Perturbative deflection angles of timelike rays

Yujie Duan,1, ∗ Weiyu Hu,1, ∗ Ke Huang,1, ∗ and Junji Jia1, 2

1School of Physics and Technology, Wuhan University, Wuhan, 430072, China

2MOE Key Laboratory of Artificial Micro- and Nano-structures,

School of Physics and Technology, Wuhan University, Wuhan, 430072, China†

(Dated: January 14, 2020)

Geodesics of both lightrays and timelike particles with nonzero mass are deflected in a gravitational

field. In this work we apply the perturbative method developed in Ref. [1] to compute the deflection

angle of both null and timelike rays in the weak field limit for four spacetimes. We obtained the

deflection angles for the Bardeen spacetime to the eleventh order of m/b where m is the ADM mass

and b is the impact parameter, and for the Hayward, Janis-Newman-Winicour and Einstein-Born-

Infeld spacetimes to the ninth, seventh and eleventh order respectively. The effect of the impact

parameter b, velocity v and spacetime parameters on the deflection angle are analyzed in each of the

four spacetimes. It is found that in general, the perturbative deflection angle depends on and only

on the asymptotic behavior of the metric functions, and in an order-correlated way. Moreover, it is

shown that although these deflection angles are calculated in the large b/m limit, their minimal valid

b can be as small as a few m’s as long as the order is high enough. At these impact parameters, the

deflection angle itself is also found large. As velocity decreases, the deflection angle in all spacetime

studied increases. For a given b, if the spacetime parameters allows a critical velocity vc, then

the perturbative deflection angle will deviate from its true value as v decreases to vc. It is also

found that if the variation of spacetime parameters can only change the spacetime qualitatively at

small but not large radius, then these spacetime parameter will not cause a qualitative change of

the deflection angle, although its value is still quantitatively affected. The application and possible arXiv:2001.03777v1 [gr-qc] 11 Jan 2020 extension of the work are discussed.

Keywords: deflection angle, gravitational lensing, weak lensing, strong lensing, neutrino mass, massive grav-

itational wave

∗These authors contributed equally to this work. †Electronic address: Corresponding Author: [email protected] 2

I. INTRODUCTION

The deflection of the lightray near a massive object played an important role in both the theoretical development of gravitational theories and the experimental observations in astrophysics. It helped establishing General Relativity as a correct description of gravity at its very early age. It is also the foundation of gravitational lensing (GL) effect, which is an important tool to deduce properties of the signal source, the signal itself and the lens it passes by. After the first observation of GL in 1979 [2], many features, such as luminous arcs [3], Einstein cross [4] and rings [5],

CMB GL [6–8] and GL of supernovas [9, 10] have been observed. These GLs can be used to study the coevolution of supermassive black holes (BHs) and galaxies [11] and cosmological parameters (for a review see [12]), properties of the supernova [13], as well as dark matter substructures [14, 15].

Traditionally, electromagnetic wave was the main kind of signal in relevant theoretical or observational studies.

However, with the discovery of SN1987A neutrinos and blazer TXS 0506+056 [16, 17], and the more recent discovery of gravitational waves (GWs) [18–21], more kinds of messengers become possible. It is known now that neutrinos have non-zero mass [22] and GW in some gravitational theories beyond GR can also be massive. To study the GL of these particles, including the apparent angles of source images and the time delay effect [23], in principle one should start from the deflection angles calculated for corresponding timelike rays. Recently some of us have examined the deflection angles of massive particles in the simple Schwarzschild and Reissner-Nordstr¨om(RN) spacetimes using exact formulas [24, 25] which are valid in both the weak and strong field limits. These results were then used to correlate the GL observations to the properties of the messengers, namely the absolute mass, mass ordering of neutrinos and velocity of GWs. For the deflection angles in more complicated spacetimes, we proposed a perturbative method which were shown to work to very high orders in the weak field limit for both null and timelike particles [1]. The results in

Schwarzschild and RN spacetimes were shown to work even when the gravitational field is not weak, and when the charge in the RN spacetime passes the extremal value [1].

In this work, we extend this perturbative method to the calculation of deflection angles of signals with general velocity in four other spacetimes, namely the Bardeen, Hayward, Janis-Newman-Winicour (JNW) and Einstein-Born-

Infeld (EBI) spacetimes. These spacetimes all carry charges of their own kind, and allow transition from BH spacetime to non-BH (possibly naked singularity) spacetime in their respective parameter space. The motivation is to see how different kinds of charges will influence the deflection angle of rays with general velocity [26, 27]. In particular, we will determine the validity of the calculated deflection angle near and beyond the critical value of the parameters.

To our best knowledge, the state of the art for the computation of the deflection angles is to the fifth order in 3

Bardeen spacetime [26, 28, 29], sixth order in Hayward spacetime [30, 31], second order in JNW spacetime[26, 28, 32] and third order in EBI BH spacetimes [28], all for only lightrays but not timelike particles. In this work, we extend these orders dramatically, to the eleventh, ninth, seventh and eleventh orders for the Bardeen, Hayward, JNW and

EBI spacetimes respectively, for both lightlike and timelike rays.

The paper is organized as the following. We first recap the general procedure to find the deflection angle in general static, spherically symmetric and asymptotically flat spacetimes for signal with arbitrary velocity in Sec.II. Then this method is applied to the above mentioned four spacetimes in subsections of the sectionIII. For each spacetime, after the deflection angles is found, the effect of the impact parameter, signal velocity and the spacetime parameters are analyzed. Finally, in Sec.IV, we compare these effects among different spacetimes. Throughout the paper we use the geometric unit G = c = 1.

II. PERTURBATIVE METHOD FOR THE DEFLECTION ANGLES

Because the computation of the deflection angle in this work are based on the perturbative method developed in

Ref. [1], here we briefly recap the procedure of the computation in weak field limit for signals with arbitrary velocity.

For any static and spherically symmetric spacetimes, we can start from a general metric

ds2 = −A(r)dt2 + B(r)dr2 + C(r) dθ2 + sin2 θdϕ2
. (1)

To be asymptotically flat, the metric functions satisfy [33]

2m A(r → ∞) = 1 − + O(r−2), (2) r 2m B(r → ∞) = 1 + + O(r−2), (3) r C(r → ∞) = r2 + O(r1), (4) where m is the ADM mass parameter of the system.

Using this metric, it is easy to find the geodesic equations, from which two first integrals can be carried out. These

first integrals define the energy E and angular momentum L at infinity for unit mass. E can be related to the velocity at infinity and L to the minimal radial coordinate r0 of the trajectory by [1]

1 (E − κA(r ))C(r ) E = √ ,L = 0 0 , (5) 1 − v2 A(r0) where κ = 0, 1 for null and timelike geodesics respectively. Then it is not difficult to show that the change of the 4 angular coordinate for a particle coming from and going back to spacial infinity is given by [1]

− 1 Z ∞ L  E2 L2  2 I(r0,E,L) =2 p κ − + dr (6) r0 C(r) B(r) A(r) C(r) Z 1 = y(u, r0, v, p)du, (7) 0

r0 where in the second step a change of variable u = r was used and we have changed L and E to the signal velocity v and r0 according to Eq. (5). The y(u, r0, v, p) denotes the integrand and p collectively stands for all parameters in the metric functions. And furthermore, r0 could be related to the impact parameter b through [1] √ 1 E2 − 1 = (8) b L √ s E2 − 1 A(r )  1  = 0 ≡ l . (9) p 2 E − κA(r0) C(r0) r0

In the weak field limit, r0  m, then the integrand in Eq. (7) can be expanded in the powers of m/r0, i.e.,

∞ n X  m  y(u, r , v, p) = y (u, v, p) . (10) 0 n r n=1 0

We emphasis that the integration for this series expansion is always feasible because of the particular form that yi(u, v, p) take (see Ref. [1]) and therefore in principle calculation to arbitrarily desired order is possible, especially if one has enough time and memory space for a symbolic computation tool to carry out these integrals. After integration, then the total change of the angular coordinate takes the form

∞ n X  m  I(r , v, p) = I (v, p) . (11) 0 n r n=0 0

Here In(v, p) should in general depend on the particle velocity v and parameters p of the spacetime. In GL however, the minimal radial coordinate r0 is still not as conveniently connected to observables, such as lens distance and apparent angles of the images, as the impact parameter b. Therefore, it is desirable to replace r0 in Eq. (11) by b.   For this purpose, we can inverse the function l 1 in Eq. (9) to express 1 as a function of 1 , also in series form, r0 r0 b

∞ n 1 X 1 = C . (12) r n b 0 n=1

Substituting this into Eq. (11) and sorting the series as powers of m/b, one obtains the change of the angular coordinate as

∞ X mn I(b, v, p) = I0 (v, p) , (13) n b n=0

0 where the In(v, p) are the new coefficients. 5

The deflection angle in the weak field limit is then α = I(b, v, p) − π. This deflection angle can be directly used in the GL equation to solve the angular positions and magnifications of images of the source, also as series expansions, as was done in Ref. [1]. However, since those computation are also lengthy, in this work we will not solve GL equations, but rather restrain ourselves to the perturbative computation and then analysis of the deflection angles in the four spacetimes mentioned in the introduction.

III. THE DEFLECTION ANGLES IN VARIOUS SPACETIMES

A. Deflection in the Bardeen spacetime

The regular Bardeen spacetime is described by the metric [34, 35]

2mr2 A(r) = B(r)−1 = 1 − ,C(r) = r2, (14) (r2 + g2)3/2 where m is the mass of the spacetime and g is the charge parameter for some nonlinear electrodynamics [35]. When g = 0, this reduces to the Schwarzschild spacetime. When g 6= 0, the spacetime is regular everywhere. This metric approaches the Schwarzschild spacetime at large r and the de Sitter spacetime at small r. The asymptotic expansion of A(r) indeed is given by

2m 3g2m 15 A(r → ∞) = 1 − + + O , (15) r r3 r from which it is clear that the Bardeen spacetime deviates from the Schwarzschild one from the third order of large r.

|g| When the dimensionless parameterg ˆ ≡ < √4 ≡ gˆ , this spacetime allows two horizons, which become extremal m 3 3 c wheng ˆ =g ˆ , and then a non-BH spacetime wheng ˆ > gˆ [36]. Forg ˆ < gˆ < gˆ ≡ 48√ , there can still exist a photon c c c p 25 5 sphere which might diverge the deflection angle if the trajectory is close to it [29].

Using metric (14) and going through the expansion and integration procedure from Eq. (7) to Eq. (13), the change of the angular coordinate IB(r0, v, gˆ) for a particle ray with the minimal radial coordinate r0 and velocity v in the

Bardeen spacetime becomes, to the eleventh order

11 n 12 X  m   m  I (r , v, gˆ) = B (v, gˆ) + O (16) B 0 n r r n=0 0 0 where the coefficients are

B0 =S0, (17a)

