The effects of finite distance on the gravitational deflection angle of light

In order to clarify effects of the finite distance from a lens object to a light source and a receiver, the gravitational deflection of light has been recently reexamined by using the Gauss-Bonnet (GB) theorem in differential geometry [Ishihara et al. 2016]. The purpose of the present paper is to give a short review of a series of works initiated by the above paper. First, we provide the definition of the gravitational deflection angle of light for the finite-distance source and receiver in a static, spherically symmetric and asymptotically flat spacetime. We discuss the geometrical invariance of the definition by using the GB theorem. The present definition is used to discuss finite-distance effects on the light deflection in Schwarzschild spacetime, for both cases of the weak deflection and strong deflection. Next, we extend the definition to stationary and axisymmetric spacetimes. We compute finite-distance effects on the deflection angle of light for Kerr black holes and rotating Teo wormholes. Our results are consistent with the previous works if we take the infinite-distance limit. We briefly mention also the finite-distance effects on the light deflection by Sagittarius A$^*$.

Most of those calculations are based on the coordinate angle. The angle respects the rotational symmetry of the spacetime. Gibbons and Werner (2008) made an attempt of defining, in a more geometrical manner, the deflection angle of light [32]. In their paper, the source and receiver are needed to be located at an asymptotic Minkowskian region. The Gauss-Bonnet theorem was applied to a spatial domain with introducing the optical metric, for which a light ray is expressed as a spatial geodesic curve. Ishihara et al. have successfully extended Gibbons and Werner's idea, such that the source and receiver can be at a finite distance from the lens object [33]. They extend the earlier work to the case of the strong deflection limit, in which the winding number of the photon orbits may be larger than unity [34]. In particular, the asymptotic receiver and source are not needed. Arakida [35] made an attempt to apply the Gauss-Bonnet theorem to quadrilaterals that are not extending to infinity and proposed a new definition of the deflection angle of light, though a comparison between two different manifolds that he proposed is an open issue. With proposing an alternative definition of the deflection angle of light, Crisnejo et al. [36] has recently made a comparison between the alternative definitions in [33][34][35] and shown by explicit calculations that the definition by Arakida in [35] is different from that by Ishihara et al. [33,34]. Their definition has been applied to study the gravitational lensing with a plasma medium [36].
The earlier works [33,34] are restricted within the spherical symmetry. Ono et al. have extended the Gauss-Bonnet method with the optical metric to axisymmetric spacetimes [37]. This extension includes mathematical quantities and calculations, with which most of the physicists are not very familiar. Therefore, the purpose of this paper provides a review of the series of papers on the gravitational deflection of light for finite-distance source and receiver. In particular, we hope that detailed calculations in this paper will be helpful for readers to compute the gravitational deflection of light by the new powerful method. For instance, this new technique has been used to study the gravitational lensing in rotating Teo wormholes [38] and also in Damour-Solodukhin wormholes [39]. This formulation has been successfully used to clarify the deflection of light in a rotating global monopole spacetime with a deficit angle [40]. This paper is organized as follows. Section II discusses the definition of the gravitational deflection angle of light in static and spherically symmetric spacetimes. Section III considers the weak deflection of light in Schwarzschild spacetime. Section IV discusses the weak deflection of light in the Kottler spacetime and the Weyl conformal gravity model. The strong deflection of light is examined in Section V. Sagittarius A * (Sgr A * ) is also discussed as an example for possible candidates. In section VI, we discuss the strong deflection of light with finite-distance corrections in Schwarzschild spacetime. Section VII proposes the definition of the gravitational deflection angle of light in stationary and axisymmetric spacetimes. Sgr A * is also discussed. The weak deflection of light is discussed for Kerr spacetime in Section VIII and for rotating Teo wormholes in Section IX. Section X is the summary of this paper.
Appendix A provides the detailed calculations for the Kerr spacetime. Throughout this paper, we use the unit of G = c = 1, and the observer may be called the receiver in order to avoid confusion between r O and r 0 by using r R .

