How to measure a black hole ’ s mass , spin , and direction of spin axis in the Kerr lens effect 1 : Test case with simple source emission near a black hole

... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . We propose a theoretical principle to measure the mass, spin, and direction of spin axis of Kerr black holes (BHs) through observing 2 quantities of the spinning strong gravitational lens effect of BHs. Those observable quantities are generated by 2 light rays emitted at the same time by a source near the BH: the primary and secondary rays that reach a distant observer, respectively, the earliest and secondary temporally. The time delay between detection times and the ratio of observed specific fluxes of those rays are the observable quantities. Rigorously, our proposal is applicable to a single burst-like (short duration) isotropic emission by the source.An extension of our principle to cases of complicated emissions may be constructed by summing up appropriately the result of this paper, which will be treated in future works. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index E31


Introduction
What is the meaning of direct measurement of BHs?To consider this question, we emphasize 2 theoretical facts about BHs (Ref.[9]) in the framework of general relativity: a BH is a very strong gravitating spacetime region that traps anything inside it, and a BH is completely characterized by 2 parameters due to the uniqueness theorem: its mass and spin angular momentum.(We expect that BHs have no electric charge in astrophysical situations.)According to these theoretical facts, let us define direct measurement of BHs as follows: It is to measure the mass and spin angular momentum of BHs by a direct observation of some strong gravitational (general relativistic) effect produced by the BH.
The first example of direct BH measurement by this definition has been succeeded by the advanced-LIGO that detected the gravitational wave of a BH-BH binary coalescence (Ref.[1]).Gravitational wave astronomy for direct BH measurement seems to be ready to start in the near future.On the other hand, no example of direct BH measurement by electromagnetic wave observations has been achieved yet.Developments of electromagnetic observations not only in technology but also in theory are expected to make collaborations of gravitational waves and electromagnetic waves in the direct observational study of BHs.
We consider the observation of light rays (massless particles) coming from a source near a single BH (not a BH binary), and propose a theoretical principle of direct BH measurement.Our principle PTEP 2017, 053E02 H. Saida makes use of some observable quantities of the strong gravitational lens effect produced by the mass and spin of a single Kerr BH (Kerr BH lens effect).
Concerning the Kerr BH lens effect for performing direct BH measurement, many ideas have been investigated so far.For example, Cunningham and Bardeen (Ref.[6]) considered a star on a circular orbit on the equatorial plane of an extreme Kerr BH and calculated the brightness of primary and secondary images detected by a distant observer.Holz and Wheeler (Ref.[13]) proposed the retro-macho, which is the multi-image of a star (the Sun in their original paper) located between the observer and a lensing BH.Bozza and Mancini (Ref.[4]) have applied the idea of retro-macho to the star S2 orbiting the massive BH candidate at the center of our galaxy.Fukumura et al. (Refs.[10,11]) proposed the BH echoes in order to discuss the so-called quasi-periodic oscillations, in which they considered X-ray sources near a Kerr BH and calculated the observable spectrum produced by multiple rays emitted randomly by the sources.Hoŕak and Karas (Ref.[14]) studied the effects of time delay of multiple rays on the polarization magnitude.James et al. (Ref.[15]) considered an observer near a BH and calculated the observed images of the BH and distant stars seen by the observer.Apart from these examples, the famous observable quantities of the Kerr BH lens effect may be the broadening of iron line emission by an accretion disk around a BH (see Kojima [16], Fanton et al. [8], and references therein) and the BH shadow (see Luminet [17], Takahashi [22], and references therein).
Although these interesting ideas have not succeeded yet in observations, improving these ideas is necessary in order to perform direct BH measurement by electromagnetic observations.In this paper, we consider a similar case with Cunningham and Bardeen (Ref.[6]), but do not restrict our discussions to the extreme Kerr BH and circular orbit on an equatorial plane.Our theoretical proposal for direct BH measurement is made under assumptions: (i) the environment around the BH is transparent for photons, at least in the frequency band of observation, (ii) a single source is moving on any orbit around the Kerr BH, (iii) the source emits a single burst-like (short duration) isotropic emission, and (iv) a distant observer detects the primary and secondary rays that reach the observer, respectively, the earliest and secondary temporally.Conditions (i) and (iv) are the same conditions as Cunningham and Bardeen (Ref.[6]), but conditions (ii) and (iii) are not.Then, if our 4 conditions are applied to a BH, our theoretical principle can let us measure not only the BH's mass and spin but also the direction angle of the spin axis seen by the observer.
In this paper, we investigate 2 observable quantities: the delay between detection times of the primary and secondary rays, and the ratio of observed specific fluxes of the 2 rays.By performing some numerical estimations of these observable quantities, we discuss the potential detectability of them by present or near future telescope capability.Thorough numerical studies on the feasibility of our principle and on the behavior of our observable quantities will be reported in another paper, since the numerical calculations will need a powerful computer.
Note that some of the existing ideas of Kerr BH lens effects, e.g., the light source radiates photons in some temporally and spatially successive, random, or continuous fashion.Such complicated emissions will be constructed by summing appropriately some burst-like emissions.Therefore, this paper concentrates on the case with single burst-like emission as the preparation for cases with complicated source emissions.The Kerr BH lens effects for such complicated source emissions are the issue of our next papers.
In Sect.2, our target phenomenon and observable quantities are specified, and the theoretical principle of direct BH measurement is proposed.In Sect.3, the formulas and our numerical procedure for calculating the observable quantities are explained so that readers can check our proposal 2/40 Downloaded from https://academic.oup.com/ptep/article-abstract/2017/5/053E02/3855709/How-to-measure-a-black-hole-s-mass-spin-and by CERN -European Organization for Nuclear Research user on 03 October 2017 PTEP 2017, 053E02 H. Saida quantitatively.However, readers who do not need the details of formulas can skip Sect.3.Then, Sect. 4 shows some numerical estimations of our observable quantities to discuss the potential detectability of them.Finally, Sect. 5 is for a summary and discussions.We use the units c = 1 and G = 1, and measure all quantities in the dimension of length.However, when MKS units are convenient for a line of thought, we use the units and show c and G explicitly.

Principle of measuring a BH's mass, spin, and spin axis direction angle 2.1. Basic assumptions
Our target phenomenon is a single burst-like (short duration) isotropic emission by a source that is much smaller than its host BH.This is modeled as a point-like source that moves near the BH and emits light rays in all directions within a duration much shorter than a typical dynamical time scale of the system composed of the source and the BH (e.g., the Keplerian time scale of the source around a BH).Further, we assume that the environment around the BH is transparent, at least in the frequency band of observation.Note that the details of the intrinsic structure of the source is outside the scope of this paper, and we assume such a simple emission occurs in the transparent environment around the BH.We focus on how the gravitational lens effect of the Kerr BH appears in observable quantities of light rays that are emitted by the source and received by a rest observer distant from the BH.
On these assumptions, let us note that an example of a transparent environment has been recognized for the massive BH candidate at the center of our galaxy, Sagittarius A * (Sgr A * ) (Refs.[3,7,21]).Radio observational data of Sgr A * for a rather wide band range indicate the disappearance of plasma scattering effects at high frequency (>230 GHz) radio waves, which implies a transparent environment around the BH for such high frequency radio photons (Refs.[3,7,21]).Further, it is also reported that intermediate mass BH candidates seem to exist in the central region of our galaxy (Ref.[19]).Such an intermediate mass BH candidate does not form a binary with some other body, but it is recognized by a significant (large acceleration) motion of gas cloud around a dark compact region, which is consistent with assuming an intermediate mass BH in the dark compact region (Ref.[19]).We may expect that such an intermediate mass BH, if it exists, wears a transparent and inactive environment.
Here note that the estimated mass of Sgr A * is about 4 × 10 6 M (Ref.[2]), and that of the intermediate mass BH is about 10 5 M (Ref.[19]), where M 2 × 10 30 kg is the mass of the Sun.Therefore, gas clouds and celestial bodies whose mass is about M are much smaller than those BHs.Then, let us consider the case that a gas cloud or a dark celestial body of stellar mass falls toward the BH and emits a short duration emission in all directions (due, e.g., to some shock formation or disruption event).Such a case may be effectively described by our assumptions or by some appropriate summation of our simple source emission.Anyway, the detail of the source's structure is outside the scope of this paper and will be properly treated in another paper.Our interest, in this paper, is in the theoretical properties of the Kerr BH lens effect under our assumptions described in the first paragraph.
Let us make a comment on the terminology.The term "strong gravitational lens effect" is already used in, e.g., the survey of galaxy distribution, dark matter distribution, and so on.The meaning of this term in those research fields is the lens effect of a cluster of masses (not necessarily BHs) that produces the multi-image of a source (e.g., galaxy) behind the cluster.This usage of "strong gravitational lens effect" includes the effect of mass, but not the effect of a BH's spin.However, we are interested in the lens effect of a spinning BH, which shall include the effect not only of the mass but also of the spin of the BH.Therefore, we use the term Kerr BH lens effect (or Kerr lens effect) for the lens effect produced by the mass and spin of a Kerr BH.

Kerr spacetime, notation in this paper
To describe Kerr spacetime, we use Boyer-Lindquist (BL) coordinates (t, r, θ, ϕ) throughout this paper.The metric components g μν are read from the line element of spacetime, where and the time coordinate t, the BH's mass M , and spin parameter a := J /M (where J is the spin angular momentum) are measured in the dimension of length.Instead of a, we use the dimensionless spin parameter χ := a/M , when it is convenient for the line of thought.In some figures shown later, we project 4-dimensional null geodesics onto 3-dimensional screen spanned by (x, y, z) := (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ). ( In this 3-dimensional screen, the z-axis corresponds to the axis of the BH's spin.The radius of the BH horizon is . The BH's spin is a rigid rotation, due to the rigidity theorem, with angular velocity ω BH = ω(r BH , π/2) = χ/(2r BH ).Without loss of generality, we can set 0 ≤ χ < 1, where the upper bound is due to the existence of the BH horizon and the nonnegativity of χ corresponds to the direction of BH spin given by ω BH ≥ 0. We do not need to consider the opposite direction of spin because of the axisymmetry of Kerr spacetime.
The zero-angular-momentum-observer (ZAMO) is defined as the one whose 4 velocity u μ zamo is perpendicular to the space-like hypersurface at t = constant, Although ZAMO has, by definition, no angular momentum with respect to the space-like hypersurface at t = constant, its 4-velocity, however, has a ϕ-component.This is understood as the effect of the rotation of spacetime itself, which is called the frame dragging effect of a Kerr BH.The angular velocity of ZAMO measured in BL coordinates u ϕ zamo /u t zamo = ω is regarded as the angular velocity of the frame dragging effect measured in BL coordinates.

