高次元・高次曲率項を含む重力理論での特異点形成

高次元・高次曲率項を含む重力理論での特異点形成
真貝寿明（大阪工大情報科学部）
鳥居隆（大阪工大工学部）
論点
＊4dim, 5dim, 6dim, … ダイナミクスはどう変化するか
＊Gauss-Bonnet項は，ダイナミクスにどう影響するか
1.
2.
2.
3.
GR 4dim vs 5dim
Field Equations (dual-null formulation)
平面対称時空：Colliding Scalar Waves
球対称時空：Wormhole-BH transition
http://www.oit.ac.jp/is/~shinkai/
2016年7月30日摂南大学数理セミナー
Introduction
一般相対論研究の面白さ
非線形性・複雑さ
ブラックホール，膨張宇宙，重力波 Einsteinも信じなかった事実
結果の美しさ
特異点はアインシュタイン方程式の解として必然として存在す
特異点定理
宇宙検閲官仮説特異点はBHホライズンの内側に隠されていて欲しい
ホライズン形成は，物質分布がある程度コンパクトな場合
フープ仮説球対称，静的，真空時空はSchwarzchild
バーコフの定理定常ブラックホール時空はKerr
BH唯一性定理 BH形成により，M, Q, J の3つだけが物理情報として残る
脱毛定理 Introduction
一般相対論研究＋高次元研究の面白さ
動機
膜宇宙論による新しいパラダイムの提示
LHCによる高次元空間の検証可能性
想定外のBH解の発見
"Black Objects"
4-dim BHs
Higher-dim BHs :
Schwarzschild Tangherlini
--- unique & stable
Kerr
Myers-Perry
--- maybe unstable in higher J
black ring (Emparan-Reall)
black Saturn
di-rings, orthogonal di-rings, ...
Introduction
Higher-dim Black Holes have Rich Structures
"Black Objects"
black hole
black string
black ring
black Saturn
di-rings, orthogonal di-rings ...
Uniqueness (only in spherical sym.)
Stability?
No Hair Conjecture?
Formation Process?
Cosmic Censorship?
Dynamical Features? ...
Hoop Conjecture?
Introduction
Dynamics in Gauss-Bonnet gravity?
• has GR correction terms from String Theory • has two solution branches (GR/non-‐‑‒GR). • has minimum mass for static spherical BH solution T Torii & H Maeda, PRD 71 (2005) 124002 • is expected to have singularity avoidance feature. (but has never been demonstrated in full gravity.) •new topic in numerical relativity. S Golod & T Piran, PRD 85 (2012) 104015 N Deppe+, PRD 86 (2012) 104011 F Izaurieta & E Rodriguez, 1207.1496 •much attentions in WH community H Maeda & M Nozawa, PRD 78 (2008) 024005 P Kanti, B Kleihaus & J Kunz, PRL 107 (2011) 271101 P Kanti, B Kleihaus & J Kunz, PRD 85 (2012) 044007 Plan of the Talk
Dynamics in 5dim GR gravity?
2. Spheroidal matter collapse
Initial data analysis, Evolutions
3. Wormhole dynamics in GR
linear stability,
dynamical stability
Yamada & HS, CQG 27 (2010) 045012
Yamada & HS, PRD 83 (2011) 064006
Torii & HS, PRD 88 (2013) 064027
HS & Torii, in preparation
Dynamics in Gauss-Bonnet gravity?
4. Wormhole dynamics in GB
5. Plane-wave collision in GB
HS & Torii, in preparation
2. Spheroidal matter collapse
A. Initial data construction
- time symmetric, asymptotically flat
- conformal flat
- non-rotating homogeneous dust
- solve the Hamiltonian constraint eq. 512^2 grids
- Apparent Horizon Search
- Define Hoop and check the Hoop Conjecture 2. Spheroidal matter collapse
B. Initial data sequence
cf. (3-dim.) Nakamura-Shapiro-Teukolsky (1988)
4+1
initial data
2. Spheroidal matter collapse
C. Evolution method
- ADM 2+1 Double Axisym Cartoon
- 130^2 x 2^2 grids - lapse function: Maximal slicing condition
- shift vectors: Minimum distortion condition
- asymptotically flat
- Collisionless Particles (5000) - the same total mass
- no rotation - Apparent Horizon Search
2. Spheroidal matter collapse
C. Evolution examples (4D, ST1991)
Apparent Horizon
appears
No AH
= naked singularity
2. Spheroidal matter collapse
C. Evolution examples (5D, ours)
2. Spheroidal matter collapse
C. Evolution examples (5D, ours)
2. Spheroidal matter collapse
D. Comparisons 4D vs. 5D
towards spindle
towards spherical
towards spherical towards spindle
2. Spheroidal matter collapse
D. Comparisons 4D vs. 5D
5D collapses
-- proceed rapidly.
