Search for exotic gravitating objects withgravitational lensing

A Thesis in Physics
by Naoki Tsukamoto©2013 Naoki Tsukamoto
Rikkyo University

2

A b s tra c t
For investigating gravitational theories, it is important to research exotic
matters theoretically and observationally. In this thesis, w e investigate two
methods with gravitational lensing to detect exotic matter objects, espe­
cially the Ellis wormhole which is an example of traversable wormholes ofthe Morris-Thorne class.
It is well known that, for the lensing by a black hole, an infinite numberof Einstein rings are formed by the light rays which wind around the blackhole nearly on the photon sphere, which are called relativistic Einstein rings.
This is also the case for the lensing by a wormhole.
First, w e study the Einstein ring and relativistic Einstein rings for the
Schwarzschild black hole and the Ellis wormhole. Given the configuration of
the gravitational lensing and the radii of the Einstein ring and relativistic
Einstein rings, w e can distinguish between a black hole and a wormhole inprinciple. W e conclude that w e can detect the relativistic Einstein rings
by wormholes which have the radii of the throat a ~ 0.5pc at a galactic
center with the distance l O M p c and which have a ~ 1 0 A U in our galaxy
using by the most powerful modern instruments which have the resolution of
10—2arcseconds such as a 10-meter optical-infrared telescope.
Second, w e investigate the signed magnification sums of general sphericallens models including the singular isothermal sphere, the Schwarzschild lensand the Ellis wormhole. W e show that the signed magnification sums are avery useful tool to distinguish exotic lens objects. Future surveys of lensed
quasars will give a strict upper bound of the number density for wormholes
or detect wormholes.
W e m a y test the hypotheses of astrophysical wormholes by using theEinstein ring systems and the signed magnification sums in the future.

3

4

A ck n o w led g m en ts
During the preparation of this thesis, I have been greatly aided by a lotof people at Rikkyo University. I would like to appreciate m y supervisor,
Tomohiro Harada, for useful encouragements, discussions and criticisms. I
have learned a lot from his sincere efforts for gravitational physics.
I thank m y collaborators, Chul-Moon Y oo and Kohji Yajima. I truly feel
grateful to U m p e i Miyamoto. H e has kept to encourage m e to investigate
m y work and to give m e valuable advices.
I wish to thank Tsutomu Kobayashi, Shingo Kameda, Noriaki Shibazaki,
Hidekazu Tanaka, Shigeaki Yahikozawa, Motoyuki Saijo, Hideki Maeda, Shunji
Kitainoto, Hiroshi Murakami, Kentaro Yaji, Tsunehide Kuroki, Tohru Eguchi,
Tsuyoshi Houri, Tetsuji Kimura, Masaliiro Shimano, Hiroya Nemoto, Satoshi
Okuda, Syo K a m a t a and K u m i k o Inagawa for their various helps. I also love
to express m y appreciation to m a n y other people w h o cheerfully set the m o o dfor theoretical physics group at Rikkyo University.
I would like to appreciate Fumio Abe, Hideki Asada, Ryuichi Takahashi,
Takao Kitaniura, Koki Nakajima，Kei Yainada, Toshifumi Futamase, Sasaki
Misao, Kenichi Nakao, Masato N o za w a and Masashi Kimura. They gave
valuable comments for m y work and kindly responded m y some rude ques­
tions.
M y work was supported in part by Rikkyo University Special Fund for
Research.
Finally I would like to thank m y family for its support.

6

C o n te n ts
Abstract
Acknowledgments
1 Introduction

35
9

1.1 Gravitational lens.........................................
9
1.2 Outline of this thesis........................................11

2 Basic equations of the gravitational lenses

13

2.1 Lens equation.............................................. 13
2.2 Deflection angle on the Schwarzschild spacetime.............. 15
2.3 Schwarzschild lens in the weak field.......................... 19

3 Ellis spacetime23

3.1 Morris-Thorne w o r m h o l e .................................. 23
3.2 Ellis spacetime in brief.................................... 26
3.3 Deflection angle in the Ellis wormhole spacetime............... 33

4 Gravitational lenses in the weak field35

4.1 T h e general spherical lens in the weak field................... 35
4.2 Directly aligned limit........................................40

5 Gravitational lenses in the strong field limit

43

5.1 Bozza’s analysis.............................................43
5.2 Strong field limit in the Schwarzschild spacetime.............. 45
5.3 Strong field limit in the Ellis spacetime...................... 48

6 Comparison of exotic lenses and mass lenses
6.1
6.2

51

Comparison the Einstein ring systems...................... 51
Comparison of the signed magnification sums ................54

7 Discussion and Conclusion

57
7

8

A Magnification in the Reissner-Nordstrom spacetime
B Microlensing
C Bozza’s analysis for the Ellis wormhole

CONTENTS
61
63
67

Chapter 1
In tr o d u c tio n

General relativity is the simplest gravitational theory which passes the clas­
sical tests that are the deflection of light, the time-delay of light and the per­ihelion shift of mercury [1]. While general relativity has w o n the confidenceof physicists and astronomers, alternative theories have been investigatedeagerly because of the possibility of renormalization, an explanation for therotation curves of galaxies, dark energy and so on.
It is well known that our universe would be filled with dark matter anddark energy. Planck measurements of the cosmic microwave backgroundand the baryon acoustic oscillation show the equation of state for the darkenergy w 三 p/p = — 1.13 + 0.13 — 0.10 (68% C.L.), where p and p are thepressure and the energy density, respectively [2]. This implies that exotic
matters with it; < — 1 m a y not be exotic in our universe. For investigating
gravitational theories, it is important to research exotic matters theoreticallyand observationally.
These something dark and exotic can be detected with gravitational lens­ing effects indirectly. In this thesis, w e will investigate the two methods todetect the exotic lens objects with the Einstein gravity. Especially w e will
research gravitational lensing by a wormhole which is a hypothetical object
which connects two or more asymptotic region.

1.1 G ravitational lensAccording to legend, w h e n Galilei researched celestial objects with his tele­scope, a m a n rejected to use telescopes because he thought that they would
be evil tools. This legend teaches us an important lesson that we must un­derstand tools.
Gravitational lensing is a very useful tool for astrophysics and cosmology.

10Chapter 1 .Introduction

Using the gravitational lensing, w e determine the cosmological constant, thedistribution of dark matter and the Hubble constant, the existence of extra­
solar planets and so on (see Schneider et al. [3] and Perlick [4, 5] and Petters
et al [6] for the detail of the gravitational lens, and references therein).
Mass lens systems in the weak field have been mainly investigated sincegravitational lensing was predicted about one hundred years ago. However,curved spacetimes such as wormhole spacetimes also cause gravitational lenseffects (see Visser [7] for the details of the wormholes). Since gravitational
lensing of the wormholes was pioneered by K i m and C h o [8] and Cramer et
al. [9], m a n y interesting aspects of gravitational lensing by various w o r m ­holes have been investigated [10,11,12,13,14,15,16]. W e m a y test somehypotheses of astrophysical wormholes by using the gravitational lensing in
the future [17,18,19, 20].
Darwin studied the orbits of particles and light rays on the Schwarzschildspacetime [21，22] and he found the relativistic images which are a series offaint images lying just outside the photon sphere [21]. For the past decade,gravitational lensing in the strong gravitational field has been studied ea­
gerly (see Virbhadra and Keeton [23], Virbhadra [24], Bozza [25], Bozza andMaiicini [26] and references therein). Frittelli et al. [27]，Virbhadra and Ellis[28, 29] and Bozza et al. [30] studied the gravitational lensing in the strong
field with the Schwarzschild spacetime and found infinite images, which are
too close to each other to separately resolve. In this thesis, w e call theseimages relativistic images. T h e gravitational lensing in the strong field on
the spherically symmetric static spacetime was investigated by Bozza [31],Hasse and Perlick [32] and Perlick [33]. They showed that the relativistic
images are formed not only in the Schwarzschild spacetime but also in theother spherically symmetric static spacetime.
Gravitational lensing is not only important for astrophysics but also in­teresting as the subject of mathematical physics. Witt and M a o investigatedthe magnifications for lensing by double lenses and found that the signed
magnification sums of the five images become unity inside a caustic [34].
Rhie gave another proof for the invariance of the signed magnification sumsof gravitational lensing by double lenses and applied it to the n-point lenssystems [35]. T he signed magnification sums of the simple galaxy models
which are variations on the singular isothermal sphere were studied by Dalai[36], and those of quadruple lenses were investigated by Witt and M a o [37].
Dalai and Rabin showed that residue integrals provide a simple explanation
for the invariance of the signed magnification sums [38]. See also Petters
et al [6] for the details of the application of the singularity theory to thegravitational lenses. Recently, Werner showed that the signed magnification
invariant is a topological invariant [39]. T h e local magnification relations

1.2. Outline of this thesis

11

with a subset of the total number of lensed images have been investigatedeagerly [40, 41 ，42, 43, 44, 45].
Gravitational lensing m a y become important to test some gravitational
theories. T h e gravitational lens effects of the braneworld gravity models
also have been investigated. M a j u m d a r and Mukherjee [46] and Eii’oa [47]studied the gravitational lens effects on a braneworld model. Keeton andPetters studied the gravitational lens effects of the Garriga-Tanaka metric in
the type II Randall-Sundrum braneworld gravity model [48].
In this thesis, we will show that the signed magnification s u m would be
a powerful tool to research the lens objects as well as the total magnificationand the magnification ratio if we observe a double image. Over than a hun­dred Multiple images including double images have been observed so far [49].
Large Synoptic Survey Telescope [50] will find about 8000 multiple imagesof lensed quasars [51]. T h e future surveys will give a strict upper bound of
the number density for wormholes or detect wormholes.
W e will also consider the Einstein ring and relativistic Einstein rings in
the Ellis wormhole spacetime and the Schwarzschild spacetime, both of which
are static and spherically symmetric ones. W e ask whether w e can distinguish
the Einstein ring systems on the Schwarzschild spacetime and on the Ellis
wormhole spacetime? T o answer this question, we focus on the relations
between the Einstein ring and the relativistic Einstein rings.

