Friedmann宇宙に於ける
ブラックホールの時空構造
Masato NOZAWA
(Waseda Univ)
Based on
arXiv:0912.281, 1003.2849
cowork with Kei-ichi Maeda
Talk @ Osaka City Univ. 4th June, 2010
Contents
Introduction
Black holes in general relativity:
9 slides
--studies of stationary black holes--
Black holes in dynamical background
6 slides
Dynamical black holes
Solution from intersecting branes
4 slides
Spacetime structure
23 slides
Concluding remarks
Summary and outlooks
4 slides
Contents
Introduction
Black holes in general relativity:
9 slides
--studies of stationary black holes--
Black holes in dynamical background
6 slides
Dynamical black holes
Solution from intersecting branes
4 slides
Spacetime structure
23 slides
Concluding remarks
Summary and outlooks
4 slides
Black holes in astrophysics
天体物理学におけるブラックホール
恒星の進化の最終状態
支えるエネルギーを失い重力崩壊
超新星爆発
ブラックホール
光さえも抜け出せない領域
Black holes: definition
ブラックホール = “no region of escape”
c.f. Hawking & Ellis 1973
= 十分遠方(漸近平坦)の観測者(光的無限遠)と
因果的に繋がる曲線なし
singularity
Black hole
事象の地平線
: 未来光的無限遠
idealized “distant observer”
観測者Oの世界線
Black holes: definition
ブラックホール = “no region of escape”
c.f. Hawking & Ellis 1973
= 十分遠方(漸近平坦)の観測者(光的無限遠)と
因果的に繋がる曲線なし
singularity
Black hole
事象の地平線
Black hole =
Event horizon =
: 未来光的無限遠
idealized “distant observer”
観測者Oの世界線
: causally disconnected region from I
: boundary of black hole
+
Black holes: definition
ブラックホール = “no region of escape”
c.f. Hawking & Ellis 1973
= 十分遠方(漸近平坦)の観測者(光的無限遠)と
因果的に繋がる曲線なし
singularity
Black hole
事象の地平線
Black hole =
Event horizon =
: 未来光的無限遠
idealized “distant observer”
観測者Oの世界線
: causally disconnected region from I
: boundary of black hole
NB. Event horizon is a null surface & a global concept
+
Black holes in general relativity
一般相対論に於けるブラックホール
• “stellar sized” ブラックホール
孤立系
•Einstein方程式の真空解(Rab=0)で近似できる
•遠方で時空は平坦 (漸近的平坦性)
• 重力波放出等でダイナミカルな変化は減衰
重力崩壊から十分経過
システムは平衡状態へ
1st step : 定常時空中の漸近平坦な真空ブラックホール
Stationary: there exists a Killing field ta which is timelike at infinity
L t gab = ∇a tb + ∇b ta = 0 ,
tata <0 at infinity
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
安定性解析
Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
安定性解析
‣ 線形摂動 Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
安定 (admits no growing modes) 重力崩壊の最終状態
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
安定性解析
‣ 線形摂動 一般的性質
Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
安定 (admits no growing modes) 重力崩壊の最終状態
Hawking 1972
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
安定性解析
‣ 線形摂動 一般的性質
Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
安定 (admits no growing modes) 重力崩壊の最終状態
Hawking 1972
‣ topology: 地平線断面はS2のみ
c.f. “topological censorship’’ by Friedman et al
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
安定性解析
‣ 線形摂動 一般的性質
Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
安定 (admits no growing modes) 重力崩壊の最終状態
Hawking 1972
‣ topology: 地平線断面はS2のみ
c.f. “topological censorship’’ by Friedman et al
‣ 面積則: 事象の地平線の表面積は減少しない
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
安定性解析
‣ 線形摂動 一般的性質
Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
安定 (admits no growing modes) 重力崩壊の最終状態
Hawking 1972
‣ topology: 地平線断面はS2のみ
c.f. “topological censorship’’ by Friedman et al
‣ 面積則: 事象の地平線の表面積は減少しない
‣ 対称性: 事象の地平線はKillingベクトルで生成される
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
安定性解析
‣ 線形摂動 一般的性質
安定 (admits no growing modes) 重力崩壊の最終状態
Hawking 1972
‣ topology: 地平線断面はS2のみ
c.f. “topological censorship’’ by Friedman et al
‣ 面積則: 事象の地平線の表面積は減少しない
‣ 対称性: 事象の地平線はKillingベクトルで生成される
唯一性
Carter 1972, Robinson 1975, Mazur 1982
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
安定性解析
‣ 線形摂動 安定 (admits no growing modes) 重力崩壊の最終状態
Hawking 1972
一般的性質
‣ topology: 地平線断面はS2のみ
c.f. “topological censorship’’ by Friedman et al
‣ 面積則: 事象の地平線の表面積は減少しない
‣ 対称性: 事象の地平線はKillingベクトルで生成される
唯一性
Carter 1972, Robinson 1975, Mazur 1982
‣ 非回転的 ‣ 回転的 Schwarzschild
Kerr
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
安定性解析
‣ 線形摂動 安定 (admits no growing modes) 重力崩壊の最終状態
Hawking 1972
一般的性質
‣ topology: 地平線断面はS2のみ
c.f. “topological censorship’’ by Friedman et al
‣ 面積則: 事象の地平線の表面積は減少しない
‣ 対称性: 事象の地平線はKillingベクトルで生成される
唯一性
Carter 1972, Robinson 1975, Mazur 1982
‣ 非回転的 Schwarzschild
‣ 回転的 Kerr
We can focus on Kerr-family for analyzing equilibrium BHs
Stationary black holes in general relativity
Schwarzschild 1915, Kerr 1963
厳密解の発見
‣ Schwarzschild解: 静的球対称 (invariant under t→-t, hole is round)
‣ Kerr解: 軸対称定常 (φ-independent and invariant under t→-t, φ→-φ)
Vishveshwara 1970, Press-Teukolsky 1973, Whiting 1989
安定性解析
‣ 線形摂動 安定 (admits no growing modes) 重力崩壊の最終状態
Hawking 1972
一般的性質
‣ topology: 地平線断面はS2のみ
c.f. “topological censorship’’ by Friedman et al
‣ 面積則: 事象の地平線の表面積は減少しない
‣ 対称性: 事象の地平線はKillingベクトルで生成される
唯一性
Carter 1972, Robinson 1975, Mazur 1982
‣ 非回転的 Schwarzschild
‣ 回転的 Kerr
We can focus on Kerr-family for analyzing equilibrium BHs
Symmetry properties
Schwarzschild解の地平線 (r=2M) はKillingベクトルで生成
ex.