A. Notation
Following Ishihara et al. [33], this section begins with considering a static and spherically symmetric (SSS) spacetime. The metric of this spacetime can be written as ds 2 = g µν dx µ dx ν = −A(r)dt 2 + B(r)dr 2 + r 2 dΩ 2 , where dΩ 2 ≡ dθ 2 + sin 2 θdφ 2 , and t, θ and φ are associated with the symmetries of the SSS spacetime. For a metric of the form (1) we always have to restrict to the domain where A(r) and B(r) are positive, such that a static emitter and a static receiver can exist. The spacetime has a spherical symmetry. Therefore, the photon orbital plane is chosen, without loss of generality, as the equatorial plane (θ = π/2). We follow the usual definition of the impact parameter of the light ray as From ds 2 = 0 for the light ray, the orbit equation is derived as Light rays are described by the null condition ds 2 = 0, which is solved for dt 2 as where I and J denote 1 and 2 and we used Eq. (1). We refer to γ IJ as the optical metric.
The optical metric can be used to describe a two-dimensional Riemannian space. This Riemannian space is denoted as M opt . The light ray is a spatial geodetic curve in M opt .
In the optical metric space M opt , let Ψ denote the angle between the light propagation direction and the radial direction. A straightforward calculation gives cos Ψ = b A(r)B(r) r 2 dr dφ .
This is rewritten as where we used Eq. (3).
We denote Ψ R and Ψ S as the directional angles of the light propagation. Ψ R and Ψ S are measured at the receiver position (R) and the source position (S), respectively. We denote φ RS ≡ φ R − φ S the coordinate separation angle between the receiver and source. By using these angles Ψ R , Ψ S and φ RS , we define This is a basic tool that was invented in Reference [33]. In the following, we shall prove that the definition by Eq. (7) is geometrically invariant [33,34]. Here, we briefly mention the Gauss-Bonnet theorem. T is a two-dimensional orientable surface. Differentiable curves ∂T a (a = 1, 2, · · · , N) are its boundaries. Please see Figure   1 for the orientable surface. We denote the jump angles between the curves as θ a (a = 1, 2, · · · , N). The Gauss-Bonnet theorem: [41] T KdS + N a=1 ∂Ta where ℓ means the line element of the boundary curve, dS denotes the area element of the surface, K means the Gaussian curvature of the surface T , κ g is the geodesic curvature of ∂T a . The sign of ℓ is chosen to be consistent with the surface orientation.
Suppose a quadrilateral ∞ R ∞ S . Please see Figure 2 for this. This is made of four lines; (1) the spatial curve for the light ray, (2) and (3) two outgoing radial lines from R and from S and (4) a circular arc segment C r that is centered at the lens with the coordinate radius r C (r C → ∞) and intersects the radial lines at the receiver or the source. We restrict ourselves within the asymptotically flat spacetime. Then, κ g → 1/r C and dℓ → r C dφ as r C → ∞ (See e.g. [32]). By using them, we find Cr κ g dℓ → φ RS . Applying this result to Therefore, α is shown to be invariant for transformations of the spatial coordinates. In addition, α is well-defined even when L is a singular point. This is because the point L does not appear in the surface integral nor in the line integral. Furthermore, α vanishes in Euclidean space. This means α is a measure of the deviation from the flat space.
Here, we explain that α defined in Eq. (7) is observable in principle. For the simplicity, let us imagine the following ideal situation. The positions of a source and receiver are known. For instance, we assume that the lens object is the Sun, the receiver is located at the Earth, and the source is a pulsar which radiates radio signals with a constant period in an anisotropic manner. In particular, we assume that the source is one of the known pulsars whose spin period and pulse signal behaviors such as pulse profiles are well-understood.
By very accurate radio observations such as VLBI, the relative positions of the Earth, Sun and the pulsar can be determined from the ephemeris. (1) From this, we can know φ RS in principle.
(2) We can directly measure the angle Ψ R at the Earth between the solar direction . In this schematic figure, the lens, receiver and source are the Sun, the Earth and a pulsar that periodically radiates radio signals in a specific anistropic manner. From the pulse profile, we can determine the radiation direction at the source. By using the ephemeris, we know the relative positions of the Sun, Earth and the pulsar. Hence, we can determine φ RS and Ψ S . By observing the pulsar, we can measure Ψ R . In principle, therefore, we can determine Ψ R − Ψ S + φ RS from these astronomical observations. and the pulsar direction. (3) More importantly, the direction of radiating the pulses that reaches the receiver can be also determined in principle, because the viewing angle of the pulsar seen by the receiver is known from the pulse profiles. The viewing angle is changing with time because of the Earth motion around the Sun. By using the pulsar position and the pulse radiation direction, we can determine Ψ S . Please see Figure 3 for this situation.
We explain in more detail how Ψ S at S can be measured by the observer at R. We consider a pulsar whose spin axis is known from some astronomical observations. A point is that the spin axis of an isolated pulsar is constant with time. The pulse shape and profile depend on the viewing angle with respect to the spin axis of the pulsar. The Earth moves around the Sun and hence the observer sees the same pulsar with different viewing angles with time.
Accordingly, the observed pulse shape changes. By observing such a change in the pulse shape, we can in principle determine the intrinsic direction of the radio emission, namely the angle between the spin axis and the direction of the emitted light to the observer. In addition, we can know the intrinsic position (including the radial direction from the lens) of such a known pulsar from the ephemeris. By using the intrinsic position (its radial direction) and emission direction at S, Ψ S can be determined in principle, though it is very difficult with current technology. As a result, we can determine in principle Ψ R − Ψ S + φ RS from astronomical observations. Namely, α in Eq. (7) is observable. Note that this procedure does not need assume a different spacetime, while such a fiducial spacetime was assumed by Arakida (2018) [35], though the receiver in our universe cannot observe the fiducial different spacetime but can assume (or make theoretical calculations of) some quantities on the different spacetime.
One can easily see that, in the far limit of the source and the receiver, Eq. (9) agrees with the deflection angle of light as Here, we define u and u 0 as as the inverse of r, the inverse of the closest approach (often denoted as r 0 ), respectively. F (u) is defined as F (u) can be computed by using Eq. (3).
The present paper wishes to avoid the far limit in the following reason. Every observed stars and galaxies are never located at infinite distance from us. For instance, we observe finite-redshift galaxies in cosmology. We cannot see objects at infinite redshift (exactly at the horizon). Except for a few rare cases in astronomy, the distance to the light source is much larger than the size of the lens. Therefore, we find a strong motivation for studying a situation that the distance from the source to the receiver is finite. We define u R and u S as the inverse of r R and r S , respectively, where r R and r S are finite. Eq. (7) is rewritten in an explicit form as [33,34] Here, we assume light rays that have the only one local minimum of the radius coordinate between r S and r R . This is valid for normal situations in astronomy. However, we should note that multiple local minimaare possible, e.g. if the emitter or the receiver(or both) are between the horizon and the lightsphere in the Schwarzschild spacetime, or if theemitter and receiver are at different sides of thethroat of a wormhole spacetime. For such a case of multiple local minima, Eq. (12) has to be modified, because it assumes only the local minimum at u = u 0 .

III. WEAK DEFLECTION OF LIGHT IN SCHWARZSCHILD SPACETIME
In this section, we consider the weak deflection of light in Schwarzschild spacetime, for which the line element becomes where r g = 2M in the geometrical unit. Then, F (u) is By using Eq. (6), Ψ R − Ψ S in the Schwarzschild spacetime is expanded as Note that Ψ R − Ψ S → π in the Schwarzschild spacetime as u S → 0 and u R → 0.

