Weak deflection gravitational lensing for photons coupled to Weyl tensor in a Schwarzschild black hole

Beyond the Einstein–Maxwell model, electromagnetic field might couple with gravitational field through the Weyl tensor. In order to provide one of the missing puzzles of the whole physical picture, we investigate weak deflection lensing for photons coupled to the Weyl tensor in a Schwarzschild black hole under a unified framework that is valid for its two possible polarizations. We obtain its coordinate-independent expressions for all observables of the geometric optics lensing up to the second order in the terms of ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} which is the ratio of the angular gravitational radius to angular Einstein radius of the lens. These observables include bending angle, image position, magnification, centroid and time delay. The contributions of such a coupling on some astrophysical scenarios are also studied. We find that, in the cases of weak deflection lensing on a star orbiting the Galactic Center Sgr A*, Galactic microlensing on a star in the bulge and astrometric microlensing by a nearby object, these effects are beyond the current limits of technology. However, measuring the variation of the total flux of two weak deflection lensing images caused by the Sgr A* might be a promising way for testing such a coupling in the future.

However, the relativistic images of Sgr A* are extremely faint [49][50][51] and, therefore, exceedingly difficult to detect. As an alternative, the primary and secondary images of weak deflection gravitational lensings are much easier to observe and they have been extensively used in astronomy and cosmology [1][2][3]. Weak deflection lensings can also provide insights on modified theories of gravity [52][53][54] and clues on the interaction between electromagnetic and gravitational fields. In this work, we will study weak deflection lensing for photons coupled to Weyl tensor in a Schwarzschild black hole, which was absent in the literature. By focusing on coordinate-invariant quantities, we obtain all of its geometric optics lensing observables, which include bending angle, image position, magnification, centroid and time delay. These observables are worked out to the second order in the perturbation parameter ε which is the ratio of the angular gravitational radius to angular Einstein radius of the lens. The results are represented in a unified form which is valid for both of two polarization directions of the Weyl coupling.
In Sect. 2, after the unified effective metric for photons coupled to the Weyl tensor in a Schwarzschild black hole with two polarization directions is briefly reviewed, we will derive its light bending angle that is expressed with invariant quantities. The lensing observables, including positions, magnifications and time delay of images, are obtained in Sect. 3 and relations between them are represents in Sect. 4. We work out practical observables of the lensing and investigate its observability for several astrophysical scenarios in Sect. 5. Finally, in Sect. 6, we summarize and discuss our results.

Effective metric
We consider a Schwarzschild black hole with mass M • as the lens, and set the observer and the source in the asymptotically flat region of its spacetime. We assume that it is vacuum outside the lens. When a photon couples to the Weyl tensor in the background of the Schwarzschild black hole, its worldline will no longer follow the null geodesic. However, it was found [39,46] that the geodesic rule can be recovered by taking an effective metric for such a coupling. This metric can be written as where r is the radial coordinate and dΩ 2 = dθ 2 + sin 2 θ dϕ 2 .
The functions A(r ), B(r ) and C(r ) are and where α is a constant with dimension of [Length] 2 characterising strength of the coupling between the photon and the Weyl tensor, m • = G M • /c 2 is the gravitational radius of the Schwarzschild black hole, and s is an constant. We use s to unify the expression of C(r ) for two different polarizations of the photon respectively along l μ (PPL) and m μ (PPM) (see [39,46] for more details): In fact, the results of weak deflection lensing that we obtain in the following parts are also valid when s takes other real numbers with different physical interpretations. Before we perform detailedly and lengthy calculation on the light bending and its resulting lensing observables, it is worth mentioning that the parameterized second-order post-Newtonian formalism for weak deflection lensing established in Refs. [52,53], which is called "Keeton-Petters formalism" for short hereafter, cannot be applied to the spacetime (1) in this paper. Keeton-Petters formalism is valid for a static, spherically symmetric and asymptotically flat spacetime, whose metric is written in the standard Schwarzschild coordinates denoted by overbar with coefficients: [52,53] Here, φ is the Newtonian potential with and a 1,2,3 and b 1,2,3 are dimensionless and numerical parameters. However, after transforming (1) from its current coordinates to the standard Schwarzschild ones, we find that b 1,2,3 depend on the radial coordinate rather than numerical ones, such as The additionalr -dependent terms have to be taken into account when evaluating the integral of the light bending angle (see next subsection for details); however, they are absent in the Keeton-Petters formalism. This issue was also recognized for Solar System tests of a scalar-tensor gravity [55]. One exception is the trivial case of α = 0 in which the metric (1) reduce to the Schwarzschild black hole in the standard coordinates. Therefore, due to the fact that such a formalism cannot be directly employed, we stick to the metric (1) and perform all of the indispensable calculation for proceeding our investigation.

