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action [1, 11–14, 26–28] and performing a canonical transformation that coordinatizes the phase space of the already linear theory with three pairs of U(1) connections and their conjugate momenta. As we shall see, the result obtained by this method coincides with that of the ”standard” route, [10, 24, 25]. In short, one could say that the processes of ”linearizing” and ”converting into Ashtekar variables” commute.

The paper is organized as follow. In section 2 we present an overview of the ADM formulation. In the next section we describe how to obtain the linearized ADM action following standard methods. In section 4 we perform a canonical transformation to parameterize the phase space in terms of canonical pairs described in a non-coordinate basis. Then, in section 5 we introduce a linearized spin connection and perform a second canonical transformation that finally yields a characterization of the phase space in terms of linearized Ashtekar variables. In the last section we present some concluding remarks.

II. ADM FORMULATION OF GRAVITY

Eintein’s equations can be derived through a variational principle from the Einstein-Hilbert action [1–3, 14]

S = R√ g d4x. (1) Z − M The space-time M is an oriented semi-Riemannian manifold, R is the Ricci scalar and g is the determinant of the metric tensor. A Hamiltonian formulation of general relativity can be achived following the ADM procedure [1, 2, 14] that we briefly summarize. Consider a separation of M in a mono-parametric family of achronals hypersufaces Στ with τ R. Thereby, the dynamics may be described in terms of changes between successive ∈ hypersurfaces. More precisely, if γ represents the world line of p Στ and ∂τ is the vector field tangent to γ, ∈ the dynamics in M is given by the changes in the ∂τ direction suffered by quantities belonging to Στ . The vector field ∂τ can be written down in terms of its tangent and normal vector fields components relative to Στ [2, 14] as

∂τ = Nn + N,~ (2)

where N is called the ”lapse function”; N~ , which belongs to the tangent space to Στ at p, is the ”shift” vector, µ ν and n is the unitary vector field normal to Στ at p, obeying gµν n n = 1 (in the case of Lorentzian gµν ) with µ,ν =0, 1, 2, 3. −

The metric components gµν can be written in terms of N and N~ and the induced 3-metric qµν on Στ as follows. 0 1 2 3 0 We assign local coordinates x , x , x , x to p Στ such that x = τ. Then ∂ = ∂τ and the vector fields ∈ 0 ∂1, ∂2, ∂3 are tangent to Στ at p. The components of the metric are then given by [14, 26]

gab = qab − gab = qab N 2N aN b − g0a = Na − g0a = N 2N a 2 a g00 = (N NaN ) − − − g00 = N 2, (3) − where a,b =1, 2, 3. From these expressions it can be seen that

√ g = N√q, (4) − 3

where q is the determinant of the induced 3-metric qab [2, 14, 26].

The Ricci scalar can be written as [1, 2, 14]

3 ab 2 R = R K Kab + K , (5) − 3 where R is the Ricci scalar induced in Στ , defined by

3 cd 3 a R = δabq Rcbd, (6)

3 a 3 a with Rcbd being the induced curvature, given in terms of the induced affine connection Γbc by 3 a 3 a 3 a 3 e 3 a 3 e 3 a R = ∂b Γ ∂c Γ + Γ Γ Γ Γ . (7) bcd cd − bd cd be − bd ce 3 a In turn, the induced affine connection Γbc can be written as

3 a 1 ad Γ = q (∂bqcd + ∂cqbd ∂dqbc). (8) bc 2 −

In equation (5) also appears the (0, 2) extrinsic curvature tensor Kab [1, 2, 14]

1 −1 3 3 Kab = N (∂ qab aNb bNa), (9) 2 0 − ∇ − ∇ ab 3 and its trace K = Kabq . Additionally is the covariant derivative operator preserving the 3-metric [2], which can be written down in terms of the∇ ”full” covariant derivative as ∇ 3 ρ σ µAν = q q ρAσ, (10) ∇ µ ν ∇ for every covariant vector field Aµ in M.

