orthogonal to the
source direction. The expected telescope properties were examined in detail by
Raymond et al. (2021) and Roelofs et al. (2023), and the baseline $(u,v)$
coverage is shown in Fig. 1 for the observational frequency of 230 GHz. The
measured visibility on a baseline is the summation of the true visibility and
systematic noise, which is assessed with a Gaussian distribution in the
complex plane with the standard deviation

$\sigma_{i,j}=\frac{1}{\eta}\sqrt{\frac{{\rm SEFD}_{i}{\rm
SEFD}_{j}}{2\Delta\nu\Delta t}},$

where $\eta\simeq 0.88$ represents the quantization efficiency (Thompson et
al. 2017), $\Delta\nu\simeq 8~{}{\rm GHz}$ refers to the bandwidth of a single
spectral window for a specific band and polarization (Roelofs et al. 2023),
and $\Delta t=10$ s denotes the integration time (Event Horizon Telescope
Collaboration et al. 2019d). We define the system equivalent flux density
(SEFD), as encompassing all thermal noises originating from the telescope
receiver chains, Earth’s atmosphere, and astronomical background (see Roelofs
et al. 2023 for a summary), where the subscripts $(i,j)$ refer to the
telescopes involved in the respective baseline.

Assuming that the time duration of a single observation is 24 hours, during
which a snapshot image does not change, we generated a data set corresponding
to a time window of ($2000t_{\rm g}\sim 720~{}{\rm days}$). Utilizing the
open-source software eht-imaging (Chael et al. 2018a), we generate synthetic
data based on the expected ngEHT $(u,v)$ coverage and source models,
incorporate systematic noises into the visibilities, and perform calibration
on the visibility phases.

We reconstructed the images using the open-source software SMILI, which
employs the regularized maximum likelihood (RML) method based on the synthetic
data (Akiyama et al. 2017a, b). For the reconstruction, we used visibilities
averaged over 6 minutes, setting the imaging field of view to
$1200~{}\mathrm{\mu as}$ and a pixel size of $4~{}\mathrm{\mu as}$. The
reconstruction process was initiated using an image characterized by a
circular Gaussian profile with a FWHM of $50~{}\mathrm{\mu as}$. Furthermore,
we utilized imaging regularizations of Total Variation (TV), with a parameter
value of $10^{6}$, and relative entropy, with a value of $1$ (Event Horizon
Telescope Collaboration et al. 2019d).

We present an example of GRRT (top panels) and reconstructed images (middle)
in Fig. 2. Each image in the top-right and middle-right panels has been
convolved with a circular Gaussian function corresponding to the typical
resolution of the EHT ($20~{}\mathrm{\mu as}$). Note that the reconstructed
image accurately captures the extended-jet features, while also exhibiting a
certain level of noise structure, which can be attributed to systematic errors
and the constrained $(u,v)$ coverage. The intensity ratio between the GRRT and
reconstructed images is shown in the bottom panels. The standard deviation of
the intensity ratio for the unconvolved images ($\sim 2.8$) is mitigated
through convolution at the nominal resolution ($\sim 0.1$). In Section 4, we
use the convolved images to investigate the common dynamic properties
consistent between GRRT and reconstructed images.

##  3 Results: fundamental properties of GRRT images

###  3.1 Time-averaged image properties

Figure 3:  The time-averaged properties of the accretion disk and the jet at $230~{}{\rm GHz}$. The left panel displays the time-averaged GRRT image in spherical polar coordinates for the case of a spin and inclination angle of $(a_{*},i)=(0.9375,160^{\circ})$. The green and cyan curves represent the disk and jet$+$disk regions, respectively, which can be detected with the typical ngEHT sensitivity ($=2\times 10^{7}~{}{\rm K}$). In the middle panel, the radial profile of the azimuthal maximum brightness temperature of the jet$+$disk is shown. The expected sensitivity of the 2017 EHT and ngEHT observations is presented as horizontal gray lines. In the right panel, the same format is followed as in the middle panel, but for the disk emission defined by the Bernoulli parameter (${\rm Be}<1.02$). The vertical gray lines in the middle and right panels indicate a radius of $100~{}{\rm\mathrm{\mu as}}$, which is the typical radius of the disk region for each spin case.  Table 2:  Time average and standard deviation of jet$+$disk and disk sizes using the ngEHT sensitivity as defined in Fig. 3, where $a_{*}$ represents the dimensionless black-hole spin, and $r_{\rm obs,,jet+disk}$ and $r_{\rm obs,,disk}$ denote the radial sizes of the jet$+$disk and disk regions, respectively.  $a_{*}$ | $r_{\rm obs,~{}jet+disk}~{}[{\rm\mathrm{\mu as}}]$ | $r_{\rm obs,~{}disk}~{}[{\rm\mathrm{\mu as}}]$
---|---|---
0.9375 | $287\pm 33$ | $81\pm 12$
0.50 | $251\pm 20$ | $96\pm 7$
0.00 | $181\pm 15$ | $110\pm 10$
-0.50 | $227\pm 25$ | $101\pm 19$
-0.9375 | $223\pm 41$ | $115\pm 18$

Identifying the spatial regions on the image plane corresponding to the jet
and the disk is essential for extracting physical information from both the
GRRT and reconstructed images. We calculated the radiation emitted from each
region, the boundary of which is defined by the Bernoulli parameter ${\rm
Be}:=-hu_{t}=1.02$, where $h$ is the specific enthalpy and $u_{t}$ is the time
component of the covariant four-velocity (Fromm et al. 2022). Along each ray
trajectory, we set the emissivity to zero outside the corresponding region
while maintaining the absorptivity to the relevant value.

