電磁真空時空における歪んだ光子面の形成
吉野裕高（大阪公立大）
H.Yoshino, arXiv:2309.14318[gr-qc]“一般相対論と幾何”

＠名古屋大学SE館

2024年2月9日

目次
イントロダクション

電磁真空時空における歪んだ光子面の形成

まとめイントロダクション

ブラックホールの特徴

r = 3M

地平面の存在光の円軌道
r = 2M

地平面の拡張概念
2

ds =

dr2
f (r)dt ++ r2 (d✓2 + sin2 ✓d 2 )
f (r)
2

地平面

f (r) = 1

2Mrr = 2M

拡張概念
事象の地平面井田, 中尾, 椎野, Hayward (1998)

見かけの地平面

Caltech-Conell Group（数値相対論）

光子球面 (Photon sphere)
光の円軌道が r = 3M に存在

r = 3M

光の円軌道の集合は
時間的超曲面を形成＝光子球面 （Photon sphere）

 

拡張概念？

r = 2M光子面（Photon surface）
Claudel, Virbhadra & Ellis, JMP42 (2001) 818 [gr-qc/0005050]

光子球面の概念の拡張

r = 3M

ある時間的超曲面 S があって, S に接する
任意の光的測地線が S 内にありつづける
時, S を Photon surface という。

静的でなくてもよいr = 2M

球対称でなくてもよい光子面（Photon surface）
Photon surface の例
Schwarzschild 時空における (r=3M)×(時間)平坦時空における (平面)×(時間)平坦時空における双曲面

Photon surface でない例
Kerr 時空における light ring
地平面

時間

Kerr の場合
Photon region ができる
Perlick, Liv. Review. Rel. 7 (2004) 9.

Teo, Gen. Rel. Grav. 35 (2003) 1909

Photon surface であるための条件

induced メトリック

Claudel, Virbhadra & Ellis, JMP42 (2001) 818 [gr-qc/0005050]

S が Photon surface である

M

外的曲率

S(hab ,ab )S の光的測地線が同時に
時空 M の光的測地線
S の任意の光的接ベクトルk a に対し

ab k

a b

k =0

S は umbilical:
ab / hab

Photon surface 条件
S=

Z

Ld ,

L=

1
gab k a k b .
2

Cederbaum が設定した問題（唯一性）
歪んだ静的 Photon surface が存在しうるか？

(外側のBC)
漸近平坦

Cederbaum, arXiv:1406.5475
静的・真空

ds2 =
(内側のBC)N 2 dt2 + gij dxi dxj

Gµ⌫ = 0

Photon surface

内側は気にしない部分的な解答(外側のBC)(再定義された) Photon sphere の唯一性定理

漸近平坦

Cederbaum, arXiv:1406.5475
静的・真空

ds2 =
(内側のBC)(Cederbaum のPhoton sphere{N 2 dt2 + gij dxi dxj

Gµ⌫ = 0

Photon surface＋
N＝(一定)

内側は気にしない

漸近的平坦な静的時空に (Cederbaum の) Photon sphere が存在し、その外)側が真空の時、時空は Schwarzschild 時空である

（再定義された）Photon sphere に対する唯一性定理
真空

Cederbaum, arXiv:1406.5475
Cederbaum & Galloway, arXiv:1504.05804

電磁場入り

Yazadjiev & Lazov, CQG32 (2015) 165021
Cederbaum & Galloway, CQG33 (2015) 075006

共形スカラー場入り 富川, 白水, 泉, PTEP2017 (2017) 033E03
富川, 白水, 泉, CQG34 (2017) 155004
篠原, 富川, 白水, 泉, PTEP2021 (2021) 093E02

その他

Yazadjiev, PRD91 (2015) 123013
Rogatko, PRD93 (2016) 064003
Yazadjiev, PRD104 (2021) 124070

電磁真空時空における光子面の唯一性
歪んだ静的 Photon surface が存在しうるか？

(外側のBC)
漸近平坦

静的・電磁真空ds2 =(内側のBC)<latexit sha1_base64="m8gLpHTSKByOI512o5/21Pzioe0=">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</latexit>

N 2 dt2 + gij dxi dxjRµ⌫ = 2 Fµ⇢ F ⇢⌫Photon surface

1gµ⌫ F⇢ F ⇢4

rµ F µ⌫ = 0<latexit sha1_base64="4sMGmD2Arp90EtLQNC/kF9uhy8k=">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</latexit>

内側は気にしないE<latexit sha1_base64="DMIlgg3nCbkblM/Xnsq8PHPmv5Y=">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</latexit>

部分的な解答(外側のBC)(再定義された) Photon sphere の唯一性定理
Yazadjiev & Lazov, CQG32 (2015) 165021
Cederbaum & Galloway, CQG33 (2015) 075006