�
�
�
�−1
2M
2M
ds2 = − 1 −
dt2 + 1 −
dr2 + r2 dΩ2 .
r
r
静的観測者(ta=(∂/∂t)a)は時空を加速運動
tb ∇b ta = κ(r)(∂/∂r)a ,
κ: 加速度
事象の地平線は特別な光的超曲面N
tb ∇b ta = κ(r = 2M)ta , on N
‣ KillingベクトルtaはNにnormal (tata=0) & tangent
‣ κ|r=2M =(4M)-1: 地平線の表面重力
ta
Kerr解では事象の地平線上でξa=ta+ΩHφa が光的 (ΩH:地平線の角速度)
N
Killing horizon
Killing地平線
Boyer 1969
• Killingベクトル ξa を法線にもつ光的超曲面N
ξaξa=0 on N,
Killing horizon
Killing地平線
Boyer 1969
• Killingベクトル ξa を法線にもつ光的超曲面N
ξaξa=0 on N,
rotation=0 on N
shear, expansion=0
Killing horizon
Killing地平線
Boyer 1969
• Killingベクトル ξa を法線にもつ光的超曲面N
ξaξa=0 on N,
• Killing地平線に流入するエネルギーなし
rotation=0 on N
shear, expansion=0
H
0 = Rab ξa ξb = 8πT ab ξa ξb .
Killing horizon
Killing地平線
Boyer 1969
• Killingベクトル ξa を法線にもつ光的超曲面N
ξaξa=0 on N,
• Killing地平線に流入するエネルギーなし
• 事象の地平線とは独立な概念
rotation=0 on N
shear, expansion=0
H
0 = Rab ξa ξb = 8πT ab ξa ξb .
Killing horizon
Killing地平線
Boyer 1969
• Killingベクトル ξa を法線にもつ光的超曲面N
ξaξa=0 on N,
• Killing地平線に流入するエネルギーなし
rotation=0 on N
shear, expansion=0
H
0 = Rab ξa ξb = 8πT ab ξa ξb .
• 事象の地平線とは独立な概念
‣対称性
Hawking 1971, Moncrief-Isenberg 1973, Sudarsky-Wald 1993
定常時空 (定常のKilling ta=(∂/∂t)aが存在) におけるBHの地平線は
(i) Killing 地平線と一致
Killing horizon
Killing地平線
Boyer 1969
• Killingベクトル ξa を法線にもつ光的超曲面N
ξaξa=0 on N,
• Killing地平線に流入するエネルギーなし
rotation=0 on N
shear, expansion=0
H
0 = Rab ξa ξb = 8πT ab ξa ξb .
• 事象の地平線とは独立な概念
‣対称性
Hawking 1971, Moncrief-Isenberg 1973, Sudarsky-Wald 1993
定常時空 (定常のKilling ta=(∂/∂t)aが存在) におけるBHの地平線は
(i) Killing 地平線と一致
定常時空では事象の地平線は時空の対称性のみで決定される
Killing horizon
Killing地平線
Boyer 1969
• Killingベクトル ξa を法線にもつ光的超曲面N
ξaξa=0 on N,
rotation=0 on N
shear, expansion=0
• Killing地平線に流入するエネルギーなし
H
0 = Rab ξa ξb = 8πT ab ξa ξb .
• 事象の地平線とは独立な概念
‣対称性
Hawking 1971, Moncrief-Isenberg 1973, Sudarsky-Wald 1993
定常時空 (定常のKilling ta=(∂/∂t)aが存在) におけるBHの地平線は
(i) Killing 地平線と一致
定常時空では事象の地平線は時空の対称性のみで決定される
(ii-a) 回転していなければ(ξa =ta), 時空は静的
(ii- b) 回転していれば(ξa ≠ta), 時空は軸対称
ξ a = t a + Ω H φa
Black hole thermodynamics
ブラックホール熱力学
• 0th law: equilibrium
Bekenstein 1971, Bardeen-Carter-Hawking 1973
c.f. Racz-Wald 1992
κ is constant on Killing horizon
κ: 表面重力
• 1st law: energy conservation
定常BHは熱力学的にも平衡状態
c.f. Gao-Wald 2001
A:地平線面積
• 2nd law: entropy increasing law
c.f. Flanagan et al 1999, Gao-Wald 2001
定常BHは面積不変 δA=0
Black hole thermodynamics
ブラックホール熱力学
• 0th law: equilibrium
Bekenstein 1971, Bardeen-Carter-Hawking 1973
c.f. Racz-Wald 1992
κ is constant on Killing horizon
κ: 表面重力
• 1st law: energy conservation
定常BHは熱力学的にも平衡状態
c.f. Gao-Wald 2001
A:地平線面積
• 2nd law: entropy increasing law
c.f. Flanagan et al 1999, Gao-Wald 2001
定常BHは面積不変 δA=0
-Taking quantum effect into account, it turns out that BH emits thermal radiation
Black hole thermodynamics
ブラックホール熱力学
• 0th law: equilibrium
Bekenstein 1971, Bardeen-Carter-Hawking 1973
c.f. Racz-Wald 1992
κ is constant on Killing horizon
κ: 表面重力
• 1st law: energy conservation
定常BHは熱力学的にも平衡状態
c.f. Gao-Wald 2001
A:地平線面積
• 2nd law: entropy increasing law
c.f. Flanagan et al 1999, Gao-Wald 2001
定常BHは面積不変 δA=0
-Taking quantum effect into account, it turns out that BH emits thermal radiation
Hawking 1973
Stationary black holes
定常ブラックホール
漸近平坦,真空というセットアップのもとでは,
• Schwarzschild解, Kerr解などの重力的に安定な厳密解が存在
• Killing地平線で表されるような熱力学平衡状態に対応
• 本質的に1種類しか存在しない(Kerr族)
N.B Einstein-Maxwell系でも同様の性質
• Kerr-Newman族: (M,J,Q)の3パラメータファミリー
Mazur 1982
2
2Mr
−
Q
Σ 2
2
2
2
2
2
2
2
2
ds = −dt +
(dt − a sin θdφ) + (r + a ) sin θdφ + dr + Σdθ2 .