To demonstrate the essence of our approach, we show in Fig. 3 the time-
averaged properties of the jet and the disk structures as averaged over the
entire simulation (i.e., from $t/t_{\rm g}=0$ to $2000$). The left panel shows
the average of the synthetic images for a Kerr black hole with $a_{*}=0.9375$
with the green and cyan curves identifying the detectable regions of the disk
and jet+disk using the typical ngEHT sensitivity ($=2\times 10^{7}$ K,
Doeleman et al. 2019; Raymond et al. 2021; Chael et al. 2021). The disk
radiation concentrates within the radial region of the image plane at
approximately $r_{\rm obs}=\sqrt{x_{\rm obs}^{2}+y_{\rm obs}^{2}}\sim
100~{}\mathrm{\mu as}\simeq 26r_{\rm g}$, while the jet radiation extends to
the radial region $100~{}\mathrm{\mu as}\leq r_{\rm obs}<300~{}\mathrm{\mu
as}$.

Each curve in the middle panel of Fig. 3 represents the radial profile of the
azimuthally maximum brightness temperature, where the black, blue, and red
curves correspond to $a_{*}=0$, $|a_{*}|=0.5$, and $|a_{*}|=0.9375$,
respectively. For $a_{*}>0$ (solid curve), the radial extent of the jet
increases with $|a_{*}|$: the jet extends towards $r_{\rm obs}\sim
500~{}\mathrm{\mu as}$ ($300~{}\mathrm{\mu as}$) with a dynamic range of
$10^{-4}$ in the spinning (non-spinning) black hole cases (solid curves). In
the case of $a_{*}<0$ (dashed curves), the radial extent of the jet is similar
with a dynamic range of $10^{-4}$. The existence of a powerful jet in the case
of a prograde black hole was also reported by other studies (e.g., Blandford &
Znajek 1977; Tchekhovskoy & McKinney 2012; Fromm et al. 2022; Narayan et al.
2022).

The right panel of Fig. 3 shows the radial profile of the time-averaged disk
emission. The profile highlights the photon-ring region, with fainter
components at the radial center and peak components at $20~{}\mathrm{\mu
as}=5.2~{}r_{\rm g}$. These features indicate the black-hole shadow and the
photon ring, respectively. The disk extends radially from $20~{}\mathrm{\mu
as}$ to $100~{}\mathrm{\mu as}$, and the brightness temperature within the
radial region of $r_{\rm obs}\sim 100~{}r_{\rm g}$ decreases. The disk extends
radially from $20~{}\mathrm{\mu as}$ to $100~{}\mathrm{\mu as}$, and the
brightness temperature within the radial region of $r_{\rm obs}\sim
100~{}r_{\rm g}$ decreases outwards.

The detectable regions of the disk and jet components can be assessed using
the sensitivities of the 2017 EHT and the expected ngEHT observations, as
represented by the gray horizontal lines in the middle and right panels. The
emission from the disk and the jet regions can be detected within the ngEHT
sensitivity at radii of $r_{\rm obs}<100~{}\mathrm{\mu as}$ and
$100~{}\mathrm{\mu as}\lesssim r_{\rm obs}\lesssim 300~{}\mathrm{\mu as}$
respectively. This would mark a significant improvement compared to the
capabilities of the 2017 EHT observations. The uncertainties related to time
variation are summarized in Table 2. In light of Fig. 3 and Table 2, we define
the radii of the disk and the jet regions on the image plane as $r_{\rm
obs}<100~{}\mathrm{\mu as}$ and $100~{}\mathrm{\mu as}\leq r_{\rm obs}$,
respectively.

###  3.2 Light-curve properties

Figure 4:  The spin dependency of the $230~{}{\rm GHz}$ light-curves and power spectral densities (PSDs). From left to right, the different panels refer to black holes with $a_{*}=-0.9375$ to $0.9375$. Top panels: the light curves emitted from both jet and disk (black), jet-only (blue), and disk-only (red) regions. Bottom panels: the power spectral densities of the light curves from the upper panels, emitted from each region, where $\tau=2000t_{\rm g}$ is the entire simulation time window.  Table 3:  Summary of the quasi-periodic oscillation (QPO). Here, $\omega$ represents the peak frequency detected in the power spectral densities (as seen in Fig. 4), $Q$ denotes the quality factor of the QPO, and $\Delta\omega$ corresponds to the full width at half-maximum (FWHM) of the peak.  $a_{*}$ | QPO frequency $\omega$ [$2\pi t_{\rm g}^{-1}$] |  $1/\omega$ [$t_{\rm g}$] | $Q=\omega/\Delta\omega$
---|---|---|---
0.9375 |  $3.1\times 10^{-3},$ $5.9\times 10^{-3}$ |  $320,$ $170$ | 8, 2
0.50 | - | - | -
0.00 | - | - | -
-0.50 | $1.9\times 10^{-3}$ | $516$ | 6
-0.9375 | - | - | -

We report in Fig. 4 the temporal variability of the emission from all regions,
including the jet and the disk. The top panels of Fig. 4 show the light curves
for different black-hole spins ($a_{*}=-0.9375,-0.5,0.0,0.5,$ and $0.9375$),
as emitted from all regions (black curve), from the jet (blue) and from the
disk (red). The light curves exhibit variability within the entire simulation
time window, defined as $\tau=2000t_{\rm g}$. The mean and standard deviation
of the total jet$+$disk light curve are comparable across each spin case
($\sim 0.96\pm 0.16~{}{\rm Jy}$) and are consistent with the core flux of M87∗
observed in previous EHT observations ($\sim 1.1-1.2~{}{\rm Jy}$, Doeleman et
al. 2012; Event Horizon Telescope Collaboration et al. 2019d).

