Off-shell Noether currents and potentials for first-order general relativity

We report off-shell currents obtained from off-shell Noether potentials for first-order general relativity described by $n$-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are obtained from the use of the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms, local $SO(n)$ or $SO(n-1,1)$ transformations, `improved diffeomorphisms', and the `generalization of local translations' of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether's theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the `half off-shell' case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local $SO(3,1)$ or $SO(4)$ Noether potentials and currents for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the on-shell case. We also study Killing vector fields in the half off-shell and on-shell cases. The current theoretical framework is illustrated in static spherically symmetric and Friedmann-Lemaitre-Robertson-Walker spacetimes in four dimensions.


Introduction
Some of the more fundamental results in mathematical physics are Noether's theorems [1,2,3], which establish a deep connection between infinitesimal symmetries of a variational principle and conservation laws. There are two Noether's theorems: the first one dealing with global (or rigid) symmetries and the second one concerning local (or gauge) symmetries. These theorems have forged and shaped the modern view of theoretical physics, furnishing a vast amount of applications in many areas of physics.
Noether's theorems provide powerful tools to calculate conserved currents and charges of physical systems. This feature has been highly exploited by the gravitational community and is an active ingredient of the modern research in general relativity and other alternative theories of gravity [4,5,6,7,8,9,10]. For instance, the definition of energy in generally covariant systems is a rather delicate issue [11] and Noether's theorem plays a central role in addressing it.
To obtain conserved charges associated to spacetime diffeomorphisms in gravitational systems, Noether's second theorem is implemented in its direct fashion (its converse also holds), leading to the construction of an associated Noether current that is conserved on-shell [4,7,9]; that is, when the equations of motion are satisfied. This is the usual viewpoint taken and one might wonder whether it is really necessary to work on-shell to obtain such conservation laws. After all, in any gauge theory there exists Noether currents that are identically off-shell conserved [5], thus leading to the definition of off-shell potentials and charges.
Some years ago, off-shell Noether currents and potentials were introduced to define quasi-local charges in any theory of metric gravity invariant under diffeomorphisms [12,13]. Later on, an analogous proposal was put forward for gravitational theories within the first-order formalism [14,15], with again off-shell Noether currents and potentials playing an essential role.
It is well-known that, within the first-order formalism, general relativity can be described by either the Palatini action or the Holst action [16], the latter one being the starting point of the loop approach to quantum gravity in its canonical and covariant versions [17,18,19,20]. In particular, the Holst action contains the so-called Immirzi or Barbero-Immirzi parameter [21,22], which plays no role when the equations of motion are satisfied, but on which the theory strongly depends when away from them (off-shell). In fact, this parameter affects the way in which fermions couple to gravity [23,24,25,26] and manifests itself in the spectra of geometric quantum operators [27,28,18] and in the black hole entropy [29,30,31,32,33] derived within the loop framework.
To expand our horizons on the off-shell effects of the Immirzi parameter, in this paper we report off-shell Noether currents and potentials for general relativity with a cosmological constant in the first-order formalism for both the Palatini Lagrangian in n dimensions and the Holst Lagrangian in four dimensions. We report these expressions for the gauge transformations that include: transformations induced by diffeomorphisms of the manifold onto itself, local SO(n) or SO(n − 1, 1) transformations, 'improved diffeomorphisms', and the so-called 'generalization of local translations' of the orthonormal frame and the connection [34]. We find that the Immirzi parameter enters in the definition of the off-shell Noether potentials and currents for the Holst Lagrangian. We also find that the currents for both 'improved diffeomorphisms' and the 'generalization of local translations' identically vanish. Nevertheless, we also show that from these symmetries we can obtain the off-shell current and potential for diffeomorphisms for both Lagrangians. The novelty of our approach is that it takes advantage of the Noether identities arising from both Lagrangians and avoids the use of Noether's second theorem in its direct version.
Additionally, we consider diffeomorphisms generated by Killing vector fields and determine their corresponding off-shell Noether currents and potentials. This leads to the introduction of an effective gauge transformation, and we report its corresponding off-shell Noether potentials and currents. We also work 'half off-shell' (in a sense made precise in Section 6) and explicitly compute the resulting Noether currents and potentials for both Palatini and Holst Lagrangians. We find that in the 'half offshell' case the resulting diffeomorphism and SO (3,1) or SO(4) Noether potentials and currents for the Holst Lagrangian still generically depend on the Immirzi parameter, even though the ones for the second-order Lagrangian (Einstein-Hilbert Lagrangian in terms of the tetrad) are independent of it. This implies that the Immirzi parameter is going to be present in these currents and potentials even 'on-shell'. Furthermore, even though the Noether potential for the effective gauge transformation for the Holst Lagrangian in the 'half off-shell' case still depends on the Immirzi parameter, the 'half off-shell' current is independent of it. Finally, the current theoretical framework is illustrated in four-dimensional static spherically symmetric and Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes.
We follow the notation and conventions of Ref. [34]. Let M be an n-dimensional Lorentzian or Riemannian manifold. In the first-order formalism, the fundamental variables are an orthonormal frame of 1-forms e I and a connection 1-form ω I J compatible with the metric (η IJ ) = diag(σ, 1, . . . , 1), dη IJ − ω K I η KJ − ω K J η IK = 0, and therefore ω IJ = −ω JI because frame indices I, J, K, . . ., which take the values 0, 1, . . . , n−1, are raised and lowered with η IJ . For σ = −1 the frame rotation group is the Lorentz group SO(n − 1, 1), whereas for σ = 1 it is the rotation group SO(n). The SO(n − 1, 1) [or SO(n)] totally antisymmetric tensor ǫ I1...In is such that ǫ 01...n−1 = 1. The symbols ∧, d, and L ζ stand for the wedge product, exterior derivative, and the Lie derivative along the vector ζ of differential forms, respectively. Furthermore, stands for the contraction of a vector field with a differential form [35], the volume form is given by η = (1/n!)ǫ I1...In e I1 ∧ · · · ∧ e In , ⋆ is the Hodge dual, and D stands for the covariant derivative with respect to ω I J . Antisymmetric tensors involving frame indices are defined by It is worth pointing out that in this paper we focus our attention on the Lagrangian n-form instead of the action principle, which in turn is defined as the integral of the Lagrangian over a determined spacetime region. Thus, the Lagrangian itself completely specifies the theory under consideration.

