Table Of ContentDiffeomorphism invariance and diffeomorphism
generation in the Hořava-Lifshitz gravity
1 F S Bemfica
1
0 Instituto de Física, Universidade de São Paulo
2 Caixa Postal 66318,05315-970,São Paulo, SP, Brazil
n
E-mail: fbemfi[email protected]
a
J
0 Abstract. This paper is intended to study diffeomorphism invariance and
2
diffeomorphism generation in the modified theory of gravity proposed by Hořava.
] Firstly, we demonstrate that the theory does not lose diffeomorphism invariance due
h tothe parameterλ, asitwaspreviouslybelieved. However,weshowthatthe presence
t
- of terms containing the Levi-Civita symbol in the original proposal of Hořava makes
p
e the theory diffeomorphism dependent. By neglecting such terms, what returns fully
h diffeomorphisminvarianceto the action,we obtainthe equations ofmotion. Secondly,
[
in the Hamiltonian formalism, we calculate the transformations generated by some
2 of the constraints of the theory. Then, we prove that all diffeomorphisms of General
v
Relativity aregenerated,onthe energyshell,by the constraintsofthe Hořava-Lifshitz
6
1 gravity.
5
5
.
0
PACS numbers: 04.50.Kd,04.20.Fy,11.10.Ef
1
0 Keywords: Diffeomorphism invariance, diffeomorphism generators, modified theories of
1
gravity.
:
v
i
X
r
a
Diffeomorphism invariance and diffeomorphism generation in the H-L gravity 2
1. Introduction
The modified theory of gravity proposed by Hořava [1], also known as Hořava-Lifshitz
gravity, has been the focus of great interest in recent times (A status report on the
subject can be found in [2]. See also [3, 4, 5, 6, 7, 8, 9, 10, 11].) In his formulation, terms
containing higher order in spatial derivatives are added with the intent of modifying
the propagator of the theory, turning it finite in the ultraviolet regime. This is done
in such a way that terms containing higher order time derivatives are not considered,
preventing theemergenceofpathologiessuchasghosts[12]. Tothisend, Hořavaimposes
an anisotropy in spacetime labeled by a dimensionful running parameter b through the
transformations t bzt, xa bxa (t , xa σ, a = 1,2,3), when defined in a
→ → ∈ ℜ ∈
foliation M = σ. In an attempt to make this anisotropy explicit, causing a partial
∼
ℜ×
lose of diffeomorphism invariance, Hořava also introduced a parameter λ in the kinetic
term of the action.
Inthisworkwewillstudydiffeomorphisminvarianceanddiffeomorphismgeneration
in the theory of Hořava. Taking into account the fact that (t,xa) are dummy variables
of integration, we can perform the inverse transformation bzt t, bxa xa and write
→ →
the Hořava-Lifshitz action [1] as
1
S = dt d3xN√q KabK λK2 +R(3) +f(R(3),R(3),...) . (1.1)
κ ab − ab
Zℜ Zσ
h i
Here, R(3) is the scalar curvature of the three-dimensional space σ, q is the 3-metric on
ab
σ, while K = (q˙ 2D N )/2N = (1/2) q is the extrinsic curvature, also known
ab ab (a b) n ab
− L ‡
as the second fundamental form, with trace defined as K = qabK . We use κ = 16πG
ab
with c = 1, where G is the Newton constant. N and Na are the Lapse function and
Shift vector of the Arnowitt-Deser-Misner (ADM) [13, 14] formalism, respectively. They
define the vector field ∂ := Nn+Na∂ . The unitary timelike vector field n is normal
t a
to σ and points to the future, thus N > 0 and √ g = N√q. We shall use the notation
−
g = det(g ), while q = det(q ). The torsion free covariant derivative compatible with
µν ab
the 3-metric q is D .
