zotero-db/storage/VJ9GMQYU/.zotero-ft-cache

arXiv:2109.03700v1 [gr-qc] 8 Sep 2021

Gravity with a Dynamical Spinning Aether
Christopher Kohler∗
Einstein-aether theory is extended by allowing for spinning degrees of freedom of the aether. In addition to the acceleration, shear, expansion, and vorticity of the aether velocity field, a spin rotation describing the dynamics of a classical intrinsic angular momentum of the aether is introduced as a kinematic quantity. The action of Einstein-aether theory is augmented by a term quadratic in the spin rotation and by coupling terms with the vorticity and the acceleration. Besides breaking the Lorentz boost invariance, the theory breaks the invariance under spatial rotations in the direction of the aether velocity. In the weak field limit, there is a linear relationship between the spin rotation, the vorticity, and the acceleration. Linearized wave solutions correspond to the ones of Einstein-aether theory where the speeds of the spin 0 and spin 1 mode are modified. The extension of Einstein-aether theory has a natural formulation in the framework of a teleparallel geometry where the kinematic quantities become torsion fields.
I. INTRODUCTION
Einstein-aether theory is a theory of gravity where a dynamical unit timelike vector field — the aether — is coupled to general relativity [1–3]. The vector field can be regarded as a kind of preferred frame violating local Lorentz invariance of the theory in that the symmetry under local Lorentz boosts is broken while the symmetry under local spatial rotations is retained. In its original formulation, the interpretation of the vector field as a four-velocity in Einstein-aether theory is achieved by a unit norm constraint imposed by a Lagrange multiplier in the action functional thus implying a spontaneous Lorentz boost symmetry breaking. If, however, the vector field in the action is implicitly assumed to be timelike and of unit norm, the approach represents a semi-tetrad formulation and the symmetry under Lorentz boosts is explicitly broken.
The timelike vector field can be seen as a fluid existing everywhere in space-time. In this work, we will consider an extension of Einstein-aether theory that is motivated by the physics of spin-fluids [4, 5]. The description of spin in this paper is purely classical. It is considered an intrinsic degree of freedom that is not quantized. The mathematical formulation of the spinning aether makes it necessary to introduce a spatial triad with respect to which the spin is fixed. In the same way as the aether in Einstein-aether theory is solely given by its four-velocity, the spinning aether will additionally be described by its triad orthonormal to the velocity. Hence, the spinning aether is described by a tetrad.
The invariance of the theory under local spatial rotations will be retained as far as possible. For this reason, we avoid the appearance of derivatives of the triad in the spatial directions in the action functional. Only derivatives of the triad in the direction of the aether velocity will be allowed which means that only the invariance under spatial rotations in this direction will be broken. In this way, isotropy of space is preserved since there is no preferred spatial direction.
In a formulation within Riemannian geometry, the aether is not a pure geometrical object. The geometrical objects of Riemannian geometry — the metric, the Christoffel symbols, and the curvature tensor — are Lorentz invariant; the Lorentz invariance breaking aether is an additional degree of freedom. Nevertheless, a pure geometrical formulation of gravity with an aether is possible using non-Riemannian geometry. A preferred frame defines a distant parallelism in that vectors with constant components with respect to the frame are considered parallel. This motivates the formulation of theories of gravity with preferred frames using teleparallel gravity. If the spatial orientation of the frames are not fixed, the natural geometry corresponds to a partial parallelization of space-time.
This paper is organized as follows. In section II, the extended version of Einstein-aether theory is introduced. The kinematic quantities of the spinning aether are defined and the action functional is chosen. The field equations are derived in section III and it is shown in which way they can be simplified using constraint equations. Section IV examines the weak field limit of the field equations. The aether excitations are explored using analogies with Maxwell’s equations. In section V, the extended Einstein-aether theory is formulated as a (semi-)teleparallel theory of gravity. The geometry is defined and a procedure to find action functionals is described.
We use the following conventions: Greek indices µ, ν, ρ, ... with the range 0, 1, 2, 3 denote space-time indices. Latin indices a, b, c, ... with the range 0, 1, 2, 3 are internal indices. Spatial indices in the range 1, 2, 3 are denoted by latin
∗ Materials Testing Institute University of Stuttgart, Pfaffenwaldring 32, 70569 Stuttgart, Germany; E-mail address: christopher.kohler@mpa.uni-stuttgart.de

2
indices from the middle of the alphabet, i, j, k, .... The metric signature is (+, −, −, −). Symmetrization is denoted by round brackets, antisymmetrization by square brackets. The totally antisymmetric pseudotensor is εµνρσ. Units are chosen in which c = 1.

II. ACTION FUNCTIONAL

We assume that the preferred frame is given by a tetrad eaµ where the four velocity of the aether is uµ = e0µ. The tetrad is assumed to be orthonormal which means that the space-time metric is given by gµν = eaµebν ηab where eaµ is the cobasis defined by eaµeaν = δνµ and ηab = diag(1, −1, −1, −1) is the Minkowksi metric.
The spatial projection tensor determined by the tetrad is given by

hµν = δνµ − uµuν = eiµeiν .

(1)

The covariant derivative corresponding to the metric gµν will be denoted by Dµ. Using the covariant derivative, we can compute the space-time components of the Ricci rotation coefficients [6],

Cµνρ = eaµDρeaν ,

(2)

which measure the deviation of the tetrad from being inertial. Given the velocity field, kinematic quantities can be defined which are projections of Cµνρ. The acceleration aµ is
given by

aµ = Duuµ = uρDρuµ = −Cµνρuν uρ.

(3)

The expansion θ is defined as

θ = Dρuρ = Cρµν uρhµν .

(4)

The shear tensor σµν being the trace-free part of the deformation tensor is given by

σµν

=

hρµhσν D(ρuσ)

−

1 3

hµν

θ

=

D(µuν)

− u(µaν)

−

1 3

hµν

θ

= Cρστ uρ

hσ(µ hτν )

−

1 3

hµν

hστ

.

(5)

The vorticity tensor ωµν is defined as

ωµν = hρµhσν D[ρuσ] = ∂[µuν] − u[µaν]

= −Cρστ uρhσ[µhτν].

