modeling scalar fields consistent with positive mass
TRANSCRIPT
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