Light bending
For a light ray propagating through spacetime (1), the distance of closet approach r 0 and the impact parameter b of the light ray satisfy the relation as [56] In the scenario of weak deflection lensing, r 0 and b are much larger than m • which leads to the Taylor expanded solution to (10) as with small parameter The first two coefficients b 1 and b 2 are where we define a dimensionless parameter as the details of other b n (n = 3, . . . , 6) can be found in Appendix A. When the coupling vanishes, i.e.,ᾱ = 0, Eq. (11) returns to the one for the Schwarzschild black hole in Einstein's general relativity (GR) [52].
Following the standard procedure, e.g. [56], the bending angle can be obtained as [49,56] which can be written in the form of a series for weak deflection lensing as [52] α with small parameter The coefficients a n for n = 1 and 2 are where the dimensionless parameterᾱ 0 is defined as For testing photons coupled to the Weyl tensor in the Solar System, the leading term ofα(h) ≈ a 1 h was calculated previously [57]; and our result is in agreement with that. Other higher-order coefficients a n (n = 3, . . . , 6) can be found in Appendix A. The deflection angle (17) can go back to the one for the Schwarzschild black hole in GR [52] when α = 0. However, such an expression depends on coordinate of r 0 and should be transformed into an gauge-invariant form. With the help of Eq. (11), we can replace the distance of closet approach r 0 with the impact parameter b and obtain wherê and higher-order coefficientsα n (n = 3, . . . , 6) can be found in Appendix A. It can be easily checked that the gaugeinvariant deflection angle (22) can return to the one for the Schwarzschild black hole in GR [52] when the coupling vanishes.
which will be used to work out the observables.

Image positions
By making the coefficients of , 2 and 3 in (31) vanish, we can find out θ n (n = 0, 1, 2). The first term gives that which leads to the zeroth-order image position for the weak deflection lensing as where It is worth mentioning that the negative solution of θ 0 is neglected due to our convention that the angular position of an image is set to be positive. The positive-and negativeparity images can respectively be found by using β > 0 and β < 0 (see next section for details). If sᾱ > 0, it will make θ 0 bigger than its corresponding values for the Schwarzschild black hole in the absence of such a coupling to the Weyl tensor; if sᾱ < 0, θ 0 will become smaller. Additionally, Eq. (33) itself also imposes a bound onᾱ. In order to ensure that η and resulting θ 0 are real, it demands that For the special case of β = 0 and s = − 1, it impliesᾱ ≤ 1/8 so that α 6 × 10 16 m 2 where b is assumed to be the radius of the Sun by considering the Sun as the lens. Such a specific bound is consistent with but looser than the one obtained by the Solar System test on the deflection of light due to the Sun [57]. The coefficient of 2 in (31) does not depend on β explicitly so that, after substituting (33), we can obtain the firstorder correction to the image position as It can be easily checked that whenᾱ = 0, θ 1 will return to its familiar value for the Schwarzschild black hole in GR [52].
Vanishing the coefficient of 3 in (31) yields the secondorder term as where the factor p and first two coefficients p 0 and p 1 are The higher-order coefficients p n (n = 2, . . . , 5) can be found in Appendix B. If the coupling to the Weyl tensor is absent, θ 2 will have its value as the same as the one for the Schwarzschild black hole [52].

Magnifications
At angular position ϑ, the signed magnification μ is [58] With scaled variables (26), we can have a series of μ expanded in terms of as where the zeroth-order, first-order and second-order terms are In the expression of μ 2 , the factor m and the first two coefficients m 0 and m 1 are and the higher-order coefficients m n (n = 2, . . . 6) can be found in Appendix B. It can be checked that μ will be divergent if sᾱ equals to either −(θ 2 0 + 1)/8 or (θ 2 0 − 1)/8. When α = 0, μ n (n = 0, 1, 2) can also return to their familiar values for the Schwarzschild black hole [52].

