nonlinear dynamics in the einstein-gauss-bonnet gravityhisaaki shinkai 真貝寿明 (osaka inst....
TRANSCRIPT
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Hisaaki Shinkai 真貝寿明 (Osaka Inst. Tech., Japan)
12017/7/6 The 13th International Conference on Gravitation, Astrophysics, & Cosmology (ICGAC-XIII)
@ Seoul, Korea
Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity
http://www.oit.ac.jp/is/~shinkai/
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 , arXiv:1706.02070
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Dynamics in Gauss-Bonnet gravity?
• has the simplest leading terms from String Theory • has two solution branches (GR/non-‐‑‒GR). • has minimum mass for static spherical BH solution
• 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
Introduction
T Torii & H Maeda, PRD 71 (2005) 124002 W-‐‑‒K Ahn, B Gwak, B-‐‑‒H Lee, W Lee, Eur. Phys. J. C75 (2015) 372
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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)
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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
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@+ @�
I(5) = RijklRijkl
GR 5d: large amplitude waves Colliding Scalar Waves
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�� �� �� �� �� �� ��
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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
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�� �� �� �� �� �� ��
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I(5) at origin
↵GB = �1
↵GB = +1
↵GB = 0
maximum of Kretschmann invariant
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BH & WH are interconvertible?
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.
S.A. Hayward, Int. J. Mod. Phys. D 8 (1999) 373
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?
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.
S.A. Hayward, Int. J. Mod. Phys. D 8 (1999) 373
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
(known fact)
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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 (known fact)
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
(known fact)
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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
(known fact)
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
)
For both models, the normal corrections (αGB > 0) work for avoiding the appearance of singularity, although it is inevitable.
We found that in the critical situation for forming a BH, the existence of the trapped region in the Einstein-‐‑‒GB gravity does not directly indicate a formation of a BH.
ΔE > ΔE6 > ΔE5 > 0 --> BH collapse ΔE < ΔE5 < ΔE6 --> Inflationary expansion