yu-ichirou sekiguchi (univ. of tokyo) 関口　雄一郎

Axisymmetric collapse simulations Axisymmetric collapse simulations of rotating massive stellar cores of rotating massive stellar cores

in full general relativity:in full general relativity: Numerical study for prompt black hole Numerical study for prompt black hole

formationformation

Yu-ichirou Sekiguchi (Univ. of Tokyo)Yu-ichirou Sekiguchi (Univ. of Tokyo)

関口　雄一郎関口　雄一郎

§1 Introduction§1 Introduction§2 Numerical Implementation§2 Numerical Implementation§3 Setting§3 Setting

§3.1 Initial conditions§3.1 Initial conditions§3.2 Parametric Equations of state§3.2 Parametric Equations of state

§4 General Feature of Collapse§4 General Feature of Collapse§5 Black Hole Formation§5 Black Hole Formation

§5.1 Criterion for prompt black hole formation§5.1 Criterion for prompt black hole formation§5.2 Dependence on parameters§5.2 Dependence on parameters§5.3 Prediction of the final system§5.3 Prediction of the final system

§6 Neutron Star Formation§6 Neutron Star Formation§6.1 Collapse dynamics§6.1 Collapse dynamics§6.2 Gravitational Waves§6.2 Gravitational Waves

§7 Summary§7 Summary

§1 Introduction§1 Introduction

§1 Introduction ①

Results of simulations for rotating stellar Results of simulations for rotating stellar core collapse in full GRcore collapse in full GR– Highly nonlinear and dynamical Highly nonlinear and dynamical

phenomenaphenomena

Numerical simulation in full GR is the Numerical simulation in full GR is the unique approachunique approach

Numerical relativity as a powerful tool of Numerical relativity as a powerful tool of exploring astrophysical phenomenaexploring astrophysical phenomena

– Black hole formation ?Black hole formation ?

– Neutron star formation ? Neutron star formation ?

In this talk, let me talk about …….

§1 Introduction ②

Black hole formation via massive Black hole formation via massive rotating rotating stellar core collapsestellar core collapse– Candidate for the central engine of the long Candidate for the central engine of the long

duration GRBsduration GRBsKnown as the Known as the collapsarcollapsar model ( model (Woosley ApJ 405, 273 Woosley ApJ 405, 273 (1993)(1993)))

– A source of the gravitational radiationA source of the gravitational radiationQuasi-normal ringingQuasi-normal ringing

Neutron star formation via massive rotating Neutron star formation via massive rotating stellar core collapsestellar core collapse– Study extensively in Newtonian gravityStudy extensively in Newtonian gravity

e.g. Zwerger and Muellar A&A 320, 209 (1997)e.g. Zwerger and Muellar A&A 320, 209 (1997)

– A promising source of the GWA promising source of the GW

§1 Introduction ③

We consider a criterion of black hole formation We consider a criterion of black hole formation in the collapse of stellar iron coresin the collapse of stellar iron cores– Performing fully general relativistic simulationsPerforming fully general relativistic simulations– On assumption of Axial symmetry On assumption of Axial symmetry – Putting emphasis on clarifying the dependence of Putting emphasis on clarifying the dependence of

black hole formation on black hole formation on mass, angular momentum, rotational velocity profile mass, angular momentum, rotational velocity profile of iron cores,of iron cores, and and equations of stateequations of state

For systematic investigation, a For systematic investigation, a parametric parametric equation of stateequation of state ( (e.g. Dimmelmeir et al. A&A 393, 523 e.g. Dimmelmeir et al. A&A 393, 523

(2002)(2002) ) is adopted (which will be introduced later)) is adopted (which will be introduced later)

§2 Numerical Implementation§2 Numerical Implementation York in “Sources of gravitational radiation” (1979)York in “Sources of gravitational radiation” (1979)

Baumgarte & Shapiro Phys.Rep. 376, 41 (2003)Baumgarte & Shapiro Phys.Rep. 376, 41 (2003)

Lehner Class. Quantum Grav. 18, R25 (2001)Lehner Class. Quantum Grav. 18, R25 (2001)

Font Living Rev. Relat. 6, 4 (2003)Font Living Rev. Relat. 6, 4 (2003)