漸近平坦

静的・電磁真空ds2 =(内側のBC)<latexit sha1_base64="m8gLpHTSKByOI512o5/21Pzioe0=">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</latexit>

N 2 dt2 + gij dxi dxjRµ⌫ = 2 Fµ⇢ F ⇢⌫(彼らのPhoton sphere{

Photon surface＋
N＝(一定)

1gµ⌫ F⇢ F ⇢4

rµ F µ⌫ = 0<latexit sha1_base64="4sMGmD2Arp90EtLQNC/kF9uhy8k=">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</latexit>

Φ＝(一定)
内側は気にしないE<latexit sha1_base64="DMIlgg3nCbkblM/Xnsq8PHPmv5Y=">AAACbXichVFNSxtRFD0ZbdWoNbUIgiLBkNZVuE+KRlcBEVx0ocaoVEOYGV90cL6YeQnYkD/gWnAhLbQgRfwZ3fQPuPAniAsXEdy48M4kpbiI3mHeO+/ce+477z3Dt61QEV0ntJ7eN2/7+geSg0PD70ZS70c3Q68WmLJkerYXbBt6KG3LlSVlKVtu+4HUHcOWW8bhUpTfqssgtDx3Qx35suzo+65VtUxdMfV119HVgVFtLDcrqQzliEgIkY6AmJ8jBgsL+VmRT4soxZFBJ1a91G/sYg8eTNTgQMKFYmxDR8jfDgQIPnNlNJgLGFlxXqKJJGtrXCW5Qmf2kMd9Xu10WJfXUc8wVpu8i81/wMo0snRFF9Siv3RJN/TYtVcj7hF5OeLZaGulXxk5Hi8+vKpyeFY4+K960bNCFfnYq8Xe/ZiJTmG29fVvp63i4nq28ZF+0S37/0nX9IdP4NbvzfM1uX6GJD/Av1tOdwebszkxlxNrnzOFL52n6McEpjHD9z2PAlawihLv6+IE3/EjcaeNaZPaVLtUS3Q0H/AstE9P65OOKw==</latexit>

漸近的平坦な静的時空に (再定義された) Photon sphere が存在し、その外)側が電磁真空の時、時空は Reissner-Nordstrom 時空である

電磁真空時空における歪んだ光子面の形成
H.Yoshino, arXiv:2309.14318[gr-qc]

唯一性の証明はまだ不十分“再定義された Photon sphere” のラプス・静電ポテンシャル一定の条件はきつい
これらの条件を外しても唯一性は証明できる？ or
これらの条件を外したら歪んだ Photon surface は存在できる？

2016年に「静的な Photon surface の摂動的唯一性」を調べた。
Schwarzschild 時空の静的摂動を考え、歪んだ時空に Photon surface があるかどうかを調べた。
HY, PRD95 (2107) 044047 [arXiv:1607.07133{gr-qc}]

時空を歪めた瞬間に Photon surface は消える。

今回は電磁真空時空の場合に摂動解析をおこなう。

静的摂動解がなければ連続変形する歪んだ解の系列はない

静的摂動解があれば、連続変形する歪んだ解の系列がある可能性がある

Reissner-Nordstrom 時空における Photon surface<latexit sha1_base64="N07A5jYD7mg+gkglKOi8OaEEvOk=">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</latexit>2

dr2
F (r)dt ++ r2 d✓2 + sin2 ✓d 2
F (r)

RN時空:

ds =

F (r) = 1

2M
Q2
+ 2
r
r

<latexit sha1_base64="AWy3APr+hNl2cHb1SSXpka83I/s=">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</latexit>

2

Q
dt
r

<latexit sha1_base64="6QMRwUHIGHOAAUHhft+OHELBqVM=">AAACi3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuoxeQmlmQkpVU71irYKujGpBUlJlcH1lYX1cYogKWKcqtTakviBZQN9AzAQAGTYQhlKDNAQUC+wHKGGIYUhnyGZIZShlyGVIY8hhIgO4chkaEYCKMZDBkMGAqAYrEM1UCxIiArEyyfylDLwAXUWwpUlQpUkQgUzQaS6UBeNFQ0D8gHmVkM1p0MtCUHiIuAOhUYVA2uGqw0+GxwwmC1wUuDPzjNqgabAXJLJZBOguhNLYjn75II/k5QVy6QLmHIQOjC6+YShjQGC7BbM4FuLwCLgHyRDNFfVjX9c7BVkGq1msEig9dA9y80uGlwGOiDvLIvyUsDU4NmM3ABI8AQPbgxGWFGeoZmeoaBJsoOTtCo4GCQZlBi0ACGtzmDA4MHQwBDKNDeHoZNDLsZ9jDxMhkzWTHZQJQyMUL1CDOgACZXAPYOmLc=</latexit>