Time delay
The time delay is the difference between the light travel time with and without the lens and it can be expressed as where R src and R obs are the radial coordinates of the source and observer and they have the relations as [52] The function T (R) has the form as and it can be integrated and expanded as where the first two terms are and higher-order terms T 3 and T 4 can be found in Appendix B. It is obvious that T 0 is not affected by the coupling to the Weyl tensor, but T 1 has the Shapiro delay term with an additional correction proportional to sᾱ 0 which is consistent with the result obtained for the Solar System test [57]. When α 0 = 0, T n (n = 1, 2, 3) can return to their values for the Schwarzschild black hole [52]. After replacing r 0 with b by using Eq. (11) and substituting (50) and (51) wherê Although it is also possible to obtain the O(ε 2 ) term forτ , that is less vital than the O(ε 2 ) corrections to θ and μ. If sᾱ > 0, it will makeτ 0 bigger than its corresponding value in the absence of the coupling to the Weyl tensor; if sᾱ < 0, τ 0 will become smaller. When such a coupling vanishes,τ 0 andτ 1 can go back to their values in GR [52].

Relations between lensing observables
Considering photons coupled to the Weyl tensor in the Schwarzschild black hole, we can find some relations between lensing observables given in the previous section.

Position relations
With Eq. (33), we can respectively obtain the positive-and negative-parity images at the leading order by specifying β > 0 and β < 0 as which also leads to and It is clear that the value of θ + 0 − θ − 0 is not affected by the coupling to the Weyl tensor and θ + 0 θ − 0 is dependent on the coupling only. When such a coupling vanishes, θ ± 0 and θ + 0 θ − 0 will have their values in GR [52]. If sᾱ > 0, then it will make θ ± 0 and θ + 0 θ − 0 bigger than their corresponding values in the absence of such a coupling to the Weyl tensor; and vice versa.
According to Eqs. (36) and (60), we can have the firstorder corrections to the image positions as They generate two relations that one is which is independent on the angular position of source; and the other is which is source-dependent. Based (37), we can obtain the second-order corrections to the image positions as where the factors P ± and the first two coefficients of P n and P n (n = 0, 1) are and higher-order coefficients P n (n = 2, . . . , 5) and P n (n = 2, 3, 4) can be found in Appendix C. They yield a relation as where the factor s and the first two coefficients s 1 and s 2 are The higher-order coefficients s n (n = 2, 3, 4) can be found in Appendix C. and where the factor M and the first two coefficients M 0 and M 1 are The higher-order coefficients M n (n = 2, 3, 4) can be found in Appendix C. They can give three simple magnification relations as Again, the sign of the magnification denotes the parity of an image so that the absolute value of μ indicate its brightness. At the zeroth-order, the difference between the fluxes of images, |μ + 0 | − |μ − 0 |, equals to the flux of the source without lensing. The relation (83) emerges because both images have the same μ 1 . These relations are immune to the coupling to the Weyl tensor.

Total magnification and centroid
If the two images can not be separated, the observables are the total magnification and magnification-weighted centroid position. The relations about magnifications (82)-(84) leads to the total magnification as The exact cancellation between μ + 1 and μ − 1 [see Eq. (83)] guarantee that μ tot does not have the O( ) term.
The magnification-weighted centroid position is defined by [52] With the results obtained in the previous parts of this section, it can be expanded in the series of as where the zeroth-order, first-order and second-order terms are 0 = |β| In the expression of 2 , the factor S and the first two coefficients S 0 and S 1 are and the higher-order coefficients S n (n = 2, 3, 4) can be found in Appendix C.

Differential time delay
The differential delay between the positive-and negativeparity images is and it has a series form as where the zeroth-order and first-order terms are Since both of them are affected by the coupling to the Weyl tensor, it is theoretically possible to test it by observing the differential time delay between two images (see next section for discussion).

Observational effects
By using the lensing relations found in the previous section, we can obtain the practical observables for the photons coupling to the Weyl tensor in the Schwarzschild black hole. After that, like Ref. [53], we will consider and discuss several astrophysical scenarios.

