25 Oct. 2017

Exploringexotic matter and energy
by gravitational lensing
浅田秀樹
Kitamura, Nakajima, Izumi, Hagiwara,
R.Takahashi,
Tsukamoto, ...

イントロ
エキゾチックな時空の現象論模型
まとめ(通常の)物質＋DM＋DE＝100%
他に新しいモノ（コト）？
（%）BHもNSも100年前は空想の話、
しかし、いまホットな天体観測現象

“Gravitational Lens” (GL)as a powerful tool

NASA/HST

Gaudi et al. Science (08)

「光学レンズ」との類似------

凸レンズ

凹レンズ

>0

<0

通常の「重力レンズ」

負の場合

『凹型重力レンズ』を探索すれば、何か新しい物質・エネルギーを伴う時空構造（天体とよべる？？）が発見できるか？

The Astrophysical Journal, 725:787–793, 2010 December 10

doi:10.1088/0004-637X/725/1/787!
C 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

GRAVITATIONAL MICROLENSING BY THE ELLIS WORMHOLE
F. AbeSolar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan; abe@stelab.nagoya-u.ac.jp
Received 2010 February 21; accepted 2010 October 7; published 2010 November 19ABSTRACT
A method to calculate light curves of the gravitational microlensing of the Ellis wormhole is derived in theweak-field limit. In this limit, lensing by the wormhole produces one image outside the Einstein ring and anotherimage inside. The weak-field hypothesis is a good approximation in Galactic lensing if the throat radius is lessthan 1011 km. The light curves calculated have gutters of approximately 4% immediately outside the Einsteinring crossing times. The magnification of the Ellis wormhole lensing is generally less than that of Schwarzschildlensing. The optical depths and event rates are calculated for the Galactic bulge and Large Magellanic Cloud
fields according to bound and unbound hypotheses. If the wormholes have throat radii between 100 and 107 km,
are bound to the galaxy, and have a number density that is approximately that of ordinary stars, detection canbe achieved by reanalyzing past data. If the wormholes are unbound, detection using past data is impossible.
Key words: gravitational lensing: micro
Online-only material: color figures

1. INTRODUCTION
A solution of the Einstein equation that connects distant pointsof space–time was introduced by Einstein & Rosen (1935).
This “Einstein–Rosen bridge” was the first solution to later
be referred to as a wormhole. Initially, this type of solution2008) on the gravitational lensing of wormholes as structuresof space–time. Such studies are expected to unveil lensing
properties directly from the space–time structure. One study(Dey & Sen 2008) calculated the deflection angle of light due tothe Ellis wormhole, whose asymptotic mass at infinity is zero.
The massless wormhole is particularly interesting because it isThe Astrophysical Journal, 740:121 (8pp), 2011 October 20

Figure 3. Sketch of the(wormhole) in the sky. Aradius θE .

Figure 2. Source and image trajectories in the sky from the position of theobserver.

image centroid posi
Schwarzschild one aof the Einstein ring
astrometric lensingSScchh
EEWWHH

Figure 4. Light curves for β̂0 = 0.2 (top left), β̂0 = 0.5 (top right), β̂0 = 1.0 (bottom left), and β̂0 = 1.5 (bottom right). Thick red lines are the light curves forwormholes. Thin green lines are corresponding light curves for Schwarzschild lenses.
(A color version of this figure is available in the online journal.)