(6)

The kinematic quantities associated with the four velocity uµ can be combined in the tensor Dµuν which can be irreducibly decomposed according to

Dµuν

=

uµaν

+

σµν

+ ωµν

+

1 3

hµν

θ

=

Cρνµuρ.

(7)

As an additional kinematic quantity, associated with the spatial triad eiµ, we define the spin rotation κµν by

κµν

=

eiν

F
Dueiµ

(8)

where

F
Du

denotes

the

Fermi

derivative

in

the

direction

of

uµ

given

by

F
DuX

µ

=

DuX µ

+

X ρaρuµ

−

X ρuρaµ

(9)

for a vector field Xµ. The spin rotation can then be written as

κµν = eiν Dueiµ + uµaν = −Cστ ρhσ[µhτν]uρ.

(10)

If the aether spin vector is assumed to be fixed with respect to the triad, the spin rotation measures the deviation (a spatial rotation) of the spin from a Fermi-Walker transported spin in the direction of the aether velocity.

3

In order to understand the role of the spin rotation κµν as a kinematic quantity, we note that the tensor Cµνρ can be decomposed according to

Cµνρ = Qµνρ + Sµνρ

(11)

into its spatial projection

Qµνρ = hλµhσν hτρ Cλστ = hσν hτρeiµDτ eiσ

(12)

and its time-space components Sµνρ. Equation (11) can be solved for Sµνρ yielding

Sµνρ = uµDρuν − uνDρuµ − κµν uρ

=

2u[µaν]uρ

+

2 3

u[µ

hν]ρ

θ

+

2u[µσν]ρ

−

2u[µων]ρ

−

κµν uρ.

(13)

The tensor Sµνρ can be seen as the generalization of the expression Dµuν in Equation (7). Equation (13) corresponds to its irreducible decomposition where the spin rotation κµν naturally appears as an irreducible part.
The vorticity tensor and the spin rotation can be represented by spatial vectors ωµ and κµ defined by

ωµ

=

1 2

εµνρωνρ

(14)

and a similar equation for κµ. Here, the totally antisymmetric spatial pseudotensor εµνρ is given by εµνρ = εµνρσuσ. We will also introduce an antisymmetric acceleration tensor aµν defined by

aµν = −εµνρaρ.

(15)

In Einstein-Aether theory, the action functional is quadratic in the expansion, shear, vorticity, and acceleration. We will add to this action a term quadratic in the spin rotation, κ2 = κµνκµν , and a term that couples the spin rotation and the vorticity, ω · κ = ωµν κµν. We will also include a parity violating term that couples the spin rotation and the acceleration, κ · a = κµνaµν . A similar term ω · a that couples the vorticity and the acceleration is a total derivative
(see Equation (33) below). We will thus consider the following action functional:

S[eaµ]

=

1 16πG

d4x√−g(R + Le)

(16)

where R is the Ricci scalar and

Le

=

1 3

cθ

θ2

+

cσ σ2

+

cω ω 2

+

2cωκω

·

κ

+

cκκ2

+

2cκaκ

·

a

−

caa2

(17)

with σ2 = σµν σµν , ω2 = ωµν ωµν , and a2 = aµν aµν and where cθ, cσ, cω, cωκ, cκ, cκa, and ca are dimensionless coupling constants. Further action terms involving matter and other fields can be added to the action (16) which we will not do in this paper. Using the tensor Sµνρ, Le can be written in the form

Le = K κλµρνσ SκνµSλσρ

(18)

where the supermetric Kκλµρνσ depends algebraically on eaµ. Since we implicitly assume that eaµ is orthonormal, the form of Kκλµρνσ is not unique. We here choose

Kκλµρνσ = uκuλ c1gµρgνσ + c2hµν hρσ + c3hµσhρν + c4uµuρgνσ

+ cωκ uκuρhσ[µhν]λ + uλuµhν[ρhσ]κ − cκuµuρhκσhλν + 2cκauµuρενσ[κuλ]

(19)

where

c1

=

cσ

+ 2

cω ,

(20)

c2

=

cθ

− 3

cσ ,

(21)

c3

=

cσ

− 2

cω ,

(22)

c4

=

2ca

−

cσ

+ 2

cω

.

(23)

4

Due to the presence of the aether velocity field uµ, the action (16) is not invariant under local Lorentz boosts. Under local spatial rotations of the tetrad field, that is, eiµ → Λijejµ where Λij is a rotation matrix, the kinematic quantities aµ, ωµν, σµν , and θ are invariant since they do not depend on eiµ. However, the spin rotation transforms
according to

κµν → κµν − eiµejν Λki∂uΛkj .

(24)

From this equation follows that if terms of the form κ2, ω · κ or κ · a are present in the action (16), there is only invariance under spatial rotations if ∂uΛij = 0. This means that the presence of the spin rotation breaks the invariance
under spatial rotations in the direction of the aether velocity field.

III. FIELD EQUATIONS

The field equations following from the action (16) are obtained by variation of the action with respect to eaµ. After contraction with eaν and taking the symmetric and antisymmetric part, the field equations read

Gµν

=

1 2

gµν

Le

− Dρ

J ρ(µν) − J(µν)ρ

−

J ρσ (µ − J(µσρ

Sν)ρσ

−

J

ρ (µ

σ

Q|ρσ|ν

)

−

1 2

N

αβλρστ

(µν)

Sασλ

Sβ

τ

ρ,

(25)

DρJ[µ

ρ ν]

−

J ρσ [µ − J[µσρ

Sν]ρσ

−

J ρ[µσ Q|ρσ|ν]

+

1 2

N

αβλρστ

[µν ] Sασλ

Sβτ

ρ

=

0

where Gµν is the Einstein tensor,

(26)

J κµν = KκλµρνσSλσρ,

(27)

and

N αβλρστ µν

=

δK αβλρστ δeaµ

eaν .

(28)

The “momentum” Jρµν is explicitly given by

J ρµν

=

1 3

cθ uρ hµν

θ

+

cσ uρ σµν

+ cωuρωµν

+ cωκ (uρκµν

+ uµωνρ) + cκuµκνρ

+cκauµ (aνρ − 2uρκν ) + 2cauρuµaν .