Most models demonstrate a dominance of jet radiation over that from the disk.
With a ratio of the jet flux to the disk flux that is $\langle F_{\rm
jet}\rangle_{t}/\langle F_{\rm disk}\rangle_{t}=2.6,1.9,1.1,0.9,$ and $1.8$
for $a_{*}=-0.9375,-0.5,0.0,0.5,$ and $0.9375$, respectively, where the
brackets $\langle~{}\rangle_{t}$ represent the time average. For both
$a_{*}\geq 0.5$ and $a_{*}<0$ cases, the ratio monotonically increases with
the absolute value of the spin, indicating the generation of a powerful jet
for rapidly rotating black holes.

Our best-bet GRMHD models have smaller fluxes in the mid-plane region around
the black hole (with a field of view of approximately $100~{}\mathrm{\mu as}$
and a polar angle range of $57.3^{\circ}\leq\theta\leq 122.7^{\circ}$)
compared to those of the thermal models based on the EHT GRMHD library (Event
Horizon Telescope Collaboration et al. 2019e). To investigate this difference,
we calculated the ratio of the total flux in the mid-plane to that in all
regions, utilizing an $80~{}\mathrm{\mu as}$ field of view, in line with the
analysis in Appendix B of Event Horizon Telescope Collaboration et al.
(2019e). The ratios of the total flux density are
$66~{}\%,~{}85~{}\%,~{}79~{}\%,~{}87~{}\%,$ and $81~{}\%$ for
$a_{*}=-0.9375,-0.5,0.0,0.5,$ and $0.9375$, respectively, whereas the MAD
thermal models yield a value of about $90~{}\%$ (see Appendix B in Event
Horizon Telescope Collaboration et al. 2019e). This discrepancy can be
attributed to the impact of non-thermal particles, which result in a
comparable continuum spectrum spanning from the radio to the near-infrared
range and influence the jet morphology at a frequency of $86~{}{\rm GHz}$ in
the case of M87∗ (see Fromm et al. 2022; Cruz-Osorio et al. 2022).

The variability characteristics of each light curve are also apparent in the
power spectral densities (PSDs) $P_{\omega}$, where $\omega$ denotes the
angular frequency (bottom panels of Fig. 4). The frequency profiles of the
PSDs, that are characterized by $P_{\omega}\propto\omega^{-2.4\pm 0.5}$, are
broadly consistent with those produced by red noise
($P_{\omega}\propto\omega^{-2}$). This property was already reported in
previous research that utilized a large library of GRMHD simulation models
without non-thermal particles (Georgiev et al. 2022).

Interestingly, some PSDs display a feature that could be associated with a
quasi-periodic oscillation (QPO). This trend is especially pronounced for
$a_{*}=0.9375$, where the PSD exhibits peak components at time scales of
$1/\omega\sim 320t_{\rm g}$ and $170t_{\rm g}$ with corresponding quality
factors of $\omega/\Delta\omega\sim 8$ and $2$, respectively. Here,
$\Delta\omega$ represents the full width at half maximum (FWHM), determined by
fitting the peaks with the Lorentzian function. This pattern is reminiscent of
QPO phenomena observed in some stellar-mass and supermassive black holes and
could be employed to deduce the properties of the background spacetime (e.g.,
Abramowicz et al. 2003; Rezzolla et al. 2003; Reynolds & Miller 2009; Kato et
al. 2008; McKinney et al. 2012; Smith et al. 2021). We intend to undertake a
detailed investigation of the QPOs in future works.

##  4 Results: movie morphologies

In what follows we discuss the analysis of the dynamics of the disk and of the
jet by employing synthetic images generated from GRRT simulations. Our
assessment of the observational detectability and uncertainty has been
achieved through the comprehensive imaging analysis of the expected ngEHT
observations, which we summarized in Section 2.2. Our goal is to demonstrate
that the structures of the jet and of the disk exhibit azimuthal and radial
fluctuations that depend on the black-hole spin.

###  4.1 Variability of the Azimuthal Distribution

Figure 5:  Azimuthal profiles of the image brightness radially averaged within
the jet region ($100~{}\mathrm{\mu as}\leq r_{\rm obs}<300~{}\mathrm{\mu
as}$). From left to right, the different panels refer to black holes with
$a_{*}=-0.9375$ to $0.9375$. The top and middle panels depict the brightness
temperature on the plane of the azimuthal angle and time $(\phi,t)$, utilizing
the GRRT and reconstructed images, respectively. The white points denote the
maximum azimuthal brightness at each time. The bottom panels exhibit the
distributions of the peak azimuthal angles obtained from the upper panels. The
blue and red histograms represent the estimates derived from the GRRT and
reconstructed images, respectively.  Figure 6:  The same as in Fig. 5, but
each profile was radially averaged only in the disk region $r_{\rm
obs}<100~{}\mathrm{\mu as}$.  Figure 7:  The spin dependency of the standard
deviation of the peak azimuthal angles in the jet (represented by circles) and
the disk (squares) regions. The blue and red markers represent the estimated
values obtained from the GRRT and reconstructed images, respectively.