IV. OTHER EXAMPLES
This section discusses two examples for a non-asymptotically flat spacetime. One is the Kottler solution to the Einstein equation. The other is an exact solution in the Weyl conformal gravity. The aim of this study is to give us a suggestion or a speculation. We note that the present formulation is limited within the asymptotic flatness, rigorously speaking.
As mentioned in Introduction, Arakida [35] made an attempt to apply the Gauss-Bonnet theorem to quadrilaterals that are not extending to infinity, though a comparison between two different manifolds that he proposed is an open issue. A more careful study that gives a justification for this speculation or perhaps disprove it will be left for future.
In this section, we do not assume the source at the past null infinity (r S → ∞) nor the receiver at the future null infinity (r R → ∞), because A(r) diverges or does not exist as r → ∞. We keep in mind that the source and receiver are located at finite distance from the lens object. Therefore, we use Eq. (12). As mentioned already, Eq. (6) is more useful for calculating Ψ R and Ψ S than Eq. (5), because Eq. (6) requires only the local quantities but not any differentiation. By straightforward calculations, we obtain the following results for the above two models.

A. Kottler solution
We consider the Kottler solution [42]. This solution is written as where the cosmological constant is denoted by Λ.
We use Eq. (6), such that Ψ R − Ψ S can be expanded in terms of 2M and Λ as Here, Ψ Sch is a pair of the terms that appear also in a case of the Schwarzschild spacetime. The above expansion of Ψ R −Ψ S has a divergent term in the limit as u S → 0 and u R → 0. The reason for this divergent behavior is that the spacetime is not asymptotically flat and therefore the limit of u S → 0 and u R → 0 is no longer allowed. Hence, the power series in Eq. (17) is mathematically valid only within a convergence radius.
For the Kottler spacetime, F (u) becomes We obtain By using Eqs. (17) and (19), α is obtained as This equation has several divergent terms as bu R → 0 and bu S → 0. The apparent divergent is problematic only in the case that the source or receiver is located at the horizon. In other words, all the terms in Eq. (20) are finite and thus harmless for astronomical situations.

B. Weyl conformal gravity case
Next, we consider Weyl conformal gravity model. This theory was originally suggested by Bach [43]. The SSS solution in this model is expressed by introducing three new parameters that are often denoted as β, γ and k. For this generalized solution in conformal gravity, Birkhoff's theorem still holds [44]. The SSS solution in the Weyl gravity model is [45] ds 2 = −A(r)dt 2 + 1 A(r) dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), Here, m ≡ β(2 − 3βγ)/2. kr 2 in the metric plays the same role as the cosmological constant in the Kottler spacetime that has been studied above. Therefore, we omit the r 2 term for simplifying our analysis.
By using Eq. (6), we expand Ψ R − Ψ S in β and γ. The result is We should note that this expansion of Ψ R − Ψ S is divergent as u S → 0 and u R → 0.
This divergent behavior is not so problematic, because the limit of u S → 0 and u R → 0 is not allowed in this spacetime. Hence, we note that, rigorously speaking, Eq. (22) is mathematically valid only within a convergence radius.
For the present case omitting k, we obtain φ RS is computed as Consequently, we obtain α as The linear terms in γ cancel out with each other and they do not appear in the final expression for the deflection angle of light. This result may suggest a correction to the results in previous papers [46][47][48] that reported non-zero contributions from γ.

C. Far source and receiver
Next, we investigate a situation of a distant source and receiver from the lens object: bu S ≪ 1 and bu R ≪ 1. Divergent terms in the deflection angle appear in the limit as bu S → 0. Therefore, We carefully investigate the leading part in a series expansion, where the infinite limit is not taken. As a result, approximate expressions for the deflection of light are obtained as follows.
(1) Kottler model: The expression for φ RS in this approximation is the same as the seventh and eighth terms of Eq. (5) in [49], the third and fifth terms of Eq. (15) in [50], and the second term of Eq.
(14) in [51]. On the other hand, they [49][50][51] did not take account of Ψ R − Ψ S . In the far This expression suggestions a correction to the earlier works [49][50][51]. For instance, only the term of φ RS was considered in Sereno (2009).
(2) Weyl conformal gravity model: Next, we consider the Weyl conformal gravity model. The deflection angle of light in the far approximation is computed as where mγ parts from Ψ R −Ψ S and from ψ RS cancel out with each other. Please see also Eqs. (22) and (24). For instance, Reference [52] gives the exact expression of the deflection angle for the asymptotic receiver and source in the Kottler and Weyl conformal gravity spacetime.

V. EXTENSION TO THE STRONG DEFLECTION OF LIGHT
In the previous sections, we considered the weak deflection of light: A light ray from the source to the receiver is expressed by a spatial curve. The curve is simply-connected. In the strong deflection limit, on the other hand, it is possible that the spatial curve has a winding number with intersection points. We thus divide the whole curve into segments. And it is easier to investigate each simple segment.

A. Loops in the photon orbit
We begin with one loop case of the light ray curve. This case is shown by Figure 4.
First, we consider the two quadrilaterals (1) and (2) in Figure 5. They can be constructed by introducing an auxiliary point (P) and next by adding auxiliary outgoing radial lines (solid line in this figure) from the point P in the quadrilaterals (1) and (2). The point P does not need to be the periastron. The direction of the two auxiliary lines in (1) and (2) is opposite to each other. The two auxiliary lines thus cancel out to make no contributions to α. Here, θ 1 and θ 2 denote the inner angle at the point P in the quadrilateral (1) and that in the quadrilateral (2), respectively. We can see that θ 1 + θ 2 = π. This is because the line from the source to the receiver is a geodesic and the point P is located in this line.
For a quadrilateral in Figure 5, the method in Section II is still applicable. By the same way of obtaining Eq. (9), we obtain Here, φ RS is divided into two parts: One is φ RS for one quadrilateral and the other is φ for the other quadrilateral.
If r S = r R , the quadrilaterals (1) and (2) are symmetric for reflection and φ (1) RS is not the same as φ (2) RS . In any case, however, φ  (π − Ψ R ) are the inner angles at S and R, respectively. Therefore, where we use θ 1 + θ 2 = π and φ (1) This result is the same as Eq. (7), though the validity domain is different.
Next, we investigate a case of two loops shown by Figure 6. For this case, we add lines in order to divide the shape into four quadrilaterals as shown by Figure 7. We immediately find where φ where we use θ 1 + θ 2 = θ 3 + θ 4 = θ 5 + θ 6 = π. Eq. (31) is obtained for the two-loop case in the same form as Eq. (7). A loop does make the contribution to α only through the terms of φ RS . Finally, we shall complete the proof. We consider the arbitrary winding number, say W .
Eq. (7) is equivalent to Eq. (12). This is shown by using the orbit equation. This expression is rearranged as We define the difference between the asymptotic deflection angle and the deflection angle for the finite distance case as δα.
The meaning of this is the finite-distance correction to the deflection angle of light. By substituting Eqs. (10) and (32) into Eq. (33), we get This expression implies two origins of the finite-distance corrections. One origin is Ψ R and Ψ S . They are angles that are defined in a curved space. The other origin is the two path integrals. They contain the information on the curved space. If we consider a receiver and source in the weak gravitational field (as common in astronomy), the finite-distance correction reflects only the weak field region, even if the light ray passes through a strong field region.

