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Equivalence between definitions of the gravitational deflection angle of light for a stationary spacetime

Kaisei Takahashi [email protected]    Ryuya Kudo [email protected]    Keita Takizawa [email protected]    Hideki Asada [email protected] Graduate School of Science and Technology, Hirosaki University, Aomori 036-8561, Japan
Abstract

The Gibbons-Werner-Ono-Ishihara-Asada method for gravitational lensing in a stationary spacetime has been recently reexamined [Huang and Cao, arXiv:2306.04145], in which the gravitational deflection angle of light based on the Gauss-Bonnet theorem can be rewritten as a line integral of two functions HH and TT. The present paper proves that the Huang-Cao line integral definition and the Ono-Ishihara-Asada one [Phys. Rev. D 96, 104037 (2017)] are equivalent to each other, whatever asymptotic regions are. A remark is also made concerning the direction of a light ray in a practical use of these definitions.

pacs:
04.40.-b, 95.30.Sf, 98.62.Sb

I Introduction

The gravitational deflection of light plays a crucial role in modern cosmology and gravitational physics SEF ; Petters ; Dodelson ; Keeton ; Will , where a conventional formulation of the gravitational deflection of light assumes the weak deflection of light in a quasi-Newtonian region that can be treated as a perturbation around Minkowski background.

Although the conventional formulation is practically useful in many situations SEF ; Petters ; Dodelson ; Keeton , it is limited. In order to discuss a more geometrical aspect of the gravitational deflection of light, Gibbons and Werner (GW) GW proposed a use of the Gauss-Bonnet theorem (GBT) Carmo ; Oprea . The GW method was initially applied to a static and spherically symmetric (SSS) spacetime GW , for which the deflection angle of light can be defined as a surface integral of the Gaussian curvature of the equatorial plane in the optical geometry. Later, Ishihara et al. generalized the GW idea for a case that an observer and source are located at a finite distance from a lens object Ishihara2016 . It was extended also to the strong deflection limit Ishihara2017 . Without assuming the asymptotic flatness, eventually, Takizawa et al. proved the equivalence between the two definitions by GW and Ishihara et al. for SSS spacetimes Takizawa2020 .

The GW method was extended by Werner to a stationary axisymmetric (SAS) case Werner2012 . This still employs asymptotically flat regions, at which the angle can be defined in a Euclid space. Furthermore, Ono, Ishihara and Asada (OIA) developed a formulation for a non-asymptotic observer and source in SAS spacetimes Ono2017 . These works assumed asymptotically flat regions. In the OIA approach, an alternative definition of the deflection angle of light was proposed in terms of a linear combination of three functions.

It was proven Ono2017 that the deflection angle of light in the OIA approach is equivalent to the GW-type definition as a two-dimensional integral of the Gaussian curvature, if the SAS spacetime has asymptotically flat regions. See e.g. Eqs. (29) and (30) in Ono2017 .

Very recently, Huang and Cao (HC) have reexamined the Gibbons-Werner-Ono-Ishihara-Asada (GWOIA) method for SAS spacetimes HC . They have found that the GW definition as a two-dimensional integral can be simplified as a line integral of two functions HH and TT. See Eq. (44) in HC .

Can the OIA definition be related with the HC line-integral definition without assuming the asymptotic flatness? The main purpose of the present paper is to prove that the two definitions are equivalent to each other for SAS spacetimes, whatever asymptotic regions are.

This paper is organized as follows. For its simplicity, first we consider a SSS spacetime to prove the equivalence in Section II. Section III extends the equivalence to SAS cases. Section IV summarizes this paper. Throughout this paper, we use the unit of G=c=1G=c=1.

II Static and spherically symmetric case

This section focuses on a SSS spacetime. The line element can be written as Ishihara2017

d​s2=−A​(r)​d​t2+B​(r)​d​r2+C​(r)​(d​θ2+sin2⁡θ​d​ϕ2).\displaystyle ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)(d\theta^{2}+\sin^{2}\theta d\phi^{2}). (1)

In the rest of this section, we assume A​(r),B​(r),C​(r)>0A(r),B(r),C(r)>0. If the spacetime represents a black hole, we study the outside of a black hole horizon.