Replacing (4) and (5) in (1) we can rewrite the action as

4 4 3 ab 2 S = d x = d x√qN( R K Kab + K ). (11) Z L Z − M M

It should be underlined that this expression depends exclusively on quantities relative to the ”space” Στ , and on the lapse and shift fields. This fact facilitates the passage to the Hamiltonian formulation. Applying the Dirac canonical procedure to the action (11) we obtain the first order ADM action [1, 2, 14, 26]

4 ab 4 ab 3 a b ab 3 ab S = d xp qab˙ dt H = d x[p q˙ab N√q( R + K K K Kab) Nb( 2 ap )], (12) Z − Z Z − − a b − − − ∇ where ∂ pab = L = √q(Kab qabK) (13) ∂q˙ab − − is the momentum conjugate to qab and

3 b H = d x[N + NbV ] (14) Z S is the Hamiltonian, with

3 ab 2 = √q( R + K Kab K ) 0 S − − ≈ b 3 ab V = 2 ap 0, (15) − ∇ ≈ 4

being the scalar and vectorial constraints respectively [1, 2, 14, 26]. The lapse and the shift are then Lagrange multipliers enforcing the constraints. These constraints are first class in Dirac’s sense, and generate time and spatial diffeomorphisms, respectively, on the phase space. The true dynamical variables are the spatial metric and its canonical conjugate, whose equations of motion are given by

q˙ab = qab,H , { } p˙ab = pab,H . (16) { } The fundamental Poisson brackets are

′ ′ ′ ′ ′ ′ a b a b a b 3 qab(x),p (y) = (δa δb + δb δa )δ (x y) { ′ ′ } − ab a b p (x),p (y) = qab(x), qa′b′ (y) =0. (17) { } { } The canonical equations (16), together with the scalar and vector constraints reproduce Einstein equations, as can be verified.

III. ADM LINEARIZED GRAVITY

In order to linearize the theory, we consider small perturbations hab around the flat metric ηab

qab = ηab + hab ab ab ab q = η h ; hab << 1, (18) − so that up to first order in hab we have [1, 3]

ac c qabq = δb . (19)

The induced 3-metric determinant, up to first order in the perturbation hab, is then given by q = =1+ h, (20)

ab ab where h = η hab is the trace of hab. In the linearized theory, indexes are raised and lowered with ηab and η ab rather than gab and g . The linearized lapsus and shift are given by N = 1+ ν; ν << 1

Na = νa; νa << 1, (21) as can be seen from their relationship with the metric components (equations (3)). Replacing (18) and (21) in (9), recalling the ”exact” expression for the covariant derivative ofa 1 form − 3 3 c aνb = ∂aνb Γ νc, (22) ∇ − ab and substituting the linearized Christoffel symbols [3]

3 a 1 a a a Γ = (∂bh c + ∂kh b ∂ hbc), (23) bc 2 − we obtain the extrinsic curvature 1 Kab = (∂ hab ∂aνb ∂bνa), (24) 2 0 − − up to first order in hab. Also, substituting (24) in (13) we obtain the linearized conjugate momenta (13)

1 1 c pab = (Kab ηabK)= (∂ hab ηab∂ h)+ (∂aνb + ∂bνa 2ηab∂ νc). (25) − − −2 0 − 0 2 − 5

To obtain the linearized equations of motion we have to keep terms up to second order in hab in the the ADM action. The second order Ricci scalar can be found from equation (6)

3 bd 3 a bd bd 3 a R = δacq R = δac(η h ) R , (26) bcd − bcd where in the exact expression for the Riemman tensor

3 a 3 a 3 a 3 e 3 a 3 e 3 a R = ∂b Γ ∂c Γ + Γ Γ Γ Γ , (27) bcd cd − bd cd be − bd ce we have to substitute (23). After that substitution we obtain

3 a 1 a a 1 a a 1 e e e a a a R = (∂b∂dh ∂ ∂bhcd) (∂c∂dh ∂c∂ hbd)+ (∂ch + ∂dh ∂ hcd)(∂bh + ∂eh ∂ hbe) bcd 2 c − − 2 b − 4 d c − e b − 1 e e e a a a (∂bh + ∂dh ∂ hbd)(∂ch + ∂eh ∂ hce), (28) − 4 d b − e c − where we have kept terms up to second order in the perturbation hbd. Replacing the above equation in (26) we obtain, to the desired order