Σ
∆
Σ = r2 + a2 cos2 θ , ∆ = r2 − 2Mr + a2 + Q2 ,
Contents
Introduction
Black holes in general relativity:
9 slides
--studies of stationary black holes--
Black holes in dynamical background
6 slides
Dynamical black holes
Solution from intersecting branes
4 slides
Spacetime structure
23 slides
Concluding remarks
Summary and outlooks
6 slides
Black holes in the universe
ダイナミカルブラックホール
定常性をはずす
応用: 原始ブラックホール
time-dependent
Carr-Hawking 1974
宇宙の初期に密度揺らぎでブラックホール形成
Hubble質量のブラックホールが生成
�c3
TB =
∼ 10−7 (M/M⊙ )−1 K , ∼ (M/1010 g)−1 TeV ,
8πGkB M
Hawking輻射が観測される可能性
宇宙論的背景の中でブラックホールを考える必要
漸近的平坦性や真空条件もはずすべき
Black holes in the universe
我々の宇宙は大きなスケールで一様等方
Robertson-Walker 計量:
Friedmann方程式:
1st step: exact black-hole solutions in FRW universe
唯一性は成り立たない
we expect much richer families of solutions
Black holes in the universe
我々の宇宙は大きなスケールで一様等方
Robertson-Walker 計量:
Friedmann方程式:
1st step: exact black-hole solutions in FRW universe
唯一性は成り立たない
Difficulties
we expect much richer families of solutions
Black holes in the universe
我々の宇宙は大きなスケールで一様等方
Robertson-Walker 計量:
Friedmann方程式:
1st step: exact black-hole solutions in FRW universe
唯一性は成り立たない
we expect much richer families of solutions
Difficulties
Putting a BH in FRW universe
Universe becomes inhomogeneous
Black holes in the universe
我々の宇宙は大きなスケールで一様等方
Robertson-Walker 計量:
Friedmann方程式:
1st step: exact black-hole solutions in FRW universe
唯一性は成り立たない
we expect much richer families of solutions
Difficulties
Putting a BH in FRW universe
Universe becomes inhomogeneous
Matter accretion
BH will grow & deform
Black holes in the universe
我々の宇宙は大きなスケールで一様等方
Robertson-Walker 計量:
Friedmann方程式:
1st step: exact black-hole solutions in FRW universe
唯一性は成り立たない
we expect much richer families of solutions
Difficulties
Putting a BH in FRW universe
Universe becomes inhomogeneous
Matter accretion
BH will grow & deform
We must solve nonlinear PDE w/ space & time simultaneously.
FRW black holes with symmetry
Schwarzschild-de Sitter
Kottler 1918
locally static (Birkhoff’s theorem)
本質的にnon-dynamical
(R+はKilling地平線)
f (R+ ) = f (Rc ) = 0, R+ < Rc
R+
Rc
t=const.
T=const.
FRW black holes with symmetry
Sultana-Dyer solution
Sultana & Dyer 2005
r=0
• sourced by dust and null dust
a=0
c.f. Saida-Harada-Maeda 2007
• Schwarzschild 計量と共形
地平線はr=2M
r=2M
RH=2Ma
• generated by a conformal Killing vector ξa=(∂/∂η)a
宇宙膨張と“同じ割合”でBHも進化
FRW black holes with symmetry
Sultana-Dyer solution
Sultana & Dyer 2005
r=0
• sourced by dust and null dust
a=0
c.f. Saida-Harada-Maeda 2007
• Schwarzschild 計量と共形
地平線はr=2M
r=2M
RH=2Ma
• generated by a conformal Killing vector ξa=(∂/∂η)a
宇宙膨張と“同じ割合”でBHも進化
• エネルギー条件の破れ
for η>r(r+2M)/2M
FRW black holes with symmetry
Sultana-Dyer solution
Sultana & Dyer 2005
r=0
• sourced by dust and null dust
a=0
c.f. Saida-Harada-Maeda 2007
• Schwarzschild 計量と共形
地平線はr=2M
r=2M
RH=2Ma
• generated by a conformal Killing vector ξa=(∂/∂η)a
宇宙膨張と“同じ割合”でBHも進化
• エネルギー条件の破れ
for η>r(r+2M)/2M
physically unacceptable
FRW black holes with symmetry
Self-similar black holes
L ξ gab = 2gab ,
自己相似性
• 減速膨張のとき,BHは存在しない
McVittie’s solution
Harada-Maeda-Carr 2006
Nolan 2002, Kaloper et al 2010
a(t) = t p ,
,t
=∞
r=M/2aは曲率特異点
r=
M
(p<1)
/2
a
a=t pのとき
r=M/2a, t:finite
FRW black holes
Black holes in FRW universe
時間依存性あり ブラックホールは時間変化
厳密解に限ってもエネルギー条件をみたすような
ブラックホールを構築するのは(数学的にも)難しい
What we have done
高次元のダイナミカルな交差ブレーン解のコンパクト化により,
4次元のダイナミカルな“ブラックホール”解を得る.
Contents
Introduction
Black holes in general relativity:
9 slides
--studies of stationary black holes--
Black holes in dynamical background
6 slides
Dynamical black holes
Solution from intersecting branes
4 slides
Spacetime structure
23 slides
Concluding remarks
Summary and outlooks
4 slides
Branes in string theory
String/M-theory
• Promising unified theory of all interactions
• 10/11 次元で定式化
• 基本的構成要素:
string (open & closed), D-brane
超重力理論のブラックpブレーン解
Horowitz-Strominger 1991
• black “holes” w/ extended into spatial p-directions
• preserves a part of supersymmetries (BPS state)
• low energy description of D-branes (and M-branes)
M-Branes in 11D supergravity
11次元超重力
F=dA: 4-form
Fが電気的(4-form)に結合 M2-brane
Fが磁気的(7-form)に結合 M5-brane
c.f. 4次元点粒子(0-dim.)
electric Fµν (0+2-dim.),
magnetic *Fµν (0+(4-2)-dim.),
extremal M2-brane
r
extended directions
y2
y1
H2: harmonic fun. on
• r=0 に点状源
• preserves 1/2-SUSY
r=0 は正則地平線
Intersecting branes in supergravity
交差ブレーン
Tseytlin 1996, Ohta 1997
e.g., M2/M2/M5/M5 branes
M2∩M2=0, M2∩M5=1, M5∩M5=3
pB
• [**] 内の計量でharmonics Hn-1がかかっている
ところにMn-braneが存在 (intersection rule)
pA
• 1/8-BPS状態
pAB
4D black hole from intersecting branes
ブレーン方向を丸めて,4Dへコンパクト化
M4xT 7
4D Einstein frame metric from M2/M2/M5/M5
• Qi ≡Qとすれば 極限 Reissner-Nordströmブラックホール
otherwise: Einstein-Maxwell (x4)-dilaton (x3)
Advantages of intersecting brane picture
ブラックホールエントロピーのミクロな導出が可能
Dブレーンは開弦のendpoint
弦の配位を数え上げ可能
S=logW=2π(N1N5Nw)1/2=A/4G10=SBH
Strominger-Vafa 1996, Callan-Maldacena 1996
• 4次元ブラックホールを得るには4電荷必要
M2/M2/M5/M5, M5/M5/M5/W, etc.
• 5次元ブラックホールを得るには3電荷必要
M2/M2/M2, M2/M5/W, D1/D5/W, etc.