Practical observables
In order to proceed the investigation, we focus on the zerothorder and first-order lensing effects. The former ones are the observables of the weak deflection limits, and the latter ones might be able to be measured in the near future. To fulfill this purpose, we need convert the scaled variables (β, θ, μ,τ ) to practical observables (B, ϑ, F, τ ). Observables of lensing usually are the positions, fluxes and time delays of the images. The fluxes are connected to the magnifications through the flux of the source, i.e., F i = |μ i |F src . Following the discussion in Ref. [53], we also construct some possibly measurable combinations of observables which are Here, we define that These combinations of observables represent our results about weak deflection lensing for photons coupling to the Weyl tensor in the Schwarzschild black hole. We will investigate the effects of the coupling on these practical observables so that we define following indicators to demonstrate its contributions: where the differences between fluxes are converted into magnitudes. Keeping the leading contributions, we can have their dominant terms as They imply that the polarization s and the strength α for photons coupling to the Weyl tensor cannot be simultaneously determined at least based on its leading observational effects. Before investigating some observational examples numerically, we need to specify the domain of the coupling constant α. In order to ensure that a photon can continuously propagate outside the event horizon of a Schwarzschild black hole, α has to satisfy two theoretical bounds [46,47] that with where M • and M are the mass of the hole and the Sun.
Thus, if it is assumed that the strength of α is irrelevant to the mass of the lens, the observational bound α obs ±1 will be much tighter than the theoretical ones α th ±1 for a supermassive system with M • 10 6 M and α th ±1 will be much more stringent for a stellar system with M • 10 M . Both of these two kinds of bounds will be adopted in the following parts.

The supermassive black hole in the Galactic Center
By monitoring stellar orbits in the Galactic Center, the mass and distance to the supermassive black hole Sgr A* was determined as M • = 4.28 × 10 6 M and d L = 8.32 kpc [59]. Its gravitational radius is m • = 6.32 × 10 9 m = 2.05 × 10 −7 pc, whose angular radius is 5.08 × 10 −6 arcsecond (as) and whose time scale is τ E = 84.3 s.
We consider a source orbiting the Sgr A* with a distance d LS d L so that d S ≈ d L . We also define a scaled distance d * LS = d LS /(1 pc). From the perspective of the observer, if the source can be close enough to Sgr A* in the sky, it can be strongly lensed. The angular Einstein radius is ϑ E = 0.0224 (d * LS ) 1/2 as and the perturbation parameter is = 2.26 × 10 −4 (d * LS ) −1/2 . Therefore, based on Eq. (99), we can make a rough estimation, in which the first-order correction is dropped since it is smaller by 4 orders of magnitude, and find that the angular separation of the two lensed images is larger than When the weak deflection lensing is considered that b m • , P tot,min even for a source with d * LS ∼ 10 −3 (and resulting ε ∼ 7.16 × 10 −3 ) is still larger than the current resolution as low as 50 microarcsecond (μas) achieved by phase referencing optical/infrared interferometry [60]. It means that P tot , ΔP, F tot , ΔF and Δτ can be obtained according to the positions, flux and timing of two separated images.
As a case study, we consider a source at d LS = 10 −3 pc with its β ranging from 10 −3 to 10. We take a wider domain of α belonging to [10 5 , 10 20 ] m 2 which can cover both α obs ±1 and |α th ±1 | for Sgr A*. It can be checked thatᾱ 1.2 × 10 −4 for both s = 1 and − 1 by using the relation of sin ϑ = b/d L and the series solutions to the image positions. Our interest is the contributions on the lensing observables due toᾱ. Estimations [see Eqs. (112)-(117)] suggest when α is sufficiently small that is satisfied for this case, s solely changes the sign of the leading effects ofᾱ. Therefore, we focus on the leading contributions for s = 1, which can nicely approximate the results for s = −1 after changing their signs. Figure 1 shows color-indexed δ P tot , δΔP, δr tot , δΔr tot and δΔτ against α and β in logarithmic scales. The coupling constant α can make the total separation of two images larger than its value in GR, i.e., δ P tot > 0. When α is around α th +1 , δ P tot can reach the level of 0.1 μas, which is still beyond the current ability. δ P tot decreases when α drops; and if α ∼ α obs ±1 , δ P tot will be down to ∼ 10 −9 μas far beyond the capability today. The existence of α can cause the angular difference between two images become smaller than the one in GR, i.e., δΔP < 0. The sub-figure of −δΔP has a similar pattern with its most significant contribution at 10 −3 μas at α ∼ α th +1 . The effects of α on δ P tot and δΔP for Sgr A* are too small to detect in the near future.
The total brightness of two images can be enhanced under the coupling to the Weyl tensor with respect to the one in GR and its difference δr tot can reach the level of ∼ 5 × 10 −4 mag, corresponding to relative flux of about 500 parts per million. Although it is within the current photometric accuracy of a dedicated space telescope, such as the Kepler mission for searching transiting exoplanets [61], the emission of Sgr A* is constantly changing and its variable flux might easily overwhelm this variation of the total flux of two images due to the coupling. But it might still be a promising way to constrain α by measuring δr tot in the future if this noise can be well understood and separated. Additionally, the coupling can reduce the brightness difference between two images, but such as reduction is less than ∼ 10 −6 mag below the current threshold. Finally, effects of α on the differential time delay between two images are represented. When α ∼ α th +1 , δΔτ can reach the level of ∼ 10 −4 s which is practically inaccessible because the exposure time for astronomical imaging is usually much longer than it. In a summary, based on the current limit of the technology and the specific circumstance of the Sgr A*, the impact of the photons coupling to the Weyl tensor in the Schwarzschild black hole is unable to be detected in the observables of weak deflection lensing, while measuring on the variation of the total flux of two images might be a promising way for testing such a coupling in the future.

