A recipe for multi-metric gravity

A recipe for
multi-metric gravity
Kouichi Nomura
Department of physics,
Theoretical Astrophysics group
2015/1

contents
1.Introduction
2.Interacting spin-2 fields
3.Ghost in tri-metric gravity
4.Ghost in multi-metric gravity
5.Conclusion
(6.Application)

1. Introduction
In modern physics, we describe phenomena
in terms of elementary particles and their interactions.
classified by mass and spin
Electromagnetism : massless spin-1
Gravity : massless spin-2
We know a lot of types of interactions, but until very recently,
examples of interactions among spin-2 fields had not been known.
Recent developments of massive gravity has opened a way.
・・・・Interaction generates unphysical modes

General relativity
Linearized
general relativity
Linear
massive gravity
Non-linear
massive gravity
Bi-metric gravity
Multi-metric gravity
non-linear
linear
(C. de Rahm,
G. Gabadadze
and A. J. Tolly,
2011)
(S. Hassan and
R. A. Rosen,
2012)
Our study

Linear-massive gravity
  ][det42
gRgxdMS pEH
  hg 
  02
 

 hm
Einstein-Hilbert action :
(In linearized general relativity, we give mass to a graviton)
We consider a perturbation
pM ][gR: Planck mass , : curvature
,














10
1
1
01
 : Minkowski metric
,
 |||| h
)2(42)2(
EHpEH LxdMS 
0 

 h
hhhhhhhhLEH







 
4
1
2
1
2
1
4
1)2(
,
EOM：
We need massive EOM:
We add quadratic terms
to the Lagrangian:
 




 hahhhm  2
4
1 ( m , a : constant )

)(
4
1 2)2( 




 hahhhmLL EHLM 
Linear Massive Gravity :
( m , a : constant )
EOM:   ,02
 

 hm
In the case a=-1:
,0 

h ,0
h
We have 10-4-1=5 degrees of freedom
・・・・・healthy theory
In the case a≠-1:
The trace part is also activated:   0
2
 


 hma
Moreover, this extra 6th mode has negative kinetic energy (ghost):
...... 2
 hhhLLM


In the following, we call such an extra ghost mode as BD-ghost
(although this terminology may be inaccurate)
unphysical

1a
)(
4
1 2)2( 




 hhhhmLL EHLM action :
In order to remove BD-ghost, we have to set
・・・Fiertz-Pauli mass term
(Fiertz and Pauli, 1939)
Non-linear extension of the Fiertz-Pauli mass term has a long history.
How to eliminate the BD-ghost even in the non-linear level is a difficult problem.
a lack of tool to detect a BD-ghost
a wrong proof that a BD-ghost cannot be removed
This problem has been resolved very recently.
(Non-linear mass term has been constructed order by order.)
(C. de Rahm and G. Gabadadze and A. J. Tolly, 2011)

Non-linear (dRGT) massive gravity
  


4
0
142242
det2][det
n
nnpp fgegxdMmgRgxdMS 
action :




fgfg 
)( 1
fgfgfg 111 

fg 1
fg 1
:a matrix with components
: square root defined by
n
nn
n
AAAAen 





 

2
2
1
121
21 ....)( 
Anti-symmetrization symbol
n : free parameters
f : fixed non-dynamical metric
non-linear mass term

Moreover, when the dynamics is given to the fixed
metric the system is proved to contain
7 degrees of freedom.
To prove the absence of the BD-ghost,
we need to show that the system contains 5 degrees of freedom.
By using the Hamiltonian analysis, this is actually proved.
(S. Hassan and R. A. Rosen, 2012)
(S. Hassan and R. A. Rosen, 2012)
,f
one massless graviton (2 DOF) and
one massive graviton (5 DOF)
A healthy theory for interacting two gravitational fields

Bimetric gravity :, fg metrics
  ][det][det 4242
fRfxdMgRgxdMS fgBi
  


4
0
14
det
n
nn fgegxda 
Einstein Hilbert term
Interaction term
:, fg MM :][],[ fRgR :aPlanck mass , coupling constant
Researches on interaction of gravitational fields have a long history.
They had suffered from the BD-ghost, but recent developments of
massive gravity have opened a way.
2+5=7 degrees of freedom
no BD-ghost is contained
}
}
Curvature,

Can we include more gravitational fields ?