1/4
�
� �
Qi 

,
RH = r 
1+

r
i
r→0

1/3
�
� �
Qi 

,
RH = r 
1+

r
i
r→0
Black hole from dynamically intersecting branes
Time-dependent branes in 11D SUGRA
Maeda-Ohta-Uzawa 2009
静的なM-ブレーンと同様な計量ansatzのもと,
時間依存性をもった交差ブレーン解を分類
ex. M2/M2/M5/M5 (4-charges) with evolving M2
• 4種のブレーンのうち,どれか1つのみ時間依存性をもつことが可能
• Qi=0とすると, 11DはKasner宇宙(空間一様性を保つ真空解)
τ ∝ t2/3
4D black hole from intersecting branes
ブレーン方向を丸めて,4Dへコンパクト化
M4xT 7
4D Einstein frame metric from dynamical M2/M2/M5/M5
• 解は時間依存+空間的非一様
• ダイナミカルブラックホールを表していると期待できる
Dynamical black hole in FRW universe?
• asymptotically (r→∞) tends to P=ρ FRW universe
• reduces to AdS2 x S2 as r→0 with t :fixed
typical near-horizon geometry of extremal BH
Kunduri-Lucietti-Reall 2007
Dynamical black hole in FRW universe?
• asymptotically (r→∞) tends to P=ρ FRW universe
• reduces to AdS2 x S2 as r→0 with t :fixed
Extremal black hole in FRW universe?
Naive Picture
extreme RN
(r~0)
P=ρ FRW
(r→∞)
Naive Picture
extreme RN
(r~0)
P=ρ FRW
(r→∞)
Naive Picture
extreme RN
P=ρ FRW
(r~0)
(r→∞)
Is this rough estimate indeed true?
Naive Picture
extreme RN
P=ρ FRW
(r~0)
(r→∞)
Is this rough estimate indeed true?
NO.
Our goal
“ダイナミカルブラックホール”の時空構造を知りたい
• 時空特異点
Singularity is naked?
• 捕捉領域
Black FLRW
hole tends
to attract
is of
expanding
an expanding
universe,
rather than a black hole.Universe
As a gooditself
lesson
above, we are requ
take special care to conclude what the present spacetime describes.
5
事象の地平線
In• this
paper, we study the above spacetime (2.1) more thoroughly [we are working mainly in E
rather than Eq. (2.4), because the former coordinates cover wider range than the latter]. We
W universe, ratherEvent
than a black hole.
As aSingularity
good lessonisofcovered?
above, we are required to
exists?
t0 > 0, viz, thehorizon
background
universe is expanding.
For simplicity and definiteness of our argum
conclude what
the
present
spacetime
describes.
will specialize to the case in which all charges are equal, i.e., QT = QS = QS � = QS �� ≡ Q (> 0).
study the above
spacetime
(2.1)
more thoroughly
[we are working mainly in Eq. (2.1)
specific,
we will be
concerned
with the metric
4), because the former coordinates cover wider range than the
We
� latter].
� assume
2
2
−1
2
2
2
kground universe is expanding. For simplicity
definiteness
our
argument,
we
ds4 =and
−Ξdt
+Ξ
drof +
r dΩ
2 ,
3
simplicity,
we assume
e case in whichFor
all charges
are equal,
i.e., QT = QS = QS � = QS �� ≡ Q (> 0). To be
whosethe
component
:膨張宇宙
concerned with
metric Ξ, Eq. (2.2), is simplified to
� 2
�
�
�
3 −1/2
2
2
−1
2
2
,
ds4 = −Ξdt + Ξ
dr + r dΩ2 , Ξ = HT HS
(2.11)
characterized by (Q, t0)
, Eq. (2.2),with
is simplified to
�
�
3 −1/2
Ξ = HT HS
,
HT =
t
Q
+ ,
t0
r
HS = 1 +
Q
.
r
(2.12)
A more general background with distinct charges are yet to be investigated. The result for the 5D
Contents
Introduction
Black holes in general relativity:
9 slides
--studies of stationary black holes--
Black holes in dynamical background
6 slides
Dynamical black holes
Solution from intersecting branes
4 slides
Spacetime structure
17 slides
Concluding remarks
Summary and outlooks
4 slides
κΦ =
ln
T
,
(3.8)
=4
ln
,
(3.8)
U(1) gauge
gravity theory.
Thefields.
torusκΦ
compactification
4 HSHS gives a set of scalar fields and the 4-form
shown
in Appendix
A, we can
thefour
following
effective
action
from
11D supergravity
U(1) gauge As
field
in 4D.
In our solution,
we derive
assume
branes,
which 4D
give
rise to
four
uge-fields
compactification,
gauge-fields
�
�
pendix A, we can derive
the following effective
4D
action
√
1 from1 11D 2supergravity via
√
4
Q
√
(T
)
Q
S = (Q
d i≡Qのとき)
x −g
R − (∇Φ) 6
(T )
Einstein-Maxwell(x2)-dilaton系
κF01
,
2
κF01= −
= −2π r2π
,
2κ
2
2 H22 2
r
H
�
�T T
�
1 �� ) �� 1√ √ 2Q Q 1 � λA κΦ (A) 2
�
4(S √
�
(S)
)
(S
(S)
(S
)
(S
)
− ,
e
(Fµν ) ,
(3
= = κF
d κF
x −g
R=−
(∇Φ)
A=T,
S,S’,S’’
κFS01
=
κF
−
2π
(3.9)
2
κF
=
=
κF
=
−
2π
,
(3.9)
01
01
16π
2
2κ 01
2
01 1
01
2H 2
r2 H
r
(Φ)
2
�S S A
Tµν = ∇µ Φ∇ν Φ − gµν (∇Φ) ,
(3.6)
�
2
(A)
� �(A)
��
� , and λA1(A = T,λAS,κΦ
2 a scalar field, four U(1) fields, and coupling constan
where
Φ,
F
S (F
, Sµν
) are
�
µν
onstants
−
e
)
,
(3.2)
g constants
1
1
(A)
• 時間依存ブレーンのみが異なる結合定数
(A) (A)ρ
(em)
eλA κΦ Fµρ
F ν16π− Agµν (Fαβ )2 .
(3.7)
Tµν
= respectively.
√ √
√√
4π
4
A above action yields the following set of basic equations,
The
�� S=
�� =
λT λ=T =6 , 6 , λS λ≡S λ≡S �λS≡� λ
− −6/3.
(3.10)
≡Sλ
6/3.
(3.10)
�
��
�
�
dallλthe
= charges,
T, S, S ,two
S )different
are a scalar
field,
four U(1)
A (A
(Φ) fields,
(em)and coupling constants,
same
coupling
constants
(3
Gµν = κ2 appear.