-- towards spherically.
-- AH forms in wider ranges.
I = Rabcd Rabcd
at I(tend )
2. Spheroidal matter collapse
C. Evolution examples
2’. Hoop Conjecture
A. Hyper-Hoop conjecture ?
Hoop Conjecture
Thorne (1972)
Hyper-Hoop Conjecture
Ida-Nakao (2002)
In 5-D, if mass gets compacted
in some area, ....
Penrose (1969)
2’. Hoop Conjecture
B. Spheroidal Cases
Define Hyper-Hoop as the surface
2’. Hoop Conjecture
C. Toroidal Cases
Hyper-Hoop
does not work for
ring horizons.
Section Summary
Dynamics in 5dim GR gravity?
2. Spheroidal matter collapse
Initial data analysis, Evolutions
Yamada & HS, CQG 27 (2010) 045012
Yamada & HS, PRD 83 (2011) 064006
（回転なしのスピンドル形状の重力崩壊）
＊5D は，4Dよりもはやく重力崩壊をおこす（局所的に強い重力）
＊5Dでの崩壊は，4Dよりも球状になりやすい（重力が多くの自由度持つ）
＊Apparent Horizonは5Dの方が形成しやすい．（球状進化の結果）
＊極端なスピンドルでは裸の特異点が出現する＊Hyper-Hoop conjectureはスピンドル形状に対して成立．
リング形状に対して不成立．
Plan of the Talk
Dynamics in 5dim GR gravity?
2. Spheroidal matter collapse
Initial data analysis, Evolutions
3. Wormhole dynamics in GR
linear stability,
dynamical stability
Yamada & HS, CQG 27 (2010) 045012
Yamada & HS, PRD 83 (2011) 064006
Torii & HS, PRD 88 (2013) 064027
HS & Torii, in preparation
Dynamics in Gauss-Bonnet gravity?
4. Wormhole dynamics in GB
5. Plane-wave collision in GB
HS & Torii, in preparation
Introduction
Why Wormhole?
They make great science ﬁction -‐‑‒-‐‑‒ short cuts between otherwise distant regions. Morris & Thorne 1988, Sagan “Contact” etc US movie 1997
Morris-Thorne’s “Traversable” wormhole
M.S. Morris and K.S. Thorne, Am. J. Phys. 56 (1988) 395
M.S. Morris, K.S. Thorne, and U. Yurtsever, PRL 61 (1988) 3182
H.G. Ellis, J. Math. Phys. 14 (1973) 104
(G. Clément, Am. J. Phys. 57 (1989) 967)
Desired properties of traversable WHs
1. Spherically symmetric and Static ⇒ M. Visser, PRD 39(89) 3182 & NPB 328 (89) 203
2. Einstein gravity
3. Asymptotically flat
4. No horizon for travel through
5. Tidal gravitational forces should be small for traveler
6. Traveler should cross it in a finite and reasonably small proper time
7. Must have a physically reasonable stress-energy tensor
⇒ Weak Energy Condition is violated at the WH throat.
⇒ (Null EC is also violated in general cases.)
8. Should be perturbatively stable
9. Should be possible to assemble
“Ellis (Morris-Thorne) wormhole”
Introduction
Why Wormhole?
They increase our understanding of gravity when the usual energy conditions are not satisﬁed, due to quantum eﬀects (Casimir eﬀect, Hawking radiation) or alternative gravity theories, brane-‐‑‒world models etc. They are very similar to black holes -‐‑‒-‐‑‒both contain (marginally) trapped surfaces and can be deﬁned by trapping horizons (TH). Wormhole = Hypersurface foliated by marginally trapped surfaces
BH and WH are interconvertible? New duality?
Introduction
BH & WH are interconvertible?
S.A. Hayward, Int. J. Mod. Phys. D 8 (1999) 373
They are very similar -‐‑‒-‐‑‒ both contain (marginally) trapped surfaces and can be deﬁned by trapping horizons (TH) Only the causal nature of the THs diﬀers, whether THs evolve in plus / minus density which is given locally. Black Hole
Wormhole
Locally
deﬁned
by
Achronal (spatial/null)
outer TH
⇨ 1-way traversable
Einstein
eqs.
Positive energy density Negative energy density normal matter (or
exotic matter
vacuum)
Appearance
occur naturally
Temporal (timelike)
outer THs ⇨ 2-way traversable
Unlikely to occur
naturally. but constructible??
Part I Wormhole dynamics in 4-dim GR
Fate of Morris-Thorne (Ellis) wormhole?