B1 =S1, (17b) 6

B2 =S2, (17c)  3  B =S − 2ˆg2 + 1 , (17d) 3 3 v2 3  72π − 48 24π − 112 B =S − gˆ2 9π + + , (17e) 4 4 16 v2 v4  3π − 52 21π − 52 6π − 22  5  B =S + 3ˆg2 −4 + + + + 2ˆg4 1 + (17f) 5 5 v2 v4 v6 v2  525π 45(24 − 31π) 3(343 − 93π) 1537 − 534π 9(41 − 12π) B =S +g ˆ2 − + + + + 6 6 64 8v2 v4 2v6 2v8 45π 135π − 91 90π − 323 +g ˆ4 + + , (17g) 16 4v2 4v4  9(67π − 928) 3(679π − 1956) 603π − 1997 1272π − 3811 B =S + 3ˆg2 −18 + + + + 7 7 16v2 4v4 v6 4v8 198 8224 − 477π 3424 − 1179π 2(81π − 263)  7  +g ˆ4 + + − − 2ˆg6 1 + , (17h) 7 16v2 4v4 v6 v2  24255π  184527π  1  707787π  1 70425 92277π  1 B =S + 3ˆg2 − + 1197 − + 18063 − + − 8 8 2048 128 v2 128 v4 2 8 v6 54843 135729π  1 73671  1 9411  1  + − + − 3024π + − 360π 2 16 v8 8 v10 8 v12 25725π 22773π 2215 1 56223π 22965 1 9105π 56017 1 +g ˆ4 + − + − + − 1024 32 4 v2 32 4 v4 4 8 v6 6351π 20253 1   175π 119 175π  1 563 175π  1  + − + 3ˆg6 − + − + − , (17i) 8 8 v8 128 8 8 v2 8 8 v4  1144 29691π 58392 1 1330839π 326454 1 1771275π  1 B =S +g ˆ2 − + − + − + − 176768 9 9 5 32 5 v2 64 5 v4 32 v6 260415π  1 66465π 531849 1  542533 1 + − 201168 + − + 8832π − 4 v8 2 5 v10 20 v12  13989 1    35223π  1 30371 − 9696π  141525π  1 + 864π − +g ˆ4 204 + 7600 − + + 56307 − 5 v14 64 v2 v4 8 v6 40542 − 13044π 10086 − 3183π   1144 1185π 8896 1 7545π  1 + + +g ˆ6 − + − + − 2968 v8 v10 21 16 7 v2 8 v4 3045π  1   9  + − 2410 + 2ˆg8 1 + , (17j) 4 v6 v2  405405π 14121 1670733π  1 351267 5894733π  1 B =S + 3ˆg2 − + − + − 10 10 8192 5 512 v2 5 256 v4 2614779 2688321π  1 815845 4111275π  1 2499479 402165π  1 + − + − + − 10 32 v6 2 32 v8 8 4 v10 4911113 77127π  1 2003647  1 173047  1  + − + − 8112π + − 672π 40 2 v12 80 v14 80 v16 654885π 8300571π 26995 1 10818879π 517357 1 3678033π 2867039 1 +g ˆ4 + − + − + − 4096 1024 4 v2 256 4 v4 32 8 v6 3953013π 3128781 1 488337π 6097593 1  1135689 1  + − + − + 11235π − 32 8 v8 8 32 v10 32 v12  1 93473 68283π  1 1221379 114111π  1 589497 23577π  1 +g ˆ6 − (15435π) + − + − + − 256 56 32 v2 56 16 v4 16 2 v6 7

286337 11343π  1  315π 1575π 539 1 525π 3179 1  + − + 9ˆg8 + − + − (17k) 16 2 v8 512 128 64 v2 32 64 v4  2210 4293165π 174858 1 18076749π 8181207 1 B =S + 3ˆg2 − + − + − 11 11 7 2048 7 v2 256 35 v4 21350013π 5271603 1 11113359π 43333089 1 5680695π 8984733 1 + − + − + − 64 5 v6 16 20 v8 8 4 v10 812145π 10128041 1  109722421 1  148573919 1 + − + 123486π − + 21408π − 2 8 v12 280 v14 2240 v16  11028373 1    12156195π  1  84259845π  1 + 1536π − +g ˆ4 1196 + 74576 − + 542955 − 2240 v18 2048 v2 512 v4  38284485π  1  3762405π  1 4318233 10945485π  1 + 1888397 − + 2938605 − + − 64 v6 4 v8 2 16 v10 6214129 993045π  1 915743  1   6460 496905π 187008 1 + − + − 36300π +g ˆ6 − + − 8 4 v12 8 v14 11 256 7 v2 1346391π 970369 1 828813π 2286525 1 398151π  1 + − + − + − 311516 32 7 v4 8 7 v6 4 v8 65019π  1  650 18510 2475π  1 55677 4815π  1 + − 102378 +g ˆ8 + − + − 2 v10 7 7 16 v2 7 2 v4  40905π  1   11 + 8049 − − 2ˆg10 1 + . (17l) 16 v6 v2

n  m  Here Sn is the coefficient of order in the change of the angular coordinate in Schwarzschild spacetime. Their r0 values are given in Ref. [1] and also listed in Eq. (A2) of AppendixA.

Using Eq. (12), the m in Eq. (16) can be expressed in powers of m . The result is r0 b

m m 1 m2  2 1  m3  4 4 3ˆg2  m4 = + 2 + 2 + 4 + 2 + 4 − 2 r0 b v b v 2v b v v 2v b  8 18 3 1  2 3  m5 + + + − − 3ˆg2 + v2 v4 v6 8v8 v2 2v4 b   1 4 2   3 6 1  15ˆg4  m6 + 16 + + − 6ˆg2 + + + v2 v4 v6 v2 v4 v6 8v2 b 32 200 200 25 5 1  16 60 30 5   4 5  m7 + + + + − + − 3ˆg2 + + + + 3ˆg4 + v2 v4 v6 v8 4v10 16v12 v2 v4 v6 4v8 v2 v4 b   1 9 15 5   1 6 6 1  99ˆg4  1 3 1  35ˆg6  m8 + 64 + + + − 120ˆg2 + + + + + + − v2 v4 v6 v8 v2 v4 v6 v8 2 v2 v4 v6 16v2 b 128 1568 3920 2450 245 49 7 5  3ˆg2 1536 13440 22400 + + + + + − + − − + + v2 v4 v6 v8 v10 4v12 8v14 128v16 16 v2 v4 v6 8400 420 7  21ˆg4 64 336 280 35  4 7  m9 + + − + + + + − 5ˆg6 + v8 v10 v12 8 v2 v4 v6 v8 v2 v4 b   1 16 56 56 14   1 12 30 20 3  + 256 + + + + − 672ˆg2 + + + + v2 v4 v6 v8 v10 v2 v4 v6 v8 v10  5 40 60 20 1  429ˆg6  1 4 2  315ˆg8  m10 +102ˆg4 + + + + − + + + v2 v4 v6 v8 v10 4 v2 v4 v6 218v2 b 512 10368 48384 70560 31752 2646 126 81 45 7  + + + + + + − + − + v2 v4 v6 v8 v10 v12 v14 8v16 64v18 256v20 8

3ˆg2 16384 258048 903168 940800 282240 14112 336 9  − + + + + + − + 32 v2 v4 v6 v8 v10 v12 v14 v16 45ˆg4 512 5760 13440 8400 1260 21  64 432 504 105 + + + + + + − 7ˆg6 + + + 16 v2 v4 v6 v8 v10 v12 v2 v4 v6 v8 15ˆg8  4 9  m11 m12 + + + O (18) 2 v2 v4 b b

m Putting this into Eq. (16), we obtain the change of the angular coordinate in powers of b for general velocity v,

11 X mn m12 I (b, v, gˆ) = B0 (v, gˆ) + O (19) B n b b n=0 where the coefficients are

0 0 B0 =S0, (20a)

0 0 B1 =S1, (20b)

0 0 B2 =S2, (20c)  3  B0 =S0 − 2ˆg2 1 + , (20d) 3 3 v2 9π 3 3 1  B0 =S0 − gˆ2 + + , (20e) 4 4 2 8 v2 v4  15 15 1   5  B0 =S0 − 12ˆg2 1 + + + + 2ˆg4 1 + , (20f) 5 5 v2 v4 v6 v2  5 15 15 1  45π 1 3 1  B0 =S0 − 105πgˆ2 + + + + gˆ4 + + , (20g) 6 6 64 8v2 4v4 v6 2 8 2v2 v4  315 1050 630 45 1  1 3 5 1   7  B0 =S0 − 6ˆg2 9 + + + + − + 198ˆg4 + + + − 2ˆg6 1 + , (20h) 7 7 v2 v4 v6 v8 v10 7 v2 v4 v6 v2 10395πgˆ2  336 1680 1792 384 735πgˆ4  1120 3360 1792 128 B0 =S0 − 7 + + + + + 35 + + + + 8 8 2048 v2 v4 v6 v8 1024 v2 v4 v6 v8 525πgˆ6  16 16 − 1 + + , (20i) 128 v2 v4 8ˆg2  9009 63063 105105 45045 3003 91 3  B0 =S0 − 143 + + + + + − + 9 9 5 v2 v4 v6 v8 v10 v12 v14  45 210 210 45 1  1144ˆg6  27 63 21  9  + 204ˆg4 1 + + + + + − 1 + + + + 2ˆg8 1 + , (20j) v2 v4 v6 v8 v10 21 v2 v4 v6 v2 135135πgˆ2  720 6720 16128 11520 2048 31185πgˆ4  1260 8400 B0 =S0 − 9 + + + + + + 21 + + 10 10 8192 v2 v4 v6 v8 v10 4096 v2 v4 13440 5760 512 2205πgˆ6  280 1120 896 128 945πgˆ8  60 80 + + + − 7 + + + + + 3 + + , (20k) v6 v8 v10 256 v2 v4 v6 v8 512 v2 v4 6ˆg2  109395 1312740 4288284 4594590 1531530 92820 3060 153 5  B0 =S0 − 1105 + + + + + + − + − 11 11 7 v2 v4 v6 v8 v10 v12 v14 v16 v18  1001 9009 21021 15015 3003 91 1  + 92ˆg4 13 + + + + + + − v2 v4 v6 v8 v10 v12 v14 6460ˆg6  55 330 462 165 11  130ˆg8  165 495 231  11 − 1 + + + + + + 5 + + + − 2ˆg10 1 + . (20l) 11 v2 v4 v6 v8 v10 7 v2 v4 v6 v2 9

(20m)

0 m
n Here Sn is the coefficient of order b in the change of the angular coordinate of the Schwarzschild spacetime given in Eq. (A2). Note that if we set v = c for lightray in Eqs. (17) and (20), their first five orders reduce to the Eqs.

(2.18) and (2.20) of Ref. [29] respectively.

From these equations, it is seen that in IB(b, v, gˆ) the deviation due to parameterg ˆ from the Schwarzschild case is only present from and above the third order of m/b. Since the deflection angle in the weak field limit is expected to be sensitive only to the asymptotic behavior of the spacetime, this implies that the metric functions at large r must only differ from the Schwarzschild spacetime starting from third order too. This is indeed confirmed in the expansion of the lapse function in Eq. (15), where the difference only appears from the third order.

To study how the deflection angle depends on various kinematic and spacetime parameters, we plot the deflection angle αB ≡ IB(b, v, gˆ) − π in Fig.1. In Fig.1 (a), we plot the partial sums defined as the sum of Eqs. (20) to the n-th order as

n X mi α = B0(v, gˆ) , (21) B,n b i=1 and the exact value of the deflection angle calculated by numerically integration Eq. (6) for the Bardeen metric. It is seen that the partial sums αB,n approaches the true value as n increases. The highest order partial sum αB,11 practically overlaps with the exact value for b greater than 7m. As b decreases from large value, the deflection angle monotonically increases. The contribution from higher orders in Eqs. (20) only becomes important when b is small enough. At b = 7m, the deflection angle reached a value of αB = 0.27π, which is rather large. Therefore this suggests that our series deflection angle works even when the gravitational field is not weak.

In Fig.1 (b), the contribution from each order of Eq. (20) and the corresponding orders for the Schwarzschild metric are plotted for v = c andg ˆ = 0.5. It is seen that for any b, as the order increases, the contribution from each order decreases linearly in this log-log plot. This suggests that the summation of these contributions will converge as they should. Moreover, it is also seen that the contributions to the deflection angle in the Bardeen spacetime only deviate from the Schwarzschild ones from the third order. This is a reflection that the asymptotic expansion (15) of the lapse function of the Bardeen metric only deviates from the Schwarzschild lapse function from the third order.

Furthermore, it is seen that comparing to the Schwarzschild case, a nonzerog ˆ decreases the deflection angle in all 10

1 1.0 1.1 1.0

n 0.100 0.8 0.9 0.8 ( m / b ) B , n 0.010 n 0.6 α 0.7 B , n 0.6

α 0.001 and S

0.4 0.5 n 0.4 10-4 7.0 7.5 8.0 8.5 9.0 9.5 10.0 ( m / b ) 0.2 n b/m B 10-5

0.0 10-6 20 40 60 80 100 8 10 12 14 b/m b/m

(a) (b)

6 1

5 0.100

4 n 0.010 B , n α

3 ( m / b ) n

B 0.001 2 10-4 1

10-5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 v/c g/m

(c) (d)

FIG. 1: The deflection angles in the Bardeen spacetime. (a) Partial sums (21) (solid curves) and the exact deflection angle

(dashed red curve) for b/m from 7 to 100 and from 7 to 14 in the inset and v = c, gˆ = 0.5. From bottom to top curve in each plot, the maximum index in the partial sum increases from 1 to 7. (b) Contribution from each order of Eq. (20) (solid lines from top to bottom curve order increases from 1 to 17) for v = c, gˆ = 0.5 and the corresponding order results in the

Schwarzschild case (dashed lines). (c) Partial sums αB,11 (solid lines) and exact values (dash lines) of the deflection angle for b = 10m andg ˆ1 = 0.5 (red lines),g ˆ2 = 0.8 (blue lines) andg ˆ3 = 1.0 (green lines). The vertical dash lines are two critical velocities. (d) Contributions from each order of Eq. (20) (dashed curves), partial sum αB,11 (top red dashed curve) and exact value (top solid curve) of the deflection angle for b = 10m, vˆ = 0.9c. 11 orders. The horizontal line at 1 [as] here represents the typical precision in the GL by galaxy or galaxy cluster. It shows that the seventh order result can reach this accuracy for b that is as small as about 12.1m.