Palatini Lagrangian
First-order general relativity in n-dimensions with (or without) a cosmological constant Λ can be described by the action principle constructed out of the ndimensional Palatini Lagrangian where R I J = dω I J + ω I K ∧ ω K J is the curvature of ω I J . Because we do not consider matter fields in this paper, we could omit the constant κ := (16πG) −1 in the previous Lagrangian. However, we will keep it for dimensional reasons (so that the Lagrangian has dimensions of action). A general variation of the Palatini Lagrangian under the corresponding variations of the frame e I and the connection ω I J takes the form where the variational derivatives E I and E IJ are given by

Off-shell current and potential for diffeomorphisms
By handling the variational derivatives E I and E IJ given in (3)-(4), we get the off-shell Noether identity [34] satisfied by the change of e I and ω I J under an infinitesimal diffeomorphism generated by ζ (converse of Noether's second theorem).
By computing the variation (2) for the change of e I and ω I J under an infinitesimal diffeomorphism generated by ζ, we obtain Using (5), we rewrite the right-hand side of last expression as that is, as an exact form. It is remarkable that the terms inside the braces can be rewritten as The meaning of the off-shell identity (8) is better appreciated by noting that it has the form where the off-shell current J ζ has been defined by with corresponding off-shell Noether potential U ζ given by It follows from (9) that J ζ is off-shell conserved dJ ζ = 0. (12) The potential (11) is the off-shell version of the one obtained in the on-shell paradigm [36], for diffeomorphism transformations in the case of the n-dimensional Palatini Lagrangian without a cosmological constant. Here we have shown that the off-shell potential (11) gives rise to the off-shell Noether current (10) without the need to invoke the equations of motion at any time.
Note that the off-shell current (10) can be further simplified and it acquires the off-shell form J ζ = (−1) n−1 ζ ω IJ E IJ + (−1) n κ ⋆ (e I ∧ e J ) ∧ D ζ ω IJ . (13) This expression is remarkable because it involves neither E I or L P , in contrast to (10).
Let us emphasize that the off-shell aspect is just one of the important features of the theoretical framework developed in this paper. The second aspect is that Noether's theorem for gauge transformations (also called Noether's second theorem) was not used at all to get (9). This is another key difference between the approach of this paper and previous ones [4,36,37,38], i.e., we did not assume (nor use) that under an infinitesimal diffeomorphism generated by ζ the change of the action is as is usually assumed when dealing with Noether's theorem for diffeomorphism transformations. In our approach, the relation (14) holds, of course, but it is deduced from the combination of (7) and (8).