3 bd 3 a bd bd 3 a a a b 1 b c a 1 a bc 1 a R = q R = (η h ) R = ∂ ∂ah ∂ ∂ hab ∂ah ∂ h + ∂ hbc∂ah + ∂ h∂ah. (29) bad − bad − − 2 c b 4 4 Here we have already neglected those terms that will produce either cubic contributions or total derivative contributions to the action. After this we are ready to write down the term N 3q 3R up to second order in the perturbation: p

3 3 1 a a b 1 b c a 1 a bc 1 a N q R = (1+ ν)(1 + h)(∂ ∂ah ∂ ∂ hab ∂ahc∂ hb + ∂ hbc∂ah + ∂ h∂ah) p 2 − − 2 4 4 1 b c a 1 a bc 1 a 1 a b a a b = ∂ah ∂ h + ∂ hbc∂ah ∂ h∂ah + ∂ h∂ hab + ν(∂ ∂ah ∂ ∂ hab) −2 c b 4 − 4 2 − a a b = T + ν(∂ ∂ah ∂ ∂ hab), (30) − where we have defined

1 b c a 1 a bc 1 a 1 a b T = ∂ah ∂ h + ∂ hbc∂ah ∂ h∂ah + ∂ h∂ hab, (31) −2 c b 4 − 4 2 ab 2 and neglected total derivative terms. Finally, the contribution coming from the term KabK K can be written as a function of the linearized conjugate momentum as −

ab 2 ab 1 2 KabK K = p pab p . (32) − − 2 Substituting (18), (20), (21), (30) and (32) in (12) we obtain the cuadratic action (that provides the linearized canonical equations of motion) as

4 ab ab 1 2 a b a ba S = d x(p q˙ab + p pab p + T ν(∂ ∂ hab ∂ ∂ah)+2νa∂bp ). (33) Z − 2 − − From this expression we read the linearized scalar and vectorial constraints, which are

a b a = ∂ ∂ hab ∂ ∂ah Sb ab − V = 2∂ap . (34) − Defining

ab 1 2 = (p pab p + T), (35) H − − 2 6

the linearized action can be written down as

4 ab a S = d x(p h˙ ab ν νaV ). (36) Z −H− S− From the above equation we see that the Hamiltonian of the theory is

3 a H = d x( + ν + νaV ). (37) Z H S Thereby, the dynamics of the linearized theory is given by

p˙ab = pab,H { } q˙ab = hab,H , (38) { } with

′ ′ ′ ′ ′ ′ a b a b a b 3 qab(x),p (y) = (δa δb + δb δa )δ (x y) { ′ ′ } − ab a b p (x),p (y) = qab(x), qa′b′ (y) =0, (39) { } { } being the canonical algebra obeyed by the linearized variables, which is readily obtained from equations (17).

IV. NON COORDINATE BASIS

Until now we have been working in a coordinate basis ∂a (a =1, 2, 3) of the tangent space at p Στ ; however, ∈ we can also associate to each p a non coordinate basis ei (i = 1, 2, 3) [3, 14]. Non coordinate basis play an important role in the Ashtekar formulation of gravity, whose linearized version is our objective. These two basis are related by ei aei = ∂a, (40) ei Following the usual practice, we shall refer both to the basis vectors ei and to the components a of the coordinate ea ei ea a eaei i basis in the new one as the triad ( i is then the inverse triad: b i = δb , j a = δj ). The scalar product of vectors is given by ei ej ei ej q(∂a, ∂b)= qab = q( aei, bej)= a bq(ei,ej), (41)

hence, if the non coordinate basis is orthonormal (q(ei,ej)= ηij , ηij being the Euclidean metric) we shall have

ei ej qab = a bηij (42) and eaeb ηij = i j qab. (43)

a The densitized triad Ei is defined as a eea Ei = i , (44) ei 2 e2 with (det a) = = q. From this we have

ab a b ij qq = Ei Ej η . (45) 7

In order to achieve our goal of obtaining linearized gravity in Ahstekar variables, we have first to write (42), (43) and (45) in terms of the linearized metric to get the linearized densitized triad. To this end we observe that from its definition, the densitized triad should be written as

a a a Ei = δi + ei ; (46)

a then, substituting in (45) and keeping terms up to second order in the perturbation ei of the densitized triad we obtain

qqab = EaEbηij (1 + h)(ηab hab) = (δa + ea)(δb + eb)ηij = δaδbηij + δaebηij + δbeaηij . i j ⇒ − i i j j i j i j j i Hence

hab + hηab = δaebηij + δbeaηij . (47) − i j j i Taking the time derivative of (47) we get