In Fig.1 (c), the dependence of the partial sum αB,11 and the exact value of the deflection angle are plotted for three representative value of parameterg ˆ,g ˆ1 = 0.5, gˆ1 = 0.8 andg ˆ3 = 1.0. It is known in the Schwarzschild and

RN BH spacetimes that for a given b, there exists a critical velocity vc below which the deflection angle diverges to infinity, i.e., the particle will enter the BH event horizon. Here forg ˆ1 which is smaller thang ˆc andg ˆ2 which is between gˆc andg ˆp, it is known that the spacetime contains a photon sphere in both cases and therefore for some fixed b there shall exist a lower limit of the velocity vc below which the true deflection angle diverges. It is seen then from the plot that this critical value forg ˆ1 is roughly 0.42c and forg ˆ2 is 0.38c when b = 10m. On the other hand, wheng ˆ =g ˆ3 which is larger thang ˆp, the photon sphere disappears and the partial sum αB,11 and the true value of the deflection angles agrees in a wider range of the velocity.

Fig.1 (d) shows the effect of the charge ˆg on the contributions to deflection angle at various orders, as well as their partial sum αB,11 and the exact value of the deflection angle. As seen from Eq. (20), comparing to the Schwarzschild metric, the effect ofg ˆ appears in the third and above orders. Therefore for a finiteg ˆ, in general its effect should be small when b is large, and it only becomes apparent when b is reasonably small, i.e., when the gravitational field is not weak anymore. To illustrate its effect, therefore we choose b = 10m in the plot. It is seen that as dictated by

Eqs. (20),g ˆ monotonically decreases the deflection angle from the third order and does the same to the partial sum.

In the entire range ofg ˆ, both below or above the criticalg ˆc andg ˆp, this partial sum agrees with the true value of the deflection angle.

B. Deflection in the Hayward spacetime

The Hayward metric is given by [37]

2mr2 A(r) = B(r)−1 = 1 − ,C(r) = r2. (22) r3 + 2l2m

3 Here m is the mass of spacetime and l characterize the central energy density 8πl2 [31, 37]. When l = 0, this reduces to the Schwarzschild spacetime. When l 6= 0 however, similar to the Bardeen case, the Hayward spacetime is regular everywhere. Moreover, one can also verify that the Hayward spacetime approaches the Schwarzschild one at large r and the de Sitter spacetime at small r by expanding A(r)

2m 4l2m2 17 A(r → ∞) = 1 − + + O , (23) r r4 r 12

r2 r5 A(r → 0) = 1 − + + O (r)8 . (24) l2 2l4m

Clearly, the Hayward spacetime deviates from Schwarzschild spacetime at large r only from the fourth order.

ˆ |l| Depending on the value of the dimensionless parameter l ≡ m , this metric corresponds to a BH or non-BH spacetime.

When ˆl is smaller than a critical value, i.e., ˆl < √4 ≡ ˆl , there are two horizons, allowing a normal BH. When ˆl = ˆl , 3 3 c c q ˆ ˆ ˆ ˆ ˆ 25 5 the BH becomes extremal and when l > lc, the spacetime becomes a non-BH one. For lc < l < lp ≡ 24 6 , there can still exist a photon sphere which might diverge the deflection angle if the trajectory is close to it [31]. Using metric

(22) and going through the expansion and integration procedure from Eq. (7) to Eq. (13), the change of the angular coordinate takes the form

13 n 14 X  m   m  I (r , v, ˆl) = H (v, ˆl) + O , (25) H 0 n r r n=0 0 0 where

H0 =S0, (26a)

H1 =S1, (26b)

H2 =S2, (26c)

H3 =S3, (26d) 1 1  H =S − 3πˆl2 + , (26e) 4 4 4 v2  32 3 4  H =S + ˆl2 − + (π − 20) + (3π − 7) , (26f) 5 5 5 v2 v4  75 4 9 1  H =S + ˆl2 − π + (12 − 19π) + (72 − 19π) + (148 − 51π) , (26g) 6 6 16 v2 2v4 v6  1 1 1 1  H =S + ˆl2 −32 + (417π − 6112) + (2217π − 6224) + (507π − 1678) + (186π − 566) 7 7 8v2 4v4 v6 v8 128ˆl4  7  + 1 + , (26h) 35 v2  11025π  46845π  1  36405π  1  15045π  1 H =S + ˆl2 − + 560 − + 7564 − + 11468 − 8 8 512 64 v2 16 v4 4 v6  9015π  1 1  3ˆl4  16 16 + 7246 − + (1854 − 600π) + 525π + (485π − 224) + (365π − 1248) , (26i) 4 v8 v10 320 v2 v4  704 30801π 31276 1 73815π  1 41355π  1 H =S + ˆl2 − + − + − 28580 + − 66034 9 9 5 64 5 v2 8 v4 2 v6 1  272133 1  55237 1  + (−3437 + 1110π) + 8550π − + 1776π − v8 10 v10 10 v12 1024 40368 555π  1 1814  9 3866  2  + ˆl4 + − + − 119π + − 243π , (26j) 21 35 8 v2 5 v4 5 v6 13

 189189π 22096 5477631π  1 500404 4153713π  1 H =S + ˆl2 − + − + − 10 10 2048 5 1024 v2 5 128 v4  6465855π  1  2145825π  1 1387972 712983π  1 + 314612 − + 425270 − + − 64 v6 16 v8 5 8 v10 923927  1 154283  1  + − 29166π + − 4944π 10 v12 10 v14 11025π 47349π 36024 1  101368 1 13833π 107984 1 + ˆl4 + − + 4635π − + − 256 32 35 v2 7 v4 2 5 v6 6363π 50224 1  63π 315π  + − − ˆl6 + , (26k) 2 5 v8 32 16v2  189189π 22096 5477631π  1 500404 4153713π  1 H =S + ˆl2 − + − + − 11 11 2048 5 1024 v2 5 128 v4  6465855π  1  2145825π  1 1387972 712983π  1 + 314612 − + 425270 − + − 64 v6 16 v8 5 8 v10 923927  1 154283  1  + − 29166π + − 4944π 10 v12 10 v14 11025π  36024 47349π  1  101368  1  107984 13833π  1 + ˆl4 + − + + − + 4635π + − + 256 35 32 v2 7 v4 5 2 v6  50224 6363π  1  63π 315π 1  + − + − ˆl6 + , (26l) 5 2 v8 32 16 v2  4160  1477124 3550617π  1  1706276 53216523π  1 H =S + ˆl2 − + − + + − + 12 12 7 35 1024 v2 5 512 v4  6798822 27563487π  1  24930075π  1  4458653 5644605π  1 + − + + −2432070 + + − + 5 64 v6 32 v8 2 8 v10  10966099 1404003π  1  40754733  1  5757121  1  + − + + − + 92136π + − + 13152π 10 4 v12 140 v14 140 v16 1120581π  451256 8388789π  1  10870808 53537775π  1 + ˆl4 + − + + − + 4096 35 512 v2 35 512 v4  36253528 5282055π  1  49805888 14449935π  1 + − + + − + 35 16 v6 35 32 v8  4569773 1166163π  1  1251989 318411π  1  + − + + − + 5 4 v10 5 4 v12  1 89344 45507π  1 1141024 58407π  1 356704 12963π  1  + ˆl6 − (5775π) + − + − + − , 128 105 32 v2 105 16 v4 35 4 v6 (26m)

 51680  5355932 358002735π  1  22483612 3857795085π  1 H =S + ˆl2 − + − + + − + 13 13 21 21 16384 v2 7 4096 v4  137874110 3199571385π  1  2338559475π  1 + − + + −57187718 + 7 512 v6 128 v8  176216073 1789998105π  1  154959447 395556735π  1 + − + + − + 2 64 v10 2 16 v12  1150481683 52187835π  1  184890359  1 + − + + − + 4215045π 28 4 v14 14 v16  1656792409  1  177870673  1  + − + 782400π + − + 84480π 672 v18 672 v20 14

 5311024 12435471π  1 9003788 48647511π  1 + ˆl4 2048 + − + − 35 1024 v2 7 128 v4 37203116 216301905π  1 70329374 6413085π  1 66486484 24143805π  1 + − + − + − 7 128 v6 7 2 v8 7 8 v10 46206759  1 2082727  1  + − 1472517π + − 331245π 10 v12 2 v14  204800  8109616 45813π  1  1888624 213273π  1 + ˆl6 − + − + + − + 429 385 32 v2 21 8 v4  2479808  1  3317952 60453π  1  32768 32768 1  + − + 52671π + − + + ˆl8 + . (26n) 15 v6 35 2 v8 3003 231 v2

Here Sn are the change of the angular coordinate at corresponding orders in the Schwarzschild spacetime given in Eq.

(A2).

To express the deflection angle in terms of the impact parameter b, we express r0 using b using Eq. (12)

m m  1  m2  2 1  m3  1 1  m4 = + 2 + 2 + 4 + 4 2 + 4 r0 b v b v 2v b v v b  8 18 3 1  1  m5   1 4 2   1 1  m6 + + + − − 2ˆl2 + 16 + + − 8ˆl2 + v2 v4 v6 8v8 v2 b v2 v4 v6 v2 v4 b 32 200 200 25 5 1  24 60 15 m7 + + + + − + − ˆl2 + + v2 v4 v6 v8 4v10 16v12 v2 v4 v6 b 320 960 576 64  4 18 12 1  m8 + + + + − 16ˆl2 + + + v8 v6 v4 v2 v2 v4 v6 v8 b 128 1568 3920 2450 245 49 7 5 + + + + + − + − v2 v4 v6 v8 v10 4v12 8v14 128v16 160 1120 1400 350 35  24 42 m9 −ˆl2 + + + + + ˆl4 + v2 v4 v6 v8 4v10 v2 v4 b   1 16 56 56 14   1 10 20 10 1  + 256 + + + + − 384ˆl2 + + + + v2 v4 v6 v8 v10 v2 v4 v6 v8 v10  1 4 2  m10 +96ˆl4 + + v2 v4 v6 b  1 131072 2654208 12386304 18063360 8128512 677376 + + + + + + 256 v2 v4 v6 v8 v10 v12 32256 2592 180 7  −896 12096 35280 29400 6615 − + − + + ˆl2 − − − − v14 v16 v18 v20 v2 v4 v6 v8 v10 441 21   1 432 504 105 1  m11 − + + 5ˆl4 64 + + + − 8ˆl6 2v12 8v14 v2 v4 v6 v8 v2 b   1 25 150 300 210 42   2 35 140 175 70 + 1024 + + + + + − 1024ˆl2 + + + + v2 v4 v6 v8 v10 v12 v2 v4 v6 v8 v10 7   1 10 20 10 1   2 5  m12 + + 960ˆl4 + + + + − 32ˆl6 + v12 v2 v4 v6 v8 v10 v2 v4 b 2048 61952 464640 1219680 1219680 426888 30492 5445 + + + + + + + − v2 v4 v6 v8 v10 v12 v14 4v16 1815 605 77 21 9ˆl2 32768 720896 3784704 + − + − − + + 16v18 64v20 128v22 1024v24 64 v2 v4 v6 15

6623232 4139520 827904 29568 528 11  21ˆl4 512 7040 + + + + − + + + v8 v10 v12 v14 v16 v18 4 v2 v4 21120 18480 4620 231  8 44 33 m13 m14 + + + + − 40ˆl6 + + + O . (27) v6 v8 v10 v12 v2 v4 v6 b b

Using this in Eq. (25), the change of the angular coordinate in terms of the impact parameter is obtained as

13 X mn m14 I (b, v, ˆl)) = H0 (v, ˆl) + O , (28) H n b b n=0 where the coefficients are

0 0 H0 =S0, (29a)

0 0 H1 =S1, (29b)

0 0 H2 =S2, (29c)