1.2 O utline of this thesis
This thesis is organized as follows. In Chapter 2, w e introduce the basic equa­tion of gravitational lenses. W e review the simple lens equation including theeffects of the strong gravitational field in Section 2.1 and the deflection angleof the light ray and the magnification under the weak field approximationin the Schwarzschild spacetime in Section 2.2. In Chapter 3, w e review theMorris-Thorne wormhole and the Ellis spacetime and the deflection angleof the light in the Ellis wormhole spacetime. In Chapter 4, w e will investi­
gate the gravitational lenses in the weak field limit. W e discuss the signed
magnification sums of general spherical lens models including the singularisothermal sphere, the Schwarzschild lens and the Ellis wormhole. W e will
number the real solutions of the lens equation because the signed magnifi­
cation sums are physical invariants only w h e n all the solutions are real. In
Chapter 5, w e discuss the gravitational lenses on the spherically s y m m e t ­ric and static spacetime in the strong field limit. In Section 5.1，we review
the magnification and the radii of the relativistic images in the strong field
limit. In Section 5.2, w e discuss the gravitational lensing in the strong field

12

Chapter 1 .Introduction

limit on the Schwarzschild spacetime in brief. In Section 5.3，w e investi­
gate the gravitational lensing in the strong field limit on the Ellis wormhole
spacetime. In Chapter 6, we will investigate two n e w methods to distinguishbetween exotic matter objects and black holes with gravitational lenses. In
Sec. 6.1，we compare the Einstein ring and the relativistic Einstein ringsin the Ellis wormhole spacetime to the ones in the Schwarzschild spacetimeand show that w e can distinguish between black holes and wormholes. In
Sec. 6.2, w e show that we can distinguish between mass lenses and exotic
matter objects with their signed magnifications. In Chapter 7, w e summarize
and discuss our result. In Appendix A, w e discuss the magnification of theReissner-Nordstrom Black hole in brief. In Appendix B, w e will review the
microlensing of exotic lenses in brief. In Appendix C, w e will recalculate thedeflection angle in the strong field limit in the Ellis wormhole spacetime on
the same way as in Bozza’s analysis. In this thesis w e use the units in which
the light speed c and N e w t o n ’s constant G are unity.

Chapter 2
B a s ic e q u a tio n s o f t h e
g r a v i t a t i o n a l le n s e sMass lens systems in the weak gravitational field have been mainly investi­
gated since gravitational lensing was predicted about one hundred years ago.
However, relativists are eager to test the effects of the gravitational lenses inthe strong gravitational field because gravitational theories including generalrelativity have never been tested bv any experiments in the strong gravita­
tional field yet.
In this chapter, w e will review the basic facts of the gravitational lenses
but our treatments for a lens equation and the deflection angle of the lightare mucli less c o m m o n than traditional ways [3, 6] since we also treat strong
gravitational field. In section *2.1, w e will introduce an effective deflection
angle of the light ray and a lens equation which are not only valid in the
weak gravitational field but also in the strong gravitational field. In section
2.2. w e review the deflect ion angle of the light ray in the Schwarzschild s])acetime. W e will express the deflec tion angle of the photon wit h an incomplete
elliptic integral of the first kind and the complete elliptic integral of the firstkind. In section 2.3. we review the Schwarzschild lens under the weak fieldapproximation [3, 6].

2.1 Lens equation
For the past decade, the lens equations which descril>e the effects of thestrong and weak gravitational field hav(、been investigated [*28, 331. In thisthesis, w e will use the simplest lens equation [52] since w e only consider thecases in weak and strong gravitational field limit.
W e will consider the case that both the observer O and the source object

13

Chapter 2. Basic equations of the gravitational lenses

14

S are far from the lensing object L, or》 b and D ls》 6, where b is theimpact parameter of the photon and Di and D ls are the separations between
the observer and lens and between the lens and source, respectively. T h e
configuration of the gravitational lensing is given in Fig. 2.1. Then, the lens
I

S

D u

Ds

Dt

Figure 2.1: T he configuration of the gravitational lensing. T h e light rays
emitted by the source S are deflected by the lens L and reach the observer O
with the angle of the lensed image Q, instead of the source angle (j). b and a
are the impact parameter and the effective deflection angle, respectively. Di
and D is are the separations between the observer and the lens and between
the lens and the source, respectively. This figure is taken from [52].
equation is given by
D lsa ^ D s(

a

(a

(2.1)

d

2tt)

9

4>

where ~
modis the effective deflection angle, and are the anglesof the lensed image I and the source S to the lens object L from the observerO, respectively, and D s = Di~\- Dis is the separation between the observer O
and source S. Note that w e have assumed \ 《 1，|^|《 1 and |^|《 1 . W ehave also assume that the light ray bends on the lens plane. This assumption
is called the thin lens approximation. T he deflection angle can be expressed\a2.2. Deflection angle on the Schwarzschild spacetime

15

a ニ（
5 + 2丌??，where n is a non-negative integer, denoting the winding numberof the light ray.

2.2 D eflection angle on the Schwarzschild space
tim e
T h e Schwarzschild spacetime is the static spherically symmetric vacuum so­lution of the Einstein equations [53]. In this section, we will review thedeflection angle on the Schwarzschild spacetime [21，22, 54, 55] since we haveto distinguish exotic lens objects from the Schwarzschild lenses, i.e., masslenses for detection exotic lens objects. T h e line element in the Schwarzschildspacetime is given in the following form:
ds2 = — ( 1 — — j dt2 +ェ咖 Tg +

r2d Q 2,(2.2)

rwhere d Q 2 = c?@2 + sin2 ㊀ゴ屮ム and rg is the Schwarzschild radius. T h e space­time has the Killing vectors= dt and ダ dp —for stationarity andaxial symmetry.
W e can concentrate ourselves on the equatorial plane because of sphericalsymmetry. Using the conservation of the energy E 三
and angularm o m e n t u m L 三ダ；^ A:ハ冷レ and た"ん" = 0 , where ん" is the photon wave number,the photon trajectory is given by$ ( ) = * - 士(1_?)，
备

（
2 .3)

where b = L / E is the impact parameter of the photon.
F r o m the photon trajectory (2.3), w e obtain(2.4)\ r2 + Veff(r) = 0,

where

Veff(r) = y

^

(i

-

7 )

-

^

( 2 -5 )and the dotrepresents the derivation with the affine parameter of the geodesic.
T h e maximal value of the effective potential Vefj isChapter 2. Basic equations of the gravitational lenses16’.)、 < |6|, while
Figure 2.2 shows that the photon is scattered if bc = •)、reaches the horizon if bc > \b\ because the photon trajectory is prohibited in
the regions Vejf < 0. Since we are interested in the scattering problem, we
assume bc < |6|.

orJ/o-UJ

-0.3
0

2

4

6

8

10

12

14

r/r9

Figure 2.2: T h e effective potentials of the photon trajectories. T h e solid,
broken and dot lines are the effective potentials with b = 5rg, b =and b = 2rg, respectively. T h e photons which are emitted by sources withミリ <

|6| reflect at Veff = 0.

Using u e 士，抑 三 ト and ug =
du\2

( J
、
姊

we getG ( u )，

(2.7)

where the function G(u) is (ldined l)yG(u) 三

+ — •

(2.8)

G(u)

In the scattering case,has three zeros v ，andu < 0 < u^. < uq < up. W e can also describe G(n) as

G{u) = ^-(u 一 u_)(i/ 一- u0).

uo which satisfy(2.9)

2/2. Deflection angle on the Schwarzschild spacetime

17

B y comparing Eq. (2.8) with Eq. (2.9), we can find一

r+ - r 9 - y ( r + - rg){r+ + 3rs)
2rsr+

Ti十 =

1
— ,

uQ =

ニ+ 一 rg + 今— ★

：+ + 3W

2r5r+(2.10)

where r+ is the closest distance of the light ray. T h e relations+ u^. + uo = u 0(2 .1 1 )is very useful. T h e impact parameter b of the light ray is described by
b2 = - 」 土— -.
r+ ~ r9(2.12)

Figure 2.3 shows the deflection of the light ray. T h e light ray which comesfrom the direction $ = — cv/2 — 7r/2 is deflected at $ = 0 and goes away in the
direction $ = a/2 + 7r/2. T h e deflection angle a of the light ray is calculated

f U+ — du狀

0 = o2 1Jo

y/G(u)(2.13)、 ノ— 7r.

This integral is expressed in terms of an incomplete elliptic integral of thefirst kind F(ip, k) which is defined by

F_= Jo V l - ^ s i n 2 ^( 2 .14)and the complete elliptic integral of the first kind K(k) which is defined as
K ( k ) 三 Fi^Kj2,k) [54, 55]. (For example, see for the details of the elliptic
integrals [56].) Thus, the deflection angle a of the light ray is given by

a

=

fu+

2^/Ug /

Jo

= 2 バ

du

■
■
■
；

= — 7rv (u — u^)(u — u + )(u — U q )du广

\f{u --duu+)(u - u0)\

Ju- \/{u — U 一
4 ^V u0 — U-[K ( ん1 ) — F ( c ^ i , んi ) ] — 丌，

7T

(2-15)

Chapter 2. Basic equations of the gravitational lenses

18

f +

f

Figure 2.3: T he deflection of the light ray. T he light ray which comes fromthe direction $ — — a/2 — 丌/2 is deflected at 3> = 0 and goes away the
direction = a/2 + 7r/2.

2.3. Schwarzschild lens in the weak field

19

where

k\

「
u+ — u —/yj (r.j- — 2m)(r+ + 6m) —

+ 6m

Uq 一

(2.16)

and(/?i三 arcsin-U-W

u + — U- —In the strong field limit b -> bc,
hence

k\

ニ

に ま

~ u 0 — )• 2ug/3

I(2 .17)^ — Ug/3 and(2.18)

1,(pi — arcsin \/-(2.19)\A ^ 0 — U-

( 2 . 20 )

and
y/Ug.