2.2.
Off-shell current and potential for local SO(n − 1, 1) or SO(n) transformations Similarly, by handling the variational derivatives E I and E IJ given in (3)-(4) we get the off-shell Noether identity satisfied by infinitesimal local SO(n − 1, 1) or SO(n) transformations of e I and ω I J with τ IJ = −τ JI being the gauge parameter (converse of Noether's second theorem).
On the other hand, by evaluating the variation (2) for an infinitesimal local SO(n − 1, 1) or SO(n) transformation of e I and ω I J , we obtain Using (15), the right-hand side of (16) acquires the form Once again, the terms inside the brackets can be written as This off-shell identity has the form where we have defined the off-shell current J τ by and the off-shell Noether potential U τ as It is clear from (19) that J τ is off-shell conserved, To close this Subsection, note that the structure of J τ in (20) resembles that of the diffeomorphism current (13).

Off-shell current for 'improved diffeomorphisms'
It is pretty obvious that we can combine the relations (9) and (19) involving the off-shell currents and Noether potentials. In particular, by adding them and taking a field-dependent local SO(n − 1, 1) or SO(n) transformation with gauge parameter τ IJ = ζ ω IJ , we get the off-shell relation −ζ L P + (−1) n−1 ζ e I E I + κ ζ R IJ ∧ ⋆ (e I ∧ e J ) = 0.
This off-shell identity is nothing but the current for an 'improved diffeomorphism' along a vector field ζ, as we show below. In fact, from the variational derivatives E I and E IJ given in (3)-(4) we derive the off-shell Noether identity satisfied by the change of e I and ω I J under an 'improved diffeomorphism', given by δ ζ e I := D ζ e I + ζ De I which are a linear a combination of a diffeomorphism transformation and a fielddependent local SO(n − 1, 1) or SO(n) transformation with gauge parameter τ IJ = ζ ω IJ . On the other hand, by equating the variation in (2) with the change of e I and ω I J under an 'improved diffeomorphism', we obtain (27) Using (24), the previous expression becomes However, the terms inside square brackets can be written as so that for an 'improved diffeomorphism' the off-shell Noether current identically vanishes: Note that (30) is precisely (23).
It is important to remark that the procedure to arrive at (30) differs from the one followed to get the off-shell Noether potential and current for diffeomorphisms found in Subsection 2.1. The difference relies in the fact that to arrive at (30) Cartan's formula was not used at all. If we use it as then the expression (29) is written as into (32) we get precisely (8), from which (9) arises. Therefore, from 'improved diffeomorphisms' we also obtain the off-shell Noether currents and potentials associated to diffeomorphisms.

Off-shell current for the 'generalization of local translations'
It was shown some years ago [34] that the variational derivatives E I and E IJ given in (3)-(4) can be handled to give the off-shell Noether identity with Here, ρ I is the gauge parameter, R IJ KL are the components of R IJ with respect to the orthonormal frame, R IJ = (1/2)R IJ KL e K ∧ e L ; R I J := R IK JK is the Ricci tensor, and R := R I I is the curvature scalar. The off-shell Noether identity (34) gives the gauge transformation of e I and ω I J named 'generalization of local translations' because it is the generalization of the socalled 'local translations' that exist in three dimensions (see [39] for a simple derivation of this symmetry in three dimensions and [40] for a straightforward derivation in four dimensions). By computing the variation (2) for a 'generalization of local translations' of e I and ω I J , we have Using (34), the right-hand side of last expression becomes If we define the vector field ρ = ρ I ∂ I , where ∂ I is the dual basis of e I (i.e., ∂ I e J = δ J I ), then ρ I = ρ e I . Using this definition, the terms inside square brackets of the previous expression can be expressed as Thus, the off-shell Noether current associated to the 'generalization of local translations' identically vanishes: Furthermore, using the off-shell identity the off-shell current (41) becomes precisely the one given in (30) with ζ replaced with ρ. However, note that the identity (42) can be used differently. If (42) is substituted into (40), we get (29) with ζ replaced with ρ, from which (8) and (9) arise as we explained in Subsection 2.3. Therefore, from the 'generalization of local translations' we also obtain the off-shell Noether current and potential associated to diffeomorphisms.

Holst Lagrangian
In four spacetime dimensions, the Holst action [16] with a cosmological constant Λ is given by the action principle determined by the Lagrangian where The variation of the Lagrangian (43) under general variations of the independent variables e I and ω I J reads where the variational derivatives E I and E IJ are given by 3.1. Off-shell current and potential for diffeomorphisms Using E I and E IJ , we obtain the off-shell Noether identity satisfied by the change of e I and ω IJ under an infinitesimal diffeomorphism generated by ζ (converse of Noether's second theorem). Then, evaluating the variation (44) for the change of e I and ω I J under an infinitesimal diffeomorphism generated by ζ, we obtain (48) which, using (47), is written as Note that the terms inside the brackets can be written as This off-shell identity has the form where we have defined the off-shell current J ζ by and the off-shell Noether potential U ζ by Then, expression (51) implies that J ζ is off-shell conserved Note that the off-shell current (52) can be further simplified off-shell, giving This expression is relevant because it involves neither E I or L H , in contrast to (52).