′ ′ ab ab ab ab a b ab ab a b b a ij pabh˙ = (Kab ηabK)h˙ = (Kabh˙ ηa′b′ Kabη h˙ )= Kab(h˙ η h˙ )= Kab(δ e˙ + δ e˙ )η − − − − − − i j j i a b ij b ˙j = 2Kabδi ej η = ej kb , (48) where we have defined

j a ij k = 2Kabδ η , (49) b − i and a total time derivative has been neglected because this expression is going to be substituted in the action (36), and total derivatives in the Lagrange density do not affect the equations of motion. It can be shown that

ea(x), kj (y) = δaδj δ3(x y) { i b } b i − ea(x),eb(y) = ki (x), kj (y) =0, (50) { i j } { a b } a i hence, ei and ka form a new set of canonical variables. In what follows, we will calculate all the quantities needed to write the first order action (36) in terms of these new canonical variables. From (47), we have

i a a i h = δaei = δi ea. (51)

Replacing this in (47) we obtain

hab = δaebηij δbeaηij + δiecηab. (52) − i j − j i c i ea Now, from equations (44) and (46) we obtain the linearized triad i 1 ea = δa + ea ebδj δa, (53) i i i − 2 j b i ei so that its inverse a is given by 1 ei = δi δj δieb + δi δj eb. (54) a a − a b j 2 a b j

Replacing (54) in (43) we can write the linearized metric qab as

i j k c k c k c qab = e e ηij = ηab ηacδ e ηbcδ e + ηabδ e , (55) a b − b k − a k c k 8

a up to first order in ei . Comparing this expression with qab = ηab + hab, we read that k c k c k c hab = ηacδ e ηbcδ e + ηabδ e , (56) − b k − a k c k hence, substituting (56) and (51) in (34) we obtain the scalar constraint b k c = 2∂ δ ∂ce 0. (57) S − b k ≈ In turn, replacing (49) in (34) we also obtain the vectorial constraint c i b i Va = δ ∂ck δ ∂ak 0. (58) i a − i b ≈ Since the extrinsic curvature is a symmetric tensor, and this is not reflected in the Poisson algebra (50), we must force this symmetry in the form of another constraint abc ε Kab =0, (59) so that, substituting equation (49) in the above expression we obtain ikl a j ε δk ηjika =0, (60) which is going to be the linearized Gauss constraint Gl when we arrive to the Ashtekar variables. This new constraint, together with equations (57) and (58), forms a set of first class constraints.

ab 1 2 Using equations (32) and (49) we can rewrite the term p pab p as − 2 ab 1 2 1 a b i j i j p pab p = δ δ (k k k k ). (61) − 2 −4 i j a b − b a Putting all this together in the action (33) we obtain

′ ′ ′ 4 a i 1 a a j j j j a i S = d x(e k˙ + δ δ ′ (k ′ k k k ′ )+ T ν νaV NiG ) (62) Z i a 4 j j a a − a a − S− − T a (the explicit expression for as a function of ei is unnecessary at this point, and we omit it for the sake of brevity). Moreover b k c = 2∂ δb ∂cek 0 S −c i b ≈ i Va = δi ∂cka δi ∂akb 0 l ikl a − j ≈ G = ε δ ηjik 0. (63) k a ≈ From the first order action we read, besides the constraints, the Hamiltonian of the theory