VI. STRONG DEFLECTION OF LIGHT IN SCHWARZSCHILD SPACETIME
In this section, we consider the Schwarzschild black hole. By using F (u) given by Eq.
where we used a logarithmic term [8] in the last term of Eq. (32). Here, the dominant terms in Ψ R and Ψ S cancel with the terms in the integrals. As a consequence, Ψ R and Ψ S do not appear in the approximate expression of Eq. (35).
As mentioned above, it follows that the logarithmic term by the strong gravity is free from finite-distance corrections such as 1 − (bu S ) 2 . By chance, δα in the strong deflection limit (See Eq. (32)) is apparently the same as that for the weak deflection case (See e.g. Eq. (29) in [34]). Therefore, the finite-distance correction in the strong deflection limit is again This is the same expression as that for the weak field case (e.g. [33]). Namely, the correction is linear in the impact parameter. The finite-distance correction in the weak deflection case (large b) is thus larger than that in the strong deflection one (small b), if the other parameters remain the same.
A. Sagittarius A * Next, we briefly mention an astronomical implication of the strong deflection. One of the most feasible candidates for the strong deflection is Sagittarius * (Sgr A * ) that is located at our galactic center. In this case, the receiver distance is much larger than the impact parameter of light and a source star may live in the bulge of our Galaxy.
The apparent size of Sgr A * is expected to be nearly the same as that of the central massive object of M87. However, the finite-distance correction to Sgr A * becomes much larger than that to the M87 case, because Sgr A * is much closer to us than M87.
For Sgr A * , Eq. (36) is evaluated as where the central black hole mass is assumed as M ∼ 4 × 10 6 M ⊙ and we take the limit of strong deflection b ∼ 3M. Rather interestingly, this correction as ∼ 10 −5 arcsec. will be reachable by the Event Horizon Telescope [31] and the near-future astronomy.
See Figure 9 for numerical estimations of the finite-distance correction by the source distance. This figure and Eq. (37) suggest that δα is ∼ ten (or more) micro arcseconds, if a source star is sufficiently close to Sgr A * , for instance within a tenth of one parsec from Sgr A * . For such a case, the infinite-distance limit does not hold, even though the source is still in the weak field. We should take account of finite-distance corrections that are discussed in this paper.
In the strong deflection case, each orbit around the black hole will have a slightly different r 0 , thereby producing a number of "ghost" images (often called relativistic images). In this paper, detailed calculations about it for the finite-distance source and receiver are not done.
It is left for future. In this section, a stationary and axisymmetric spacetime is considered, for which we shall discuss how to define the gravitational deflection angle of light especially with using the Gauss-Bonnet theorem [37]. The line element in this spacetime is [53][54][55] Here, p, q mean 1 and 2, γ pq is a two-dimensional symmetric tensor, µ, ν take from 0 to 3, t and φ coordinates respect the Killing vectors. We rewrite this metric into a form, such that γ pq can be diagonal. We prefer to use the polar coordinates rather than the cylindrical ones, because the Kerr metric and the rotating Teo wormhole one are usually expressed in the polar coordinates. In the polar coordinates, Eq. (38) is rewritten as [56] where a local reflection symmetry is assumed with respect to the equatorial plane θ = π 2 . This assumption is expressed as The functions are A(r, θ) > 0, B(r, θ) > 0, C(r, θ) > 0, D(r, θ) > 0 and H(r, θ) > 0. This assumption by Eq. (40) is needed for the existence of a photon orbit on the equatorial plane.
Note that we do not assume the global reflection symmetry with respect to the equatorial plane.
The null condition ds 2 = 0 is solved for dt as [59,60] where i, j denote from 1 to 3, γ ij and β i are defined as This spatial metric γ ij ( = g ij ) is used in order to define the arc length (ℓ) along the photon orbit as for which we define γ ij by γ ij γ jk = δ i k . γ ij defines a 3-dimensional Riemannian space (3) M, where the photon orbit is a spatial curve. In the Appendix of Ref. [60], they show that ℓ is an affine parameter of a light ray.
If the spacetime is static, spherically symmetric and asymptotically flat, β i is zero and γ ij is nothing but the optical metric. The photon orbit follows a geodesic in a 3-dimensional Riemannian space. In this section and after, we refer to γ ij as the generalized optical metric.
Note that the metric γ ij has been called the Fermat metric and the one-form β i the Fermat one-form by some authors.
We apply Gauss-bonnet theorem to a surface (See Figure 1). The Gauss-Bonnet theorem is expressed as where we note that the geodesic curvatures of the path from S to S ∞ and the path from R to R ∞ are both 0, because these paths are geodesic. κ g is the geodesic curvature of the photon orbit andκ g is the geodesic curvature of the circular arc segment with an infinite radius.