T(A)
+
T
,(A)
µν(A) µν
(A)
(A)
(A)
hows
that
the
above
basic
equations
(3.3),
(3.4),
and
(3.5)
are
satisfied
by
our
−
∇
A
as,
are
expressed
in
terms
of
the
electrostatic
potentials
F
=
∇
A
expressed in terms of the electrostatic potentials F�
ν µ µ as,
µν ν − ∇ν A
µν µν= ∇µ A
κ
λ
κΦ
(A)
2
•
Maxwell場とdilatonがソース
n
yields
the
following
set
of
basic
equations,
provided
dilaton profile
✷Φ −
λA e A (Fµν ) = 0 ,
(3
√
√
16π
�) �2π 2π �
√(T ) (T�
A
2
(Φ)
(em)
�
�
κA
=
,
6
H
κA
=
,
(3.3)
G
=
κ
,
T + Tµν
0 Tµν
µν
0
ν
λ
κΦ
(A)
A
H
κΦ =
ln
,:massless
(3.8)
scalar
H
∇
e
F
=
0
,
(3
T
“P=ρ
universe”
T
µν
4 κ �
HS
��
��
Matter fields
√ √ λ κΦ (F (A) )2 = 0 ,
(S) (S) = λA e2πA1 1
✷ΦκA
−κA
(3.4)
µν
− 1, ,
(3.11)
2π
−
1
(3.11)
0=
016π
lds
H
HS S
A
�
�
ν QλA κΦ (A)
3 )If Q √
(T
different
QS , the
result still holds. iIt is because
such a difference amou
(S)
e(S)from
Fother
=→0same
,∞ charges
(3.5)
T is ∇
h present
(S)
(S)
µνasthree
κF
=
−
2π
,
→
0
r
using
a
gauge
freedom.
Therefore
the
present
uned
A
to
assure
A
01
2 0 as r → ∞2 using a gauge
−1Therefore
0→
freedom.
d A0 0to assure
A0r2 H
2
2
2 the present
to the trivial
conformal
change ds4 = (QT /QS )1/2 −Ξ
∗ dt∗ + Ξ∗ (dr + dΩ2 ) with simple parameter redefiniti
T
is the
exact
solution
of the
Einstein-Maxwell-dilaton
system
(3.2).One
Onemay
may
verifythat
that
the
exact
solution
the
Einstein-Maxwell-dilaton
system
verify
3 ]−1/2
−1/2 t, (3.2).
1/2
√
�of
��
Ξ
=
[(t
/t
+
Q
/r)(1
+
Q
/r)
,
t
=
(Q
/Q
)
and
t
=
(Q
/Q
)
t
.
Q
∗
∗
∗
∗0
∗0
0
S
S
T
S
T
S
(S)
(S )
(S )
large
charge
satisfying
•=QはMaxwell電荷
κF01
κF01 = κF01 = − 2π 2 2 ,
(3.9)
satisfying
r HS
�
�
Q
1
m other three same charges
result
still holds.
S
λA κΦ
(A)
µν
Q √ Q=
1, the
h present
iIt is because such a difference amounts
λ
κΦ
(A)
µν
e
F
dS
,
(3.12)
ts
A
µν
−1
1/2 −Ξ
2 +
2+
2 ) with simple parameter redefinitions
=
e
F
dS
,
(3.12)
rmal change ds24 = √
(QT /Q
)
dt
Ξ
(dr
dΩ
µν
4π
∗
∗
GS4π
2
S ∗
• 優勢エネルギー条件を満足
G
√
√
S
/r)(1 +λQ
/r)36
]−1/2
(QλTS/Q
)−1/2
t, −
and6/3.
t∗0 = (QT /QS )1/2 t0 .
� ≡
, , tλ
(3.10)
∗ S=≡
Sλ
T S=
S �� =
nd
sphere
surrounding
the
source.
This
expression
is
obtainable
by
thefirst
firstintegral
integralofof
sphere surrounding the source. This expression is obtainable by the
take special care to conclude what the present spacetime describes.
In this paper, we study the above spacetime (2.1) more thoroughly [
rather than Eq. (2.4), because the former coordinates cover wider r
t0 > 0, viz, the background universe is expanding. For simplicity an
will specialize to the case in which all charges are equal, i.e., QT = Q
時空特異点
specific, we will be concerned with the metric
� 2
�
2
2
−1
2
2
すべての曲率不変量 (RabcdRabcd, CabcdCabcd,Ψ2 etc)が発散
ds4 = −Ξdt + Ξ
dr + r dΩ2 ,
Singularities
whose component Ξ, Eq. (2.2), is simplified to
�
�
3 −1/2
Ξ = HT HS
,
with
これらの特異点は timelike & central (面積半径0)
HT =
t
Q
+ ,
t0
r
HS = 1 +
Q
.
r
c.f. Christdoulou
there exist an infinite
number
A more
generalofbackground with distinct charges
are yet to 1984
be investiga
(C1) will be
given in Appendix
C.
ingoing null geodesics
terminating
into singularities
outgoing null geodesics emanating from singularities
III.
MATTER FIELDS AND THEIR PRO
It seems to be a good starting point to draw our attention to the ma
the 4D metric components, we can read off the total energy-momentum
4D Einstein equations,
κ2 Tµν = Gµν ,
where κ2 = 8πG is the gravitational constant and Gµν = Rµν − (R/2
the metric (2.4) vanishes, is not singular at all since the curvature invarian
Singularities
follows that the big bang singularity t̃ = 0 is smoothed out due5 to a nonvanis
Hence, one has also to consider the t̃ < 0 region in the coordinates (2.11).
= 0 surface
is neither
thetospacetime acro
verse, rather
than ar̃ black
hole. As
a goodsingular,
lesson ofthereby
above,we
wecan
areextend
required
すべての曲率不変量は
ude what the present
spacetime
describes.
Since
the allowed
region is where HT HS3 > 0 is satisfied, we shall focus atten
t=0で有限(2.1) more thoroughly
t=0 はBig-bang
特異点ではない(t<0
に接続可能)
the above spacetime
[we are working
mainly in Eq. (2.1)
≥ t̃s (r̃),We assume
r̃ ≥ −1 ,
cause the former coordinates cover wider range than the t̃latter].
nd universer=0で有限
is expanding. For simplicity
and definiteness
our argument, we
地平線の候補
(staticofbraneではr=0が地平線)
in which all chargesinare
i.e., Qanalysis.