•
•
•
•
“Dynamical wormhole” defined by local trapping horizon
spherically symmetric, both normal/ghost KG field
apply dual-null formulation in order to seek horizons
Numerical simulation
ghost/normal Klein-Gordon fields
!
Tµν = Tµν (ψ) + Tµν (φ) = ψ,µ ψ,ν − gµν
%
ψ=
dV1 (ψ)
,
dψ
"
1
(∇ψ)2 + V1 (ψ)
2&'
normal
φ=
dV2 (φ)
.
dφ
#$
(
!
+ −φ,µ φ,ν − gµν
%
"
#$
1
− (∇φ)2 + V2 (φ)
2
&'
ghost
(Hereafter, we set V1 (ψ) = 0, V2 (φ) = 0)
(
Initial data on x+ = 0, x− = 0 slices and on S
Generally, we have to set :
(Ω, f, ϑ± , φ, ψ) on S: x+ = x− = 0
(ν± , ℘± , π± ) on Σ± : x∓ = 0, x± ≥ 0
Grid Structure for Numerical Evolution
xminus
xplus
S
wormhole throat
dual-null formulation, spherically symmetric spacetime (4D)
• The spherically symmetric line-element:
ds2 = −2e−f dx+ dx− + r2 dS 2 , where r = r(x+ , x− ), f = f (x+ , x− ), · · ·
• To obtain a system accurate near ℑ± , we introduce the conformal factor Ω = 1/r . We also define
first-order variables, the conformally rescaled momenta
expansions
inaﬃnities
momenta of φ
momenta of ψ
ϑ± = 2∂± r = −2Ω−2 ∂± Ω
(θ± = 2r−1 ∂± r)
ν± = ∂± f
℘± = r∂± φ = Ω−1 ∂± φ
π± = r∂± ψ = Ω−1 ∂± ψ
(1)
(2)
(3)
(4)
The set of equations (remember the identity: ∂+ ∂− = ∂− ∂+ ):
2
∂± ϑ± = −ν± ϑ± − 2Ωπ±
+ 2Ω℘2± ,
(5)
∂± ν∓ = −Ω2 (ϑ+ ϑ− /2 + e−f − 2π+ π− + 2℘+ ℘− ),
(7)
∂± π∓ = −Ωϑ∓ π± /2.
(9)
∂± ϑ∓ = −Ω(ϑ+ ϑ− /2 + e−f ),
(6)
∂± ℘∓ = −Ωϑ∓ ℘± /2,
(8)
4d GR
↵GB = 0
HS-Hayward PRD 66(2002) 044005
Wormhole evolutionの結果
⌃(t)
⌃(t)
#+ = 0
# =0
#+ = 0
Inflationary
Expansion
Black Hole
# =0
x
# =0
#+ = 0
x
+
Positive Energy Input
= add normal scalar field
= subtract ghost field
x
#+ = 0
# =0
x+
Negative Energy Input
= add ghost scalar field
Ghost pulse input -- Bifurcation of the horizons（4d)
↵GB = 0
4d GR
HS-Hayward PRD 66(2002) 044005
Wormhole evolutionの結果
⌃(t)
⌃(t)
#+ = 0
# =0
#+ = 0
Inflationary
Expansion
Black Hole
# =0
x
# =0
#+ = 0
x+
Positive Energy Input
= add normal scalar field
= subtract ghost field
x
#+ = 0
# =0
x+
Negative Energy Input
= add ghost scalar field
E<0
E>0
Travel through a Wormhole
-- with Maintenance Operations!
Summary of Part I
HS & Hayward, PRD66 (2002) 044005
Dynamics of Ellis (Morris-Thorne) traversible WH
WH is Unstable
(A) with positive energy pulse ---> BH
---> conﬁrms duality conjecture between BH and WH.
(B) with negative energy pulse ---> Inﬂationary expansion
---> provides a mechanism for enlarging a quantum WH
to macroscopic size
(C) can be maintained by sophisticated operations
---> a round-trip is available for our hero/heroine
The basic behaviors has been conﬁrmed by
A Doroshkevich, J Hansen, I Novikov, A Shatskiy, IJMPD 18 (2009) 1665
J A Gonzalez, F S Guzman & O Sarbach, CQG 26 (2009) 015010, 015011
J A Gonzalez, F S Guzman & O Sarbach, PRD80 (2009) 024023
O Sarbach & T Zannias, PRD 81 (2010) 047502
Part 2 WH in higher-dim. (1) Exact Solution
(1) Exact Solution : Basic eqns.