B. Gaussian curvature
In this subsection, we examine whether or not the rotational part (β i ) of the spacetime makes a contribution to the Gaussian curvature. The Gaussian curvature on the equatorial plane is expressed by using the 2-dimensional Riemann tensor (2) R rφrφ as where (2) R rφrφ and (2) Γ ı jk are defined by using the generalized optical metric γ ij on the equatorial plane. det γ (2) ij is the determinant of the generalized optical metric in the equatorial plane.
The surface integration of the Gaussian curvature in Eq.(45) is rewritten explicitly as where r OE means the solution of the orbit equation. The geodesic curvature in the vector form is defined as (see e.g. [57]) where, for a parameterized curve, T denotes the unit tangent vector for the curve by reparameterizing the curve using its arc length, T ′ means its derivative with respect to the parameter, and N indicates the unit normal vector for the surface. The geodesic curvature of a curve vanishes, if the curve follows the geodesic. This zero is because the acceleration vector T ′ vanishes.

D. Photon orbit with the generalized optical metric
In this subsection, we discuss geometrical aspects of a photon orbit in terms of the generalized optical metric. The unit vector tangent to the spatial curve is generally expressed where a parameter ℓ is defined by Eq.(44).
The flight time T of a light from the source to the receiver is obtained by performing the integral of Eq.(41), The light ray follows the Fermat's principle, namely δT = 0 [58]. The Lagrangian for a photon can be expressed as From this, We obtain d dℓ ∂L ∂e k =γ ik e i ,l e l + γ ik,l e i e l + β k,i e i , where we used γ ij e i e j = 1 and the comma (,) defines the partial derivative. The Euler-Lagrange equation is calculated as e j ,l e l + γ kj γ ik,l e i e l − This leads to the equation for the light ray as [60] de Therefore, the geodesic equation is equivalent to where we define | as the covariant derivative with respect to γ ij . (3) Γ i jk means the Christoffel symbol by γ ij .
The acceleration vector a i is defined by By using the Levi-Civita symbol ε ijk , we express the cross (outer) product of A and B in the covariant manner The Levi-Civita tensor ǫ ijk is defined by ǫ ijk ≡ √ γε ijk , where and ε ijk is the Levi-Civita symbol (ε 123 = 1). The Levi-Civita tensor ǫ ijk in a three-dimensional satisfies ǫ sjk ǫ s lm = γ jl γ km − γ jm γ kl .
By using Eqs. (58), (59) and (60), Eq. (57) is rewritten as The vector a i is the spatial vector representing the acceleration due to β i . In particular, a i is caused in gravitomagnetism [62]. To be more precise, the gravitomagnetic vector has an analogy to the Lorentz force in electromagnetism ∝ v × (∇ × A m ), in which A m denotes the vector potential. The vector potential is defined as B = ∇ × A m , E = −∇φ − ∂Am ∂t , where E and B are the electric and magnetic fields, respectively, and the electric potential is φ. γ ij is not an induced metric but the generalized optical metric. If β i is non-vanishing, the photon orbit may be different from a geodesic in (3) M with γ ij , even though the light ray in the four-dimensional spacetime follows the null geodesic.
In a stationary and axisymmetric spacetime, it is always possible to find out coordinates, such that g 0i can vanish and a i = 0. In this case, the photon orbit is considered a spatial geodesic curve in (3) M.
We study axisymmetric cases, which allow g 0i = 0. Therefore, geodesic curvature κ g does not always vanish in the photon orbit in the Gauss-Bonnet theorem, because the geodesic curvature κ g for a photon orbit is owing to the gravitomagnetic effect. This non-vanishing κ g for the photon orbit leads to a crucial difference from the SSS case [33,34].

E. Geodesic curvature of a photon orbit
Eq. (49) is rearranged to be in the tensor form as where T and T ′ are corresponding to e k and a j , respectively.
In this paper, the acceleration vector of the photon orbit depends on β i . Hence, the geodesic curvature for the photon orbit also depends on it. A non-vanishing integral of the geodesic curvature along the light ray appears in the Gauss-Bonnet theorem Eq. (8).
Substituting Eq. (57) into a i in Eq. (62) leads to where we used γ ij e i e j = 1 and γ ij e i N j = 0. The unit vector normal to the equatorial plane is where the upward direction is chosen without loss of generality.

F. Geodesic curvature of a circular arc segment
In a flat space, the geodesic curvature κ of the circular arc segment of radius R is obtained The geodesic curvatureκ g of a circular arc segment of radius R c = R ∞ is obtained as where the radius R c is sufficiently larger than r R and r S , and the circular arc segment is in the asymptotically flat region.
Eq.(44) becomes dℓ 2 = dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), because we assume an asymptotically flat spacetime. Hence, the line element in the path integral ofκ g is obtained as where we choose θ = π/2 and r = R c for the circular arc segment.
Therefore, the path integral ofκ g in Eq.(45) is rewritten as where we denote the angular coordinate values of the receiver and the source as φ R and φ S , respectively.

