# Schwarzschild black hole and redshift rapidity:
A new approach towards measuring cosmic distances

Mehrab Momennia<EMAIL_ADDRESS><EMAIL_ADDRESS>Instituto de
Física, Benemérita Universidad Autónoma de Puebla,
Apartado Postal J-48, 72570, Puebla, Puebla, Mexico  Pritam Banerjee
<EMAIL_ADDRESS>bpritamphy@ gmail.com Indian Institute of Technology,
Kharagpur, 721302, India  Alfredo Herrera-Aguilar<EMAIL_ADDRESS>Instituto de Física, Benemérita Universidad Autónoma de Puebla,
Apartado Postal J-48, 72570, Puebla, Puebla, Mexico  Ulises Nucamendi
<EMAIL_ADDRESS><EMAIL_ADDRESS>Instituto de Física y
Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo,
Edificio C–3, Ciudad Universitaria, CP 58040, Morelia, Michoacán, Mexico

###### Abstract

Motivated by recent achievements of a full general relativistic method in
estimating the mass-to-distance ratio of supermassive black holes hosted at
the core of active galactic nuclei, we introduce the new concept redshift
rapidity in order to express the Schwarzschild black hole mass and its
distance from the Earth just in terms of observational quantities. The
redshift rapidity is also an observable relativistic invariant that represents
the evolution of the frequency shift with respect to proper time in the
Schwarzschild spacetime. We extract concise and elegant analytic formulas that
allow us to disentangle mass and distance to black holes in the Schwarzschild
background and estimate these parameters separately. This procedure is
performed in a completely general relativistic way with the aim of improving
the precision in measuring cosmic distances to astrophysical compact objects.
Our exact formulas are valid on the midline and close to the line of sight,
having direct astrophysical applications for megamaser systems, whereas the
general relations can be employed in black hole parameter estimation studies.

Keywords: Schwarzschild black hole, black hole rotation curves, redshift and
blueshift, redshift rapidity.

###### pacs:

11.27.+d, 04.40.-b, 98.62.Gq

##  I Introduction

Black holes first have been found just as mathematical solutions to Einstein
field equations of the general relativity theory by Karl Schwarzschild whereas
their existence in the cosmos has been recently approved through gravitational
wave detections GW  and electromagnetic wave observations EHTM87 ; EHTSgr .
However, investigating the motion of stars around the center of our galaxy has
already provided convincing evidence for the presence of a supermassive black
hole hosted at the core of the Milky Way galaxy Ghez ; Morris ; Eckart ;
Gillessen .

Even though the aforementioned methods are fairly accurate in obtaining the
black hole parameters, they are not always applicable to existing data from
black holes and black hole environments. Therefore, we have been inventing and
developing an independent general relativistic approach to measure the black
hole and cosmological parameters PRDhn ; PRDbhmn ; KdS . This method is based
on the observational frequency-shifted photons emitted by massive geodesic
particles orbiting central black holes and has been initially suggested in
PRDhn , generalizing previous Keplerian models Herrnstein2005 ; Argon2007 ;
Humphreys2013 . Recently, this approach has been developed to analytically
express the mass and spin parameters of the Kerr black hole in terms of a few
directly observational quantities PRDbhmn . Besides, it has been demonstrated
that the Kerr black hole in asymptotically de Sitter background allows us to
measure the Hubble constant and black hole parameters, simultaneously KdS .

The initial method of PRDhn  has been employed to obtain the kinematic
redshift and blueshift in terms of black hole parameters for several black
hole solutions, such as Kerr-Newman black holes in de Sitter spacetime KNdS ,
the Plebanski-Demianski black holes PlebanskiDemianski , higher dimensional
Myers-Perry spacetime MyersPerry , and regular black holes in nonlinear
electrodynamics RegularBH . Furthermore, a similar relation for the kinematic
redshift has been found for black holes in modified gravity MOG , black holes
coupled to nonlinear electrodynamics NED , black holes immersed in a strong
magnetic field SMF , and the boson stars BosonStar .

These studies have been performed based on the kinematic frequency shift which
is not an observable quantity, unlike the total frequency shift of photons.
Hence, this fact has motivated us to consider the total frequency shift as a
directly observable element and extract concise and elegant analytic formulas
for the mass and spin of the Kerr black hole in terms of few observable
quantities PRDbhmn . With the help of this general relativistic method, the
free parameters of polymerized black holes have been expressed in terms of the
total redshift as well FuZhang . More recently, this approach has been
extended to general spherically symmetric spacetimes in order to extract the
information of free parameters of black holes in alternative gravitational
theories beyond Einstein gravity Diego .