Einstein equationsEinstein equations– ADM (3+1) decomposition of the spacetimeADM (3+1) decomposition of the spacetime

e.g. York in Sources of gravitation (1979) Cambridge; Baumgarte e.g. York in Sources of gravitation (1979) Cambridge; Baumgarte & Shapiro Phys.Rep. 376, 41 (2003)& Shapiro Phys.Rep. 376, 41 (2003)

– Shibata-Nakamura (BSSN) reformulationShibata-Nakamura (BSSN) reformulationShibata and Nakamura PRD 52, 5428 (1995), Baumgarte & Shibata and Nakamura PRD 52, 5428 (1995), Baumgarte & Shapiro PRD 59, 024007 (1999)Shapiro PRD 59, 024007 (1999)

– Cartoon method (solving 2D problem in Cartesian grid)Cartoon method (solving 2D problem in Cartesian grid)Alcubierre et al. Int.J.Mod.Phys. D10, 273 (2001)Alcubierre et al. Int.J.Mod.Phys. D10, 273 (2001)

Gauge conditionsGauge conditions– Approximate maximal slicing condition (Shibata Prog.Theor.Phys. Approximate maximal slicing condition (Shibata Prog.Theor.Phys.

101, 251 (1999))101, 251 (1999))– Dynamical gauge (shift) condition (Shibata ApJ 595, 992 (2003))Dynamical gauge (shift) condition (Shibata ApJ 595, 992 (2003))– e.g. Baumgarte & Shapiro (2003)e.g. Baumgarte & Shapiro (2003)

Apparent horizon finderApparent horizon finder (Shibata PRD 55, 2002 (1997))(Shibata PRD 55, 2002 (1997))

§2 Numerical Implementation

§3 Setting§3 Setting §3.1 Initial conditions§3.1 Initial conditions

§3.2 Parametric equations of state§3.2 Parametric equations of state

Initial conditions– Rotating iron cores of massive stars– Modeled by rotating polytrope in

equilibrium Central density

Mass range

angular momentum

§3.1 Setting -- Initial conditions --

10 31.0 10 g/cmc

/ 2.0 3.0M M

20 1.1 (spherical to mass shedding limit)

Rotation law (Komatsu et al. MNRAS 239, 153 (1989) )

– In Newtonian limit Cylindrical rotation

– Differential rotation parameter : A

§3.1 Setting -- Initial conditions --

20( )t

/ , 1.0, 0.5d eA

: radius at equator e

, rigid rotationA

0 2 2d

0, larger degree of differential rotationA

Parametric equations of state

Parameters of EOS :– Parameters of EOS are so chosen that the maximum mass

of the cold spherical polytropes is almost identical

– We set for simplicity

§3.2 Setting -- Parametric EOS (1) --

poly thP P P

polyP 1 nuc ( )K 2

2 nuc ( )K

4 / 3 Unstable due to the photo-dissociation and electron capture

1 2 2.0 Sudden stiffening due to nuclear force

th th th( 1)P

1 2 nuc th( , , , )

Thermal and shock heating effects

max NS 1.6M M

We set for simplicity

Note : Collapse dynamics is less sensitive Note : Collapse dynamics is less sensitive to the value of .to the value of .

– As long as As long as

Shibata & Sekiguchi PRD 69, 084024 (2004)Shibata & Sekiguchi PRD 69, 084024 (2004)

th1.3 5 / 3

Parameters of EOSParameters of EOS

1 2 nuc 14 th 1( , , / ),

EOS- : (1.32,2.25,2.0)

EOS- : (1.30,2.5,2.0)

EOS- : (1.30,2.22,1.0)

EOS- : (1.28,2.75,2.0)

14 314 10 g/cm

max NS 1.6M M

Note: EOS-c is stiffer than EOS-b

§4 General Feature of Collapse§4 General Feature of Collapse

Infall phase

Bounce phase

Ring-down phase

§4 General feature of the collapse (1)

Infall phase :Infall phase :– Core becomes unstable due to sudden softening of EOSCore becomes unstable due to sudden softening of EOS

Photo-dissociation, electron capture Photo-dissociation, electron capture

Outer core :

The inner region which collapses at subsonic velocity

Inner core :

The outer region in which the matter falls at supersonic velocity

Bounce phase :Bounce phase :– Sudden stiffening of EOS decelerates the inner core at Sudden stiffening of EOS decelerates the inner core at

supra-nuclear densitysupra-nuclear density– (a) mass of the inner core is very large → collapse to a black (a) mass of the inner core is very large → collapse to a black

holehole

– (b) mass of the inner core is not too large → bounce(b) mass of the inner core is not too large → bouncePart of stored internal energy at bounce is releasedPart of stored internal energy at bounce is released