QS � = Q
0).3 t̃To
theequal,
subsequent
Another
permitted
region
> be
t̃s and r̃ < −1
T = QS =
S �� ≡ Q (>
(degenerate
null surface)
ned with the
metrichere,
since it turns out to be causally disconnected to the outside region, as
� 2
� are depicted in Figure 1.
allowed
2
2
−1 coordinate
2 ranges
2
ds = −Ξdt + Ξ
dr + r dΩ ,
(2.11)
4
2
(2.2), is simplified
to
•許される座標範囲は
t
�
�
3 −1/2
Ξ = HT HS
,
HT =
t
Q
+ ,
t0
r
HS = 1 +
(2.12)
Q
.
r
ts(r)
allowed
-Q
(2.13)
s(r)
forbidden
nd with distinct charges are yet to be investigated. The
result for the 5D tsolution
endix C.
forbidden
I.
MATTER FIELDS AND THEIR PROPERTIES
r
How to find event horizon
N.B 事象の地平線は大域的概念
地平線上の各点は局所的に
Black hole
なんら他の点と区別はない
Event horizon
Strategy
(i) 捕捉領域を調べる(局所的構造)
(ii) 地平線の候補を探す
(iii) “地平線近傍”の幾何を解析
(iv) 測地線を数値的に解いて本当にEHか否か確認
we can sketch the Penrose diagram
Trapped surface
捕捉領域/みかけの地平線 (à la Penrose)
Penrose 1967
• a two-dimensional compact surface on which
ougoing null rays have negative expansion θ+ < 0
in asymptotically flat spacetimes
• causally disconnected from I + if asymptotically flat
Hawking 1971
→ trapped regions must be contained within BH region
trapped region
Trapped surface
捕捉領域/みかけの地平線 (à la Penrose)
Penrose 1967
• a two-dimensional compact surface on which
ougoing null rays have negative expansion θ+ < 0
in asymptotically flat spacetimes
• causally disconnected from I + if asymptotically flat
Hawking 1971
→ trapped regions must be contained within BH region
trapped region
Trapped region characterizes strong gravity
Trapped regions in spherical symmetry
捕捉領域/捕捉地平線 (à la Hayward)
Hayward 1993
• a 2D compact surface on which θ+θ- > 0
Schwarzchild BH (θ+ <0, θ-<0), WHの内部 (θ+ >0, θ->0) はtrapped
• a 3D surface foliated by marginal surface θ+θ- =0 is called a trapping horizon
球対称時空における捕捉領域
線素は次のように書ける:
R(y) : 面積半径(Area =4πR2)
:null normalに沿った面積変化率
trapped: θ+θ- >0 (∇R)2<0
R=const. is spacelike
Future trapping horizons
(i) black hole type (future θ+ =0)
trapped
ex. 光的ダストの重力崩壊
TH θ+ =0
0
v=
M(v)=
m(v) (0<v<v0): Vaidya
v 0 ya
v= Vaid
0 (v<0): flat space
untrapped
M≡m(v0) (v0<v): Schwarzschild
Trapping horizon occurs at r=2M(v)
Trapping horizon is spacelike
Past trapping horizons
(ii) cosmological/white hole type (past θ- =0)
ex. P=ρのFriedmann宇宙
H=a’/a: Hubble パラメータ
untrapped
R=ar: 面積半径
trapped
TH θ- =0
Trapping horizon occurs at R=1/H (Hubble horizon)
Trapping horizon is spacelike
Trapping horizons
Future and past trapping horizons occur at
t
r<0
forbidden
t=-t0Q/r
- 1.0
r=-Q
-1.2
- 0.8
- 0.6
- 0.4
t
4
r>0
20
2
r
Trapped
(θ+θ->0)
- 0.2
0
Trapped (θ+θ->0)
0.5
1.0
1.5
2.0
t=-t0Q/r
-20
-2
forbidden
- 40
-4
2.5
3.0
r
Trapping horizons
Future and past trapping horizons occur at
t
r<0
forbidden
t=-t0Q/r
- 1.0
r=-Q
-1.2
- 0.8
- 0.6
- 0.4
t
4
r>0
20
2
r
Trapped
(θ+θ->0)
- 0.2
0
Trapped (θ+θ->0)
0.5
1.0
1.5
2.0
2.5
3.0
r
t=-t0Q/r
-20
-2
forbidden
- 40
-4
attractive due to `BH’
repulsive due to `expanding univ.’
Trapping horizons
Future and past trapping horizons occur at
t
r<0
forbidden
t=-t0Q/r
- 1.0
r=-Q
-1.2
- 0.8
- 0.6
- 0.4
t
4
r>0
20
2
r
Trapped
(θ+θ->0)
- 0.2
0
Trapped (θ+θ->0)
0.5
1.0
1.5
2.0
2.5
3.0
r
t=-t0Q/r
-20
-2
forbidden
- 40
-4
attractive due to `BH’
repulsive due to `expanding univ.’
捕捉地平線のr→0 極限 が事象の地平線のlikely-candidate
Trapped surface
捕捉地平線の性質はr=0の面を境に変わる
捕捉地平線のr→0 極限 が事象の地平線のlikely-candidate
• 無限大の赤方(青方)変位面に対応
• 面積半径は一定値に漸近
R
2.0
R+
1.5
Trapped (θ+θ->0)
1.0
R-
0.5
-1
0
1
2
3
r
Near horizon geometry
捕捉地平線の“地平線近傍”
(±)
tTH
→ c± /r , r → 0 ,
スケーリング極限によりwell-defined なnear-horizon limit
L ξ gNH
µν = Dµ ξν + Dν ξµ = 0 , :Killingベクトル
ξ[µ Dν ξρ] = 0 ,
R±をKilling地平線に持つ静的ブラックホール
:超曲面直交
Near horizon geometry
“horizon-candidate”は静的ブラックホールのKilling地平線で記述される
• R+>R-
地平線は非縮退
�
�−1/2
tr
Ξ̄ = (r/Q)−2 1 +
, f (R) = (R4 − R4+ )(R4 − R4− )
t0 Q
t→+∞
‣ 温度はノンゼロ
‣ 流入するエネルギーなし
throat
I
I’
II
• Near-horizon計量の大域構造はRN-AdSと同じ
t→±∞でR± R+はBHとWHの`外側’の地平線 III
III’
R-はBHとWHの`内側’の地平線 • t:有限,r→0とすればスロート(AdS2xS2)
N.B もとの時空でξµ がKillingとなるのは地平線上のみ
t→-∞
Global spacetime structure
Outside the horizon r>0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
t=0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
R→∞ as r→∞ on t ≥0
t=0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
R→∞ as r→∞ on t ≥0
t=0
i0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
R→∞ as r→∞ on t ≥0
• For t/t0>0, metric asymptotes to P=ρ FRW as r→∞
t=0
i0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
I+
R→∞ as r→∞ on t ≥0
• For t/t0>0, metric asymptotes to P=ρ FRW as r→∞
t=0
i0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
I+
R→∞ as r→∞ on t ≥0
• For t/t0>0, metric asymptotes to P=ρ FRW as r→∞
• Singularity ts(r) = -t0Q/r (<0) is timelike
t=0
i0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
I+
R→∞ as r→∞ on t ≥0
• For t/t0>0, metric asymptotes to P=ρ FRW as r→∞
• Singularity ts(r) = -t0Q/r (<0) is timelike
t=0
i0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
I+
R→∞ as r→∞ on t ≥0
• For t/t0>0, metric asymptotes to P=ρ FRW as r→∞
• Singularity ts(r) = -t0Q/r (<0) is timelike
• r→0 with t: fixed is an infinite throat
t=0
i0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
I+
R→∞ as r→∞ on t ≥0
• For t/t0>0, metric asymptotes to P=ρ FRW as r→∞