From a practical point of view, the mass-to-distance ratio of several
supermassive black holes hosted at the core of the active galactic nuclei
(AGNs) NGC 4258 ApJL , TXS-2226-184 TXS , and an additional $15$ galaxies
TenAGNs ; FiveAGNs  has been estimated by employing this general relativistic
approach. These supermassive black holes enjoy circularly orbiting water vapor
clouds within accretion disks that emit redshifted photons toward a distant
observer. This allows us to utilize the general relativistic method in order
to estimate the $M/D$ ratio and quantify the gravitational redshift produced
by the spacetime curvature which is a general relativistic effect.

In the parameter estimation studies of the supermassive black holes performed
in ApJL ; TXS ; TenAGNs ; FiveAGNs , just the mass-to-distance ratio $M/D$ of
these compact objects has been estimated with the help of observational
redshifted and blueshifted photons due to the fact that this ratio is
degenerate in this general relativistic formalism. In this paper, we are going
to introduce a new general relativistic invariant observable quantity, the
redshift rapidity, which is the derivative of the redshift with respect to
proper time. With the aid of the redshift rapidity, we shall disentangle $M$
and $D$ in the Schwarzschild background to extract concise and elegant
analytic formulas for mass and distance to the black hole just in terms of
observational elements. Thus, these formulas will help us to break the
degeneracy in the $M/D$ ratio of the Schwarzschild black hole and compute the
mass and distance to the black hole separately. We would like to remark that
these new formulas enable us to compute the distance to black holes and other
astrophysical compact objects in a completely general relativistic framework
with the aim of improving the precision in measuring cosmic distances with
respect to previous Keplerian approaches that compute the angular-diameter
distance to several megamaser galaxies like UGC 3789 MCP2 , NGC 6264 MCPV ,
NGC 5765b MCPVIII , NGC 4258 Humphreys2013 ; Reid2019 , CGCG 074-064 MCPXI ,
among others.

The outline of this paper is as follows. The next section is devoted to a
brief review of our general relativistic formalism in the Schwarzschild black
hole spacetime. Then, we express the mass-to-distance ratio of the
Schwarzschild black hole in terms of observable frequency shifts for three
spacial points of the circular motion of orbiting massive particles, namely,
on the midline and close to the line of sight. In Sec. III, we define redshift
rapidity as the proper time evolution of the frequency shift in the
Schwarzschild background and employ this observable quantity in order to
express the black hole mass and distance just in terms of observational
elements on the midline and close to the line of sight. We finish our paper
with some concluding remarks.

##  II Schwarzschild black hole and the frequency shift

In order to describe our formalism to measure cosmic distances, we start with
a brief review on frequency shift formulas of massive probe particles in the
Schwarzschild background based on PRDbhmn . The Schwarzschild black hole
metric is given by the following line element

$ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\varphi\varphi}d\varphi^{2},$
(1)

with the metric components

$g_{\mu\nu}=diag\left[-\left(1-\frac{2M}{r}\right),\left(1-\frac{2M}{r}\right)^{-1},r^{2},r^{2}\sin^{2}\theta\right],$
(2)

where $M$ is the total mass of the black hole and the event horizon is located
at the Schwarzschild radius $r_{+}=2M$.

The massive geodesic particles revolving the Schwarzschild black hole feel the
curvature of spacetime produced by the black hole mass and keep memory of it.
Besides, the observers located on these particles can exchange redshifted
photons that have information about the Schwarzschild black hole mass in their
frequency from emission till detection. Thus, the shifts in the frequency of
photons along with the orbital parameters of the emitter can be used to
determine the black hole parameters ApJL ; TXS ; TenAGNs ; FiveAGNs .
Therefore, this formalism allows one to compute the black hole parameters in
terms of directly measured observational quantities and the orbital parameter
of the emitter PRDbhmn ; FuZhang ; Diego .

Within General Relativity, the frequency of a photon that is emitted/detected
by an emitter/observer with the proper $4$-velocity
$U_{p}^{\mu}=(U^{t},U^{r},U^{\theta},U^{\varphi})\mid_{p}$ at some position
$x_{p}^{\mu}=(x^{t},x^{r},x^{\theta},x^{\varphi})\mid_{p}$ reads

$\omega_{p}=-\left(k_{\mu}U^{\mu}\right)\mid_{p}\,,$ (3)

where $k_{p}^{\mu}=\left(k^{t},k^{r},k^{\theta},k^{\varphi}\right)\mid_{p}$ is
the $4$-momentum of the photons with the null condition $k_{\mu}k^{\mu}=0$ and
the index p refers to either the emission point $x_{e}^{\mu}$ or detection
point $x_{d}^{\mu}$ of the photons. Additionally, the $4$-velocity of
particles is normalized to unity $U_{\mu}U^{\mu}=-1$, and when the detector’s
orbit is located far away from the emitter-black hole system, $U_{d}^{\mu}$
reduces to

$U_{d}^{\mu}=(1,0,0,0)\,.$ (4)

The frequency shift of light signals emitted by massive geodesic particles in
equatorial circular orbits revolving the spherically symmetric background (1)
is given by PRDhn ; PRDbhmn

