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22017/10/12 @ Einstein Toolkit, Mallorca, Spain
Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity
http://www.oit.ac.jp/is/~shinkai/
Hisaaki Shinkai (Osaka Inst. Tech., Japan)
Who am I ?
1996-99 Postdoc at WashU.
Newman-Penrose scalar in
3+1 language
Cactus developer at the very initial
phase
Formulation Problem of Einstein equations
Higher-dimensional
numerical relativity
Alternative Gravity Theories, Approaches
Chair
of the Board, KAGRA Scientific Congress
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32017/10/12 @ Einstein Toolkit, Mallorca, Spain
Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity
http://www.oit.ac.jp/is/~shinkai/
Hisaaki Shinkai (Osaka Inst. Tech., Japan)
4-dim, 5-dim, 6-dim,
… how dimensionality affects to dynamics?
Gauss-Bonnet terms
… how higher-order curvature terms affects to
dynamics? 2 models
Colliding scalar pulses / Fate of wormholes Ref:
HS & Torii , PRD 96(2017)044009 [arXiv:1706.02070]
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Spheroidal matter collapse 4D vs 5D
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5D collapses -- proceed rapidly. -- towards spherically. -- AH
forms in wider ranges.
I = RabcdRabcd
at I(tend)
Spheroidal matter collapse 4D vs 5D
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Dynamics in Gauss-Bonnet gravity?
E>O PEB OFI HBOP HB>AFJD PB IO C I P FJD EBE>O PS O H
PF J >J EBO 9 J J 9 E>O IFJFI I I>OO C OP>PF O EB F
>H -3 O H PF J
FO BT B PBA P E>RB OFJD H> FP >R FA>J B CB>P B
P
E>O JBRB BBJ ABI JOP >PBA FJ C HH D >RFP
JBS P F FJ J IB F >H BH>PFRFP
A 7R RH B MTE 4 .+ (& (
& & +
4ISSI 4 ., (& ( & &
6 9 EWTMI E 5 RHTMKWI (&-
,
I E >PPBJPF JO FJ 3 II JFP
8 EIHE R EYE 4 -. (&&.
&( &&+
E M 2 IMLEW : W &- (& (- &
E M 2 IMLEW : W 4 .+
(& ( & &&-
Introduction
B BRTMM 8 EIHE 4 - (&&+ ( &&(C 1L 2 7YEN 2 8 II
C II 5WT L : 3-+ (& + )-(
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Formulation for evolution [N+1] Introduction (2)
JFPF>H >H B . JOP PF J RF> . JC I>H > > E
2
EGN LR I M M ME HE E0 8 DR LM R 4 .) (& & & &
BP C 0M >PF JO
TIEH FW GRPS MGE IH
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Formulation for evolution [dual null] Field Equations (1)
x
+x
�
⌃0
Evolution
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Formulation for evolution [dual null] Field Equations (1)
-
matter variables Field Equations (2)
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evolution equations (1) Field Equations (3)
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evolution equations (2) Field Equations (4)
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@+ @�
I(5) = RijklRijkl
GR 5d: small amplitude waves Colliding Scalar Waves
flat background, normal scalar field
x
+x
+x
�x
�
-
@+ @�
I(5) = RijklRijkl
GR 5d: large amplitude waves Colliding Scalar Waves
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��
����
��
����
��
����
��
�� �� �� �� �� �� ��
������������������������������������������������������������
GR 5d GaussBonnet 5d
I(5) = RijklRijkl
I(5) at origin
x
�
I(5) = RijklRijkl
GaussBonnet 5d (negative α)
↵GB = +1
↵GB = �1
↵GB = 0
↵GB = �1
↵GB = +1
↵GB = 0
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Colliding Scalar Waves
max (RijklRijkl
)
*4dim, 5dim, 6dim,… higher dim *Gauss-Bonnet coupling (α>0)
̶> less growth of curvature
↵GB = �1
↵GB = +1
↵GB = 0
��
����
��
����
��
����
��
�� �� �� �� �� �� ��
������������������������������������������������������������
I(5) at origin
↵GB = �1
↵GB = +1
↵GB = 0
maximum of Kretschmann invariant
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BH & WH are interconvertible?