Off-shell current and potential for local SO(3, 1) or SO(4) transformations
Similarly, using E I and E IJ , we obtain the off-shell Noether identity satisfied by infinitesimal local SO(3, 1) or SO(4) transformations of e I and ω IJ with τ IJ = −τ JI being the gauge parameter (converse of Noether's second theorem). Now, computing the variation (44) for an infinitesimal local SO(3, 1) or SO(4) transformation of e I and ω I J , we get Using (56), the right-hand side of the previous expression takes the form The terms inside the parenthesis can be written as This off-shell identity has the form where we have defined the off-shell current J τ by and the off-shell Noether potential U τ by It follows from (60) that J τ is off-shell conserved, Notice that the structure of the SO(3, 1) or SO(4) current (61) resembles that of the diffeomorphism current (55).

Off-shell current for 'improved diffeomorphisms'
By combining the variational derivatives E I and E IJ , we obtain the off-shell Noether identity satisfied by the change of e I and ω I J under an 'improved diffeomorphism'. By calculating the variation (44) for the change of e I and ω I J under an 'improved diffeomorphism', we have Using (64), the right-hand side of (65) can be written as Notice that the terms inside the brackets can be written as which means that for an 'improved diffeomorphism' the off-shell Noether current identically vanishes: To close this Subsection, we remark that like for the n-dimensional Palatini Lagrangian, we can also use Cartan's identity (31) to rewrite (67) as into (69), we obtain (50), and then (51) arises. Therefore, from 'improved diffeomorphisms' we also obtain the off-shell Noether current and potential associated to diffeomorphisms.

Off-shell current for the 'generalization of local translations'
By handling the variational derivatives E I and E IJ , we get the off-shell identity [34] with where we have defined for R I J ∧ e J =: (1/3!)B I JKL e J ∧ e K ∧ e L , and ρ I is the gauge parameter. The off-shell Noether identity (71) gives the gauge transformation of e I and ω IJ named 'generalization of local translations.' By equating the variation in (44) with a 'generalization of local translations' of e I and ω IJ , we have Then, using (71), the right-hand side of this expression takes the form Defining the vector field ρ = ρ I ∂ I , then ρ I = ρ e I and the terms inside the parenthesis of (78) can be written as This implies that for the 'generalization of local translations' the off-shell Noether current identically vanishes: It is worth noting that using the off-shell identity the off-shell current (80) becomes precisely the one given in (68) with ζ replaced by ρ. However, note that the identity (81) can be used differently. If (81) is substituted into (79), we get (67) with ζ replaced with ρ, from which (50) and (51) arise as we explained in Subsection 3.3. Therefore, from the 'generalization of local translations' we also obtain the off-shell Noether current and potential associated to diffeomorphisms.

Off-shell Noether charges
An advantage (and a possible use) of the identities satisfied by the off-shell Noether currents and potentials reported in Sections 2 and 3 is that they always hold because no restrictions or specific hypotheses were imposed to obtain them. Therefore, these identities lead naturally to the definition of off-shell Noether charges via where Σ is an (n − 1)-dimensional surface and ∂Σ its boundary in the case of the Palatini Lagrangian in n-dimensions, whereas Σ is a three-dimensional surface for the Holst Lagrangian.
It is important to remark that the off-shell Noether charges (82) are also kinematical in the sense that the variational derivatives E IJ and E I are not set to zero. Nevertheless, the off-shell currents and potentials are constructed using the ndimensional Palatini and Holst Lagrangians, so they capture or encode the dynamical information contained in the Lagrangians through the way the frame e I and the connection ω I J couple to each other. After all, Palatini and Holst Lagrangians lead to the equations of motion for general relativity via the action principle (see [12,13] for the construction of an off-shell Noether current and potential in the metric secondorder formalism).
The right hand-side of (82) can be computed on-shell too, of course, because the off-shell identities between the off-shell potentials and currents are general. Due to the fact that the off-shell potentials for the Holst Lagrangian studied in Section 3 depend on the Immirzi parameter, we expect the resulting charges generically depend on this parameter too. Moreover, in Sections 6 and 7 we consider the 'half off-shell' case for the Holst Lagrangian (in the sense defined there) and the Immirzi parameter is present in the resulting expressions for the diffeomorphisms and SO(3, 1) or SO(4) potentials and currents. Therefore, from this it is deduced that the Immirzi parameter will generically appear in these expressions even 'on-shell' too. In fact, the on-shell case is illustrated in Section 7 and the Immirzi parameter is present.