Therefore, the deflection angle of the light ray in the strong field limit isgiven bya

K (ki) — F I arcsin i/-, k\— 丌( 2 -21)log oo.

Ihis implies that the light ray winds around the black hole nearly on thephoton sphere in the strong field limit.
In the weak field approximation b 》 rg, from Eq. (2.15), the deflection
angle is given by [53, 55]

a

2rg

2 22)

( .

2.3 Schwarzschild lens in the weak field
In this section w e review the Schwarzschild lens under the weak field approx­
imation [3，6]. This review in brief would help us to understand Sec. 4.1,
Sec. 4.2 and Sec. 6.2.

Chapter 2. Basic equations of the gravitational lenses

20

In this case, the effective deflection angle \sa — a or the winding numberof light is n = 0. Using by the deflection angle (2.22) and b = Did, the lensequation (2.1) becomes沪— 刼 - 1 = 0 ,(2.23)

where(2.24).

飞and(2.25)
W h e n ailobserver has a linear view of a source and a lens object or (/)= 0, thering image which iscalled Einstein ring with the image angle 20o is formed
for the spherical symmetry. T h e solutions of the lens equation are
L =

+ 4J.

(2.26)

So the separation of two images is given by§+- L =

+ 4.

(2.27)

T h e solutions have some relations which given by
2
2
2

—
\
—
\0
— /3
8/ ヽ
29/ 2

/\
_ '/ V—/—9+ 9^ = 0,
6 ^ - 0 - = — 1,
- l く i く e+.

-

3

1
<
0
+

If the source object is far away from the lens object on the source plain0 》 1，the image angles are
|<彡

<彡

Then,
(2.32)
^

=

2

±

2

^

2

+

4

)

4

2.3. Schwarzschild lens in the weak field

21

and the signed magnifications are

。

" ±4

，

= •

小細

Z

士
“ 士

+

学丨.

♦

4 \ ^ 2+ 4

(2,3)

T h e signed magnification s u m 人i0+ +/io— is always unity. T h e absolute valuesof the magnifications are given byl^o±l = 7 I —f == + — 1— - I 士

，
ゾが+ 4

( 2 .3 4 )中

T h e total magnification is obtained byMo

三1^0+1+ IMo-1 — ~=1.

(2.35)

も /タ + 4

Thus, the total image always is magnified. T h e ratio of the magnification isChapter 2. Basic equations of the gravitational lenses

Chapter 3
E llis s p a c e tim e
The Ellis spacetime was investigated as a geodesically complete particle
model by Ellis [57] and turned out to describe a wormhole connecting two
Minkowski spacetimes. The Ellis spacetime is a static, spherically symmetric,
asymptotically flat solution of the Einstein equation with a massless scalar
field with a wrong sign as a matter field. It is well known that such a matterfield violates energy conditions and it could represent the negative energy
density from quantum effects, such as the Casimir effect. This spacetimeis the simplest and earliest example of wormholes proposed by Morris andThorne [58, 59]. This is a traversable wormhole in the sense that an observer
can cross this wormhole in both directions.
In this chapter, we briefly review Morris-Thorne wormhole [58], the Ellis
spacetime [57] and the deflection angle in the Ellis spacetime [52, 60, 61].

3.1 M orris-Thorne worm holeIn this section, we will briefly review the Morris-Thorne wormhole [58]. Theline element of the static and spherically spacetime can be described by

ds2 —e2A^ d t 2 H—

1r+ r'2^

2 + ^ sin2 ㊀帅

2 ，

(3.1)

where B(r) and A(r) are arbitrary functions of r. If we set A(r) = 0 andB(r) = a2/r, where a is a positive constant, the spacetime is the Ellis w o r m ­hole spacetime. W e use a set of orthonormal basis vectors

Chapter 3. Ellis spacetime

24

(3.4)
(3.5)

r sin ㊀The non-zero components of the Einstein tensor

r

are

_B'

(3.6)

B + 2 1'^3rJ r
^ e © = ^4>l> = ( 1

(3.7)

A!'

Bマ

2r(r-B) A，+(ズ尸 + 7 ~ 2今一.■可 ,

From the symmetry, the non-zero components of the stress-energy tensor
areTit = 0 ( 0 ， ^rr — ~ T(r)-

T もら=

=

p(r),

(3-9)

where p(r) is the total density of mass-energy that static observers measure,r(r) is the tension per unit area that they measure in the radial direction
and p(r) is the pressure that they measure in the directions orthogonal toradial.
From the Einstein equations

G (ip(3.10)

the Einstein tensor (3.6)-(3.8) and the stress-energy tensor (3.9), we obtain
the equations

B = 8ttGt2p,
G r r 3+ B
A! : S n2r(r— B)
p+r{p — t )A' — 2
丨(3.11)(3.12)(3.13)

or

P

P

Bf
8?rGr2’B — 2(r — B )rA l
8丌G r 3’^[(p - t ) A ' - T 1) - T.

(3.14)(3.15)(3.16).(3.8)

3.1. Morris-Thorne wormhole25

W e define a dimensionless function ( as
_

い

丁 一

B — r B l — 2(r — B ) r A

p

I

vw\

=

(3.17)'

Embedding diagram is a useful tool to understand wormhole spacetimes.
W e embed the two dimensional surface with the line element

ds2 = ェ drB[r) + r2d ^ 2:

(3.18)rwhich is the line element (3.1) by setting t = constant and ㊀ = !
in theEuclideanspace. W e introduce cylindrical coordinatesz 、r and $ and theline element of the embedding space is described by

ds2 — dz2 + dr2 + r2d ^ 2.

(3.19)The line element of the embedded two-dimensional surface can be describedrd z \ 2
dr2 + r2挪 2,
1 + \drds2(3.20)

where z — z(r) is an embedding function. If we identify the coordinates r
and $ of Eq. (3,18) with the ones of Eq. (3.20)，the embedding function z
should satisfy
dz , I 1, [ B
dr 1 会 - 1\V ^B '
(3.21)
Wormholes must have a m i ni mu m radius r = B q which is called the throatof the wormhole. Since the embedded surface is vertical at the throat, from
Eq. (3.21),B = r. Thus, r = B — B q at the throat. The inverse of theembedding function r(z) have to satisfy

%

=

〉 °'

( 3 .2 2 )at or near the throat r = B = B q to connect two asymptotically flat regions.
From Eqs. (3.17) and (3.22), the dimensionless function ( is given by
C ニ 蒜

⑶

- 与

(3,3)

W e assume the energy density p is finite. Prom Eq. (3.11),B ’ should be finite
and hence (r — B )A l must vanish at the throat. Therefore，the dimensionlessChapter 3. Ellis spacetime26

function ぐ0 s C ( r = B = 5 0)is positive at the throat. Prom the definitionof the dimensionless function し (See Eq. (3.17)，）we realize that wormholes) > /?o 三
need to be composed of an exotic matter with ro 三 丁 {r — B ~ 5 。
[){t = B = Bo). This implies that we need exotic matters near the wormholethroat at least which violate the null energy condition T^uk^ku > 0, wherehf. is an arbitrary, future-directed null vector.
However, if we consider gravitational theories with an adding term H^uand the modified Einstein equations which are described by
Gp.u +— r",レ，(3.24)

the analysis for the Morris-Thorne wormhole only implies that the condition(T叫 — H^u) h fk u > 0 is violated near the throat. Thus, we does not always
need exotic matters violating the null energy condition for wormholes in thiscase.

Form ation of worm holes
Various wormhole solutions have been found still now but the formation of
wormholes is still a open question. Recently, a simple analytic model ofwormhole formation from the initial data with a massless ghost scalar fieldhas been investigated by Maed a [70]. It m a y be important to note that acurvature singularity appears when the wormhole is formed. Cosmologicalwormholes which are formed from initial singularities also have been investi­
gated [71，72, 73, 74].

3.2 Ellis spacetim e in brief
In this section, we review the Ellis spacetime which is the simplest and earliest
solution of the Morris-Thorne class [57]. The line element of the static andspherically symmetric spacetime is described by

ds2 = —dt2 + [dp — f(p)dt]2 + r2(p)(dQ2 + sin2 Q d^2)
= — (1 — f(p )2)dt2 + dp2 — 2f(p)dtdp + r2(p) (d㊀2 + sin2 © d $ 2) , (3.25)
where r(p) is non-negative and the range of the coordinate is given by — oo く
t < oo, — oo く p < o o , 0 < ㊀ < 丌 and — 7r < $ < tt. The determinant g of
the metric tensor g^u is obtained by

g = —r4 s in 2 ㊀.

(3 .2 6 )

3.2. Ellis spacetime in brief

27

Ellis introduced the vector field u which is orthogonal to the hypersurfacet = constant
む"3" = - d t - f(p)dp.
(3.27)
W e use an orthonormal frame system {eM}

e 0 = u = - d t - f{p)dp,(3.28)^ 1 = dpi
62 —
63 =

(3.29)

f ヽ5© 7r(p), 、 .

(3.30)

へ

r(p) sin 0

d 否.

(3.31)The non-zero components of the Ricci curvature tensor

Roo
Roi
Rn
122

^ are obtained by(r2f f Y(3.32)

2r " /

で

r J(r2f f j

2r"(3.34)’

で-^33 =(3.33)- [び ’（1 一 f 2)}'r2

(3.35)•

W e consider the Lagrangian described by

L = R% + I < x ^ ,where K is the non-zero coupling constant anddescribed by(3-36)
is a scalar field which is

X = 1(P),(3-37)

where 7 (p) is a non-constant, differentiable, real-valued function. Prom thevariational principle, we obtain

0=5{R% + K x ^ X ；,) d4x.