In Fig. 5, we show the azimuthal profiles of the jet, where each image is
radially averaged within the jet region; i.e., $100~{}\mathrm{\mu as}\leq
r_{\rm obs}<300~{}\mathrm{\mu as}$ (corresponding to $26\leq r_{\rm
obs}/r_{\rm g}<79$). The azimuthal profile corresponding to a negative spin
exhibits substantial variability along the temporal direction, while profiles
with positive spins display relatively stable behavior over time. The white
points $\phi_{\rm obs,~{}peak}(t)$ illustrate the temporal evolution of the
azimuthal angle at maximum brightness. In the instance of $a_{*}=-0.9375$,
intermittent azimuthal rotations of $\phi_{\rm obs,~{}peak}(t)$ occur in the
same direction as the accretion flow (clockwise) with a yearly timescale
($\sim 1000t_{\rm g}\simeq 1.0~{}\rm{year}$ for $M=6.5\times
10^{9}M_{\odot}$). For example, the white points within the range of
$-180^{\circ}\leq\phi_{\rm obs}\leq 60^{\circ}$ and $170\leq t/t_{\rm g}\leq
880$ exhibit an angular velocity of $d\phi_{\rm obs}/dt\sim-0.4^{\circ}/t_{\rm
g}$. The substantial variation in the azimuthal angle on a yearly scale,
$\delta\phi_{\rm obs}\sim 240^{\circ}$, is a distinctive characteristic of the
case with $a_{*}=-0.9375$, separating it from the jets produced by black holes
with $-0.5\leq a_{*}\leq 0$ and much more evidently from black hole with
$a_{*}=0.9375$.

The azimuthal profiles derived from the reconstructed images shown in the
middle panels of Fig. 5 clearly highlight that the azimuthal variability is
observationally detectable. The comparison between the top and middle panels
reveals consistent time development of $\phi_{\rm obs,~{}peak}(t)$ in both the
GRRT and reconstructed images. The time-aggregated distributions of $\phi_{\rm
obs~{}peak}(t)$ are reported in the bottom panels using GRRT (blue histogram)
and reconstructed images (red histogram). Note how the negative spin cases
exhibit a wide range of peak azimuthal angles, whereas the positive spin cases
display sharply peaked histograms.

Figure 6 is similar to Fig. 5, but in it we show the azimuthal variability in
the disk region only, $r_{\rm obs}<100~{}\mathrm{\mu as}$ (corresponding to
$r_{\rm obs}/r_{\rm g}<26$). The temporal evolutions of $\phi_{\rm
obs,~{}peak}(t)$ (white points) show more substantial variability than those
in the jet region (Fig. 5) and a similar trend is evident in the jet region,
where the variations of the white points are larger for the negative spin
cases compared to the positive spin cases.

In Fig. 7, we summarize the standard deviation of the azimuthal angle at peak
brightness $\phi_{\rm obs,~{}peak}(t)$ in the jet (circle points) and the disk
regions (square points), utilizing both GRRT (blue) and reconstructed images
(red). The peak azimuthal angle of the disk region is determined by the
gravitational lensing and beaming effects, as well as the brightness
distribution of the accretion flow. The transition value between $a_{*}\leq 0$
and $a_{*}>0$ is approximately $\sigma(\phi_{\rm obs,~{}peak})\sim 30^{\circ}$
in both regions, which allows us to constrain the direction of the black-hole
rotation. The standard deviation of the peak azimuthal angles exhibits similar
values across nearly all spin cases in both the jet and the disk images.

Moreover, we can see that the disk component $\sigma(\phi_{\rm obs,peak})\sim
90^{\circ}$ has an azimuthal variation that is larger than that of the jet
component ($\sim 40^{\circ}$) in the case of $a_{*}=-0.5$, which can be
explained with the different rotational direction between the black hole and
accretion flow. For $a_{*}\geq 0$, the gravitational lensing and beaming
effects cause the mean peak position angle to be distributed around
$\sim-60^{\circ}$ (see Fig. 6). On the other hand, at $a_{*}=-0.9375$, the
mean peak azimuthal angle is distributed around $\sim 63^{\circ}$ due to the
strong frame dragging effect of the counter-clockwise rotation of the black
hole, which disrupts the rotation direction of the accretion flow in the
opposite direction. In the case of $a_{*}=-0.5$, the effect of the black
hole’s frame dragging is not as dominant over the opposing flow of the
accretion disk as it is when $a_{*}=-0.9375$, and thus, large fluctuations
occur due to the competition between these two effects. This unique property
of azimuthal variation seen in the case of $a_{*}=-0.5$ has potential utility
in distinguishing between observational results for $a_{*}=-0.9375$, $-0.5$,
and $0.0$.

###  4.2 Radial wave propagation of jet inhomogeneities

Figure 8:  The radial propagation of wave inhomogeneity within the jet region
in the case of $a_{*}=0.9375$. The top and bottom panels represent the
distributions using the GRRT and reconstructed images, respectively. The three
panels on the left-hand side show the polar distribution of temperature
inhomogeneity at corresponding timestamps ($t/t_{\rm g}=1280,~{}1290,$ and
$1300$). The panels on the right-hand side display spacetime diagrams
illustrating the inhomogeneity distributions at an azimuthal angle of
$\phi_{\rm obs}=-20^{\circ}$.

In this section, we shift our focus to the radial variability in the jet
region and propose observational quantities to estimate the magnitude of the
black-hole spin. To this scope, we investigate the time development of radial
jet propagation. We introduced a distribution of the relative brightness
temperature

$\displaystyle f(t,r_{\rm obs},\phi_{\rm obs})$ $\displaystyle:=$
$\displaystyle{T_{B}(t+\Delta t,r_{\rm obs},\phi_{\rm obs})\over
T_{B}(t,r_{\rm obs},\phi_{\rm obs})}-1,$

where $T_{B}(t,r_{\rm obs},\phi_{\rm obs})$ represents the brightness
temperature in polar coordinates at the observational time $t$, and $\Delta
t=10t_{\rm g}$ denotes the time interval of the GRRT simulations. We
calculated the relative brightness in regions where $T_{B}(t,r_{\rm
obs},\phi_{\rm obs})$ and $T_{B}(t+\Delta t,r_{\rm obs},\phi_{\rm obs})$
exceed the sensitivity threshold of the ngEHT, $T_{B,{\rm thres}}=2\times
10^{7}~{}{\rm K}$.