0 0 H3 =S3, (29d) 3π 3π  H0 =S0 − ˆl2 + , (29e) 4 4 4 v2 1 2 1  H0 =S0 − 32ˆl2 + + , (29f) 5 5 5 v2 v4 15πˆl2  90 120 16 H0 =S0 − 5 + + + , (29g) 6 6 16 v2 v4 v6  28 70 28 1  128ˆl4 1 1  H0 =S0 − 32ˆl2 1 + + + + + + , (29h) 7 7 v2 v4 v6 v8 5 7 v2 64ˆl2  594 3465 4620 1485 66 1  1024 9216 3072 1024 H0 =S0 − 11 + + + + + − + ˆl4 + + + , (29i) 9 9 5 v2 v4 v6 v8 v10 v12 21 7v2 v4 v6 63063π  210 1680 3360 1920 256 H0 =S0 − ˆl2 3 + + + + + 10 10 2048 v2 v4 v6 v8 v10 1575π  280 1120 896 128 63π  10 + ˆl4 7 + + + + − ˆl6 1 + (29j) 256 v2 v4 v6 v8 32 v2 64  5720 60060 168168 150150 40040 1820 40 1  H0 =S0 − ˆl2 65 + + + + + + − + 11 11 7 v2 v4 v6 v8 v10 v12 v14 v16 3840  55 330 462 165 11  8192  22 33 + ˆl4 1 + + + + + − ˆl6 1 + + (29k) 11 v2 v4 v6 v8 v10 231 v2 v4 2297295π  1188 15840 59136 76032 33792 4096 H0 =S0 − ˆl2 11 + + + + + + 12 12 65536 v2 v4 v6 v8 v10 v12 4851π  16632 138600 295680 190080 33792 1024 + ˆl4 231 + + + + + + 4096 v2 v4 v6 v8 v10 v12 5775π  36 120 64 − ˆl6 1 + + + (29l) 128 v2 v4 v6 160  41990 692835 3325608 5819814 3879876 881790 38760 969 H0 =S0 + ˆl2 −323 − − − − − − − + 13 13 21 v2 v4 v6 v8 v10 v12 v14 v16 38 1   91 1001 3003 3003 1001 91 1  − + + 2048ˆl4 1 + + + + + + + v18 v20 v2 v4 v6 v8 v10 v12 v14 16

40960  260 1430 1716 429 32768  13 − ˆl6 5 + + + + + ˆl8 1 + . (29m) 429 v2 v4 v6 v8 3003 v2

0 Here the coefficients Sn are the coefficients in the change of the angular coordinate in the Schwarzschild spacetime given in (A6). Note that if we set v = c for lightray in Eqs. (26) and (29), the first sixth orders agree with Eq. (18) and (19) of Ref. [31] respectively.

From these equations, it is seen that the modification comparing the Schwarzschild case due to parameter ˆl is only present from and above the fourth order of m/b. Since the deflection angle in the weak field limit is expected to be sensitive only to the asymptotic behavior of the spacetime, this implies that the metric functions at large r must only differ from the Schwarzschild spacetime starting from high orders too. This is indeed confirmed in the expansion of the lapse function in Eq. (23), where the difference only appears starting from the fourth order.

To study how the deflection angle depends on various kinematic and spacetime parameters, we plot the deflection

ˆ angle αH ≡ IH (b, v, l) − π in Fig.2. In Fig.2 (a), the partial sum of the deflection angle defined as the sum of Eqs.

(29) to the n-th order

n X mi α = H0 (v, ˆl) , (30) H,n n b i=1 as well as the exact value of the deflection angle obtained using numerical integration are shown. It is seen that in general, the deflection angle decreases monotonically as b increases, becoming completely determined by the leading order formula (29b) and overlaps with the exact value when b is large. As b decreases, the contribution from higher orders start to manifest (see the insert for a zoom-in at small b). However, the partial sum to the thirteenth order still overlaps with the true value of the deflection angle, even when b is as small as 7m, at which point αH ≈ 0.51π.

This shows that the deflection angle we calculated perturbatively can even work when the gravitational field is not weak or when deflection angle is not small anymore.

In Fig.2 (b), the contribution from each order of Eqs. (29) are plotted (solid lines), together with the values at the corresponding orders for the Schwarzschild spacetime for comparison (dashed lines). It is seen that the contribution from each order decreases roughly by the same factor as the order increases, suggesting that the summation (28) is convergent for the given range of b. It is also clear that as dictated by the asymptotic expansion (23) and Eq.

(29e), the contribution starts to deviate from that Schwarzschild spacetime result from and above the fourth order.

Moreover, from these deviations, one sees that an increase of ˆl indeed decreases the deflection angle. Furthermore, 17

1.4 1.4 1.2 1.2 0.100 n 1.0 1.0 H , n ( m / b ) n 0.8 α 0.8 0.001 H , n or S α

0.6 0.6 n

0.4 0.4 -5 ( m / b ) 10

7.0 7.5 8.0 8.5 9.0 9.5 10.0 n H 0.2 b/m

0.0 10-7 20 40 60 80 100 8 10 12 14 b/m b/m

(a) (b) 10 1

8 0.100

0.010

6 n

α 0.001 ( m / b ) 4 n H 10-4 2 10-5

0 10-6 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.5 1.0 1.5 2.0 v b/m

(c) (d)

FIG. 2: Deflection angles in the Hayward spacetime. (a) Partial sums (30) (solid curves) and the exact value (red dashed curve) of the exact deflection angle for b/m from 7 to 100 and from 7 to 10 in the inset and v = c, ˆl = 0.5. From bottom to top curve in each plot, the maximum index in the partial sum increases from 1 to 13. (b) Contribution from each order in Eq. (29) (from top to bottom curve the order increases from 1 to 13) for v = c, ˆl = 0.5 and corresponding results in the Schwarzschild case

(dashed lines). (c) Partial sum αH,13 (solid curves) and the exact deflection angle (dashed curves) for b = 10m and ˆl1 = 0.5

(red lines), ˆl2 = 0.85 (blue lines) and ˆl3 = 1.4 (green lines). (d) Contribution from each order (dashed curves), partial sum

αH,13 (top solid curve) and exact deflection angle (top red dashed curve) for b = 10m, v = 0.9. the horizontal line at 1 [as] shows that the thirteenth order result can reach this resolution for the impact parameter b as small as 10.7m.

In Fig.2 (c), the deflection angle is plotted against the velocity v for some fixed b and three representative values of

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ l, i.e., l1 = 0.5 < lc and lc < l2 = 0.85 > lp and l3 = 1.4 > lp. Similar to the Bardeen BH spacetime case, for Hayward ˆ ˆ ˆ ˆ spacetime with b = 10m, their exist a critical velocity around 0.43c for l = l1 and 0.41c for l = l2. From Fig.2 (c) it is ˆ ˆ ˆ seen that for l = l1 or l2, as velocity decreases from c, the deflection angle calculated both from the partial sum αH,13 18 and numerical integration increase monotonically. As v approaches vc, the deflection angle αH,13 calculated using the ˆ ˆ perturbative method starts to deviate from the true deflection angle around 0.50c for l1 and 0.48c for l2. Below these, the true deflection angles in both cases diverge as v → vc from above and the perturbative deflection angle becomes ˆ ˆ invalid. On the other hand, when l = l3, the photon sphere disappears and the partial sum αH,13 still roughly agree with the true value even when the velocity is as small as 0.38c.

ˆ In Fig.2 (d), we examine the effect of parameter l on the deflection angles. It is seen that the partial sum αH,13 and the numerical value overlaps for the entire range of ˆl from 0 to 2, which shows that our result also works after the parameter ˆl passes through the critical value. The deflection angle monotonically decreases as ˆl increases in the whole

ˆ ˆ ˆ range. In particular, when l passes lc and lp, the deflection angle does not experience any qualitative change, which is a reflection that the perturbative expansion depends only on the asymptotic expansions of the metric functions.

C. Deflection in the JNW spacetime

We consider now the most general spherically symmetric asymptotically flat exact solution to the Einstein massless scalar equations, i.e., JNW metric [38]

 β λ  β 1−λ A(r) = B(r)−1 = 1 − ,C(r) = r2 1 − , (31) r r where β is the location of the naked curvature singularity and β < r < ∞. β, λ are related to the ADM mass m and scalar charge q by

p 2m β = 2 m2 + q2, λ = . (32) β

Clearly, when q = 0, we will have λ = 1 and the JNW spacetime reduces to the Schwarzschild spacetime. In order for m ≥ 0, it is required that 0 ≤ λ ≤ 1 [27, 39]. Moreover, it was known that in the strong field limit, the deflection

1 angle as well as the GL features of the JNW metric with 2 < λ < 1 are quite similar to the case of Schwarzschild metric, because of the existence of a photon sphere at

1 + 2λ r = β (33) J,p 2

1 in this case [39]. While if 0 < λ < 2 , the above photon sphere is within the naked singularity at r = β and therefore not accessible. Consequently, the GL will only have finite number of images [27]. We would like to see whether the

1 critical value λc = 2 , i.e. q = m will play any role in the deflection angle in the weak field limit. 19

Using the metric functions (31), and going through the procedure from Eq. (7) to Eq. (13), the change of the angular coordinate for a particle ray with velocity v and impact parameter b in the JNW spacetime becomes

7 n 8 X  β   β  I (r , v, λ) = J (v, λ) + O , (34) J 0 n r r n=0 0 0 where the coefficients are

J0 =π, (35a)  1  J =λ + 1 , (35b) 1 v2 π 1 1  π 1 3π  1 1  J = − + λ + + λ2 − + − 1 − , (35c) 2 16 2 2v2 4 2 4 v2 2v4 π  1 π  1 π  1  π 1 3π  1 1  J = − + λ + + + + λ2 − + − 1 − 3 16 24 16 24 16 v2 4 2 4 v2 2v4 7 π 101  1 13 3π  1 7  + λ3 − + − π + − + , (35d) 8 4 24 v2 8 4 v4 24v6 55π 3π 3 3π 3  1   1 π  1 21π  1  1 9π  1  J = − + λ − + − + λ2 + + + + − 4 1024 32 16 32 16 v2 48 16 24 64 v2 48 64 v4 21 3π 101 3π  1 39 9π  1 7  + λ3 − + − + − + 16 8 16 2 v2 16 8 v4 16v6 3π 55 81π 175 1 75π 47 1 3π 15 1 3  + λ4 − + − + − + − − , (35e) 8 48 32 24 v2 32 6 v4 4 8 v6 16v8 23π 149π 1513 149π 1513 1  13 π 13 3π  1 13 9π  1  J = − + λ − + − + λ2 − + − + − 5 512 1536 5760 1536 5760 v2 24 8 12 32 v2 24 32 v4 323 13π 2291 35π  1  9 7π  1 7π 37  1  + λ3 − + − + − + − 576 96 576 48 v2 64 16 v4 32 192 v6 3π 55 81π 175 1 75π 47 1 3π 15 1 3  + λ4 − + − + − + − − 4 24 16 12 v2 16 3 v4 2 4 v6 8v8 1975 13π 87683 227π  1 2287 59π  1 255 3π  1 83  + λ5 − + − + − + − + , (35f) 1152 24 5760 48 v2 192 16 v6 128 4 v8 640v10 617π 265π 601 265π 601  1  J = − + λ − + − 6 16384 3072 2304 3072 2304 v2  9413 2815π 9413 1389π  1  9413 713π  1  + λ2 − + − + − 11520 12288 5760 4096 v2 11520 2048 v4 55π 905 65π 665  1 25π 475 1 35π 155 1  + λ3 − + − + − + − 192 1152 96 1152 v2 32 128 v4 64 128 v6 1511π 1687 4409π 6799 1 1519π 1535 1  17 103π  1 157 19π  1  + λ4 − + − + − + + + − 3072 1152 1024 576 v2 512 144 v4 192 256 v6 384 64 v8 9875 65π 87683 1135π  1 7645 305π  1 11435 295π  1 1275 15π  1 + λ5 − + − + − + − + − 2304 48 2304 96 v2 128 16 v4 384 32 v6 256 8 v8 83  635π 29863 2365π 166063 1 2683π 756193 1 + + λ6 − + − + − 256v10 768 11520 256 5760 v2 128 11520 v4 1031π 9709 1 161π 12383 1 3π 263 1 73  + − + − + − − , (35g) 64 192 v6 32 768 v8 4 128 v10 768v12 20