(3.38)

U K > 0, then r is convex while if< 0, then r is concave [57]. W e takeX > 0 to remove the Schwarzschild singularity and to make a throat andwe will set K = 2 with rescaling 入. From Eq. (3.38)，we get the EinsteinequationsRp —\9/ju — — 2 (

~ oA'

)

(3 .3 9 )Chapter 3. Ellis spacetime28

and the scalar wave equation= 0.

(3.40)The Einstein equations (3.39) are also expressed by

Rfit^ = -レ.

(3-41)

The wave equation (3.40) yields[r2 (/2 - l ) y ] ，= 0-(3-42)The (0 — 0), (0 — 1)，( 1 — 1 ) and (2 — 2) components of the Einsteinequations (3.41) are obtained by{r2frP\ 2^ f = 2 f j \(3.43)

2, 1
r

2/ ザ ，(3.44)B / 7+ ど =2デ(3.45)j,2

and
K ( i - / 2)r

r2

■= 0,

(3.46)

respectively. The (3 — 3) component is equal to the (2 — 2) component forthe spherical symmetry. Equations (3.43)-(3.46) are equivalent to the threeequations
ち

=

(

3

-

4

7

)

(r2//')’= 0,(3.48)[rr'(l — /2)]' = 1_

(3.49)

W e can integrate Eqs. (3.48) and (3.49) and we getr2(l — / 2)' = 2 m ，

（
3.50)

(r2) ' ( l - / 2) = 2(p — m )，(3.51)

where m is an integration constant and the zero point of p has been adjusted
to equalize two integration constants. From Eqs. (3.50) and (3.51), it followsthat
r 2( l - / 2) = p 2 + C ,(3 .5 2 )

3.2. Ellis spacetime in brief

29

where C is an integration constant. From Eqs. (3.42) and (3.52), we obtain-I-I

r2(l — /2)

p2 + C ，(3.53)

where I is an integration constant. From Eqs. (3.51) and (3.52), we getr’ (j — mr p2 + C(3.54)andr"

C + m2(p2 + C)2 ’r(3.55)

From Eqs. (3.47)，(3.53) and (3.55), it follows thatC = I2 — m 2.

(3.56)

Equations (3.53)，(3.54) and (3.56) imply thatl(p ) = レ2 + l2 - m 2\ exp 2 m ? (p)(3.57)

Here we introduce a new coordinate T satisiying

d T = dt + T ^ f W ) dp-(3.58)The line element (3.25) is described by

ds2 = _ [1 _ f ( p ) } dT2

dp2
1 - 尸 (P)+ r2(p) (dQ2 + sin2 ㊀帅 2) . (3.59)

From Eqs. (3.52)，(3.56) and (3.57)，the line element (3.59) is expressed by

ds2 = sgn(p2 + I2 — m 2)e^

dT2 + e — '^where sgii(x) is a sign function defined by[dp2 + (p2 + /2 — m 2) dQ2] | .(3.60)

Chapter 3. Ellis spacetime

30

In the following, we will make an analysis for case where I2 > m 2. (SeeEllis [57] for the cases where I2 < m 2 and I2 = m 2.) To simplify the dis­cussion, we assume that the integration constants m and I are non-negative.
From Eqs. (3.52), (3.53), (3.56) and (3.57)，it follows that，

(3.62)

r2(p) = (p2 + a 2)e 2^ ^(3.63)/ 2(p) = l - e ~ ^ ,(3.64)

7Tp'——
arctaii一2a.

(3.65)

伽-六and hence
7(P)= -

where a 三 \/Z2 — m 2 and we have applied the boundary condition(3.66)

lim x = lim 7(p) = 0.
Under this boundary condition, 7 is everywhere positive.
The derivatives of / 2(p) and r2(p) are given by
d(P)

dp

2m

(3.67)

p2 + a2

and

d(r2)
dp

(3.68)e甲 (2p - 2m),

respectively. Under the assumption m > 0 and I > 0, f 2(p) is monotonicallydecreasing with respect to the coordinate p and changes from 1 —to 0
as p increases from —00 to 00 and r2 has a minimal value at /? — m.
The asymptotic behavior is given byr(p) = p + m + 0 \-j:(3.69)(3.70)n p ) = 7 ^+ 0 ( kas p ^ 00 andr(p)=-e °(p + m) + O

f\p )l - e - 竽

1 -(3.71)

2m() + m ノ+ 0(3.72)

3.2. Ellis spacetime in brief31

as p — > — oo. Equations (3.69) and (3.70) and the line element (3.59) imply
that the Ellis spacetime is asymptotic to Schwarzschild spacetime with themass parameter m as p ^ oo. As p -> — oo, from Eqs. (3.71) and (3.72), theline element (3.59) is asymptotic to

ds2 = — (1 H------- ) d (e_ 〒 r) 2 H

す ^
+

r

+ r2 (d㊀2 + sin2 0 d $ 2) .

(3.73)

This implies that the Ellis spacetime is asymptotic to Schwarzschild space­time with the mass parameter —me « as p ^ —oo.
The case m — 0 ，f(p) vanishes from Eq. (3.64). From Eq. (3.63), it
follows thatr2(p) ニ "2 + a2 = p2 + Z2

(3.74)and the line element (3.59) is obtained by

dr2 = —dT + dp2 + (p2 + a2) (dQ2 + sin2 0 d $ 2) .
(3.75)In this thesis, we will concentrate on the case m = 0 and we will namethe Ellis spacetime with m = 0 the Ellis wormhole spacetime tentatively.
Introducing= p2 + a2, we can rewrite this into
ds2 = - d T 2 + ^ 1 - を )dR2 + R 2d n 2,

(3.76)

where R = 士a corresponds to the wormhole throat.

U pper bound of the num ber density
It seems that wormholes cannot be bright unlike stars. Thus, gravitational
lensing is a useful tool to search wormholes. The Sloan Digital Sky Survey
Quasar Lens Search [62, 63, 64, 65] has the largest quasar lens sample in the
Sloan Digital Sky Survey [66]. There are 50,836 quasars with the redshift
2：= 0.6 — 2.2 and the apparent magnitude brighter than i =19.1 and 19
lensed quasars are found. However, there is no lensed image by the unseen
lens objects including the Ellis wormhole. Recently, Takahashi and Asada[51] presented the upper bound of the number density < 10—4/i3M p c J of the
Ellis wormholes for a range of the throat radius 101 < a < 104pc by using
the Sloan Digital Sky Survey Quasar Lens Search.
The femtolensing by the gamma-ray bursts can constrain the abundanceof compact objects [67]. Yoo et al. [68] gave the upper bound of the number
density < 10_9A U °of the Ellis wormholes with the throat ~ lcm by dataof the Fermi Gamma-ray Burst Monitor [69] _32Chapter 3. Ellis spacetime

Instability
The Ellis wormhole is unstable with respect to unrestricted linear perturba­
tions of the metric and the scalar field [75] and Armendariz-Picon showed that
the Ellis wormhole is stable with a restricted perturbation which is required
to vanish at the throat [76]. Nonlinear instabilities of the Ellis wormhole
were also investigated [77, 78, 79]. Needless to say，it does not imply thatstable wormholes do not exit. The stability of a phantom static wormholewith thin shells for the linearized spherically symmetric perturbation was
found by Lobo [80]. Static wormholes are too simple and realistic wormholes
with the cosmological evolution should be described in dynamical situations[71,72, 73, 74, 81,82, 83, 84]. Although the Ellis wormhole is unstable, wewill research the gravitational lensing effect by the Ellis wormhole becauseof the simplicity of the gravitational lensing configuration in this thesis.

Other m atters
Ellis wormhole geometry is obtained with a Kalb-Ramond axion [85] andwith tachyon matter and a positive cosmological constant [86] as a source.

G ravitational lensing
Chetouani and Clement derived the deflection angle of light in the Ellis w o r m ­hole spacetime and calculated the scattering cross-section [60]. Perlick in­vestigated the gravitational lensing effects of the light ray through the Ellis
wormhole throat by using the full lens equation [33] and Nandi et ai [14]applied the analysis of the strong field limit [28, 29, 30, 31].
It was pointed out that the qualitative features of gravitational lensing
in the Ellis wormhole spacetime are similar to the ones in the Schwarzschildspacetime [14, 33 ，87]. However, A b e showed that one can distinguish be­
tween the Ellis wormholes and mass lens objects with their light curves in
the weak field limit [88] (see Appendix B). The Ellis wormholes could be de­tected with the astrometric image centroid trajectory in the weak field limit[89] _
Recently, Nakajima and Asada [61] recalculated the deflection angle oflight on the Ellis wormhole spacetime and proved that Dey and Sen [16]’scalculation is only correct at the lowest order in the weak field limit, while
the conclusions by Abe [88] and Toki et al. [89] are still valid.

3.3. Deflection angle in the Ellis wormhole spacetime33

3.3 D eflection angle in the Ellis worm hole space
tim e
In this section, we review the deflection angle on the Ellis wormhole spacetime[60, 61].
W e can concentrate ourselves on the equatorial plane because of sphericalsymmetry. The photon trajectory with the metric (3.76) is given by

1 ( dR、
R4 \d ^

b2

1

b2
R2

1

R2

(3.77)

where b is the impact parameter of the photon.
W e can see that the photon is scattered if |6| > a, while reaches thethroat if |6| < a. Since we are interested in the scattering problem, we
assume |6| > a. Using u 三 1//?，we findf du—( 1 - a2u2) ( l - b2u2).

(3.78)

PuttingG(u) = a2(a~2 — u2)(b~2 — u2),
the azimuthal angle $ can be given as a function of u by
パ_1

du(3.79)(3.80)VG (u)

Here we have set $(6_1) = 0 . The deflection angle a is then calculated to
give
1 du
a = 2 /
7T.
(3.81)ゾo舞)In the present case, we find

r b~l

du
V g (u )

r /2

K
-(!)

(3.82)

sin2 ^

where we have transformed u = b—1 sini9. Since K{k) admits a power series

K(k)

E

' ( 2 n - 1)!!'.

(2n)!!