Killing vector fields
If the vector field ζ is a Killing vector field, then the Lie derivative of the metric tensor along it vanishes, Since g = η IJ e I ⊗ e J , equation (83) implies that which means that the Lie derivative of the orthonormal frame e I equals an infinitesimal local SO(n − 1, 1) or SO(n) transformation of itself, for some suitable gauge parameter τ IJ (ζ). From this relation we obtain, in particular, the field-dependent gauge parameter τ IJ (ζ) (86) On the other hand, the Lie derivative of the connection ω I J with respect to a Killing vector is more involved because we are working off-shell, and we need to consider separately Palatini and Holst Lagrangians.

Palatini Lagrangian
Using (4) we express De I in terms of the variational derivatives E IJ , From this relation and (85) we obtain that the Lie derivative of the connection with respect to a Killing vector field equals a local SO(n) or SO(n − 1, 1) transformation plus a 'trivial gauge transformation' W IJ (see [41] for the definition of trivial gauge transformations) where τ IJ (ζ) is given by (86) and Thus, expressions (85) and (88) are the corresponding changes of the frame e I and the connection ω I J when the vector field ζ is a Killing vector field, for the Palatini Lagrangian (1). Equation (88) expresses the fact that the action of a Killing vector on the connection is compensated by a local SO(n) or SO(n − 1, 1) transformation and the 'trivial gauge transformation' given by the term W IJ proportional to the variational derivative E IJ according to (89) and (90); thus giving rise to an effective transformation as shown below. By substituting (85) and (88) into (5) we get Using (15) the previous expression acquires the form which involves the gauge transformation that leaves the frame e I unchanged, while the connection ω I J undergoes a 'trivial gauge transformation'. By taking the gauge transformation (93) as the starting point and applying the same procedure developed in Section 2, we get the off-shell relation with the off-shell potential U P and the off-shell current J P given by The potential and current can, alternatively, be written as where U ζ is given by (11) and J ζ is given by (13). Similarly, U τ (ζ) is given by (21) and J τ (ζ) is given by (20) with τ IJ (ζ) given by (86).

Holst Lagrangian
Using (46) we get De I in terms of the variational derivatives E IJ with From this relation and (85) we obtain that the Lie derivative of the connection with respect to a Killing vector field equals a local SO(3, 1) or SO(4) transformation plus a 'trivial gauge transformation' W IJ (= −W JI ) where τ IJ (ζ) is given by (86) and with Thus, expressions (85) and (101) are the corresponding changes of the frame e I and the connection ω I J when the vector field ζ is a Killing vector field, for the Holst Lagrangian (43). Again, the action of a Killing vector on the connection is compensated by a local SO(3, 1) or SO(4) transformation and the 'trivial gauge transformation' given by the term W IJ proportional to the variational derivative E IJ according to (102) and (103); thus giving rise to an effective transformation as shown below. In fact, by substituting (85) and (101) into (47) we get (104) Using (56) the previous expression acquires the form which involves the gauge transformation that leaves the frame e I invariant, while the connection ω I J undergoes a 'trivial gauge transformation'. By taking the gauge transformation (106) as the starting point and applying the same procedure developed in Section 3, we get the off-shell relation with the off-shell potential U H and the off-shell current J H given by The potential U H and current J H can, alternatively, be written as where U ζ is given by (53) and J ζ is given by (55). Similarly, U τ (ζ) is given by (62) and J τ (ζ) is given by (61) with τ IJ (ζ) given by (86). Note that the Noether potential U ζ for diffeomorphisms (53) when ζ is a Killing vector field can off-shell, alternatively, be rewritten as where here U n=4 ζ is in fact the corresponding Noether potential (11) for the Palatini Lagrangian in four-dimensional spacetimes.
Moreover, the potential U H (108) can be further simplified and it acquires the off-shell expression with U P given by (97). Alternatively, using (99), it can off-shell be written as Therefore, with J P given by (98).