ab a
The essential difference between the action in (1.1) and the action of General
Relativity (GR) in the ADM formalism is the presence of the dimensionless parameter λ
and the function f(R(3),R(3),...). The just mentioned function must contain terms like
ab
(R(3))2, R(3)abR(3), CabC , among others. Weneglect terms depending ontheconstant
ab ab §
symbol of Levi-Civita such as εabcR D Rd, which does not transform as a tensor and
ad b c
make the theory completely diffeomorphism dependent. In this article we will not stick
in the details of f, unless it depends on the scalar of curvature R(3), the Ricci tensor
(3)
R , and their covariant derivatives D . Moreover, the most important property of f is
ab a
this being a scalar. It is worth saying that λ and f must satisfy the conditions f 0
→
n is the Lie derivative and generates passive diffeomorphisms in the direction of n. We use the
‡noLDtaetsiponiteLvtThbea11···f··a·bamcntttohsahtorttheenC(LovttTo)nba11··t···e·banmns,owrhdeerpeeTndiss aonnatrhbeitrcaornystteannstorLfievelid-C.ivita symbol Cab =
ε§acdDc(Rd(3)b δdbR(3)), the contraction CabCab does not.
−
Diffeomorphism invariance and diffeomorphism generation in the H-L gravity 3
and λ 1 in the limit where GR apply.
→
In the section that follows, we will study the diffeomorphism invariance of (1.1).
We shall demonstrate that the presence of the parameter λ does not affect the
diffeomorphism invariance of the theory (See also [6]). The presence of terms containing
theconstantLevi-Civitasymbolcompletelydestroythisinvariance, while, intheabsence
of such terms, the theory is diffeomorphism invariant under Diff(M). In section 3, we
calculate the equations of motion for the action in (1.1). The Hamiltonian formalism
of the theory is reviewed in section 4, where we show that the new Hamiltonian and
vector constraints are directly related to the equations of motion of the theory. Then,
we dedicate ourselves to calculate the transformations generated by these constraints.
We prove that, on the energy shell, they are the generators of the diffeomorphisms of
M, as in GR. In section 5 are the conclusions.
2. Diffeomorphism invariance
It has been argued [1] that the modified theory given in (1.1), in the presence of
λ, is restricted to a subgroup Diff (M) Diff(M) of diffeomorphisms. We shall
F
⊂
demonstrate that the theory described by (1.1) is, indeed, invariant throughout all the
diffeomorphisms of the GR group Diff(M). In other words, the theory is independent
of coordinates. In [1], the argument imposed is that a theory must have a Lagrangian
density invariant under active diffeomorphisms (that changes the points but not the
L
differentiable structure of M) of the type p p′ = p+δp δxµ(p) = xµ(p′) xµ(p) :=
→ ⇒ −
ǫµ(t(p),x(p)) [xµ = (t,x).] Here, p p′ is an active diffeomorphism M M. We must
→ →
be aware that the Lagrangian density of any theory may not be invariant under such
active diffeomorphisms, that is to say ∆ = (xµ+ǫµ) (xµ) = ǫµ∂ (xµ) = 0. This
ǫ µ
L L −L L 6
property, in turns, prevents the Lagrangian density of being a constant, independent
L
of the spacetime coordinates xµ = (t,x) and, as a consequence, of the points p M.
∈
In fact, a diffeomorphism is an isomorphism in the category of smooth manifolds.
A bijective map ϕ between differentiable manifolds is a diffeomorphism if ϕ and its
inverse are differentiable [15]. In the particular case of GR, Diff(M) is the set of maps
ϕ : M M. Passive diffeomorphisms, or simply diffeomorphisms, includes thechanges
→
of parametrization (coordinate systems) of the points of M. Thus, the invariance we
are talking about must be present in the transformation ϕ : (xµ) ′(x′µ)(p) =
∗
L → L
(xµ)(p), keeping the action (1.1) invariant for different parametrizations x and x′ of
L
each point p M. It is important to say that, although the Lagrangian may not be
∈
invariant under active diffeomorphism, its action is [13].