_'2n(3.83)

Chapter 3. Ellis spacetime

34

where n!! denotes the double factorial of n and (— 1)!!=1. The deflection(3.84)

丌.

W e get the deflection angleE( 2 n - 1)!
(2n)!!

Q2n.

(3-85)

Thus, the deflection angle is approximately given in the weak-field regime
H 》 a bya(3.86)
4 \b，
In general, the deflection angle is always greater than its weak-field approx­
imation and is diverging as |6| — >■a.

Chapter 4
G r a v i t a t i o n a l le n s e s i n t h e
w e a k fie ld
It is well known that the sum of signed magnifications is invariant in the weak
field limit for mass lens systems. If gravitational lenses of the exotic lens
object break the magnification invariance, the su m of signed magnifications
would be a powerful tool to detect exotic lens objects.
In this chapter, we will discuss the signed magnification sums of gen­eral spherical lens models, including the singular isothermal sphere, theSchwarzschild lens and the Ellis wormhole in the weak field approximation[90]-

4.1 T he general spherical lens in the weakfield
It is well known that the sum of signed magnifications is invariant for masslens systems in the maximal-image domains. In this section, we will calcu­
late the signed magnification s um of the general spherical lens model andwe will number the real solutions of the lens equation because the signed
magnification sums are physical invariants only when all the solutions arereal [90].
W e consider the general spherical lens model with the deflection angle,
parametrized by
ひ= 士= 士^ ^ - '(4.1)

where C is a positive constant, n is a non-negative integer, b is the impact
parameter of light rays emitted by a source and 6 is the image angle and we35

36

Chapter 4. Gravitational lenses in the weak field

have used the relation b = Did. If n is odd, then the sign is only the upper
one, while if n is even, then the sign is the upper one for 0 > 0 and the lower
one for 0 < 0. Thus, we have to treat two lens equations when n is even. Thislens model describes the singular isothermal sphere, the Schwarzschild lensand the Ellis wormhole for n = 0,1 and 2, respectively. The case where n > 3
would describe some exotic lens objects and the gravitational lens effect ofmodified gravitational theories. The following discussion in this chapter does
not depend on the value of C.
The lens equation is given by
r +1-

t

i = o,

(4.2)

where

e^T0

mid

れ

T0 '

(4.3)

and

% = V D JO f)(4.4)

is the Einstein ring angle. W e can concentrate ourselves on the case where
the source angle (f) is positive for symmetry. The solutions...，^n+l Of
the lens equation (4.2) of (n + l)-th degree satisfyn ^ - ^ ) = o .
i=l

(4-5)

For n > 1 we compare Eq. (4.2) with Eq. (4.5) and obtain(4.6)? ,= i

and
》:
iく j

— — Jln，(4.7)

where Jln = 0 for n > 2 and Sin = 1 for n = 1 . Using the both equations,
we obtain

4.1. The general spherical lens in the weak field

37

This implies

E (j) d(j)n+lOi dOi(4.9)

Note that these solutions 6{ m a y be complex and not all the magnifications
are always physical and that Eq. (4.9) is satisfied regardless of the sign ofEq. (4.2).
N o w we will count the number of the images. W e can express the lensequation (4.2) as follows

H(4.10)In the following, we will make analysis for cases (i) where n is odd, (ii) where
n is even and positive and (m) where n = 0, separately.

(i) n is odd.
In the case where n is odd, the lens equation is given by

e~n = e -(4.ii)The solutions are given by the intersections of y = l / x n and y = x —(p.
Figure 4.1 shows the left-hand side and the right-hand side of the lensequation and the intersections for n =1.
W e find a positive solution 6+ and a negative solution
regardlessof the value for <j). W e also can see the positive solution~
andthe negative solution
2 0 for 0 》 1 . W e also can see that n = 1
is the only case where all the solutions of the lens equation are real
and the physical signed magnification s u m is always unity. The lenswith n = 1 and C = 2r0 is the Schwarzschild lens. Thus, its signed
magnification s um is always unity.

38

Chapter 4. Gravitational lenses in the weak field

Figure 4.1: The solid (red) lines y = l / x 11 and the broken (green) line y =x—respectively, correspond to the left-hand side and the right-hand side ofthe lens equation (4.11) for r?= 1 and 0 = 0.5. The intersections correspondto the real solutions of the lens equation. This figure is taken from [!)()].
(ii) n(> 2 ) is even.
W e consider the case where n > 2 and is even. The lens equation is
obtained by
土 卜=(4.12)The solutions are given by intersections of y = \ / x n for x > 0 and
y = - \ / x n for x < 0 and y = x - み . This gives a figure which is very
similar to Fig. 4.1 and we obtain a positive solution 0+ and a negative
solution 0 regardless of the value for 4>. The signed magnifications um (4.9) is not a physical quantity because it includes one or more
non-real solutions in this case. The lens with n = 2 and C = no2/A
is the Ellis wormhole.
(iii) n = 0.
For n = 0, the lens equation is given by士1 = 6 - み(4.13)The solutions are given by one or two intersections of y = 1 forx > 0 and iv = — 1 for x < 0 and y = x — (j). Figure 4.2 shows theleft-hand side and the right-hand side of the lens equations and theintersections.

4.1. The general spherical lens in the weak field

39

W e obtain only one positive solution
in the range> \ while
we get a positive solution 0+ and a negative solution 0 in llie range
0 < 0 <

1.

In the range (f) > I, by a straight forward calculation, we get0+ d0+
4> d(j)
111 the range 0 < d) <(4.14)

1+

we obtaine± d§±
4> d(f)1 土

4>(4.15)

I'herefore the signed magnification sum is 2 in this range.
Only in the case n = 0, the number of images is not always 2. The
singular isothermal sphere lens is given by setting n = 0 and C =4艽び2, where o is the velocity dispersion of particles.

Figure 4.2: The solid (red) lines y = 1 for x > 0 and y = — 1 for x < 0 andthe broken (green) lines y = x —respectively, correspond to the left-handside and the right-hand side of the lens equation (4.13). W e plot the lines in
the case (j) = 0.5 and cf>= 3. The one or two intersections correspond to thereal solutions of the lens equations. This figure is taken from |9()|.
For 0 > 0. the positive solution 々+($) and the negative solution 0 (0) of
the lens equation (4.2) represent an outer image angle and an inner image
angle while 6^.((f>) and 0-((f>) are an inner one and an outer one for 0 < 0,respectively. The positive solution
monotonicallv increases as 0 increases.

Chapter 4. Gravitations] lenses in the weak field40

The signed magnifications of the images in the weak field limit are given by"o±(み 三 l ず dな (み .

(4.16)The lens equation (4.2) has symmetry with respect to the point $ = 9 = 0,
so that
L(^>) = - 9 +( -$ )(4.17)“ ).

(4-18)

and=令

Thus, the relation of the magnifications is given by叫- 淡 = ^ ^ ( 0 ) == " 。十( - 办(4.19)

From the lens equation (4.2), we obtain"0 士=

起"て2 —.
(的 + 1 干 1)(の + 1 ±n)(4.20)For 必 》 1，Eq. (4.20) implies that /i0+ — 1 because 0+ 2 公 》 1，while
I_iq_ 《 1 because Q— 《 1. In other words, if it is far from the alignment,
the positive image is as luminous as the unlensed image，while the negative
image is extremely faint. Thus, we can ignore the gravitational lensing effects
and the signed magnification sum //0+ + Po—becomes almost unity for 本》 1
regardless of the values of n.

4.2 D irectly aligned lim it
In this section we will discuss the signed magnification sums of the general
spherical lens model for n $ 1 in the directly aligned limit (4> ~ 0).
The positive image angle and magnification in the directly aligned limit{(j) ~ 0) are given byん(み ^ 1 +

+ 2( 1 + n)2#

(4.21)

4.2. Directly aligned limit

41

and
へ

1 1 + 6

パ0+ ( 0 - i + n

01(4 .2 2 )respectively. From the symmetry, we can easily obtain the negative image
angle and the signed magnification in the directly aligned limit
允(み ニ —l + I ^ $ - ^

^

^

2

(4.23)

andMo-(0) ^ - n 上 ...l- 2 气(4.24)
丄 十 れ q)respectively. Therefore the total magnification and the ratio of the magnifi­
cations in the directly aligned limit are given by^o(^) 三 "o+($) +— t丄十
——n 0j(4.25)

andM o + (み26

" 0 — (みrespectively.
The difference of the reduced image angle in the directly aligned limit isgiven by0+ - L~ 2 + .(1 + n ) 2 $ .

Thus, the Einstein ring angle is given by

in terms of— Q_ and 4>.

(4 .2 7 )Chapter 4. Gravitational lenses in the weak fieldChapter 5
G r a v i t a t i o n a l le n s e s i n
s tr o n g fie ld

th elim itIn this chapter, we will review and investigate gravitational lenses on thespherically symmetric and static* spacetime in the strong field limit. Grav­
itational lensing is on the strong field has been investigated eagerly for the
Schwarzschild black hole [21,22. 23, 24, 27, 28, 30, 31,55, 91 ，92, 93. 94, 95],
the Kerr black hole [96, 97, 98], the Rdssner-Nordstrom black hole [31，91.94.
95 ，99], the Giblxms-Maeda-Garfinkle-Horowitz-Strominger charged blackhole [100], Einstein-Born-Infeld black holes [94, 101], braiieworld black holes[47, 48,102, 103], a static spherically symmetric spacetime of Braiis-Dicke
scalar-tensor theory [104]. naked singularities [23. 29], the Janis-NevvniaiiWinicour solution [16. 31.94, 95], the Barriola-Vilenkin monopole [33], andwormholes [14，16, 33, 87] because it can give clues for verification of gravita­
tional theory in the strong filed. In Sec. 5.1, we review the magnification andthe radii of relativistic images in the strong field limit [30. 31]. In Sec. 5.2,
we review gravitational lensing in the strong field limit on the Schwarzschildspacetime in brief [30, 31]and in Sec. 5.3. we investigate gravitational lensing
in the strong field limit on the Ellis wormhole spacetime.