The GRRT images clearly show the propagation of inhomogeneous wave-like
features in the radial direction. In Fig. 8, we report the inhomogeneous
radial wave observed in the case of $a_{*}=0.9375$. The relative distributions
between the neighboring time stamps (top and left three panels) demonstrate
the propagation of inhomogeneity in the radial direction. Comparing the
relative brightness distributions between the GRRT (top left panels) and the
reconstructed images (bottom left panels) provides evidence that similar wave-
like features propagating outwards in the radial direction can be detected
through expected ngEHT observations.

To better highlight the temporal evolution of the radial wave and measure its
speed, we show in the right panels of Fig. 8 the time development of the
radial profile at the azimuthal angle of $\phi_{\rm obs}=-20^{\circ}$.
Intermittent wave propagation is clearly observed, as evidenced by the
presence of five red curves in the spacetime diagram within the time range of
$1200\leq t/t_{\rm g}<1400$ based on both the GRRT (top panel) and
reconstructed images (bottom panel). Each red/blue stripe exhibits a linear
behavior, indicating the intermittent radial propagation of wave inhomogeneity
with an almost continuous velocity. This feature sets the stage for our
forthcoming examination of the spin dependence on the radial wave velocity in
the subsequent section.

###  4.3 Radial velocity of jet inhomogeneity

Figure 9:  The number of detected waves with respect to the relative brightness $f$ across the entire time and azimuthal domains (from left to right: $a_{*}=-0.9375,-0.5,0.0,0.5,$ and $0.9375$). The top and middle panels show the distributions of $N/{\rm max}(N)$ at the residual radius $r_{\rm obs}-r_{\rm obs,~{}fit}$ and the relative brightness of each wave, utilizing the GRRT and reconstructed images, respectively. Here, $r_{\rm obs,~{}fit}$ represents the fitting radius of each wave in the spacetime diagram, the white curve represents the boundary where $N/{\rm max}(N)=0.1$, and the leftmost number $N_{\rm tot}$ denotes the total number of detected waves. The bottom panels depict the vertical integration of $N$ from the upper panels, with normalization to its maximum value.  Figure 10:  Azimuthal and temporal distributions of the radial wave velocities $v^{r}_{\rm true}$ for each spin. The top and middle panels show the distribution of radial velocity using the GRRT and reconstructed images, respectively. The lower panels present the median values of the radial velocity in the temporal direction, derived from both the GRRT (blue) and reconstructed (red) images, with the error bars indicating the corresponding median absolute deviations.  Figure 11:  Spin dependency of the azimuthally maximum value of radial wave velocity $v^{r}_{\rm true}$ (refer to the bottom panels in Fig. 10), where the blue and red points represent the estimated values obtained from the GRRT and reconstructed images, respectively. The top labels in the orange and cyan-colored regions indicate potential characteristics that can be used to constrain the spin within each spin range. For clarity, the red points are slightly shifted to the right ($0.05$) in the horizontal direction.  Table 4:  Summary of the spin dependency of the disk-jet dynamic. In this table, $\sigma_{\rm jet}(\phi_{\rm obs})$ and $\sigma_{\rm disk}(\phi_{\rm obs})$ denote the standard deviation of the peak azimuthal angles in the jet and the disk regions, respectively (refer to Fig. 7). Furthermore, $v^{r}_{\rm true}$ represents the maximum radial wave velocity along the azimuth (refer to Fig. 11). The values enclosed in parentheses are the estimates derived from the reconstructed images.  $a_{*}$ | $\sigma_{\rm jet}(\phi_{\rm obs})~{}[^{\circ}]$ | $\sigma_{\rm disk}(\phi_{\rm obs})~{}[^{\circ}]$ | $v^{r}_{\rm true}$
---|---|---|---
0.9375 | 4 ( 4) | 13 ( 11) | 0.49$\pm$ 0.02 (0.49$\pm$ 0.03)
0.5 | 5 ( 6) | 16 ( 13) | 0.34$\pm$ 0.06 (0.34$\pm$ 0.02)
0.0 | 53 (52) | 53 ( 49) | 0.26$\pm$ 0.06 (0.26$\pm$ 0.02)
-0.5 | 42 (40) | 91 ( 89) | 0.31$\pm$ 0.04 (0.31$\pm$ 0.02)
-0.9375 | 72 (72) | 73 ( 73) | 0.40$\pm$ 0.03 (0.33$\pm$ 0.06)

By utilizing the spacetime diagram in the right panels of Fig. 8, we conducted
an investigation into the temporal and radial development of the inhomogeneous
waves in the jet region. We estimated the time and azimuthal distribution of
the number of waves $N$, relative brightness $f$, and radial velocity $v^{r}$
of each wave through the following steps:

  1. 1.

Given an azimuthal angle $\phi$, we collected radial waves with $f(t,r_{\rm
obs})>0.1$, that is, with an excess of $10\%$ in the relative temperature
brightness. Continuity of wave propagation is defined as a set of temporally
and radially continuous points, spanning a time duration of more than
$30~{}t_{\rm g}$. Based on this process, we obtained a set of events
$(t,r_{\rm obs})$ and the corresponding relative brightness $f$ of each wave
train.

  2. 2.

We repeated this procedure at each azimuthal angle, resulting in a set of
waves $(t,r_{\rm obs},\phi_{\rm obs},f)_{j}$, where $i$ denotes the wave index
$1\leq j\leq N_{\rm tot}$ and $N_{\rm tot}$ is the total number of detected
waves.

  3. 3.

