Modeling scalar fields consistent with positive mass

Modeling scalar fields

consistent with positive mass

Tetsuya ShiromizuDepartment of Physics, Kyoto University

Yukawa Institute 7th Feb. 2014

Nozawa and Shiromizu, Physical Review D89, 023011(2014)

With Masato Nozawa(KEK)

Content

1. Introduction

2. Positive mass theorem

3. Einstein-scalar system

4. Future issues

1. Introduction

Positive mass theorem

~ Positive mass theorem

Schoen&Yau 1981, Witten 1981, Gibbons et al 1983,…

M ≧ 0

M =0 ⇔ Minkowski/anti-deSitter

for GR, SUGRA, regular spacetimes, energy condition, …

The existence of ground state

Restriction on theories

Scalar potentials consistent with positive mass Boucher 1984, Townsend 1985

)()(2

1 2 URL

22

)(12)(

8)(

W

d

dWU

Cf) SUGRA ,W superpotential

Summary of our work

matterLXKRgxdS ),(24

gX2

1

The cases consistent with positive mass are

(i)

(ii)

22/1 )(12)()(

24

WX

d

dWK

22

)(12)(

8)(

W

d

dWXUXK

actionNozawa & Shiromizu 2014

No cosmological solution

Canonical form with “superpotential”

・strong restriction

・classical stability is automatically guaranteed

2. Positive mass theorem

Back to Witten 1981

Positivity: essence

0 i

i

i

Si dSM

~

di

i )(

d 22||

dT 2

00

2 ||||~ 0

If the energy-momentum tensor satisfies the energy condition,

we can prove the positivity of mass.

spinor :

Rigidity

0

0||||~ 2

00

2 dTM

0 abcdR

Minkowski spacetime

Precisely

iN :

VGiN 2

0 i

i

duNdSN

2

1

vector tonormalunit directed future : u

0 i

dVGTdSNGM i

0

2 8||22

18 0

iV :

≧0 (energy condition)

3. Einstein-scalar system

Nozawa & Shiromizu 2014

Model

matterLXKRgxdS 2),(24

gX2

1

action

KgKT X )(

)()( matterTTG

does not satisfy the (dominant) energy condition in general

Mass expression

0ˆ ,)(ˆ i iA

RgRG 2

1:

FiS :

)(2 ][][ AAAF

iV

uSVGidGM ˆˆ28

Required condition

)(ˆˆ)(

)(2

ˆˆ2ˆ

AAiAAi

FFi

VGiN

)(2 ][][ AAAF

AA

We imposed

Otherwise, non-controllable terms appear

Strategy

spinor: 0ˆ ii

i

SidSM

ˆ~

di

i )ˆ(

dSTT matter 0)(

0000

2|ˆ|

KgKT X )(

2||

Einstein eq.

Look for the theory for scalar field to have the form for

Look at detail more

)(WA

22

22

2

)(12)(

8

W

d

dWfXfK

fKX

d

dWXfXf

)(),(4),(

2

1: 1

)(ˆ A

uSVGidGM ˆˆ28

222222

2

12)(8)(2

1

124

WWfffVi

WVWi

FiS

)(2 ][][ AAAF

VTiS )(

If

Then

22

22

2

)(12)(

8

W

d

dWfXfK

fKX

d

dWXfXf

)(),(4),(

2

1: 1

2

2

)(128

WK

WKXK

X

X

088

2

22

X

XX

X

XXK

WXK

K

WKXK

0)( XXKi

08

)(2

2

XK

WXii

22

)(12)(

8)(

W

d

dWXUXK

22/1 )(12)()(

24

WX

d

dWK

Case (ii)

22/1 )(12)()(

24

WX

d

dWK

For homogeneous-isotropic spacetimes,

)(t 02/2 X

not work does (ii) case the,)( offactor the toDue 2/1X

Summary

matterLXKRgxdS ),(24

gX2

1

(i)

(ii)

22/1 )(12)()(

24

WX

d

dWK

22

)(12)(

8)(

W

d

dWXUXK

actionNozawa & Shiromizu 2014

No cosmological solution

Canonical form with “superpotential”

4. Future issues

Future issues

Extension to more general

cases/modified gravity?

AA

)(WA general enough?

Some basics

Covariant derivative

ˆˆ

ˆˆ

4

1

ˆˆ

ˆˆ

4

1 ,

ˆˆ

ˆˆ

ˆˆ

ˆˆ

ˆˆ

4

1)(

Local Lorentz transformationˆˆ

ˆˆˆˆ

ˆ

ˆ

ˆˆ

ˆ

ˆ

ˆˆ

0:ˆ

ˆˆˆˆ

ˆ

ˆˆˆˆ

eeeeeeD

0 gD

Witten spinor

0 i

iD

],[)()(8

1 ,)( ˆˆ

ˆˆ)1()1(

klj

l

i

jk

i

n

i

n

ii eDeD

ji

l

j

k

ikl eeg ˆ̂ˆˆ )()(

S

cehypersurfa spacelike dim.-1)-(n :),( q

(Witten equation)

0 r

We have solutions which are asymptotically

approaches a constant spinor

Proof

2)1(22 ||4

1||||

2

1 RDDD n

i

i

2)1(22

0 ||4

1||.).(

2

1||8 RDccDdSM ni

iADM

0 0)1(

ADM

n MR

0],[ ],[ 0 0 )1( lk

ijkl

n

jiiADM RDDDM

∑ is flat space

Surface integral

00

00

)1(

11

)1(

0

11

1

)(4

1

)(.).(2

1

j

ji

j

ij

i

A

A

i

i

j

njn

hhdS

dSdSccDdS

01

)1(

0101 nD

0

)1(

101010

0

)1(

10

0

)1(

0)(

0)( 0

i

ni

A

A

i

n

i

i

i

n

i

i

i

iD

1

)1( 1],)[(

16

1njk

j

i

kk

i

j

i

n

rOhh

ki

kjj

i

j

i

ijijij eheehg )(2

1)( )( ,

ˆ)0(ˆ(0)ˆ

1