Without loss of generality, a photon orbit can be chosen as the equatorial plane (θ=π/2\theta=\pi/2) because of the spherical symmetry. From the null condition, we obtain Ishihara2017

d​t2\displaystyle dt^{2} =γi​j​d​xi​d​xj\displaystyle=\gamma_{ij}dx^{i}dx^{j}
=B​(r)A​(r)​d​r2+C​(r)A​(r)​d​ϕ2,\displaystyle=\frac{B(r)}{A(r)}dr^{2}+\frac{C(r)}{A(r)}d\phi^{2}, (2)

which defines the optical metric on the equatorial plane AK ; GW ; Ishihara2016 . We examine a light ray with an impact parameter bb, which is related to the specific energy EE and specific angular momentum LL of a photon as b≡L/Eb\equiv L/E. The null condition of a photon orbit becomes Ishihara2017

(d​rd​ϕ)2+C​(r)B​(r)=C​(r)2b2​A​(r)​B​(r).\displaystyle\left(\frac{dr}{d\phi}\right)^{2}+\frac{C(r)}{B(r)}=\frac{C(r)^{2}}{b^{2}A(r)B(r)}. (3)

As a solution to Eq. (3), rr along the light ray is a function of ϕ\phi.

In the optical geometry, the angle from the radial direction to the light ray tangent is denoted as Ψ\Psi, which is expressed in terms of the metric components as Ishihara2016

cos⁡Ψ\displaystyle\cos\Psi =b​A​(r)​B​(r)C​(r)​d​rd​ϕ,\displaystyle=\frac{b\sqrt{A(r)B(r)}}{C(r)}\frac{dr}{d\phi}, (4)
sin⁡Ψ\displaystyle\sin\Psi =b​A​(r)C​(r).\displaystyle=\frac{b\sqrt{A(r)}}{\sqrt{C(r)}}. (5)

See also Figure 1.

Refer to caption
Figure 1: Ψ\Psi as the angle from the radial direction to the light ray tangent. ΨO\Psi_{O} and ΨS\Psi_{S} are Ψ\Psi at the observer and source, respectively.

By using Ψ\Psi and ϕ\phi, Ishihara et al. Ishihara2016 ; Ishihara2017 defines the deflection angle of light αI\alpha_{I} as

αI≡ΨO−ΨS+ϕO​S,\displaystyle\alpha_{I}\equiv\Psi_{O}-\Psi_{S}+\phi_{OS}, (6)

where ΨO\Psi_{O} and ΨS\Psi_{S} are Ψ\Psi at the observer (O) and source (S), respectively, ϕO​S=∫SO𝑑ϕ\phi_{OS}=\int_{S}^{O}d\phi is the longitude from S to O, and ΨO\Psi_{O} equals to ΨR\Psi_{R} in the notation of Ishihara2016 ; Ishihara2017 . For the later convenience, this definition is rewritten as

αI=∫SO𝑑ϕ​(d​Ψd​ϕ+1).\displaystyle\alpha_{I}=\int_{S}^{O}d\phi\left(\frac{d\Psi}{d\phi}+1\right). (7)

By differentiating Eq. (5) with respect to ϕ\phi, we obtain

d​Ψd​ϕ=C​(r)A​(r)​B​(r)​dd​r​A​(r)C​(r),\displaystyle\frac{d\Psi}{d\phi}=\frac{C(r)}{\sqrt{A(r)B(r)}}\frac{d}{dr}\sqrt{\frac{A(r)}{C(r)}}, (8)

where we use Eq. (4) and |d​r/d​ϕ|<+∞|dr/d\phi|<+\infty for a non-radial photon orbit.

See Eq. (4.25) in Reference HC for the HC definition of the deflection angle of light. The HC definition is

αH​C=∫SO𝑑ϕ​[1+H+T].\displaystyle\alpha_{HC}=\int^{O}_{S}d\phi[1+H+T]. (9)

For the SSS case, HH and TT simply become

H\displaystyle H ≡−12​γ​d​(γϕ​ϕ)d​r\displaystyle\equiv-\frac{1}{2\sqrt{\gamma}}\frac{d(\gamma_{\phi\phi})}{dr}
=−A​(r)2​B​(r)​C​(r)​dd​r​(C​(r)A​(r)),\displaystyle=-\frac{A(r)}{2\sqrt{B(r)C(r)}}\frac{d}{dr}\left(\frac{C(r)}{A(r)}\right), (10)
T\displaystyle T =0,\displaystyle=0, (11)

for γ≡det(γi​j)\gamma\equiv\det(\gamma_{ij}).