523π 17477π 368839 17477π 368839  1  J = − + λ − + − 7 16384 245760 1612800 245760 1612800 v2 3253 1023π 3253 1671π  1 3253 699π  1  + λ2 − + − + − 3840 4096 1920 4096 v2 3840 2048 v4 38473π 144349 56761π 1028999 1 35933π 34481 1 4799π 90137 1  + λ3 − + − + − + − 61440 76800 30720 230400 v2 20480 5120 v4 6144 46080 v6 171 409π 67 267π  1 115 1443π  1 617 651π  1 + λ4 − + + + − + − 128 1024 64 1024 v2 16 512 v4 64 256 v6 277 57π  1  296099 18617π 9510937 97451π  1 90829 19389π  1 + − + λ5 − + − + − 128 64 v8 76800 15360 230400 7680 v2 1536 1024 v4 94747 9373π  1  2501 57π  1 3π 9857  1  + − + − − + − 4608 1536 v6 3072 256 v8 8 15360 v10 635π 29863 7095π 166063 1 8049π 756193 1 + λ6 − + − + − 256 3840 256 1920 v2 128 3840 v4 3093π 9709 1 483π 12383 1 9π 789 1 73  + − + − + − − 64 64 v6 32 256 v8 4 128 v10 256v12 930367 4933π 88329593 11137π  1 2553593 67613π  1 8364067 22201π  1 + λ7 − + − + − + − 230400 3840 1612800 640 v2 15360 1280 v4 46080 384 v6 269615 1791π  1 312923 51π  1 10713 3π  1 523  + − + − + − + . (35h) 3072 64 v8 15360 8 v10 5120 4 v12 7168v14

The β/r0 in Eq. (35) can be expanded as a power series of β/b using Eq. (12), given by  2     3      β β β 1 1 1 β 1 1 1 2 3 1 1 = + − + λ 2 + + + λ − 2 − + λ + 2 + 4 r0 b b 2 2v 2 b 8 2v 2 8 v 8v β 4   1 1  1 5 1  1 19 3  + λ + + λ2 − − − + λ3 + + b 6v2 6 2 4v2 4v4 3 12v2 4v4 β 5  1  1 1   35 7 7  + − + λ + + λ2 + + b 128 48v2 48 192 16v2 64v4  25 109 21 1  125 37 149 1 1  +λ3 − − − − + λ4 + + + − 48 48v2 16v4 16v6 384 16v2 64v4 4v6 128v8 β 6   1 1   1 7 1   3 37 1 1  + λ − − + λ2 + + + λ3 + + + b 40v2 40 16 48v2 24v4 16 48v2 2v4 24v6  9 175 97 3  27 781 11 41  +λ4 − − − − + λ5 + + + 16 48v2 24v4 4v6 80 240v2 2v4 24v6 β 7  1  3 3   2387 341 341  + + λ − − + λ2 − − − b 1024 1280v2 1280 46080 2880v2 9216v4  49 193 45 5  1715 325 6125 125 25  +λ3 + + + + λ4 + + + + 384 384v2 128v4 128v6 9216 288v2 4608v4 384v6 3072v8  2401 21121 1245 505 55 1  +λ5 − − − − − + 3840 3840v2 128v4 128v6 256v8 256v10 16807 12931 103291 2575 1835 3 1  β 8 +λ6 + + + + − + + O (36) 46080 2880v2 9216v4 384v6 3072v8 128v10 1024v12 b

Substituting this into Eq. (34), we obtain the change of angular coordinates as a series of β/b for general velocity

7 n 8 X β  β  I (b, v, λ) = J 0 (v, λ) + O (37) J n b b n=0 21 where the coefficients for the first few orders are

0 J0 =π, (38a)  1  J 0 =λ + 1 , (38b) 1 v2 π 1 3  J 0 = − + λ2π + , (38c) 2 16 4 4v2 1 1  3 49 5 1  J 0 = − λ + + λ3 + + − , (38d) 3 3 3v2 4 12v2 4v4 12v6 9π  5 25 5  1 65 55  J 0 = − λ2π + + + λ4π + + , (38e) 4 1024 32 64v2 64v4 4 32v2 32v4  4 4  25 109 7 1  125 6841 J 0 =λ + − λ3 + + + + λ5 + 5 45 45v2 36 36v2 4v4 12v6 144 720v2 119 65 3 1  + + − + , (38f) 8v4 24v6 16v8 80v10 25π  259 1813 259   315 6125 3395 105  J 0 = − + λ2π + + − λ4π + + + 6 16384 4096 12288v2 6144v4 1024 3072v2 1536v4 256v6  81 3367 4081 315  + λ6π + + + , (38g) 256 768v2 384v4 64v6  4 4  343 1351 21 7  J 0 = − λ + + λ3 + + + 7 175 175v2 900 900v2 20v4 60v6  2401 21121 83 101 11 1  − λ5 + + + + + − 1800 1800v2 4v4 12v6 24v8 120v10 16807 1979713 21319 17671 1375 437 13 1  + λ7 + + + + − + − (38h) 14400 100800v2 320v4 320v6 192v8 960v10 320v12 448v14

We plot the deflection angle αJ ≡ IJ(b, v, λ) − π in Fig.3. In Fig.3 (a), we fix λ = 3/4 so that a photon sphere

10 is located using Eq. (33) at r = 3 m and plot the partial sum of the deflection angle as the sum of Eqs. (38) to the n-th order

n i X β  α = J 0 (v, λ) , (39) J,n n b i=1 and the true value of the deflection angle obtained by numerically integrating Eq. (6) for the JNW metric. It is seen that similar to the previous Bardeen and Hayward cases, the deflection angle monotonically decreases as b increases, and the contribution from lower order in Eq. (38) dominates. If b decreases instead, the contribution from the fourth order, Eq. (38b) becomes more apparent. In the whole range of b, the partial sum of the highest order αJ,7 overlaps with the true value of the deflection angle, even when b is as small as 7m. At this point, the deflection angle reaches about 1.04 and the gravitational field is not weak anymore. 22

1 1.0 1.1 1.0 0.100 n 0.8 0.9 0.8 0.010 J , n ( m / b ) n 0.6 α 0.7 J , n 0.6 0.001 α and S

0.5 n 0.4 10-4 0.4 7.0 7.5 8.0 8.5 9.0 9.5 10.0

( m / b ) -5 0.2 n 10 b/m J 10-6 0.0 20 40 60 80 100 10 15 20 25 30 b/m b/m

(a) (b) λ 1. 0.8 0.6 0.4 1 5

4 0.100 n

J ,7 3 α 0.010 ( m / b ) n 2 J 0.001 1

10-4 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.5 1.0 1.5 2.0 2.5 v/c q/m

(c) (d)

FIG. 3: Deflection angles in the JNW spacetime. (a) Partial sums (39) (solid curves) and the exact deflection angle (dashed red curve) for b/m from 7 to 100 and from 7 to 14 in the inset and v = c, λ = 3/4. From bottom to top curve in each plot, the maximum index in the partial sum increases from 1 to 7. (b) Contribution from each order (from top to bottom curve the order increases from 1 to 7) for v = c, λ = 3/4. (c) Partial sums and the true value of the deflection angle for λ = 3/4 and 2/3 and b = 10m. (d) Contributions from each order of Eq. (38), the partial sum αJ,7 and the true deflection angles for b = 10m, v = 0.9c.

In Fig.3 (b), we plot the contribution to the deflection angle from each order in Eq. (38) and the corresponding orders in the Schwarzschild spacetime. As expected, the contribution decreases linearly in the logarithmic plot as

β the order increases, suggesting that the sum as a series of b will converge. Furthermore, the difference between the 23

JNW and the Schwarzschild cases appears from the very first order, in contrast to the Bardeen and Hayward cases which differs from the Schwarzschild case from much higher orders. The fundamental reason of this is that the JNW spacetime although can reduce to Schwarzschild one when q = 0, is not a perturbation of the Schwarzschild spacetime in the usual sense [40]. Nevertheless, a comparison between the JNW and Schwarzschild results shows that a nonzero scalar charge q typically lowers the contribution from each order.

In Fig.3 (c), the effect of velocity v is studied for two value of λ, λ1 = 0.8 > λc and λ2 = 0.4 < λc. Again, similar to the case of Bardeen and Hayward, smaller velocity will increase the deflection angle. For λ1 = 0.8, it is known that there exists a photon sphere and therefore for a fixed b, the true deflection angle is expected to diverge when v approaches the critical value from above. From the plot, it is clear that for λ1, the critical value vc = 0.42c, around and below which clearly the deflection angle αJ,7 obtained perturbatively becomes invalid. For λ = λ2 however, since this spacetime does not have a photon sphere, clearly for the entire range of v considered, the partial summation αJ,7 agrees very well with the true deflection angle.

In Fig.3 (d), the dependence of the deflection angle on the parameter ˆq or equivalently λ is shown. Clearly at √ 1 q the critical λ = λc = 2 , i.e., m = 3, the deflection angle at all order, or its partial sum αJ,7 and true value, are continuous and smooth. This suggests basically that similar to the Bardeen and Hayward cases, the deflection angle should depend asymptotically on the metric function and therefore insensitive to the appearance/disappearance of the photon sphere at small r.

D. Deflection angle in the EBI spacetime

The EBI BH metric describing a nonlinear electrodynamics takes the form [41, 42]

Z ∞ −1 2m 2 p 4 2 2 2 2 A(r) = B(r) = 1 − + 2 ( ρ + β q − ρ )dρ, C(r) = r , (40) r β r r

p 2 2 where m is the mass and q = qE + qM is the charge of the spacetime. In the limit β → 0, this reduces exactly to the RN spacetime. The asymptotic behavior of the EBI solution also resemble that of the RN spacetime at large r.

The integral in Eq. (40) can be re-written using an elliptical function F of the first kind

( p " √ #) 2m 2 p |βq|3 r2 − |βq| 2 A(r) = 1 − + r2 − r4 + β2q2 + F arccos , (41) r 3β2 r r2 + |βq| 2 24

Expanding this function at large r, one can see that A(r) deviate from the metric function of the RN spacetime starting from the sixth order

2m q2 β2q4 110 A(r → ∞) = 1 − + − + O . (42) r r2 20r6 r

ˆ β |q| Depending on the values of the dimensionless parameters β ≡ m andq ˆ ≡ m , there can be zero, one or two nonzero solutions for A(r) = 0 (see Fig.4 for the partition of this parameter space). When the dimensionless parameter ˆq is small, i.e., 0 ≤ qˆ ≤ qˆc1(β) (the green region), there is one solution which labels the horizon of a regular BH. When the charge is at a medium value,q ˆc1(β) < qˆ < qˆc2(β) (the blue region), similar to the RN BH spacetime, there will be two event horizons. Asq ˆ further increases to beyondq ˆ > qˆc2(β) (the brown region), the two horizons merge and disappear so that there is only a naked singularity at r = 0 left. One of these critical lines,q ˆc1(β) can be solved analytically from the metric function (41) as

1 " # 3 1024πβˆ qˆc1 = . (43) 3
4 9Γ − 4

The other critical lineq ˆc2(β) can only be solved numerically from the metric function. These two curves meet at the √ ˆ 64 2π value of β = 3 2 , beyond which theq ˆc1(β) becomes the single boundary separating the one from the zero event 3Γ(− 4 ) horizon regions in this parameter space.

