Deection in higher dimensional spacetime and asymptotically non-at spacetimes Jinhong HeQianchuan WangQiyue Hu and Li Feng

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Deﬂection in higher dimensional spacetime and asymptotically

non-ﬂat spacetimes

Jinhong He,∗Qianchuan Wang,∗Qiyue Hu, and Li Feng

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

Junji Jia†

Center for Astrophysics &MOE Key Laboratory of Artiﬁcial Micro- and Nano-structures,

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

(Dated: February 16, 2023)

Abstract

Using a perturbative technique, in this work we study the deﬂection of null and timelike signals in

the extended Einstein-Maxwell spacetime, the Born-Infeld gravity and the charged Ellis-Bronnikov

(CEB) spacetime in the weak ﬁeld limit. The deﬂection angles are found to take a (quasi-)series

form of the impact parameter, and automatically takes into account the ﬁnite distance eﬀect of

the source and observer. The method is also applied to ﬁnd the deﬂections in CEB spacetime

with arbitrary dimension. It’s shown that to the leading non-trivial order, the deﬂection in some

n-dimensional spacetimes is of the order O(M/b)n−3. We then extended the method to spacetimes

that are asymptotically non-ﬂat and studied the deﬂection in a nonlinear electrodynamical scalar

theory. The deﬂection angle in such asymptotically non-ﬂat spacetimes at the trivial order is found

to be not πanymore. In all these cases, the perturbative deﬂection angles are shown to agree with

numerical results extremely well. The eﬀects of some nontrivial spacetime parameters as well as

the signal velocity on the deﬂection angles are analyzed.

Keywords: deﬂection angle, perturbative method, timelike signal, high dimensional spacetime

∗These authors contributed equally to this work.

†Corresponding author: junjijia@whu.edu.cn

arXiv:2210.00938v2 [gr-qc] 15 Feb 2023

I. INTRODUCTION

The deﬂection and gravitational lensing (GL) of light rays are important tools in astron-

omy. The former historically contributed signiﬁcantly to the acceptance of general relativity

(GR) by scientists [1]. And the latter is being used to ﬁnd exoplanets [2, 3], to measure

the mass distribution of galaxies and their clusters [4, 5], and to test theories beyond GR

[6, 7]. In recent years, with the observation of gravitational wave (GW) [8] and the black

hole shadow [9–11], the deﬂection and GL of GW, as well as the deﬂection and GL of light

rays in the strong ﬁeld limit, have drawn enormous amount of attention.

On the other hand, after the discovery of cosmic rays [12] and the SN 1987A neutrinos

[13, 14], especially after the conﬁrmation of the extrasolar origin of the former [15, 16] and

the nonzero mass of the latter [17, 18], people become aware that massive signals such as

cosmic rays and neutrinos from various sources can also experience gravitational deﬂection

and be messengers of GL. One particularly encouraging progress in this direction is the

discovery of GLed supernovas in recent years [19, 20].

To theoretically investigate the deﬂection of these signals, recently two methods have

been intensively used. One is to use the Gauss-Bonnet theorem method [21–23], which has

been developed to handle both null and timelike signals [23–26], and to take into account

the ﬁnite distance eﬀect of the source and detector [27–29], as well as the electromagnetic

force [26, 29, 30]. The other method is the perturbative method developed by some authors

of this paper. The method can also calculate the deﬂection of both null and timelike signals

[31] and has been extended to include the ﬁnite distance eﬀect [32] and the extra kind of

force [33, 34] too. Moreover, the perturbative method can be used in arbitrary stationary

and axisymmetric spacetimes [31, 32], as well as in the strong ﬁeld limit [35].

There are two folds of motivations of this work. The ﬁrst is to apply the perturbative

method previously developed to other interesting spacetimes, namely the extended Einstein-

Maxwell spacetime [36], the Born-Infeld gravity [37] and the charged Ellis-Bronnikov (CEB)

spacetime [38], to study the eﬀect of the various spacetime parameters on the deﬂection

of signals in these spacetimes. The second motivation is to test whether the perturbative

method can be extended to treat rare spacetimes, particularly the higher dimensional ones

and asymptotically non-ﬂat ones. As we will see, it turns out the extension is actually

quite simple and we will apply it to the higher dimensional CEB spacetime and a nonlinear

electrodynamical scalar (NES) theory.

The paper is organized as follows. In Sec. II we extend the perturbative method that

was previously developed to arbitrarily high dimensional spacetime. In Sec. III we apply

the method to three four-dimensional spacetimes that are asymptotically ﬂat, and study the

eﬀects of the spacetime parameters on the deﬂection of both null and timelike signals. We

emphasize that to our knowledge, the deﬂections in these spacetimes in the weak ﬁeld limit

were not studied before. Moreover, in Subsec. III C, we also use the perturbative method to

study the deﬂection in the higher dimensional CEB spacetime. In Sec. IV, the perturbative

method is extended to the asymptotically non-ﬂat case and used to study the NES theory.

We also point out a prominent feature of the deﬂection angle in this kind of spacetimes. We

conclude the work with a short discussion in Sec. V. Throughout this work, we adopt the

natural unit system G=c= 1 and the most plus metric convention.

