5.4 Modified Versions of HL Gravity
53
5.4 Modified Versions of HL Gravity
While the HL gravity seems to solve the problem of renormalizability, and the most
important classical solutions in it reproduce those ones for the GR in certain limits, the
consistent description of degrees of freedom in HL gravity turns out to be problematic.
This fact has been firstly described in [102]. Following that paper, the main problem of
the HL gravity is as follows: the full-fledged general covariance group is broken up to
the subgroup which leaves the space-time foliation to be invariant. In other words,
since there is no more symmetry between space and time, one has the reduced gauge
group for spatial coordinates only. Thus, the gauge symmetry is partially broken,
which implies in arising of new degrees of freedom which can imply unstable vacuum,
strong coupling and other unusual effects [103]. It was claimed in [102] that, actually,
the extra mode appears to satisfy the first-order equation of motion and hence does
not propagate.
To illustrate this fact, let us consider the equations of motion (5.6), (5.9), (5.10). As
we already noted, they are invariant under three-dimensional gauge transformations
in linearized case looking like δg i j = ∂ i ξ j + ∂ j ξ i . These transformations allow to
impose the gauge N i = 0 [102, 103]. Afterwards, the Eq. (5.3) takes the form:
˙
g i j = 2N K i j . However, in the system (5.6), (5.9), (5.10) there is no equation for
the evolution of N ! And since N is separated from all other dynamical variables, it
cannot be fixed by gauge transformations. As a result, one concludes that N describes
the new degree of freedom. To study it we take the time derivative of (5.7), combine
it with other equations, and arrive at
∇ i
N
2
ξ(λ − 1)∇
i K + F
i
(K jk , R jk , K )
= 0.
(5.20)
It is easy to see that we have 13 dynamical variables (K i j , g i j , N ), five constraints
given by (5.6), (5.9), (5.20), so, we rest with 8 independent variables. Using three
gauge parameters ξ i we can eliminate three variables more. For five remaining ones,
we have four initial conditions for two helicities of h i j . So, we stay with one extra
degree of freedom!
More detailed analysis performed in [102] shows that if we consider k i j , a small
fluctuation of K i j , its trace κ = k
i
i does not propagate since ∇
2
κ = 0. So, we can
conclude that this extra mode is non-physical.
Returning to dynamics of N , we can fix N through the additive term in the
action given by S n =
d
3 xdt
√ g N
ρ
2
(N
−2
− 1), which implies strong coupling
(roughly speaking, due to the presence of the constraint). Under some tricks like
covariant extension (i.e. introducing of a Lorentz-covariant analogue), it appears
to be equivalent to the Einstein-aether action (with φ is a Stuckelberg field) S n =
d
3 xdt
√ g
ρ
2
(∇ μ φ∇
μ
φ − 1) [104], φ is called chronon since there is a gauge in
which this field is equal to a time coordinate, φ = t.
53
5.4 Modified Versions of HL Gravity
While the HL gravity seems to solve the problem of renormalizability, and the most
important classical solutions in it reproduce those ones for the GR in certain limits, the
consistent description of degrees of freedom in HL gravity turns out to be problematic.
This fact has been firstly described in [102]. Following that paper, the main problem of
the HL gravity is as follows: the full-fledged general covariance group is broken up to
the subgroup which leaves the space-time foliation to be invariant. In other words,
since there is no more symmetry between space and time, one has the reduced gauge
group for spatial coordinates only. Thus, the gauge symmetry is partially broken,
which implies in arising of new degrees of freedom which can imply unstable vacuum,
strong coupling and other unusual effects [103]. It was claimed in [102] that, actually,
the extra mode appears to satisfy the first-order equation of motion and hence does
not propagate.
To illustrate this fact, let us consider the equations of motion (5.6), (5.9), (5.10). As
we already noted, they are invariant under three-dimensional gauge transformations
in linearized case looking like δg i j = ∂ i ξ j + ∂ j ξ i . These transformations allow to
impose the gauge N i = 0 [102, 103]. Afterwards, the Eq. (5.3) takes the form:
˙
g i j = 2N K i j . However, in the system (5.6), (5.9), (5.10) there is no equation for
the evolution of N ! And since N is separated from all other dynamical variables, it
cannot be fixed by gauge transformations. As a result, one concludes that N describes
the new degree of freedom. To study it we take the time derivative of (5.7), combine
it with other equations, and arrive at
∇ i
N
2
ξ(λ − 1)∇
i K + F
i
(K jk , R jk , K )
= 0.
(5.20)
It is easy to see that we have 13 dynamical variables (K i j , g i j , N ), five constraints
given by (5.6), (5.9), (5.20), so, we rest with 8 independent variables. Using three
gauge parameters ξ i we can eliminate three variables more. For five remaining ones,
we have four initial conditions for two helicities of h i j . So, we stay with one extra
degree of freedom!
More detailed analysis performed in [102] shows that if we consider k i j , a small
fluctuation of K i j , its trace κ = k
i
i does not propagate since ∇
2
κ = 0. So, we can
conclude that this extra mode is non-physical.
Returning to dynamics of N , we can fix N through the additive term in the
action given by S n =
d
3 xdt
√ g N
ρ
2
(N
−2
− 1), which implies strong coupling
(roughly speaking, due to the presence of the constraint). Under some tricks like
covariant extension (i.e. introducing of a Lorentz-covariant analogue), it appears
to be equivalent to the Einstein-aether action (with φ is a Stuckelberg field) S n =
d
3 xdt
√ g
ρ
2
(∇ μ φ∇
μ
φ − 1) [104], φ is called chronon since there is a gauge in
which this field is equal to a time coordinate, φ = t.