1.4

1.2

1.0

0.8  q

0.6

0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0  β

FIG. 4: The partition of the parameter space spanned by βˆ andq ˆ for numbers of event horizons. Green, blue and brow regions allow one, two and zero event horizons. 25

Using this metric, and going through the procedure from Eq. (7) to Eq. (13) for the EBI metric, one can find the change of angular coordinate as a series of r0 up to the eleventh order,

11 n 12 X  m   m  I (r , v, β,ˆ qˆ) = E (v, β,ˆ qˆ) + O (44) E 0 n r r n=0 0 0 with the coefficients given by

E0 =R0, (45a)

E1 =R1, (45b)

E2 =R2, (45c)

E3 =R3, (45d)

E4 =R4, (45e)

E5 =R5, (45f)  π 3π  E =R + + qˆ4βˆ2, (45g) 6 6 128 64v2  12  91 3π  1 3 9π  1  E =R + + − + − qˆ4βˆ2, (45h) 7 7 175 100 64 v2 4 32 v4 105π 1407π 17 1 1239π 677  1 57π 109 1  E =R + + − + − + + − qˆ4 8 8 2048 1280 25 v2 640 100 v4 32 20 v6  21π 41π π   + − − + qˆ6 βˆ2, (45i) 2048 320v2 640v4 16 1099 957π  1 3157 1707π  1 29209 1217π  1 1123  1  E =R + + − + − + − + − 9π qˆ4 9 9 45 100 1280 v2 100 160 v4 600 80 v6 40 v8  16  2537 289π  1  1063 139π  1  13 7π  1   + − + − + + − + + − + qˆ6 βˆ2, (45j) 105 700 1280 v2 200 80 v4 40 80 v6 3969π  193 422013π  1  13919 111201π  1 E =R + + − + + − + 10 10 16384 25 40960 v2 100 2560 v4 124977π 30371 1 56103π 55411 1 2463π 4823 1  + − + − + − qˆ4 1280 100 v6 640 200 v8 64 40 v10  1323π 618 27873π  1 35601 5193π  1 12587 1293π  1 129 309π  1  + − + − + − + − + − qˆ6 8192 175 5120 v2 700 320 v4 200 64 v6 40 320 v8  189π 1987π 181π 17π    7π 35π  + + + + qˆ8 βˆ2 − + qˆ6βˆ4, (45k) 16384 8192v2 5120v4 2560v6 4096 2048v2 8 8731 273177π  1 51361 3307761π  1 314957 638751π  1 E =R + + − + − + − 11 11 5 100 40960 v2 100 20480 v4 200 1280 v6 420367 429933π  1 1060213 134619π  1 14693 4683π  1  + − + − + − qˆ4 200 640 v8 800 320 v10 32 32 v12  16 100883π 35128 1 47233π 1251673 1 127821π 110218 1 + − + − + − + − 11 20480 525 v2 512 4200 v4 640 175 v6  589141 25133π  1  4831 997π  1   8 9493 25653π  1 + − + + − + qˆ6 + + − 1200 160 v8 240 160 v10 33 1050 40960 v2 26

3774 129483π  1 3753 3609π  1  11 21π  1   + − + − + − − qˆ8 βˆ2 175 20480 v4 800 2560 v6 160 1280 v8  32  491 35π  1  83 175π  1  + − + − + + − + qˆ6βˆ4, (45l) 2079 1512 2048 v2 168 1024 v4 where the Rn are the coefficient of order n in the change of the angular coordinate in RN spacetime. Their values are given in Ref. [1] and also listed in Eq. (A8).

Using Eq. (12), the expansion parameter m can be expressed as a power series of m , given by r0 b

11 m X mn m12 = C + O (46) r E,n b b 0 n=1 with the coefficients given by

CE,1 =CR,1, (47a)

CE,2 =CR,2, (47b)

CE,3 =CR,3, (47c)

CE,4 =CR,4, (47d)

CE,5 =CR,5, (47e)

CE,6 =CR,6, (47f) qˆ4βˆ2 C =C − , (47g) E,7 R,7 40v2  1 3  C =C + − + qˆ4βˆ2, (47h) E,8 R,8 10v2 40v4  3 9 3   1 3   C =C + − + − qˆ4 + − qˆ6 βˆ2, (47i) E,9 R,9 10v2 20v4 16v6 20v2 80v4  4 9 3 7   3 9 3   C =C + − + − + qˆ4 + − + qˆ6 βˆ2, (47j) E,10 R,10 5v2 5v4 2v6 16v8 10v2 20v4 16v6  2 6 15 35 63   6 27 9 21  C =C + − + − + − qˆ4 + − + − qˆ6 E,11 R,11 v2 v4 2v6 8v8 64v10 5v2 10v4 4v6 32v8  3 9 3   qˆ6βˆ4 + − + − qˆ8 βˆ2 + . (47k) 40v2 80v4 64v6 144v2 where the CR,i are the corresponding coefficients for the RN spacetime listed in Eq. (A4). Substituting this expansion into Eq. (44), we finally obtain the change of the angular coordinate in the EBI spacetime for general velocity, as

11 X mn m12 I (b, v, β,ˆ qˆ) = E0 (v, β,ˆ qˆ) + O (48) E n b b n=0 with coefficients

0 0 E0 =R0, (49a) 27

0 0 E1 =R1, (49b)

0 0 E2 =R2, (49c)

0 0 E3 =R3, (49d)

0 0 E4 =R4, (49e)

0 0 E5 =R5, (49f) π  6  E0 =R0 + 1 + qˆ4βˆ2, (49g) 6 6 128 v2 4  42 35 E0 =R0 + 3 + + qˆ4βˆ2, (49h) 7 7 175 v2 v4 21π  120 240 64  16 16  E0 =R0 + 5 + + + qˆ4 − 1 + + qˆ6 βˆ2, (49i) 8 8 2048 v2 v4 v6 v2 v4 16  36 126 84 9  16  27 63 21  E0 =R0 + 1 + + + + qˆ4 − 1 + + + qˆ6 βˆ2, (49j) 9 9 45 v2 v4 v6 v8 105 v2 v4 v6 63π  3150 16800 20160 5760 256 E0 =R0 + 63 + + + + + qˆ4 10 10 16384 v2 v4 v6 v8 v10  280 1120 896 128  30 80 32  7π  10 −6 7 + + + + qˆ6 + 3 1 + + + qˆ8 βˆ2 − 1 + qˆ6βˆ4, (49k) v2 v4 v6 v8 v2 v4 v6 4096 v2 8  66 495 924 495 66 1  E0 =R0 + 1 + + + + + + qˆ4 10 11 5 v2 v4 v6 v8 v10 v12 16  55 330 462 165 11   8 32 48 224 8   − 1 + + + + + qˆ6 + + + + + qˆ8 βˆ2 11 v2 v4 v6 v8 v10 33 3v2 v4 5v6 v8 32  22 33 − 1 + + qˆ6βˆ4. (49l) 2079 v2 v4

0 The Rn are the corresponding expansion coefficients in the RN spacetime listed in Eq. (A10). It is seen from these equations that the change of the angular coordinate in the EBI spacetime shows difference from the RN spacetime from the sixth order. Similar to the Bardeen and Hayward vs. the Schwarzschild case, this difference in the deflection angle at the sixth order also should imply that the metric lapse functions of the EBI and RN spacetimes are asymptotically different only from the sixth order. This is indeed verified from the expansion (42).

To study how the deflection angle depends on various kinematic and spacetime parameters, we plot αE in Fig.5.

In Fig.5 (a), we plot the partial sums αE,n defined using Eqs. (49) as

n X mi α = E0(v, β,ˆ qˆ) . (50) E,n i b i=1

It is seen that similar to the cases in the Bardeen, Hayward and JNW spacetime, the partial sum αE,n approaches the true value obtained numerically as n increases. The highest order partial sum αE,11 agrees with the true value 28

1 1.0 1.1 1.0 0.100 0.9 0.8 0.010

0.8 n E , n 0.6 α 0.7

E , n 0.001

0.6 ( m / b ) α n

0.5 E 0.4 10-4 0.4 7.0 7.5 8.0 8.5 9.0 9.5 10.0 0.2 -5 b/m 10 0.0 10-6 20 40 60 80 100 8 10 12 14 b/m b/m

(a) (b) 8 v=vc 1

6 0.100

n 0.010

E ,11 4 α ( m / b ) n

E 0.001 2 10-4

0 10-5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 v/c q

(c) (d) 10-6

10-7 n 10-8 ( m / b ) n 6th order -9 - R 10

n 7th order 8th order 10-10 ( m / b ) 9th order n

E 10th order -11 10 11th order

10-12 0.01 0.05 0.10 0.50 1 β

(e)

FIG. 5: Deflection angles in the EBI spacetime. (a) Partial sums (50) (from bottom to top curve in both plots, the maximum index of the partial sum increases from 1 to 11) and the exact deflection angle (dashed red curve) for v = c, βˆ = 1 andq ˆ = 0.43.

(b) Contribution from each order (from top to bottom curve n increases from 1 to 11) for v = c, βˆ = 1 andq ˆ = 0.43. (c) Partial sum αE,11 (solid curves) and the true value (dashed curves) of the deflection angles forq ˆ = 0.43 (red curves) andq ˆ = 1.4 (blue curve) respectively and b = 10m and βˆ = 1. (d) Contributions from each order (dashed curves), the partial sum αE,11 (solid line) and the true value (dashed curve overlapping the solid line) for b = 10m, v = 0.9c and βˆ = 1. (e) The difference between contribution from each order in the EBI and RN spacetimes for b = 10m, v = 0.9c and βˆ = 1. 29 for b as small as 7m, at which the gravitational field is not weak anymore and deflection angle is not small.

The Fig.5 (b) shows the contribution from each order of Eq. (49). It is clear that for a fixed b, as the order increases, the contribution from each order decreases by a constant small factor, therefore the summation in Eq. (48) will converge.

In Fig.5 (c), the deflection angle is plotted against the velocity v for fixed b = 10m, βˆ = 1 andq ˆ = 0.43. As v decreases, the partial sum αE,11 and exact deflection angle overlap and increase until about v ≈ 0.55c from which the true deflection angle starts to diverge due to the existence of one event horizon in the spacetime for the given choice of

ˆ β andq ˆ and the consequent existence of an critical vc ≈ 0.42c. This shows again that the partial sum to high order as an asymptotic expansion although can work when the gravitational field is not weak, is still not accurate near event horizon.

ˆ ˆ ˆ In Fig.5 (d), the effect of charge ˆq is plotted for two representative values of β, β1 = 1 and β2 = 3. For the first, it is seen from Fig.4 that there might exist zero, one or two event horizons as ˆq increases, while for the second βˆ, there can only be zero or one event horizon. As seen from the plot, the deflection angles decreases monotonically and smoothly asq ˆ increases, regardless whether it is near the critical values or not. This shows that our perturbative deflection angle works for all the cases whether there is one, two or no event horizons.

Comparing to the Bardeen, Hayword and JNW spacetimes, the EBI spacetime contains one more parameter, βˆ.

However comparing the same order contribution from Eqs. (48) and Eqs. (A8), we see that the effect of βˆ to deflection angle is much smaller than the effect of chargeq ˆ. For this reason, in Fig.5 (e), we are not plotting the contributions from each order in Eq. (48), but the difference between each order of the deflection angles of the RN and EBI spacetimes, against parameter βˆ. It is clear that this difference only starts to appear from the sixth order. Moreover,

ˆ4 0 although differences of order β or above are present in order E10 and above, all the lines in this log-log plot has a slope of value 2, which suggests that the βˆ2 terms dominate higher orders.

IV. DISCUSSIONS

In this paper we used the perturbative expansion of the integrand in the change of the angular coordinate, (6), and integrated it to obtain the deflection angles in powers of the closest radial coordinate r0 and impact parameter b for signals with general velocity v in the Bardeen, Hayward, JNW and EBI spacetimes to very high orders (indeed they can reach any desired order).

There are a few findings that apply to all general static and spherically symmetric spacetimes worth mentioning 30 here. The first is that the deflection angles in the weak field limit is determined by the asymptotic behavior of the

m metric functions solely. Our finding shows that a deviation of the metric function at the n-th power of b from a given base metric, will result in a deviation of the deflection angle at the same order from that of the base metric.

One important consequence of this is the continuous dependence of the deflection angle on (charge) parameters of the spacetime. This is even so when these parameters pass their critical values at which a BH spacetime becomes non-BH one or other critical transit happens, because these dramatic changes usually happen at small r but not asymptotic

m r. In this sense, the deflection angle found perturbatively as a power series of b will qualitatively be the same for spacetimes that are drastically different due to the variation of the metric parameter(s) [43]. This is in clear contrast with the deflection angle in the strong field limit, which experiences drastic change when the parameters pass their critical values for the existence/non-existence of the event horizon or photon sphere [32].

The second finding, obtained by comparing with the exact deflection angles calculated numerically, is that the deflection angles found perturbatively in the weak field limit, i.e., large r0 or b limit, can work for surprisingly small b as long as the order is high enough. The deflection angle for the smallest valid b can also be large, at least 0.3π for the four spacetime we tested.