$1+z_{{}_{Schw}}\\!=\frac{\omega_{e}}{\omega_{d}}=\frac{(E_{\gamma}U^{t}-L_{\gamma}U^{\varphi})\mid_{e}}{(E_{\gamma}U^{t}-L_{\gamma}U^{\varphi})\mid_{d}},$
(5)

where the conserved quantities $E_{\gamma}$ and $L_{\gamma}$ stand for the
total energy and axial angular momentum of the photons. By considering Eq.
(4), this relation reduces to

$1+z_{{}_{Schw}}\\!=(U^{t}-b_{\gamma}U^{\varphi})\mid_{e},$ (6)

for distant observers and $b_{\gamma}=L_{\gamma}/E_{\gamma}$ is the deflection
of light parameter that takes into account the light bending generated by the
gravitational field in the vicinity of the Schwarzschild black hole. The non-
vanishing components of the $4$-velocity of particles in the equatorial
circular motion read Diego

$U_{e}^{t}=\left.\sqrt{\frac{2}{2g_{tt}-rg_{tt}^{\prime}}}\right|_{r=r_{e}}=\sqrt{\frac{r_{e}}{r_{e}-3M}},$
(7)
$U_{e}^{\varphi}=\left.\frac{1}{r}\sqrt{\frac{rg_{tt}^{\prime}}{2g_{tt}-rg_{tt}^{\prime}}}\right|_{r=r_{e}}=\frac{1}{r_{e}}\sqrt{\frac{M}{r_{e}-3M}},$
(8)

where a prime denotes $\partial_{r}$ and $r_{e}$ is the radius of the emitter.
In addition, the ($\varphi+\delta$)-dependent light bending parameter is given
by [see Eq. (42) of the Appendix]

$\displaystyle b_{\gamma}$ $\displaystyle=$
$\displaystyle\frac{-g_{\varphi\varphi}\sin\left(\varphi+\delta\right)}{\sqrt{-g_{tt}}\sqrt{g_{\varphi\varphi}\sin^{2}\left(\varphi+\delta\right)+r_{e}^{2}g_{rr}\cos^{2}\left(\varphi+\delta\right)}}$
(9) $\displaystyle=$
$\displaystyle-\frac{r_{e}^{3/2}\sin\left(\varphi+\delta\right)}{\sqrt{r_{e}-2M\sin^{2}\left(\varphi+\delta\right)}},$

where $\varphi$ is the azimuthal angle ranging $0\leq\varphi\leq 2\pi$ and
$\delta$ is the aperture angle of the telescope (angular distance) that is a
measurable quantity. Now, by substituting Eqs. (7)-(9) into the redshift
formula (6), we can find the frequency shift of photons emitted by massive
geodesic particles on an arbitrary point of equatorial circular orbits in the
Schwarzschild background as follows

$\displaystyle 1+z_{Schw}\\!\\!=\sqrt{\frac{r_{e}}{r_{e}-3M}}\times$
$\displaystyle\left(1+\sqrt{\frac{Mr_{e}}{r_{e}-2M\sin^{2}\left(\varphi+\delta\right)}}\sin\left(\varphi+\delta\right)\right),$
(10)

with the contribution of the gravitational redshift $z_{g}$ and the kinematic
redshift $z_{kin}$ as

$z_{g}=-1+\sqrt{\frac{r_{e}}{r_{e}-3M}},$ (11)
$z_{kin}=\sqrt{\frac{r_{e}}{r_{e}-3M}}\sqrt{\frac{Mr_{e}}{r_{e}-2M\sin^{2}\left(\varphi+\delta\right)}}\sin\left(\varphi+\delta\right),$
(12)

satisfying $z_{{}_{Schw}}=z_{{}_{g}}+z_{kin}$. In the Newtonian limit
$M/r_{e}\rightarrow 0$, the frequency shift in the Schwarzschild spacetime
(10) reduces to the projection of the Keplerian velocity of a particle in
circular motion on the line of sight in the following way

$z_{N}=\sqrt{\frac{M}{r_{e}}}\sin(\varphi+\delta)+\mathcal{O}\left(\frac{M}{r_{e}}\right),$
(13)

as we expected. Besides, the azimuthal angle $\varphi$ is a function of the
observable $\delta$ for an arbitrary point on the circular orbit as follows
[see Eq. (47) of the Appendix]

$\delta=\arccos\left(\frac{D-r_{e}\cos\varphi}{\sqrt{D^{2}+r_{e}^{2}-2Dr_{e}\cos\varphi}}\right),$
(14)

where $D$ is the radial distance between the black hole center and the
observer.

Fig. 1 illustrates the redshift formula (10) versus the azimuthal angle
$\varphi$. This figure shows how the frequency shift in the Schwarzschild
spacetime changes with the motion of the massive test particle and the non-
vanishing value of the frequency shift at the line of sight indicates the
gravitational redshift. Besides, on the midline where $\varphi=\pm\pi/2$, the
total frequency shift is maximal, hence easier to be identified
observationally.