The shock wave is generated at the outer edge of the inner coreThe shock wave is generated at the outer edge of the inner core

(a) (b)

Ringdown phase : (after the bounce)Ringdown phase : (after the bounce)– The inner core oscillation damps via The inner core oscillation damps via PdVPdV works (This works (This

process powers shocks)process powers shocks)

– (a) Shock is strong enough → a neutron star is left(a) Shock is strong enough → a neutron star is left

– (b) Shock is not strong enough → fallback induced (b) Shock is not strong enough → fallback induced collapse to a black holecollapse to a black hole

(b)(a)

Fall back

§5 Black Hole Formation§5 Black Hole Formation AA criterion for black hole formationcriterion for black hole formation

Dependence on parametersDependence on parameters Dependence of EOSDependence of EOS

Effects of shocksEffects of shocks

Effects of rotationEffects of rotation

Effects of differential rotationEffects of differential rotation

Predicting the final systemPredicting the final system

§5.1 A criterion for prompt black hole formation － in M-q plane －

: (1.32,2.25,2.0)

: (1.30,2.5,2.0)

: (1.30,2.22,1.0)

: (1.28,2.75,2.0)

■ : BH for all EOS

☆ : BH for EOS-b (-d)

× : BH for EOS-a

□ : NS for all EOS

§5.2 Black hole formation － Dependence on EOS －

EOS-a2.7M M

: (1.32,2.25,2.0)

: (1.30,2.5,2.0)

: (1.30,2.22,1.0)

: (1.28,2.75,2.0)

Direct CollapseDirect Collapse14

1 2( , , /10 ) (1.32,2.25,2.0), 1.32nuc th

22.7 , / 1.0M M cJ GM

Neither sudden stiffening of EOS nor Rotational effect cannot halt the collapse

Direct collapse to a BH

A black hole is formed directly without any distinct A black hole is formed directly without any distinct bouncebounce

1 2 nuc 14 thEOS- : ( , , / ) (1.32,2.25,2.0), 1.32a

Mass of the inner core at bounce is large

inner core NS maxM M

No shock propagates outward

BH is more likely to be formed

2.4M M

§5.2 Black hole formation － Dependence on EOS －EOS-b 2.7M M

: (1.32,2.25,2.0)

: (1.30,2.5,2.0)

: (1.30,2.22,1.0)

: (1.28,2.75,2.0)

Fallback Induced Collapse (1) Fallback Induced Collapse (1) 14

1 2( , , /10 ) (1.3,2.5,2.0), 1.3nuc th

22.7 , / 1.0M M cJ GM

The shock wave propagate outward ……, however,

Fallback Induced Collapse (2) Fallback Induced Collapse (2) 14

1 2( , , /10 ) (1.3,2.5,2.0), 1.3nuc th

22.7 , / 1.0M M cJ GM

The shocked matters fall back to the inner core and a black hole is eventually formed

Inner cores experience a bounce before BH formationInner cores experience a bounce before BH formation

1 2 nuc 14 thEOS- : ( , , / ) (1.3,2.5,2.0), 1.3b

1 2 nuc 14 thEOS- : ( , , / ) (1.28,2.75,2.0), 1.28d

EOS-binner core NS maxM M

・ Shocks propagate outward

thP Contributes to support the core

Threshold mass of prompt BH formation is larger than for the cases with EOS-a

・ Mass of the inner core at bounce is large

Dependence on – For larger

Cores collapse more homologously– Mass of the inner core at the bounce is larger

Shocks (if generated) heat less fraction of the core

Degree of overshooting at the bounce is larger

– For smaller The initial pressure reduction is lager (in particular at central region)

– The central region collapses first– Mass of the inner core at bounce is smaller

Shocks heat larger fraction of the core

| 4 / 3 | 0

Dependence on– For smaller

Equation of state for proto-neutron star is softer

Degree of overshooting is larger

Larger inner core mass

Larger degree of overshooting– Compactness at maximum compression is

larger

BHs are more liable to form promptly

A black hole is formed directly without any distinct A black hole is formed directly without any distinct bouncebounce