(29)

Equations (25) represent the Einstein field equations with the aether stress-energy tensor on the right hand side. Equations (26) are the field equations of the aether field. In the case cκ = cωκ = cκa = 0, Equations (25) and (26) are equivalent with the field equations of Einstein-aether theory.
If we are working at the level of the kinematic quantities — and not at the level of the tetrad — the field equations have to be supplemented by the constraint and evolution equations for the kinematic quantities which follow from the Ricci identity

DρDν eaµ − Dν Dρeaµ = Rσµνρeaσ

(30)

where Rσρµν is the Riemann curvature tensor (see also [7]). This equation can be solved for Rσρµν in terms of the tensor Cµνρ yielding

Rσµνρ = DρCσµν − Dν Cσµρ + CσλρCλµν − Cσλν Cλµρ.

(31)

The first Bianchi identity then leads to the following 16 constraint equations for Cµνρ.

Cτ σ = εµνρτ DρCσµν + CσλρCλµν = 0.

(32)

Only the projections Cτ σuσ contain solely the kinematic quantities. These four equations are the constraint and evolution equations for the vorticity,

Dρωρ = ω · a,

(33)

Duωρ

=

1 2

Dσ

aσρ

+

ω σ σσ ρ

−

2 3

θωρ.

(34)

5

Furthermore, from Equation (31), we can compute the time-time and time-space projections of the Ricci tensor Rµν = Rρµρν which can be expressed by the kinematic quantities,

Rµν uµuν

=

−∂uθ

−

1 3

θ2

+

Dρaρ

−

σ2

+

ω2,

Rµν uµhνρ

=

Dσ σσ ρ

+

uρσ2

+

σρσ aσ

−

Dσ ωσ ρ

−

uρω2

−

ωρσ aσ

−

2 3

∂ρ

θ

+

2 3

uρ

∂u

θ.

(35) (36)

By projecting the field equations (25) onto the aether velocity and its orthogonal spatial directions using uµ
and hµν , they can be split into three groups of equations consisting of a temporal-temporal equation [E00] obtained by contraction with uµuν, temporal-spatial equations [E0i] obtained by contraction with uµhνρ, and spatial-spatial equations [Eij] as a result of a contraction with hµρ hνσ. We first note that by taking the trace of Equation (25), the curvature scalar is given by

R = cθ

∂uθ

+

2 3

θ2

− cσσ2 − cωω2 − 2cωκω · κ − cκκ2 + 2cκa (Dρκρ − κ · a) − ca 2Dρaρ − a2 .

(37)

The

temporal

part

of

Equation

(25),

which

contains

Gµν uµuν

=

Rµν uµuν

−

1 2

R,

can

then

be

expressed

solely

in

terms

of the kinematic quantities by using Equations (35) and (37). The result is

[E00]

1

+

cθ 2

∂uθ

+

1 3

θ2

+ (1 − cσ) σ2 − (1 − cω + 2cωκ) ω2 − cκ κ2 + 2ω · κ

−cκa [Dρ (ωρ + κρ) + (ω + κ) · a] − (1 − ca) Dρaρ = 0.

(38)

In a similar way, the temporal-spatial part of the Ricci tensor Rµνuµhνρ can be eliminated from [E0i] using Equation (36) resulting again in equations containing only the kinematic quantities. The Ricci tensor is only present in the
spatial-spatial part [Eij ] in the form Rµνhµρ hνσ. The long equations [E0i] and [Eij ] are given in Appendix A. The antisymmetric aether field equations (26) can be analogously split in two groups [A0i] and [Aij ] by projections
using uµ and hµν . The spatial-spatial equations, which are empty in Einstein-aether theory, can be written in the form

[Aij ]

cκ

Duκµ

+

θκµ

−

1 2

uµκ

·

a

+ cωκ

Duωµ

−

κµρωρ

+

θωµ

−

1 2

uµω

·

a

+cκa

Duaµ

−

κµρaρ

+

θaµ

−

1 2

uµa2

= 0.

(39)

These equations relate the time evolution of κµ, aµ, and ωµ. The long equations [A0i] are given in Appendix A. Due to the constraint (35), only the spatial part Rµνhµν of the curvature scalar appears in the gravitational action
besides the kinematic quantities. This can be made more transparent by expressing R by Cµνρ using Equation (31). To this end, the spatial projection Qµνρ of Cµνρ can be decomposed in the following way. Since Qµνρ has only spatial components, we can define a second order tensor Qµν by

Qµν

=

1 2

εµρσ

Qρσ

ν

.

(40)

This tensor can be irreducibly decomposed into its antisymmetric part Ωµν , its trace Θ which is the totally antisymmetric part of Qµνρ, and its symmetric trace-free part Σµν :

Defining the vector

Ωµν

=

Q[µν]

=

1 2

ερσ[µ Qρσ

ν]

,

Θ

=

Qρρ

=

1 2

εµν ρ Qµν ρ

,

Σµν

=

Q(µν)

−

1 3

hµν

Θ

=

1 2

ερσ(µ Qρσ

ν)

−

1 3

hµν

Θ.

(41) (42) (43)

Ωµ

=

1 2

εµρσ

Ωρσ

=

1 2

Qµρ

ρ,

(44)

the irreducible decomposition of Qµνρ reads

Qµνρ

=

2Ω[µhν]ρ

−

εµνσ Σσρ

−

1 3

εµνρ

Θ.

(45)

6

Using Equation (31), the Ricci scalar is then given by

R

=

−

2 3

Θ2

+

Σ2

−

Ω2

+

2Ω

·

a

+

2 3

θ2

−

σ2

+

ω2

+

2ω

·

κ

−

2Dρ

(2Ωρ

+

uρθ

−

aρ)

(46)

where Σ2 = Σµν Σµν, Ω2 = Ωµν Ωµν, and Ω · a = Ωµνaµν . Inserting this equation into the Lagrangian L = R + Le, we obtain

L

=

−

2 3

Θ2

+

Σ2

−

Ω2

+

2Ω

·

a

+

2 3

1

+

cθ 2

θ2 −(1 − cσ) σ2 +(1 + cω) ω2 +2 (1 + cωκ) ω ·κ+cκκ2 +2cκaκ·a−caa2 (47)

where total derivatives have been omitted. While the quantities Ωµ, Σµν , and Θ appear together with the kinematic

quantities in the decomposition of Cµνρ, they have a different status than the kinematic quantities since they depend

on the arbitrary orientation of the triad in space. The kinematic quantities can be seen as field strengths with the

tetrad as potentials.