A=E<latexit sha1_base64="1+A0Q4iF1RcBXsbr6HE4A7C6kq8=">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</latexit>r = 3M

3.0
r(+)
p

2.5
2.0

Q
<latexit sha1_base64="9SEduqlq4DWiNsrFlSNrVj7RL3o=">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</latexit>r+

r/M 1.5

r = 2M

r(-)
p
1.0

光子面
0.5
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r-

0.0
0.0

0.2

0.4

0.6
Q/M

rp =
0.8

1.0

3M
2

1±

r

1

8Q2
9M 2

!

今回考える摂動問題
歪んだ帯電ブラックホール<latexit sha1_base64="AWy3APr+hNl2cHb1SSXpka83I/s=">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</latexit>

F (r) = 1
<latexit sha1_base64="PV4xHyJUNUhP5QGp7h8lKS/QQUk=">AAACsXichVFNaxNRFD2dqq3xo7HdFNwMhkoCNryJYkvBUnDThYt+pR9kkjgzuUkeefPBzEughv4B/0AXXSmIiDv/ghv/gGCXupMuK7hx4c1kQLSod3jzzjv3nvvOe8+NlEy0EKcTxuSly1empq/mrl2/cXMmf2t2Nwn7sUdVL1RhvO86CSkZUFVLrWg/isnxXUV7bu/xKL83oDiRYbCjDyOq+04nkG3pOZqpZv5JK2lUTNt8xGOxXYxLtqK2rlmLFZuiRKowMNcbw6JVOuLcQdMmpRp+0dZd0s49O+rKkh3LTlfXW7pRaeYLoizSMC8CKwMFZLER5l/DRgshPPThgxBAM1ZwkPBXgwWBiLk6hszFjGSaJxwhx9o+VxFXOMz2+N/hVS1jA16Peiap2uNdFI+YlSYWxEfxRpyLD+Kt+Cp+/LXXMO0x8nLIszvWUtSceT6//f2/Kp9nje4v1T89a7SxnHqV7D1KmdEpvLF+8Oz4fHtla2F4V7wUZ+z/hTgV7/kEweCb92qTtk6Q4wew/rzui2C3UrYelq3NB4W11ewppnEbd1Dk+17CGtaxgSrv+w6f8BlfjPvGgfHUcMelxkSmmcNvYfR+AscDpQY=</latexit>

ds

2

h
fF(r)
(r) 1
<latexit sha1_base64="VJDMJrCJIfzOXsZ5rzTgBsQsrmA=">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</latexit>

=

2✏H

2M
Q2
+ 2
r
r
(1)

(r)Y`m (✓,

i

) dt2

i
1 h
(1)
m
+
1 + 2✏H (r)Y` (✓, ) dr2
fF(r)
(r)
h
i
2
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m
+r 1 + 2✏K (r)Y` (✓, ) (d✓2 + sin2 ✓d 2 )
✓
◆
Q
A =
+ ✏'(1) (r)Y`m (✓, ) dt
r<latexit sha1_base64="vGtZgvzZI3Qf25PBMKtjsWvmUXs=">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</latexit>E<latexit sha1_base64="1+A0Q4iF1RcBXsbr6HE4A7C6kq8=">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</latexit><latexit sha1_base64="VJDMJrCJIfzOXsZ5rzTgBsQsrmA=">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</latexit>r = 3M<latexit sha1_base64="58oHBGFfIAbITZyWX8YKnce0UsY=">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</latexit><latexit sha1_base64="7ini+u5Hd09nB8wm2KAQHqa5K2g=">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</latexit>

Q<latexit sha1_base64="9SEduqlq4DWiNsrFlSNrVj7RL3o=">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</latexit>r = 2M