EB > B RB OFIFH> PE JP>FJ I> DFJ>HH P > BAO
C> BO >JA >J B ABWJBA P > FJD E FV JO 3
7JH PEB > O>H J>P B C PEB 3O AF B O SEBPEB 3OBR HRB FJ
H O IFJ O ABJOFP SEF E FO DFRBJ H >HH
, 3> S> A JP A 8E O / ) (
Black Hole WormholeLocally defined
by
Achronal (spatial/null) outer TH ⇨ 1-way traversable
Temporal (timelike) outer THs
⇨ 2-way traversable
Einstein eqs.
Positive energy density
normal matter (or vacuum)
Negative energy density
“exotic” matter
Appear-ance occur naturally
Unlikely to occur naturally.
but constructible??
Wormhole Evolution
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BH & WH are interconvertible?
EB > B RB OFIFH> PE JP>FJ I> DFJ>HH P > BAO
C> BO >JA >J B ABWJBA P > FJD E FV JO 3
7JH PEB > O>H J>P B C PEB 3O AF B O SEBPEB 3OBR HRB FJ
H O IFJ O ABJOFP SEF E FO DFRBJ H >HH
, 3> S> A JP A 8E O / ) (
Black Hole WormholeLocally defined
by
Achronal (spatial/null) outer TH ⇨ 1-way traversable
Temporal (timelike) outer THs
⇨ 2-way traversable
Einstein eqs.
Positive energy density
normal matter (or vacuum)
Negative energy density
“exotic” matter
Appear-ance occur naturally
Unlikely to occur naturally.
but constructible??
Wormhole Evolution
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Wormhole evolution
#+ = 0 #� = 0#+ = 0#� = 0
x
+x
�x
+x
�
Positive Energy Input= add normal scalar field= subtract ghost
field
Negative Energy Input= add ghost scalar field
Black Hole InflationaryExpansion
4d GR ↵GB = 0 HS-Hayward PRD 66(2002) 044005
⌃(t)
#+ = 0 #� = 0
⌃(t)
#+ = 0#� = 0
N RY JEG
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�E > 0
�E < 0
4d GR ↵GB = 0 HS-Hayward PRD 66(2002) 044005
#+ = 0 #� = 0#+ = 0#� = 0
x
+x
�x
+x
�
Positive Energy Input= add normal scalar field= subtract ghost
field
Negative Energy Input= add ghost scalar field
Black Hole InflationaryExpansion
⌃(t)
#+ = 0 #� = 0
⌃(t)
#+ = 0#� = 0
Wormhole evolution N RY JEG
真貝著 タイムマシンと時空の科学
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4-dim.
5-dim.
6-dim.
x
+x
�
Sn�2
#+ = 0#� = 0
Positive Energy
Black Hole
In higher dim, large instability. (linear perturbation
analysis)
Wormhole evolution in n-dim n-dim GR Torii-HS PRD 88 (2013)
064027↵GB = 0
N RY JEG
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4-dim.
5-dim.
6-dim.