II. THE PERTURBATIVE METHOD

The perturbative method to calculate the deﬂection angle in the static and spherically

symmetric spacetimes in the weak ﬁeld limit was initiated in Ref. [31] and further formalized

in Ref. [32]. There, the metric functions were assumed to allow asymptotic expansions into

integer power series of the radius. In this section, we will ﬁrst extend the main procedure

of this method to arbitrarily high dimensional spacetimes and then further extend it to

asymptotically non-ﬂat metrics in Sec. IV.

Static and spherically symmetric spacetimes in n-dimensional (n≥4) spacetimes can

always be described by the line element

ds2=−A(r)dt2+B(r)dr2+C(r)(dθ2+ sin2θdφ2+ cos2θdΩ2

n−4),(1)

where

dΩ2

n−4= dχ2

1+ sin2χ1dχ2

2+··· +

n−5

i=1

sin2χidχ2

n−4(2)

and (t, r, θ, φ, χ1, χ2,··· , χn−4) are the coordinates and A(r), B(r) and C(r) are

the metric functions of ronly. The asymptotic ﬂatness of the spacetime often allows the

following asymptotic expansion of the metric functions

A(r) = 1 + X

n=1

rn, B(r) = 1 + X

n=1

rn,C(r)

r2= 1 + X

n=1

rn,(3)

where an,bnand cnare ﬁnite constants. Although locally we can always set C(r) = r2,

there are occasions that C(r) is transformed to other forms and therefore we will keep its

general form as in (3) for now. Due to spherically symmetry of the spacetime, we need only

to consider the particle trajectory on the equatorial plane (θ=π/2 and χi=constant). The

geodesic equations associated with a test particle in this plane then can be readily obtained

t=E

A(r),(4a)

φ=L

C(r),(4b)

˙r2=1

B(r)κ−E2

A(r)+L2

C(r),(4c)

where the dot stands for the derivative with respect to the aﬃne parameter. Land Eare

respectively the conserved angular momentum and energy per unit mass of the signal, and

κ= 0 and 1 respectively for null and timelike signals.

θs

θd

∆ϕ

FIG. 1: The trajectory from the source S at radius rsto the detector D at rd, passing by the lens

L. The impact parameter and closest approach are marked as band r0respectively.

From these equations, one can express the deﬂection angle ∆φfrom the source at coor-

dinate (rs, φs) to the detector at (rd, φd) (see Fig. 1) as [32, 39]

∆φ=Zrs

+Zrd

r0rB

p(E2/A −κ)C−L2dr, (5)

where r0is the closest approach of the trajectory and can be solved from the condition

˙r|r=r0= 0. Using Eq. (4c), this condition can also establish a relation between Land r0

L=p[E2−κA(r0)]C(r0)/A(r0).(6)

In an asymptotically ﬂat spacetime, Land Ecan also be expressed as

L=|r×p|=v

√1−v2b, E =1

√1−v2,(7)

where vis the signal velocity at inﬁnity and bis the impact parameter. Solving bfrom Eq.

(7) and using Eq. (6), we have

b=√E2−κ

pE2−κA(r0)sA(r0)

C(r0)(8)

≡p1

r0(9)

where in the last step the right-hand side of Eq. (8) is deﬁned as a function pof 1/r0. For

later purpose, we will denote the inverse function of p(x) as q(x). We point out that as long

as the metric functions are known, both functions p(x) and q(x) can be known too (at least

perturbatively).

To calculate ∆φin (5), in Ref. [32, 39] we proposed the change of variable from rto u

using the relation

r= 1/q u

b.(10)

Substituting Eq. (10) into Eq. (5), it is not too diﬃcult to verify that it is transformed to

∆φ=Z1

sin θs

+Z1

sin θdyu

bdu

√1−u2,(11)

where

yu

b=sB(1/q)

C(1/q)

p0(q)q2

b,(12)

and

θs= arcsin b·p1

rs, θd= arcsin b·p1

rd (13)

are actually the apparent angles of the signal at the source and detector respectively.

In the weak ﬁeld limit, y(u/b) in Eq. (11) can be expanded into a power series of u/b,

i.e.,

yu

b=

∞

n=0

ynu

bn,(14)

where yncan be expressed in terms of coeﬃcients in the asymptotic expansions (3) of the

metric. The ﬁrst few of them are

y0=1,(15a)

y1=b1

2−a1

2v2,(15b)

y2=4(c2+b2)−(c1−b1)2

8+a1(2a1−c1−b1)−2a2

2v2.(15c)

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摘要：

Deectioninhigherdimensionalspacetimeandasymptoticallynon-atspacetimesJinhongHe,QianchuanWang,QiyueHu,andLiFengSchoolofPhysicsandTechnology,WuhanUniversity,Wuhan,430072,ChinaJunjiJiayCenterforAstrophysics&MOEKeyLaboratoryofArticialMicro-andNano-structures,SchoolofPhysicsandTechnology,WuhanUniversi...

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Deection in higher dimensional spacetime and asymptotically non-at spacetimes Jinhong HeQianchuan WangQiyue Hu and Li Feng

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