Figure 1: The frequency shift $z_{Schw}$ (left panel) and the redshift
rapidity $\dot{z}_{Schw}$ (right panel) versus the azimuthal angle in the
Schwarzschild background for $M=1$, $r_{e}=6$, and $D=10^{3}r_{e}$. The shift
in the frequency is maximal on the midline where $\varphi=\pm\pi/2$, whereas
the maximum redshift rapidity is at the line of sight $\varphi=0$. In Eqs.
(10) and (23), we replaced $\delta$ with Eq. (14) to plot these curves.

In order to express the mass-to-distance ratio of the Schwarzschild black hole
in terms of observational quantities, it is convenient to consider two special
important cases useful for describing accretion disks revolving supermassive
black holes hosted at the core of AGNs.

###  II.1 Midline case

The first case is related to highly shifted frequency photons emitted from
particles located at the midline where $\varphi=\pm\pi/2$. The position vector
of these particles with respect to the black hole center is orthogonal to the
observer’s line of sight. Hence, by substituting $\varphi=\pm\pi/2$ in the
frequency shift formula (10), one can find the highly redshifted
($\varphi=+\pi/2$) and blueshifted ($\varphi=-\pi/2$) photons as below

$1+z_{{}_{Schw_{1,2}}}^{m}\\!=\sqrt{\frac{r_{e}}{r_{e}-3M}}\left(1\pm\sqrt{\frac{Mr_{e}}{r_{e}-2M\cos^{2}\delta_{m}}}\cos\delta_{m}\right),$
(15)

where the index “$m$” means the aperture angle $\delta$ should be measured on
the midline and the plus (minus) sign refers to the redshifted (blueshifted)
photons denoted by $z_{{}_{Schw_{1}}}^{m}$ ($z_{{}_{Schw_{2}}}^{m}$) on the
midline. By multiplying $R_{m}:=1+z_{{}_{Schw_{1}}}^{m}$ and
$B_{m}:=1+z_{{}_{Schw_{2}}}^{m}$, it is straightforward to obtain the mass-to-
distance ratio of the Schwarzschild black hole in terms of observational
quantities on the midline as follows

$\frac{M}{D\delta_{m}}=\frac{R_{m}B_{m}-1}{2R_{m}B_{m}}-\frac{\left(1-R_{m}B_{m}\right)\left(3-4R_{m}B_{m}\right)}{6R_{m}B_{m}(3-R_{m}B_{m})}\delta_{m}^{2},$
(16)

where we used the approximation (45) where $D\gg r_{e}$ is the radial distance
between the black hole and the detector. Additionally, we considered the limit
$\delta_{m}\rightarrow 0$ in consistency with the condition (4) for distant
observers as well. As we can see from Eq. (16), the mass-to-distance ratio of
the Schwarzschild black hole is expressed in terms of directly observational
elements $\left\\{R_{m},B_{m},\delta_{m}\right\\}$.

###  II.2 Line of sight case

The second case is describing the frequency shift of photons emitted close to
the line of sight where $\varphi\rightarrow 0$. Hence, by substituting
$\varphi=\varphi_{s}\sim 0$ in the frequency shift formula (10), we find the
expressions for slightly redshifted and slightly blueshifted photons as below

$1+z_{{}_{Schw_{1,2}}}^{s}\\!=\sqrt{\frac{r_{e}}{r_{e}-3M}}\pm\sqrt{\frac{M}{r_{e}-3M}}\left(\varphi_{s}+\delta_{s}\right),$
(17)

where the index “$s$” means the measurement should be performed for systemic
particles (particles close to the line of sight) and we applied the limit
$\delta_{s}\rightarrow 0$ simultaneously. Additionally, we have just the
gravitational redshift $z_{g}$ exactly at the line of sight with
$\varphi_{s}=0=\delta_{s}$. In this relation, the angles $\delta_{s}$ and
$\varphi_{s}$ should be measured close to the line of sight, and the plus
(minus) sign refers to the redshifted (blueshifted) photons close to the line
of sight denoted by $z_{{}_{Schw_{1}}}^{s}$ ($z_{{}_{Schw_{2}}}^{s}$),
respectively. Now, by multiplying $R_{s}:=1+z_{{}_{Schw_{1}}}^{s}$ and
$B_{s}:=1+z_{{}_{Schw_{2}}}^{s}$, we can obtain the mass-to-distance ratio of
the Schwarzschild black hole close to the line of sight as follows

$\frac{M\varphi_{s}}{D\delta_{s}}=\frac{R_{s}B_{s}-1}{3R_{s}B_{s}}+\mathcal{O}\left(\varphi_{s}^{2}\right),$
(18)

in which we used the approximation $r_{e}\sim D\delta_{s}/\varphi_{s}$ from
Eq. (46) valid close to the line of sight and considered the limits
$\left\\{\delta_{s}\rightarrow 0,\varphi_{s}\rightarrow 0\right\\}$
simultaneously to derive this equation. Furthermore, this relation is a
function of a set of purely observational quantities
$\left\\{R_{s},B_{s}\right\\}$.