′ ′ ′ 3 1 a a j j j j a i H = d x( δ δ ′ (k ′ k k k ′ ) T + ν + νaV + NiG ). (64) − Z −4 j j a a − a a − S The equations of motion are given by e˙a = ea,H i { i } k˙ i = ki ,H , (65) a { a } with ea(x), kj (y) = δaδj δ3(x y) { i b } b i − ea(x),eb(y) = ki (x), kj (y) = 0 (66) { i j } { a b } being the canonical algebra. The dynamics must be complemented with the constraints (57), (58) and (60). We have then attained a description of linear gravity in terms of the linearized densitized triad and the linearized extrinsic curvature as conjugate variables, starting from the ADM formulation of linearized gravity. This setting provides a starting point to obtain the Ashtekar formulation of linearized gravity, which is our next step. 9

V. ASHTEKAR VARIABLES

In order to carry out the canonical transformation towards the linearized Ahstekar variables we must introduce the linearized spin connection, which allows to express the covariant derivative in the non coordinate basis. To i this end let us recall some definitions of the ”full” non-linear theory. If = ei is a contravariant vector field, we have [3] A A

i a i i k j a = ( a )dx ei = (∂a + εj kΓ )dx ei, (67) ∇A ∇ A ⊗ A aA ⊗ k where Γa is the spin connection, that can be thought as a SU(2) connection [2, 14, 17, 18]. To relate the spin connection with the affine connection we rewrite the last expression as

i a i i k j a = ( a )dx ei = (∂a + εj kΓ )dx ei ∇A ∇ A ⊗ A aA ⊗ i b i k j b a c = (∂a(e )+ εj kΓ e )dx (e ∂c) bA a bA ⊗ i c b c i c i k j a = (∂a + (e ∂ae + e εj kΓ e ))dx ∂c. (68) A A i b i a b ⊗ On the other hand, in the coordinate basis one has

b a b 3 b c a = ( a )dx ∂b = (∂a + Γ )dx ∂b, (69) ∇A ∇ A ⊗ A acA ⊗ 3 a where Γbc are the Christoffel symbols [2, 3]. Comparing both expressions we obtain ec ei i ecej k 3 c i ∂a b + εj k i bΓa = Γab. (70) The linearized version of this expression is obtained with the aid of equations (53) and (54)

i k i b 3 c i c 1 i k d εj kΓ = δ δ Γ + δ ∂ae δ δ ∂ae , (71) a c j ab c j − 2 j d k 3 c i where Γab must be substituted by the linearized Christoffel symbols given by equation (23), while Γa is now the linearized spin connection. From this expression we have

j k a j c 3 b j c 1 j l c a εi kΓ δ = (δ δ Γ + δ ∂ae δ δ ∂ae )δ a j b i ac c i − 2 i c l j a = ∂aei , (72) a 1 1 i c where we have used Γac = 2 ∂ah = 2 δc∂aei . ˜j In terms of the linearized spin connection we define the U(1) connections Aa (one for each internal index) ˜j j j Aa =Γa + βka, (73) where β is an arbitrary constant that corresponds to the Immirzi parameter of the ”full” theory [15, 17, 18]. ˜j a a ˙ i The linearized variables Aa and ei are canonical, as we shall show. First, observe that the term ei ka of the linearized action becomes 1 eak˙ i = ea(A˜˙i Γ˙ i )=˜eaA˜˙ i e˜aΓ˙ i , (74) i a β i a − a i a − i a

a −1 a withe ˜i = β ei . Now, from the ”exact” theory we know that [12]

a i 1 abc j i E Γ˙ = ε ∂a(e˙ e ηij ), i a −2 b c whose linearized version, using (46) (53) and (54), is given by

a i β abc j i a i e˜ Γ˙ = ε ∂a(e ˙ e ηij ) βδ Γ˙ . (75) i a − 2 b c − i a 10

a ˙ i Hence, ei Γa can be written down as the sum of total derivative terms which do not contribute to the equations of motion. Therefore, under appropriate boundary conditions, we can make the substitution

a ˙ i a ˜˙ i ei ka =e ˜i Aa (76)

a ˜i in the action. On the other hand, it can be shown that the Poisson algebra betweene ˜i and Aa is given by

e˜a(x), A˜j (y) = δaδj δ3(x y) { i b } b i − e˜a(x), e˜b(y) = A˜i (x), A˜j (y) =0, (77) { i j } { a b } a ˜i wherebye ˜i and Aa form a pair of canonical variables. Hence, the passage to these new variables constitutes a canonical transformation. In what follows we shall rewrite all the relevant quantities of the previous section in terms of the new canonical variables. Using (72) the scalar constraint can be written as