凹レンズによる減光！

Exotic Object捜しにおいてワームホールに限定する必要はないとはいえ、何か理論模型があると便利イントロエキゾチックな時空の現象論模型
まとめthe weak-field approximation. We consider light propHe
atingthroughafour-dimensional
spacetime,
thoughthe
時空構造の新しい理論模型を提唱した$ðholespacetimemay
behigher-dimensional.
Thefour（逆ベキ則、３パラメタの数式）
themensional spacetime metric is expressed as
Kitamura, Nakajima, HA(2013)!
!
"""1 2"222
ds ¼ $ 1 $ n dt þ 1 þ n drrr
222222
þ r ðd! þ sin !d" Þ þ Oð"1 ; "2 ; "1 "2 Þ; (1)here
the circumference radius and "1 and "2 are((11))r is遠方で平坦、かつ静止mall bookkeeping parameters in the following iterative((22)) 弱い場の近似lculations.
Here, "1 and "2 may be either positive orgative,respectively.
Anegative"and"forn¼
112((33))  nn==11  ::  通常の天体
rresponds to a negative mass (in the linearized          nn==22  ::  ワームホール
hwarzschild metric).
（時空のトンネル）
For investigating light propagation, it is useful below
連星への拡張：BBoozzzzaa++  ((22001166))  
makea conformal transformation with a factor ofsid

whDSfrosou光の曲がり角  ¯(b) = nbnn==00  ::  SSiinngguullaarr  IIssootthheerrmmaall  SSpphheerree  ((SSIISS))  
nn==11  ::  SScchhwwaarrzzsscchhiilldd  
nn==22  ::  EElllliiss  WWoorrmm  HHoollee  ((EEWWHH))  
同じ曲がり角が独立に提唱          TTssuukkaammoottoo  aanndd  HHaarraaddaa  ((22001133))  
高次元時空模型との関連          TTssuukkaammoottoo,,  KKiittaammuurraa,,  NNaakkaajjiimmaa,,  HHAA  ((22001144))  

ns that the deflection (" ¼ "=b!
) and m

物理的な解釈ed here correspondto a   convergences density) of theform
質量面密度"ð1! & nÞ 1#ðbÞ ¼:
nþ12bε>>00  かつ  nn>>11ならば,,    
herefore,
the effective surface-mass de
負の質量((エネルギー))面密度を表す
ect is interpreted as negative in the frard lens theory. This means that the mad to be exotic if n > 1.

κ>0NNoonn--vvaacc..
RRiiccccii--ffooccuussiinngg
VVaacc..

κ=0
WWeeyyll--ffooccuussiinngg

κ<0NNoonn--vvaacc..
RRiiccccii--ddeeffooccuussiinngg

ε > 0 & n < 1 SSIISS
ε<0&n>1n=1SScchh

ε > 0 & n > 1 EEWWHH
ε<0&n<1source,respectively.
source,respectively.
or">0,
thereis alwaysa"D
positiverootcorredefined
as[36]! LS nþ1!
"
#%
:
1
E
e Einstein
ringfor!
¼0.
The
EinsteinringFor""D">
0,thereisalwaysapositinFor>0,thereisalwaysposit! LS nþ1DS DL!
#%:
(5)
ned
as[36]
En
theEinstein
Einsteinringfor!
¼
0.
TheE
totothe
ring
for!
¼
0.
The
"D!
DS DL!
"<defined0, on theas
otherhand,Eq.
(4)hasnoposit1#%[36]"D! LS nþ1 E
definedas[36]
D
¼0.
This
is
becausethiscase
describes
th#%:Ener hand, Eq. (4) has nopositive
root
for!
"DD1SL!
"e.
For
later
convenienceinnormalizing"D!
nþLSause this case describes
therepulsive"D!
nþLS
#%
E
ation,
wedefinethe(tentative)
Einstein
n
#%
<0,on
the
otherhand,Eq.
(4)
has
nopositiv
If"
<0,on
the
otherhand,EqE
DD
onvenienceinnormalizingthelensnS
L
DS DLthe"0.<This
0 as is because this case describes
e the(tentative)
Einsteinring radius!
¼0.
Thisisbecause
this!
"e. IfFor
later
convenienceinnormalizing
1 Eq. (4) h" < 0, on the otherhand,j"jD!
nþ1LSIftion,"<0,ontheotherhand,Eq.
(4)hforce.
Forlaterconvenienc
wedefinethe(tentative)
Einsteinrin#%;
E! ¼!0. This" is because
n this case deDDS
1 becauseL this case d"!<¼0as0.
Thisisequation,
we define the (tenj"jD!

nþ1force. ForLS later
convenience in n

Kitamura+ (2013)