x
+x
�
Sn�2
#+ = 0#� = 0
Positive Energy
Black Hole
In higher dim, large instability. (linear perturbation
analysis)
Wormhole evolution in n-dim n-dim GR Torii-HS PRD 88 (2013)
064027↵GB = 0
N RY JEG
Confirmed Numerically
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↵GB = +0.001
Wormhole evolution in Gauss-Bonnet のおさらい 5d Gauss-Bonnet WH :
positive energy injection
MSmass, H.Maeda-Nozawa, PRD77 (2008) 063031
(a)
0.0
0.5
1.0
1.5
2.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Circumference radius of throat [5dim GB alpha=+0.001](with
perturabation of positive-energy pulse)
alpha=0.001, c1=0.1, E=+0.11alpha=0.001, c1=0.2,
E=+0.46alpha=0.001, c1=0.3, E=+1.04alpha=0.001, c1=0.5,
E=+2.85alpha=0.001, c1=0.7, E=+5.49
circ
umfe
renc
e ra
dius
of t
he th
roat
proper time at the throat
(a)
ΔE > ΔE5 > 0 --> BH collapse ΔE < ΔE5 -->
Inflationary expansion
�E < +0.5
�E > +0.5
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↵GB = +0.001
Wormhole evolution in Gauss-Bonnet のおさらい 6d Gauss-Bonnet WH :
positive energy injection
MSmass, H.Maeda-Nozawa, PRD77 (2008) 063031
ΔE > ΔE6 > ΔE5 > 0 --> BH collapse ΔE < ΔE5 <
ΔE6 --> Inflationary expansion
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Circumference radius of throat [6dim GB alpha=+0.001](with
perturabation of positive-energy pulse)
alpha=0.001, c1=0.50, E=+1.5276alpha=0.001, c1=0.60,
E=+2.1954alpha=0.001, c1=0.65, E=+2.5790alpha=0.001, c1=0.70,
E=+2.9896
circ
umfe
renc
e ra
dius
of t
he th
roat
proper time at the throat
(b)(b)
�E > +2.2
�E < +2.2
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0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
horiz
on lo
catio
ns
(alph
a=+0
.01, w
ith p
ertur
batio
n of
posit
ive e
nerg
y)
theta_
+=0 :
alph
a=0.0
1, a=
0.0
theta_
-=0 : a
lpha=
0.01,
a=0.0
theta_
+=0 :
alph
a=0.0
1, a=
0.5
theta_
-=0 : a
lpha=
0.01,
a=0.5
theta_
+=0 :
alph
a=0.0
1, a=
0.6
theta_
-=0 : a
lpha=
0.01,
a=0.6
theta_
+=0 :
alph
a=0.0
1, a=
0.7
theta_
-=0 : a
lpha=
0.01,
a=0.7
theta_
+=0 :
alph
a=0.0
1, a=
0.8
theta_
-=0 : a
lpha=
0.01,
a=0.8
xminu
sxp
lus
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
horiz
on lo
catio
ns
(alph
a=+0
.01, w
ith p
ertur
batio
n of
posit
ive e
nerg
y)
theta_
+=0 :
alph
a=0.0
1, a=
0.77
theta_
-=0 : a
lpha=
0.01,
a=0.7
7
theta_
+=0 :
alph
a=0.0
1, a=
0.78
theta_
-=0 : a
lpha=
0.01,
a=0.7
8
theta_
+=0 :
alph
a=0.0
1, a=
0.79
theta_
-=0 : a
lpha=
0.01,
a=0.7
9
theta_
+=0 :
alph
a=0.0
1, a=
0.80
theta_
-=0 : a
lpha=
0.01,
a=0.8
0
xminu
sxp
lus
critical behavior
existence of trapped surface ̶> not necessary to form a
BH
↵GB = 0.01
5d Gauss-Bonnet WH : trapped surface
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Colliding Scalar Waves
Summary
Wormhole Evolution
max (RijklRijkl
)
1 PE I ABHO PEB J I>H B PF JO - S G C>R FAFJD PEB >
B> >J B C OFJD H> FP >HPE DE FP FO FJBRFP> HB
CI JRW H LE M LI GTM MGE M WE MR JRTJRTPM K E 28 LI I M I GI RJ
LI TESSIHTIKMR M LI 5M IM 72 KTE M HRI RHMTIG M HMGE I E JRTPE MR
RJ E 28
ΔE > ΔE6 > ΔE5 > 0 --> BH collapse ΔE < ΔE5 <
ΔE6 --> Inflationary expansion