##  III Disentangling mass and distance

The mass-to-distance ratio formulas (16) and (18) of the Schwarzschild black
hole can be employed to estimate the $M/D$ ratio of real astrophysical systems
as it was accomplished for the central supermassive black holes of $17$
galaxies in ApJL ; TXS ; TenAGNs ; FiveAGNs . As the next step in this
direction, we are going to disentangle the mass $M$ and the distance to the
black hole $D$ to obtain closed formulas for each parameter in terms of the
set of observational elements
$\left\\{R_{m},B_{m},\delta_{m};R_{s},B_{s},\delta_{s}\right\\}$. To do so, we
define the redshift rapidity as the proper time evolution of the frequency
shift $z_{Schw}$ (6) in the Schwarzschild background as follows

$\dot{z}_{Schw,e}=\frac{dz_{Schw}}{d\tau}=\frac{d}{d\tau}(U_{e}^{t}-b_{\gamma}U_{e}^{\varphi}),$
(19)

which is the redshift rapidity at the emission point. Notwithstanding, we
should note that the redshift rapidity needs to be measured from the Earth.
Therefore, by making use of the chain rule, we rewrite Eq. (19) at the
observer position as

$\dot{z}_{Schw}=\frac{dz_{Schw}}{dt}=\frac{d\tau}{dt}\frac{dz_{Schw}}{d\tau}=\frac{1}{U_{e}^{t}}\frac{d}{d\tau}(U_{e}^{t}-b_{\gamma}U_{e}^{\varphi}),$
(20)

which is an observable quantity that we measure here on the Earth. For massive
geodesic particles circularly orbiting the black hole in the equatorial plane,
the redshift rapidity (20) reduces to

$\dot{z}_{Schw}=-\frac{db_{\gamma}}{d\tau}\
\frac{U_{e}^{\varphi}}{U_{e}^{t}},$ (21)

since the $t$-component $U_{e}^{t}$ (7) and $\varphi$-component
$U_{e}^{\varphi}$ (8) of the $4$-velocity are constant quantities for circular
orbits whereas the light bending parameter (9) is time dependent through
$\delta$ and $\varphi$. By employing the chain rule, we have

$\dot{z}_{Schw}=-\left(\frac{\partial
b_{\gamma}}{\partial\varphi}+\frac{\partial
b_{\gamma}}{\partial\delta}\frac{\partial\delta}{\partial\varphi}\right)\frac{\left(U_{e}^{\varphi}\right)^{2}}{U_{e}^{t}},$
(22)

where we used $U_{e}^{\varphi}=\left.\frac{d\varphi}{d\tau}\right|_{r=r_{e}}$.
Now, we perform the aforementioned derivatives by taking into account the
light bending parameter (9) and the $\delta\left(\varphi\right)$-relation (14)
to obtain the redshift rapidity for an arbitrary point on the circular orbit
as below

$\displaystyle\dot{z}_{Schw}\\!\\!=\frac{1}{\left[r_{e}-2M\sin^{2}(\varphi+\text{$\delta$})\right]^{\frac{3}{2}}}\times$
$\displaystyle\frac{DM\left(D-r_{e}\cos\varphi\right)\cos(\varphi+\text{$\delta$})}{\sqrt{r_{e}-3M}\left(D^{2}+r_{e}^{2}-2Dr_{e}\cos\varphi\right)}\\!.$
(23)

Fig. 1 shows the behavior of the redshift rapidity versus the azimuthal angle
$\varphi$. As one can see from this figure, the redshift rapidity at the line
of sight is maximal, hence it is easier to measure this quantity at
$\varphi=0$.

It is worth noting that the redshift rapidity (23) reduces to the projection
of the Keplerian acceleration of a particle in circular motion on the line of
sight in the Newtonian limit $M/r_{e}\rightarrow 0$ and large distances
$r_{e}/D\rightarrow 0$

$\dot{z}_{N}=\frac{M}{r_{e}^{2}}\cos(\varphi+\delta)+\mathcal{O}\left(\frac{M^{2}}{r_{e}^{3}}\right),$
(24)

as it should be.

As the final step, based on the emitter-black hole system configuration and
observational data availability, we can employ the redshift rapidity (23) as
well as either Eq. (16) or Eq. (18) to disentangle the Schwarzschild black
hole mass $M$ and the distance to the black hole $D$. Here, we are going to
disentangle $M$ and $D$ for two important cases describing astrophysical
photon sources within accretion disks circularly orbiting central supermassive
black holes at the core of AGNs.