5.1 B ozza’s analysis
We briefly review Bozza’s analysis of the magnification and the angles ofrelativistic images [30, 31|. \V(、consider that the deflection angle of the light
ill the strong field limit is written by
ひ(の = 一" log

一 1 ) + も+ O(0Di 一 bc),
43

(5.1)

Chapter 5. Gravitational lenses in the strong field limit

44

where bc is the critical impact parameter, a is a positive parameter and b
is a regular parameter. We will calculate the parameters a and b for the
Schwarzschild lens and the Ellis wormhole in Sec. 5.2 and Sec 5.3, respec­tively.
When the winding number n > 1 .we define an angleby= 2皿 .

(5-2 )

From Eqs. (5.1) and (5.2), we obtainbe ( 1,

We expand the deflection angle W の around 6 = 9^>l to obtain the effective
deflection angle a. We define the small angle^ n > l 三 沒 ぬ 1 — ^,>1^(5-4)

where 0n>\ is the solution of the lens equation (2.1 ) with the winding numbern > 1 . Prom Eqs. (5.1) and (5.3)，the effective deflection angle of the lightin the strong field limit is given by
a = —— ae 6で nA0n>i.

(5.5)

We substitute the effective deflection angle (5.5) into the lens equation(2. 1) and obtaine l>x + 1 + ギ^ 手を6’尝尝) A0nw

(5.6)

From Eqs. (5.6) and (5.4), We get the angles of the relativistic images 0n>\
6-2

(5-7)

11

where we have used bcj D \ 《 1 .
The magnification ^n>i of the relativistic image is obtained by•- +b —2 7 rn \b —2 ttti(5.8)<j—> dcj)~ k ぺ ； »Dls
The sum of the magnifications of all the relativistic images is given by
1 b^,1 Ds e 言
2tt
ハ
(5.9)
l ^n>l 〜~ares + 1 + eaJ

5.2. Strong field limit in the Schwarzschild spacetime45

and the sum of the magnifications of the relativistic images excluding theoutermost relativistic image is given by
£ ル^
71=2

(e f + e ¥ + el) ■1( 5 .io)

From Eqs. (5.10) and (5.8), the ratio of the magnification of the outermost
relativistic image divided by the sum of the magnification of the other rela­tivistic images is given byny^ — ^
Z^n=2 ^ln(e 誓 — l ) ( e ^ +

,

L

パ .

e « + e « + e«

(5.11)

From Eqs. (5.3) and (5.7), the innermost relativistic image angle is obtained by
^oo —

—瓦 .

（5.12)

Prom Eqs. (5.3). (5.7) and (5.12)，the difference of the angles between theoutermost relativistic image and innermost one is given bye l - e 00^ e \ - e 00 = 900e ^ .

(5 .1 3 )Here, we ignored the second term of the right-nand side of the Eq. (5.7).

5.2 Strong field lim it in the Schwarzschild space
tim e
In this section, we consider the strong field limit in the Schwarzschild space­time [30, 31]. We use the trajectory equation of the photon (2.3) with themetric tensor (2.2). Prom Eq. (2.13), the deflection angle a is given by
a = 7(r+) — 7T,(5.14)

where
where r+ is the closest distance of the photon orbit and
G ( r ，r + )

三

3 (1

_

；£

) +

p

(_i 7
+

)

, 15)

•

(5 .1 6 )Chapter 5. Gravitational lenses in the strong field limit

46

Using
2= 1 - —,(5.17)The integral 7(r+) is obtained byI{r+) = 2 [ f ( z , r +)dz,
Jo(5.18)

where
1/0,r+)y M G ( z , r +)(5.19)1い+ (l — ザ— 3[-1 + ^ ( 1 - z )We will divide the integral /(r +) into the divergent part iu (r+) and theregular partorJ(r+) = /£>(r+) + /i?(r+ ).
(5.20)The divergent part / の(厂+) is defined byID{r+) =k { z . r +)dz,(5-21)

where
h{z.r+ )■s/K,(r+)z + P(r+)zJ '(5.22)

where+(5.23)+)三 2 一^ ?

卢(ホち-- l1 ++ r+ .

(5.24)The divergent part /r>(r+) is calculated in a straight way and obtained byん㈧=v m logK(r+) + 2/3 (r+) + 2^/[K(r+) + 0(r+))p{r+)命 +)5.2. Strong field limit in the Schwarzschild spacetime

47

Thus, the divergent part lD(r+) in the strong field limit r+ —> 3rg/2 + 0 is
obtained by

From the relation between the impact parameter b and the closest distancer+ ( 2. 12), it follows thatiog( H •iog(表ィ)+1iog(書).

(5 .27)

Thus, the divergent part In{b) in the strong field limit b —> 3v^rp + 0 is
obtained by
Io {b )= —log Q 〜 — 1 ) + log 6 + O ( 6 ---- 2~~^^ _(5.28)The regular part I r (t^) is defined byIR(r+) =g{z,r+)dz,(5.29)

where
g{z,r+) = f ( z , r +) - fo{z,r+).

(5.30)In the strong field limit r+ —^ 3rff/2 + 0, the regular part In(r^) becomesIn.(r+) = 2

广 f rl _ _ - } ：) d z + o ( r + - ^ j/0

W

1 一 f2

2 log 6 (2 一 \/^)+ O (r + 一 孕 )

（
5.31)

Thus, the deflection angle of the light on the Schwarzschild spacetime inthe strong field limit is obtained bya (の =

Id + Ir—

瓦- 1) + i o g

216 (7 - 4 ^ )- 7T + O

0A

Sy/3rg0.32)

Chapter 5. Gravitational lenses in the strong fie ld limit

48

where we have used the image angle 6 = b/Di and we get the parameters
a = 1 and b = log [216 (7 —4\/3)] — tt — 0.40.
The magnification of all the relativistic images on the Schwarzschildspacetime is obtained by
7

2
1

0

6
|

4o

1
1
<0.
1

⑵ と

2'

( 5 .3 3 )The ratio of the magnification of the weak field image divided by the sum oi
the magnification of all the relativistic images is given by

( (

響

が

妙

， 5

= ⑵ 3…》1.， )鉍Thus，we can ignore the effect of the relativistic images on the light curve in
the Schwarzschild spacetime. The ratio shows that the relativistic images arealways fainter than images in the weak field. However, this does not meanthat we cannot observe the relativistic images since they can get bright when
the source angle is small. We may measure the relativistic images of thegalactic black hole of our Galaxy (~ 20/xas) with interferometry instruments
which are now under development [105, 106, 107, 108].
The sum of the magnification of all the relativistic images become unity
when the source angle isゎ ？( 著) 吾/ - 277= vS ( 急) ’ 2'

( 5 .3 5 )Note that the cross section of gravitational lensing with the galactic blackhole in our Galaxy is very tiny because the Einstein ring angle is 6q ~ 10-6.

5.3 Strong field lim it in the Ellis spacet im e
The deflection angle of the light on the Ellis wormhole spacetime in the strong
field limit is given byひ( の =

—

log (

l )

+

3 log 2 — 丌 + O { 0 D i —a ) .

(5.36)

Here, we have used the relation
lim K (k ) = log — + 0 ( k a log た’)，（5.37)

5.3. Strong field limit in the Ellis spacetime

49

where ん’ = ( 1 —k2)^ and 0 < A: < 1 . Therefore, a — I and b = 3 log2 — n.
The magnification of the all relativistic images on the Ellis wormhole
spacetime is obtained by
2

3

8

0
7

d
1—
\
/

^

6

~ /

D
1 D"

\

2

|彡

n

~l

a A

1

§

as

The ratio of the magnification of the weak field image divided by the sum of
the magnification of the all relativistic images is given by

A 傭

( 職

=U⑶

パ

》 1 (5 .39)

Thus, we can also ignore the effect of the relativistic images on the light
curve in the Ellis wormhole spacetime. The sum of the magnification of all
the relativistic images become unity when the source angle is)2急’2K 是)2’2'_

We notice that the cross section of the gravitational lensing in the strong
field limit is different from the one in the Schwarzschild spacetime.

Chapter 5. Gravitational lenses in the strong field limitChapter 6
C o m p a r is o n o f e x o t ic
andle n s e s

m a s s le n s e s

In this chapter, we will investigate two new methods to distinguish between
exotic matter objects and black holes with gravitational lenses. In Sec. 6.1.
we compare the Einstein ring and the relativistic Einstein rings in the Ellis
wormhole spacetime to the ones in the Scliwarzscliild spacetime mid showthat we can distinguish between black holes and wormholes. In Sec. 6.2, weshow that we can distinguish between mass lenses and exotic matter objects
with their signed magnifications.

6.1 Com parison the E instein ring system s
In this section we compare the Einstein ring and the relativistic Einstein ringsin the Ellis wormhole spacetime to the ones in the Schwarzschild spacetimeand show t hat we can distinguish between black holes and wormholes.
The Einstein ring and the relativistic Einstein ring images correspond to
the the solution 0n of the lens equation (2 . 1) with the winding number n forvanishing source angle 0 = 0 .
First, we consider the Einstein ring and the relativistic Einstein rings on
the Ellis wormhole spacetime. From the lens equation (2.1), the deflection
angle (3.84) and b = DiO, we find that the ring image is given by
9n = ^b(61)

where kn € (0 . 1) is a unique root of the transcendental equation
2K{k) - k'i = (2n + 1)tt.