Note that while the quantities Ωµ, Σµν , and Θ are not invariant under space dependent spatial rotations, the special

combination

−

2 3

Θ2

+

Σ2

−

Ω2

+

2Ω

·

a

is invariant

under

such

transformations up

to

a

total

derivative.

By

adding

general terms quadratic in Ωµ, Σµν, and Θ as well as cross terms with the kinematic quantities to the Lagrangian,

the invariance under the full Lorentz group could be broken which we will not do in this paper.

IV. WEAK FIELDS

In this section, we will study the linearized version of the field equations (25) and (26). In the weak field approximation, the vectors eaµ and eaµ can be expanded around Minkowski space-time up to first order terms according
to

eaµ = δaµ + χaµ, eaµ = δµa + ψaµ

(48) (49)

where indices of χaµ and ψaµ are raised and lowered with the Minkowski metric. In the following, we will therefore not differentiate between internal and spacetime indices of first order fields. The condition δνµ = eaµeaν leads to the relation χµν = −ψνµ which allows to use only ψµν in the following. ψµν can be decomposed into its symmetric and
antisymmetric part according to

ψµν

=

1 2

γµν

+ ζµν

(50)

where γµν = ψµν + ψνµ and ζµν = ψ[µν]. The metric tensor is then given up to first order by

gµν = ηµν + γµν .

(51)

Under infinitesimal coordinate transformations xµ → xµ + ξµ, the tetrad transforms as

δeaµ = Lξeaµ = ξν Dν eaµ − eaν Dν ξµ = −∂aξµ.

(52)

This leads to the following gauge transformations of γµν and ζµν .

γµν → γµν + ∂µξν + ∂ν ξµ, ζµν → ζµν − ∂[µξν].

(53) (54)

In order to simplify the notation, in the following all fields will be first order quantities in this section. The tensor Cµνρ in first order approximation is

Cµνρ = ∂ρζµν + ∂[µγν]ρ.

(55)

It can be checked that Cµνρ is gauge invariant implying that all kinematic quantities in first order approximation are gauge invariant.
The field equations (25) and (26) in first order are given by

Gµν = ∂ρJ(µν)ρ − ∂ρJ ρ(µν),

∂ρ

J

ρ [µ

ν]

=

0.

(56) (57)

7

Since the first order approximation of the kinematic quantities as well as the first order curvature tensor are gauge invariant, it is convenient to write the first order field equations in terms of the kinematic quantities. The first order approximations of the aether field equations [Aij] and [A0i] are

[Aij ] cωκω˙ + cκκ˙ + cκaa˙ = 0,

(58)

[A0i]

1 3

cθ∇θ

−

cσ ∇

·

σ

−

cω ∇

×

ω

−

cωκ∇

×

κ

−

2cκaκ˙

+

2caa˙

=

0.

(59)

Here and in the following, we use vector notation. For example (a)i = ai, (∇ × ω)i = ǫijk∂jωk = ∂jωji, ∇ · κ = ∂iκi, θ˙ = ∂0θ, (σ)ij = σij , (R)ij = Rij , (1)ij = δij. (ǫijk is the totally antisymmetric symbol with ǫ123 = 1.) The first
order approximations of the field equations [E00] and [E0i] lead to

[I0]

cκa∇ · κ + (1 − ca) ∇ · a = −

1

+

cθ 2

θ˙,

(60)

[Ii]

cω

+

(1

+

cωκ)

1

cσ − cσ

∇×ω+

cωκ

+

cκ

1

cσ − cσ

∇×κ

+

2cκaκ˙

+

cκa 1

cσ − cσ

∇

×

a

−

2caa˙

=

2cσ 3 (1

+ cθ − cσ)

∇θ

(61)

where in Equation (61) the field equation [A0i] has been substituted in order to eliminate ∇ · σ. Finally, the weak field version of the field equation [Eij ] is

[Iij ]

R

=

−

1 6

cθ1θ˙

−

cσ σ˙

+

cκa1∇

·

κ

−

ca1∇

·

a.

(62)

Since we are working at the level of the kinematic quantities, we have to take the corresponding constraint equations into account. The linear approximations of the constraint equations (33) and (34) are

[H0] ∇ · ω = 0,

(63)

[Hi] ∇ × a + 2ω˙ = 0.

(64)

The spin rotation κ can be eliminated from the weak field equations using Equation (58). For this, we can split the fields into time independent and time varying parts. The static part corresponds to the zero frequency Fourier mode of the fields while the dynamic part represents the finite frequency contribution. We here consider only the dynamic case. Equation (58) can then be integrated resulting in

κ

=

−

cωκ cκ

ω

−

cκa cκ

a.

(65)

Substituting this equation into the field equations [I0], [Ii], and [Iij ], we obtain

[I0]

∇

·

a

=

−

1 1

+ −

cθ 2
c¯a

θ˙,

(66)

[Ii]

1 2c¯a

c¯ω

+

1

cσ − cσ

∇ × ω − a˙ =

cσ

+

cθ 2

∇θ,

3c¯a (1 − cσ)

(67)

[Iij ]

R = −1

cθ − 3c¯a

1

+

cθ 2

32

1 − c¯a

1θ˙ − cσσ˙

(68)

where

c¯ω

=

cω

−

c2ωκ cκ

,

(69)

c¯a

=

ca

+

c2κa cκ

.

(70)

Furthermore, combining Equations (59) and (67) yields

[A0i]

∇×

ω

=

2 3

1

+

cθ 2

∇θ + (1 − cσ) ∇ · σ.

