On Exact Solutions

and

the Consistency of 3D Minimal Massive Gravity

Emel Altas, Bayram Tekin

Phys. Rev. D 92, 025033 (2015)

25.05.2016

Introduction

Research Problem/Open Questions

1) Consistency of Field Equations

A)Source Free Case

B)Matter Coupled Case

2) Constant Scalar Curvature Solutions

A)Type-N

B)Type-D

Conclusion

References

INTRODUCTION

TMG [1] modifies the field equations of general relativity by

adding a new term with three derivatives.

The TMG action is:

The TMG field equations derived from this action as:

for generic values of the parameters the corresponding equations of

motion have several solutions including AdS, BTZ .

For TMG, the central charge of a holographically dual conformal

field theory (CFT) is negative whenever the bulk spin-2 mode has

positive energy, implying a non-unitary CFT.

[1] S. Deser, R. Jackiw and S. Templeton;’’Topologically Massive Gauge Theories’’, Ann. Phys. (N.Y.) 140 372 (1982)

The field equation of MMG [2] includes the additional,curvature squared, symmetric tensor-J.

The MMG field equation is:

MMG field equations can not be obtain from metric formalism.

In contrast to TMG, MMG has both a positive energy graviton and positive central charges for the asymptotic AdS-boundary conformal

algebra.

[2] E. Bergshoeff, O. Hohm, W. Merbis, A. J. Routh and P. K. Townsend, “Minimal Massive 3D Gravity,” Class. Quant. Grav. 31, 145008 (2014).

Consistency of the field equations requires that the first divergence

vanishes but from direct substitution we get

which means the MMG field equations does not obey the Bianchi

Identity and therefore cannot be obtained from an action with the metric

being the only variable.

But the covariant divergence vanishes for metrics that are solutions

to the full MMG equations.

Therefore, one has an ”on-shell Bianchi Identity”.

RESEARCH PROBLEM / OPEN QUESTIONS

The questions we aim to research are as follows:

Are the field equations of MMG consistent for both source-free

and matter-coupled cases?

Are there any constant scalar curvature solutions for the theory?

Is it possible to find a relation between constant scalar curvature

solutions of MMG and TMG?

1) CONSISTENCY OF MMG FIELD

EQUATIONS

A)SOURCE FREE CASE

This is necessary condition for the consistency of the classical field

equations but not a sufficient condition, since the the rank- two tensor

equations are susceptible to double-divergence.

When we calculate double divergence we get,

We show that for the source-free case the double-divergence of the field

equations vanish for the solutions of the field equation.

Einstein tensor can written in terms of Schouten tensor as

And we can write Cotton tensor in terms of Einstein tensor J-tensor, R and metric tensor by using MMG field equation.

By using,

For the source free case MMG field equations are consistent.

B)MATTER-COUPLED CASELet us now consider the consistency of matter-coupled MMG

equations [3].

Given a covariantly conserved energy-momentum tensor ,the field equations become :

where the source term reads,

[3] A. S. Arvanitakis, A. J. Routh and P. K. Townsend, “Matter coupling in 3D minimal massive gravity,” Class. Quant. Grav. 31, no. 23, 235012 (2014)

For the consistency of the matter-coupled MMG, one should require

the covariant divergence of the left-hand side and the right-hand side

to be equal to each other when the field equations are used.

So first condition for the consistency of the matter coupled case is

satisfied ,this is necessary but not sufficient and one should also

check the double divergence.

RHS

LHS

By using the equalities

The double divergence of the left hand-side and the right hand side

of the field equations are equal to each other on shell.

For the matter-coupled case MMG field equations are

consistent.

2)CONSTANT SCALAR CURVATURE

SOLUTIONS

In three dimensions, classification of space-times can be done

either using the

Cotton-tensor (analogous to the four dimensional Petrov

classification )

or using the;

Traceless Ricci tensor (analogous to the four dimensional Segre

classification ).

For Segre classification, one needs the following two scalar curvature

invariants:

To search for solutions , let us rewrite the source-free field equations

as a trace part and a traceless part.

The trace part of TMG equations simply says that:

while the traceless part reads :

trace part of MMG equations

and traceless part

where the traceless part of the J-tensor is

A)TYPE-N For Type-N space-times traceless Ricci tensor can be written as

where ρ is a scalar function which will not play a role and is a null vector: For Type-N space-times, since and,

,from the trace part of the MMG field equations, Ricci scalar is constant with two possible values.