Observable quantities
By the Kerr lens effect, the light ray winds around the BH before reaching the distant rest observer.An arbitrary number of windings around the BH are possible by tuning the values of the impact parameters of the light ray.In this paper, we focus on the 2 light rays that reach the distant observer earliest and secondary temporally.We call these rays, respectively, the primary ray (p-ray) and secondary ray (s-ray).Some numerical examples of the orbits of these rays (solutions of null geodesic equations) are shown in Fig. 1, which depicts not only the p-ray and the s-ray but also 2 other rays of higher winding number that reach the same rest observer.(The detail is explained in Sect.3.) As explained in Sect. 1, the observable quantities that we focus on are the delay t obs between the detection time of the p-ray t obs(p) and that of the s-ray t obs(s) , and the ratio R obs of the observed specific flux of the s-ray F obs(s) to that of the p-ray F obs(p) : The dimensions of F obs are W/(m 2 Hz) in MKS units, where 1/m 2 denotes per unit spatial area perpendicular to the observer and 1/Hz denotes per unit observed frequency which is the meaning of "specific".Under the assumptions introduced in Sect.2.1, the time delay t obs and specific flux ratio R obs are regarded as functions of some parameters.The time delay t obs is regarded as a function of following parameters: M , χ : parameters characterizing Kerr spacetime, (θ obs , ϕ obs ) : angular coordinates of the distant observer, x μ emi = (0, r emi , θ emi , 0) : spacetime position of the source at emission, where the emission time is set to 0 (t emi = 0) and the distant limit of the observer (r obs → ∞) is supposed as mentioned in Sect.2.1.The parameters (θ obs , ϕ obs ), and x μ emi determine the relative location among the BH, source, and distant observer.
Note that, by the axisymmetry of Kerr spacetime, we can rotate the BL coordinates in the ϕ-direction so as to make the new azimuthal angles of the observer and source ϕ obs = 0 and ϕ emi = −ϕ obs .However, for the convenience of our numerical calculation, we fix the source's azimuthal coordinate at ϕ emi = 0 and let the observer's azimuthal coordinate ϕ obs be a free parameter.Also, it is worth emphasizing that the zenithal angle θ obs is the direction angle of the BH's spin axis seen from the observer (the angle between the line of sight and the spin axis).
Once the relative location is fixed, the null geodesic equation can be solved and the orbits of the p-ray and the s-ray are specified.Then, the detection times t obs(p) and t obs(s) are determined and the time delay t obs can be read from the numerical result as shown in the right-hand panel of Fig. 1.This t obs may be roughly estimated as t obs ∼ α2πr emi , where 0 α 1, and α ∼ 1 for ϕ obs ∼ 0 (the emission event is in front of the BH seen from the observer) and α ∼ 0 for ϕ obs ∼ π (the emission event is behind the BH seen from the observer).For the example in Fig. 1, (M , χ, r emi ) = (1, 0.8, 2.2r BH ), this estimation formula gives t obs ∼ 4.4πM (1+(1−χ 2 ) 1/2 ) 22, which is roughly consistent with the value of t obs read from the right-hand panel of Fig. 1.
Next, the observed specific fluxes of the p-ray and the s-ray, F obs(p) and F obs(s) , are regarded as functions of the parameters in list (5) and u μ emi : 4-velocity of the source at emission, ν obs : observed frequency of the p-ray or s-ray, I emi (ν emi ) : intrinsic specific intensity of the source at emission (ν emi is the emission frequency measured by the source), where, because the isotropic emission is assumed as mentioned in Sect.2.1, the emission intensity I emi (ν emi ) depends only on the emission frequency ν emi and not on the emission direction.The dimensions of I emi (ν emi ) are W/(m 2 Hz ste-rad) in MKS units, where 1/m 2 denotes per unit spatial area perpendicular to the source, 1/ste-rad denotes per unit solid-angle seen from the source, and 1/Hz denotes per unit frequency measured at the source.Further, F obs 's dependence on u μ emi is due to the kinetic (special relativistic) Doppler effect on the light ray.Since the kinetic Doppler effect is determined by the relative velocity between the source and ray, the strength of the kinetic Doppler effect is different between the p-ray and the s-ray.However, since the p-ray and the s-ray are emitted by the same source at the same time, the strength of gravitational Doppler effects on those rays are the same because the gravitational Doppler effect is determined by the gravitational potential at the emission.
There are 2 basic types of observation of the specific flux ratio R obs : Type LD (line detection): Detect the p-ray and the s-ray at the same observation frequency ν obs .In this case, generally, the emission frequency of the p-ray ν emi(p) and that of the s-ray ν emi(s) are different (ν emi(p) = ν emi(s) ) because the strength of the kinetic Doppler effect on the p-ray is generally different from that on the s-ray.Type LE (line emission): Detect the p-ray and the s-ray at different observation frequencies, respectively ν obs(p) and ν obs(s) , so that the difference between ν obs(p) and ν obs(s) offsets the difference between the kinetic Doppler effects on those rays.This means that the emission frequencies of We are studying a method of measuring the value of (M , χ, θ obs ) by observing quantities ( t obs , R obs ), which are functions of many parameters in lists (5) and (6).Our theoretical principle of measuring (M , χ, θ obs ) is simple, consisting of the following 3 steps: Step 1 (BH measurement): This step consists of theoretical and observational tasks.
Task of theory: Theoretically calculate the values of ( t obs , R obs ) for various values of the parameters ( 5) and (6).Task of observation: Using an appropriate telescope, observe the values of ( t obs , R obs ) for 1 BH candidate as many times as possible.
Step 2 (BH measurement): From the results of step 1, construct a table such as Table 1.In each row of this table, all sets of the values of parameters ( 5) and ( 6) in the right-hand slot predict the same value of observable quantities ( t obs , R obs ) in the left-hand slot.
Step 3 (BH measurement): It is a fact that the parameters (M , χ) and the direction angle θ obs will not change for all observational data ( t obs , R obs ) since the target BH candidate is the same for all data.Hence, if we find only 1 set of values of (M , χ, θ obs ) that is shared by all rows in Table 1, then the common values should be the true values of the mass, spin parameter, and direction angle of the BH candidate under observation.
The above method consisting of 3 steps is our principle for measuring the value of a BH's parameters (M , χ) and the direction angle θ obs of a BH's spin axis.Because the value of (M , χ) is determined by the observable quantities of the Kerr BH lens effect (one of the general relativistic effects), our method is regarded as one principle of direct BH measurement under the definition given at the beginning of Sect. 1.Note that, in the task of observation, we do not need the repetition of observation under the same relative location among the BH, source, and observer.In general, the value of ( t obs , R obs ) for 1 relative location differs from that for the other relative location.The point is that the number of rows in Table 1 becomes large by observing the value of ( t obs , R obs ) as many times as possible for various relative locations.The larger the number of rows in Table 1, the more certain the true value (M true , χ true , θ true ).
Also note that, when the values of parameters ( 5) and ( 6) are given, the flux ratio R (LD) obs of the type LD observation differs from R (LE) obs of the type LE observation in general (but the time delay t obs is common to both types of observation).Therefore, in the task of theory, we need to perform 2 calculations of R (LD) obs and R (LE) obs for each given value of parameters ( 5) and (6).Then, one of them should be selected (or their combination should be constructed) according to the real observational method used in the task of observation.

Formalism to calculate t obs and R obs
Under the condition that the parameters in Eqs. ( 5) and ( 6) are given, we construct the formalism for calculating ( t obs , R obs ).

Numerical setup of the observer
We consider the rest observer (r obs , θ obs , ϕ obs ) = constant, whose velocity is u As mentioned in Sect.2.1, we consider the distant observer (r obs → ∞) whose velocity becomes The velocity of our observer coincides with the time-like Killing vector: u obs = ξ (t) .
Although our ideal setup is the observer at r obs → ∞, our numerical calculation will be performed with the rest observer at r obs = 100r BH which gives 100/99 ( 1.01) ≤ 1/ √ |g tt | ≤ 50/49 ( 1.02).On the other hand, we will use Eq. ( 7) for u μ obs in our numerical calculation.This implies that our numerical results of observable quantities, shown in Sect.4, include an intrinsic error of 1% ∼ 2%.However, this error does not affect the numerical results showing the potential detectability of our observable quantities.