(71)

8

Comparing Equations (66)-(68) with the limiting case cκ = cωκ = cκa = 0 of Equations (60)-(62), that is, Einsteinaether theory, we conclude that the introduction of the spin rotation amounts in the dynamic case to the substitutions cω → c¯ω and ca → c¯a.
Equations (63), (64), (66), and (67) have a close resemblance with Maxwell’s equations where a and 2ω play the role of electric and magnetic fields, respectively, and where the electric charge density ρ and current j are given by

ρ

=

−1

+

cθ 2

θ˙,

(72)

1 − c¯a

j

=

cσ

+

cθ 2

3c¯a (1 − cσ

)

∇θ.

(73)

A further useful equation involving the kinematic quantities follows from the first order limit of the contracted second Bianchi identity and the field equation [Iij ] (see Appendix B for a derivation):

∆σ

−

(1

−

cσ )

σ¨

−

2

(∇ (∇

·

σ))sym

−

1 3

1∆θ

+

1 3

1

+

cθ 2

1

−

3c¯a 1 − c¯a

1θ¨ −

1 3

∇∇θ

+

(∇a˙ )sym

=

0

(74)

where the suffix sym denotes symmetrization. The linearized wave solutions of Einstein-aether theory were given in [8]. Since the weak field equations of the
extended theory are effectively the same as in Einstein-aether theory, the wave solutions are the same but with different wave speeds and field content. The analogy of the weak field equations with Maxwell’s equations suggests the existence of spin 1 acceleration-vorticity waves similar to electromagnetic waves. Indeed, in the case θ = 0 Equations (64) and (67) lead to the wave equations

a¨ − s2aω∆a = 0, ω¨ − s2aω∆ω = 0

(75)

with the wave speed saω given by

s2aω

=

1 4c¯a

c¯ω

+

1

cσ − cσ

.

(76)

From Equations (63), (66), and (64) follows that a and ω are transverse and perpendicular to each other. Equation (65) with Equations (75) leads to a wave equation for κ with a polarization determined by the coupling constants cωκ, cκa, and cκ.
Furthermore, the conservation equation ρ˙ + ∇ · j = 0 for the charge density (72) and the current (73) corresponds to a wave equation for spin 0 expansion waves,

θ¨ − s2θ∆θ = 0

(77)

where the wave speed sθ is given by

s2θ

=

1 − c¯a 3c¯a

cθ

2

1

+

cθ 2

+

1

cσ − cσ

.

(78)

From Equation (66) follows that these waves are accompanied by longitudinal acceleration waves. Equation (64) then shows that for these solutions ω = 0. From Equation (65) follows that the spin rotation waves are also longitudinal.
Finally, in the case θ = 0, a = ω = κ = 0 it follows from Equation (71) that ∇ · σ = 0 and Equation (74) leads to a wave equation for spin 2 transverse shear waves,

σ¨ − s2σ∆σ = 0

(79)

with the wave speed squared

s2σ

=

1

1 −

cσ

.

(80)

The plane wave solutions for the three cases are given in Appendix C. If the weak field equations are expressed by the tetrad variables, we have to take the gauge transformations (53)
and (54) into account. In order to identify the independent wave modes, a gauge fixing has to be applied. In the present case, we use the gauge fixing conditions

ψi0 = 0,

(81)

∂iψ0i = 0.

(82)

9

Given a field configuration ψµν , these conditions can be reached by applying the gauge transformation

ξ0 =

d3x′ 4π|x −

x′

|

∂iψ0i,

(83)

ξi = − dt′ψi0.

(84)

In terms of the decomposition (50), the gauge fixing reads

ζi0

=

−

1 2

γi0,

(85)

∂iζ0i = 0, ∂iγ0i = 0.

(86)

In the following, we will again use vector notation to simplify equations. We define (A)i = 2ζi0 = −γi0, (ζ)i =

−

1 2

ǫijk

ζjk

,

γ

=

1 2

γkk

,

(γ)ij

=

1 2

γij

−

1 3

δij

γ

,

φ=

1 2

γ00.

The

gauge

fixing

conditions

can

then

be

written

as

a

kind

of

Coulomb gauge,

∇ · A = 0.

(87)

Taking the gauge fixing into account, the kinematic quantities in terms of the tetrad variables are

a = −A˙ − ∇φ,

(88)

ω

=

1 2

∇

×

A,

(89)

κ

=

−ζ˙

−

1 2

∇

×

A,

(90)

σ = γ˙ ,

(91)

θ = −γ˙ .

(92)

The tetrads for plane waves are given in Appendix C.

V. GEOMETRICAL CONSIDERATIONS

In this section, we resume the discussion of the nonlinear theory of the spinning aether in sections II and III. The formulation given there is entirely within Riemannian geometry, that is, gravity is described by the metric part of the tetrad. The aether, in contrast, is described by the full tetrad. Moreover, the symmetries of the aether — the broken Lorentz invariance — are imposed at the dynamical level through the action functional. In the following, we will argue for a pure geometrical formulation of the aether which implements the gravitational nature of the aether and its symmetries already at the kinematical level.
∗
For a start, we observe that we can define a covariant derivative Dµ by

∗
DµX

ν

=

DµX ν

+

Sν ρµX ρ

(93)

where Xµ is a vector field and Sµνρ is given by Equation (13). Since Sµνρ is antisymmetric in the first two indices,

∗

∗

the connection corresponding to Dµ is metric compatible, that is, Dµgνρ = 0. With the help of Equations (7), (10),

and (13), we can show that for a preferred frame eaµ

D∗ µuν = 0,

(94)

D∗ ueiµ = 0.

(95)

These equations may be viewed as trivial rearrangements of Equations (7) and (10). However, they can also be interpreted as defining a non-Riemannian geometry in which the aether velocity uµ is a parallel vector field and the triad eiµ is parallel in the direction of uµ. In this geometry, Sµνρ is the contortion tensor with the torsion tensor

Tµνρ = Sµρν − Sµνρ.

(96)

Thus, in this geometry, the kinematic quantities are torsion fields which realizes mathematically the interpretation of the kinematic quantities as field strengths.

10

∗

In order to gain insight into the geometry of Dµ, we can compute the corresponding spin connection. The spin

connection

ω∗

a bµ

is

related

to

the

connection

coefficients

∗
Γ

µ ν

ρ

of

the

derivative

∗
Dµ

by

ω∗ abµ

=

eaν ∂µebν

+ eaν

∗
Γ

ν

λµ

ebλ

=

eaν

∗
Dµ

ebν

.