( 6 .2 )Chapter 6. Comparison of exotic lenses and mass lenses

52

where 7} = Dsa/ (DiDis) and k 三 ajb.
We should note that 2K(k)~7]/k is monotonically increasing with respect
to k and changes from —oo to oo as たincreases from 0 to 1 . The uniquenessof the root follows from the monotonicity. Moreover, we can conclude thatkn monotonically increases and approaches 1 as n —)• oo and hence the image
angle 6n monotonically decreases and approaches a jD い
In the weak-field regime \b\ 》 a, the winding number n should be n = 0.
Using the deflection angle (3.86)，we can solve the transcendental equation(6.2) approximately and get the diameter angle of the Einstein ring0q ~f i IX
4 D sD f
2 .0 arcseconds

Du

f 20M pc\ 1 ( lOMpc'

lOMpc ノ 乂

ノ 、 A

) 乂0_5ロら(6.3)

This approximation is good for D / 》 a and Dl s 》 a. The relative error is~ 10-2 for a = 0.5pc and Di = Dis = 1 0 Mpc.
In the especially strong-field regime, where the winding number n be­comes n > 1 , we can easily check that a b or kn 1 satisfies the transcen­
dental equation (6.2) in numerical calculations. Physically this means thatthe light rays which wind around the wormhole nearly on the photon spheremake the relativistic Einstein rings [31, 33]. Then the diameter angles of therelativistic Einstein rings are approximately given by9n^ - 最~ 1.0 x 10- 2 arcseconds

.

(6.4)

Regardless of the values of D[s, Di and a, the relative error of the above
anproximation to the direct numerical solution of the outermost relativistic
Einstein ring (n = 1 ) is 2 10 3 and those of the other relativistic Einsteinrings (n > 2) are smaller than 10-5. This implies that it is difficult to resolve
each relativistic Einstein ring separately.
Thus, we conclude that there are one Einstein ring image and count ably
infinite relativistic Einstein ring images, the latter of which accumulate toform the apparently single ring image of the throat with the diameter a jD ぃ
This conclusion does not depend on the value ofIf we are given the distance Ds to the source from the observer, thedistance Di to the lens from the observer and the radius 0。of the Einsteinring, we can determine the radius of the throat a from Eq. (6.3). Then, we
can use 9n>\ (6.4) to test the assumption that the lens object is a wormhole.

6.1. Comparison the Einstein ring systems

53

From Eqs. (6.3) and (6.4) we obtain the relation between 0。

0n>\ by0n^ - ( ^ ) 01

(6-5)

This relation generally holds in astrophysical situations, as long as a 《and a 《 Dls are satisfied.
Second, we consider the Einstein ring and the relativistic Einstein ringson the Schwarzschild spacetime.
In the weak-field regime b 》 rg, the winding number n should vanish
and the deflection angle is given by a ~ 2rp/6. See Eq. (2.22). From thedeflection angle (2.22) and the lens equation ( 2 . 1), the diameter angle of theEinstein ring is given by

(為 )5

―

.(6 .6)

We can determine rg in the same way as the radius of the throat a.
In the especially strong-field regime, where the winding number n be­comes n > 1, the impact parameter b that satisfies the lens equation shouldbe nearly the critical impact parameter bc = (3\/3/2)r 0 (see [28, 30, 31]).
Then the diameter angles of the inseparable relativistic Einstein rings 6n>\are given by
3\/3
^n>\

Tg

了

A

5.1 x 10-5 arcseconds

.

(6.7)

Regardless of the values of Dls, Di and rg, the relative error of the above
approximation to the direct numerical solution of the outermost relativistic
Einstein ring (n = 1 ) is ~ 10-3 and those of the other relativistic Einsteinrings (n > 2) are smaller than 10-5.
It is useful to remember that the leading term of the deflection angle in
the weak-field regime is the second order of the small amount a /6 on the Ellis
wormhole geometry (3.86), while it is the first order of the small amount rg/b
on the Schwarzschild geometry (2.22). So the relation between do and Qn>\for the Schwarzschild spacetime
0n>i ^ 手

を

2。

(6.8)

Chapter 6. Comparison of exotic lenses and mass lenses

54

is different from (hat on the Ellis wormhole spacetime (6.5). Figure (i.1 shows
the angle of the relativistic Einstein ring 0n \ versus the angle of the Einsteinring Oo for D\ = Dis = lOMpc*. Thus, we can distinguish between black holes
and wormholes in principle if we are given D H/D\, 0^ and 01t い

一p u ? m p i 4 < ^

itr5

icr4

io*3

icr2

io_

^o[arcsecund]

,

i o°

101

Figure 6.1: The angle of the relativistic Einstein ring 0r, i versus the angleof the Einstein ring 0o for = Dls = lOMpc. The broken (green) and solid(red) lilies plot the cases where the lens objects are a wormhole and a blackhole, respectively. This figure is taken from [52].

6.2 Com parison of the signed m agnificat ionsum s
In this section, we show that we can distinguish between mass lenses andexotic matter objects with their signed magnificat ions.
Figure 6.2 shows that one can distinguish the general spherical lens models
with their signed magnification sums "0+ +"o which are less than unity. We
also can see that one can distinguish the general spherical lens with n = 1
from n = 2, 3 and 4 but one (annot distinguish between n = 2. 3 and 4 forNote that we have to know the imlensed luminosity to calculate the signed
magnification sum パ + パo—. If we observe a double lensed image witli> 2 and the unlensed image with 0 》 1，we can obtain the the signed
magnification sum "0+ +Uo • We also can see that one can distinguish n = 1

6.2. Comparison of the signed

01

s u 川s

2

3

4

55

5

6

Figure 6 .2: The signed magnification sums of some general spheric al lensmodels. The solid, broken, dot and dot-dashed lines are the general sphericallens models for n = 1 .2 , 3 and 4 ，respectively. This shows that the signed
magnification sums is a useful tool to detect the exotic objects with n > 1.
This figure is taken from |90|.
from n = 2，3 and 4 but one cannot distinguish between n = 2, 3 and 4 for
0 ^ 2 . The miniiiium value of the signed magnification sums is given l>y，
from Eqs. (4.22) and (4.24 )，lin i(/ 如+(み + "o ( み ) =

(

6

.

The lower bound of the total magnification /iq is given by2~^ M0+ +

IMi- ^ |/^o+1 + l ^ o - l = IM)-

9)

(6 .1 0 )Therefore, gravitational lensing necessarily gives amplified light curves forn = 1, while it does not necessarily for n > 1 (see Appendix B for microlensFor n = 0 these analyses are not valid in the region \ < 4> because of the
non-existence of the negative image angle 0
discussed in the chapter 4.
However, it is valid in the region 0 < (^ < 1 . For 1 < <^, the magnification is

1 < M o + (0) < 2 .

(6.11)and the tot al magnification パo+(4 ) + /’o (0 ) is always 2 in the range 0 < 0 <1. So one can also distinguish the case n = 0 from the other cases.

Chapter 6. Comparison of exotic lenses and mass lenses

C h a p te r 7
D is c u s s io n a n d

C o n c lu s io n

Einstein ring systems

It is understood that the qualitative features of the gravitational lensing on
the Ellis wormhole sparetime are very similar to the ones on tlie Schwarzschikl
spacetiiiK1 for their photon spheres and their asymptotic flatness 111.33. 87|.
However, we realize that their quantitative features are very different due totheir different weak-fiekl behaviors.
We consider the experimental situation where we know the separation
D s between the observer and the source and the separation D\ between theobserver and the lens. We assume that we do not know whether the lens
object is a black bole or a wormhole and do not its parameter, i.e., the mass
M or the radius a of the throat in advance.
We need at least two observable quantities to determine whet her the lens
object is a black hole or woiiiihole since the lens system has one parameterin this situation. First, we observe an Einstein ring and detennine the pa­
rameter for both possibilities. Second, we observe relativistic Einstein rings
and tell the wormhole from the black hole. If the predicted relativistic ringangles by the black hole and by the wormhole were of similar size, we could
not discern the difference. However, Eqs. (6.5) and (6.8 ) and Fig. 6.1 showthat we do not confuse them.
We conclude that we can detect the relativistic Einstein rings bv worm­
holes which have a ~ 0.5pc at a galactic center with the distance D/ =Dis = lOMpc and which have a ヒ 10AU in our galaxy with the distance
Di = Dis = lOkpc using the most powerful modern instrunients wliich havethe resolution of 10 2arcseconds such as a 10-meter optical-infrared telescope.
Note that the corresponding black holes which have the Einstein rings of thesame size are galactic supermassive black holes with 1010A/. and l()7i l/ ..
respectively, and that the relativistic Linstein rings bv these black holes are57

58

Chapter 7. Discussion and Conclusion

too small to measure with the current technology.
In fact, our results imply that we can distinguish between slowly rotating
Kerr-Newman black holes and the Ellis wormholes with their Einstein ringsystems. This is because the leading term of the deflection angle for thelensing by the Kerr-Newman black holes in the weak-field regime is equal
to the one for the lensing by the Schwarzschild black holes, while the blackhole charge and small spin only slightly change the radii of the relativistic
Einstein rings [31，96, 99,109, 110]. Moreover, this also suggests that it ismuch more challenging to determine the charge and/or small spin of black
holes than to distinguish between black holes and the Ellis wormholes.
We assumed that the observer, the lensing object and the source object
are directly-aligned, though such a configuration is fairly rare. In general thestrong gravitational lensing effect is observed as broken-ring images which arecalled relativistic images [28]. Therefore, more realistic problem is to size therelativistic images. Our result suggests that we can distinguish black holes
and wormholes by using the relativistic images. To observe the relativistic
images is one of the challenging works with many difficulties. Bozza et al
pointed out that relativistic images are always very faint with respect to theweak field images [30]. The Very Large Telescope Interferometer (VLTI) has
high resolution [111 , 112] but it will not work because of this demagnifying
effect.
We also assumed point-like sources, although astrophysical sources havetheir own size. If the source object is a galaxy, it may conceal the relativistic
Einstein rings, especially, in the case where the lens object is a black hole.
Testing some hypotheses of astrophysical wormholes by using the relativistic
Einstein rings and the Einstein ring is left as future work.
Tejeiro and Larranaga [87] investigated the gravitational lensing effect of
the wormhole solution obtained by connecting the Ellis wormhole solution
as ail interior region and the Schwarzschild solution as an exterior region[113]. They concluded that we cannot distinguish the Schwarzschild blackhole and the wormhole unless the discontinuity of the magnification curve atthe boundary is observed. This does not contradict our results because theirwormhole solution behaves as the Schwarzschild solution in the weak-fieldregime.
Signed m agnification sum s
It is well known that the signed magnification sum of the Schwarzschild lensis always unity in the weak field limit. We realize that one can distinguishthe exotic lenses with the parameter n > 1 of the general spherical lens suchas Ellis wormhole from mass lens systems because the signed magnification