And the traceless part of the field equation is:

which is nothing but the field equations of TMG with the simple replacement of the parameters as

Type-N solutions of TMG are also solutions of the MMG with

the change in the parameters.

Let us give an example of Type-N solution [4] which is locally equivalent to most Type-N solutions of TMG, including the AdS-ppwave solutions [5]

this is a solution to TMG for arbitrary functions fi(u). This solution also solves MMG after the replacement

[4] S. Olmez, O. Sarioglu and B. Tekin, “Mass and angular momentum of asymptotically ads or flat solutions in the topologically massive gravity,” Class. Quant. Grav. 22, 4355 (2005)

[5] D. D. K. Chow, C. N. Pope and E. Sezgin, “Classification of solutions in topologically massive gravity,” Class. Quant. Grav. 27, 105001 (2010).

6

for Type-N space-t imes can be writ ten as [11]

Rµν = ρξµξν , (36)

where ρ is a scalar function which will not play a role and ξµ is a null vector: ξµξµ = 0. For Type-N space-t imes,

since I 1 = 0, from (28) one concludes that the Ricci scalar is constant with two possible values

R± =12µ

γ(µσ̄ ± m), m ≡ µ2σ̄2 − γΛ̄0. (37)

Note that m = 0 point is a special point ( ”merger point” [2]) where two roots coalesce and needs seperate attention,which we note below. The trace-free part of the J -tensor becomes

Jµν = −1

12RRµν , (38)

reducing the MMG field equations to

1

µCµν + σ̄ −

γR

12µ2Rµν = 0, (39)

which is nothing but the field equations of TMG (31) with the simple replacement of the parameters as

µσ̄→µσ̄ −γR

12µ. (40)

Hence all Type-N solut ions of TMG solve MMG once this replacement is taken into account together with the values6Λ = R± . For m = 0, observe that the traceless part of the MMG equation simply reduces to the vanishing of theCotton tensor since R = 12µ2σ̄/ γ at this point . So all such solut ions are conformally flat spaces.

Let us give an example of Type-N solut ion [25] which is locally equivalent to most Type-N solut ions of TMG,including the AdS-pp wave solut ions [21]

ds2 = dρ2 + e2ρ/ ℓdudv + e(1/ ℓ+ µσ̄)ρf 1(u) + e2ρ/ ℓ f 2(u) + f 3(u) du2 (41)

with R = − 6/ ℓ2, this is a solut ion to TMG for arbit rary funct ions f i (u). This solut ion also solves MMG after thereplacement (40). At the merger point, one can show that (41) becomes a conformally flat metric but not an Einsteinmetric.

B . T ype-D solut ions

Depending on the t ime-like or space-like nature of the eigenvectors of the traceless Ricci tensor, Type-D solut ionssplit into two as Type-D t and Type-Ds [20] and both types have the traceless Ricci tensor as

Rµν = p gµν −3

aξµξν , (42)

where ξµξµ ≡ a = ± 1 and p is a scalar funct ion. If the following equation is satisfied [20]

∇ µξν =µσ̄

3ηµνρξ

ρ, (43)

then TMG equat ion is solved with a constant p, as long as

p =1

9µ2σ̄2 +

Λ̄0

σ̄(44)

For both Type-D cases, one has

RµρRνρ = p2 gµν +

3

aξµξν , I 1 = 6p2, I 2 = − 6p3. (45)

B)TYPE -D

Type-D solutions split into two as Type-Dt and Type-Ds and both types

have the traceless Ricci tensor as

Where and p is a scalar function. The traceless part of

the J-tensor becomes and since from the

trace part of field equations again we have constant curvature scalar

with two possible solutions:

reducing the MMG equation to the TMG equation as,

Type-D solutions of TMG are also solutions of the MMG with the

change in the parameters.

Let us now give two examples of such solutions (time-like squashed AdS3 and space-like squashed AdS3)

For these two solutions of TMG also solve MMG with the replacement of the parameters:

Also the general Kundt solution of TMG reported in [6] also solves MMG.