(97)

This equation corresponds to a local linear transformation of the connection coefficients from a holonomic (coordinate) basis to an anholonomic basis given by the tetrad. In the case of the derivative (93), it can be shown that

ω∗ i0µ = 0, ω∗ ijρuρ = 0, ω∗ ijρhρµ = ωijρhρµ.

(98) (99) (100)

where ωabµ is the torsion-free Levi-Civita connection. Equations (98)-(100) can be summarized as

ω∗

a bµ

=

−Qabµ

(101)

where Qabµ = eaρebσQρσµ.

Equation

(98)

means

that

the

connection

ω∗

a bµ

is

a

SO(3)

connection.

Equation (99)

means

additionally

that

ω∗

a bµ

is

a

trivial

1

connection

along

uµ.

According

to

Equation

(100),

the

spatial

geometry

is

Riemannian. The connection (98) was used in [9] to formulate a version of Einstein-aether theory employing Weinberg’s

quasi-Riemannian gravity. The full connection (98)-(100) was derived from the symmetries of the spinning aether

in [10]. The corresponding geometry was referred to as semi-teleparallel since only the temporal part of space-time is

parallelized.

From Equation (98) follows that the time-space components of the curvature tensor Rabµν = 2∂[µωabν] +2ωac[µωcbν] vanish,

∗
R

i0

ρσ

=

2∂[ρ

ω∗

i0 σ]

+

2

ω∗

i j[ρ

ω∗

j0 σ]

=

0.

(102)

Using space-time indices, Equation (102) means

uµ

∗
R

µν

ρσ

=

0.

(103)

From the Ricci identity for the derivative (93),

∗∗
DρDν

eaµ−

∗
Dν

∗
Dρ

eaµ

+

T

σ ρν

∗
Dσ

eaµ

∗
=R

σ

µνρ

eaσ

,

(104)

∗
follows a relation between the curvature tensors of the derivatives Dµ and Dµ,

∗
R σµνρ

=

Rσµνρ

−

DρSσµν

+

Dν Sσµρ

−

Sσλρ S λ µν

+

Sσλν Sλµρ.

(105)

Using Equations (103) and (105), the relations (35) and (36) can be derived which can be seen to be a direct consequence of the semi-teleparallel geometry. Moreover, the first Bianchi identity for the semi-teleparallel connection is equivalent with the constraint equations (32).
The action functional (16) can be directly interpreted in terms of the semi-teleparallel geometry. A more general action, which is not equivalent with (16), can be obtained if we relax the implicit assumption that the spatial geometry is Riemannian, that is, has vanishing torsion. In this case, however, we go beyond the formulation of Einstein-aether theory within Riemannian geometry in that we start with a Riemann-Cartan geometry. In this geometry, we use a general Lorentz connection

ω˜

a bµ

=

ωabµ

+

K˜

a bµ

(106)

where

K˜

a bµ

=

eaρ

eb

ν

K˜

ρ νµ

is

the

contortion

tensor.

In

order

to

formulate

a

theory

of

gravity

with

a

spinning

aether

using Riemann-Cartan geometry, the action (16) has to be extended to an Einstein-Cartan theory. The natural way

is to define the kinematic quantities with respect to the connection ω˜abµ and to use the curvature scalar belonging to

11

ω˜abµ in the Lagrangian. The kinematic quantities corresponding to the derivative D˜ µ are (see also [11])

a˜µ = aµ + K˜ µρσuρuσ,

ω˜µν = ωµν + K˜ ρ[µν]uρ − u[µK˜ ν]ρσuρuσ,

θ˜ = θ − K˜ρuρ,

σ˜µν

=

σµν

− K˜ ρ(µν)uρ

+ u(µK˜ ν)ρσuρuσ

+

1 3

hµν

K˜ ρ

uρ,

κ˜µν = κµν + K˜ µνρuρ + 2u[µK˜ ν]ρσuρuσ

(107) (108) (109) (110) (111)

where

K˜ µ

=

K˜

µρ ρ

is

the

trace

of

the

contortion

tensor.

Furthermore,

the

curvature

scalar

corresponding

to

D˜ µ

is

R˜ = R + 2DρK˜ ρ − K˜ ρK˜ ρ + K˜ σλρK˜ ρλσ.

(112)

The generalization of the action (16) to an Einstein-Cartan-aether theory thus is

S[eaµ, ω˜abµ]

=

1 16πG

d4x√−g R − K˜ ρK˜ ρ + K˜ σλρK˜ ρλσ

+

1 3

c˜θ

θ˜2

+

c˜σ σ˜ 2

+ c˜ωω˜2

+

2c˜ωκω˜

· κ˜

+ c˜κκ˜2

+

2c˜κaκ˜ · a˜ −

c˜aa˜2

(113)

up to a surface term.
In order to get an action functional for a semi-teleparallel geometry, simply replacing the Lorentz connection in the action for Einstein-Cartan theory (113) by a semi-teleparallel connection is not suitable since the corresponding kinematic quantities (107)-(111) are zero in this case and the scalar curvature contains only the spatial part of the curvature tensor. A method to obtain an action functional for a semi-teleparallel geometry was proposed in [10]. It was shown there that starting from a general Lorentz connection ω˜abµ, there is a unique decomposition

ω˜ a bµ

=

ω∗

a bµ

+

H

a bµ

(114)

where

ω∗ abµ

is

a

semi-teleparallel

connection

with

respect

to

a

tetrad

eaµ

and

where

H

a bµ

is

a

tensor

field

satisfying

Habµ

= −Hbaµ

and which has vanishing

spatial components,

Hijµ hµν

= 0.

Actually,

H

a bµ

is the

difference

between

the

contortion

tensors

of

ω˜

a bµ

and

ω∗ abµ,

H

µ νρ

=

K˜

µ ν

ρ

−

∗
K

µνρ.

(115)

The basic idea in constructing an action for the semi-teleparallel geometry is the following:

1. Start with an action functional S[eaµ, ω˜abµ] that depends on the tetrad and an arbitrary Lorentz connection ω˜ abµ.

2.