k b c k c m b l = 2δ ∂ ∂ce =2δ δ εkl ∂ Γ . (78) S − b k b m c Substituting equation (73) in the last expression we have

k c m b l k c m b l l k c m b l 2δ δ εkl ∂ Γ =2δ δ εkl ∂ (A˜ βk )=2δ δ εkl ∂ A˜ , (79) b m c b m c − c b m c where we have neglected terms proportional to the Gauss constraint. Then, by defining

l l l f = ∂bA˜ ∂cA˜ (80) bc c − b the scalar constraint can be written as

k c m l = δ δ εkl f 0. (81) S b m bc ≈ In turn, the vectorial constraint

c i c i Va = δ ∂ck δ ∂ak 0 (82) i a − i c ≈ can be written as

−1 c i i −1 c i i Va = β δ (∂cA˜ ∂aA˜ ) β δ (∂cΓ ∂aΓ ) 0, (83) i a − c − i a − c ≈ where equation (73) has been used. The second term in this equation vanishes. In fact, from (71), and up to a first order in ei we have

i 1 ijk b l j c 1 j l c l j c 1 j l c 1 j l c c l a Γ = ε δ [ δ δ ∂be + δ δ ∂be + δ δ ∂ae δ δ ∂ae + δ δ ∂be δ δ ∂be ]. (84) a 2 k − a c l 2 a c l b c l − 2 b c l 2 a c l − j c l Hence,

d i i 1 dcb l c 1 dcb l c 1 dcb l c 1 cdb l c δ (∂dΓ ∂aΓ ) = ε δ ∂d∂ae + ε δ ∂a∂be ε δ ∂d∂ae + ε δ ∂b∂ae =0. (85) i a − d 2 b l 2 d l − 2 b l 2 d l In view of this, the vectorial constraint can be finally cast in the form

c i Va = δ˜ f 0, (86) i ca ≈ ˜a −1 a with δk = β δk . Regarding the Gauss constraint, it can be written as

k j a k j a Gi = εij k δ = εij βk δ˜ 0. (87) a k a k ≈ 11

But from equation (72) we have a ˜j ˜a a k j ˜a ∂ae˜i + εijkΓaδk = ∂ae˜i + εij Γaδk =0, (88) since the spin connection is invariant under a re-scaling of the triad [12]. Introducing the above result in the expression for the Gauss constraint we arrive to

a k j a k j a a k j a Gi = ∂ae˜ + εij Γ δ˜ + εij βk δ˜ = ∂ae˜ + εij A˜ δ˜ 0. (89) i a k a k i a k ≈ Our final step will be to write T as a function of the new canonical variables. From equation (30) we can write

3 a b a N√q R = T ν(∂ ∂ hab ∂ ∂bh), (90) − − which, up to second order, yields 1 (1 + h)3R = T. (91) 2 Here, 3R is the second order Ricci scalar, which must be written in terms of the new set of variables. To this end, we recall the following exact expressions. The curvature tensor is a function of the Christoffel symbols

3Ra = ∂3Γa ∂3Γa + 3Γe 3Γa 3Γe 3Γa , (92) bcd b cd − c bd cd be − bd ce and from equation (70) we have

3 a ea ei eaej i k Γcd = i ∂c d + i dεj kΓc . (93) Substituting this in equation (92) we obtain

3 a a j i l R = e e ε lj , (94) bcd i d Fbc with

′ l l l l k k = ∂bΓ ∂cΓ + εk k′ Γ Γ . (95) Fbc c − b c b i Now we take Γa as the linearized connection and replace it according to the canonical transformation that i defines the U(1) connection Aa

Γi = A˜i k˜i , a a − a ˜i i with ka = βka. This yields, up to the second order in the linear canonical variables

′ l l l k k l = F + εk k′ k˜ k˜ + D k˜ , (96) Fbc bc c b [c b] with

′ l l l l k k F = ∂bA˜ ∂cA˜ + εk k′ A˜ A˜ , (97) bc c − b c b and ˜l ˜l i j ˜k Dckb = ∂ckb + ε jkΓckb . (98) The next step consist in using equations (53), (54) and (96) into (94) to build 3R up to the desired order. We have