###  III.1 Midline case

The first important case is related to the emitters on the midline where their
emitted photons are highly redshifted/blueshifted. The mass-to-distance ratio
of the central black hole in terms of these frequency shifted photons is given
in Eq. (16). In order to find the redshift rapidity of the same redshifted
particles, we substitute the approximation (45) for $r_{e}$ in the redshift
rapidity relation (23) and consider the limit $\delta_{m}\rightarrow 0$ to get

$\displaystyle\dot{z}_{Schw}^{m}$ $\displaystyle=$
$\displaystyle\frac{\frac{M}{D\delta_{m}}}{D\sqrt{1-3\frac{M}{D\delta_{m}}}\left(1-2\frac{M}{D\delta_{m}}\right)^{\frac{3}{2}}}\times$
(25)
$\displaystyle\left(1+\frac{11-28\frac{M}{D\delta_{m}}-12\left(\frac{M}{D\delta_{m}}\right)^{2}}{6\left(1-3\frac{M}{D\delta_{m}}\right)\left(1-2\frac{M}{D\delta_{m}}\right)}\delta_{m}^{2}\right),$

up to second order in $\delta_{m}$. Finally, one can find the explicit form of
the Schwarzschild black hole mass and distance to the black hole by solving
Eqs. (16) and (25) as below

$M=\frac{\left(R_{m}B_{m}-1\right)^{2}}{2\dot{z}_{Schw}^{m}\sqrt{6-2R_{m}B_{m}}}\delta_{m}+\mathcal{O}\left(\delta_{m}^{3}\right),$
(26) $\displaystyle
D\\!\\!=\frac{R_{m}B_{m}\left(R_{m}B_{m}-1\right)}{\dot{z}_{Schw}^{m}\sqrt{6-2R_{m}B_{m}}}\times$
(27)
$\displaystyle\left(1+\frac{R_{m}B_{m}(3R_{m}B_{m}-7)(8R_{m}B_{m}-27)-36}{6(R_{m}B_{m}-3)^{2}}\delta_{m}^{2}\right)\\!,$

fully expressed in terms of observational quantities
$\left\\{R_{m},B_{m},\dot{z}_{Schw}^{m},\delta_{m}\right\\}$ that should be
measured on the midline.

###  III.2 Line of sight case

The second important case belongs to the photon sources that lie close to the
line of sight where their emitted photons are slightly redshifted/blueshifted
but have maximum redshift rapidity. The mass-to-distance ratio of the central
black hole in terms of these frequency shifted photons is given by Eq. (18).
Thus, we substitute $r_{e}=D\delta_{s}/\varphi_{s}$ from the approximation
(46) into the redshift rapidity formula (23) and apply the limits
$\left\\{\delta_{s}\rightarrow 0,\varphi_{s}\rightarrow 0\right\\}$ to get

$\dot{z}_{Schw}^{s}=\frac{1}{M}\left(1-3\frac{M\varphi_{s}}{D\delta_{s}}\right)^{-\frac{1}{2}}\left(\frac{M\varphi_{s}}{D\delta_{s}}\right)^{2},$
(28)

for the photon sources close to the line of sight. Now, by substituting Eq.
(18) in this relation, we can find the Schwarzschild black hole mass as
follows

$M=\frac{\left(R_{s}B_{s}-1\right)^{2}}{9\dot{z}_{Schw}^{s}\left(R_{s}B_{s}\right)^{\frac{3}{2}}},$
(29)

fully in terms of observational quantities
$\left\\{R_{s},B_{s},\dot{z}_{Schw}^{s}\right\\}$ that should be measured
close to the line of sight. As the next step, we take advantage of this
relation and Eq. (16) in order to combine observations from both midline and
line of sight to get the distance to the black hole as

$D=\frac{\left(R_{s}B_{s}-1\right)^{2}}{9\dot{z}_{Schw}^{s}\left(R_{s}B_{s}\right)^{\frac{3}{2}}}\frac{2R_{m}B_{m}}{\delta_{m}\left(R_{m}B_{m}-1\right)},$
(30)

which is a significant relation because it contains information on both high-
frequency shifted particles on the midline $\left\\{R_{m},B_{m}\right\\}$ and
maximum redshift rapidity close to the line of sight $\dot{z}_{Schw}^{s}$ (see
Fig. 1 and related discussion). From an observational point of view, this
formula is important due to the fact that it is easier to identify frequency
shifts on the midline and the redshift rapidity close to the line of sight.

##  IV Discussion and final remarks

In this paper, we have presented two relations for the mass-to-distance ratio
of the Schwarzschild black hole in terms of observable frequency shifts for
three spacial points of the circular motion of orbiting massive particles: on
the midline and close to the line of sight. Then, we calculated the derivative
of the frequency shift in the Schwarzschild background with respect to proper
time in order to introduce the redshift rapidity. Then, we employed the
redshift rapidity to express the Schwarzschild black hole mass and its
distance from the Earth just in terms of observational quantities. In this
formalism, the redshift rapidity is also a general relativistic invariant
observable element, indicating the proper time evolution of the frequency
shift in the Schwarzschild spacetime.