Insert

the

decomposition

(114)

to

obtain

an

action

functional

S[eaµ

,

ω∗

a bµ

,

H

a bµ

].

3.

Find

a

stationary

point

of

this

action

with

respect

to

H

a bµ

resulting

in

an

action

functional

S′[eaµ, ω∗ abµ]

that

is defined for a semi-teleparallel connection ω∗ abµ.

In the present case, we start with the action (113). Insertion of the decomposition (114) and variation with respect to Hµνρ leads to algebraic equations for Hµνρ. Solving these equations and inserting Hµνρ back into the action (113) results in a new action of the form

S

′

[ea

µ,

ω∗

a bµ

]

=

1 16πG

d4x

R+

1 3

cθ

θ2

+

cσ σ2

+

cω ω 2

+

2cωκω

·

κ

+

cκκ2

+ (2aρ + cωZ ωρ + cκZ κρ) Zρ + cZ ZρZρ + ZσλρZρλσ

(116)

where

Z µν ρ

=

hµσhντ hρλ

∗
K

στ λ

(117)

12

is the spatial part of the contortion tensor of the semi-teleparallel geometry and Zµ = Zµρρ is its trace. The coupling constants cσ and cθ in (116) are given by

cσ

=

c˜σ c˜σ +

1,

cθ

=

2

2c˜θ − c˜θ

.

(118) (119)

The remaining coupling constants cω, cωκ, cκ, cωZ, cκZ, and cZ are functions of the coupling constants c˜ω, c˜ωκ, c˜κ, c˜κa, c˜a.
The action (116) differs from the original action (16) in the presence of terms quadratic in the spatial torsion and
coupling terms of the acceleration, vorticity, and spin rotation with the trace of the spatial torsion. Furthermore, there are no terms of the form cκaκ · a and caa2 in the action. Nevertheless, in the source-free case, the dynamics following from action (116) is effectively the same as the one following from the action (16). To see this, we first note
that — analogously to the tensor Qµνρ in Equation (45) — the spatial contortion tensor Zµνρ can be decomposed into irreducible parts according to

Zµνρ

=

Z[µhν]ρ

−

εµνσ Zσρ

−

1 3

εµν

ρ

Z.

(120)

where Z is a scalar and Zµν a symmetric trace-free tensor. Variation of the action (116) with respect to the semi-

teleparallel

connection

ω∗

a bµ

leads

to

the

field

equations

Z = 0,

Zµ

=

−1

1 + 2cZ

(2aµ

+

cωZ ωµ

+

cκZ κµ) ,

Zµν = 0.

(121)

Inserting these field equation back into the action (116) leads to an action of the form (16).

Although the coupling constants in the action (116) are by construction not independent, the action can be used as

a starting point for a semi-teleparallel theory of gravity with independent coupling constants. Moreover, it is possible

to add terms of the form θZ and σµν Zµν, which couple the spatial torsion with the expansion and the shear, as well

as a further term quadratic in the spatial torsion of the form ZµνρZµνρ.

∗
Since K µνρ = Zµνρ + Sµνρ, the general

action has the form

S[eaµ, ω∗ abµ]

=

1 16πG

d4x

R + Kκλµρνσ

∗
K κνµ

∗
K λσρ

(122)

with a suitable choice of the supermetric Kκλµρνσ .

VI. CONCLUSIONS
In this paper, an extension of Einstein-aether theory has been proposed which incorporates internal rotational degrees of freedom of the aether. The main objectives of the paper are
(1) The introduction of the spin rotation κµν as an additional kinematic quantity. The fact that this tensor naturally appears besides the acceleration, vorticity, shear, and expansion in the time-space decomposition of the Ricci rotation coefficients gives this approach a certain completeness.
(2) The formulation of a theory of gravitation that incorporates couplings of the spin rotation with the vorticity and the acceleration.
(3) The study of the theory in the weak field limit. In the case of dynamical fields, the spin rotation, the vorticity, and the acceleration are linearly related which allows to eliminate one of them from the field equations. The field equations acquire the simplest form if the spin rotation is eliminated. In that case, the linearized theory has the form of Einstein-aether theory with rescaled coupling constants.
(4) The formulation of the theory as a (semi-)teleparallel theory of gravitation. The geometry in this approach is adapted to the symmetries of the spinning aether. As a result, the kinematic quantities are part of the torsion tensor. In the matter-free case, the spatial torsion is algebraically related to the kinematic quantities which makes the approach effectively equivalent to the Riemannian formulation. However, if matter fields are present, the semiteleparallel formulation may be different from the Riemannian one.
There are many open questions connected with the Einstein-spin-aether theory which concern the physical implications of the spin rotation. These could be elucidated by studying exact solutions of the nonlinear theory such as black

13
hole solutions and cosmological solutions. It is to be expected that the simple relation between the kinematic quantities found in the linearized theory will no longer hold in solutions of the nonlinear theory. Further open questions concern the coupling to matter, in particular to spinning matter, where a semi-teleparallel formulation may lead to different interactions than the Riemannian formulation, the Hamiltonian formulation of the extended Einstein-aether theory, and the study of the observational constraints on the coupling constants.

Appendix A: Field Equations
In this appendix we will list the projections [E0i], [Eij ], and [A0i] of the field equations (25), (26) not given in Section III.