• Singularity ts(r) = -t0Q/r (<0) is timelike
• r→0 with t: fixed is an infinite throat
t=0
i0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
I+
R→∞ as r→∞ on t ≥0
• For t/t0>0, metric asymptotes to P=ρ FRW as r→∞
• Singularity ts(r) = -t0Q/r (<0) is timelike
• r→0 with t: fixed is an infinite throat
• Killing horizons R± develop from the throat
t=0
i0
Global spacetime structure
Outside the horizon r>0
• t=0 is a regular slice
I+
R→∞ as r→∞ on t ≥0
• For t/t0>0, metric asymptotes to P=ρ FRW as r→∞
• r→0 with t: fixed is an infinite throat
• Killing horizons R± develop from the throat
t=0
∞
t=
• Singularity ts(r) = -t0Q/r (<0) is timelike
t=
∞
R+
R-
i0
Global spacetime structure
Inside the horizon r<0
Global spacetime structure
Inside the horizon r<0
• there exist Killing horizons R± as
Global spacetime structure
Inside the horizon r<0
• there exist Killing horizons R± as
R-
t=
∞
∞
t=
R+
vanishes, is not singular at all since the curvature invariants remain finite at t̃ = 0. It
big bang singularity t̃ = 0 is smoothed out due to a nonvanishing Maxwell charge Q (> 0).
also to consider the t̃ < 0 region in the coordinates (2.11). In addition, we find that the
neither singular, thereby we can extend the spacetime across the r̃ = 0 surface to r̃ < 0.
3
InsideHthe
r<0 we shall focus attention to the coordinate domain
d region is where
> 0 is satisfied,
T HShorizon
Global spacetime structure
t̃ ≥ t̃s (r̃),
horizonsr̃ R≥± −1
as ,
• there exist Killing
(4.6)
∞
t=
nt analysis. Another permitted region t̃ > t̃s and r̃ < −1 is not our immediate interest
RFor
t
>t
,
singularity
t=-t
Q/r
is
visible
0 outside region, as we shall show below. Possible
ns out to be
• causally0 disconnected to the
ate ranges are depicted in Figure 1.
• For t <t0, singularity r=-Q is visible
t
ts(r)
t=
∞
t
t0
r
ts(r)
allowed
-Q
R+
r
oordinate ranges. The grey zone denotes the forbidden region, and the dashed curves correspond
ularities.
cetime is spherically symmetric, electromagnetic and gravitational fields do not radiate.
vanishes, is not singular at all since the curvature invariants remain finite at t̃ = 0. It
big bang singularity t̃ = 0 is smoothed out due to a nonvanishing Maxwell charge Q (> 0).
also to consider the t̃ < 0 region in the coordinates (2.11). In addition, we find that the
neither singular, thereby we can extend the spacetime across the r̃ = 0 surface to r̃ < 0.
3
InsideHthe
r<0 we shall focus attention to the coordinate domain
d region is where
> 0 is satisfied,
T HShorizon
Global spacetime structure
t̃ ≥ t̃s (r̃),
horizonsr̃ R≥± −1
as ,
• there exist Killing
(4.6)
∞
t=
nt analysis. Another permitted region t̃ > t̃s and r̃ < −1 is not our immediate interest
RFor
t
>t
,
singularity
t=-t
Q/r
is
visible
0 outside region, as we shall show below. Possible
ns out to be
• causally0 disconnected to the
ate ranges are depicted in Figure 1.
ts(r)
t
t
t=t0
t0
t=
∞
• For t <t0, singularity r=-Q is visible
r
ts(r)
allowed
-Q
R+
r
oordinate ranges. The grey zone denotes the forbidden region, and the dashed curves correspond
ularities.
cetime is spherically symmetric, electromagnetic and gravitational fields do not radiate.
vanishes, is not singular at all since the curvature invariants remain finite at t̃ = 0. It
big bang singularity t̃ = 0 is smoothed out due to a nonvanishing Maxwell charge Q (> 0).
also to consider the t̃ < 0 region in the coordinates (2.11). In addition, we find that the
neither singular, thereby we can extend the spacetime across the r̃ = 0 surface to r̃ < 0.
3
InsideHthe
r<0 we shall focus attention to the coordinate domain
d region is where
> 0 is satisfied,
T HShorizon
Global spacetime structure
t̃ ≥ t̃s (r̃),
horizonsr̃ R≥± −1
as ,
• there exist Killing
t=-t0Q/r
(4.6)
∞
t=
nt analysis. Another permitted region t̃ > t̃s and r̃ < −1 is not our immediate interest
RFor
t
>t
,
singularity
t=-t
Q/r
is
visible
0 outside region, as we shall show below. Possible
ns out to be
• causally0 disconnected to the
ate ranges are depicted in Figure 1.
• For t <t0, singularity r=-Q is visible
t
r=-Q
ts(r)
t0
r
ts(r)
allowed
-Q
t=
∞
t=t0
t
R+
r
oordinate ranges. The grey zone denotes the forbidden region, and the dashed curves correspond
ularities.
cetime is spherically symmetric, electromagnetic and gravitational fields do not radiate.
vanishes, is not singular at all since the curvature invariants remain finite at t̃ = 0. It
big bang singularity t̃ = 0 is smoothed out due to a nonvanishing Maxwell charge Q (> 0).
also to consider the t̃ < 0 region in the coordinates (2.11). In addition, we find that the
neither singular, thereby we can extend the spacetime across the r̃ = 0 surface to r̃ < 0.
3
InsideHthe
r<0 we shall focus attention to the coordinate domain
d region is where
> 0 is satisfied,
T HShorizon
Global spacetime structure
t̃ ≥ t̃s (r̃),
horizonsr̃ R≥± −1
as ,
• there exist Killing
t=-t0Q/r
(4.6)
∞
t=
nt analysis. Another permitted region t̃ > t̃s and r̃ < −1 is not our immediate interest
RFor
t
>t
,
singularity
t=-t
Q/r
is
visible
0 outside region, as we shall show below. Possible
ns out to be
• causally0 disconnected to the
ate ranges are depicted in Figure 1.