Torii & HS, PRD88 (2013) 064027
‣ general relativity, n-dimension
Z
S=
dn x
p
g

massless scalar ﬁeld (ghost)
1
R
2
2⇥n
1
(⌅⇤)2
2
V (⇤) ,
✏=
1
R
‣ static, spherical sym., asymptotically ﬂat
ds2n =
f (r)dt2 + f (r)
1
dr2 + R(r)2 hij dxi dxj
(k = 1)
‣ Basic equations
(t, t) :
n
2
2
f
(KG) :
f 00
+ (n
2
h1
i
2)(n 3)kf
2
02
= ⇥n f
f ⇤ + V (⇤) ,
2R2
2
2R00
f 0 R0
(n 3)R02
(n
+
+
+
R
fR
R2
n
(r, r) :
(i, j) :
2

✓

2 R0 f 0
(n 3)R0
+
2 R f
R
2◆
R00
f 0 R0
n 4 R0
3)f
+
+
R
fR
2 R2
1
Rn
2
R
n 2
f⇥
0 0
=
dV
.
d⇥
(n
2)(n 3)k
⇥2n h 1
02
=
f
⇤
2f R2
f 2
(n
3)(n
2R2
4)k
0
=
⇥2n
C
=
f Rn
i
V (⇤) ,
h1
i
02
f ⇤ + V (⇤) ,
2
constant
2
r
Part 2 WH in higher-dim. (1) Exact Solution
Solution
r=0
‣ regularity at the throat ( )
throat radius
R=a
R0 = 0,
f = f0 ,
f 0 = 0,
★ from the scaling rule
a=1
=0
2 2
nC
Basics eqns.
= (n
3)a2(n
2)(n
f0 = 1
3)
‣ Exact solution
1
f
r(R) =
mBz
(n
⇥=
2)(n
m,
3)
n
m=
1
2(n
3)
,
z=R
m
1
2
an
[1
[m(n
3
1
R(r)n
m]
4)]
2
dr
z
Bz (p, q) :=
tp
1
(1
t)q
1
dt
Incomplete Beta func.
0
★ in another metric form: V. Dzhunushaliev+, 2013
Part 2 WH in higher-dim. (1) Exact Solution
Configurations
‣ conﬁgurations
3
2.5
3
n=4
2.5
2
2
n = 10
1.5
1.5
r
!
n=4
r
2.5
2
1.5
1
1
1
n = 10
0.5
0.5
0
0.5
0
0.5
1
1.5
R
2
2.5
3
0
-2
0
2
r
expansion is 0
trapping horizon
★ large curvature near the throat.
★ scalar ﬁeld goes steep if n is large.
3
3
★ In the n ! 1 limit
2.5
2
R=r+1
2.5
⇥ = 0 (r = 0)
2
1.5
r
r
n=4
1.5
2
(r > 0)
n=4
4
6
8
0.5
Part 2 WH in higher-dim. (2) Linear Stability
(2) Linear Stability: Master eqn.
Torii & HS, PRD88 (2013) 064027
metric‣ metric
metric
−2δ(t,r) 2
−1 2
ds2 2n = −f (t, r)e−2δ(t,r)
dt2 + f (t, r)−1
dr2 + R(t, r)22 hij dxii dxjj
dsn = −f (t, r)e
dt + f (t, r) dr + R(t, r) hij dx dx
‣ linear perturabation
—————————Einstein equations
static solution
(1)
(1)
(t, t) : (r, r) : (t, r) :
f = f0 (r) + f1 (r)eiωt , R = R0 (r) + R1 (r)eiωt ,
(2)
(2)
δ = δ0 (r) + δ1 (r)eiωt ,
(3)
φ = φ0 (r) + φ1 (r)eiωt .
!
‣ master equation
′ r)′ :
(t, t)R: +(r,
(t, nr)−: 3 R−n+3 φ′ = ω 2 R
R1′′ − (n − 3)R0−2n∗4
R
δ
−
2
1
1
0 1
1
n−2 0
—————————00
+ W (r) 1 = 2 1 ,
1
Einstein equations
!
n − 3 −n+4
! i′
W becomes
negative!
′h 3(n ′ 2)2
′′
′
′
1
(n
−
3)f
+
R
R
f
+
2R
R
+
2(n
−
3)R
RR
−
2
R
φ
=
0
1
0
0
0
1
1
1
0
(n
4)(n
6)n .0− 3 1 −n+3 n
W (r) ′′=
−
2
2
2(n
3)
−2n∗4
′
′
′
2
4R−0 3)R
R1 − (n
R 1 + R 0 δ1 − 2
R
φ1 = ω R1
R00
n−2 0
(n − 3)f1 + R0 R0′ f1′ + 2vR0−n+4 φ′1 − 2R0 R0′ δ1′
!