After the no-ghost proof of bimetric gravity,
two prominent researches have been published.
1: tri-metric gravity
2 : multi-vielbein gravity
(Nima Khosravi, Nafiseh Rahmanpour, Hamid Reza Sepangi, and Shahab Shahidi (2012))
・A naive extension of bimetric gravity
・Whether or not BD-ghosts are contained is unanswered there.
・A theory of interacting multiple spin-2 fields written only by vielbeins
・It is believed that there is no BD-ghost.
(Kurt Hinterbichler and Rachel A. Rosen (2012))
2.Interacting spin-2 fields

Our first purpose is to resolve
the ghost problem in tri-metric gravity.
Then, investigate more general
multi-metric cases.
But prior to them,
we introduce some notations.

Preparation 1 : graphical expression
g f
a
g f
0a
In the case of bimetric Gravity
  


4
0
14
det
n
nn fgegxda 
n
nn
n
AAAAen 





  2
2
1
121
21
....
)( 
anti-symmetrization
 fg,No interaction between

  ][det][det][det 424242
hRhxdMfRfxdMgRgxdMS hfgTri
       






4
0
1
3
4
0
1
2
4
0
1
1 detdetdet
n
nn
n
nn
n
nn ghehahfefafgega 
g
f h
1a
2a
3a
The originally proposed tri-metric gravity is
(Nima Khosravi, Nafiseh Rahmanpour, Hamid Reza Sepangi, and Shahab Shahidi (2012))
(metrics g,f,h)

g
f h
1a
02 a
3a
We can cut one of interactions
  ][det][det][det 424242
       






4
0
1
3
4
0
1
2
4
0
1
1 detdetdet
n
nn
n
nn
n

4
4
3
3
2
2
1
14321
4321
),,,( 







  DCBADCBA 
anti-symmetrization for matrices A,B,C,D
)(),,,( 4 AeAAAA 
)()1,,,( 3 AeAAA 
)()1,1,,( 2 AeAA 
)()1,1,1,( 1 AeA 
1 : unit matrix
neRelation to
Preparation 2 : anti-symmetrization

Multi-vielbein theory
JI
IJ EEg  I
EVielbein :














10
1
1
01
 : Minkowski metric
)),2(),2(),2(),1(( EEEE
)),2(),2(),1(),1(( EEEE )1(E )2(E
(Kurt Hinterbichler and Rachel A. Rosen (2012))
a box represents a vielbein
・bi-vielbein E(1), E(2) case
))2(),1(),1(),1(( EEEE

))2(),2(),2(),1(( EEEE
))3(),3(),3(),1(( EEEE

)1(E
)2(E )3(E
Examples of tri-vielbein interactions
)1(E
)2(E )3(E

))2(),2(),2(),1(( EEEE
))3(),3(),3(),1(( EEEE
))3(),3(),3(),2(( EEEE

)1(E
)3(E)2(E
))3(),2(),2(),1(( EEEE
We can also consider
They are believed to be ghost-free.


In general, vielbein theories do not
overlap with metric theories.
JI
IJ EEg   )2()2()2( 
 )2(,)1()2()1( ggEE JI
IJ 
JI
IJ EEg   )1()1()1( 
metrics and vielbeins are related as
For instance, if we have
we cannot go to a metric formulation

More precisely, we can show relation such as
)),2()1(()1(det))2(),2(),2(),1(( 1
3 EEegEEEE 

)).2()1(()1(det))2(),2(),1(),1(( 1
2 EEegEEEE 

),2()1()2()1( 11
EEgg 

Therefore, If the system contains a constraint
vielbein theory is translated into the metric language.
This is only when
)1(E
)2(E )3(E
)1(E )2(E or
or

Tri-metric gravity vs Tri-vielbein gravity
g
f h
g
f h
g
f h
)1(E
)2(E )3(E
)1(E
)2(E )3(E
)1(E
)3(E)2(E
Tri-metric Tri-vielbein
Overlap
3.Ghost in tri-metric gravity

g
f h
g
f h
g
f h
)1(E
)2(E )3(E
)1(E
)2(E )3(E
)1(E
)3(E)2(E
No BD-ghostunknown
We investigate

g
f h ),,,,(det 1111
hghgfgfgg 

Here, we included a metric interaction given by
An example is

  ][det][det][det 424242
       






4
0
1
3
4
0
1
2
4
0
1
1 detdetdet
n
nn
n
nn
n
g
f h
1a
2a
3a
・Eventually, we found that
the left pattern cannot exclude a BD-ghost
・To show the existence of a BD-ghost,
we have only to show such a situation
We assume spatial homogeneity
to simplify the analysis