[6] D. D. K. Chow, C. N. Pope and E. Sezgin, “Kundt spacetimes as solutions of topologically massive gravity,” Class. Quant. Grav. 27, 105002 (2010)

7

Then the two roots of the trace-part of the field equat ion become

R± =12µ

γ(µσ̄ ± M ), M ≡ µ2σ̄2 − γΛ̄0 +

γ2p2

µ2. (46)

The new merger point is given by M = 0 which is generically sat isfies by two possible Λ0’s. On the other hand, thetraceless part of the J -tensor becomes

J̃µν = − p +R

12Rµν , (47)

reducing the MMG equat ion to the TMG equat ion as

1

µCµν + σ̄ −

γ

µ2(p +

R

12) Rµν = 0, (48)

which means all Type-D solut ions of TMG solve MMG once the following replacement is made

µσ̄→ µσ̄ −γ

µ(p +

R

12) = µσ̄ −

γ

µ

1

9µ2σ̄2 +

Λ̄0

σ̄+

R

12. (49)

Let us note that at the merger point and for the part icular value of γσ̄ = − 9, the traceless part of the MMG equat ionbecomes the sign-reversed version

1

µCµν − σ̄Rµν = 0, (50)

which in the TMG language refers to a change of helicity from + 2 to -2 keeping the mass intact .Let us now give two examples of such solut ions. In [21], almost all Typ-D solut ions in the literature were shown to

be locally equivalent to the time-like squashed AdS3

ds2 =λ2 − 4

2R− λ2(dτ + coshθdφ)2 + dθ2 + sinh2 θdφ2 , (51)

or the space-like squashed AdS3

ds2 =λ2 − 4

2R− cosh2 ρdτ 2 + dρ2 + λ2(dz + sinhρdτ )2 , (52)

with the squashing parameter λ, which for TMG reads

λ2 =8σ̄2µ2

2σ̄2µ2 − 9R. (53)

For these two solut ions of TMG to also solve MMG, the squashing parameter changes according to to the replacementrecipe (49) which we do not depict explicit ly as it is clear. For the sake of completeness, let us note that for (51), onefinds the traceless part of the J -tensor as

Jµν = −R

4

(3λ2 − 4)

(λ2 − 4)Rµν , (54)

while the square of the Cot ton tensor reads

CµνCµν =12R3λ2(λ2 − 1)2

(λ2 − 4)3, (55)

for λ = 1 one has the round AdS3 metric and λ = 2 corresponds to the flat space.Finally, let us note that the following restricted version of the general Kundt solut ion of TMG reported in [26]

ds2 = 2dudv +1

2R −

1

9µ2σ̄2 v2du2 + dρ+

2

3µσ̄vdu

2+ du2, (56)

also solves MMG since the traceless part of the J -tensor reads

Jµν = −1

4R +

4

9µ2σ̄2 Rµν . (57)

7

Then the two roots of the trace-part of the field equat ion become

R± =12µ

γ(µσ̄ ± M ), M ≡ µ2σ̄2 − γΛ̄0 +

γ2p2

µ2. (46)

The new merger point is given by M = 0 which is generically sat isfies by two possible Λ0’s. On the other hand, thet raceless part of the J -tensor becomes

J̃µν = − p +R

12Rµν , (47)

reducing the MMG equat ion to the TMG equat ion as

1

µCµν + σ̄ −

γ

µ2(p +

R

12) Rµν = 0, (48)

which means all Type-D solut ions of TMG solve MMG once the following replacement is made

µσ̄→ µσ̄ −γ

µ(p +

R

12) = µσ̄ −

γ

µ

1

9µ2σ̄2 +

Λ̄0

σ̄+

R

12. (49)

Let us note that at the merger point and for the part icular value of γσ̄ = − 9, the traceless part of the MMG equat ionbecomes the sign-reversed version

1

µCµν − σ̄Rµν = 0, (50)

which in the TMG language refers to a change of helicity from + 2 to -2 keeping the mass intact .Let us now give two examples of such solut ions. In [21], almost all Typ-D solut ions in the literature were shown to

be locally equivalent to the t ime-like squashed AdS3

ds2 =λ2 − 4

2R− λ2(dτ + coshθdφ)2 + dθ2 + sinh2 θdφ2 , (51)

or the space-like squashed AdS3

ds2 =λ2 − 4

2R− cosh2 ρdτ 2 + dρ2 + λ2(dz + sinhρdτ )2 , (52)

with the squashing parameter λ, which for TMG reads

λ2 =8σ̄2µ2

2σ̄2µ2 − 9R. (53)

For these two solut ions of TMG to also solve MMG, the squashing parameter changes according to to the replacementrecipe (49) which we do not depict explicit ly as it is clear. For the sake of completeness, let us note that for (51), onefinds the traceless part of the J -tensor as

Jµν = −R

4

(3λ2 − 4)