1 2 nuc 14 thEOS- : ( , , / ) (1.32,2.25,2.0), 1.32a

Mass of the inner core at bounce is large

is larger

No shock propagates outward

BH is more liable to be formed

2.4M M

Inner cores experience a bounce before BH formationInner cores experience a bounce before BH formation

1 2 nuc 14 thEOS- : ( , , / ) (1.3,2.5,2.0), 1.3b

1 2 nuc 14 thEOS- : ( , , / ) (1.28,2.75,2.0), 1.28d

is smaller

Shocks propagate outward

thP Contributes to support the core

Threshold mass of prompt BH formation is larger than for the cases with EOS-a

: (1.32,2.25,2.0)

: (1.30,2.5,2.0)

: (1.30,2.22,1.0)

: (1.28,2.75,2.0)

■ : BH for all EOS

☆ : BH for EOS-b (-d)

× : BH for EOS-a

□ : NS for all EOS

The pressure near nuclear density is larger for EOS-cThe pressure near nuclear density is larger for EOS-c

1 2 nuc 14 thEOS- : ( , , / ) (1.3,2.5,2.0), 1.3b

1 2 nuc 14 thEOS- : ( , , / ) (1.3,2.22,1.0), 1.3c

Shocks are stronger for EOS-cThreshold mass is larger

§5.3 Black hole formation 　－ Effect of shocks －

Contribution of thP

Maximum mass of the cold spherical polytrope

max NS 1.6M M

For spherical models

crit sphe 2.1 2.3M M

Thermal effects increase the threshold mass by 20 ～ 40 %

Effect of shock is stronger for EOS-c

§5.4 Black hole formation － Effects of rotation －

2crit rot crit sphe rotM M C J

Threshold mass for rotating models may be written as (Shibata (2000) PThP 104, 325)

Rotational effects increases the threshold mass at most by 17 ～ 20 %

Rotational effects (i) Effectively supply additional pressure(ii) Reduce the amount of matters which eventually fall into inner core

§5.5 Black hole formation － Effect of differential rotation －

0.89, NS, rigidq

0.79, BH, rigidq

The threshold for BH formation locates between these curves

0.63, NS, 0.5q A

0.54, BH, 0.5q A

As the degree of differential rotation increases, a black hole is less liable to form

The inner region which is responsible to black hole formation “rotates” more rapidly

2.4M M

Fluid elements of smaller specific angular momentum will fall into the black hole

§5.6 Estimation of mass of disk

・ Consider the innermost stable circular orbit (ISCO) around a formed BH

ISCOj j

・ If increases as a result of the accretion, more fluid elements fall into the BH

・ Thus, if evolution of has a maximum, the dynamical growth of BH will terminate there

Cf. Shibata and Shapiro ApJ 572, L39 (2002)

・ Define mass and spin parameter in terms of the specific angular momentum : q(j) and m(j)

2* *'( ) 2 (cos )

j jm j r drd

( ) 2 ' (cos )j j

J j j r drd

( )( )

J jq j

・ Approximating the spacetime as Kerr spacetime, Jisco can be expressed by m(j) and q(j) e.g. Shapiro and Teukolsky Chap.12

Search the maximum of Jisco (j)

Mass of the formed disk will be < 10% of the initial mass

§6 Neutron star Formation§6 Neutron star Formation 6.1 Dependence of EOS6.1 Dependence of EOS

6.2 Gravitational waves6.2 Gravitational waves– WaveformsWaveforms– Energy spectraEnergy spectra

§6.1 Dependence on EOS

1 2 nuc 14 thEOS- : ( , , / ) (1.32,2.25,2.0), 1.32a

1 2 nuc 14 thEOS- : ( , , / ) (1.3,2.5,2.0), 1.3b

1 2 nuc 14 thEOS- : ( , , / ) (1.28,2.75,2.0), 1.28d

The collapse dynamics depends strongly on the The collapse dynamics depends strongly on the adopted EOS adopted EOS

The effects are reflected in Gravitational wavesThe effects are reflected in Gravitational waves

2.5M M

rig. rot.

2.5M M

rig. rot.

2.5M M

rig. rot.

2.5M M

rig. rot.