Let us follow the steps in [13, 14] by performing an immersion of σ into M through
an arbitrary diffeomorphism X : σ M; σ Σ ; (t,x) X(t,x) := X (x). The
t t
ℜ× → → →
spacetime is foliated in a spacelike hypersurface Σ . Any diffeomorphism ϕ Diff(M)
t
∈
has the form ϕ = X′ X−1, where X and X′ are two distinct foliations related by
◦
′
the diffeomorphism X = ϕ X. Thus, since the action in (1.1) is invariant under the
◦
immersion X, it will be invariant under any diffeomorphism ϕ Diff(M). We define
∈
Diffeomorphism invariance and diffeomorphism generation in the H-L gravity 4
the parametrization
∂Xµ(t,x)
tµ(X) := (∂ )µ = = N(X)nµ(X)+Nµ(X) := Xµ(X), (2.1)
t ∂t t
(cid:12)X(t,x)=X
(cid:12)
where nµN = 0 and nµv =(cid:12) 0 if v is a covector on Σ . The inverse of (2.1), subtracted
µ µ (cid:12) t
from the relation dt(∂ ) = tµ t = 1, is given by t(X) = n (X)/N(X) := (X−1)t,
t ∇µ ∇µ − µ µ
which is the covector normal to Σ . Here, is the torsion free metric preserving
t µ
∇
covariant derivative compatible with g instead of q . It is convenient to define the
µν µν
following three spacial vector fields of Σ
t
∂Xµ
Xµ := . (2.2)
a ∂xa
(cid:12)X(t,x)=X
(cid:12)
Being X a diffeomorphism(cid:12), (2.2) must have an inverse. Then, we can write
(cid:12)
qab(t,x) = (X−1)a(X−1)bqµν(X(t,x)),
µ ν
K (t,x) = XµXνK (X(t,x)). (2.3)
ab a b µν
This, in turns, enables one to find
K(x,t) = (qabK )(t,x) = (X−1)a(X−1)bXαXβ(qµνK )(X(t,x))
ab µ ν a b αβ
= (qµνK )(X(t,x)) = K(X(t,x)),
µν
(KabK )(t,x) = (X−1)a(X−1)bXαXβ(K Kµν)(X(t,x))
ab µ ν a b αβ
= (K Kµν)(X(t,x)). (2.4)
µν
The above calculations where performed by taking into account the identity
n
(X−1)bXβ = δβ (X−1)tXβ = δβ + νtβ, (2.5)
ν b ν − ν t ν N
togetherwithn qµν = n Kµν = 0. Inordertosavespace, fromnowonwewilleventually
µ µ
leave aside the label X. The first fundamental form is written as
q = g +n n . (2.6)
µν µν µ ν
One can easily show that K = (X−1)a(X−1)bK = qαqβ n = q /2 [The
µν µ ν ab (µ ν)∇β α Ln µν
indexes symmetrization T U := (1/2)(T U + T U ).] Here, the Lie derivative
(a b) a b b a v
L
generates active diffeomorphisms in the direction of the vector field v. The covariant
derivative compatible with q is defined by D f = qα f, and D v = qαqβ v ,
µν µ µ∇µ µ ν µ ν∇β α
where f must be a smooth function on Σ and nµv = 0. The same result holds for
t µ
arbitrary tensor fields defined on this hypersurface. The reader may verify that
D v = XµXνD v = XµXνqαqβ v , v T Σ
a b a b µ ν a b µ ν∇α β X ∈ X t
(3)c
= ∂ v Γ v . (2.7)
a b − ab c
This result can be extended to any tensor field in Σ , what enables one to write
t
R(3)(t,x) = R(3)(X(t,x)), R(3) = XµXνR(3)(X(t,x)), andso on. Details regarding these
ab a b µν
and other calculations are in [13, 14]. We argue that the function f(R(3),R(3),...), in
ab
the absence of terms containing the constant Levi-Civita symbol, is a scalar. Thus,
f(R(3),R(3),...)(t,x) = f(R(3),R(3),...)(X(t,x)).
ab µν
Diffeomorphism invariance and diffeomorphism generation in the H-L gravity 5
What remains to be done is to write N√q = g(t,x) on the pushforward
−
X (dtd3x g(t,x)) = d4X g(X). Collecting all the above results, the action in
∗
− − p
(1.1) takes the form
p p
1
S = d4X√ g KµνK λK2 +R(3) +f(R(3),R(3),...) , (2.8)
κ − µν − µν
ZM
for any diffeomorphism X. This d(cid:2)emonstrates that the introduction of λ in(cid:3)the theory
of Hořava does not affect its diffeomorphism invariance.
At thismoment, it isimportantto emphasize thata termlikeεabcR D Rd isclearly
ad b c
dependent of the diffeomorphism we choose. In other words, although the Ricci tensor
transforms as a tensor of rank 2, the Levi-Civita symbol εabc, which is a constant, does
not, and the theory loses all its diffeomorphism invariance, even Diff (M), as argued
F
by Hořava [1].