We conducted a linear fitting to each wave $j$ using the least square method,
such that the slope of the linear function $r_{\rm obs,~{}fit}(t_{j})$
provides the radial velocity $v^{r}$. Note that because the estimated velocity
is projected to the image plane, the deprojected relativistic velocity
$v^{r}_{\rm true}$ is calculated with the observed apparent one $v^{r}$ (e.g.,
Falla & Floyd 2002):

$v^{r}_{\rm true}=\frac{v^{r}}{\sin(180^{\circ}-i)+v^{r}\cos(180^{\circ}-i)}.$

  4. 4.

Finally, the velocity was averaged both temporally and azimuthally across
sections with widths $\Delta t$ and $\Delta\phi$. For the formers, we selected
$\Delta t=200t_{\rm g}$, which corresponds to the propagation time in the
radial jet region ($100~{}\mathrm{\mu as}\leq r_{\rm obs}<300~{}\mathrm{\mu
as}$) when $v^{r}\geq 0.1$. For the latter, we have chosen an azimuthal
section, $\Delta\phi=30^{\circ}$, which encompasses the range spanned by
azimuthal jet brightness for $a_{*}\geq 0.5$ (see Fig.7), and is consistent
with the width of the bottom histograms in Fig.5. The validity of the radial
velocity estimate was inspected using geometric jet models shown in Appendix
A.

Following steps (1)-(3), we investigated the radial wave properties in the jet
region, and Fig. 9 shows the number of detected waves $N$ with respect to the
relative brightness $f$. The top and middle panels show the distributions of
$N/{\rm max}(N)$ at the residual radius $r_{\rm obs}-r_{\rm obs,~{}fit}$ and
the relative brightness of each wave, utilizing the GRRT (top panels) and
reconstructed images (middle panels), respectively. Clearly the maximum value
of the relative brightness $f$ depends on the absolute value of the spin, and
for $N/\max(N)=0.1$ (white curves), $f$ extends to approximately
$1.5,1.4,1.4,1.3,$ and $1.6$ in the cases of $a_{*}=-0.9375,-0.5,0.0,0.5,$ and
$0.9375$, respectively. This behavior suggests the inhomogeneity of the radial
waves increases with the magnitude of the black-hole spin. The number of
radial waves, $N_{\rm tot}$ (left value), monotonically increases with the
absolute value of the spin in the case of the GRRT images (top panels), and a
similar trend is seen also for the reconstructed images (middle panels).
However, they do not quite accurately capture the spin dependence observed in
the GRRT images due to the uncertainties in the synthetic observations. The
bottom panels show the wave width of the fit lines. Each component is radially
symmetric and has a FWHM of $\sim 20~{}\mathrm{\mu as}$, which is the
restoring beam size of the image (see Section 2.2). The symmetric profiles,
characterized by the FWHM, indicate the accurate detection of the center of
jet waves using linear fitting. This also provides insight into the quality of
the fitting process, with the validation detailed in Appendix A.

Figure 10 presents the estimated radial velocities $v^{r}_{\rm true}$ for each
spin. The top and middle panels display the distributions of $v^{r}_{\rm
true}$ at each azimuthal angle and time using the GRRT and reconstructed
images, respectively. The estimated velocities exhibit significant variations
in both time and azimuthal directions for $a_{*}<0.5$, while demonstrating
more robust statistics for $a_{*}\geq 0.5$. This behavior is consistent with
the spin dependence of the azimuthal variability of the jet discussed in
Section 4.1. The bottom panels show the median velocity with respect to the
temporal direction for each azimuthal angle, where the error bars correspond
to the median absolute deviations. Note that the velocity peaks around the
rotation axis of the black hole $(\phi=18^{\circ})$ for all spins.

Finally, Fig. 11 summarizes many of the results presented so far and the
relationship between spin and the maximum radial velocity in the azimuthal
direction. The estimated velocity, as indicated by the GRRT (blue) and
reconstructed (red) images, demonstrates a monotonic dependency on the
absolute spin value. These estimates allow us to distinguish between the cases
of $a_{*}=0.5$ and $a_{*}\geq 0.9375$. As discussed in Section 4.1, the
azimuthal variability of the jet and the disk enables differentiation in the
case of $a_{*}\leq 0$ (the orange region in Fig. 11). As a result, Fig. 11
provides evidence that the azimuthal and radial variations of the jet and the
disk depend on the black-hole spin, thus enabling us to extract spacetime
information (Table 4). Similar spin dependences are also observable at
different inclination angles $i$, with the measurement of radial velocity and
azimuthal variation providing constraints on $a_{*}$ and $i$ (Appendix B).

##  5 Physical interpretation of jet radial waves

Figure 12:  Radial distribution of the fluid velocity ($|v|$, red curve) and
Alfvén velocity ($v_{\rm Alf}$, blue curve) averaged over time and around the
jet for each spin value. The dashed and solid curves correspond to the average
around the vertical jet axis ($\sigma_{\rm B}>3$) and in the jet region (${\rm
Be}>1.02$ and $\sigma_{B}<3$).  Figure 13:  Spin dependency of the fluid and
Alfvén velocities is shown as averaged values in the jet region (represented
by red and blue points, respectively), along with the radial wave velocity
$v^{r}_{\rm true}$ (depicted in black, as detailed in Fig. 11). For clarity,
the positions of the red and blue points are slightly shifted to the right by
increments of $0.05$ and $0.1$ in the horizontal direction, respectively.