Time delay t obs : null geodesic equations
Since we assume the duration of emission is short (burst-like), the p-ray and the s-ray are both regarded as 1 pulse made of photons.The 4-dimensional orbit of a light ray is a null geodesic of spacetime.To emphasize the behavior of a light ray as a pulse, we use the term "light ray".To emphasize the 4-dimensional orbit of a light ray, we use the term "null geodesic".Then, we can say that any light ray propagates on a null geodesic.
Let h denote an affine parameter measured in the length dimension, and let x μ ng (h) be the spacetime position of a light ray on the null geodesic.The wave vector of a light ray is given by K μ = dx μ ng (h)/dh, which must satisfy the null condition K μ K μ = 0. Due to the stationarity and axisymmetry of Kerr which are conserved along the null geodesic K μ Q ;μ = 0, where Q = ν obs , l ϕ , l nor .
Our setup (7) of u μ obs means that the quantity ν obs is calculated from u μ obs as ν obs = −K μ u μ obs .This ν obs is the dimensionless frequency of a light ray observed by a rest observer at infinity (r obs → ∞), where "dimensionless" denotes that ν obs is normalized by a fiducial frequency.The quantity l ϕ is the ϕ-component (toroidal component) of the orbital angular momentum of a light ray, and the quantity l nor relates to the norm of the orbital angular momentum of a light ray.These components, l ϕ and l nor , are measured per unit energy of the ray, since the dimension of them is the length in natural units (c = 1, G = 1).
Note that the fiducial frequency for the normalization of ν obs can be chosen arbitrarily.For example, in MKS units, if we choose 300 GHz as the fiducial frequency, then the unit of length size in our numerical calculation becomes c/(300 GHz) = 10 −3 m.If, conversely, we set the unit of length size to r BH , then the fiducial frequency becomes c/r BH .
Hereafter, for simplicity of calculations, we normalize the null geodesic by ν obs as where η := ν obs h is the dimensionless affine parameter.Further we define the toroidal impact parameter b and the "normic" impact parameter q by b := l ϕ ν obs , q := l nor ν obs .
The spacetime position of a light ray x μ ng (η) is determined by the null geodesic equations k α k μ ;α = 0, whose components in BL coordinates, together with the null condition k μ k μ = 0, can be arranged as where the auxiliary functions are X (r) = (r 2 + a 2 − ab)/ (r) and Y (θ) = a sin 2 θ − b, and the radial and zenithal effective potentials are Note that the normic impact parameter q appears only in the squared form q 2 .This means that the solution of the null geodesic equation, k μ (η) and x μ ng (η), does not depend on the signature of q.We set q ≥ 0 throughout this paper.Also, in our numerical calculation, we set η = 0 at the emission event of light rays: We can recognize from Eq. ( 10) that, given the emission event x μ emi and the initial null vector k μ emi at the emission, the values of impact parameters (b, q) corresponding to x μ emi and k μ emi are determined by solving Eq. ( 10) algebraically for (b, q).In our numerical calculation, we use Eq. ( 10) for the following 2 purposes: (i) to evaluate the values of impact parameters (b, q) for given x μ emi and k μ emi , and (ii) to judge whether the light ray of given x μ emi and k μ emi is to be absorbed eventually by the BH.The criterion of the judgment is given by the effective potentials (11) as explained in Appendix A.
If one tries to integrate Eq. ( 10), then there arises the problem of how to choose the signatures We avoid this problem by using the Hamiltonian formalism of geodesics instead of Eq. (10).Define the Hamiltonian of a light ray by where we regard the position x μ ng (η) and the null 1-form k μ (η) as the dynamical variables in the present Hamiltonian formalism.In Kerr spacetime, the components of the 1-form in BL coordinates are where the t-component and ϕ-component are constant because of Eqs. ( 8) and (9b).The Hamilton equations dx μ ng (η)/dη = ∂H/∂k μ and dk μ (η)/dη = −∂H/∂x μ ng are reduced to the following 6 simultaneous equations: In these equations, no square root appears.Note that the Euler-Lagrange equations given by the Lagrangian corresponding to the Hamiltonian (12) can be arranged to geodesic equations of the form (10). Also, note that the value of the Hamiltonian, H ∝ k 2 ≡ 0, is conserved along the geodesic.Therefore, the null geodesic x μ ng (η) is obtained automatically by solving Eq. ( 14) with the initial condition satisfying k 2 emi = 0.In our numerical calculation, the initial null 1-form k emi μ is constructed by making use of Eq. ( 10) as explained in Appendix A. Then, the null geodesics of the p-ray x μ ng(p) (η) and the s-ray x μ ng(s) (η), which connect the source and observer, are obtained by solving Eq. ( 14) numerically, and we can plot those solutions in the t-r plane and read out the time delay t obs as shown in Fig. 1.

Total Doppler factor
Given the observed frequency ν obs of a light ray detected by the observer, the emission frequency ν emi of the ray can be given by where Eq. ( 13) is used.When the source is moving near the BH, we cannot identify which term in Eq. ( 15) is responsible for the gravitational or kinetic Doppler effect.Let us define the total Doppler factor D as the ratio of ν obs to ν emi , This can be regarded as a function of the emission event x

Specific flux F obs : Jacobi equations 3.4.1. Primitive form of F obs
In real situations, each winding ray (p-ray, s-ray, and higher winding rays) is detected by the telescope as a beam made of neighboring light rays.The 4-dimensional track of the beam in spacetime is a bundle made of neighboring null geodesics of light rays composing the beam.In order to calculate the specific flux, which is the surface density of the observed power of light rays on the telescope's face, we focus on the beam incident to 1 point on the telescope's face, as illustrated in Fig. 2.
Because we assume that the source of light rays is much smaller than the BH, the beam of light rays is narrow.Hence, the orbits (null geodesics) and the spectrum of all light rays in 1 beam can be regarded approximately as the same ones, which are represented by 1 constituent light ray in the beam.We call such a ray the representative light ray, and its null geodesic the representative null geodesic.
Let I obs (ν obs ) denote the observed specific intensity of the representative light ray, and let δ obs denote the visible solid-angle of the source measured by the observer.Then, the observed specific flux of the null geodesic bundle is given by where is the solid-angle around the telescope.
Our numerical calculation of F obs (ν obs ) is performed under 2 suppositions: (i) the parameters ( 5) and ( 6) are already given, and (ii) the representative null geodesic has already been obtained numerically by the procedure given in Sect.3.2.Under supposition (i), we derive I obs (ν obs ) from I emi (ν emi ).Under supposition (ii), we derive the formula for the visible solid-angle δ obs in order to calculate it from the narrow null geodesic bundle including the given representative null geodesic (see Fig. 2).ng (η obs ).The quantity Y μ is the Jacobi vector field on the representative null geodesic.By considering a small celestial sphere around the observation point, the visible solid-angle δ obs is calculated from the area swept by Y μ on the celestial sphere.

Relation between I obs (ν obs
null geodesic bundle of the beam possesses a conserved quantity, where I (ν) is the specific intensity measured by an arbitrary observer at an arbitrary spacetime point inside the null geodesic bundle (Ref.[18]).The quantity I (ν)/ν 3 keeps the same value from observer to observer and from spacetime point to spacetime point inside the null geodesic bundle.Then, we find the relation between I obs (ν obs ) and I emi (ν emi ): where D is the total Doppler factor of Eq. ( 16).The value of D is uniquely determined by the representative null geodesic.

Formulation of δ obs
Let us construct the formula of visible solid-angle δ obs so as to be calculated from the narrow null geodesic bundle including the given representative null geodesic.The setup of our formulation is shown in Fig. 3. Consider a small celestial sphere of radius δr centered at the observation point, and let δS denote the area on the celestial sphere that is penetrated by the beam of light rays.The visible solid-angle is given by We evaluate δr and δS in the following discussions of this Sect.3.4.3.
For the first, in order to evaluate the radius δr of the observer's celestial sphere, let η obs denote the value of the affine parameter η of the representative null geodesic when it reaches the observer, and let η obs − δη denote the value of η when the representative null geodesic crosses the observer's celestial sphere.In terms of the position of the representative light ray x μ ng (η), the observation event is x μ ng (η obs ) and the crossing event is x μ ng (η obs − δη).Then, the following vector connects these spacetime events (see the right-hand panel of Fig. 3): where k μ obs := k μ (η obs ) is the null tangent vector of the representative null geodesic at the observation event.The projection of this vector onto the space-like hypersurface perpendicular to the observer's velocity u μ obs is where g λμ obs +u λ obs u μ obs is the projection tensor onto the hypersurface perpendicular to u μ obs , and Eqs. ( 7) and ( 13) are used in the second equality.The radius δr measured by the observer is the norm of this spatial vector, Next, in order to evaluate δS, we make use of the Jacobi vector field Y μ (η) (the geodesic deviation vector field) that connects the representative null geodesic to a neighboring null geodesic in the bundle of null geodesics.The Jacobi field can be defined so as to be space-like (Y μ Y μ > 0), perpendicular to the representative null geodesic (Y μ k μ = 0), and invariant under the Lie transport along the geodesic (L k Y μ = 0) (Ref.[12]).The Lie-invariant condition, L k Y μ = 0, can be arranged into the Jacobi equations (the geodesic deviation equations) in terms of covariant derivatives, where R μ αλβ (η) is the Riemann curvature tensor at the spacetime point x μ ng (η) on the representative null geodesic.Furthermore, since we are now considering the null geodesic bundle converging at the observation event (see Fig. 2 and the left-hand panel of Fig. 3), the Jacobi equations are to be solved under the "initial" condition where Y The significance of Jacobi fields is that, as illustrated in the left-hand panel of Fig. 3 and explained in detail in Appendix B, the 2-dimensional cross-sectional area δS of the narrow null geodesic bundle is regarded as the area swept by the Jacobi vectors surrounding the representative null geodesic at the crossing event x μ ng (η obs − δη).Thus, the Jacobi equations (24) can be rearranged into a form that describes the evolution of cross-sectional area from the emission event x From condition (25), we can set for sufficiently small δη, Therefore, we find the relation where A obs is the space-like area swept by the space-like vector k α obs Y μ obs;α which surrounds the representative null geodesic at the observation event x μ obs .(Since Y μ is space-like, Eq. ( 26) denotes that k α Y μ ;α is also space-like.)Because the Jacobi equations (24) describe the evolution of the Jacobi vector Y μ (η) along the representative null geodesic, the evolution of the cross-sectional area of the null geodesic bundle along the representative null geodesic is also described by the Jacobi equations (24).As explained in detail in Appendix B, an appropriate rearrangement of Eq. ( 24) provides us with the transformation of the cross-sectional area of the null geodesic bundle at x μ emi , which is denoted by A emi hereafter, to the area A obs at x μ obs appearing in Eq. ( 27), where .Also, it has to be noted that the area A emi can be regarded as the cross-sectional area of the source of light rays seen from the emission direction of the light ray.Now, we have obtained the ingredients for calculating the visible solid-angle δ obs .Combining Eqs. ( 20), ( 23), (27), and (28), we find This is the formula for the visible solid-angle we use in our numerical calculation.This formula does not depend on the observation frequency ν obs .
3.4.4.Our formulas for F obs and R obs Substituting Eqs. ( 19) and (29) into Eq.( 17), the formula for the observed specific flux used in our numerical calculation is where the total-Doppler factor D is given in Eq. ( 16), and the area-transfer coefficient C is given in Eq. (B.32b) of Appendix B. It is important to specify which factors in Eq. ( 30) include the dependence on the observation frequency ν obs and on the choice of p-ray or s-ray: The ν obs -dependence of F obs arises from only the intrinsic specific intensity of the source I emi (ν obs /D).And the dependence of F obs on the choice of p-ray or s-ray (dependence on the winding number around the BH) arises from the factors D(x emi , u emi , k emi ) and C(x obs , u obs , k obs , x emi , u emi , k emi ).Here, the arguments k μ emi , k μ obs , and t obs (= x t obs ) depend on the choice of p-ray or s-ray, while the other arguments are shared by the p-ray and the s-ray.
Note that, in our numerical calculation, the cross-sectional area of the source A emi , which is seen from the emission direction of the ray in the hypersurface perpendicular to u μ emi , is treated as a given parameter.For simplicity, we assume that the shape of the source is spherical when it is seen by an observer comoving with the source.Then, the value of A emi is the same for the p-ray and the s-ray: (31) This is consistent with our model that the source is point-like (see Sect. 2.1).Under this assumption, the ratio of observed specific flux is given by the formula which is independent of A emi .Then, as explained in Sect.2.3, the ratio R obs of type LE (line emission) is calculated under the condition ν obs(s) /D (s) = ν obs(p) /D (p) (⇔ ν emi(s) = ν emi(p) ), and we find which is independent of I emi .Also, the ratio R This R (LD) obs depends on I emi (ν emi ) except for the white-noise-type emission I emi (ν emi ) = constant.For white-noise emission, the observed flux ratio of type LD and that of type LE are the same and given by Eq. (32b).