Half off-shell case
There are essentially three different cases when dealing with gauge symmetries: the first case is defined by E I = 0 and E IJ = 0 and it is named 'off-shell' case. The second case is defined by E I = 0 and E IJ = 0 and it is named 'on-shell' case. The third case is defined by E I = 0 and E IJ = 0, and we named it 'half off-shell' case (other possible names for this case are 'half on-shell' and 'semi on-shell'). The 'half off-shell' case is the central focus of this section and will be illustrated with examples in Section 7 to appreciate the explicit expressions for the currents and potentials in spacetimes having particular symmetries. The 'on-shell' case is also illustrated in Section 7.
In the 'half off-shell' case E IJ = 0 (and so W IJ = 0). Therefore, ω I J becomes the spin connection Γ I J , which is defined by and has the explicit expression Therefore, both (88) and (101) become Palatini Lagrangian. In the half off-shell case, the expressions for the Noether potentials and currents for the Palatini Lagrangian in n-dimensional spacetimes can simply be obtained by replacing E IJ = 0 and ω I J with Γ I J in the expressions found in Section 2.
Holst Lagrangian. In the half off-shell case, the expressions for the Noether potentials and currents for the Holst Lagrangian can simply be obtained by replacing E IJ = 0 and ω I J with Γ I J in the expressions found in Section 3. Nevertheless, note that the resulting expressions for the potentials and currents for both diffeomorphisms and local SO(3, 1) or SO(4) transformations still carry the Immirzi parameter γ. Therefore, this case is very different from the one we would get if we imposed the 'half off-shell" condition E IJ = 0 from the very beginning and we replaced ω I J with Γ I J in the Holst Lagrangian because in such a case the term involving the Immirzi parameter γ in L H would vanish as a consequence of the Bianchi identity, and the Lagrangian L H would reduce to the Einstein-Hilbert Lagrangian in terms of the frame e I rather than the metric: with R I J = dΓ I J +Γ I K ∧Γ K J being the curvature of Γ I J . The action principle defined by this Lagrangian is S[e] = M L EH and we are in the second-order formalism.
Einstein-Hilbert Lagrangian. For the sake of completeness, if we redo the calculations for the Lagrangian L EH for spacetimes in n-dimensions, we find the following off-shell Noether identities, potentials, and currents: Diffeomorphisms. From the Noether identity for the change of e I under an infinitesimal diffeomorphism generated by ζ and applying the same off-shell procedure, we get Local SO(n − 1, 1) or SO(n) transformations. From the Noether identity for local SO(n − 1, 1) or SO(n) transformations and applying the same off-shell approach, we obtain These expressions can alternatively be obtained in an easier way by just setting E IJ = 0 (and replacing ω I J with Γ I I ) in the corresponding ones reported in Section 2 of this paper.
Killing vector fields and half off-shell case. In the half off-shell case, we simply have to substitute the corresponding E IJ = 0 which implies K IJ = 0 and W IJ = 0 in both Subsections 5.1 and 5.2. Let us analyze more carefully the results for the Holst Lagrangian contained in Subsection 5.2. In particular, the Noether potential U ζ for diffeomorphisms (112) when ζ is a Killing vector field becomes where here U n=4 ζ is in fact the corresponding Noether potential (11) for the Palatini Lagrangian in four-dimensional spacetimes in the half off-shell case too.
Moreover, in the half off-shell case, the expressions for the potential U H and the current J P for the effective transformation considered in the Subsection 5.2 become We emphasize again that these expressions have been obtained by substituting E IJ = 0 at the end of the computations. Notice, however, that the potential U H for the Holst Lagrangian still carries a γ dependence inside the total differential. As we explained before, this half off-shell potential is very different from the potential we would obtain if the condition E IJ = 0 were used from the very beginning in the Holst Lagrangian and we replaced ω I J with Γ I J in the Holst Lagrangian, because in such a case the term involving the Immirzi parameter would disappear. Notice that if the potential (127) is integrated over a two-dimensional compact surface without boundary, then the last term in (127) vanishes and the charge for the Holst Lagrangian coincides with the charge for the Palatini Lagrangian. Finally, we would like to remark that the potentials (126) and (127) depend on the Immirzi parameter even in the on-shell case.

Examples
In this section we apply the theoretical framework developed in sections 2, 3, 5, and 6, i.e., we choose relevant spacetimes and their Killing vector fields and work in the half off-shell case.
The orthonormal frame is given by The isometry group of the metric (129) is R × SO(3) and has associated the following Killing vector fields as generators [42]: The vector ζ 1 is the generator of time translations, whereas the vectors ζ 2 , ζ 3 , and ζ 4 correspond to the components of the angular momentum, that is, to the generators of SO (3). From now on, we take f (r) = e 2a(r) and h(r) = e 2b(r) to simplify the calculations. With this at hand, the 'half off-shell' potentials and currents for diffeomorphisms associated with the Killing vector fields, local SO(3, 1) transformations induced by Killing vector fields, and the effective transformation acquire the explicit forms contained in the following Subsections 7.1.1 and 7.1.2.