There are a few directions along which the results in this work are usable or can be extended. The first is to substitute these deflection angles in the lensing equation to use the corresponding GL effects to examine the properties of the central spacetime. Although the qualitative effects of the spacetime (charge) parameters to the deflection angle are not large, quantitatively the deflection angles and consequently the images in the corresponding GL effects are still influenced by these parameters. Therefore they can be put to constrain these parameters. We point out that there are already many works along this line. The GL effect in the Bardeen spacetime has been considered in Refs. [29, 44–46] and in the Hayward spacetime in Refs. [30, 47], in JNW spacetime in Refs. [26, 32, 33, 48] and in the EBI spacetime in Ref. [42]. Since these works are all for null rays, the results here can still be applied to timelike particles such as neutrino and massive GWs.

The second application is to use the GL effect in these spacetime to constrain the velocity of the messenger. For neutrinos, the velocity can be simply connected to their neutrino absolute mass and mass hierarchy. For GW, its velocity itself plays a key role in distinguishing some modified gravity models. For the particular four spacetimes

m studied in this work, because the velocity appears in the very first b order of the deflection angle rather than the third or higher orders in which the spacetime (charge) parameters appear, the application in this direction is expected to be more practical. 31

Acknowledgments

This work is supported in part by the NNSF China 11504276 and MOST China 2014GB109004.

Appendix A: Formula for the deflection angles in the Schwarzschild and RN spacetimes

In Ref. [1], the change of the angular coordinate for signal with velocity v and impact parameter b in the

Schwarzschild spacetime with mass m were found. In terms of the closest coordinate r0, this takes the form

∞ n X  m  I (r , v) = S (v) , (A1) S 0 n r n=0 0 where the coefficients to the thirteenth order are

S0 =π, (A2a)  1  S =2 1 + , (A2b) 1 v2 3π 1 1 S = + (3π − 2) − 2 , (A2c) 2 4 v2 v4 10  3π  1 1 7 1 S = + 26 − − 3(2π − 3) + , (A2d) 3 3 2 v2 v4 3 v6 105π 93π  1 69π  1 1 1 S = + − 18 + − 86 + (12π − 23) − 3 , (A2e) 4 64 4 v2 4 v4 v6 v8 42 201π  1 1 1  207 1 83 1 S = − − 174 − (117π − 285) − (63π − 255) − 24π − + , (A2f) 5 5 16 v2 v4 v6 4 v8 20 v10 1155π 8787π  1 10851π  1 2007π 4207 1 S = + − 114 + − 1246 + − 6 256 64 v2 32 v4 4 3 v6 1  443 1 73 1 + (183π − 687) + 48π − − , (A2g) v8 4 v10 12 v12 858 4866 9897π  1  42105π  1  35535π  1 S = + − + 3809 − + 7431 − 7 35 5 128 v2 32 v4 16 v6 20861 3585π  1 34731  1 9253  1 523 1 + − + − 480π + − 96π + , (A2h) 4 2 v8 20 v10 40 v12 56 v14 225225π 185637π 3138 1 963063π 61062 1 S = + − + − 8 16384 256 5 v2 256 5 v4 331515π  1 670665π  1  67199 1 + − 31487 + − 34265 + 5670π − 32 v6 64 v8 4 v10  84097 1  19071 1 119 1 + 1188π − + 192π − − , (A2i) 20 v12 40 v14 8 v16 4862 174858 858633π  1 179357 363561π  1 1873517 1248837π  1 S = + − + − + − 9 63 35 2048 v2 5 32 v4 15 32 v6 733511 119295π  1 533009 329445π  1 1979077  1 + − + − + − 16524π 4 2 v8 4 8 v10 40 v12 32

2765857  1 2183697  1 14051 1 + − 2832π + − 384π + , (A2j) 280 v14 2240 v16 576 v18 2909907π 59009547π 112174 1 131274411π 3405538 1 37060149π 2249681 1 S = + − + − + − 10 65536 16384 35 v2 4096 35 v4 256 5 v6 71222277π 4427723 1 17880387π 17282663 1 1148721π 9235633 1 + − + − + − 256 5 v8 64 20 v10 8 20 v12  5480777 1  6354781 1  4442329 1 13103 1 + 45444π − + 6576π − + 768π − − , (A2k) 40 v14 280 v16 2240 v18 320 v20 8398 2560186 69502047π  1 9762087 687200823π  1 53657057 992311497π  1 S = + − + − + − 11 33 105 32768 v2 35 8192 v4 35 2048 v6 76784969 157756977π  1 98956459 199770417π  1 141049413 36346527π  1 + − + − + − 20 128 v8 20 128 v10 40 32 v12 58958723 920367π  1 812709823  1 344536399  1 + − + − 119736π + − 14976π 40 2 v14 2240 v16 6720 v18 54036263  1 98601 1 + − 1536π + , (A2l) 13440 v20 1408 v22 156165009π 1127476893π 1640018 1 15337199865π 14309566 1 S = + − + − 12 1048576 65536 105 v2 65536 21 v4 811056015π  1 40234680075π 108659839 1 2069467071π 505097751 1 + − 4948511 + − + − 512 v6 8192 7 v8 256 20 v10 1897384125π 1406267327 1 66690645π 103772803 1 5541675π 247796467 1 + − + − + − 256 60 v12 16 8 v14 4 56 v16  1247909311 1  109722263 1  109282609 1 15565 1 + 305280π − + 33600π − + 3072π − − , 1344 v18 960 v20 13440 v22 128 v24 (A2m)

371450 132929354 2692560531π  1 201718541 9148875279π  1 S = + − + − 13 429 1155 262144 v2 105 16384 v4 322639187 79680420705π  1 1640587857 38369599155π  1 + − + − 21 16384 v6 28 2048 v8 3320156743 77018937885π  1 5524434529 2824294095π  1 + − + − 28 2048 v10 40 64 v12 3935582091 1994399505π  1 19772726781 113269545π  1 + − + − 40 64 v14 448 8 v16 17027542177  1 6218405311  1 37267872227  1 + − 3974940π + − 758400π + − 74496π 1344 v18 2688 v20 147840 v22 9707157937  1 1423159 1 + − 6144π + . (A2n) 591360 v24 6656 v26

When expressing 1/x0 in powers of 1/b in the Schwarzschild spacetime, Eq. (12) can be used, and the result is

∞ m X mn = C , (A3) r S,n b 0 n=1 where the coefficients are

CS,1 =1, (A4a) 33

1 C = , (A4b) S,2 v2 2 1 C = + , (A4c) S,3 v2 2v4  1 1  C =4 + , (A4d) S,4 v2 v4  8 18 3 1  C = + + − , (A4e) S,5 v2 v4 v6 8v8  1 4 2  C =16 + + , (A4f) S,6 v2 v4 v6 32 200 200 25 5 1  C = + + + − + , (A4g) S,7 v2 v4 v6 v8 4v10 16v12  1 9 15 5  C =64 + + + , (A4h) S,8 v2 v4 v6 v8 128 1568 3920 2450 245 49 7 5  C = + + + + − + − , (A4i) S,9 v2 v4 v6 v8 v10 4v12 8v14 128v16  1 16 56 56 14  C =256 + + + + , (A4j) S,10 v2 v4 v6 v8 v10 512 10368 48384 70560 31752 2646 126 81 45 7 C = + + + + + − + − + , (A4k) S,11 v2 v4 v6 v8 v10 v12 v14 8v16 64v18 256v20  1 25 150 300 210 42  C =1024 + + + + + , (A4l) S,12 v2 v4 v6 v8 v10 v12 2048 61952 464640 1219680 1219680 426888 30492 5445 1815 605 C = + + + + + + − + − S,13 v2 v4 v6 v8 v10 v12 v14 4v16 16v18 64v20 77 21 + − . (A4m) 128v22 1024v24

If we express the change of angular coordinate in terms of impact parameter, it becomes

∞ X mn I (b, v) = S0 (v) , (A5) S n b n=0 where the coefficients to the ninth order are

0 S0 =π, (A6a)  1  S0 =2 1 + , (A6b) 1 v2 3π  4  S0 = 1 + , (A6c) 2 4 v2 5 15 5 1  S0 =2 + + − , (A6d) 3 3 v2 v4 3v6 105π  16 16 S0 = 1 + + , (A6e) 4 64 v2 v4 21 105 210 42 3 1  S0 =2 + + + − + , (A6f) 5 5 v2 v4 v6 v8 5v10 1155π  36 30 4  S0 = 1 + + + , (A6g) 6 256 v2 v4 v6 34

429 3003 3003 3003 429 143 13 1  S0 =2 + + + + − + − , (A6h) 7 35 5v2 v4 v6 v8 5v10 5v12 7v14 45045π  320 2240 3584 1280 S0 = 5 + + + + , (A6i) 8 16384 v2 v4 v6 v8 2431 21879 29172 68068 43758 4862 884 204 17 1  S0 =2 + + + + + − + − + , (A6j) 9 63 7v2 v4 v6 v8 v10 3v12 7v14 7v16 9v18 2909907π  100 1200 3840 3840 1024 S0 = 1 + + + + + , (A6k) 10 65536 v2 v4 v6 v8 v10 4199 46189 230945 969969 1385670 646646 58786 3230 323 95 S0 =2 + + + + + + − + − 11 33 3v2 v4 v6 v8 v10 v12 v14 v16 3v18 7 1  + − , (A6l) 3v20 11v22 22309287π  1008 18480 98560 190080 135168 28672 S0 = 7 + + + + + + , (A6m) 12 1048576 v2 v4 v6 v8 v10 v12 185725 2414425 4828850 10623470 26558675 26558675 9657700 742900 S0 =2 + + + + + + + 13 429 33v2 3v4 v6 v8 v10 v12 v14 37145 10925 1150 1150 25 1  − + − + − + . (A6n) v16 3v18 3v20 33v22 11v24 13v26

The change of the angular coordinate for signal with velocity v and impact parameter b in the RN spacetime with mass m and Q was also found in Ref. [1]. In terms of the closest coordinate r0 and to the eleventh order, it takes the form

∞ n X  m  I (r , v, qˆ) = R (v, qˆ) , (A7) R 0 n r n=0 0 where the coefficients are

R0 =S0, (A8a)

R1 =S1, (A8b) π π  R =S − + qˆ2, (A8c) 2 2 4 2v2  π  1 1  R =S − 2 − − 11 − (π − 1) qˆ2, (A8d) 3 3 2 v2 v4 45π 121π  1  29π  1 1  9π 7π π  R =S − + − 10 − 37 − + (2π − 3) qˆ2 + + − qˆ4, (A8e) 4 4 32 8 v2 4 v4 v6 64 8v2 8v4 28 473 85π  1 153π  1 665  1  43 1  R =S − + − − − 184 + − 27π − 4π − qˆ2 5 5 3 3 8 v2 2 v4 6 v6 6 v8   25π  1 11π 47 1 π 1 1  + 2 + 26 − − − + − qˆ4, (A8f) 16 v2 2 4 v4 2 4 v6 1575π 20101π 368 1 3397 4921π  1 2655π  1 R =S − + − − − + − 922 6 6 256 128 3 v2 3 16 v4 8 v6 1801  1  95 1  525π 1329π  1  441π  1 − − 79π + 8π − qˆ2 + + − 28 − 204 − 6 v8 6 v10 256 32 v2 8 v4 35