Signed magnification sums

59

sums of exotic lenses are less than unity.
The signed magnification sum is a powerful tool to find exotic lens objects
because it only depends on the reduced source angle (p and n and we justhave to observe the images for ^ < 1 and for 》 1 to determine thesigned magnification sum. However, we need a high resolution to observe thedouble images. We would also distinguish the lens objects with the ratio of
magnifications of the double images and the total magnification. If we alsomeasure the difference — Q— of the image angles, one can determine theEinstein ring angle 6q and the source angle <j>= 0q4>.
Observing double lensed images is a practical idea to search the exotic
lenses. Over than a hundred strong lensing including many double images
have been found so far [49]. Future surveys by Pan-Stan's [114], Dark Energy
Survey [115], Subaru Hyper Suprime-Cam [116] and Large Synoptic Survey
Telescope [50] will find more multiple images of lensed quasars. These surveys
will give the stricter upper bound of the number density for wormholes ordetect wormholes.
The method to detect the Ellis wormholes and other exotic lenses byobserving the light curves with the characteristic demagnification by Abe[88] and Kitamura et al. [117]. Notice that the method to distinguish thelens objects with the signed magnification sums would be used in both themagnification and demagnification phases. Thus, we do not have to rely
on only the demagnification to detect the Ellis wormholes and other exotic
lenses.
Our method with the signed magnification sums is complementary to themethods to detect exotic lens objects with the light curves [88, 117] and theastrometric image centroid displacements [89].
In this thesis, we only consider gravitational lensing effects of the sim­plest wormhole. Analyses of gravitational lensing effects by most wormhole
solutions which have been found thus far are left as future works.

Chapter 7. Discussion and Conclusion

A p p e n d ix

A

M a g n ific a tio n

in

th e

R e is s n e r - N o r d s tr o m

s p a c e tim e

In this appendix, we consider the Reissiier-Nordstrom Black hole which is a
black hole solution with the electrical charge.[99. 109.118] The line elementof the Reissner-Nordstrom spacetime is given by

2…(レ，幻ゎ(レ宇4 )、心+ r 2(sin2 e 帅 2 + d©2).
(A.l)
We consider the weak field approximation. If the amount of the charge issmall Q /M < 1 ,the metric is given by, in the quasi-Minkowski ail coordinatest and X .
2M

ds2

i g 2\

2M2

2 J p ) "X2-

~Y

dt2

1 Q2 \

3M 2 d X 1
3APJ I T 1

2M

~Y

(A.2)

T h e deflection angle a in the weak field limit is given by

AM I— 37r-

Q2\ M 2

(X — — —(A.3)
J p )b \b \ywhere 6 is the impact parameter of the photon. Thf signed magnifications"o土 are given by
37T
/ な)土

32 (タ + 4

w つ61

A/D, 土.

DiD\s

0 +2
20 V ^ 2 + 4

(A.4)

62

Appendix A. Magnification in the Reissner-Nordstrom spacetime

We see that we cannot detect the effect of the electrical charge with microlensing at 0 ( M 2/b2) of the deflection angle a; because of the cancelling[118].

A p p e n d ix B
M ic r o le n s in g
In this appendix, we will review the niicrolensing of exotic lenses in brief[88. 117, 119]. We consider the situation that a lens object exists in ourgalaxy and a source goes across near the lens object on the lens plane. We
define the smallest source angle as Figure B.l shows the source trajectorywith けq ~ 0.4^n on the lens plane.

Figure B .l: A source trajectory on a lens plane with the smallest source
angle ~ O.40o* The large circle expresses the Einstein ring angle.
rig\ire B.2 shows the light curves of the microlensing for the mass lensand the Ellis wormhole [88]. We realize that the Ellis wormhole lens shows
the characteristic demagnification and the light curves is useful tool to detect
the Ellis wormholes.
r lgiire B.3 shows the light curves for the general spherical lens with the
reduced smallest source angle ん 三 0。/^) = 0.1[117]. We can determine n63

64

Appendix B. Microlensing

from the c harac teristic dips.

Figure B.2: The light curves for the = 0 .2 0 。
(top left), Pq = 0.50。(topright), = l.O0o (bottom left) and /3o = 1.50。(bottom right). The thin
(green) lines and the thick (red) lines, respectively, correspond to the masslens and the Ellis wormhole lens. These figures are taken from |88].

Tim e、magnification

Dehiagnrfication

Figure B.3: The light curves for the general spherical lens with n = 1 (topleft), n = 2 (top right), n = 3 (bottom left) and n = 10 (bottom right) for
the reduced smallest source angle 0o 三
= 0.1. These figures are takenfrom [117].

Appendix B. MicrolensingA p p e n d ix

C

B o z z a ’s a n a ly s is f o r t h e

E llisw o r m h o leBozza calculat(Ml the deflect ion angle of the light on the static and spheri­cal black holes spacet ime with asymptotic flatness in the strong field limit
and showed that some observable physical quantities are determined by thedeflection angle in the strong field limit [31].
However, Bozza’s calculation of the deflection angle of the light cannot
directly be applied to the Ellis wormhole because Bozza’s calculation does
not work the ultra static coordinate. Nandi, Zhang and Zakharov might haveused Bozza’s calculat ion and get a wrong deflection angle in the strong field
limit [14]. Their mistake affects measurable physical quantities such as thedifference of the angles between outermost relativistic image and innermost
one (5.13) and the ratio of the magnification of the outermost relativistic*
image divided by the sum of the magnification of t he other relativistic images(5.11)In this appendix, we will recalculate the deflect ion angle in the strong field
limit in the Ellis wormhole spacetime on a similar way to Bozza’s analysis.
The deflec t ion angle a is then calculated to givea =1(b) - Tr

(C.l)

where
(C.2)
and
W e transformG(u ) 三 a2(a 2 一 w2)( 6—2 — u2

(C.3)

z = 1 —bu

(C.4)

67

Appendix C. Bozza’s analysis for the Ellis wormhole

68
and then^m= j o K z ；k)dz,

(C.5)

A:三 ■

(C.6)

where
and

f(z , k ) 三

2
=■
(C.7)
— k2)z + (—1 + 5k2)z2 —Ak2z° + k2z4
We divide I(k) into the divergent part Io(k) and the regular part IR(k)，ort

I ( k ) ^ I D{k) + IR{k).

(C.8 )The divergent part ん (ん）is defined byID(k) =

f 0(z,k)dz,

(C.9)

where
（
c .10)

K (k) 三 2( 1 —k2) and 0 ( k ) = —1 + 5ん2.
K(k) is always positive in the scattering case (0 < た < 1) . P(k) is negative
for 0 くんく l/\/5 , zero for k = l/\/5 and positive for l/y/5 < k < 1. Now
we can concentrate in a strong field case (1/a/5 < A: < 1 )where P(k) should
be positive because we will consider the strong field limit (k —)• 1) . Whenboth K(k) and /3(k) are positive, the divergent part Io(k) can be calculated[31],lD{k) = w

log—

n —

, V - l + 5fc2 + V 1 + 3A:2
log
%Al + 5A:2
^ ( l - k ) { k + l)
(C.ll)

K(k) and /3(k) in the strong field limit yield,
lim K(k) = 0,

(C.12)

lim P(k) = 4.

(C .1 3 )fc— lfc—)•!

69

So, the divergent part /^(A:) in the strong field limit is obtained by
I d { ^ ) — —lo g (l— ん）+ 2 log 2 + 0 (k — 1)
The regular part /^(/c) is defined by
lR(k

)

/

Jo

三

where

g{z,k)dz,

d iz ：た）三 _/い，た）- fo(z ，た)•We can expand Ir(k) in powers of (k — 1)，
w

Now,

= E ^
„=o n

-

!

1)n r

ム dkn k=ldz.

/o(: ，1 ) — —i/CM)z(2 2— z) ’

(c.

4)

(c.

5)

(c.

6)

(c.

7)

(c.
(c.

8)
9)

5( も 1 ) = /(^, 1 ) - / 0(2,1)

So, the regular part I r (Ic) is!R(k)=

2-z(C.20)g(z, \)dz + 0 (k — 1)

dz + 0 ( k - l )Jo= log2 + 0{k - 1).
2 - 2(C.21)
Thus, the deflection angle of the light on the Ellis wormhole spacetime inthe strong field limit is obtained bytt(ん） = ID(k) + IR(k) - 7r
= —lo g (l—k) + 31og2 —7T+ 0 (k — 1).
(C.22)
Since b ~ 6Dh
a (の = —log

_ 1) + 3 log 2 — 7r + 0 (k — 1).
(C .2 3 )Appendix C. Bozza’s analysis for the Ellis wormholeB ib lio g r a p h y[1 ]C. M. Will, Theory and Experiment in Gravitational Physics, (Cam­bridge University Press, Cambridge, England, 1993).
[2] Planck Collaboration, arXiv: 1303.5076vl.
[3] P. Schneider, J. Ehlers and E. E. Falco, Gravitational Lenses (SpringerVerlag, Berlin, 1992).
[4] V. Perlick,
Living Rev. Relativity 7,
9 (2004),
http: / / relativity.livingreviews.org/Articles/lrr-2004-9.
[5] V. Perlick, arXiv:1010.3416[gr-qc].
[6] A. O. Petters, H. Levine and J. Wambsganss, Singularity Theory andGravitational Lensing (Birkliauser, Boston, 2001).
[7] M. Visser, Lorentzian Wormholes: From Einstein to Hawking (Ameri­can Institute of Physics, Woodbury, NY, 1995).
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