We have extracted concise and elegant analytic formulas that allow us to
disentangle mass and distance to the black hole in the Schwarzschild
background and compute these parameters separately, not the mass-to-distance
ratio that we have estimated before in ApJL ; TXS ; TenAGNs ; FiveAGNs . Our
exact formulas have been obtained on the midline and close to the line of
sight whereas the general relations could be employed in black hole parameter
estimation studies. The next step in this research direction would be
estimating the mass of supermassive Schwarzschild black holes hosted at the
core of AGNs and their distance to the Earth with the help of the general
relativistic formalism developed in the present paper. A task that we shall
address in future works.

## Acknowledgements

We thank D. Villaraos and A. González-Juárez for fruitful discussions. All
authors are grateful to CONACYT for support under Grant No. CF-MG-2558591;
M.M. also acknowledges SNI and was supported by CONACYT through the
postdoctoral Grant No. 31155. P.B. acknowledges financial support from the
Science and Engineering Research Board, Government of India, File Number
PDF/2022/000332. A.H.-A. and U.N. thank SNI and PRODEP-SEP and were supported
by Grants VIEP-BUAP and CIC-UMSNH, respectively. U.N. also acknowledges
support under Grant No. CF-140630.

*

##  Appendix A ($\varphi+\delta$)-dependent light bending parameter and the
angular distance $\delta$ as a function of the azimuthal angle $\varphi$

Figure 2: The geometrical illustration of the bidimensional vector $K$, and
the relation between the azimuthal angle $\varphi$ and the angular distance
$\delta$. Figure 3: The profile of the angular distance versus the azimuthal
angle for $D=3r_{e}$. The vertical dashed lines denote
$\phi_{\pm}=\pm\arccos(1/3)$ where $d\delta/d\varphi$ vanishes.

In a similar manner to PRDbhmn , we obtain the light bending parameter for a
general point of the circular orbit in the equatorial plane. However, unlike
the previous work where the angular distance $\delta$ was neglected due to the
large distances between the black holes and the Earth, we consider the
contribution of this measurable parameter in our analysis. This is because
even though the angular distance is very small, its variation with respect to
time, especially close to the line of sight, is quite important. The other aim
of this Appendix is to show the mathematical relation between the observable
angular distance $\delta$ and the unobservable azimuthal angle $\varphi$.

In order to obtain the light bending parameter of photons coming from a
general point of the circular orbit in the equatorial plane (see Fig. 2), we
take into account the equation of motion of photons in the equatorial plane
($k^{\theta}=0$) of the Schwarzschild spacetime (1)

$g_{\mu\nu}k^{\mu}k^{\nu}=g_{tt}\left(k^{t}\right)^{2}+g_{rr}\left(k^{r}\right)^{2}+g_{\varphi\varphi}\left(k^{\varphi}\right)^{2}=0,$
(31)

where $k^{t}$ and $k^{\varphi}$ can be found through the temporal Killing
vector field $\xi^{\mu}=\delta_{0}^{\mu}$ and the rotational Killing vector
field $\psi^{\mu}=\delta_{3}^{\mu}$ as follows

$E_{\gamma}=-\xi_{\mu}k^{\mu}=-g_{tt}k^{t},$ (32)
$L_{\gamma}=\psi_{\mu}k^{\mu}=g_{\varphi\varphi}k^{\varphi},$ (33)

which $E_{\gamma}$ and $L_{\gamma}$ are the energy and angular momentum of the
photons, respectively.

By introducing (32) and (33) into (31), the equation of motion takes the
following form

$g_{rr}\left(k^{r}\right)^{2}+\frac{E_{\gamma}^{2}}{g_{tt}}+\frac{L_{\gamma}^{2}}{g_{\varphi\varphi}}=0,$
(34)

that can be employed to express $k^{r}$ in terms of the constants of motion
and metric components as below

$\left(k^{r}\right)^{2}=-\frac{1}{g_{rr}}\left(\frac{E_{\gamma}^{2}}{g_{tt}}+\frac{L_{\gamma}^{2}}{g_{\varphi\varphi}}\right).$
(35)

Now, by considering the non-vanishing angular distance $\delta\neq 0$, we
geometrically introduce the auxiliary bidimensional vector $K$ defined by the
following decomposition (see Fig. 2 for more details)

$k^{r}=K\cos\left(\varphi+\delta\right),$ (36)
$r_{e}k^{\varphi}=K\sin\left(\varphi+\delta\right),$ (37)

with

$K^{2}=\left(k^{r}\right)^{2}+r_{e}^{2}\left(k^{\varphi}\right)^{2}.$ (38)