Selection rule for the p-ray and the s-ray
Suppose that we have found numerically some null geodesics that connect the given emission event x μ emi and the given observation position (r obs , θ obs , ϕ obs ).This means that we have obtained some numerical solutions of the null geodesic equations ( 14) for different values of the initial 1-form k emi μ .Further, suppose that we have not recognized which solutions are the p-ray and the s-ray.This is the case that we confront in our numerical calculation, as will be explained in Sect.3.6.The issue in this subsection is how to select the p-ray and the s-ray from the set of numerical solutions of null geodesics.
We should emphasize that, in our numerical calculation, we cannot always regard the numerical solution of the null geodesic possessing the earliest (or second earliest) observation time as the p-ray (or the s-ray).The reason is that our numerical search for the solution of null geodesic equations ( 14) is the shooting method with discretely varying value of the initial 1-form k emi μ , and that the initial 1-forms appropriate for the p-ray k emi(p) μ and the s-ray k emi(p) μ may be omitted in the discrete variation of k emi μ .Therefore, we need the criterion to judge whether the p-ray and the s-ray exist in the set of numerical solutions of null geodesics, and to select the p-ray and the s-ray when they exist in the set of numerical solutions of null geodesics.To construct the selection rule of the p-ray and the s-ray, let us notice that the light ray can propagate only in the spacetime region where the radial and zenithal effective potentials are nonpositive (V eff (r) ≤ 0 and U eff (θ) ≤ 0), as indicated by the geodesic equations (10).Hence, the θ -coordinate θ ng (η) of the null geodesic is confined to the interval 0 < θ ng (η) < π (positions on the spin axis θ = 0 and π are excluded) for nonzero toroidal impact parameter b = 0, because U eff → +∞ as θ → 0 and π for b = 0. Therefore, for the case b = 0, the winding number of the null geodesic around the BH can be counted by the ϕ-coordinate ϕ ng (η) of the null geodesic.We define the winding number W of the null geodesic as the integer given by where δϕ := ϕ ng (η obs ) − ϕ emi .Using this winding number, we offer the selection rule for the p-ray and the s-ray as follows: Selection rule for the p-ray and the s-ray.For b = 0, our selection rule consists of 2 parts: • The p-ray is the null geodesic of the winding number W = 0.There exists only 1 p-ray once the emission event x μ emi and the observation position (r obs , θ obs , ϕ obs ) are specified.• Collect the null geodesics of the winding number W = 1 and −1.Then, the s-ray is the null geodesic of the earliest observation time among them.Also, for b = 0, we can define the winding number W in the θ -direction by the same form as Eq.(33).Then, the p-ray may be the null geodesic of W = 0 and W = 0.The s-ray may be the null geodesic of W = ±1 or W = ±1 and the earliest observation time.However, since the impact parameters (b, q) in our numerical procedure are not the input parameters but the parameters determined from the initial 1-form as explained in Appendix A, the case b = 0 has not occurred so far in our numerical calculations.Hence, we focus on the case b = 0 in the following discussion.
Let us note again that, in our numerical calculation, the p-ray and/or the s-ray may be omitted in our numerical setup of the initial 1-forms.If the true s-ray has not been obtained in the set of numerical solutions for given x μ emi and (r obs , θ obs , ϕ obs ), then the null geodesic of the earliest observation time among the numerical solutions of winding number W = ±1 is not the true s-ray.Therefore, we need a supplemental rule to check whether such a null geodesic selected by the above-mentioned rule is the true s-ray or not.We have searched numerically the solutions of null geodesic equations ( 14) and the Jacobi equations (24), we have found the following rule: Supplemental rule for selecting the s-ray.Among numerical solutions of the winding number W = ±1, the s-ray is the ray that passes through only 1 caustic before reaching the observer.Here, as explained at the end of Sect.B.1, the caustic is the spacetime point at which the cross-sectional area of the null geodesic bundle becomes 0.
Furthermore, we have found numerically that the p-ray passes through no caustic.An example of this statement is shown in Fig. 1.We find in Fig. 1 that the p-ray passes through no caustic and the s-ray passes through only 1 caustic.

Steps of numerical calculation of t obs and R obs
Combining the discussions given so far, our numerical calculation consists of the following steps: Step 1 (our calculation): Specify the values of the BH parameters M and χ.Also, specify the emission event x μ emi , the velocity u μ emi , and the intrinsic specific intensity I emi (ν emi ) of the source of light rays.
Step 2 (our calculation): Consider the sphere of radius r obs , on which the point is described by (θ obs , ϕ obs ).Then, create the set of observers (detectors) as the grid points on the sphere.We assume the velocity u μ obs of each observer is given by Eq. ( 7).
Step 3 (our calculation): Create the set of values of the initial direction angles (α emi , β emi ) of k μ emi as the grid points on the parameter region 0 ≤ α emi ≤ π and 0 ≤ β emi < 2π, where the definition of (α emi , β emi ) is given in Appendix A.
Step 4 (our calculation): For each value of (α emi , β emi ) created in the previous step, calculate the components of the initial 1-form k emi μ by the procedure given in Sect.A.1.Further, check whether the light ray of the given initial 1-form is to be absorbed eventually by the BH or not by following the selection rule of the initial 1-form given in Sect.A.2.
Step 5 (our calculation): Solve the null geodesic equations ( 14) for the initial 1-forms that are not absorbed by the BH.Those solutions of the null geodesics arrive at different points on the observation sphere of radius r obs .Then, for each null geodesic, the nearest grid point on the observation sphere, which is created in step 2, is regarded as the position of the observer who detects the null geodesic; i.e., the relative location among the BH, source, and observer is (approximately) determined for each null geodesic.
Step 6 (our calculation): Count the winding number W by Eq. ( 33) for all null geodesics obtained in the previous step.Then, for each observer on the observation sphere, there can exist some null geodesics that possess the same winding number W .Among such null geodesics of the same value of W , let us select the one that arrives at the nearest point to the observer on the observation sphere, and delete the others of the same W from the numerical data.
Step 7 (our calculation): The null geodesics selected in the previous step are regraded as the representative null geodesics of null geodesic bundles.Calculate the observed specific flux F obs of each null geodesic bundle by the formula (30) under the assumption (31), where the procedure for calculating the area-transfer coefficient C is given in Sect.B.3.Also, during the calculation of F obs , count the number of zeros of det J (η) given from Eq. (B.12) on each null geodesic bundle, which is the number of caustics on the bundle as explained at the end of Sect.B.1.
Step 8 (our calculation): For each observer on the observation sphere, search the null geodesics detected by the observer for the p-ray and the s-ray by following the selection rules give in Sect.3.5.Then, if the p-ray and the s-ray are found for the given observer, the time delay t obs and the flux ratio R obs are obtained by definition (4).
If the grid points on the parameter plane of (α em , β emi ) are not well prepared in step 3, then the p-ray and/or the s-ray are not found at some observation points in step 8.

Numerical results and potential detectability of t obs and R obs
This section is for a discussion of the potential detectability of our observable quantities t obs and R obs .After summarizing an estimation of telescope capability, we show some results of our numerical calculations, performed by following the procedure given in Sect.