The HC definition is thus reduced to

αH​C≡∫SO𝑑ϕ​(1+H).\displaystyle\alpha_{HC}\equiv\int_{S}^{O}d\phi\left(1+H\right). (12)

By direct calculations for Eqs. (8) and (10), we find

H\displaystyle H =C​(r)​A′​(r)−C′​(r)​A​(r)2​A​(r)​B​(r)​C​(r)\displaystyle=\frac{C(r)A^{\prime}(r)-C^{\prime}(r)A(r)}{2A(r)\sqrt{B(r)C(r)}}
=d​Ψd​ϕ,\displaystyle=\frac{d\Psi}{d\phi}, (13)

where the prime denotes the differentiation with respect to rr. Therefore, Eq. (7) equals to Eq. (12). In the SSS case, the two definitions are thus equivalent to each other.

III Stationary and axisymmetric case

In this section, we consider a SAS spacetime. The line element can be written as Ono2017

d​s2=\displaystyle ds^{2}= −A​(r,θ)​d​t2+B​(r,θ)​d​r2+C​(r,θ)​d​θ2+D​(r,θ)​d​ϕ2\displaystyle-A(r,\theta)dt^{2}+B(r,\theta)dr^{2}+C(r,\theta)d\theta^{2}+D(r,\theta)d\phi^{2}
−2​W​(r,θ)​d​t​d​ϕ.\displaystyle-2W(r,\theta)dtd\phi. (14)

The null condition is rewritten in a form as Ono2017

d​t=γi​j​d​xi​d​xj+βi​d​xi,\displaystyle dt=\sqrt{\gamma_{ij}dx^{i}dx^{j}}+\beta_{i}dx^{i}, (15)

where

γi​j​d​xi​d​xj=\displaystyle\gamma_{ij}dx^{i}dx^{j}= B​(r,θ)A​(r,θ)​d​r2+C​(r,θ)A​(r,θ)​d​θ2\displaystyle\frac{B(r,\theta)}{A(r,\theta)}dr^{2}+\frac{C(r,\theta)}{A(r,\theta)}d\theta^{2}
+A​(r,θ)​D​(r,θ)+[W​(r,θ)]2[A​(r,θ)]2​d​ϕ2,\displaystyle+\frac{A(r,\theta)D(r,\theta)+[W(r,\theta)]^{2}}{[A(r,\theta)]^{2}}d\phi^{2}, (16)
βi​d​xi=\displaystyle\beta_{i}dx^{i}= −W​(r,θ)A​(r,θ)​d​ϕ.\displaystyle-\frac{W(r,\theta)}{A(r,\theta)}d\phi. (17)

In the rest of this section, we focus on the equatorial plane (θ=π/2\theta=\pi/2) for a photon orbit, where we assume A​(r,π/2),B​(r,π/2),D​(r,π/2)>0A(r,\pi/2),B(r,\pi/2),D(r,\pi/2)>0 and a local reflection symmetry with respect to θ=π/2\theta=\pi/2 as implicitly assumed in Reference Ono2017 ; HC . Henceforth, A​(r,π/2),B​(r,π/2),D​(r,π/2),W​(r,π/2)A(r,\pi/2),B(r,\pi/2),D(r,\pi/2),W(r,\pi/2) are denoted simply as A,B,D,WA,B,D,W, respectively.

On the equatorial plane in the SAS spacetime, the OIA definition of the deflection angle of light is Ono2017

αO​I​A≡ΨO−ΨS+ϕO​S,\displaystyle\alpha_{OIA}\equiv\Psi_{O}-\Psi_{S}+\phi_{OS}, (18)

where Ψ\Psi in the SAS metic satisfies

cos⁡Ψ\displaystyle\cos\Psi =BA​A​(A​b+W)A​D+W2​d​rd​ϕ,\displaystyle=\sqrt{\frac{B}{A}}\frac{A(Ab+W)}{AD+W^{2}}\frac{dr}{d\phi}, (19)
sin⁡Ψ\displaystyle\sin\Psi =A​b+WA​D+W2.\displaystyle=\frac{Ab+W}{\sqrt{AD+W^{2}}}. (20)