Reissner-Nordstrom 時空の静的摂動の方程式<latexit sha1_base64="ZR81Dutiu41GVyu4SBxZper6Rmw=">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</latexit>(0)(1)
Rµ⌫ = Rµ⌫+ ✏ Rµ⌫+ ···<latexit sha1_base64="/cC9dic/J82OaWBuHjrZEfCsnFA=">AAACrXichVHLShxBFD22xsck6kQ3QjaNg0FRxtshRAkYhGxc+hoVbB26e8qxsF90Vw+YZn7AH3CRlYJICPgTbvyAZOHC7EOWCm5ceKenQaIk3qarTp17z61TVXboylgRXXZonV0vunt6+wovX/UPDBZfD63FQRI5ouIEbhBt2FYsXOmLipLKFRthJCzPdsW6vfe5lV9viCiWgb+q9kOx5Vl1X+5Ix1JMVYsLq9XU9BLTT5q6OTVnTukPxHY6ThNNfVI3RRhLN/AfZ4121qkFKq4WS1SmLPSnwMhBCXksBsVTmKghgIMEHgR8KMYuLMT8bcIAIWRuCylzESOZ5QWaKLA24SrBFRazezzWebWZsz6vWz3jTO3wLi7/ESt1jNFP+kbXdEHf6Tfd/bNXmvVoednn2W5rRVgdPBhZuX1W5fGssPug+q9nhR3MZl4lew8zpnUKp61vfDm8Xvm4PJa+pWP6w/6P6JLO+QR+48Y5WRLLX1HgBzAeX/dTsPaubHwoG0vvS/Of8qfoxRuMYpzvewbzWMAiKrzvGX7gCr+0aa2imdp2u1TryDXD+Cu0+j1jZKR7</latexit>(0)(1)
Tµ⌫ = Tµ⌫+ ✏ Tµ⌫+ ···<latexit sha1_base64="pr1BlPrBKrd5BqUat/7WicQ2yB4=">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</latexit>(0)(1)
Fµ⌫ = Fµ⌫+ ✏ Fµ⌫+ ···Einstein 方程式<latexit sha1_base64="ggonBAswW75efAcJyXZb+vDoPtw=">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</latexit>

0次

(0)
(0)
Rµ⌫
= 8⇡Tµ⌫<latexit sha1_base64="5DdtjK9NzCP/WQ2/A77oCbFpKPw=">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</latexit>

1次

(1)
(1)
Rµ⌫
= 8⇡Tµ⌫Maxwell 方程式<latexit sha1_base64="+Kcr0XnqppQEnHW6xFBt1O+jRyo=">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</latexit>(rµ F µ⌫ )

(0)

= 0

(1)

= 0

<latexit sha1_base64="mRoW/o3psxtbe8vyNN1gt221QS8=">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</latexit>(rµ F µ⌫ )

Einstein-Maxwell 方程式<latexit sha1_base64="BgQDrxEZWoCdN3biMUdiHGM/ytY=">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</latexit>

Einstein Eq.

r2 e2⌫

(0)

(1)
H,rr
+ 2r(re2⌫(0)(1)),r H,rr2 (e2⌫(0)(1)),r K,r<latexit sha1_base64="ukmNWtq7ZBesvJj7lsMMghUWnjE=">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</latexit>`(` + 1)H<latexit sha1_base64="5qY8J+uIyFwDNw/7LfO4NsCdxjQ=">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</latexit>

2 2⌫ (0)r e<latexit sha1_base64="HKkGsLOrrx+RUJj+U4caw3vXYFw=">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</latexit>

⇣

(1)
H,rr

r[r(e2⌫<latexit sha1_base64="vGsk3y8uy2NIV5lQS0P6UeFOCGk=">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</latexit>e2⌫ (0)

⇣

(0)

(1)
K,rr

⌘

),r + 4e2⌫
⌘

(1)
H,r

(1)
K,r

(1)
K,rr

2re2⌫

+ 2r(re2⌫
(0)

(0)

(1)

2Q2 (1)+ 2 H= 2Q'(1),rr(1)),r H,r(1)]K,r+ `(` + 1)H (1) ++ (e2⌫(0)),r H (1) =2Q (1)
H= 2Q'(1),r
2r

2Q (1)'
r2<latexit sha1_base64="px9NeVVxvVtHa7zQH7IamLBR4mw=">AAAC1nichVFNS8NAEJ3Gr1o/WvUieBGL0qKWSREVT4IXwYu2Vgu1LUlcazBN4iYt1BBvInjy5sGTgoj4M7z4B0T8CeJRwYuC0zSgVdRZdmf2vXmzs7uyqamWjfgQEFpa29o7gp2hru6e3nCkr3/NMipcYRnF0AyelSWLaarOMrZqayxrciaVZY2tyzsLdX69yrilGvqqXTNZviyVdHVLVSSboGJklxeSrOAkN/RKwYlh3HWXyItxt+hMcO5OJnkzu/jJuuM8x2PNdLxBTDWj+S813WIkign0bPhnIPpBFHxbNiKXsAGbYIACFSgDAx1sijWQwKKRAxEQTMLy4BDGKVI9noELIdJWKItRhkToDq0l2uV8VKd9vablqRU6RaPJSTkMo3iHV/iMt3iNj/j2ay3Hq1HvpUZebmiZWQwfDaZf/1WVyduw/an6s2cbtmDW61Wl3k0Pqd9CaeireyfP6bnUqDOG5/hE/Z/hA97QDfTqi3KxwlKnEKIPEL8/989gLZkQpxPiylR0ftb/iiAMwQjE6L1nYB4WYRkydO49vAfaAx1CVtgXDoTDRqoQ8DUD0GTC8Qc9zbIS</latexit>

r2 e2⌫

(0)