Example of telescope capability
As explained in Sect.2.1, we assume a transparent environment around the BH, at least in the frequency band of observation, and a few candidates for the BH with such a transparent environment are at present recognized by radio observations (Refs.[3,7,19,21]).Then, as an example of a telescope, let us consider the radio telescope of 34 m diameter operated by the Space-Time Measurement Group at the National Institute of Information and Communications Technology (NICT), Japan.In general, the signal-to-noise ratio R sn of a radio telescope is given by (Ref.[23]) where F obs is the specific flux of the signal received by the telescope and F sefd /(2 δB δt) 1/2 is the total noise of the telescope, where δt is the duration of the observation time, δB is the bandwidth of the observation frequency and F sefd is the system equivalent flux density, which measures the system noise of the telescope in the dimensions of specific flux.For the radio telescope of NICT, the bandwidth is δB 1024 MHz, and the system noise is F sefd 300 Jy, where 1 Jy = 10 −26 W/m 2 Hz is the unit of specific flux.Here, as an example of a target BH, consider the massive BH candidate at the center of our galaxy, Sgr A * of mass M SgrA * 4 × 10 6 M (Refs.[3,7,21]).The Newtonian (Keplerian) dynamical time scale t dyn near the horizon radius of Sgr A * is t dyn := (r 3  SgrA * /GM SgrA * ) 1/2 60 sec, where r SgrA * = 2GM SgrA * /c 2 .Then, let us assume a short duration of emission is δt = t dyn /100 0.6 sec, and the criterion for signal detection by telescope is R sn > 5. Note that since no precise and high resolution observation in the vicinity of the BH horizon has been performed, we do not have relevant observational data for estimating δt.Therefore our assumption δt = t dyn /100 is a simple assumption that should be investigated properly in future studies; however, we expect that 1% of t dyn may not be bad as a short duration emission near the BH horizon.On the other hand, the criterion for signal detection, R sn > 5, is consistent with real astronomical radio observations.Then, under the above assumptions, the signal flux F obs detectable by the NICT telescope should satisfy Note that a typical observed radio flux F SgrA * coming from Sgr A * is F SgrA * ∼ 0.1 Jy (Ref.[7]), which comes from a region with an approximate size of a few r SgrA * .Then, in order to estimate the value of R obs that is detectable by the NICT telescope, let us consider 2 cases.One is that the flux of the p-ray is stronger than that of the s-ray (F obs(p) > F obs(s) ), and the other is the inverse case (F obs(p) < F obs(s) ).Note that, as mentioned in Sect. 1, our setup introduced in Sect.2.1 is similar to the setup considered by Cunningham and Bardeen (Ref.[6]), which considered a star on a circular orbit on the equatorial plane of an extreme Kerr BH.Cunningham and Bardeen (Ref.[6]) had already shown that the brightness of the primary image can be stronger and weaker than the brightness of the secondary image, due to frame-dragging by the Kerr BH and the beaming and kinetic Doppler effects on the light rays emitted by the source star.Therefore, in our situation where the source is not a star (radiating lights continuously) but a short duration emission, it may be expected that both cases F obs(p) > F obs(s) and F obs(p) < F obs(s) can be found.Indeed, at the end of this section, it will be shown by our numerical calculation that these 2 cases are possible for a Kerr BH but not for a Schwarzschild BH.
For the case F obs(p) > F obs(s) , we can set F obs(p) = F SgrA * ∼ 0.1 Jy.In order to let the s-ray be detectable, its specific flux F obs(s) needs to satisfy condition (35).Then, we obtain the condition on the flux ratio detectable by the NICT telescope: And, for the case F obs(p) < F obs(s) , we can set F obs(s) = F SgrA * ∼ 0.1 Jy.In this case, the detectable p-ray needs to satisfy condition (35), and we obtain a condition for the detectable flux ratio: Hence, although some assumptions are introduced, the above estimation with Sgr A * seems to suggest that present or near future telescopes can detect the p-ray and the s-ray from Sgr A * if there occur some emission events near the BH candidate that result in the observed flux ratio in the interval, The interesting issue is whether general relativity permits the flux ratio R obs within this interval.

Some results of the numerical estimation of t obs and R obs
To discuss whether the flux ratio R obs can take values in the interval (37), we estimate the values of t obs and R obs numerically.The procedure is given in Sect.3.6, and our numerical results are obtained with Mathematica version 10.The numerical results shown in this section are some typical results obtained with the following values of the parameters (see Fig. 4): • The BH's mass is set to unity (M = 1) and all quantities are calculated with this unit.
• The emission position of the source of light rays is set to (r emi , θ emi ) = (2.2rBH , π/2).This means that the emission event is outside (but near) the ergosurface r erg (π/2) = 2M .• Two cases of the source's velocity are calculated: the radial falling case (u t emi , −1, 0, 0) and the ZAMO-like case u The numerical results with θ obs = 16π/31 are plotted, where ϕ obs is varied from 0 to 2π .However, in our numerical program, numerical calculations at some values of ϕ obs could not produce an appropriate initial value for the geodesic equations.The values of ϕ obs for which the numerical calculation was completed are shown in the left-hand panel.The colors of the points in each of the panels denote the variation in the value of ϕ obs .In the right-hand panels, the data points corresponding to ϕ obs < π do not appear since those data points degenerate to the data points corresponding to ϕ obs ≥ π in the case χ = 0.
• For simplicity, we suppose the observation type LE, R (LE) obs , given in Eq. (32b).Alternatively, this can also be understood as type LD with white-noise-type emission, I emi (ν emi ) = constant, as explained at the end of Sect.3.4.4.
We do not show all the numerical results of the possible combinations of the values of χ, u μ emi , θ obs , and ϕ obs .However, we do show some typical results in order to discuss whether the flux ratio R obs can take values in the interval (37).Further, note that although we have tried to calculate the observable quantities t obs and R obs for all values of ϕ obs , we could not obtain the numerical values of t obs and R obs for some values of ϕ obs because the appropriate initial condition for the geodesic equations could not be created.(See the comment at the end of Sect.3.6.)To obtain the values of t obs and R obs for all given values of the input parameters, we may need a more sophisticated calculation procedure than the present one and a more powerful computer than the author uses at present.

Radial falling source toward a BH of spin χ = 0
Figure 5 shows numerical results for t obs , R obs , and ν obs(s) /ν obs(p) with the parameter values listed in the figure's caption.The source is radially falling toward the BH.As shown in the left-hand panel, we were able to complete the numerical calculations for 29 values of ϕ obs at θ obs = 16π/31 ( 0.516π).
Note that (or return here after reading Sect.4.2.2), in comparison with the results in Fig. 6, the time delay t obs should extend up to t obs ∼ 30.However, our numerical calculations for t obs > 20 were not successful (see the end of Sect.3.6).Also note that the values of t obs , R obs , and ν obs(s) /ν obs(p) for 0 < ϕ obs ≤ π seem to degenerate to those for π < ϕ obs ≤ 2π.This seems to be the typical situation for the Schwarzschild BH case.
It can be read from the lower-right panel that the flux ratio R obs can take values in the interval (37).However, note that if the observed frequency of the p-ray ν obs(p) and that of the s-ray ν obs(s) are  different, so that the bandwidth of the telescope does not include both of them, the flux ratio R obs made of such a p-ray and s-ray cannot be detected even when R obs takes a detectable value (37).Hence, numerical results with ν obs(s) /ν obs(p) 1 are desirable for expecting safe detectability of both of the p-ray and the s-ray by 1 telescope.From the upper-right panel, we find that a flux ratio R obs in the interval (37), together with a frequency ratio around unity ν obs(s) /ν obs(p) 1, is realized for ϕ obs π. 6 shows a modification of Fig. 5 by increasing the BH's spin from χ = 0 to 0.3 while keeping the other parameters fixed at the same value as those in Fig. 5.The source is radially falling toward the BH.As shown in the upper-left panel, we could complete numerical calculations for 41 values of ϕ obs at θ obs = 16π/31 ( 0.516π).It can be read from the upper-right and lower panels that a flux ratio R obs in the interval (37), together with a frequency ratio around unity ν obs(s) /ν obs(p) 1, is realized for π < ϕ obs < 16π/15 ( 1.066π).7 shows a modification of Fig. 5 by increasing the BH's spin from χ = 0 to 0.8 while keeping the other parameters fixed at the same value as those in Fig. 5 and adding another parameter value of the direction angle of the BH's spin.The source is radially falling toward the BH.As shown in the upper-left panel, we were able to complete the numerical calculations for 19 values of ϕ obs at θ obs = 4π/31 ( 0.129π) and 44 values of ϕ obs at θ obs = 16π/31 ( 0.516π ).It can be read from the upper-right and lower panels that a flux ratio R obs in the interval (37), together with a frequency ratio around unity ν obs(s) /ν obs(p) 1, is realized for θ obs = 16π/31 ( 0.516π) and 1.1π < ϕ obs < 17π/15 ( 1.133π).Note that, in comparison with the results in Fig. 6, the time delay t obs should extend up to t obs ∼ 30.However, our numerical calculations for t obs > 25 were not successful (see the end of Sect.3.6).8 shows a modification of Fig. 7 by replacing the source's velocity from the radial falling case to the ZAMO-like case while keeping the other parameters fixed at the same values as those in Fig. 7. Since the same numerical data for the solution of the null geodesic equations are used in Figs.7  and 8, the number of data for each value of θ obs is the same as indicated in Fig. 7.But the behaviors of R obs and ν obs(s) /ν obs(p) in this case are somewhat different from those in Fig. 7  effect and the kinetic Doppler effect are different, due to the difference in the source's velocity.It can be read from the lower and upper-right panels that a flux ratio R obs in the interval (37), together with a frequency ratio around unity ν obs(s) /ν obs(p) 1, is realized for θ obs = 16π/31 ( 0.516π ) and 1.1π < ϕ obs < 17π/15 ( 1.133π).