We perform the Hamiltonian analysis
under the spatial homogeneity ansatz
ji
ij dxdxtdttNdxdxg )()( 22

 
ji
ij dxdxtdttLdxdxf )()( 22

 
ji
ij dxdxtdttQdxdxh )()( 22

 
By using spatial coordinate transformation, one of metrics can be diagonalized.
QLN QCLCNCH Hamiltonian :
:,, QLN Lagrange multipliers
0,0,0  QLN CCCPrimary Constraints :
dynamical variables : 15663 

Constraints must be preserved along the time evolution
0},{},{},{  QCCLCCHCC QNLNNN

0},{},{},{  QCCNCCHCC QLNLLL

0},{},{},{  LCCNCCHCC LQNQQQ

3},{ aCCC QNNQ 
,},{ 1aCCC LNNL 
,},{ 2aCCC QLLQ 
0
0
0
0























Q
L
N
CC
CC
CC
LQNQ
LQNL
NQNL Lagrange multipliers
are determined
from the left equation
We can find

g
f h
0,0  NQNL CC },{, QNLQNL CCC Secondary Constraints :
・one of Lagrange multiplies is left undetermined (one gauge freedom)
・In total, we have 3+2=5 constraints
The total number of
degrees of freedom : 122/)15215( 
masless graviton : １
massive graviton : ２
BD ghost：0
0,,,  QNLLNLNNLNL QCLCNCdtdC
0,,,  QNQLNQNNQNQ QCLCNCdtdC
Consistency condition
(for comparison)
0
00
00
0























Q
L
N
C
C
CC
NQ
NL
NQNL
N
L Q

Constraints ：３
Gauge fixing ：１
132/)13215( 
Masless graviton : １
Massive graviton : ２
BD ghost : １
g
f h
No secondary constraints
,Q
C
C
N
NL
LQ
 ,Q
C
C
L
NL
NQ
 Q : undetrmined
The total number of
degrees of freedom :
N
L Q
0
0
0
0























Q
L
N
CC
CC
CC
LQNQ
LQNL
NQNL

g
f h
The structure is the same as that in the previous case
132/)13215( 
Masless graviton : １
Massive graviton : ２
BD ghost : １
The total number of
0
0
0
0























Q
L
N
CC
CC
CC
LQNQ
LQNL
NQNL
Constraints ：３
Gauge fixing ：１

g
f h
g
f h
g
f h
)1(E
)2(E )3(E
)1(E
)2(E )3(E
)1(E
)3(E)2(E
No BD-ghostBD-ghost exists
Conclusion in tri-metric case
Overlap

4.ghost in multi-metric gravity
metrics :
vielbeins :
nggg ,......,, 21
)(),......,2(),1( nEEE
We generalize tri-metric gravity
to more general multi-metric gravity
We include n metrics
and n vielbeins
Metric theories and vielbein theories coincide only partially.

Multi-metric Multi-vielbein
No BD-ghostunknown
1g
2g
3g
4g
ng
1g
2g
3g
4g
ng
1g
2g ng
)1(E
)(nE)2(E
)1(E
)2(E
)3(E
)4(E
)(nE
)1(E
)2(E
)3(E
)4(E
)(nE
Overlap

Moreover, we have
2g
ng1g
3g
1E
2E 3E
nE
Overlap

Under the spatial homogeneity ansatz
,)()( 22 ji
ijI dxdxIdtNdxdxIg 
 
nnCNCNCNH  2211the Hamiltonian is
n-primary constraints : 0,,0,0 21  nCCC 
Consistency conditions :
0},{
1
 
n
J
JIJII NCHCC
},{ JIIJ CCC 

these patterns can exclude BD-ghosts
1g
2g
3g
4g
ng
2g
ng1g
3g
0
0
0
2
1
1
12
112































nn
n
N
N
N
C
C
CC


0,,0 112  nCC (n-1)-secondary constraints
Consistency conditions 0},{},{
1
111  
n
J
JJIII NCCHCC
One of Lagrange multipliers is left undetermined
)1(52
2
1)1())1(63(2


n
nnn Masless graviton : １
Massive graviton : n-1
BD ghost : 0
The total number of

below patterns contain BD-ghosts
1g
2g
3g
4g
ng
1g
2g ng
0
























JIJ NC
anti-symmetric matrix ・・・・rank must be even number
When n=odd, rank=n-1
There is no secondary constraint, and one of Lagrange multipliers is left undetermined.
2
1
)1(52
2
1))1(63(2 

 n
n
nnThe total number of
BD-ghosts

2/)1( nthe number
of BD ghosts:
(n:odd）
2/)2( n (n:even）
In any case
When n=even, rank=n
We have a secondary constraint ,0det IJC
0},{detdet  
I
IIIJ NCCC
dt
d
Which reduces the rank to n-2.
Two Lagrange multipliers are left undetermined.
But, we must further impose the consistency condition on the secondary constraint
Only one Lagrange multiplier is left undetermined.
The total number of
degrees of freedom : 2
2
)1(52
2
11))1(63(2 

 n
n
nn
BD-ghosts

5.conclusion
In multi-metric gravity,
only tree type interaction is allowed
Loop is forbidden