The origin of radial jet waves can primarily be attributed to the motion of
relativistic plasma along the jet, a phenomenon both confined and accelerated
by the magnetic field. By examining the fluid velocities
($|v|=\sqrt{v_{i}v^{i}}$, where $v_{i}$ and $v^{i}$ are the covariant and
contravariant three velocities of the accretion flow) and Alfvén velocities
($v_{\rm Alf}=\sqrt{\sigma_{B}/(\sigma_{B}+h)}$), we investigate the origin of
the radial jet wave velocity derived in the previous section. Here,
$\sigma_{B}=b^{2}/\rho$ is the plasma magnetization, $b$ is the norm of the
magnetic field in the fluid frame, and $\rho$ is the rest-mass density. Figure
12 reports radial distribution of the velocities averaged over time and around
the jet; the red and blue curves denote the fluid and Alfvén velocities,
respectively. The dashed and solid lines indicate the average values around
the jet axis ($\sigma_{B}>3$) and in the jet region ($\sigma_{B}<3$ and $\rm
Be>1.02$), respectively. In each region and each spin case, the fluid velocity
(blue curves) exhibits a monotonic increase with the radius due to the
diminishing gravitational attraction. Conversely, the Alfvén velocity (red
curves) decreases monotonically with the radius as it moves farther from the
center where the magnetic field is most concentrated.

Both velocities have a monotonic dependency on the spin around the jet axis
(see dashed curves). The fluid and Alfvén velocity in the region averaged in
the radial direction, are given by $|v|=0.81,~{}0.74,~{}0.70,~{}0.82,~{}0.87$
and $v_{\rm Alf}=0.85,~{}0.93,~{}0.94,~{}0.93,~{}0.87~{}$, corresponding to
the black-hole spins of $a_{*}=-0.9375,~{}-0.5,~{}0.0,~{}0.5$, and $0.9375$,
respectively. These results indicate that the accretion flows around the jet
region accelerated with the strong magnetic fields represented by the Alfvén
velocity ($v_{\rm Alf}\sim 0.90$) and the fluid velocity monotonically
increases with the magnitude of the spin.

We further investigate the spin dependency of the fluid and Alfvén velocities
within the jet region. In Fig. 13, we present a comparison between the
spatially averaged fluid and Alfvén velocities in the jet region (red and blue
points) and the estimated velocity derived from the jet radial waves (Section
4.3, depicted in black). The accompanying error bars for the fluid and Alfvén
velocities represent the standard deviation. The fluid velocities align
closely with the estimates from the jet radial waves. In contrast, the Alfvén
velocities do not exhibit a clear spin dependency. It is further noted that
the fluid velocity in the jet region is predominantly radial, with ratios of
the radial three-velocity $\sqrt{v^{r}v_{r}}$ and $|v|$ averaged in the jet
region are $85~{}\%,~{}90~{}\%,~{}93~{}\%,~{}92~{}\%,~{}95~{}\%$ for
$a_{*}=-0.9375,-0.5,0.0,0.5$, and $0.9375$, respectively. The results indicate
that inhomogeneity within the jet serves as an effective marker, tracing the
contribution of spin to the relativistic fluid motion.

##  6 Summary and future prospects

We investigated the horizon-scale dynamics of the jet and the accretion disk
and demonstrated the potential of black-hole spin measurement. We focused on
the best-bet GRMHD models whose continuum spectra are consistent with that of
the M87∗ observations within a small set of model parameters (Fromm et al.
2022; Cruz-Osorio et al. 2022). The azimuthal variation of the jet and disk
brightness reveals information about the spin direction, with the azimuthal
angle of peak brightness showing temporal stability in cases of $a_{*}>0$ and
instability when $a_{*}\leq 0$. Furthermore, this variation has the potential
to distinguish negative spin cases. In the jet region ($100~{}\mathrm{\mu
as}\lesssim r_{\rm obs}\lesssim 300~{}\mathrm{\mu as}$), inhomogeneous waves
propagate radially with relativistic velocities ($0.26\lesssim v^{r}_{\rm
true}\lesssim 0.50$). The radial wave velocity, as estimated from the GRRT
images, shows a monotonic dependence on the absolute value of the black-hole
spin. The reconstructed images from the ngEHT have demonstrated the
detectability of these dynamics and the accuracy of the estimated black-hole
spin. This paper provides the essence of the spacetime measurement with the
horizon-scale dynamics and future theoretical and observational projects.

A survey with a broader library of GRMHD simulations will provide a more
comprehensive view of spin measurement under various possibilities of
spacetime, magnetic field, accretion flow properties, and inclination angles.
For a survey with an expanded library of GRMHD simulations, it is essential to
investigate each regime of parameter constraints with individual properties:
continuum spectrum, visibility fitting, polarization, and jet-disk dynamics
(Fromm et al. 2022; Event Horizon Telescope Collaboration et al. 2019f, 2021a;
Davelaar et al. 2023). In addition to the expansion of model parameters, the
dynamics survey has the potential to provide additional information on tilted
disk parameters depending on the Lense-Thirring precession of the accretion
disk and jet (e.g., Liska et al. 2018; Chatterjee et al. 2020). We also
focused on the simplest case without gain uncertainty and performed image
reconstruction. To investigate the fiducial morphology under more realistic
conditions, we need to conduct imaging surveys with numerous imaging
parameters (Event Horizon Telescope Collaboration et al. 2019d, 2022c) with
the more realistic data including gain uncertainties, polarization leakages,
and instrumental effects.

Alongside the prospective utility of ngEHT observations, it is worth noting
that the annual azimuthal variation in the photon ring, as observed through
current and impending EHT studies, is also synergetic with our proposed
method. The annual EHT observations, which offer less dense $(u,v)$ coverages
compared to ngEHT, are either under analysis or in the planning stages to
investigate variations of the photon ring. Particularly, RML imaging, as
demonstrated with the 2017 EHT data, has furnished an azimuthal angle
assessment of the photon ring, achieving an accuracy of approximately $\sim
20^{\circ}$ (Event Horizon Telescope Collaboration et al. 2019d). By
continuous observations of the azimuthal angle variabilities and capturing the
temporal evolution of images within the disk region, discerning differences in
the azimuthal angle variation correlated to the black hole’s spin direction
(roughly $60^{\circ}$) will become attainable. Conversely, the task of
differentiating small differences in azimuthal angle variabilities between
($a_{*}=0.5$) and ($a_{*}=0.9375$) (around $2-3^{\circ}$, as outlined in Table
4) is challenging. The observation of the jet’s temporal progression,
augmented by the denser $(u,v)$ coverage facilitated by ngEHT observations,
will be an essential requirement for constraining the magnitudes of the spin.