Palatini Lagrangian
Half off-shell potentials and currents for diffeomorphisms generated by the Killing vector fields.
(i) For the Killing vector field ζ 1 , the potential (11) acquires the form and therefore (ii) Likewise, for the Killing vector field ζ 2 , the potential (11) becomes and so (iii) For the Killing vector field ζ 3 , the potential (11) acquires the form and then (iv) For the Killing vector field ζ 4 , the potential (11) becomes and therefore We can do more. We can compute the integral of the potential U ζ1 over a sphere S 2 of constant radius r, which defines the conserved charge inside it. We obtain This result is general. For instance, using in particular the explicit expressions with M the "mass parameter" and Λ the cosmological constant, which correspond to the Schwarzschild-de-Sitter or the Schwarzschild-anti-de-Sitter solution depending on the sign of Λ, we arrive at Note that the term involving Λ has a very appealing behavior. Such a term is added to M if Λ < 0 while it is subtracted from M if Λ > 0, thus indicating an attractive effect in the former case (effective mass increases) and a repulsive effect in the latter case (effective mass decreases). Of course, the region of spacetime and its boundary must be clearly defined to calculate the Noether charges using the potentials computed in this paper, and the current calculations can be used to achieve that goal. In particular, it would be interesting to use the current expressions to compute masses, energies, and entropy of the Schwarzschild-de-Sitter black hole, and compare with the results of Ref. [43]. Similarly, the Schwarzschild-anti-de-Sitter black hole can be analyzed and compared with Refs. [44,45].
Half off-shell potentials and currents for local SO(3, 1) transformations induced by Killing vector fields.

Holst Lagrangian
Half off-shell potentials and currents for diffeomorphisms generated by the Killing vector fields.
(i) For the Killing vector field ζ 1 , the potential (126) acquires the form and thus Note that there is no γ in the current J ζ1 despite the fact that it appears in the potential U ζ1 .
(ii) For the Killing vector field ζ 2 , the potential (126) becomes and so Note that there is no γ in the current J ζ2 in spite of the fact it is in the potential U ζ2 .
(iii) For the Killing vector field ζ 3 , the potential (126) acquires the form and then (iv) For Killing vector field ζ 4 , the potential (126) becomes and therefore Notice that in the cases (iii) and (iv) the Immirzi parameter shows up in the corresponding expressions for potentials and currents. Even if the explicit expressions (141) are used, the Immirzi parameter will be present.
Half off-shell potentials and currents for local SO(3, 1) transformations induced by the Killing vector fields.
(iii) For the gauge parameter τ IJ (ζ 3 ), the potential (62) acquires the form and thus (iv) For the gauge parameter τ IJ (ζ 4 ), the potential (62) becomes and thus Again, the Immirzi parameter shows up in the potentials and in their associated currents, and it is not possible to get rid of it even in the 'on-shell' case.
Notice that U P is given by the first term in the last equality in (163).
(ii) For ζ 2 Notice that U P is given by the first and second terms in the last equality in (165).
Notice that U P is given by the first three terms in the last equality in (167).
(iv) For ζ 4 Notice that U P is given by the first three terms in the last equality in (169).

Friedmann-Lemaitre-Robertson-Walker cosmology
For the sake of simplicity, let us consider Lorentzian (σ = −1) spacetimes in four dimensions with homogeneous and isotropic spacelike slices. In local coordinates x µ = (x 0 , x 1 , x 2 , x 3 ) = (t, r, θ, φ) adapted to these symmetries, the general form of the metric is given by the FLRW metric where k = 0, 1, −1 is the spatial curvature and a(t) is the scale factor. Do not confuse k with κ in the expressions of this Subsection. From (171) we read off the orthonormal frame given by e 0 = dt, e 1 = a(t) √ 1 − kr 2 dr, e 2 = a(t)rdθ, e 3 = a(t)r sin θdφ.

Palatini Lagrangian
Half off-shell potentials and currents for diffeomorphisms generated by the Killing vector fields.

Holst Lagrangian
Half off-shell potentials and currents for diffeomorphisms generated by the Killing vector fields.
(i) For the Killing vector field χ 1 , the potential (126) acquires the form and thus (ii) For the Killing vector field χ 2 , the potential (126) becomes and so (iii) For the Killing vector field χ 3 , the potential (126) acquires the form and therefore J ζ1 = dU ζ1 = 2κ ra cos φ cot θ csc θ + 2kr 2 cos θ + 2r 2 da dt 2 cos θ + r 2 a d 2 a dt 2 cos θ e 0 ∧ e 1 ∧ e 2 − 2κ ra sin φ −1 − cot 2 θ + 2kr 2 + 2r 2 da dt 2 + r 2 a d 2 a dt 2 e 0 ∧ e 1 ∧ e 3 + 4κ γa da dt csc θ cos φ e 0 ∧ e 2 ∧ e 3 + 4κ γra 1 − kr 2 csc θ cos φ e 1 ∧ e 2 ∧ e 3 . (205) (v) For the Killing vector field ζ 2 , the potential (126) acquires the form and therefore Notice that the potentials and currents that depend on the Immirzi parameter will still depend on it even in the 'on-shell' case, except for the current (209), which actually coincides with the current (186) for the Palatini Lagrangian. This is a consequence of the fact that the terms on the second row of the potential (208) (that involve the Immirzi parameter) can be cast as the exact form d(κγ −1 ar sin θ e 3 ) and hence do not contribute to the current (209).
Half off-shell potentials and currents for local SO(3, 1) transformations induced by the Killing vector fields.
In this case all the nonvanishing potentials and currents depend on the Immirzi parameter, which also holds in the 'on-shell' case.
Notice that U P is given by the first three terms in the last equality in (231).