SScchhEEWWHH

増光率
光源の位置^

Kitamura+ (2013)

REPORTS

増光率時刻SScchh

EEWWHHPHYSICAL REVIEW D 87, 027501

減光はワームホール以外でも可能cut of deriving
lengthy calculawith the x and yng the radial and^ 0Þthat ð#^i Þ ¼ ð#;
hanges in !^ and'Þ and ðd!^ i Þ ¼
uthal angle. The
e independent of
rms vanish in the
mediately obtain

PHYSICAL REVIEW D 88, 024049 (2013)ZUMI et al.

ttaannggeennttiiaallllyy  eelloonnggaatteedd

IG. 1 (color online). #, "þ , and "& for " > 0. They are denoted by solid (blue), dotted (purple), and dashed (red) curves,spectively. The horizontal axis denotes the image position ! in the units of the Einstein radius. Top left: n ¼ 0:5. Top right: n ¼ 1.
ottom left: n ¼ 2. Bottom right: n ¼ 3.

show $" > $þ .
angentially elon-Finally, we mention the dependence on the exponent n.

ðn þ 1Þðn þ 2Þwhere the x and y coordinates are chosen along the radia
BByy  aaxxiissyymmmmeettrryy  aalloonngg  ll..oo..ss..  
and tangential directions, respectively, such that ð#^i Þ ¼^ 0Þ and ð!^ i Þ ¼ ðPHYSICAL^ 0Þ. Hence,REVIEW. . . tthhee  mmaaggnniiffiiccaattiioonn  mmaattrriixx  iiss      88, 024049factor(201ð#;!;
the radialDelongation!
is 1=$" , while the tangential one is!1=$þ .
1"%"&
0$" ^ 0ðAijFirst,(1Þ ¼ let us investigate the primary%image (# > 0).; By
0 obtain1 " % þ &0 $þ
using Eq. (7), we

where the x and y coordinates
are chosen
along the rad1!^$þ ¼ ¼ 1 " nþ1 ;(11)^^^and tangential directions,
respectively,suchthatð
#iÞ##^ 0Þ and ð!^ i Þ ¼ ð!;^ 0Þ. Hence, the radial elongation fac
ð#;
is 1=$" , while the tangential
one
is1=$.
^þd!
n¼ 1primaryþ nþ1 :image (#^ > 0).
$" ¼
(12)B
First, let us investigate
thed#^#^
using Eq. (7), we obtain
To reach Eqs. (11) and (12), we need several steps^!
where first the Jacobian matrix is 1computed and next

d"^n!! ¼¼ 1 ! nþ1 :d#^#^(21)

only if n > !1. Hence,w that !! < !þ , if andrraaddiiaallllyy  eelloonnggaatteedd
FIG. 4 (color online). $, #þ , and #% for " < 0. They are denoted by solid (blue), dotted (purple) and dashed (red) curves,respectively. The horizontal axis denotes the image position ! in the units of the Einstein radius. Top left: n ¼ 0:5. Top right: n ¼ 1.
Bottom left: n ¼ 2. Bottom right: n ¼ 3.

NNoo  lleennss
あくまでイメージ図MM  >>  00

MM  <<  00

courtesy of Koji Izumi

First cosmological upper bound
The Astrophysical Journal Letters, 768:L16 (4pp), 2013 May 1

doi:10.1088/2041-8205/768/1/L16!
C 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

OBSERVATIONAL UPPER BOUND ON THE COSMIC ABUNDANCES OF NEGATIVE-MASS COMPACTOBJECTS AND ELLIS WORMHOLES FROM THE SLOAN DIGITAL SKY SURVEY QUASAR LENS SEARCH
Ryuichi Takahashi and Hideki Asada

Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan
Received 2013 March 6; accepted 2013 April 4; published 2013 April 17ABSTRACT
The latest result in the Sloan Digital Sky Survey Quasar Lens Search (SQLS) has set the first cosmological constraints
on negative-mass compact objects and Ellis wormholes. There are no multiple images lensed by the above two
exotic objects for ∼50,000 distant quasars in the SQLS data. Therefore, an upper bound is put on the cosmic
abundances of these lenses. The number density of negative-mass compact objects is n < 10−8 (10−4 ) h3 Mpc−3 atthe mass scale |M| > 1015 (1012 ) M$ , which corresponds to the cosmological density parameter |Ω| < 10−4 at thegalaxy and cluster mass range |M| = 1012–15 M$ . The number density of the Ellis wormhole is n < 10−4 h3 Mpc−3
for a range of the throat radius a = 10–104 pc, which is much smaller than the Einstein ring radius.
Key words: cosmology: observations – gravitational lensing: strong
Online-only material: color figures

1. INTRODUCTION
In theoretical physics, negative mass is a hypothetical conceptof matter whose mass is of opposite sign to the mass ofnormal matter. Negative mass could thus generate the repulsive
gravitational force. Although possible negative-mass ideas havethroat radius of 100–107 km could be constrained (or detected)
by using the Galactic microlensing. Although its abundance isquite unknown, some authors speculated that it is as abundant asstars in the universe (Krasnikov 2000; Abe 2010). More recently,Yoo et al. (2013) gave the rough upper bound of its number
density as n ! 10−9 AU−3 for the throat radius a ∼ 1 cm from(A color version of this figure is available in the online journal.)
Takahashi & Asadais figure is available in the online journal.)
5

mi = 1017 (orange), 18 (blue) and 19 (green),−5

10Ellis wormholen (Mpc/h)−3n (Mpc/h)−3ve is the result without thenegative−mass
magnification bias.
objecttted for the fixed number density of the lens
upperboundand the probability
scales
as P ∝ n.
aluate the observational upper bound on then of the lenses,
we use the likelihood function
0! NQ
10chanek (1993), ln L # − j =1 P (zs,j , mi,j ),i,j are the redshift and the apparent magnitude
NQ is the total number of statistical samples|Ω|=1= 50836. The data of zs,j and−2 mi,j were|Ω|=10m the SQLS Web site.2

100upper bound95%−4|Ω|=104. RESULTS

95%

10−568%68%s the upper bound on the cosmologicalnumbergative-mass compact objects. The vertical axis
nsity n (h3 Mpc−310), while thehorizontal
axis is16024121418101010−11010 curves 1010e of the mass |M|(hM10are$ ). The two
throat radius a (pc/h)
at 68% and 95% confidence
levels.
Itturnsoutmass M (M /h)Figure 3. Same as Figure 2, but for the Ellis wormhole. The horizontal axi
density is less than n < 10−7 (10−5 ) h3 Mpc−3
ure 2.
12Upper bound on the cosmological number density of negative-mass
the−3throat radius a (h−1 pc).
(10)
M.
The
bluedashed
lines
show
the3
$ The vertical axis denotes the density n (h Mpc ). The
pact objects.
fzontalthe axis
cosmologicaldensityvalue
parameter,|Ω|
denotes the absoluteof the mass|M|=(h−1 M$ ). The twod1.
curvesto upper boundsof 68% andThecorresponddensity parameteris definedas 95%
the confidence levels.
blue dashedlines cosmologicaldenote the absolutevalue ofthe cosmological density
videdby thecriticaldensity,−4 , 10−2 , and 1.
Figure 3 shows the Ellis wormhole cases. The horizon
meterforthenegativemass,|Ω|=
10s shown in the figure, the density parameter−1axisisthethroatradiusa(hpc). As shown in the figu
−4 of this figure is available in the online journal.)
olorversion< 10 for galactic and cluster mass scale
the number density is n < 10−4 h3 Mpc−3 for a = 10–104
5イントロ
エキゾチックな時空の現象論模型

まとめ1) Exotic matter/energyを伴う重力レンズ現象の基礎論を展開した
2) 実際の天文観測につなぐ部分は今後の課題

e.g.『(通常の)凸型重力レンズ』における有限距離の光源と観測者の効果
Ishihara+(1604.08308, 1612.04044), Ono+(1704.05615)