By differentiating Eq. (20) with respect to ϕ\phi, we obtain

d​Ψd​ϕ=A​D+W2A​B​((A​b+W)′A​b+W−(A​D+W2)′2​(A​D+W2)),\displaystyle\frac{d\Psi}{d\phi}=\sqrt{\frac{AD+W^{2}}{AB}}\left(\frac{(Ab+W)^{\prime}}{Ab+W}-\frac{(AD+W^{2})^{\prime}}{2(AD+W^{2})}\right), (21)

where we use Eq. (19) and |d​r/d​ϕ|<+∞|dr/d\phi|<+\infty for a non-radial orbit.

The HC definition in the SAS case is HC

αH​C≡∫SO𝑑ϕ​(1+H+T),\displaystyle\alpha_{HC}\equiv\int_{S}^{O}d\phi\left(1+H+T\right), (22)

where HH and TT are defined as

H\displaystyle H ≡−12​γ​d​(γϕ​ϕ)d​r,\displaystyle\equiv-\frac{1}{2\sqrt{\gamma}}\frac{d(\gamma_{\phi\phi})}{dr}, (23)
T\displaystyle T ≡−d​(βϕ)d​r​1γϕ​ϕ(d​rd​ϕ)2+1γr​r.\displaystyle\equiv-\frac{d(\beta_{\phi})}{dr}\sqrt{\frac{1}{\gamma_{\phi\phi}}\left(\frac{dr}{d\phi}\right)^{2}+\frac{1}{\gamma_{rr}}.} (24)

In terms of the SAS metric components, HH and TT become

H=\displaystyle H= A​D+W2A​B​A′​D−D′​A−2​W​W′+2​W2​A′​A−12​(A​D+W2),\displaystyle\sqrt{\frac{AD+W^{2}}{AB}}\frac{A^{\prime}D-D^{\prime}A-2WW^{\prime}+2W^{2}A^{\prime}A^{-1}}{2(AD+W^{2})}, (25)
T=\displaystyle T= A​D+W2A​B​W′−W​A′​A−1A​b+W.\displaystyle\sqrt{\frac{AD+W^{2}}{AB}}\frac{W^{\prime}-WA^{\prime}A^{-1}}{Ab+W}. (26)

By combining Eqs. (21) (25) and (26), one can show

H+T=d​Ψd​ϕ.\displaystyle H+T=\frac{d\Psi}{d\phi}. (27)

Therefore, αO​I​A=αH​C\alpha_{OIA}=\alpha_{HC}.

Before closing this section, we mention the direction of a photon orbit. The sign convention of ΨO\Psi_{O}, ΨS\Psi_{S} and ϕO​S\phi_{OS} in this paper is counterclockwise (See also Figure 1). Hence, we should pay attention to the sign convention when we wish to distinguish prograde and retrograde motion. This issue seems a bit obscure in the HC line-integral definition, because HH and TT in Eqs. (23) and (24) are functions of the metric components and hence they do not directly manifest the direction of a photon (e.g. prograde or retrograde). The sign of A​b+WAb+W in TT of Eq. (26) can distinguish prograde and retrograde.

IV Summary

We proved the equivalence between the OIA and HC definitions without assuming any property of the asymptotic regions in SAS cases, for which the GW-type definition also is equivalent to the HC one HC . By combining the two results, the three definitions by GW, OIA and HC GW ; Ono2017 ; HC are equivalent to each other, whatever asymptotic regions are.

The essential part of the present proof relies upon the photon orbit but not upon any two-dimensional integration domain. This point agrees with the HC finding that the deflection angle in the Gauss-Bonnet method is independent of integration domains if the photon orbit is fixed HC . The present proof thus deepens our understanding of the GBT-inspired definitions GW ; Werner2012 ; Ishihara2016 ; Ishihara2017 ; Takizawa2020 ; Ono2017 ; HC . Further study along this direction is left for future.

Acknowledgements.
We are grateful to Marcus Werner for stimulating conversations. We thank Yuuiti Sendouda and Ryuichi Takahashi for the useful conversations. This work was supported in part by Japan Science and Technology Agency (JST) SPRING, Grant Number, JPMJSP2152 (K.T.), and JPMJSP2152 (R.K.) and in part by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research, No. 20K03963 (H.A.).

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