Galactic and astrometric microlensing
In a scenario of microlensing, a foreground gravitational object lenses a more distant background star. Plenty of projects [62][63][64][65][66][67] on Galactic microlensing measure change of the total fluxes of stars in the bulge with time, while astrometric microlensing on centroid shifts of a remote source by a nearby lens is also discussed [68][69][70] and practiced [71][72][73].
For Galactic lensing, masses and distances of the lens and source are respectively scaled by the solar mass and by 8 kpc as M * • = M • /M and d * S = d S /(8 kpc). We suppose a situation that the lens is located at rough midpoint between the observer and the source, i.e. d L ∼ d LS ∼ d S /2. We can, therefore, have that ϑ • ∼ 2.5 × 10 −12 (M * • /d * S ) as, ϑ E ∼ 10 −3 (M * • /d * S ) 1/2 as and ε ∼ 2.4 × 10 −9 (M * • /d * S ) 1/2 . We concentrate on the contribution of the coupling to the Weyl tensor on F tot which is indicated by δr tot .
For astrometric lensing, masses and distances of the lens and source are respectively scaled by the solar mass and by 2 kpc as M * • = M • /M and d * S = d S /(2 kpc). We assume that a nearby lens is located at 10 pc from the observer, i.e. d * L = d L /(10 pc), so that d S ≈ d LS . We can have that ϑ • ∼ Astrometric lensing by a nearby star has an Einstein ring radius almost 30 times larger then the one of Galactic lensing. The centroid position S cent is the observables, in which δS cent indicates the contributions of α.
The left panel of Fig. 2 shows color-indexed δr tot for a specific case of Galactic lensing, where we assume that M • = M and 2d L = d S = 8 kpc. The coupling constant α is set belonging to the domain [10 6 , 10 12 ] m 2 , covering both α obs ±1 (dashed line) and α th +1 (dash-dot line) with s = 1 for the lens. The right panel of Fig. 2 represents color-indexed δS cent for a case of astrometric lensing with M • = 0.676 M , d L = 5.55 pc and d S = 2 kpc based on the microlensing event caused by a near white dwarf [73]. We take α ∈ [10 5.5 , 10 12 ] with s = 1 in order to contain the observational and theoretical bounds. However, the contributions of α in these two microlensing δr tot and δS cent are far beyond the current observational limits. The contributions from the other polarization for the coupling, i.e. s = −1, can be sufficiently approximated by changing the signs of the ones shown in this figure; and they are too small to detect as well.

Conclusions and discussion
In order to provide one of the missing puzzles of the whole physical picture of photons coupled to Weyl tensor in a Schwarzschild black hole, we investigate its weak deflection lensing, as an extension of the previous works on its strong deflection lensing [46,47]. Under a unified framework valid for both two polarization directions of the coupling, we obtain its bending angle, image position, magnification, centroid and time delay in the coordinate-invariant forms upon to the second order in the perturbation parameter of the ratio of the angular gravitational radius to angular Einstein radius of the lens. The contributions of such a coupling on some astrophysical scenarios are also studied. We find that, in the weak deflection lensing on a star orbiting the Sgr A*, Galactic microlensing on a star in the bulge and astrometric microlensing by a nearby lens, these effects caused by coupling are beyond the current limits of technology. However, measuring the variation of the total flux of two images caused by the Sgr A* might be a promising way for testing such a coupling in the future.
In this work, as an astrophysically important ingredient, the spin of a black hole has not been taken into account. A self-consistent treatment should move from the present model we considered to photons coupled to the Weyl tensor in a Kerr black hole, in which the strong deflection lensing was studied [48]. As expected, the spin can make the light propagation more complicated and effectively causes the caustic shifted and distorted [74]. We will leave the detailed investigation on its weak deflection lensing for future works.