G. Impact parameter and light rays
By using Eq. (39), we study the orbit equation on the equatorial plane. The Lagrangian for a photon in the equatorial plane is obtained aŝ where the dot denotes the derivative with respect to the affine parameter and the functions A(r), B(r), D(r), H(r) mean, to be rigorous, A(r, π/2), B(r, π/2), D(r, π/2), H(r, π/2) respectively.
The metric (or the LagrangianL in the 4-dimensional spacetime) is independent of t and φ. Therefore, Then, associated with the two Killing vectors ξ µ = (1, 0, 0, 0) andξ µ = (0, 0, 0, 1), respectively, where k µ = dx µ dℓ is the vector tangent to the light ray in the four-dimensional spacetime. There are two constants of motion where E denotes the energy of the photon and L means the angular momentum of the photon. The impact parameter of the photon is defined as In terms of the impact parameter b,L = 0 can be considered as the orbit equation where we used Eq.(39). By introducing u ≡ 1/r, we rewrite the orbit equation as where F (u) is We examine the angles (Ψ R , Ψ S in figure 10) at the receiver position and the source one.
The unit vector tangent to the photon orbit in (3) M is e i . Its components on the equatorial plane are expressed as where ξ satisfies This can be derived also from γ ij e i e j = 1 by using Eq. (77).
In the equatorial plane, the unit radial vector is where the outgoing direction is chosen for a sign convention.
By using the inner product between e i and R i , therefore, we define the angle as where Eqs. (80), (81) and (82) are used. This is rewritten as where Eq. (77) is used. We should note that sin Ψ in Eq. (84) is more useful in practical calculations, because it needs only the local quantities. On the other hand, cos Ψ by Eq.
(83) needs the derivative as dr/dφ. In addition, the domain of this Ψ is 0 ≤ Ψ ≤ π and hence sin Ψ is always positive.
By substituting r R and r S into r of Eq.(84), we obtain sin Ψ R and sin Ψ S , respectively.
We note that the range of the principal value of y = arcsin x is − π 2 ≤ y ≤ π 2 as usual. However, the range of Ψ R (Ψ S ) is 0 ≤ Ψ R (Ψ S )≤ π. By using the usual principal value, Eq.(84) for (Ψ R ) and (Ψ S ) become respectively, because Ψ R is an acute angle and Ψ S is an obtuse angle as shown by figure 10.

H. Gravitational deflection light in the axisymmetric case
We define rearranged as Here, dℓ is positive when the photon is in the prograde motion, whereas it is negative for the retrograde case. Eq. (88) means that α is coordinate-invariant for the axisymmetric case. Up to now, we do not use any equation for gravitational fields. Therefore, the above discussion and results still stand not only in the theory of general relativity but also in a general class of metric theories of gravity, only if the light ray in the four-dimensional spacetime is a null geodesic.

A. Kerr spacetime and γ ij
In this section, we focus on the weak deflection of light in the Kerr spacetime as an axisymmetric example. Kerr metric in the Boyer-Lindquist form is expressed as where Σ and ∆ are defined as Using the Gauss Bonnet theorem, the deflection angle of light in the Kerr spacetime was calculated for the asymptotic source and receiver by Werner [61]. However, his method based on the osculating metric is limited within the asymptotic case. Later, Ono et al.
developed a different approach using the Gauss-Bonnet theorem that enables to calculate the deflection angle for the finite distance case in the Kerr spacetime [37].
By using Eqs. (42) and (43), the generalized optical metric γ ij and the gravitomagnetic term β i for the Kerr metric are obtained as Note that γ ij has no linear terms in the Kerr spin parameter a, because only g 0i in g µν has a linear term in a and g 0i ∝ H contributes to γ ij through a quadratic term g 0i g 0j ∝ H 2 as shown by Eq. (42).
In order to calculate the Gaussian curvature K of the equatorial plane, the geodesic curvature κ g of the light ray and the geodesic curvatureκ g of the circular arc of an infinite radius and the angles Ψ R and Ψ S , we use two approximations for the weak field and slow rotation, where M and a play a role of book-keeping parameters though they are dimensional quantities.
By using Eq.(77), we obtain the orbit equation where the weak-field and slow-rotation approximations are used in the last line. There are no M-squared terms in the last line. The orbit equation becomes We solve iteratively Eq.(95). In order to find the zeroth order solution, we solve the truncated Eq.(95) The zeroth order solution for this equation is where we use du dφ φ=π/2 = 0 as the boundary condition. This condition means that the closest approach of the photon orbit is expressed as r = r 0 = 1/u 0 , φ = π/2. We assume that the linear-order solution with M is u = sin φ b + u 1 (φ)M. In order to obtain u 1 (φ), we substitute this expression of u into the Eq.(95) with terms linear in M du dφ where we used the boundary condition mentioned above. The solution with a is in a form of u = sin φ b + M b 2 (1 + cos 2 φ) + u 2 (φ)a . Since Eq.(95) does not include any linear term in a, we find u 2 (φ) = 0. The solution with aM is u = sin φ b + M b 2 (1 + cos 2 φ) + u 3 (φ)aM. We substitute this solution into Eq.(95) Hence, u 3 (φ) is obtained as Bringing the above results together, the iterative solution of Eq.(95) is expressed as Next, we solve Eq.(102) for φ. We obtain φ as where we can choose the domain of φ to be −π ≤ φ < π without loss of generality. In the following, the range of the angular coordinate value φ S at the source point is − π 2 ≤ φ S < π 2 and the range of the angular coordinate value φ R at the receiver point is |φ R | > π 2 . We find |bu| < 1, because the square root in Eq.(103) must be real and nonzero, and the value of b and u are positive. Therefore, bu satisfies 0 < bu < 1 in our calculation.

B. Gaussian curvature on the equatorial plane
Let us explain how to compute the Gaussian curvature by using Eq. (46). In the Kerr case, it becomes where the weak-field and slow-rotation approximations are used in the last line.
Next, we discuss the area element on the equatorial plane by using Eq. (47). In the Kerr case, the area element of the equatorial plane is expressed as By using Eqs. (104) and (105), the surface integral of the Gaussian curvature in Eq. (88) is performed as where the weak-field and slow-rotation approximations are used in the last line. We stress that the terms of a n M (n ≥ 2) do not exist in this expression.
The line element for the path integral by Eq.(67) becomes where Eq.(102) was used for a relation between r and φ.
By using (107) and Eqs.(108), the path integral of κ g in Eq.(88) is performed as Here, we assumed dℓ > 0, such that the orbital angular momentum can be parallel with the spin of the black hole and we used a linear approximation of the photon orbit as 1/r = u = In the retrograde case, dℓ becomes negative and the magnitude of the above path integral thus remains the same but the sign of the integral is opposite.
The displacement of the angular coordinate φ in Eq.(87) is computed as where the orbit equation by Eq.(78) was made use of. We substitute Eq.(95) into F (u) in Eq.(110) to obtain where the prograde case is assumed. In the retrograde motion, the sign of the linear term in a is opposite. In Eq.(111), the impact parameter b is rewritten in terms of the closest approach u 0 for the integration from u S (or u R ) to u 0 . Namely, Eq.(95) tells us the relation between the impact parameter b and the inverse of the closest approach u 0 as b = u −1 0 + M − 2aMu 0 + O(M 2 u 0 , a 2 u 0 ) in the weak field and slow rotation approximations. By making use of this relation, Eq. (111) is rearranged as The first line of this equation recovers Eq. (32) of Reference [33].