In the next sections we will turn to the problem of diffeomorphism transformations
and show that the theory is not only diffeomorphism invariant, as seen here, but also
possess its constraints as the generators of the active diffeomorphisms of GR.
3. The equations of motion
From now on we set, for simplicity,
κ = 1. (3.1)
To calculate the equations of motion for the action in (1.1), we first rewrite (2.8) using
the reverse procedure of the ADM formalism, namely,
S = d4X√ g R(4) +(1 λ)K2 +f(R(3),R(3),...) . (3.2)
− − µν
ZM
The identity (cid:2) (cid:3)
R(4) = R(3) + KµνK K2 2 (nν nµ nµ nν) , (3.3)
µν µ ν ν
− − ∇ ∇ − ∇
from the Gauss-Codazzi equ(cid:2)ation [14], wa(cid:3)s taken into account. The scalar curvature
R(4), defined on M, stems from the definition for the curvature tensor [ , ]v =
µ ν α
∇ ∇
(4)β
R v . The last termin(3.3) isa surfacetermandhasbeen neglected. Byperforming
µνα β
the inverse diffeomorphism X−1 : M σ on (3.2) we obtain
→ ℜ×
S = dt d3x g(t,x) R(4) +(1 λ)K2 +f(R(3),R(3),...)(t,x) . (3.4)
− − ab
Zℜ Zσ p h i
Let us recall some important relations. In the coordinate system (t,xa), the normal
vector field n has components nµ = (1/N, Na/N), while n = ( N,0). The metric on
µ
− −
σ is given by q (qa = δa), while qµ may project any vector field on M into σ. Again,
ab b b ν
we write √ g = N√q, with q = det(q ). The remaining components of q (qµν) are:
ab µν
−
q = N , q = NaN (qtt = qta = 0,) being nµq = 0 and g = q n n . The
ta a tt a µν µν µν µ ν
−
properties of the projectors q and n n enable us to set down any vector field v on M
⊗
as
v = (v +g(v,n)n) g(v,n)n := vˆn+vˆa∂ , (3.5)
a
−
Diffeomorphism invariance and diffeomorphism generation in the H-L gravity 6
where
vˆ = n vµ = Nvt, (3.6a)
µ
−
vˆa = qavµ = va Navt. (3.6b)
µ −
Above, vˆ is the v component normal to σ, while its tangent component is vˆa, both
written in the nonholonomic basis e = (n,∂ ).
µ a
We return to the variation of the action in (3.4) and write down
dS[g +sδg]
δS :=
ds
(cid:12)s=0
(cid:12)
= dt d3x(cid:12) R(4) +(1 λ)K2 +f(R(3),R(3),...) √qδN
(cid:12) − ab
Zℜ Zσ nh i
1
+ R(4) +(1 λ)K2 +f(R(3),R(3),...) N√qqabδq
2 − ab ab
h i
+ √ g Rµνδg +2(1 λ)KδK +δf(R(3),R(3),...) . (3.7)
− − µν − ab
h io
The relation above has been calculated by means of the equality δ√q =
(1/2)√qqµνδq = (1/2)√qqabδq , together with qta = qtt = 0. After we take the
µν ab
variation δR(4) = R(4)δg + α δ(µ ν) gµν δg , we discarded the surface term
µν α α µν
− ∇ ∇ − ∇
in parenthesis. (cid:16) (cid:17)
We must express the total variation of S in terms of the independent variables N,
Na, and q , separately. Thereunto, it is necessary an analyzes of each term in the last
ab
equality in (3.7). We may begin writing
R(4)µνδg = R(4)αβδµδνδg = R(4)αβ(qµ nµn )(qν nνn )δg
− µν − α β µν − α − α β − β µν
2
= R(4)αβqaqb +2R(4)αβqan nb δq + R(4)αβqan δN
− α β α β ab N α β a
(cid:0)2 (cid:1)