Accuracy errors in numerical calculations of t obs and R obs
We may be able to estimate the errors in our numerical calculations of t obs and R obs that arise from a numerical uncertainty in the position of the observer in our numerical procedure.In our setup, given at the beginning of this Sect.4.2, the angular uncertainty (δ obs , δϕ obs ) in the position of the observer is δθ obs π/31 ∼ 0.1 and δϕ obs π/30 ∼ 0.1 for θ obs = 16π/31.This uncertainty corresponds to the angular separation between neighboring grid points on the observation sphere of radius r obs , which are prepared in step 2 of our numerical procedure given in Sect.3.6.The length size δl of the position uncertainty of the observer is δl ∼ r obs δθ obs 0.1r obs .
Let us estimate the error δ t obs in our numerical calculation of the time delay t obs .The direction of the observer's position uncertainty δl is tangent to the observation sphere of radius r obs .Also, t obs can be roughly estimated by the spatial length of a path on which a light ray propagates, t obs ∼ r obs .Hence, the numerical error δ t obs may be estimated by considering a right triangle whose legs are sides of length r obs and δl.(the angle between these legs is the right angle).Our estimation is δ t obs (r 2 obs + δl 2 ) 1/2 − r obs 1 2 (δl) 2 /r obs ∼ 0.005r obs .The relative error is δ t obs / t obs ∼ 0.005.Further, including another error in our numerical setup estimated in Sect.3.1, we expect that the total numerical error in the accuracy of t obs in our numerical calculation may be several percent.
Next, the error δR obs in our numerical calculation of the specific flux ratio R obs may be estimated by the uncertainty of the curvature tensor R obs − (r obs + δ t obs ) −2 ] ∼ 2 δ t obs /r obs ∼ 0.01.This may be the origin of the numerical error in the specific flux of the p-ray F obs(p) and that of the s-ray F obs(s) .Hence, because R obs := F obs(s) /F obs(p) , we expect δR obs ∼ δR μ ναβ ∼ 0.01.Further, including another error in our numerical setup estimated in Sect.3.1, we expect that the total numerical error in the accuracy of R obs in our numerical calculation may be several percent.
These errors in t obs and R obs do not seriously affect our conclusion derived from the numerical results shown in Figs. 5 to 8.

Summary of numerical results
Numerical results similar to those shown in Figs. 5 to 8 have been obtained for various parameters, as far as the author has checked.Hence, it may be reasonable to expect detectability of the observable quantities t obs and R obs by present or near future telescope capability, at least, for the case that the source of light rays and the observer are located near the equatorial plane of a BH's spin (θ obs θ emi = 0.5π) and the emission event is behind the BH seen from the observer (ϕ obs π).
Let us note that both cases log 10 R obs > 0 (F obs(s) > F obs(p) ) and log 10 R obs < 0 (F obs(s) < F obs(p) ) appear for Kerr BH cases, Figs. 6 to 8. As mentioned in Sect.4.1, because our setup is similar to the setup considered in Cunningham and Bardeen (Ref.[6]), which predicted that the brightness of the primary image of a star orbiting a Kerr BH can be stronger and weaker than that of the secondary image, it was expected that our numerical results would show both R obs > 1 and R obs < 1 cases.This expectation is supported by our numerical results.Further note that the case F obs(s) > F obs(p) does not appear in the Schwarzschild BH case in Fig. 5. Therefore, the case F obs(s) > F obs(p) may be mainly due to the frame-dragging effect of a spinning BH.More detailed numerical study will be reported in another paper.
We find from all the presented figures, Figs. 5 to 8, that the flux ratio R obs has local minima about t obs ∼ 15GM /c 3 in the plots of log 10 R obs versus t obs .This may reflect some universal property of BH spacetime, since our numerical results in Figs. 5 to 8 include some cases of BH spin χ , its direction angle θ obs , and source velocity u μ obs .However, we could not specify physical reasons for the appearance of local minima in the R obs -t obs relation.This behavior of R obs remains an open issue for future works.

Summary and discussions
The main theoretical proposal in this paper is in Sect.2.4, i.e., the principle to measure the mass M , spin parameter χ, and direction angle θ obs of BHs through observing the time delay t obs and specific flux ratio R obs created by the p-ray and the s-ray.This principle is a method of direct BH measurement under the definition given in Sect. 1, since t obs and R obs are quantities created by the Kerr BH lens effect (a general relativistic effect).And, following the numerical procedure for calculating t obs and R obs constructed in Sect. 3 and Appendices A and B, we showed in Sect. 4 the potential detectability of t obs and R obs by present or near future telescope capability.However, since our assumptions on the source of light rays are very simple, cases of complicated source emissions are to be studied by appropriately summing the results of this paper.
The conditions assumed in this paper are described in Sect.2.1, and the numerical setup is given at the beginning of Sect.4.2.These assumptions and related issues are summarized as follows: Source of light rays: We have assumed that the source of light rays is point-like and emits light rays isotropically in its comoving frame, and also that the emission duration is much shorter than a typical dynamical time scale of the system composed of the source and the BH.On the other hand, in astrophysical situations, not only such simple emissions, but also other complicated emissions, would occur.Complicated source emissions, such as spatially and temporally continuous or random emissions, will be constructed by summing appropriately some simple emissions.Such an extension of the source's structure is a task for future works.Environment around the BH: We have assumed a transparent environment around the BH, at least in the frequency band of observation.Some observational evidence of such an environment around massive BH candidates in the central region of our galaxy has been reported (Refs.[3,7,19,21]).
On the other hand, in astrophysical situations, it is also expected that a BH is surrounded by dense plasmas that form some opaque environment around the BH.Inclusion of such opaque effects in our study is also an interesting issue for future works.(We may give priority not to the opaque effects but to the complicated source emissions, since some observational evidence of a transparent environment around a BH is already known.)Numerical calculation: Under the numerical setup given at the beginning of Sect.4.2, our numerical procedure, which is summarized in Sect.3.6, could not produce some desired null geodesics, that should connect the source and observer, within the numerical error evaluated in Sect.4.2.5.Therefore, we need a more sophisticated numerical procedure or technique to obtain all the desired null geodesics.At present, the author is modifying the numerical procedure in order to obtain all the desired null geodesics.After completing the modification, a complicated source case will be reported in a future work.

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We have found 2 by-products of our numerical study.The first by-product, explained in Sect.3.5, is that the p-ray passes through no caustics and the s-ray passes through only 1 caustic before reaching the observer.Note that this statement is nothing but a conjecture based on our numerical calculation, and theoretical exact proof of this statement remains to be constructed.The second by-product, explained at the end of Sect.4, is that a large flux ratio R obs > 1 seems to occur due to the framedragging effect of a spinning BH.However, other detailed properties of t obs and R obs , such as the local minima of R obs at t obs ∼ 15M in the R obs -t obs relation, can hardly be analyzed by our present numerical results shown in Sect. 4.More detailed numerical study of t obs and R obs will be reported in future works.
Let us make a comment on the significance of our principle of direct BH measurement.The existing famous observable quantities of the Kerr BH lens effect may be the BH shadow (see Takahashi [22] and references therein) and the broadening of the iron line emission by an accretion disk around a BH (see Kojima [16], Fanton et al. [8], and references therein).Although these quantities have not been clearly detected so far, intensive observational approaches are now developing.While the imaging of a BH shadow needs many radio telescopes in order to compose a very-long-baseline-interferometer system, our method of measuring (M , χ, θ obs ) can be carried out, in principle, by just 1 telescope.And, while the broadening of the iron line depends, by definition, on the accretion disk model, our observable quantities t obs and R obs do not depend so largely on the disk model.We expect that the combination of the BH shadow, the broadening of the iron line, and our proposal will strengthen the observational study of BHs by astronomical methods.
Finally, from the viewpoint of general relativity, it is necessary to recognize exactly what we can extract from the observation of the Kerr BH lens effect.Each of the BH shadow, the broadening of the iron line, and our proposal observe the light rays emitted by sources moving around the BH.Those light rays wind sometimes around the BH before reaching the observer.Here, it must be emphasized that if the source of the light rays is located outside the so-called photon sphere (which is a sphere of radius r = 3M , for a non-spinning BH, where the peak of the photon's radial potential is located), any light ray entering the inside of the photon sphere can never escape from the photon sphere but will be absorbed by the BH eventually.Therefore, any light ray connecting the source and the distant observer can never approach nearer than the radius of the photon sphere before reaching the observer.This theoretical fact implies that, even if the BH shadow, the broadening of the iron line, and our observable quantities t obs and R obs are observed clearly, the direct conclusion of these observational data is never the existence of the BH horizon but the existence of the photon sphere.Hence, whenever we aim to observe the BH through the Kerr BH lens effect, we are faced with the theoretical issue of whether the existence of the photon sphere denotes the existence of the BH horizon.At present, the final answer to this issue has not been obtained.However, there is some positive theoretical evidence for this issue, e.g., by Cardoso et al. [5] and Saida et al. [20] (see also references therein).Thus, although we still do not have complete theoretical support for the existence of the BH horizon under the existence of the photon sphere, the existence of the BH horizon seems probable if the existence of the photon sphere is shown by the observation of the Kerr BH lens effect.
PTEP 2017, 053E02 H. Saida In order to calculate k sn and k t emi , let us make use of the relation k μ emi = g μν k emi ν , where the components of the inverse of the metric g μν in BL coordinates are From these components, together with k emi μ in Eq. ( 13), we find where the subscript "emi" denotes the value at the emission event.Comparing the ϕ-component k This relation determines the value of the toroidal impact parameter b after obtaining the value of k sn .Substituting Eq. (A.4) into the t-component k t emi in Eq. (A.3), we find the relation By this relation, the unknown quantities k t emi and k sn in Eq. (A.1) are reduced to only 1 unknown quantity k sn .Then, the spatial norm k sn is determined by the null condition It must be noted that Eq. (A.6) is a second-order algebraic equation in k sn , and the appropriate solution of k sn should satisfy k t emi > 0 (future-pointing vector).From the above discussions, the initial 1-form k emi μ = (−1, k emi r , k emi θ , b) is given by the following procedure in our numerical calculation: Step 1 (k emi μ ): Specify the value of the emission angles (α emi , β emi ) as shown in Fig Step 2 (k emi μ ): Solve the second-order algebraic equation (A.6) for k sn , and adopt the solution satisfying k t emi > 0.
Step 4 (k emi μ ): Calculate the r-component k emi r and the θ-component k emi θ using the following formulas, given by comparing Eqs.(A.1) and (A.3): Step 5 (k emi μ ): Calculate the normic impact parameter q from the θ -component of Eq. ( 10): By these 5 steps, we can calculate the impact parameters (b, q) and the components of the initial 1-form k emi μ from given values of the emission angles (α emi , β emi ).3-dimensional region perpendicular to k μ (η).In such a 3-dimensional region in the neighborhood of x μ ng (η), let us introduce a reference 2D surface as a 2-dimensional space-like surface inside the 3-dimensional region.Obviously, there can be infinitely many reference 2D surfaces in the neighborhood of x μ ng (η) since we have not specified the normal direction to the reference 2D surface in the 3-dimensional region perpendicular to k μ (η).
Once a reference 2D surface is specified in the neighborhood of x μ ng (η), we can define the crosssectional area of the null geodesic bundle measured on the reference 2D surface at x μ ng (η) as the intersection area of the reference 2D surface with the null geodesic bundle.Under this definition, the value of the cross-sectional area changes as the normal direction to the reference 2D surface changes in the neighborhood of x μ ng (η).This change in the value of the cross-sectional area can be understood as the Lorentz transformation of the cross-sectional area in the neighborhood of x μ ng (η).Given the definitions of the reference 2D surface and the cross-sectional area of the null geodesic bundle, our derivation of the area-transfer coefficient C consists of the following parts: Part 1 of C: We focus on the cross-sectional area of the null geodesic bundle measured on a temporal reference 2D surface, which is useful for our numerical calculation and defined exactly in Sect.B.1.Then, by making use of the Jacobi equations, we calculate the relation between 2 cross-sectional areas: (i) the cross-sectional area δ S on the celestial sphere around the observer, and (ii) the crosssectional area A emi of the source of the light rays.Here the tilde, such as in A, denotes the value evaluated on the temporal reference 2D surface.Part 2 of C: We construct the Lorentz transformation from the temporal reference 2D surface to the appropriate reference 2D surface, which is defined exactly in Sect.B.2, so that it provides us with the values of t obs and R obs that our telescope measures.Such a Lorentz transformation lets us obtain the detailed form of C.
An illustrative summary of these parts is shown in Fig. B.1, and the details of these parts are explained in the following subsections.Some items in this appendix may already be well known to readers familiar with the application of general relativity to astrophysics and astronomy.However, we describe the detail of the derivation of Eq. ( 28) so that it becomes accessible to as many researchers as possible.Also, a detailed explanation of the formula (28) may be useful for future improvements of the numerical calculation of observable quantities t obs and R obs .For part 1 of deriving C, suppose that the representative null geodesic x μ ng (η) is given.Then, in order to derive the details of the formula (28), it is useful to rearrange the Jacobi equations (24) into some simultaneous ordinary differential equations by using appropriate tetrad components.
In our numerical calculation, we construct double null tetrad basis vectors on the representative null geodesic, {k μ (η), l μ (η), e where there should hold the orthonormal condition Here, the space-like tetrad index is denoted with parentheses, (i) = (1), ( 2) and (j) = (1), (2).The vector k μ is the given null vector tangent to the representative null geodesic, and the null vector l μ and 2 space-like vectors e where the tilde, such as in ẽ, denotes the value evaluated with this ansatz, and the nonzero components are determined by the orthonormal condition (B.1b).
Let the temporal tetrad basis {k μ (η), lμ (η), (η)} denote the ones constructed from the ansatz (B.2).We assign the tilde, such as in Q, to the value of quantity Q when it is evaluated on the temporal tetrad basis.And, given the temporal tetrad basis, we can introduce the space-like 2dimensional surface spanned by the 2 space-like vectors ẽμ (i) (η) in the neighborhood of x μ ng (η).We call this surface the temporal reference 2D surface.Note that every spacetime point x μ ng (η) (0 ≤ η ≤ η obs ) on the representative null geodesic possesses 1 temporal reference 2D surface in its neighborhood, and, due to Eq. (B.1a), all such temporal reference 2D surfaces are generated by transporting parallel the temporal reference 2D surface at x μ emi along the representative null geodesic.Therefore, the cross-sectional area of the null geodesic bundle δ S (= δη 2 A obs due to Eq. ( 27)) measured on the temporal reference 2D surface at x μ obs and the cross-sectional area of the bundle A emi measured on the temporal reference 2D surface at x μ emi can be related by tracing the parallel transport of area A obs 1 When the initial condition satisfies the orthonormal condition (B.1b), the orthonormality of the tetrad basis is automatically preserved at all spacetime points on the representative null geodesic because the inner product of any 2 tetrad basis vectors is invariant under parallel transport along the representative null geodesic due to Eq. (B.1a).In order to rearrange the Jacobi equations (24), we decompose the Jacobi vector by the temporal tetrad basis (η), where the tetrad components are the scalar quantities calculated from the orthonormal condition (B.1b) as