6.Applications
There are a large number of applications of multi-metric gravity
(especially bimetric gravity)
For example,
Gravitational waves, inflationary universe, ……etc
Here, we consider an application
to the AdS/CFT correspondence

the AdS/CFT correspondnce
One kind of holography between
(d+1)-dimensional gravity theory ↔ d-dimensional matter field theory
A lot of examples are known, but the most classic one is
five-dimensional general relativity ↔ four-dimensional Yang-Mills theory
Especially, in the first order hydrodynamic limit (derivative expansion),
we can easily calculate the transport coefficients such as sheer viscosity.
We can investigate complicated (quantum) matter field theory,
thorough the rather simple (classical) gravity theory
We consider an extension of this method to the case of bimetric gravity

(Schwarzschild AdS Black-Hole)
the background metric is set to be
0u
1u
:AdS-boundary
:Black Hole Horizon
Lr ,0
: constant
u=0: AdS-Boundary (t,x,y,z)
Asymptotic AdS
Space Time
u
u=1: Black Hole Horizon (t,x,y,z)
We consider five-dimensional pure general relativity
The case of general relativity (a short review)

We take a perturbation
We substitute the solution into the original action
: the value on the AdS boundary
and solve the EOM
......
4
)( 4)0(
0
2
)0(
 u
r
Li
u  


Fourier transform (t → ω)
)0()0(
 u

Through the AdS/CFT prescription (GKP-Witten relation), we obtain
the (perturbed) energy-momentum tensor for the boundary field theory
On the other hand, if the boundary space-time
is slightly distorted from the flat background
the linear response of the energy momentum tensor can be written as
Therefore,
we conclude that
pressure：
sheer viscosity:
and the ratio to the entropy density is
from the background( )

the case of dRGT massive gravity
(16πG=1,L=1)
mass term is
introduced
g : fixed background
given outside the theory
In linear level

n
nn AeAe )()( 

We take a perturbation
and solve the EOM
Fourier transform (t → ω)
 BA , :ω dependent coefficient,
This solution is substituted to the action, and we encounter divergences

In order to remove the divergence,
we introduce a new counter term
which reduces to the Fiertz-Pauli form in the linear level
Then, the leading divergence is removed
but other divergent terms remain.
We eliminate them by the BF-bound like condition ( )
and obtain the finite on-shell action

We fix the remaining constant  BA ,
the solution of the EOM should coincide with that of general relativity in the massless limit.
by the condition that
Then, the action is
and the energy momentum tensor
for the boundary field is
Comparing to the linear response formula,
we find that the pressure is zero .
However, the pressure can be calculated from the background metric
which contradicts with our result.
:Euclidean on-shell action
ES

We give dynamics to the fixed background
the case of bimetric gravity
Bimetric gravity: two metrics ｇ、ｆ
gG
],[][][],[ int fgSfSgSfgS GRGRBi 
Each sector has the same set ,
Gravitational constant and fG
（one massless graviton +one massive graviton）

We take a perturbation around Schwartzschild AdS BH
We encounter additional divergences, which can be canceled by
introducing a new counterterm and recalling the BF-bound

We solve the EOM and obtain the on-shell action
We interpret the result as two-component fluid

To interpret this result, we assume that there are two AdS-boundaries at u=0,
which correspond to metric g and f respectively. Focusing on the boundary
for g, the field sourced by φ has the energy momentum tensor
and the field sourced by ψ has the energy momentum tensor
we are focusing on the boundary not for fsourced by φ
xy
T )(
xy
T )(

xy
T )(
xy
T )(
We compare
to the linear response formula
Pressure Sheer viscosity
)(T
)(T

Total pressure coincides with the value calculated from the
background
＋＝

s
 ,
On the other hand
,
s

if

summary
However, what we studied is only the simplest setting.
Further investigation is needed to clarify the boundary theory
We attempted an application of bimetric gravity
to the AdS/CFT correspondence.
In massive gravity, how to interpret the result was unclear,
but in bimetric gravity, the result suggested
the emergence of two component fluid.

A recipe for multi-metric gravity

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