+ R(4)αβn n δN . (3.8)
α β
N
Now, it is needed to calculate the variation of the trace K. This can be easily achieved
expressing K as K = nµ [13, 14], then
µ
∇
1
δK = δnµ + nαgλβ δg . (3.9)
µ α λβ
∇ 2 ∇
The identity n δnα = nαδn helps us to find
α α
−
δnµ = qµδnα nµn δnα
α − α
nµ
= nαqµaδq δN . (3.10)
aα
− − N
Equations. (3.9) and (3.10) are essential to calculate the integral that follows,
dt d3x√ gKδK
−
Zℜ Zσ
1
= dt d3x√ g δnµ K gλβ K2 + K δg
µ n λβ
− − ∇ − 2 ∇
Zℜ Zσ (cid:20) (cid:21)
1 (cid:0) (cid:1)
= dt d3xN√q nαδq DaK + δN K
aα n
N ∇
Zℜ Zσ (cid:20)
Diffeomorphism invariance and diffeomorphism generation in the H-L gravity 7
1
K2 + K gλβδg
n λβ
− 2 ∇
(cid:21)
(cid:0) (cid:1) δN δN
= dt d3xN√q ( K2)+ a(DaK)
N − N
Zℜ Zσ (cid:26)
qab
+ δq n(aDb) K2 + K . (3.11)
ab n
− 2 ∇
(cid:20) (cid:21)(cid:27)
(cid:0) (cid:1)
In the first equality above we have performed an integral by parts, neglecting surface
terms. We also used the equality qµa K = DaK and, afterwards, applied the relation
µ
∇
for g given by
gλβδg = qλβδq +nαnβδ(n n )
λβ λβ α β
2
= qabδq 2nαδn = qabδq + δN . (3.12)
ab α ab
− N
The remaining term to be analyzed in (3.7) involves f(R(3),R(3),...) and needs a
ab
special attention. We do not want nor need to treat the exact form of f. So, we begin
by studying the particular case w = w(R(3)). We know that R(3) depends only on q
ab
(3)d
an D through [D ,D ]v = R v , then, the integral
a a b c abc d
d3xN√qδw(R(3))
Zσ
∂w(R(3))
= d3xN√q R(3)ab +DaDb qabDcD δq
c ab
∂R(3) − −
Zσ
(cid:0) ∂w(cid:1)(R(3))
= d3x√qδq R(3)ab +DaDb qabDcD N
ab c
− − ∂R(3)
Zσ (cid:18) (cid:19)
(cid:0) (cid:1)
:= d3x√qδq [ (N)]ab , (3.13)
ab w
D
Zσ
where the effect of the variation of w(R(3)) inside the integral was summarize, up to
surface terms, by the tensor of rank 2
∂w(R(3))
[ (N)]ab = R(3)ab +DaDb qabDcD N . (3.14)
w c
D − − ∂R(3)
(cid:18) (cid:19)
(cid:0) (cid:1)
ab µν
Observe that [ (N)] transforms as a tensor and n (X)[ (N(X))] = 0 in any
w µ w
D D
diffeomorphism X. This result will be extended to the case of (3.7). As we imposed, the
function f(R(3),R(3),...) is a covariant combination of the Ricci tensor, scalar curvature
ab
and their covariant derivatives. This enables us to write
d3xN√qδf(R(3),R(3),...)
ab
Zσ
∂f(R(3),R(3),...) ∂f(R(3),R(3),...)
= d3xN√q ab δR(3) + ab δR(3)
Zσ ∂R(3) ∂Ra(3b) ab
∂f(R(3),R(3),...)
+ ab δ(D R(3))+
∂D R(3) c ab ···!
c ab
:= d3x√qδq [ (N)]ab , (3.15)
ab f
D
Zσ
Diffeomorphism invariance and diffeomorphism generation in the H-L gravity 8
ab µν
where [ (N)] must be a tensor of rank 2 with the property n (X)[ (N(X))] in
f µ f
D D
any diffeomorphism X. The above result takes into account that any variation of the
(3)a
Ricci tensor, Ricci scalar, or of the Christoffer symbol Γ , will generate covariant
bc
derivatives of δq , that can be eliminated by performing some integrations by parts and
ab
by dropping surface terms.