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Here, the component Y (l) is constant along the representative null geodesic as indicated by d Y (l) /dη = −k λ k μ Y μ ;λ = 0, where the last equality is obtained from the geodesic equations k λ k μ ;λ = 0, the original form of the Jacobi equations and the constancy of the norm k μ k μ = constant.Hence, without loss of generality, we require the simplification Then, the decomposition of Y μ by the temporal tetrad basis becomes This decomposition guarantees Y μ to be space-like, since ) 2 ≥ 0, where the equality Y μ Y μ = 0 holds if and only if the space-like components vanish (Y (i) = 0).Substituting the decomposition (B.5) into the Jacobi equations (24), we obtain where equations (B.1) are used in deriving this expression.The inner products of this equation with l μ and e μ (i) give, respectively, the following simultaneous ordinary differential equations: where k β are the tetrad components of the Riemann tensor.Here, since the tetrad indices (i) and (j) take only the space-like values (1) and (2), the latter Eq. (B.8) is independent of the former Eq.(B.7).Hence, in order to calculate the cross-sectional area on the temporal reference 2D surface, we focus on the projection of the Jacobi vector (B.5) onto the temporal reference 2D surface, The "initial" condition of Eq. (B.8) at x μ obs is read from condition (25) as where the latter condition is obtained from the relation . The latter condition Z (i) (η obs ) = 0 guarantees that the area A obs is nonzero, since A obs is swept by the nonzero vector Z where is the Jacobi matrix.The evolution equation of the Jacobi matrix along the given representative null geodesic is obtained by substituting Eq. (B.11) into Eq.(B.8), and the "initial" condition is given by substituting Eq. (B.11) into Eq.(B.10), Once these equations are solved, we obtain the relation between the tetrad components of the Jacobi vector at x μ obs and those at x μ emi , emi(j) Z (j) (η obs ), (B.13) where J emi = J (η = 0) is the Jacobi matrix at x μ emi .Then, by supposing the value of Z (i) is sufficiently small, we can understand that the matrix J emi transforms the areal element on the temporal reference 2D surface at x μ obs into that at x μ emi .Here let us remember that the area A obs is swept by the vector Z μ obs = Z (i) ẽμ (i) η=η obs and that the area A emi is swept by the vector Y μ (0) = Y (i) ẽμ (i) η=0 .This fact, together with the relation (B.13), implies the relation of areas Here, A emi is the area measured on the new reference 2D surface spanned by {ê This relation is used in the next subsection.
Here, let us make a comment on the caustic.The caustic on the null geodesic bundle is the spacetime point x μ ng (η c ) where det J (η c ) = 0 holds.Equation (B.11) indicates that the cross-sectional area vanishes at caustics.Further, by condition (B.10), the observation event x μ obs is interpreted as a caustic of the null geodesic bundle.

B.2. Part 2 of C: Lorentz transformation to the observational value A obs
The area-transfer coefficient C is given by formula (28), which is similar to Eq. (B.14).Our desired area A emi , which appears in Eq. ( 28), is the cross-sectional area of the source measured on the appropriate reference 2D surface that is perpendicular to u μ emi at x emi .Another desired area A obs , which appears in Eq. (28), is the area measured on the appropriate reference 2D surface perpendicular to u μ obs at x obs .On the other hand, the areas A obs and A emi in Eq. (B.14) are the areas evaluated on the temporal reference 2D surface.Therefore, we need some appropriate Lorentz transformations in order to obtain the detailed form of Eq. (28).Our procedure for calculating the appropriate Lorentz transformations consists of 3 subparts: i)obs } in the sense that both tetrad bases include the same space-like basis vectors êμ (i)obs that span the reference 2D surface on which the area A obs is measured.
Remember that our desired area A obs is related to the cross-sectional area of the null geodesic bundle as δS = δη2 A obs (see Eq. ( 27)), which is measured at the intersection event x μ ng (η obs − δη) of the representative null geodesic with the celestial sphere of radius δη (see Eq. ( 23) and Fig. 3).Also, the area A obs should satisfy the relation δ S = δη 2 A obs , where δ S is the cross-sectional area measured on the reference 2D surface spanned by the parallel transform of êμ (i)obs from x μ obs to x μ ng (η obs − δη).Here note that, for a sufficiently small δη, any reference 2D surface at x μ ng (η obs − δη) can be regarded as the reference 2D surface at x μ obs . 2 Hence, the construction of the Lorentz transformation between δS and δ S can be carried out at x Here it should be noted that the shape of the beam of light rays appearing on the hypersurface in Fig. B.2, which is shown by the shaded or colored area, need not necessarily express the real shape determined by the real distribution of photons in the hypersurface.The point is that, in the hypersurface perpendicular to u μ obs , the area δS has to be the observable value of the cross-sectional area of the beam measured by the observer of velocity u μ obs .And, because the other areas δ S and δS c are not the observable quantities in our setup, we can define those areas so that they give the observable value δS through the formula (B.32) derived below.To do so, we consider the imaginary distribution of the null geodesic bundle that is illustrated as a quadrangular prism in Fig. B.2.As explained below, given such an imaginary distribution of null geodesics, the areas δ S and δS c will be determined automatically once the reference 2D surface spanned by êμ (i)obs and the crossing 2D surface are specified.
The relation between the areas δS c and δS in the hypersurface perpendicular to u μ obs can be derived by considering the projection of the area δS c , which is measured on the crossing 2D surface, into the appropriate reference 2D surface, which is spanned by e μ (i)obs .The projection tensor h μν obs into the appropriate reference 2D surface is given by h

B.3. Steps for numerical calculation of A obs
In our numerical calculation after the solution of the null geodesic equations x μ ng (η) has been obtained, the procedure for calculating the area A obs is as follows: Step 1 (A obs ): Construct the double null temporal tetrad basis {k μ (η), lμ (η), ẽμ Step 2 (A obs ): Solve the evolution equations of the Jacobi matrix (B.12).Then, we obtain the Jacobi matrix at the emission event J Step 5 (A obs ): Given the above 3 bases, the Lorentz transformation (B.31) from A obs to A obs is calculated, where the matrices h (i,j)obs and ĥ(i,j)obs are given by Eqs.(B.27) and (B.30).
Step 6 (A obs ): Finally, collecting the results of steps 2 and 5, the relation (B.32) between A obs and A emi is calculated, where the value of A emi is assumed to satisfy condition (31) in our numerical calculation.