(λ2 − 4)Rµν , (54)

while the square of the Cotton tensor reads

CµνCµν =12R3λ2(λ2 − 1)2

(λ2 − 4)3, (55)

for λ = 1 one has the round AdS3 metric and λ = 2 corresponds to the flat space.Finally, let us note that the following restricted version of the general Kundt solut ion of TMG reported in [26]

ds2 = 2dudv +1

2R −

1

9µ2σ̄2 v2du2 + dρ+

2

3µσ̄vdu

2+ du2, (56)

also solves MMG since the traceless part of the J -tensor reads

Jµν = −1

4R +

4

9µ2σ̄2 Rµν . (57)

7

Then the two roots of the trace-part of the field equat ion become

R± =12µ

γ(µσ̄ ± M ), M ≡ µ2σ̄2 − γΛ̄0 +

γ2p2

µ2. (46)

The new merger point is given by M = 0 which is generically sat isfies by two possible Λ0’s. On the other hand, thetraceless part of the J -tensor becomes

J̃µν = − p +R

12Rµν , (47)

reducing the MMG equat ion to the TMG equat ion as

1

µCµν + σ̄ −

γ

µ2(p +

R

12) Rµν = 0, (48)

which means all Type-D solut ions of TMG solve MMG once the following replacement is made

µσ̄→ µσ̄ −γ

µ(p +

R

12) = µσ̄ −

γ

µ

1

9µ2σ̄2 +

Λ̄0

σ̄+

R

12. (49)

Let us note that at the merger point and for the part icular value of γσ̄ = − 9, the traceless part of the MMG equat ionbecomes the sign-reversed version

1

µCµν − σ̄Rµν = 0, (50)

which in the TMG language refers to a change of helicity from + 2 to -2 keeping the mass intact .Let us now give two examples of such solut ions. In [21], almost all Typ-D solut ions in the literature were shown to

be locally equivalent to the t ime-like squashed AdS3

ds2 =λ2 − 4

2R− λ2(dτ + coshθdφ)2 + dθ2 + sinh2 θdφ2 , (51)

or the space-like squashed AdS3

ds2 =λ2 − 4

2R− cosh2 ρdτ 2 + dρ2 + λ2(dz + sinhρdτ )2 , (52)

with the squashing parameter λ, which for TMG reads

λ2 =8σ̄2µ2

2σ̄2µ2 − 9R. (53)

For these two solut ions of TMG to also solve MMG, the squashing parameter changes according to to the replacementrecipe (49) which we do not depict explicit ly as it is clear. For the sake of completeness, let us note that for (51), onefinds the traceless part of the J -tensor as

Jµν = −R

4

(3λ2 − 4)

(λ2 − 4)Rµν , (54)

while the square of the Cot ton tensor reads

CµνCµν =12R3λ2(λ2 − 1)2

(λ2 − 4)3, (55)

for λ = 1 one has the round AdS3 metric and λ = 2 corresponds to the flat space.Finally, let us note that the following restricted version of the general Kundt solut ion of TMG reported in [26]

ds2 = 2dudv +1

2R −

1

9µ2σ̄2 v2du2 + dρ+

2

3µσ̄vdu

2+ du2, (56)

also solves MMG since the traceless part of the J -tensor reads

Jµν = −1

4R +

4

9µ2σ̄2 Rµν . (57)

7

Then the two roots of the trace-part of the field equat ion become

R± =12µ

γ(µσ̄ ± M ), M ≡ µ2σ̄2 − γΛ̄0 +

γ2p2

µ2. (46)

The new merger point is given by M = 0 which is generically sat isfies by two possible Λ0’s. On the other hand, thetraceless part of the J -tensor becomes

J̃µν = − p +R

12Rµν , (47)

reducing the MMG equat ion to the TMG equat ion as

1

µCµν + σ̄ −

γ

µ2(p +

R

12) Rµν = 0, (48)

which means all Type-D solut ions of TMG solve MMG once the following replacement is made

µσ̄→ µσ̄ −γ

µ(p +

R

12) = µσ̄ −

γ

µ

1

9µ2σ̄2 +

Λ̄0

σ̄+

R

12. (49)

Let us note that at the merger point and for the part icular value of γσ̄ = − 9, the traceless part of the MMG equat ionbecomes the sign-reversed version

1

µCµν − σ̄Rµν = 0, (50)

which in the TMG language refers to a change of helicity from + 2 to -2 keeping the mass intact .Let us now give two examples of such solut ions. In [21], almost all Typ-D solut ions in the literature were shown to

be locally equivalent to the t ime-like squashed AdS3

ds2 =λ2 − 4

2R− λ2(dτ + coshθdφ)2 + dθ2 + sinh2 θdφ2 , (51)

or the space-like squashed AdS3

ds2 =λ2 − 4

2R− cosh2 ρdτ 2 + dρ2 + λ2(dz + sinhρdτ )2 , (52)

with the squashing parameter λ, which for TMG reads

λ2 =8σ̄2µ2

2σ̄2µ2 − 9R. (53)