E. Ψ parts
In the Kerr spacetime by Eq.(89), Eq. (85) is and Eq.(86) is calculated as where r R = 1/u R , r S = 1/u S and we used the weak-field and slow-rotation approximations.
By combining Eqs. (113) and (114), we obtain Ψ R and Ψ S as By combining these relations, we obtain the Ψ part in Eq.(87) as

F. Deflection of light in Kerr spacetime
On the equatorial plane in the Kerr spacetime, the deflection angle of light is described by Eq.(87) and Eq.(88). Let us examine whether the two results agree with each other.
First, we substitute Eqs. (112) and (116) into Eq. (87). We obtain the deflection angle of light as where the prograde orbit of light is assumed. For the retrograde motion, we obtain Next, we substitute Eqs.(106) and (109) into Eq.(88). Then, we obtain the deflection angle of light in the prograde motion as and the deflection angle for the retrograde case as Note that a 2 terms in the deflection angle in Eq.(87) cancel out thanks to Eq.(88).
Here, we consider the limit as u R → 0 and u S → 0. In this limit, we get This shows that Eqs. (117) and (118) agree with the asymptotic deflection angles that are known in earlier works [4,[63][64][65]67].
If we wish to consider the deflection angle of light in a case where the receiver point is closer to the source point than the closest approach point, Eqs. (117) and (118) become If we wish to consider the deflection angle of light in such a case that the source point is closer to the receiver than the closest approach point, Eqs.(117) and (118) become

G. Finite-distance corrections
In the previous subsections so far we discussed an effect of the spin of the lens object to the deflection of light. In particular, we do not require that the receiver and the source are located at the infinity. The finite-distance correction to the deflection angle of light is defined as δα. This is the difference between the asymptotic deflection angle α ∞ and the deflection angle for the finite distance case. Namely, Equations (117) and (118) tell us the magnitude of the finite-distance correction to the gravitomagnetic bending angle due to the spin. The result is where bu R , bu S < 1 is assumed, J ≡ aM denotes the spin angular momentum of the lens and the subscript GM means the gravitomagnetic part. We introduce the dimensionless spin parameter as s ≡ a/M. Hence, Eq. (124) is rearranged as This implies that δα GM is of the same order as the second post-Newtonian effect (with the dimensionless spin parameter).
The second-order Schwarzschild contribution to α is 15πM 2 /4b 2 . This contribution can be obtained also by using the present method, especially by using a relation between b and r 0 in M 2 in calculating φ RS . Appendix A provides detailed calculations at the second order of M and a. We explain detailed calculations for the integrals of K and κ g in the present formulation. Note that δα GM in the above approximations is free from the impact parameter b. We can see this fact from Figure 11 and Figure 9 below.

H. Possible astronomical applications
What are possible astronomical applications? As a first example, we consider the Sun, in which its higher multipole moments are ignored for its simplicity. Its spin angular momentum denoted as J ⊙ is ∼ 2 × 10 41 m 2 kg s −1 [68]. This means GJ ⊙ c −2 ∼ 5 × 10 5 m 2 , for which the dimensionless spin parameter becomes s ⊙ ∼ 10 −1 .
Here, our assumption is that a receiver on the Earth observes the light deflected by the Sun, while the distant source is safely in the asymptotic region. For the light ray passing near the Sun, Eq. (125) allows us to make an order-of-magnitude estimation of the finite-distance correction. The result is where 4M ⊙ /R ⊙ ∼ 1.75 arcsec. ∼ 10 −5 rad., M odot means the solar mass and R ⊙ denotes the solar radius. This correction is nearly a pico-arcsecond. Therefore, the correction is beyond the reach of present and near-future technology [69,70]. See Figures 12 and 13 for the deflection angle with finite-distance corrections for the prograde motion and retrograde one, respectively, where we choose r S ∼ 1.5 × 10 8 km and r R ∼ ∞. The finite-distance correction reduces the deflection angle of light. As the impact parameter b increases, the finite-distance correction also increases.
As a second example, we discuss Sgr A * that is located at our galactic center. This object is a good candidate for measuring the strong gravitational deflection of light. The distance to the receiver is much larger than the impact parameter of light. On the other hand, some of source stars may live in our galactic center.
For Sgr A * , Eq. (125) becomes where we assume that the mass of the central black hole is M ∼ 4 × 10 6 M ⊙ . This correction is nearly at a sub-microarcsecond level. Therefore, it is beyond the capability of present technology (e.g. [31]).
See Figure 9 for the finite-distance correction due to the source location.

A. Rotating Teo wormhole and optical metric
In this section, we consider a rotating Teo wormhole [71] in order to examine how our method can be applied to a wormhole spacetime. The spacetime metric for this wormhole is where we denote Here, b 0 means the throat radius of this wormhole,ā is corresponding to the spin angular momentum, and d is a positive constant.
For the rotating Teo wormhole Eq.(128), the components of the generalized optical metric are [38] γ ij dx i dx j = r 7 (r − b 0 ) r 4 − 4ā 2 sin 2 θ (16dā 2 cos 2 θ + r) 2 dr 2 + r 6 r 4 − 4ā 2 sin 2 θ dθ 2 + r 10 sin 2 θ Here, γ ij is not the induced metric in the ADM formulation. The components of β i are obtained as In this section, we restrict ourselves within the equatorial plane, namely θ = π/2. On the equatorial plane, the constant d in the metric always vanish, because d is always associated with cos θ.
We employ the same way for the Kerr case, we first derive the orbit equation on the equatorial plane from Eq.(77) as where b denotes the impact parameter of the light ray and we use the weak field and slow rotation approximations in the last line. There are no b 0 squared terms in the last line. The orbit equation thus becomes This equation is iteratively solved as Solving Eq.(135) for φ S and φ R , we obtain φ S and φ R as

B. Gaussian curvature
In the weak field approximation, the Gaussian curvature of the equatorial plane is where we use sin (137) and (136) in the last line.