(0)

(1)
H,r
+ r[r(e2⌫(0)),r + 4e2⌫(0)(`2 + `

2)K (1) = 2Q'(1)
,r

(1)
]K,r

<latexit sha1_base64="Gc8ZiJXehOfjLjFi1gzsdJqAQIc=">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</latexit>2(re2⌫<latexit sha1_base64="DfH3w0S/oGJSpheWYPLEEF2iLDA=">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</latexit>

Maxwell Eq.

r2 e2⌫

(0)

✓

2 (1)
'(1)
+
'
,rr
r ,r

(0)

◆

),r H (1)`(` + 1)'(1) = 2Qe2⌫

(0)

(1)
K,r

解析解が求まる<latexit sha1_base64="2txHEovXWIMl5XubtjWn/m29u+c=">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</latexit>(簡単のため) Extremal casex = r/M<latexit sha1_base64="7bkbTgD0a5EdW9CswT1Hew5R3lA=">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</latexit><latexit sha1_base64="uQ1WFdwfbPKCkjInau9WgNGbdAE=">AAACcHichVHLSsNAFD2N7/po1U3BhdFSURdlIqIiKIKbLvuwKrRVkjitwTSJSVqopT/gD3ThRgUR8TPc+AMu+gniTgU3LrxNA6Ki3mFmzpy5586ZGcXSNcdlrBUQurp7evv6B4KDQ8MjofDo2LZjVmyVZ1VTN+1dRXa4rhk862quznctm8tlRec7ytFme3+nym1HM40tt2bxQlkuGVpRU2WXqEJirz4rzTXEvLiWF/fDURZnXog/geSDKPxImuFr5HEAEyoqKIPDgEtYhwyHWg4SGCziCqgTZxPSvH2OBoKkrVAWpwyZ2CMaS7TK+axB63ZNx1OrdIpO3SaliBh7YDfshd2zW/bI3n+tVfdqtL3UaFY6Wm7th04jmbd/VWWaXRx+qv707KKIFc+rRt4tj2nfQu3oqyfNl8xqOlafYZfsifxfsBa7oxsY1Vf1KsXTZwjSB0jfn/sn2F6IS0txKbUY3Vj3v6IfE5jGLL33MjaQQBJZOvcYTZzjIvAsRIRJYaqTKgR8zTi+hDD/AckHjW8=</latexit>

H (1) =

A2(` + 2)[(2` + 3)x `]2(2` + 3)x(x 1)`+2+B22x(x 1)`+B2
2x(x 1)`<latexit sha1_base64="cwxl8wo5Cwl8uEwOOLNW/7rfGTU=">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</latexit><latexit sha1_base64="jGsItzRxAM2cAeX0AlpYxJwRLQo=">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</latexit><latexit sha1_base64="cGQ2RzgFWB3z+rMO4VPKmoK6C04=">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</latexit>

K (1) =

A2`[(2` + 3)x (` + 2)]2(2` + 3)x(x 1)`+2<latexit sha1_base64="cwxl8wo5Cwl8uEwOOLNW/7rfGTU=">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</latexit><latexit sha1_base64="aK4ZOfiqMRmKSY/NRl/qsb/NcEg=">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</latexit>

A1 `1電荷の多重極子を質量の多重極子を加えたことによる摂動

加えたことによる摂動<latexit sha1_base64="7r15rXDq9LYb5QLhIwu4KkeSPIo=">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</latexit>(`

2)<latexit sha1_base64="gFAVT+NxzAnOEGF8jWCrPPIDZHU=">AAACbHichVHLSgMxFD0d3/VVHwtBhGKx1E3JiKgIiuDGpVpbK20pM2Oqg5kHM9OCFn/ArQsX6kJBRPwMN/6Ai36CCG4quHHh7XRAtKg3JDk5uefmJFFtobseY7WQ1Nbe0dnV3RPu7esfGIwMDWdcq+xoPK1ZwnKyquJyoZs87eme4Fnb4YqhCr6tHqw29rcr3HF1y9zyDm1eMJQ9Uy/pmuIRtZPIcyGiS/J0MRJjSeZHtBXIAYghiHUrcos8dmFBQxkGOEx4hAUUuNRykMFgE1dAlTiHkO7vcxwjTNoyZXHKUIg9oHGPVrmANWndqOn6ao1OEdQdUkYxxZ7YHauzR3bPntnHr7Wqfo2Gl0Oa1aaW28XBk7HU+78qg2YP+1+qPz17KGHB96qTd9tnGrfQmvrK0Vk9tbg5VY2za/ZC/q9YjT3QDczKm3azwTfPEaYPkH8+dyvIzCTluaS8MRtbWQ6+ohvjmESC3nseK1jDOtJ0roFTXOAy9CqNSuPSRDNVCgWaEXwLKf4JLZuMSg==</latexit>