105π 267 1 3π 5 1  25π 157π 5π π  + − − − qˆ4 − + − + qˆ6, (A8g) 4 4 v6 2 4 v8 256 128v2 32v4 16v6  198 13381π 20413 1 96527π 13091 1 16249π 40831 1 R =S + − + − + − + − 7 7 5 128 15 v2 64 3 v4 8 6 v6 4777π 41507 1  91307 1  4049 1  + − + 208π − + 16π − qˆ2 4 12 v8 120 v10 120 v12  1565 4903π  1 14803 6811π  1 5147 3033π  1 + 18 + − + − + − 3 128 v2 12 16 v4 4 8 v6 6349 197π  1  103 1   411π  1 1143π 205 1 + − + 4π − qˆ4 + −2 + − 47 + − 24 2 v8 24 v10 128 v2 64 4 v4 11 15π  1 3π 1 1  − + + − qˆ6, (A8h) 8 16 v6 8 8 v8  105105π 14938 608893π  1 85417 1348113π  1 109123 766485π  1 R =S + − + − + − + − 8 8 4096 15 512 v2 5 256 v4 3 64 v6 189181 308185π  1 44805  1 221449  1 2821  1  + − + − 3795π + − 516π + − 32π qˆ2 6 32 v8 4 v10 120 v12 40 v14 121275π 301055π 1396 1 1081745π 20473 1 113345π 128557 1 + + − + − + − 8192 512 3 v2 512 3 v4 32 12 v6 117695π 73183 1  21215 1 301  1  + − + 320π − + − 10π qˆ4 64 12 v8 24 v10 24 v12  11025π  44751π  1  28461π  1 2105 5757π  1 81 105π  1 + − + 60 − + 710 − + − + + 4096 512 v2 128 v4 4 32 v6 8 32 v8 7 3π  1  1225π 803π 39π 11π 5π  + − qˆ6 + + + + − qˆ8, (A8i) 8 2 v10 16384 512v2 512v4 64v6 128v8  1144 398377π 993439 1 1193677π 884186 1 1756347π 1757143 1 R =S + − + − + − + − 9 9 7 512 105 v2 64 15 v4 32 10 v6 277265π 638909 1 151885π 2952011 1  662131 1 + − + − + 11094π − 4 3 v8 4 24 v10 20 v12  2430129 1  81603 1  572 29178 479403π  1 + 1232π − + 64π − qˆ2 + + − 560 v14 560 v16 5 5 1024 v2 605559 610641π  1 290011 1447125π  1 513933 83985π  1 + − + − + − 20 64 v4 4 64 v6 8 4 v8 963711 58815π  1 426581  1  5363 1  + − + − 948π + 24π − qˆ4 40 8 v10 160 v12 160 v14  88 51385π 3878 1 101435π 15251 1 20495π 66211 1 + − + − + − + − 3 512 3 v2 64 3 v4 8 8 v6 9275π 10331 1 2321 35π  1  187 1  + − − + + 5π − qˆ6 8 3 v8 48 4 v10 48 v12   11177π  1 297 2837π  1 97 303π  1 11 13π  1 5π 5  1  + 2 + 74 − + − + − + − + − qˆ8, 2048 v2 2 64 v4 4 64 v6 64 8 v8 16 64 v10 (A8j)

 6891885π 140900 252273505π  1 19342783 62200733π  1 R =S + − + − + − 10 10 65536 21 32768 v2 105 1024 v4 11143912 122414497π  1 37511813 100535597π  1 6043237 41702245π  1 + − + − + − 15 512 v6 30 256 v8 6 128 v10 36

10252363 530825π  1 459593  1 5586893  1 167371  1  + − + − 30578π + − 2864π + − 128π qˆ2 24 4 v12 5 v14 560 v16 560 v18 2837835π 11553985π 14306 1 39435901π 1745708 1 16127133π 7822791 1 + + − + − + − 32768 2048 3 v2 1024 15 v4 128 20 v6 21081145π 6302849 1 1601545π 7400429 1 207435π 674701 1 + − + − + − 128 12 v8 16 24 v10 8 8 v12  1201277 1 13547  1   945945π  26840595π  1 + 2636π − + − 56π qˆ4 + − + 1312 − 160 v14 160 v16 32768 16384 v2  18934095π  1 140245 23203875π  1 502879 5010525π  1 + 27705 − + − + − 2048 v4 2 1024 v6 8 256 v8 69555 740715π  1 3007 75π  1 225  1  218295π 2587225π  1 + − + + + − 15π qˆ6 + + − 110 4 128 v10 16 4 v12 16 v14 65536 16384 v2 2671225π  1 425675π 10105 1 37975π 613 1 1195π 131 1 + − 1910 + − + − + − 4096 v4 512 4 v6 512 2 v8 128 64 v10 45 25π  1  3969π 62417π 1507π 381π 77π 7π  + − qˆ8 − + + + − + qˆ10, (A8k) 64 16 v12 65536 32768v2 2048v4 1024v6 512v8 256v10  41990 163920193π 18362717 1 2939299557π 20900769 1 R =S + − + − + − 11 11 63 32768 315 v2 16384 35 v4 471598515π 122425265 1 523008717π 381772261 1 282754857π 280188473 1 + − + − + − 512 42 v6 256 60 v8 128 40 v10 84979881π 164824687 1  2294376137 1  109249577  1 + − + 426027π − + 80700 − π 64 40 v12 1680 v14 448 v16  909081107 1  24593209 1  4420 748997 69100279π  1 + 6528π − + 256π − qˆ2 + + − 40320 v18 40320 v20 7 15 16384 v2 188316941 137496109π  1 22599985 76127315π  1 408829123 70065547π  1 + − + − + − 420 1024 v4 12 128 v6 120 64 v8 35713057 59988655π  1 611485309 3294017π  1 303602051 167543π  1 + − + − + − 12 64 v10 480 8 v12 1120 2 v14 1288199  1  460919  1   24935133π 91848 1 + − 7000π + − + 128π qˆ4 + −260 + − 64 v16 2240 v18 16384 5 v2 339175455π 556915 1 153001887π 18959671 1 105099615π 5096833 1 + − + − + − 8192 4 v4 1024 40 v6 512 8 v8 14695425π 5865715 1 1567767π 11871201 1 41269  1 + − + − − + 30π 128 16 v10 64 160 v12 64 v14  14363 1  130 8080 7128915π  1 31615 37868925π  1 + 42π − qˆ6 + + − + − 320 v16 3 3 32768 v2 2 8192 v4 896225 23959375π  1 4992479 2175535π  1 147215 156075π  1 + − + − + − 24 2048 v6 192 256 v8 64 256 v10 5305 2695π  1 25π 1475 1   272293π  1 1524629π 685 1 + − + − qˆ8 + −2 + − 107 + − 384 64 v12 4 384 v14 32768 v2 16384 2 v4 36479π  1 2897π 191 1  35 449π  1 35π 7  1  + − 134 + − + − + − qˆ10, (A8l) 1024 v6 512 64 v8 128 256 v10 128 128 v12

The expansion of 1/r0 in power series of 1/b in the RN spacetime is

CR,1 =CS,1, (A9a) 37

CR,2 =CS,2, (A9b) qˆ2 C =C − , (A9c) R,3 S,3 2v2  2 1  C =C − + qˆ2, (A9d) R,4 S,4 v2 v4  2 3 1  1  1 3  C =C − 3 + + qˆ2 + + qˆ4, (A9e) R,5 S,5 v2 v4 4v6 2 v2 4v4  1 3 1   3 6 1  C =C − 16 + + qˆ2 + + + qˆ4, (A9f) R,6 S,6 v2 v4 v6 v2 v4 v6 40 200 150 25 5  C =C − + + + − qˆ2 R,7 S,7 v2 v4 v6 2v8 16v10 12 45 45 15   1 5 5  + + + + qˆ4 − + + + qˆ6, (A9g) v2 v4 2v6 16v8 2v2 4v4 16v6  2 15 20 5   1 6 6 1   4 18 12 1  C =C − 48 + + + qˆ2 + 40 + + + qˆ4 − + + + qˆ6, (A9h) R,8 S,8 v2 v4 v6 v8 v2 v4 v6 v8 v2 v4 v6 v8 224 2352 4900 2450 735 49 7  C =C − + + + + − + qˆ2 R,9 S,9 v2 v4 v6 v8 4v10 8v12 32v14 120 1050 1750 2625 525 35  + + + + + − qˆ4 v2 v4 v6 4v8 16v10 64v12 20 140 175 175 35   1 21 35 35  − + + + + qˆ6 + + + + qˆ8, (A9i) v2 v4 v6 4v8 32v10 2v2 8v4 16v6 128v8  1 14 42 35 7   1 12 30 20 3  C =C − 512 + + + + qˆ2 + 336 + + + + qˆ4 R,10 S,10 v2 v4 v6 v8 v10 v2 v4 v6 v8 v10  1 10 20 10 1   5 40 60 20 1  − 80 + + + + qˆ6 + + + + + qˆ8, (A9j) v2 v4 v6 v8 v10 v2 v4 v6 v8 v10 1152 20736 84672 105840 39690 2646 189 81 45  C =C − + + + + + − + − qˆ2 R,11 S,11 v2 v4 v6 v8 v10 v12 2v14 16v16 256v18 896 14112 49392 51450 15435 3087 147 63  + + + + + + − + qˆ4 v2 v4 v6 v8 v10 4v12 8v14 128v16 280 3780 11025 18375 33075 2205 105  − + + + + + − qˆ6 v2 v4 v6 2v8 16v10 32v12 128v14 30 675 1575 7875 4725 315   1 9 63 105 63  + + + + + + qˆ8 − + + + + qˆ10, (A9k) v2 2v4 2v6 16v8 64v10 256v12 2v2 2v4 8v6 32v8 256v10

(A9l)

Finally the change of the angular coordinate in terms of the impact parameter in the RN spacetime takes the form

0 0 R0 =S0, (A10a)

0 0 R1 =S1, (A10b) π  2  R0 =S0 − 1 + qˆ2, (A10c) 2 2 4 v2  6 1  R0 =S0 − 2 1 + + qˆ2, (A10d) 3 3 v2 v4 38

45π  12 8  3π  24 8  R0 =S0 − 1 + + qˆ2 + 3 + + qˆ4, (A10e) 4 4 32 v2 v4 64 v2 v4 7 140 70 28 1   15 15 1  R0 =S0 − 4 + + + − qˆ2 + 2 1 + + + qˆ4, (A10f) 5 5 3 3v2 v4 3v6 3v8 v2 v4 v6 1575π  30 80 32 105π  120 240 64 5π  90 120 16 R0 =S0 − 1 + + + qˆ2 + 5 + + + qˆ4 − 5 + + + qˆ6, 6 6 256 v2 v4 v6 256 v2 v4 v6 256 v2 v4 v6 (A10g)

6  1386 5775 4620 495 22 1   315 1050 630 45 1  R0 =S0 − 33 + + + + − + qˆ2 + 2 9 + + + + − qˆ4 7 7 5 v2 v4 v6 v8 v10 v12 v2 v4 v6 v8 v10  28 70 28 1  − 2 1 + + + + qˆ6, (A10h) v2 v4 v6 v8 105105π  56 336 448 128 17325π  336 1680 1792 384 R0 =S0 − 1 + + + + qˆ2 + 7 + + + + qˆ4 8 8 4096 v2 v4 v6 v8 8192 v2 v4 v6 v8 1575π  280 1120 896 128 35π  1120 3360 1792 128 − 7 + + + + qˆ6 + 35 + + + + qˆ8, (A10i) 4096 v2 v4 v6 v8 16384 v2 v4 v6 v8 8  10296 84084 168168 90090 8008 364 24 1  R0 =S0 − 143 + + + + + − + − qˆ2 9 9 7 v2 v4 v6 v8 v10 v12 v14 v16 4  9009 63063 105105 45045 3003 91 3  + 143 + + + + + − + qˆ4 5 v2 v4 v6 v8 v10 v12 v14 8  594 3465 4620 1485 66 1   45 210 210 45 1  − 11 + + + + + − qˆ6 + 2 1 + + + + + qˆ8, (A10j) 3 v2 v4 v6 v8 v10 v12 v2 v4 v6 v8 v10 6891885π  90 960 2688 2304 512 R0 =S0 − 1 + + + + + qˆ2 10 10 65536 v2 v4 v6 v8 v10 315315π  720 6720 16128 11520 2048 315315π  210 1680 3360 1920 256 + 9 + + + + + qˆ4 − 3 + + + + + qˆ6 32768 v2 v4 v6 v8 v10 32768 v2 v4 v6 v8 v10 10395π  1260 8400 13440 5760 512 + 21 + + + + + qˆ8 65536 v2 v4 v6 v8 v10 63π  3150 16800 20160 5760 256 − 63 + + + + + qˆ10, (A10k) 65536 v2 v4 v6 v8 v10 10  461890 6235515 23279256 29099070 11639628 881790 38760 R0 =S0 − 4199 + + + + + + − 11 11 63 v2 v4 v6 v8 v10 v12 v14 2907 190 7  + − + qˆ2 v16 v18 v20 4  109395 1312740 4288284 4594590 1531530 92820 3060 153 5  + 1105 + + + + + + − + − qˆ4 7 v2 v4 v6 v8 v10 v12 v14 v16 v18  5720 60060 168168 150150 40040 1820 40 1  − 4 65 + + + + + + − + qˆ6 v2 v4 v6 v8 v10 v12 v14 v16 10  1001 9009 21021 15015 3003 91 1  + 13 + + + + + + − qˆ8 3 v2 v4 v6 v8 v10 v12 v14  66 495 924 495 66 1  − 2 1 + + + + + + qˆ10, (A10l) v2 v4 v6 v8 v10 v12 39

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