C. Geodesic curvature of photon orbit
We study the geodesic curvature of the photon orbit on the equatorial plane in the stationary and axisymmetric spacetime by using the generalized optical metric. It generally becomes [37] κ g = − 1 γγ θθ β φ,r .
In the Teo wormhole, this expression is rearranged as We compute the path integral of the geodesic curvature of the photon orbit. The detailed calculations and result are for the retrograde orbit of the photon. In the last line, we used sin φ R = bu R +O(āb −2 , b 0 b −1 ) and sin φ S = bu S + O(āb −2 , b 0 b −1 ) from Eq. (135). The above result becomes 4ā/b 2 , as r R → ∞ and r S → ∞. The sign of the right hand side in Eq. (142) is opposite, if the photon is in the prograde motion.
The rotating Teo wormhole is an asymptotically flat spacetime as seen from Eq.(128).
Therefore, the integral of the geodesic curvature of the circular arc segment with an infinite radius can be expressed simply as φ RS . By using Eqs. (136) and (137), φ RS is obtained as and Eq.(86) becomes where the slow rotation approximation is not needed. Therefore, we obtain Ψ R and Ψ S as where we used the slow rotation approximation.

F. Deflection angle of light
We combine Eqs. (139) and (142) to obtain the deflection angle of light in the prograde orbit as The deflection angle of the retrograde light is Next, by using Eqs. (143), (146) and (147), we obtain the deflection angle of the prograde light as The deflection angle of light in the retrograde orbit is The deflection of light in the prograde (retrograde) orbit is weaker (stronger) with increasing the angular momentum of the Teo wormhole. The reason is as follows. The local inertial frame in which the light travels at the light speed c in general relativity moves faster (slower). Hence, the time-of-flight of light becomes shorter (longer). On the light propagation A similar explanation is done by using the dragging of the inertial frame also by Laguna and Wolsczan [73]. They discussed the Shapiro time delay. The expression of the deflection angle of light by a rotating Teo wormhole is similar to that by Kerr black hole.
This implies that it is hard to distinguish Kerr black hole from rotating Teo wormhole by the gravitational lens observations.
In Eqs. (150) and (151), the source and receiver can be located at finite distance from the wormhole. In the limit as r R → ∞ and r S → ∞, Eqs. (148) and (149) become They are in complete agreement with Eqs. (39) and (56) in Jusufi andÖvgün [72], where they restrict themselves within the asymptotic source and receiver (r R → ∞ and r S → ∞).

G. Finite-distance corrections in the Teo wormhole spacetime
To be precise, we define the finite-distance correction to the deflection angle of light as the difference between the asymptotic deflection angle α ∞ and the deflection angle for the finite distance case. It is denoted as δα.
We consider the following situation. An observer on the Earth sees the light deflected by the solar mass. The source o light is located in a practically asymptotic region. In other words, we choose b 0 = M ⊙ ,ā = J ⊙ , r R ∼ 1.5 × 10 8 km, r S ∼ ∞. See Figure 14 for the finite-distance correction due to the impact parameter b. In Figure 14, the green curve means the difference between the asymptotic bending angle and the deflection angle with finite-distance corrections, the blue curve denotes the asymptotic deflection angle and the orange curve is the deflection angle with finite-distance corrections by the rotating Teo wormhole. The deflection angle is decreased by the finite-distance correction. If the impact parameter b increases, the finite-distance correction also increases.
See also Figure 15 for numerical calculations of the finite-distance correction due to the impact parameter b. In Figure 15, the blue curve is the deflection angle with finite-distance correction by a Kerr black hole and the red curve is the deflection angle with finite-correction by a rotating Teo wormhole. The deflection of light is stronger in a Kerr black hole case for the chosen values.

X. SUMMARY
In this paper, we provided a brief review of a series of works on the deflection angle of light for a light source and receiver in a non-asymptotic region. [33,34,37,38]. The validity and usefulness of the new formulation come from the GB theorem in differential geometry. First, we discussed how to define the gravitational deflection angle of light in a static, spherically symmetric and asymptotically flat spacetime, for which we assume the finite-distance source and receiver. We examined whether our definition is invariant geometrically by using the GB theorem. By using our definition, we carefully computed finite-distance corrections to the light deflection in Schwarzschild spacetime. We considered both cases of weak deflection and strong one. Next, we extended the definition to stationary and axisymmetric spacetimes.
This extension allows us to compute finite-distance corrections for Kerr black holes and rotating Teo wormholes. We verified that these results are consistent with the previous works in the infinite-distance limit. We mentioned also the finite-distance corrections to the light deflection by Sagittarius A * . It is left as future work to apply the present formulation to other interesting spacetime models and also to extend it to a more general spacetime structure.
formed as where we use, in the second line, an iterative solution for the orbit equation by Eq. (77) in the Kerr spacetime.
Next, we study the geodesic curvature. On the equatorial plane, we find Note that a 2 terms do not exist. Therefore, we obtain where we use sin φ S = √ r S 2 − b 2 /r S + O(M/r S ) and sin φ R = − √ r R 2 − b 2 /r R + O(M/r R ).
By combining Eqs. (A3) and (A5), we obtain Note that a 2 terms and a 3 ones do not appear in α for the finite distance situation as well as in the infinite distance limit. If we assume the infinite distance limit u R , u S → 0, Eq. (A6) This agrees with the known results, especially on the numerical coefficients at the order of M 2 and aM.