⌘ 2
0
′ ′¯0
n −2n+5
2⇣
d
+2(n
−
3)R
R
+
2(n
−
3)R
R
=
−2ω
0
1
0′ 11
′
′′1 = R0 20
′ 1 R′1 , R0nR−
⇥
=−
D+3)f
DR
1
1(n
11 +
+ =
0
R
f
+
2R
R
+
2(n
−
3)R
RR
−
2
0 1 ¯1 0 1
0 dr
0R0 1
!
n−3 3 ′
n ★
′
n ′ invariant
：Gauge
inR
spherical
R0 R0 f1 +1 2R0 R′ 1 −
−n+4
′ 2
′φ1 = 0 sym.
′ ′
0
(n − 3)f1 + R0 R0 f1 + 2vR
φ
−
2R
R
n−
2
0
1
0 δ1
0
′
′
−2n+5
2
3 −n+4 ′
R
φ1 = 0W
n − 2 ★0potential
(4)
(3)
(4)
(5)
(5)
(6)
(6)
(7)
(7)
Part 2 WH in higher-dim. (2) Linear Stability
Unstable!
‣ exist negative mode
n
4
5
6
7
8
9
10
11
20
50
100
2
1.39705243371511
2.98495893027790
4.68662054299460
6.46258414126318
8.28975936306259
10.1535530451867
12.0442650147438
13.9552091676647
31.5751101285105
91.3457759137153
191.283017729717
eigenvalues of negative mode
eigenfunction of negative mode
★ In all dimensions, we found negative modes.
Ellis s wormhole is unstable
★ Higher dimension, instability appears in short time scale
Torii-HS PRD 88 (2013) 064027
n-dim GR ↵GB = 0
Wormhole evolution in n-dim の線形解析結果
4-dim.
Black Hole
5-dim.
6-dim.
x+
x
#+ = 0
# =0
Positive Energy
Sn
2
次元が大きいほど，不安定モードを拾う．
（線形解析）
0
0
double trapping horizon
0
0
1
1
1
1
2
2
2
2
3
3
3
us
pl
m
us
n
i
us
pl
x
x
x
x
us
n
i
m 3
4
4
4
4
d
6-
6
4d
im
4d
d
4-
st
ho
-g
5
5
6d
5d
,c
1=
im
,c
im
5d
-0
.0
1
1=
he
,t
01
-0
.
01
-0
.
1=
,c
1=
,c
im
4d
im
4d
he
,t
st
ho
-g
positive energy input --> BH formation
6
,c
im
d
4-
)
1)
-0
.0
1
(c
_1
=0.
(c
_1
=
lse
pu
lse
pu
ta +
=0
t
,
0
a =im
th
1=
.0
et
1,
,c
-0
a+ 0
th
1=
.0
=0
e
1
-0
ta ,t
.0
he
=
1,
ta + 0
th
=
et
0
a =0
6d
d
6-
6
,c
i
1=
5d m,
-0
c1
im
.1
=5d
,t
,c
0.
he
im
1
1
=
6d
t
,
,c
t
0.
he a =+
im
1
1
=
0
6d
t
,t
,c
-0
he a =- 0
im
1=
.1
t
,t
,c
-0
he a =+
1=
.1
0
t
,t
-0
he a =- 0
5
.1
t
,t
he a =+
0
ta =0
5
6
4d 5d 6d GR
ghost pulse (negative amp.) input
0
0
double trapping horizon
0
0
1
1
1
1
2
x
m
2
x
x
pl
u
s
s
3
3
us
pl
u
in
2
2
x
i
m
3
3
s
nu
4
4
d
4
6-
4
im
4d
gh
6
6
m
+
pu
lse
d
6-
st
gh
o
1)
c_
1=
0.
d
4-
(c
_1
=0
.0
1)
ul
se
(
os
tp
,c
1=
,c
+
1
m
=+ 0.1,
5d
,c
th
0.
i
1=
et
6d m,
+0 1, t
a+
c1
im
h
=0
.
e
=
1
6d
,c
t
,
+
a =th
0.
im
1=
et
0
,c
+0 1, t
he a =+
1=
.1
0
t
,t
+0
he a =- 0
.1
ta +
,t
he
=0
ta =
6
0
5d
i
4d
i
d
4-
+
4d
i
4d m,
c1
i
=+
5d m,
c
0.
i
1=
5d m,
+0 01,
c1
im
th
.0
=+
6d
,c
et
1,
0
im
a+
1
t
.
h
=
0
6d
=0
,c
et
1,
+0
im
a
1=
t
.
h
=
0
,c
et
1,
+0
a+ 0
1=
th
.0
=0
et
1,
+0
a =th
.