• For t <t0, singularity r=-Q is visible
t
r=-Q
ts(r)
t0
t=
∞
t=t0
t
R+
r
ts(r)
allowed
-Q
r
•
oordinate ranges. The grey zone denotes the forbidden region, and the dashed curves correspond
t→-∞ with r(<0): fixed is a past infinity
ularities.
→∞
cetime is spherically symmetric, electromagnetic and gravitational fields do not radiate.
Global spacetime structure
trapped: θ+θ- >0 (∇R)2<0
R=const. is spacelike
R
2.0
1.5
R+
R=const. is spacelike
1.0
0.5
-1
R0
R=const. is timelike
1
2
3
r
Consistency has been checked by solving geodesics numerically
The solution indeed turns out to describe a BH in FRW cosmology
Extensions
Extensions
Extensions
• Patched region corresponds to t0<0
Extensions
• Patched region corresponds to t0<0
but...
extension is not unique due to nonanaliticity
Contours
t=const.
r=const.
Φ=const.
(∇Φ)2<0 at infinity
ds2 = −Ξ̄D−3 dt̄2 + a2 Ξ̄−1 δIJ dxI dxJ ,
the equation of state
Extension to arbitrary power-law FRW 2(D − 3)n
where
��
1+
a=
�
Ξ̄ =
(2.25)
H̄T
a2(D−2)/nT
�nT
�
1 + H̄S
� nS
�−1/(D−2)
P = wρ , with w =
,
(2.26)
and
t̄
t̄0
�p
, with p =
nT
.
(D − 3)nS
(2.27)
Since we imposed the boundary condition
��such that the
I 2
harmonics H̄T and H̄S fall off as r :=
I (x ) → ∞,
S
(D − 1)nT
− 1.
(2
It turns out that the parameter nT (or nS ) is associ
to the expansion law of the universe (2.27). Notably
can obtain an accelerating universe (p ≥ 1) by set
nT ≥ 2 or equivalently nS ≤ 2/(D − 3). In partic
the exponential expansion (the de Sitter universe) is
derstood to be p → ∞ (nS → 0). Figure 1 depicts
conformal diagrams of the FLRW universe. The asy
totic regions of the present spacetime (2.12) resemble
corresponding shaded regions in Figure 1.
Background:
p<1/2
p=1/2
1/2<p<1
p=1
1<p
FIG. 1: Conformal diagrams of a flat FLRW universe a = (t̄/t̄0 )p for (1) 0 < p < 1/2, (2) p = 1/2, (3) 1/2 < p
(4) p = 1 and (5) p > 1. The dotted and dotted-dashed lines denote the trapping horizon, rTH (t̄) = (da/dt̄)−1 , and
big-bang singularity at a = 0, respectively. The cases (2) and (4) correspond respectively to the radiation-dominant uni
P = ρ/(D − 1) and the marginally accelerating universe driven by the curvature term ρ ∝ a−2 . The cosmological hor
accelerating
rCH (t̄) = (p − 1)−1 t̄0 (t̄/t̄0 )1−p , is abbreviated
to CH, which exists only in the strictly accelerating case (p > 1). The sh
decelerating
regions corresponding to r → ∞ approximate our original spacetime.
Black hole in power-law FRW universe
Einstein-Maxwell-dilaton theory with a Liouville potential
(t0,Q,p=nT/nS)の3パラメータ族
• 遠方で power-law FRW universe に漸近
• 弱エネルギー条件 を満足
• 事象の地平線は Killing地平線 と一致
Gibbons-Maeda 2009,
Maeda-M.N 2010
nT=1: Maeda-Ohta-Uzawa solution
nT=4: M=Q RN-de Sitter solution
Global structures
(I) decelerating universe: p<1
TH
admits two horizons
(II) Milne universe: p=1
TH
no event horizon
Global structures
(III) accelerating universe: p>1
admits three horizons
admits two horizons
(degenerate)
no event horizon
Contents
Introduction
Black holes in general relativity:
9 slides
--studies of stationary black holes--
Black holes in dynamical background
6 slides
Dynamical black holes
Solution from intersecting branes
4 slides
Spacetime structure
23 slides
Concluding remarks
Summary and outlooks
4 slides
Summary
We explore the global structure of a “dynamical black hole candidate’’
derived from 11D intersecting branes & its generalizations
• asymptotes to FRW universe
• satisfies suitable energy conditions
• additional symmetry appears at the event horizon (=Killing horizon)
• ambient matters do not fall into the hole
Summary
We explore the global structure of a “dynamical black hole candidate’’
derived from 11D intersecting branes & its generalizations
• asymptotes to FRW universe
• satisfies suitable energy conditions
• additional symmetry appears at the event horizon (=Killing horizon)
• ambient matters do not fall into the hole
The solution describes an equilibrium BH in dynamical background
Further generalizations
Higher-dimensional and/or rotating generalizations
• describes a BMPV black hole in FRW
Breckenrige et al 1996
• possesses CTCs around singularities (gψψ<0)
Black hole thermodynamics
• Can we define meaningful mass function in FRW universe?
??
Multiple generalizations
c.f. Kastor-Traschen 1993
• Multi-center metric is expected to describe BH collisions in FRW universe
Why superposition is possible?
Analogue of supersymmetric solutions
The solution inherits properties of supersymmetric black holes
• BPS solutions satisfy the `no force’ condition
gravitational attractive force
c.f. Majumdar-Papapetrou sol.
electromagnetic repulsive force
• However, supergravity admits only AdS vacua
e.g. Minimal gauged SUGRA coupled to U(1)N vector fields with scalars
U=CIJKVIVJXK >0
I, J,...=1,...,N; A,B,..=1,...,N-1
CIJK :intersection numbers of CY
(N: Hodge number h1,1 of CY)
g :(inverse) AdS radius
SUSY transformation
Embedding into supergravity
Wick rotation (g → iλ) gives an inverted potential
V=2λ2CIJKVIVJXK >0
“Fake supergravity”
Our 5D metric is a solution of fake supergravity with C123=1
hij : hyper-Kähler space
V1=V2=(6λt0)1, V3=0
e.g.
‣ “Killing spinor” equation is satisfied for
1/2-“BPS” state
‣ 4D solution is obtainable via Gibbons-Hawking space
We expect all BPS solutions can be obtained using Killing spinors
M.N. in work
Black holes in FRW universe
Black hole in “Swiss-Cheese Universe”
•glue Schwarzschild BH w/ FRW universe
Einstein-Straus 1945
S
•Israel’s junction condition at Σ:
Schwarzschild
r=0
Ȉ
a=0
dust FRW
-Schwarzschild portion is static
-matters do not accrete onto the hole
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