ゲージモード(` = 1)

光子面条件を満たせるか<latexit sha1_base64="wDr7Jr8PEeaQ7abEzsNp3h6suG4=">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</latexit>r = rp + ✏ f (1) (✓, ) + · · ·Photon surface の位置:
Photon surface 条件

ab / hab<latexit sha1_base64="sONI7spdvtb/rrGsgavCIXBiHbQ=">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</latexit>f (1) = C0 + C11 Y11 (✓, )(1)(1)
f,✓ = cot ✓f, ,+C10 Y10 (✓) + C1 1 Y1 1 (✓, )<latexit sha1_base64="PWN11tv5DM7a8WkhkntSgJA2fy8=">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</latexit>(1)(1)
f, = sin2 ✓f,✓✓⇣<latexit sha1_base64="i1i60rxLO1Z4oRIIgHYp7Fe+ceQ=">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</latexit>

(1)
(1)
H,r
+ K,r

⌘

(1)
sin ✓ cos ✓f,✓ ,r=rpY`m (✓, ) =

"✓

F,r
2F

◆

+
,r

1
r2

#

f (1) +
r=rp

1
(1)
f
,✓✓
r2 F (rp )= 0<latexit sha1_base64="Kk5sGt3oqnBdz5rCi6g8s0yG+BQ=">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</latexit><latexit sha1_base64="FXHi24FON/Gks4643rojgWCC0pQ=">AAACtnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQsEKsSkFSUmVzvFG9VWOwKJGB0FW4UYHS4FBV2IjLFGTGpOjraRpoYRiBFnpG0KFjDRrK2GCGkbQ+W0DTVrdeIFlA30DMBAAZNhCGUoM0BBQL7AcoYYhhSGfIZkhlKGXIZUhjyGEiA7hyGRoRgIoxkMGQwYCoBisQzVQLEiICsTLJ/KUMvABdRbClSVClSRCBTNBpLpQF40VDQPyAeZWQzWnQy0JQeIi4A6FRhUDa4arDT4bHDCYLXBS4M/OM2qBpsBckslkE6C6E0tiOfvkgj+TlBXLpAuYchA6MLr5hKGNAYLsFszgW4vAIuAfJEM0V9WNf1zsFWQarWawSKD10D3LzS4aXAY6IO8si/JSwNTg2YzcAEjwBA9uDEZYUZ6hmZ6hoEmyg520KjgYJBmUGLQAIa3OYMDgwdDAEMo0N7tDHcZnjE8Z7JgimNKZUqHKGVihOoRZkABTAUAfSekCQ==</latexit><latexit sha1_base64="ZFiy0H04+ae6AAomort3OmR/Psc=">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</latexit>

`

2 のとき

B2
=
A2

3(` + 2)(2`2 + 5` + 4),
(2` + 3)(2` + 1)

２つの摂動解の振幅の比をうまく選べば歪んだ光子面を作ることができる。

光子面条件を満たす B2/A20rp(+) (l = 2)
rp(+) (l = 3)

-10

-10

B2 /A2 -20

-15

-20

rp(-) (l = 2)

-30
-25

rp(-) (l = 3)

-30

-40

1.040

0.0

1.045

0.2

1.050

0.4

1.055

1.060

0.6
Q/M

1.065

0.8

1.0

1.2

(1)

(1)

光子面上での H , K , φrp(+) (l = 3)

0

(1)

0

rp(+) (l = 2)
-1
0

-2

-10

0

-5

rp(+) (l = 3)

-10rp(-) (l = 2)H1 /A2

rp(+) (l = 2)K1 /A2 -20

-10

-20

-3

rp(-) (l = 3)

-15rp(-) (l = 2)

-30

-30

-4
-20
1.030

rp(-) (l = 3)

-40
1.035

1.040

1.045

1.050

1.055

1.060

1.065

1.040

1.045

1.050

1.055

1.060

1.065

-40

-5
0.0

0.2

0.4

0.6

0.8

1.0

0.0

1.2

0.2

0.4

0.6

0.8

1.0

1.2

1.0

1.2

Q/M

Q/M
0

<latexit sha1_base64="PV4xHyJUNUhP5QGp7h8lKS/QQUk=">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</latexit>

ds2 =

h
fF(r)
(r) 1
<latexit sha1_base64="VJDMJrCJIfzOXsZ5rzTgBsQsrmA=">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</latexit>i2✏H (1) (r)Y`m (✓, ) dt2

i
1 h
(1)
m
+
1 + 2✏H (r)Y` (✓, ) dr2
fF(r)
(r)
h
i
+r2 1 + 2✏K (1) (r)Y`m (✓, ) (d✓2 + sin2 ✓d 2 )<latexit sha1_base64="vGtZgvzZI3Qf25PBMKtjsWvmUXs=">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</latexit>-50rp(+) (l = 2)