For these two solut ions of TMG to also solveMMG, the squashing parameter changes according to to the replacementrecipe (49) which we do not depict explicit ly as it is clear. For the sake of completeness, let us note that for (51), onefinds the traceless part of the J -tensor as

Jµν = −R

4

(3λ2 − 4)

(λ2 − 4)Rµν , (54)

while the square of the Cotton tensor reads

CµνCµν =12R3λ2(λ2 − 1)2

(λ2 − 4)3, (55)

for λ = 1 one has the round AdS3 metric and λ = 2 corresponds to the flat space.Finally, let us note that the following restricted version of the general Kundt solut ion of TMG reported in [26]

ds2 = 2dudv +1

2R −

1

9µ2σ̄2 v2du2 + dρ+

2

3µσ̄vdu

2+ du2, (56)

also solves MMG since the traceless part of the J -tensor reads

Jµν = −1

4R +

4

9µ2σ̄2 Rµν . (57)

7

Then the two roots of the trace-part of the field equation become

R± =12µ

γ(µσ̄ ± M ), M ≡ µ2σ̄2 − γΛ̄0 +

γ2p2

µ2. (46)

The new merger point is given by M = 0 which is generically sat isfies by two possible Λ0’s. On the other hand, thetraceless part of the J -tensor becomes

J̃µν = − p +R

12Rµν , (47)

reducing the MMG equation to the TMG equation as

1

µCµν + σ̄ −

γ

µ2(p +

R

12) Rµν = 0, (48)

which means all Type-D solut ions of TMG solve MMG once the following replacement is made

µσ̄→ µσ̄ −γ

µ(p +

R

12) = µσ̄ −

γ

µ

1

9µ2σ̄2 +

Λ̄0

σ̄+

R

12. (49)

Let us note that at the merger point and for the part icular value of γσ̄ = − 9, the traceless part of the MMG equationbecomes the sign-reversed version

1

µCµν − σ̄Rµν = 0, (50)

which in the TMG language refers to a change of helicity from + 2 to -2 keeping the mass intact.Let us now give two examples of such solut ions. In [21], almost all Typ-D solut ions in the literature were shown to

be locally equivalent to the t ime-like squashed AdS3

ds2 =λ2 − 4

2R− λ2(dτ + coshθdφ)2 + dθ2 + sinh2 θdφ2 , (51)

or the space-like squashed AdS3

ds2 =λ2 − 4

2R− cosh2 ρdτ 2 + dρ2 + λ2(dz + sinhρdτ )2 , (52)

with the squashing parameter λ, which for TMG reads

λ2 =8σ̄2µ2

2σ̄2µ2 − 9R. (53)

For these two solut ions of TMG to also solveMMG, the squashing parameter changes according to to the replacementrecipe (49) which we do not depict explicit ly as it is clear. For the sake of completeness, let us note that for (51), onefinds the traceless part of the J -tensor as

Jµν = −R

4

(3λ2 − 4)

(λ2 − 4)Rµν , (54)

while the square of the Cotton tensor reads

CµνCµν =12R3λ2(λ2 − 1)2

(λ2 − 4)3, (55)

for λ = 1 one has the round AdS3 metric and λ = 2 corresponds to the flat space.Finally, let us note that the following restricted version of the general Kundt solut ion of TMG reported in [26]

ds2 = 2dudv +1

2R −

1

9µ2σ̄2 v2du2 + dρ+

2

3µσ̄vdu

2+ du2, (56)

also solves MMG since the traceless part of the J -tensor reads

Jµν = −1

4R +

4

9µ2σ̄2 Rµν . (57)

CONCLUSION

We checked consistency of the field equations of MMG for both

source free and matter coupled cases. We show that for the source-free

case the double-divergence of the field equations vanish for the

solutions of the field equation. In the matter-coupled case, we show

that the double-divergence of the left-hand side and the right-hand side

are equal to each other for the solutions of the theory. This

construction proves the consistency of the field equations.

We have also found a large class of solutions to MMG equations

that are also solutions to the TMG equations. These solutions have

constant scalar curvature. Type- N solutions and Type-D solutions in

the Segre classification. We provided some explicit metrics that are

called squashed AdS3 .

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