141 2EOS- : ( , , /10 ) (1.32,2.25,2.0)nuca

2.5 , 1.0M M q

Since mass of the inner core and the degree of overshooting of the Inner core at bounce is larger…..,

A steep density gradient is formed around the rotational axis

Aspherical shock wave generation and propagation

Since the density of matters in front of the shock “pole” is much smaller,

The shock velocity is higher in this direction

§6.1 Dependence on EOS 14

1 2EOS- : ( , , /10 ) (1.32,2.25,2.0)nuca

2.5 , 1.0M M q

141 2EOS- : ( , , /10 ) (1.3,2.5,2.0)nucb

2.5 , 1.0M M q

Difference of the centrifugal force between along the rotational axis and around the equator is not very large

The density gradient along the rotational axis is not outstanding

A slightly prorate shock wave is generated

141 2EOS- : ( , , /10 ) (1.3,2.5,2.0)nucb

2.5 , 1.0M M q

A slightly prorate shock wave is generated

141 2EOS- : ( , , /10 ) (1.28,2.75,2.0)nucd 2.5 , 1.0M M q

Since the shock wave generated at the bounce is rather weak, the shocked matters around the rotational axis fall and beat the inner core.

The inner core oscillates and subsequent shock waves are generated.

§6.2 Gravitational Waves – dependence on EOS -

2.5M M

rig. rot.

20 210 kpc3 10 sin

1000 cmzz xxI I

2.5M M

rig. rot.

20 2046

/ 10kpc5 10 ~ 5 10 at 10kpc

10 erg/Hzeff eff

dE dfh h r

§6.2 Gravitational Waves

2.5M M

rig. rot.

20 210 kpc3 10 sin

1000 cmzz xxI I

2.5M M

rig. rot.

20 2046

/ 10kpc5 10 ~ 5 10 at 10kpc

10 erg/Hzeff eff

dE dfh h r

2.5M M

rig. rot.

20 210 kpc3 10 sin

1000 cmzz xxI I

2.5M M

rig. rot.

Several oscillation modes

§6.2 Dependence on Mass§6.2 Dependence on Mass14

1 2EOS- : ( , , /10 ) (1.3,2.5,2.0)nucb 2/ 1.0q cJ GM

2.2M M

2.7M M

Amplitudes of GW at bounce increase as the inner core mass becomes larger

inner core outer core/M M

Due to large fraction of outer core falling into inner core, the oscillation of inner core is prevented

§6.2 Effect of rotation§6.2 Effect of rotation

2.5M M

The amplitudes of GW increase as the value of q does

141 2EOS- : ( , , /10 ) (1.3,2.5,2.0)nucb

No saturation is observed in the amplitudes

§6.2 Effect of rotation§6.2 Effect of rotation

141 2EOS- : ( , , /10 ) (1.28,2.75,2.0)nucd

2.5M M

No saturation is observed in the amplitudes

In the Newtonian simulation……..In the Newtonian simulation……..

( )zz xx

rotation ↑

| | zz xxI I

∵ quadrupole formura

Mass and radius of inner core at bounce ↑

Yamada and Sato (1994) ApJ 434, 268; (1995) ApJ 450, 245

The amplitude of GW spike saturates for q=0.5

The core less contract due to centrifugal force

The value of h saturates for a certain value of q

In the present simulation…………In the present simulation…………

Mass of the cores are much larger

Gravity is not Newtonian but full general relativity

2bounce bounce1 ( / ) 0.3GM Rc

Due to the stronger gravity, the inner core contract more.

This leads to the higher central density and the more deformation of the inner core.

Thus, no saturation is observed for q<1

Indeed, the modulation of the waveforms is more outstanding in the GR simulation

In rotating stellar core collapse to a black holeIn rotating stellar core collapse to a black hole– Thermal effects (in particular shock) increase the Thermal effects (in particular shock) increase the

threshold mass by 20 threshold mass by 20 ～～ 40 %40 %– Rotational effects increase the threshold at most Rotational effects increase the threshold at most

by 17 by 17 ～～ 20 %20 %– These effects depend sensitively of the equations These effects depend sensitively of the equations

of stateof stateDirect black hole formation and fallback-induced Direct black hole formation and fallback-induced collapsecollapse

– Differential rotation further increases the Differential rotation further increases the thresholdthreshold

Black hole + Disk system Black hole + Disk system – The predicted mass of the disk is at most ~10% of The predicted mass of the disk is at most ~10% of

the initial massthe initial mass– BH excision techniqueBH excision technique

§7 Summary and Discussion

In rotating stellar core collapse to a In rotating stellar core collapse to a neutron starneutron star– The collapse dynamics and the shock The collapse dynamics and the shock

generation and propagation strongly depend generation and propagation strongly depend on EOSon EOS