-10<latexit sha1_base64="VJDMJrCJIfzOXsZ5rzTgBsQsrmA=">AAACZ3ichVHLSgMxFD0dX7U+WhWk4EYtim5KRkTFhRQEcVmrVUFFZsa0DZ0XM2mhFn/AhVsFVwoi4me48Qdc+AniUsGNC+9MB0RFvSHJyck9NyeJ7prCl4w9xpS29o7Ornh3oqe3rz+ZGhjc9J2aZ/Ci4ZiOt61rPjeFzYtSSJNvux7XLN3kW3p1OdjfqnPPF469IRsu37O0si1KwtBkQK1MedP7qQzLsjBGfwI1AhlEkXdS19jFARwYqMEChw1J2IQGn9oOVDC4xO2hSZxHSIT7HEdIkLZGWZwyNGKrNJZptROxNq2Dmn6oNugUk7pHylFMsAd2w17YPbtlT+z911rNsEbgpUGz3tJydz95nF5/+1dl0SxR+VT96VmihIXQqyDvbsgEtzBa+vrh6cv6YmGiOcku2TP5v2CP7I5uYNdfjas1XjhHgj5A/f7cP8HmTFady6prs5ncUvQVcYxgHFP03vPIYRV5FOncCk5wirPYk5JUhpV0K1WJRZohfAll7AOK6Iqp</latexit>

rp(+) (l = 3)-10<latexit sha1_base64="58oHBGFfIAbITZyWX8YKnce0UsY=">AAACwXichVHLbtNAFL01j5bwaIANUjcRUZGjoGhsIUAsUKVuKrHpg7RFmTiynZtk1LE9Gk8iFas/wA+wYNVKFar4C9j0B7roH4BYQZHYsODasYSgAq7lmXPP3HPnzEygpEgNY6cz1oWLly7Pzl2pXL12/cZ89eatzTQZ6xDbYSITvR34KUoRY9sII3FbafSjQOJWsLOcr29NUKciiZ+bXYXdyB/GYiBC3xDVq/Km9lwucWA6TtPlqFIhk7j2zMtsp7Fn68aLHkcpvcjmZoTGv8/VSDS4FsOR6dr9Kem5TZ6KmBoVaa2fF3luo1etsxYronYeOCWoQxmrSfUtcOhDAiGMIQKEGAxhCT6k9HXAAQaKuC5kxGlColhH2IMKacdUhVThE7tD45CyTsnGlOc900Id0i6Sfk3KGiyyE3bEztgxe8c+sR9/7ZUVPXIvuzQHUy2q3vyrOxvf/6uKaDYw+qX6p2cDA3hceBXkXRVMfopwqp+8fH228WR9MbvHDthn8r/PTtkHOkE8+RYeruH6G6jQAzh/Xvd5sOm2nIctZ+1Bfelp+RRzsAB3wab7fgRLsAKr0KZ938NH+AJfrWVLWMrS01JrptTcht/Cyn4C7NSsdQ==</latexit>φ1 /A2 -15
-20

-30

-20

rp(-) (l = 2)<latexit sha1_base64="7ini+u5Hd09nB8wm2KAQHqa5K2g=">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</latexit>

A =

✓

Q
+ ✏'(1) (r)Y`m (✓, )
r

◆

-40

dt

-25

rp(-) (l = 3)
-50
1.030

-30

0.0

0.2

1.035

1.040

0.4

1.045

1.050

0.6
Q/M

1.055

1.060

0.8

1.065

光子面条件を満たせるか<latexit sha1_base64="wDr7Jr8PEeaQ7abEzsNp3h6suG4=">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</latexit>

Photon surface の位置:
Photon surface 条件

r = rp + ✏ f (1) (✓, ) + · · ·ab / hab<latexit sha1_base64="sONI7spdvtb/rrGsgavCIXBiHbQ=">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</latexit>f (1) = C0 + C11 Y11 (✓, )(1)(1)
f,✓ = cot ✓f, ,+C10 Y10 (✓) + C1 1 Y1 1 (✓, )<latexit sha1_base64="PWN11tv5DM7a8WkhkntSgJA2fy8=">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</latexit>(1)(1)
f, = sin2 ✓f,✓✓⇣<latexit sha1_base64="i1i60rxLO1Z4oRIIgHYp7Fe+ceQ=">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</latexit>