0
5
et
1,
a+ 0
th
=0
et
a =0
5
5
5
6
4d 5d 6d GR
ghost pulse (positive amp.) input
negative energy input --> throat inﬂates
nDynamics in 5dim
GR gravity?
x
Section Summary
2. Spheroidal matter collapse
Initial data analysis, Evolutions
3. Wormhole dynamics in GR
linear stability,
dynamical stability
Yamada & HS, CQG 27 (2010) 045012
Yamada & HS, PRD 83 (2011) 064006
Torii & HS, PRD 88 (2013) 064027
HS & Torii, in preparation
Ellis (Morris-Thorne) traversable WH解線形摂動＆時間発展
WH は不安定である
高次元ほど不安定
(A) 正のエネルギーパルス ---> BH
(B) 負のエネルギーパルス ---> Inﬂationary expansion
(C) 頑張ればメンテナンス可能
Plan of the Talk
Dynamics in 5dim GR gravity?
2. Spheroidal matter collapse
Initial data analysis, Evolutions
3. Wormhole dynamics in GR
linear stability,
dynamical stability
Yamada & HS, CQG 27 (2010) 045012
Yamada & HS, PRD 83 (2011) 064006
Torii & HS, PRD 88 (2013) 064027
HS & Torii, in preparation
Dynamics in Gauss-Bonnet gravity?
4. Wormhole dynamics in GB
5. Plane-wave collision in GB
HS & Torii, in preparation
Introduction
Dynamics in Gauss-Bonnet gravity?
• has GR correction terms from String Theory • has two solution branches (GR/non-‐‑‒GR). • has minimum mass for static spherical BH solution T Torii & H Maeda, PRD 71 (2005) 124002 • is expected to have singularity avoidance feature. (but has never been demonstrated in full gravity.) •new topic in numerical relativity. S Golod & T Piran, PRD 85 (2012) 104015 N Deppe+, PRD 86 (2012) 104011 F Izaurieta & E Rodriguez, 1207.1496 •much attentions in WH community H Maeda & M Nozawa, PRD 78 (2008) 024005 P Kanti, B Kleihaus & J Kunz, PRL 107 (2011) 271101 P Kanti, B Kleihaus & J Kunz, PRD 85 (2012) 044007 Field Equations (1)
Formulation for evolution [dual null]
Evolution
x
x
⌃0
+
Field Equations (1)
Formulation for evolution [dual null]
Field Equations (2)
matter variables
Field Equations (3)
evolution equations (1)
Field Equations (4)
evolution equations (2)
Wormhole Evolution
BH & WH are interconvertible?
S.A. Hayward, Int. J. Mod. Phys. D 8 (1999) 373
They are very similar -‐‑‒-‐‑‒ both contain (marginally) trapped surfaces and can be deﬁned by trapping horizons (TH) Only the causal nature of the THs diﬀers, whether THs evolve in plus / minus density which is given locally. Black Hole
Wormhole
Locally
deﬁned
by
Achronal (spatial/null)
outer TH
⇨ 1-way traversable
Einstein
eqs.
Positive energy density Negative energy density normal matter (or
exotic matter
vacuum)
Appearance
occur naturally
Temporal (timelike)
outer THs ⇨ 2-way traversable
Unlikely to occur
naturally. but constructible??
Torii-HS PRD 88 (2013) 064027
n-dim GR ↵GB = 0
Wormhole evolution in n-dim のおさらい
4-dim.
Black Hole
5-dim.
6-dim.
x+
x
#+ = 0
# =0
Positive Energy
Sn
2
次元が大きいほど，不安定モードを拾う．
（線形解析）
5d Wormhole
Gauss-Bonnet
evolution
WH : positive
in Gauss-Bonnet
energy injection
のおさらい
(1)
↵GB = +0.001
MSmass, H.Maeda-Nozawa, PRD77 (2008) 063031
E < +0.5
E > +0.5
coupling 正（通常のGaussBonnet）⇨ BHを形成しにくい
ある程度以上の正エネルギーを追加 ⇨ BH形成に転じる
5d Wormhole
Gauss-Bonnet
evolution
WH : positive
in Gauss-Bonnet
energy injection
のおさらい
(2)
↵GB = +0.01
MSmass, H.Maeda-Nozawa, PRD77 (2008) 063031
E < +6.9
E > +6.9
alpha 大きければ，閾値大きい
coupling 正（通常のGaussBonnet）⇨ BHを形成しにくい
ある程度以上の正エネルギーを追加 ⇨ BH形成に転じる
0
0
0.
0.
5
5
0.
0.
0
0
1.
1.
1.
us
2.
0
2.
5
ha
=+
0.