Collecting the results of (3.8), (3.11), and (3.15) we obtain
δS = dt d3x√q
Zℜ Zσ (cid:26)
2R(4)αβn n +R(4) (1 λ)K2 +f(R(3),R(3),...) δN (3.16a)
α β − − ab
h i
+ R(4)αβqaqb +2R(4)αβqan nb
− α β α β
(cid:20)
1 1
+ qabR(4) + (1 λ)K2 +f(R(3),R(3),...)
2 2 − − ab
(cid:16) 1 (cid:17)
+(1 λ) 2n(aDb)K qab K + [ (N)]ab Nδq (3.16b)
n f ab
− − ∇ N D
(cid:21)
(cid:0) (cid:1)
+2 R(4)αβqan +(1 λ)DaK δN . (3.16c)
α β − a
(cid:27)
(cid:2) (cid:3)
The terms in brackets must cancel separately so that δS = 0. We claim that the above
result can be compactified as follows
1
G˜αβ := Gαβ gαβ (1 λ)K2 +f(R(3),R(3),...)
− 2 − − µν
(1 (cid:2)λ)n(αDβ)K +qαβ(1 λ) K (cid:3) 1 [ (N)]αβ = 0, (3.17)
n f
− − − ∇ − N D
where Gαβ = R(4)αβ (1/2)gαβR(4) is the Einstein tensor. The reader may verify that
−
the term in brackets in (3.16a) corresponds to 2n n G˜αβ, while the one in (3.16b) is
α β
equals to (qaqb 2q(anb)n )G˜αβ, and, the remaining term in (3.16c) can be achieved
α β − α β
by performing 2qan G˜αβ. Thus, as we had claimed, (3.17) is the covariant equation of
α β
motion of the Hořava-Lifshitz theory.
4. Hamiltonian formalism and diffeomorphism generation
Before looking at the transformations generated by the Hamiltonian and vector
constraints, we must write the Hamiltonian formalism of the theory. The procedure
is similar to the λR theory (where f(R(3),R(3),...) = 0) studied in [9, 10, 11]. Without
ab
going into the details, we write the total Hamiltonian [16, 13]
H = H(N)+H (Na)+C(γ)+C (γa)
a a
= d3x(N +Na +γ +γa ) , (4.1)
a a
H H C C
Zσ
where
1
= πabπ λ˜π2 √q R(3) +f(R(3),R(3),...) , (4.2a)
H √q ab − − ab
h i (cid:16) (cid:17)
Diffeomorphism invariance and diffeomorphism generation in the H-L gravity 9
= 2D πb (4.2b)
Ha − b a
δL
πab = = √q Kab λqabK , (4.2c)
δq˙ −
ab
δL (cid:0) (cid:1)
= Π = 0, (4.2d)
C δN˙ ≈
δL
= Π = 0. (4.2e)
Ca a δN˙a ≈
˜
We use the definition λ := λ/(3λ 1). Following the formalism of Dirac [16], and
− C
are primary constraints, and their persistence in time lead us to the secondary
a
C
constraints 0 and 0, also known as the Hamiltonian and vector constraints,
a
H ≈ H ≈
respectively. In (4.1), N and Na depends on the spacetime variables (t,x), being
coordinates of the phase space Γ = q ,N,Na,πab,Π,Π , while γ and γa are Lagrange
ab a
{ }
¯
multipliers. WewilldenotebyΓthereducedphasespace, wheretheconstraintequations
= = = = 0 must hold. We also define the equal time Poisson brackets for
a a
H H C C
′
arbitrary functions F, F Γ
∈
′ ′
δF δF δF δF
F,F′ := d3y . (4.3)
{ } δQ(y)δP(y) − δP(y)δQ(y)
Zσ (cid:18) (cid:19)
The reader may perceive thatthe short handnotationQ = (q ,N,Na) for thefields and
ab
P = (πab,Π,Π ) for their canonically conjugate momenta has been applied, together
a
with an implicit index summation. At this point, we can identify the relation of
the equations of motion with the Hamiltonian and vector constraints. The identities
n n Gµν = (1/2)(KabK K2 R(3)) and nµqaGµν = D (Kab qabK) [13, 14],
µ ν − ab − − ν b −
together with (4.2c) and the equations of motion G˜µν in (3.17), enable us to find
1
n n G˜µν = , (4.4a)
µ µ
− 2√qH
1
qan G˜µν = a. (4.4b)
µ ν − 2√qH
In other words, just as in GR, the Hořava-Lifshitz theory on shell defines the reduced
phase space Γ¯.