(1)
(1)
H,r
+ K,r

/ A2
<latexit sha1_base64="VSXUAxxcXmsVWZQAJM6RE6rxPRA=">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</latexit>

⌘

r=rp

(1)
sin ✓ cos ✓f,✓ ,Y`m (✓, ) =` = 1 のとき

F,r
2F

◆

+
,r

1
r2

#

f (1) +
r=rp

/ C10
<latexit sha1_base64="40T81J+yoLDvKCpSOCo5rHC7vhY=">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</latexit><latexit sha1_base64="L3X4gr4JsGgDUoAEnDHQSzkGvYc=">AAACh3ichVG7SsRAFD0b3+tr1UawWVwUq/VGRUVQlG1sBF+rgo+QxFkNZpOQZBc0pBd/wMJKQUQEW+1t/AELP0EsFWwsvJsNiIp6h5k5c+aeO2dmNMc0PJ/oMSHV1NbVNzQ2JZtbWtvaUx2dK55dcnWR123Tdtc01ROmYYm8b/imWHNcoRY1U6xqe7nK/mpZuJ5hW8v+viM2i+qOZRQMXfWZUlK9GwVX1YOcIm9RGMwoQ2F6Ml3lhofDQKZwTkllKEtRpH8COQYZxDFvpy6wgW3Y0FFCEQIWfMYmVHjc1iGD4DC3iYA5l5ER7QuESLK2xFmCM1Rm93jc4dV6zFq8rtT0IrXOp5jcXVam0UcPdEkvdE9X9ETvv9YKohoVL/s8a1WtcJT2o+6lt39VRZ597H6q/vTso4DxyKvB3p2IqdxCr+rLB8cvSxOLfUE/ndEz+z+lR7rjG1jlV/18QSyeIMkfIH9/7p9gZSgrj2blhZHM9FT8FY3oQS8G+L3HMI1ZzCPP5x7iGje4lZqkQWlUGq+mSolY04UvIc18ACt5lZI=</latexit><latexit sha1_base64="eVqkJFyB81cM50XnCMx4Y2rCvn0=">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</latexit>

"✓

C10
33
=
M
A2
10

光子面の位置をうまく選べば歪んだ光子面を作ることができる。

1
(1)
f
,✓✓
r2 F (rp )光子面条件を満たす C1/D
4

2

rp(+) (l = 1)00-2(C01 /D)/M

-500

-4
-1000

rp(-) (l = 1)

-6

-1500

-2000

-8

-2500

-10

1.00

0.0

1.01

0.2

1.02

0.4

1.03

1.04

0.6
Q/M

1.05

1.06

0.8

1.0

1.2

^ (1)^(1)

光子面上での H , φ00rp(+) (l = 1)

rp(+) (l = 1)
-5

-1
0

0

-10
-2

-100
-100

 1 /DH

1φ /D -15

-200

-200

rp(-) (l = 1)

-3
-300

-300

-20

rp(-) (l = 1)

-400
-400

-4

-25
-500
1.00

-5
<latexit sha1_base64="Meiml7Ctmn4ZAuMW9E2cXngiSLI=">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</latexit>

0.0

1.01

0.2

1.02

1.03

0.4

1.04

0.6

1.05

-500

1.06

0.8

1.0

1.2

-30

0.0

0.2

1.02

1.03

0.4

ds2 =

h
fF(r)
(r) 1
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i
1 h
1 + 2✏H (1) (r)Y`m (✓, ) dr2
fF(r)
(r)
h
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A =

✓

0.6

<latexit sha1_base64="Meiml7Ctmn4ZAuMW9E2cXngiSLI=">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</latexit><latexit sha1_base64="xkSoq/JjZqZjaw1dd2c0Xi0ULhA=">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</latexit><latexit sha1_base64="58oHBGFfIAbITZyWX8YKnce0UsY=">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</latexit><latexit sha1_base64="+nxQUJzpV0b9XAs1y0hRgie3U3I=">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</latexit>

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1.06

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F (r) 1

1.04

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1.01

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rp + C10 Y`0◆Q+ ✏ 'ˆ(1) Y`0 (✓, ) dt
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まとめ

まとめと議論
摂動的非唯一性（電磁真空時空）
歪んだ光子面を持ち、球対称帯電ブラックホール解に連続につながるような解の系列を作ることができる。

光子面の唯一性は成立しない（？）

完全にアインシュタイン方程式を解いて歪んだ光子面を持つ時空を作れるか？
時空が光子面を持つ/持たないことによる面白い現象はあるか？

ブラックホールシャドウではすでに議論がある
光子面がない時の光子のカオス的運動

カオスが生じる際のブラックホールシャドウ