00 h
1 , or
w izo
ith n
p e loc
r t at
u r io
b a ns
tio
n [6d
of im
po ]
si
ti
4.
0
th
th eta
e
th ta _+=
th eta _-= 0 :
th eta _+= 0 : alp
th eta _-= 0 : alp ha
th eta _+= 0 : alp ha= =0
th eta _-= 0 : alp ha 0. .00
1
et _+ 0
h
= 0
a_ = : alp a= 0. 01, , a
=0
+= 0 : alp ha 0.0 00
a
=
0
0 1, =0 .5
a h
: a lph a=0 0.0 1, a= .50 0
lp a= .0 01 a= 0.6
ha 0 01 , a 0.
0
=0 .00 , a =0 60
3.
.0 1, =0 .6
0
01 a
5
.
, a =0 65
.
=0 70
3.
.7
5
0
lp
ve
en
er
gy
)
↵GB = 0.001
xp
l
5
5
(a
5d
5
.0
2.
0
5
0
↵GB = 0.001
1.
us
in
xm 2
3.
3.
4.
5d, 6d Gauss-Bonnet WH
6d
need more positive energy for transition to BH in 6dim
5d, 6d Gauss-Bonnet WH
↵GB = 0.001
5d
6d
↵GB = 0.001
circumference radius of the throat [GB 6dim]
(alpha=+0.001, with perturabation positive Energy)
2.0
E < +0.5
E > +0.5
circumference radius of the throat
1.5
E < +2.2
1.0
E > +2.2
0.5
alpha=0.001,
alpha=0.001,
alpha=0.001,
alpha=0.001,
alpha=0.001,
a=0.00,
a=0.50,
a=0.60,
a=0.65,
a=0.70,
E=+0.0000
E=+1.5276
E=+2.1954
E=+2.5790
E=+2.9896
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
proper time at the throat
need more positive energy for transition to BH in 6dim
0
0
0.
0.
5
5
0.
0.
0
0
1.
1.
xp
5
5
1.
1.
lu
s
us
in
xm
0
0
2.
2.
5
5
2.
2.
0
0
3.
3.
5
0
0
th
th
et
et
a_
a_
+=
et
0
a_
:a
0
,
+=
01
-=
et
0.
th
=+
th
ha
th
lp
th
(a
:a
w ho
ith r
i
pe zon
rt l
ur oc
ba a
tio tio
n ns
of
po
et
a_
a_
-=
0
:a
lp
lp
ha
=0
h
0
a
et
: a lph = 0 .01
th
a +
et _ - =0
.
l
a
th
a_ = 0 : a p h a = 0 0 1 , , a =
e
: a lph = 0 .01 a = 0.0
th t a _ +=
e
0
t h ta_ - = 0 : a l p h a = 0 . 0 1 , a = 0 . 0
et
: a lph a = 0 .01 , a = 0.5
a _ +=
, a 0.
lp
a= . 0
-= 0 :
5
1
=
0
0
a h
: a lph a = 0 .01 , a = 0.6
, a 0.
lp
a= . 0
6
1
ha
=
0
3.
= 0 .01 , a = 0.7
5
.0
, a 0.
7
1,
=
a = 0.8
0.
4.
8
3.
4.
si
tiv
e
en
er
gy
)
5d Gauss-Bonnet WH : trapped surface
↵GB = 0.01
↵GB = 0.001
existence of trapped surface
̶> not necessary to form BH
0
0
0.
5
5
5
0.
0.
5
1.
0
0
0
1.
1.
0
5
5
1.
lu
s
2.
xp
lu
s
2.
0
0
0
0
us
in
xm 2 .
xp
5
1.
1.
5
2.
5
0
0
3.
3.
0
et
a_
+=
a_
-=
0
a_
:a
0
,
+=
01
:a
w ho
ith r
i
pe zon
rt l
ur oc
ba a
tio tio
n ns
of
po
3.
5
4.
0
lp
a_
(a
et
-=
ha
a_
=+
,
lp
01
:a
0.
0
4.
0
e
e
tiv
tiv
si
si
w ho
ith r
i
pe zon
rt l
ur oc
ba a
tio tio
n ns
of
po
lp
ha
=0
h
0
a
et
: a lph = 0 .01
th
a +
et _ - =0
.
l
a
th
a_ = 0 : a p h a = 0 0 1 , , a =
e
: a lph = 0 .01 a = 0.0
th t a _ +=
e
0
t h ta_ - = 0 : a l p h a = 0 . 0 1 , a = 0 . 0
et
: a lph a = 0 .01 , a = 0.5
a _ +=
, a 0.
lp
a= . 0
-= 0 :
5
1
=
0
0
a h
: a lph a = 0 .01 , a = 0.6
, a 0.
lp
a= . 0
6
1
ha
=
0
3.