– ReflectingReflecting this, the gravitational waveforms this, the gravitational waveforms also depend on EOSalso depend on EOS

– No amplitude saturation is shown in the No amplitude saturation is shown in the present simulations ⇔ the previous present simulations ⇔ the previous Newtonian simulationNewtonian simulation

Possibility for onset of dynamical Possibility for onset of dynamical nonaxisymmetric instabilities ?nonaxisymmetric instabilities ?– which occur for T/W > 0.27 or highly differentially which occur for T/W > 0.27 or highly differentially

rotating case (even for T/W < 0.1)rotating case (even for T/W < 0.1)– Unlikely to occur for rigidly and moderately diff. Unlikely to occur for rigidly and moderately diff.

rot. cases rot. cases – Set in during the collapse ?Set in during the collapse ?

Yes, but the conditions for onset is – Initially rapidly rotating, highly differentially

rotating (A<0.1)

– Also necessary is large initial pressure depletion

Γ1< ～ 1.28

Initial

MaxModels of diff.rot.

Models of rigid rot.

－ Dynamical Instability －

Relations of initial and maximum value of T/W

Candidates for the onset of dynamical instabilities

YS and Shibata 05

A=0.1 models

(T/W)_init increases

Candidates/T W

init/ | 0.0127T W

2.5M M

強微分回転

init/ | 0.0177T W

2.5M M

強微分回転

, ,21 1810Mpc 10kpc10 3 10

0.31km 0.1km

Quadrupole formula

Gauge inv.

init/ | 0.0127T W

2.5M M

, ,21 1810Mpc 10kpc10 3 10

0.31km 0.1km

Quadrupole formula

Gauge inv.

init/ | 0.0177T W

2.5M M

22 1847

/ 10Mpc2 10 ~ 3 10 at 10kpc

10 erg/Hzeff eff

dE dfh h r

~ 1 kHz

init/ | 0.0177, 0.0127T W

2.5M M

1.5M M

非軸対称変形により、 Axi. Sym. の～１０倍以上の振幅

§2.1 §2.1 ３＋１３＋１ decomposition (1)decomposition (1) 　　

ab ab a bg n n

Foliation of the spacetime 4D manifold into a family of 3D hypersurface Σ

： lapse function： shift vector

： metric on Σ

a a at n

： unit normal to Σ

timelike coordinate vector

§2.1 §2.1 ３＋１３＋１ decomposition (2)decomposition (2) 　　2 2 2( ) 2k i i j

i ijds dt dtdx dx dx

ab ab a bg n n

： lapse function： shift vector

： 3D metric on Σ

： extrinsic curvature of Σ

Represent gauge freedom

2ab a b n abK n L

which describes how Σ is embedded in the spacetime

Corresponds to the “velocity” of the 3D metric

Dynamical variable is the pair (γ, K)

Cf. York (1979) in Sources of Gravitational Radiation

§2.1 §2.1 ３＋１３＋１ decomposition (3)decomposition (3)

8a b a bab abG n n T n n

8a aab abG n T n

8ab abG T

(3) 2 16ijij hR K K K

( ) 8ij ij ijD K K j

2t ij ijK L

4 [ ( ) 2 ]ij h ijS S 2 ]l

t ij i j ij il j ijK D D R K K KK L

Hamiltonian constraint

momentum constraint

Evolution equation for abK

abEvolution equation for

Definition of abK

a a bbj T n

ab abS T

a bh abT n n

(3) 2 16ijij hR K K K

( ) 8ij ij ijD K K j

2 ]lt ij i j ij il j ijK D D R K K KK L

4 [ ( ) 2 ]ij h ijS S

2t ij ijK L

Decomposed Einstein equations

a bh abT n n a a b

bj T n

ab abS T

Evolution equations

Constraint equations

§2.1 §2.1 ３＋１３＋１ decomposition (4)decomposition (4) 　　

§2.2 Shibata-Nakamura reformulation§2.2 Shibata-Nakamura reformulation

3ij ij ijA e K K

ijkijt kK D D A A K

2 ijijt A L

TF4 2k

ij ij ik jt ij i jA e R D D K A A A L

jkiji kF Add as a independent variable

4ijij e

Decompose geometrical variablesCf. Shibata and Nakamura (1995) PRD 52, 5428

yu-ichirou sekiguchi (univ. of tokyo) 関口　雄一郎

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