[E0i]
[Eij ] [A0i]

1 3

1 2

cθ

+

2

(∂µθ − uµ∂uθ) +

1 2

cσ

−

1

Dρσρµ + uµσ2 + σµρaρ −

1 2

cω

−

1

Dρωρµ + uµω2 + ωµρaρ

+ cωκ

1 2

Dρ

(2ωρµ

−

κρµ)

+

1 2

uµω

·

(2ω

−

κ)

−

1 2

κµρaρ

+

1 2

Qρσµ ω ρσ

+ cκ

Dρκρµ

+

uµω

·

κ

+

1 2

Qρσµ κρσ

+ cκa

Du

(2ωµ

+

κµ)

+

2 3

θ

(2ωµ

+

κµ)

−

σµ ρ

(2ωρ

+

κρ)

+

1 2

uµ

(2ω

−

κ)

·

a

+

3ωµρκρ

+

1 2

Qρσµ aρσ

− ca

Duaµ

−

1 2

uµa2

−

σµρaρ

+

3ωµρaρ

+

2 3

aµθ

=0

(A1)

hρµhσν Rρσ

=

1 6

cθ

hµν

∂uθ + θ2

− cσ

Duσµν + 2u(µσν)ρaρ + σµν θ + 2σ(µρων)ρ

− 2cωωρµωρν

+ 2cωκ ωρ(µσρν) + ωρµωρν − ωρ(µκρν) + 2cκ κρ(µσρν) + κρ(µωρν)

+ 2cκa

1 2

hµν

Dρ

(ωρ

+

κρ)

+

a(µ

ων) + κν)

+ σρ(µaρν)

− ca (hµν Dρaρ + 2aµaν )

(A2)

1 6

cθ

(∂µθ

−

uµ∂uθ)

+

1 2

cσ

Dρσρµ + uµσ2 + σµρaρ

+

1 2

cω

Dρωρµ + uµω2 + ωµρaρ

+

1 2

cωκ

(Dρκρµ

+

uµω

·

κ

+

κµρaρ

−

2ωµρaρ

+

Qρσµ ω ρσ )

−

cκ

κµρaρ

−

1 2

Qρσµ

κρσ

− cκa

Duκµ

−

1 2

uµκ

·

a

−

σµρ κρ

−

ωµρκρ

+

2 3

κµ

θ

−

1 2

Qρσµ

aρσ

+ ca

Duaµ

−

1 2

uµa2

−

σµρaρ

−

ωµρaρ

+

2 3

aµ

θ

=0

(A3)

Appendix B: Derivation of Equation (74)

Starting with the second Bianchi identity,

contracting with gστ leads to

DµRνσρτ + DρRνστ µ + Dτ Rνσµρ = 0,

(B1)

For weak fields, it follows

DµRνρ + DσRσνρµ − DρRµν = 0.

(B2)

Rij,0 = R0i,j + Ri0j0,0 − Rj0ik,k

(B3)

where commas denote partial derivatives. The terms on the right hand side can be obtained from the Ricci identity (30) in the following way. Using the first order form of Equation (36) yields

R0i,j

=

−

2 3

θ,ij

− σki,jk

+ ωki,jk.

(B4)

14

Furthermore, from Equation (31), we obtain

Ri0j0,0

=

1 3

δij

θ,00

− σij,00

+

ωij,00

+ ai,j0,

Rj0ik,k

=

−

1 3

θ,ij

+

1 3

δij

θkk σkj,ik

−

σij,kk

+ ωkj,ik

− ωij,kk.

(B5) (B6)

Inserting Equations (B4) - (B6) into Equation (B3) and taking the symmetric part yields

R˙

=

(∇a˙ )sym

−

2

(∇ (∇

·

σ))sym

+

∆σ

−

σ¨

−

1 3

∇∇θ

−

1 1
3

∆θ − θ¨

(B7)

where vector notation has been used. The time derivative of R can also be obtained from the field equation (68) resulting in

R˙ = − 1

cθ − 3c¯a

1

+

cθ 2

32

1 − c¯a

1θ¨ − cσσ¨.

(B8)

Equating (B7) and (B8) yields Equation (74).

Appendix C: Linearized plane wave solutions

We assume that the kinematic quantities and the tetrad in first order approximation are given by plane waves,

Sµνρ = Sˆµνρeikσxσ , ψµν = ψˆµν eikσxσ

(C1) (C2)

where kµ is the wavevector with (k)i = ki and k = nk. The spatial vector n is the normal of the wave front and s = k0/k is its speed. In the following, we will give the solutions to the field equations which are obtained by insertion of (C1) and (C2) into the field equations and Equation (74).

1. Spin 0 expansion waves

We assume that the expansion is given by θ = θˆeikσxσ . The other kinematic quantities are then given by

a

=

−sθ

1 1

+ −

cθ 2
c¯a

θn,

ω = 0,

κ

=

sθ

cκa cκ

1 1

+ −

cθ 2
c¯a

θn,

σ

=

1 −1

+ −

cθ 2
cσ

θ

nn

−

1 1
3

.

(C3) (C4) (C5)
(C6)

Using the tetrad, the spatial trace of the first order metric is given by γ = γˆeikσxσ . The other components of ψµν in the gauge (87) then are

A = 0,

φ

=

−s2θ

1 1

+ −

cθ 2
c¯a

γ,

ζ

=

sθ

cκa cκ

1 1

+ −

cθ 2
c¯a

γn,

γ

=

1 1

+ −

cθ

2
cσ

γ

nn

−

11 3

.

(C7) (C8) (C9) (C10)

15

2. Spin 1 acceleration-vorticity waves

If the acceleration is given by a = aˆeikσxσ with the transversality condition n · a = 0, the kinematic quantities are

θ = 0,

ω

=

−

1 2saω

n

×

a,

κ

=

−

cκa cκ

a

+

cωκ 2saω cκ

n

×

a,

σ

=

saω

1 (1 −

cσ )

(na)sym

.

(C11) (C12) (C13) (C14)

Analogously to electromagnetic waves, the condition of transversality reduces the number of independent modes to two. In terms of the tetrad, the potential A is given by A = Aˆeikσxσ with the gauge condition n · A = 0. The other components of the first order tetrad are then given by

φ = 0,

ζ

=

1 2saω

cωκ cκ

−

1

n

×

A

−

cκa cκ

A,

γ

=

− saω

1 (1 −

cσ )

(nA)sym

,

γ = 0.

(C15) (C16) (C17) (C18)

3. Spin 2 shear waves
The shear waves are given by σ = σˆ eikσxσ with the transversality condition n · σ = 0 and all other kinematic quantities vanishing, a = ω = κ = 0, θ = 0. Since σ has five independent components and the transversality condition consists of three equations, there are two independent shear modes. If the tetrad is used, the solutions are the gravitational waves of Einstein gravity with γ = γˆeikσxσ and n · γ = 0. All other components of the first order tetrad are zero, φ = γ = 0, A = ζ = 0.

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