= 0 .01 , a = 0.7
5
.0
, a 0.
7
1,
=
a = 0.8
0.
4.
8
th
th
et
et
et
0.
th
=+
th
ha
th
lp
th
(a
th
th eta
th et _+
th eta a_- =0
th et _+ =0 :
th eta a_- =0 : a alp
th et _+ =0 : lph ha
th eta a_- =0 : a alp a= =0.
et _ =0 : lp ha 0. 01
a_ += : al ha = 01 ,
-= 0 alp ph =0 0.0 , a a=
2.
0 : a h a= .0 1, = 0.
5
: a lp a= 0 1 a 0. 77
lp ha 0. .01 , a =0 77
ha = 01 , =0 .7
3.
=0 0.0 , a a= .7 8
0
.0 1, =0 0.7 8
1, a .7 9
a= =0 9
3.
0. .80
5
80
2.
5
5
2.
2.
0
5
0
)
)
gy
gy
er
er
en
en
↵GB = 0.01
0.
0.
0
0
0.
0.
0.
1.
1.
us
in
xm
3.
3.
4.
5d Gauss-Bonnet WH : trapped surface
critical behavior
existence of trapped surface
̶> not necessary to form a BH
Section Summary
Dynamics in 5dim GR gravity?
2. Spheroidal matter collapse
Initial data analysis, Evolutions
3. Wormhole dynamics in GR
linear stability,
dynamical stability
Yamada & HS, CQG 27 (2010) 045012
Yamada & HS, PRD 83 (2011) 064006
Torii & HS, PRD 88 (2013) 064027
HS & Torii, in preparation
Dynamics in Gauss-Bonnet gravity?
4. Wormhole dynamics in GB
HS & Torii, in preparation
Gauss-Bonnet重力の特色
正のcouplingでは，同じ初期条件でも特異点形成は遅くなる．
正のcouplingでは，同じ初期条件でもBHは形成しにくい．
エネルギー底上げ・特異点回避の傾向がある．
高次元になるほど，不安定性は拡大する．
trapped surfaceの存在は，必ずしもBH形成を意味しない．
Plan of the talk
Dynamics in 5dim GR gravity?
2. Spheroidal matter collapse
Initial data analysis, Evolutions
3. Wormhole dynamics in GR
linear stability,
dynamical stability
Yamada & HS, CQG 27 (2010) 045012
Yamada & HS, PRD 83 (2011) 064006
Torii & HS, PRD 88 (2013) 064027
HS & Torii, in preparation
Dynamics in Gauss-Bonnet gravity?
4. Wormhole dynamics in GB
5. Plane-wave collision in GB
HS & Torii, in preparation
Colliding Scalar Waves
GR 5d: small amplitude waves
flat background, normal scalar field
@+
@
I (5) = Rijkl Rijkl
Colliding Scalar Waves
GR 5d: large amplitude waves
@+
@
I (5) = Rijkl Rijkl
I (5) = Rijkl Rijkl
↵GB = 0
I (5) = Rijkl Rijkl
GR 5d
↵GB = +1
GaussBonnet 5d
��
↵GB =
I
1
(5)
��
��
��
at origin
��
��
��
↵GB =
1
��
↵GB = 0
��
GaussBonnet 5d (negative α)
��
↵GB = +1
��
��
��
��
��
��
x
��
Colliding Scalar Waves
massless scalar waveの衝突による特異点形成
max (Rijkl Rijkl )
5,6,7次元 Gauss-Bonnet
↵GB =
1
↵GB = 0
↵GB = +1
＊4dim, 5dim, 6dim,… 高次元化
＊Gauss-Bonnet項（正αの項）
は，どちらも特異点形成条件を緩くさせる
Summary
✅ 平面スカラー波の衝突 ✅ 球対称ワームホールのBHへの転移現象
4dim, 5dim, 6dim, …
高次元になるほど，同じ初期条件でもBHは形成しにくい
Yamada-HS (2011) [naked singularity形成]とconsistent
1
高次元になるほど，不安定性は拡大する
F ⇠ n 2
Torii-HS (2013) [WH不安定性]とconsistent
r
Gauss-Bonnet重力の特色
正のcouplingでは，同じ初期条件でも特異点形成は遅くなる．
正のcouplingでは，同じ初期条件でもBHは形成しにくい．
エネルギー底上げ・特異点回避の傾向がある．
高次元になるほど，不安定性は拡大する．
trapped surfaceの存在は，必ずしもBH形成を意味しない．
（面積定理が成立しない解系列があることに対応か）

Download Report