4 / 40 Fig. 1 .
Fig.1.Examples of winding rays (green curves) for a BH of (M , χ) = (1, 0.8).All rays shown here reach the same observer at (r obs , θ obs , ϕ obs ) = (100r BH , 15π/31, 7π/30).The left-hand panel is the 3D screen spanned by the variables (2).The ergoregion is colored gray, the equator of the ergosurface is the red circle, and the BH horizon is the black sphere.The source is at (r emi , θ emi , ϕ emi ) = (2.2rBH , π/2, 0).The right-hand panel is the t-r(t) graph from which the time delay t obs is read.In calculating the flux ratio R obs , the caustics that each ray passes through are found as by-products (see Sect. 3).

μFig. 2 .
Fig.2.Illustration of the specific intensity I obs (ν obs ), the visible solid-angle δ obs , and the specific flux F obs .The dimensions of them are shown in MKS units, where 1/m 2 in the dimensions of F obs denotes per unit area on the telescope, and 1/ste-rad in the dimensions of I obs denotes per unit solid-angle seen from the telescope.

Fig. 3 .
Fig.3.The setup for calculating the visible solid-angle δ obs .The left-hand panel is a 3D schematic illustration, and the right-hand panel is a spacetime diagram near the observation event x μ ng (η obs ).The quantity Y μ is the Jacobi vector field on the representative null geodesic.By considering a small celestial sphere around the observation point, the visible solid-angle δ obs is calculated from the area swept by Y μ on the celestial sphere.

μ
obs := Y μ (η obs ) and Y μ obs; α := Y μ ;α (η obs ) are the Jacobi vector and its covariant derivative at the observation event.The value of Y μ obs; α is not specified uniquely.Different values of Y μ obs; α correspond to the different Jacobi fields connecting the representative null geodesic to different neighboring null geodesics in the bundle.

μ
emi = x μ ng (0) to the observation event x μ obs = x μ ng (η obs ) along the representative null geodesic.The following paragraphs are the outline of the calculation procedure of δS, and the details are given in Appendix B.

Fig. 5 .
Fig.5.Numerical results for t obs , R obs , and ν obs(s) /ν obs(p) for a radial falling source near a BH with χ = 0.The numerical results with θ obs = 16π/31 are plotted, where ϕ obs is varied from 0 to 2π .However, in our numerical program, numerical calculations at some values of ϕ obs could not produce an appropriate initial value for the geodesic equations.The values of ϕ obs for which the numerical calculation was completed are shown in the left-hand panel.The colors of the points in each of the panels denote the variation in the value of ϕ obs .In the right-hand panels, the data points corresponding to ϕ obs < π do not appear since those data points degenerate to the data points corresponding to ϕ obs ≥ π in the case χ = 0.

Fig. 6 .
Fig. 6.Numerical results for t obs , R obs , and ν obs(s) /ν obs(p) for a radial falling source near a BH with χ = 0.3.The numerical results with θ obs = 16π/31 are plotted, where ϕ obs is varied from 0 to 2π , but some values of ϕ obs could not produce an appropriate initial value for the geodesic equations.The values of ϕ obs for which the numerical calculation was completed are shown in the upper-left panel.The colors of the points denote the variation of ϕ obs .

Fig. 7 .Fig. 8 .
Fig. 7. Numerical results for t obs , R obs , and ν obs(s) /ν obs(p) for a radial falling source near a BH with χ = 0.8.Two cases, θ obs = 4π/31 and 16π/31, are plotted.For each case, ϕ obs is varied from 0 to 2π , but some values of ϕ obs could not produce an appropriate initial value for the geodesic equations.The values (θ obs , ϕ obs ) for which the numerical calculation was completed are shown in the upper-left panel.The colors of the points denote the variation of ϕ obs .

Fig. B. 1 .
Fig. B.1.The evolution of the cross section is described by the Jacobi equations.The areas A obs and A emi are measured on the temporal reference 2D surface.Our desired areas A obs and A emi are obtained by an appropriate Lorentz transformation of A obs and A emi .

μ
obs := k α obs Y μ obs; α = Z (i) ẽμ (i) | η=η obs = 0, as implied by Eq. (27).In order to calculate the area A obs as it is swept by Z μ obs , we transform the Jacobi equations (B.8) into the other form.The appropriate transformation of Eq. (B.8) is given by the well-known theory of simultaneous first-order ordinary differential equations.According to the theory, the general solution of the simultaneous equations (B.8) under the condition (B.10) is expressed as

Fig. B. 2 .Fig. B. 3 .
Fig. B.2.Geometrical illustration of the relation between the cross-sectional areas of the narrow null geodesic bundle δ S (= δη 2 A obs ) and δS (= δη 2 A obs ).The null geodesic bundle is illustrated as a quadrangular prism, but only 2 null geodesics are illustrated on the side faces of the prism.
μν obs := g μν obs − p μ obs p ν obs + u μ obs u ν obs .For any vector v μ , the components perpendicular to p μ obs and u μ obs are deleted by operating this tensor as h μν obs v ν .Thus, the projection of basis vectors c μ (i)obs into the appropriate reference 2D surface is given by (see Fig. B.3) hc μ (i)obs := h μν obs c (i)obs ν = c μ (i)obs − (p ν obs c (i)obs ν )p μ obs , (B.25) where condition (B.24) is used in the last equality.Since this hc μ (i)obs is the vector on the appropriate reference 2D surface, we can expand it as hc μ (i)obs = j=1,2 h (i,j)obs e μ (j)obs , (B.26) where the expansion coefficient is given by h (i,j)obs = e (i)obs μ hc μ (j)obs = e (i)obs μ c μ (j)obs , (B.27) 38/40 Downloaded from https://academic.oup.com/ptep/article-abstract/2017/5/053E02/3855709/How-to-measure-a-black-hole-s-mass-spin-and by CERN -European Organization for Nuclear Research user on 03 October 2017 PTEP 2017, 053E02 H. Saida where Eq. (B.25) and condition (B.23b) are used in the last equality.As indicated in Fig. B.3, the parallelogram of area det h obs on the appropriate reference 2D surface spanned by e μ (i)obs corresponds to the unit square on the crossing 2D surface, where det h obs is the determinant of the matrix made of the expansion coefficients (B.27).This indicates the relation δS c /δS = 1/ det h obs , which denotes δS c = (det h obs )δS.(B.28)Further, by replacing the reference 2D surface spanned by e μ (i)obs with the reference 2D surface spanned by êμ (i)obs in the derivation of Eq. (B.28), we can derive the relation between the areas δS c and δ S in the hypersurface perpendicular to ûμ obs .The resultant relation is δS c = (det ĥobs )δ S, (B.29) where det ĥobs is the determinant of the matrix ĥ(i,j)obs = ê(i)obs μ c μ (j)obs .(B.30) Given the relations (B.28) and (B.29), we obtain the Lorentz transformation between the areas A obs (= δS/δη 2 ) and A obs (= δ S/δη 2 ), A obs = det ĥobs det h obs A obs .(B.31) Finally, substituting the Lorentz transformation of area (B.31) into the relation (B.17), we obtain the detailed form of formula (28), A obs = CA emi , (B.32a) where the area transformation coefficient is C = det ĥobs det h obs 1 det J emi , (B.32b)where the matrices h (i,j)obs and ĥ(i,j)obs are, respectively, in Eq. (B.27) and (B.30).We can recognize that C is determined by x μ 1) (η), ẽμ (2) (η)} along the given representative null geodesic by solving equations (B.1a) under the initial condition (B.2) whose nonzero values are determined by the algebraic equations (B.1b).
, any light ray possesses 3 constants of motion (ν obs , l ϕ , l nor ) given by K μ and Killing fields, spacetime the precise form of the coefficient C is given in Eq. (B.32b) of Appendix B. We call this coefficient C the area-transfer coefficient.The point of this relation is that the transformation between A obs and A emi is a linear relation, and the area-transfer coefficient C is determined by the set of quantities x μ emi , and k μ emi Saida telescope capability and the numerical results implies that, if the assumptions introduced in Sect.2.1 hold, then our observational quantities t obs and R obs can, in principle, be measured by present or near future telescope capability.The cases that modify some of our assumptions will be discussed in other papers.
Part 2-1 of C: For the first, at the emission event x Finally, calculate the Lorentz transformation of the area A obs to the desired area A obs at x μ obs spanned by êμ (i)obs is given by Eq. (B.17).Part 2-3 of C: μ obs .Then, we will arrive at the detailed form of Eq. (28), which is Eq.(B.32).34/40 Downloaded from https://academic.oup.com/ptep/article-abstract/2017/5/053E02/3855709/How-to-measure-a-black-hole-s-mass-spin-and by CERN -European Organization for Nuclear Research user on 03 October 2017 C is the calculation of the Lorentz transformation from the area A obs measured on the reference 2D surface spanned by êμ (i)obs to our desired area A obs measured on the appropriate reference 2D surface that is perpendicular to u μ obs .The rest of this subsection is for this step.For the preparation of this Lorentz transformation, we introduce the imaginary observer at x (i)emi }.Also, we find from Eq. (B.17),A emi = (det J emi ) A obs , (B.21) where our desired area A emi = A emi by the construction of (b) (a) .35/40 Downloaded from https://academic.oup.com/ptep/article-abstract/2017/5/053E02/3855709/How-to-measure-a-black-hole-s-mass-spin-and by CERN -European Organization for Nuclear Research user on 03 October 2017