′ ac bd a a j ac bd i l i a b a b a b l i l a b k k 3 3 e e ′ ˜ ˜ R = δ q Rbcd = i dδ q ε lj bc = ε lj (δi δj + δi ej + ei δj )Fba + ε lj εk k δi δj kakb F ′ ′ i a b l i l ′ a b ˜k ˜k i l ′ a b˜k ˜k = 2ε lj δi ej fba + ε lj εk k δi δj AaAb + ε lj εk k δi δj ka kb , (99) 12

l ˜l ˜l where terms proportional to the scalar constraint have been neglected, and we have defined fba = ∂bAa ∂aAb. Substituting equation (99) in (91) we get −

1 3 i a b l a b i j j i a b i j j i T = (1+ h) R =2ε lj δ e f + δ δ (A˜ A˜ A˜ A˜ )+ δ δ (k˜ k˜ k˜ k˜ ). (100) 2 i j ba i j a b − a b i j a b − a b Finally, the linearized action of the theory in terms of the linear Ashtekar variables results to be

S = d4x(˜ei A˜˙ i νdV L νSL N iGL), (101) Z a a −H− d − − i where

k c m l = δ δ εkl f 0 S b m bc ≈ c i Va = δ˜ f 0 i ca ≈ a k j a Gi = ∂ae˜ + εij A˜ δ˜ 0 (102) i a k ≈ are the constraints and

2 1 i a b l a b i j j i (β 4 ) a b i i j j j j i i =2ε jlδ e f δ δ (A˜ A˜ A˜ A˜ ) − δ δ [(Γ A˜ )(Γ A˜ ) (Γ A˜ )(Γ A˜ )]. (103) H i j ba − i j a b − a b − β2 i j a − a b − b − a − a b − b is the Hamiltonian density.

VI. CONCLUDING REMARKS

The expressions for the Hamiltonian and the constraints obtained in the present article coincide with those of reference [24], were the procedure for attaining the linearized Ashtekar formulation was different to the one we followed. In our case, we first made the linearization from the first order ADM action and then performed the passage to ”linear” Ashtekar variables. In the previous works, instead, the linearization was performed after having formulated the ”full” theory in the Ashtekar new variables. Nevertheless, there is a subtle conceptual difference between both approaches, regarding the linearization point, which is worth mentioning. One might wonder about where comes the linear Hamiltonian from, since in the full theory there is no Hamiltonian at all, but just constraints. The answer is that the Hamiltonian comes from the multiplication of the 0 th order lagrange multipliers (the lapsus and the shift functions) times the quadratic part of the scalar and− vectorial constraints. Now, at the level of the ADM linear action, it is obvious what these 0 th orders should be: it suffices to see how the lapse and the shift relate with the metric components (equations− (3)) to get the answer (equations (21)). We believe that this point can be better understood within the approach discussed in this article than in the standard one. In fact, one could conceive different linearizations starting from the ”full” Ashtekar formulation, in which there is no Hamiltonian at all [10]. This amounts to taking a linear theory different from the Fierz-Pauli one, which could also be consistent, but that could lead to different physical predictions. Finally, it is interesting to notice that the linear theory, unlike the ”full” one, could admit different versions of the ”Loop Representation”, in the following sense. Being an Abelian theory (like the Maxwell theory), there exist the possibility of both an ”electric” and a ”magnetic” representation. In the former, the linearized triad would act as the loop form factor (i.e. the ”loop coordinate”), the linear Ashtekar connection taking the role of the ”path derivarive” and its curl acting as a ”loop derivative”. But since the linear theory is dual (in the ”electric-magnetic” sense) (see reference [9]), these roles could be interchanged, just as in Maxwell theory. On the other hand, and closely related with the previous discussion, it seems possible to consider the introduction of a ”Loop Representation” at stages previous to the introduction of the linearized Ashtekar variables. For 13

a i instance, the canonical pairs (ei , ka) could serve as a starting point for doing this. These aspects are currently under work.

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