We shall beconcentrated now onthe transformationsgenerated by the Hamiltonian
and vector constraints. We must verify that these constraints are the generators of
the diffeomorphism from Diff(M). Let us begin by quoting the active diffeomorphism
δxµ = ǫµ(t,x) = (ǫt,ǫa). We may write ǫ in the nonholonomic basis e = (n,∂ ) as
µ a
ǫ = ǫˆn+ǫˆa∂ , where ǫˆ= Nǫt and ˆǫa = ǫa Naǫt. From the active diffeomorphism on
a
−
g given by g = ǫα∂ g +2g ∂ ǫα, it is easy to show that [13, 14]
µν ǫ µν α µν α(µ ν)
L
q = ǫˆ q = 2ǫˆK , (4.5a)
ǫˆn ab n ab ab
L L
Lǫˆc∂cqab = 2D(aǫˆb). (4.5b)
Also, it can be verified that
δ q := q ,H(ǫˆ)
ǫˆn ab ab
{ }
2ǫˆ
= π λ˜q π = 2ǫˆK , (4.6a)
ab ab ab
√q −
(cid:16) (cid:17)
Diffeomorphism invariance and diffeomorphism generation in the H-L gravity 10
δǫˆc∂cqab := {qab,Hc(ǫˆc)}
= 2D ǫˆ . (4.6b)
(a b)
The above equations reflect the fact that the Hamiltonian and vector constraints are
the generators of the diffeomorphisms given in (4.5a) and (4.5b). It now remains to
verify the active diffeomorphisms for the canonically conjugate momenta πab, since the
transformations that generate the active diffeomorphisms on N and Na are performed
bythesmeared functionsC( N) andC ( Na), respectively. Byconsidering theactive
ǫ a ǫ
L L
diffeomorphism
Lǫˆc∂cKab = ǫˆcDcKab −2Kc(aDb)ǫˆc, (4.7)
together with Lǫˆc∂c√q = (1/2)√qqabLǫˆc∂cqab, Lǫˆc∂cqab = −qacqbdLǫˆe∂eqcd and πab =
√q(Kab λqabK), easily we reach the result
−
Lǫˆc∂cπab = πabDcˆǫc +√q ǫˆeDeKab −2Ke(aDeˆǫb) +2λKD(aˆǫb)
2λqabKcdD ˆǫ λqabǫˆeD K +2λqabKcdD ǫˆd
(c (cid:0)d) e c
− −
= πabD ˆǫc +ǫˆeD πab 2πe(aD ǫˆb). (4.8)
c e e (cid:1)
−
Taking into account that the constraint in (4.2b) is the same vector constraint of GR,
(4.8) reduces to [13]
δǫˆc∂cπab := πab,Hc(ǫˆc)
= L(cid:8)ǫˆc∂cπab. (cid:9) (4.9)
Sofar,allthediffeomorphismsofGRaregeneratedbytheconstraintsoftheHořava-
Lifshitz theory. But things must change radically for the remaining case. It is already
known from GR in the ADM formalism that the active diffeomorphism may be
Nn
L
generated by the constraint H(N) only on shell. We shall show that it will also happen
in the present case. The property
= ǫˆ = ǫ0 (4.10)
ǫˆn n Nn
L L L
tells us that any diffeomorphism in the direction of n may be generated by . By this
Nn
L
reason, we may study the transformation δ πab only. The following straightforward
Nn
result
δ πab := πab,H(N)
Nn
N
= (cid:8) qab 2N(cid:9)√q KacKb λKabK
2 H− c −
+ Nqab R(3) +f(R(cid:0)(3),R(3),...) +√(cid:1)q DaDb qabDcD N
ab − c
+ √q[ h(N)]ab N√qR(3)ab, i (cid:0) (cid:1) (4.11)
f
D −
is obtained using the equality πacπb λ˜ππab = q(KacKb λKabK) after computing the
c − c −
Poisson bracket. In order to find πab = N πab, we must calculate K first ,
Nn n n µν
L L L k
Most of the details of these calculations can be found in [13].
k