The quality of horizon-scale images has improved significantly due to recent
advancements in imaging techniques (e.g., Chael et al. 2022; Mueller & Lobanov
2022; Arras et al. 2022). These developments potentially pave the way for the
application of our spin measurement methodology to EHT observations from 2021
onwards. These results, in conjunction with EHT observations, are expected to
synergize with current and future multi-wavelength observations (Lu et al.
2023). By utilizing our methodology in future EHT observations, theoretical
and observational studies will be able to achieve a much more comprehensive
measurement of the black-hole spacetime of M87∗.

Acknowledgements. We thank Prashant Kocherlakota and Indu Kalpa Dihingia for
their insightful comments on this research. This research is supported by the
European Research Council for the Advanced Grant “JETSET: Launching,
propagation and emission of relativistic jets from binary mergers and across
mass scales” (Grant No. 884631). CMF is supported by the DFG research grant
“Jet physics on horizon scales and beyond” (Grant No. FR 4069/2-1). YM is
supported by the National Natural Science Foundation of China (Grant No.
12273022), the Shanghai Municipality orientation program of basic research for
international scientists (Grant No. 22JC1410600), and the National Key R&D
Program of China (Grant No. 2023YFE0101200). LR acknowledges the Walter
Greiner Gesellschaft zur Förderung der physikalischen Grundlagenforschung e.V.
through the Carl W. Fueck Laureatus Chair. The simulations were performed on
GOETHE-HLR LOEWE at the CSC-Frankfurt, Calea and Iboga at ITP Frankfurt and
Pi2.0 and Siyuan Mark-I in Shanghai Jiao Tong University. Software: BHAC
(https: //bhac.science/) (Porth et al. 2017), BHOSS (Younsi et al. 2020), eht-
imaging (https://achael.github.io/eht-imaging/) (Chael et al. 2018b), and
SMILI (https://github.com/astrosmili/smili) (Akiyama et al. 2017a, b).

##  Appendix A Inspection of radial velocity estimates with geometrical jet
models

Figure 14:  The radial velocity of the geometric jets as defined in Section A.
The gray curve represents the input (or ’true’) radial velocity, while the red
curve represents the estimated radial velocity.

In Section 4.3, we introduced a method for estimating the radial wave velocity
of jet inhomogeneity. To validate these velocity estimates, we applied the
method to simple geometric jet models and evaluated the accuracy of the
resulting velocity estimates. The intensity distribution of the geometric
model is given by

$\displaystyle I(t,r,\phi)$ $\displaystyle=$ $\displaystyle
I_{0}(t)G(t,r,\phi)\exp\left[-\left(\frac{r-v^{r}t}{\Delta
r}\right)^{2}\right]~{}(|\phi|<\Delta\phi),$

where $v^{r}$ is the input velocity, $\Delta r$ and $\Delta\phi$ represent the
radial and azimuthal width of the geometric wave, and $G(t,r,\phi)$ is a
random Gaussian distribution with a mean and standard deviation of one. The
intensity $I_{0}(t)$ is defined such that the total flux equals $1.0~{}{\rm
Jy}$ at each time.

Figure 14 compares the estimated velocities to the input (true) velocities,
showcasing typical examples with $\Delta r=20~{}\mathrm{\mu as}$ and
$\Delta\phi=40^{\circ}$. The velocity estimates are derived from the input
images following the same strategy described in Section 4.3. Across the input
velocity space in this paper ($0.2\leq v^{r}\leq 1$), the estimated velocities
accurately reproduce the ground-truth values with an error of about $\Delta
v^{r}\sim 0.01$.

##  Appendix B Inclination angle dependency

Figure 15:  The inclination angle dependence of the estimated azimuthal
variation (top panel) and radial velocity (bottom panel). For clarity, the
curves in the case of $a_{*}\geq 0$ are slightly ($1^{\circ}$) shifted in the
horizontal direction.

The main text focuses on the best-bet model parameters, which reproduce the
continuum spectrum of M87∗ at a fixed viewing angle ($i=160^{\circ}$). In this
appendix, we demonstrate that the broad spin dependency of both the azimuthal
variability of the jet (Section 4.1) $\sigma(\phi_{\rm obs,~{}peak})$ and
radial wave velocity $v^{r}_{\rm true}$ (Section 4.3) is observable at
different inclination angles.

In Fig. 15, we show the inclination angle dependency of the azimuthal
variation $\sigma(\phi_{\rm obs,~{}peak})$ and the radial wave velocity
$v^{r}_{\rm true}$ for each spin (the red, blue, and green curves represent
the cases of $|a_{*}|=0.9375,0.5,$ and $0$, respectively) with the GRRT
images. The transition value of $\sigma(\phi_{\rm obs,~{}peak})$ between
$a_{*}\leq 0$ and $a_{*}>0$ is $\sigma(\phi_{\rm obs,~{}peak})\sim 30^{\circ}$
in the case of $150^{\circ}\leq i\leq 170^{\circ}$. The radial velocity
monotonically increases with $|a_{*}|$, indicating that the inhomogeneous jet
profile propagates more rapidly with the absolute value of the spin. These two
features are consistent across all inclination angles. Thus, the measurement
of the radial velocity and azimuthal variability could potentially constrain
both the spin value and the viewing angle.

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