Hence, by substituting Eqs. (33) and (35) into (38), one can find $K^{2}$
versus the constants of motion and metric components as follows

$K^{2}=-\frac{1}{g_{rr}}\left(\frac{E_{\gamma}^{2}}{g_{tt}}+\frac{L_{\gamma}^{2}}{g_{\varphi\varphi}}\right)+r_{e}^{2}\frac{L_{\gamma}^{2}}{g_{\varphi\varphi}^{2}}.$
(39)

On the other hand, by introducing Eq. (36) into (35), we can find a similar
relation for $K^{2}$ as

$K^{2}=-\frac{1}{g_{rr}\cos^{2}\left(\varphi+\delta\right)}\left(\frac{E_{\gamma}^{2}}{g_{tt}}+\frac{L_{\gamma}^{2}}{g_{\varphi\varphi}}\right).$
(40)

Equating previous relations gives an equation for the
$\left(\varphi+\delta\right)$-dependent light bending parameter $b_{\gamma}$
as below

$\left(g_{tt}b_{\gamma}^{2}+g_{\varphi\varphi}\right)g_{\varphi\varphi}\tan^{2}\left(\varphi+\delta\right)+r_{e}^{2}g_{rr}g_{tt}b_{\gamma}^{2}=0,$
(41)

that leads to the following solution for $b_{\gamma}$

$b_{\gamma}=\frac{-g_{\varphi\varphi}\sin\left(\varphi+\delta\right)}{\sqrt{-g_{tt}}\sqrt{g_{\varphi\varphi}\sin^{2}\left(\varphi+\delta\right)+r_{e}^{2}g_{rr}\cos^{2}\left(\varphi+\delta\right)}},$
(42)

which is the light bending parameter for an arbitrary point of the circular
orbit on the equatorial plane.

On the other hand, we assume the emitter is far enough from the black hole and
consider the geometrical configuration illustrated in Fig. 2 to obtain a
relation between the observable angular distance $\delta$ and the unobservable
azimuthal angle $\varphi$. This is necessary to express $\varphi$ in terms of
the observable quantity $\delta$ in order to obtain $M$ and $D$ just in terms
of observational elements, hence being able to break the degeneracy of the
$M/D$ ratio.

By taking into account the right triangle in Fig. 2, one can immediately
identify the following relation between the set
$\left\\{D,r_{e};\varphi,\delta\right\\}$

$D\tan\delta=r_{e}\left(\sin\varphi+\cos\varphi\tan\delta\right),$ (43)

where only $\delta$ is observable and the rest of the parameters are unknown
in the case of black holes since we cannot identify the black hole position by
observations. After doing some straightforward simplifications, this relation
reduces to

$D\sin\delta=r_{e}\sin\left(\varphi+\delta\right),$ (44)

in which for the far away detectors where $D\gg r_{e}$ and $\delta\rightarrow
0$, we have

$r_{e}\approx D\delta_{m}\left(1+\frac{\delta_{m}^{2}}{3}\right),\text{ \ for
\ }\varphi=\pm\frac{\pi}{2},$ (45)
$r_{e}\approx\frac{D\delta_{s}}{\varphi_{s}},\text{ \ for \
}\varphi\rightarrow 0,$ (46)

for the midline $\varphi=\pm\pi/2$ and close to the line of sight
$\varphi\rightarrow 0$, respectively.

Now, we solve the relation (44) in order to express $\delta$ in terms of the
rest of the parameters. This equality has four solutions, and we choose the
physical one as below

$\delta=\arccos\left(\frac{D-r_{e}\cos\varphi}{\sqrt{D^{2}+r_{e}^{2}-2Dr_{e}\cos\varphi}}\right),$
(47)

since as $\varphi$ decreases/increases, $\delta$ should also decrease/increase
in the same direction (see Fig. 2).

Fig. 3 illustrates the behavior of $\delta$ versus $\varphi$. As one can see
from this figure, if we solve the azimuthal angle $\varphi$ in terms of
$\delta$, we shall have four expressions as follows

$\varphi=\left\\{\begin{array}[]{cc}\pm\delta\mp\arcsin\left(\frac{D\sin\delta}{r_{e}}\right),&\phi_{-}\leq\varphi\leq\phi_{+}\\\
\pm\pi\mp\delta\mp\arcsin\left(\frac{D\sin\delta}{r_{e}}\right),&\left\\{\begin{array}[]{c}-\frac{\pi}{2}\leq\varphi\leq\phi_{-}\\\
\phi_{+}\leq\varphi\leq\frac{\pi}{2}\end{array}\right.\end{array}\right.,$
(48)

to cover the whole semicircular path from $\varphi=-\pi/2$ to $\varphi=\pi/2$.
In these relations, $\phi_{+}$ ($\phi_{-}$) denotes a special point where
$d\delta/d\varphi$ vanishes for $\varphi>0$ ($\varphi<0$), as indicated in
Fig. 3.

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