Conclusion
In this paper we have constructed off-shell Noether currents and potentials for the n-dimensional Palatini Lagrangian and the four-dimensional Holst Lagrangian, which embody first-order formulations of general relativity with a cosmological constant. A remarkable aspect of our approach is that the currents are off-shell conserved.
More precisely, we have derived off-shell expressions for the currents and potentials associated to diffeomorphisms and local SO(n − 1, 1) or SO(n) transformations, the two underlying gauge symmetries of gravity in this kind of formulations. To derive them, we have appealed to the use of the Noether identities satisfied by the variational derivatives of each formulation, which combined with the variation of the Lagrangian under the infinitesimal versions of the above symmetries, lead to the appropriate identification of these off-shell currents and potentials. In addition, we have computed the associated off-shell currents for the so called 'improved diffeomorphisms' and for the 'generalization of local translations' reported in Ref. [34], showing that they identically vanish for both first-order formulations of general relativity. However, we have also found that the off-shell current and potential associated to diffeomorphisms emerge from these symmetries. We have also studied how these off-shell Noether currents and potentials simplify in a spacetime with symmetries generated by Killing vector fields. In particular, we have found that the action of a Killing vector field on the orthonormal frame and the connection equals a local SO(n − 1, 1) or SO(n) transformation plus a trivial gauge transformation that only affects the infinitesimal transformation of the connection. The resulting off-shell potentials and currents for this effective gauge transformation have also been reported and they can be expressed, respectively, as the difference of the off-shell potentials and currents associated to Killing vectors and their induced SO(n) or SO(n − 1, 1) transformations.
To simplify things a bit, we have considered the half off-shell case, which is defined by the condition E IJ = 0 (so, we work on solutions of the equation of motion for the connection) for both formulations of general relativity and thus the aforementioned trivial transformation of the connection is set to zero. We have found that the potentials and currents for diffeomorphisms and local SO(3, 1) or SO(4) transformations for the Holst Lagrangian generically depend on the Immirzi parameter, which is also true in the 'on-shell' case. Furthermore, in the half off-shell case, the potential associated to the effective transformation for the Holst Lagrangian differs from that for the Palatini Lagrangian by an exact differential form depending on the Immirzi parameter. Such a contribution is not expected from the point of view of the second-order formalism for general relativity in terms of the tetrad, which is what the Holst Lagrangian collapses to when the condition E IJ = 0 is satisfied and does not depend on the Immirzi parameter whatsoever. To illustrate our approach, we have explicitly computed the half off-shell potentials and currents discussed above, for Killing vector fields, their induced local SO(3, 1) transformations, and the associated effective gauge transformations, in four-dimensional static spherically symmetric and FLRW spacetimes, for both Palatini and Holst Lagrangians. For the Holst Lagrangian, the resulting potentials and currents generically depend on the Immirzi parameter, except for the current associated to the effective gauge transformation.
Although we did not consider adding boundary terms to the Lagrangians in this paper, they can be handled with our theoretical techniques and we expect the addition of boundary terms to the action principles defined by the Palatini and Holst Lagrangians generically contribute to the off-shell currents and their associated potentials. The understanding of such terms in gravity is essential to appropriately define quantities such as asymptotic charges and black hole entropy, and will be one of the main focuses of our forthcoming studies. In addition, those studies might help to clarify the role of the Immirzi parameter in the definition of conserved charges and entropy as well. We expect to confront our results with those obtained in the literature following alternative approaches within the first-order formalism [14,36,37,47,48,15,38,49,50,51].
Even though we have constructed the off-shell Noether currents and potentials for general relativity in the first-order formalism, it is obvious that the same theoretical framework can be extended to any gauge theory and, in particular, to any diffeomorphism invariant theory of gravity in the first-order formalism. In particular, similar off-shell Noether currents and potentials can be obtained using the formalism developed in this paper for f (R) theories [40], matter fields coupled to general relativity [39], and any other alternative theory of gravity such as for instance Lovelock gravity [52] in the first-order formalism. Moreover, it would also be interesting to study other gravitational models within the first-order formalism including some background structure into play, such as unimodular gravity [53,54] and extensions thereof.