Nonlinear massive spin–two field generated by higher derivative gravity
Abstract
We present a systematic exposition of the Lagrangian field theory for the massive spin–two field generated in higher–derivative gravity upon reduction to a second–order theory by means of the appropriate Legendre transformation. It has been noticed by various authors that this nonlinear field overcomes the well known inconsistency of the theory for a linear massive spin–two field interacting with Einstein’s gravity. Starting from a Lagrangian quadratically depending on the Ricci tensor of the metric, we explore the two possible second–order pictures usually called “(Helmholtz–)Jordan frame” and “Einstein frame”. In spite of their mathematical equivalence, the two frames have different structural properties: in Einstein frame, the spin–two field is minimally coupled to gravity, while in the other frame it is necessarily coupled to the curvature, without a separate kinetic term. We prove that the theory admits a unique and linearly stable ground state solution, and that the equations of motion are consistent, showing that these results can be obtained independently in either frame (each frame therefore provides a self–contained theory). The full equations of motion and the (variational) energy–momentum tensor for the spin–two field in Einstein frame are given, and a simple but nontrivial exact solution to these equations is found. The comparison of the energy–momentum tensors for the spin–two field in the two frames suggests that the Einstein frame is physically more acceptable. We point out that the energy–momentum tensor generated by the Lagrangian of the linearized theory is unrelated to the corresponding tensor of the full theory. It is then argued that the ghost–like nature of the nonlinear spin–two field, found long ago in the linear approximation, may not be so harmful to classical stability issues, as has been expected.
1 Introduction
A consistent theory of a gravitationally interacting spin–two field could not be developed until a significant progress was made in an apparently unrelated subject, i.e. higher–derivative metric theories of gravity. It is well known that a single linear spin–two field cannot be consistently coupled to gravity. It is therefore a common belief that Nature avoids the consistency problem by simply not creating fundamental spin–two (nor higher spin) fields except gravity itself. Nevertheless the subject has remained fascinating over decades and some authors have studied various aspects of linear spin–two fields [1, 2, 3], in particular their dynamics in Einstein spaces [4, 5].
On the other hand, higher–derivative metric theories of gravity, where the Lagrangian is a scalar nonlinear function of the curvature tensor (hence in this paper they are named nonlinear gravity theories, NLG) have attracted much more attention. Most work was centered on quadratic theories, i.e. on Lagrangians being quadratic polynomials in the Ricci tensor and the curvature scalar [6, 7], but several authors studied more general Lagrangians [8]. These theories turned out to be inadequate as candidates for foundations of quantum gravity since they are non–unitary, but recently play a role as effective field theories. What is more relevant here, it was found that their particle spectrum contains a massive spin–two field. The dynamics of this field can be described and investigated by recasting the fourthorder NLG theory into a standard nonlinear second–order Lagrangian field theory. The procedure entails a decomposition of the dynamical data, consisting of the metric field and its derivatives up to the third order, into a set of independent fields describing the physical state by their values and their first derivatives only. In this peculiar sense, one may say that the single “unifying” field is replaced by (or decomposed into) a multiplet of gravitational fields.
An adequate mathematical tool for this purpose is provided by a specific Legendre transformation [9, 10, 11]. Although the transformation has been known for more than a decade, it is not currently used in a systematic way. Instead, most papers on the nonlinear spin–two field have employed various ad hoc tricks adjusted to quadratic Lagrangians [12, 13, 14] (actually equivalent to the Legendre transformation for this particular case), but such approach does not allow one to fully exhibit the structure of the theory.
Although the Legendre transformation is essentially unique, the various fields of the resulting multiplet can be given different physical interpretations; the different choices are traditionally called ‘‘frames”^{1}^{1}1The use of the word “frame” in this sense should be deprecated, because it does not refer to the choice of a physical reference frame, but this abuse of terminology is now so universally adopted that we feel that trying to introduce here a more appropriate term, for instance “picture” as it is used in Quantum Mechanics, would only lead to confusion.. In general, the original “Jordan frame” (JF; the name is borrowed from scalar–tensor theories) consisting of only the unifying metric , can be transformed into frames including fields of definite spin in two ways. A first possibility is that the field remains the spacetime metric, now carrying only two d.o.f., while the other degrees of freedom (previously carried by its higher derivatives) are encoded into auxiliary (massive) fields of definite spin: this is the Helmholtz–Jordan frame (HJF). Alternatively, one introduces (via an appropriate redefinition of the Legendre transformation) a new spacetime metric , while the symmetric tensor is decomposed into spin2 and spin0 fields, forming in this way the massive, nongeometric components of the gravitational multiplet; these variables form together the ‘‘Einstein frame” (EF). Both frames are dynamically equivalent and very similar on the level of the field equations: the equations of motion are secondorder Lagrange field equations, and in each frame the corresponding spacetime metric satisfies Einstein’s field equations^{2}^{2}2Some authors seem instead to believe that Einstein equations can be obtained only after redefinition of the spacetime metric, i.e. in EF., thus the theory looks like ordinary general relativity, with the non–geometric components of the multiplet acting as specific matter fields. The two frames differ however in the action integral. In HJF the spin–0 and spin–2 fields are nonminimally coupled to gravity (to curvature) while there are no kinetic terms for those fields in the Lagrangian; only the metric has the standard Einstein–Hilbert Lagrangian . In consequence, propagation PDEs for the fields with spin zero and two arise from the action in a more involved way (through the variation of the metric tensor), so the theory in this frame cannot be obtained by minimal coupling to ordinary gravity of some additional fields already possessing a definite dynamics in a fixed background spacetime. Yet it is remarkable that in the EF variables the theory has fully standard form [9, 10]: one recovers the Lagrangian for the metric and universal kinetic terms for the spin–0 and spin–2 fields (independently of the form of the original Lagrangian in JF), only the potential part of the action being affected by the actual form of . The EF variables are uniquely characterised by these features, while in HJF different ad hoc redefinitions of the variables can be intertwined with the Legendre transformation (the latter being itself sometimes disguised as a mere change of variables) [12, 13].
Though matematically equivalent, the two frames are physically inequivalent; the difference is most clearly visible while defining the energy since the latter is very sensitive to redefinitions of the spacetime metric [15]. Both mathematical similarity to ordinary general relativity and physical arguments indicate that the EF is physical [15]: in HJF the energy–momentum tensor is unphysical, being linear in both non–geometric fields, while in EF the stress–energy tensor for the scalar field has the standard form and that for the spintwo field seems also more acceptable than in HJF.
Anyhow, in both frames NLG theories provide a consistent description of a self–gravitating massive spin–two field. The field is necessarily nonlinear and in quantum theory it is ghostlike. The latter defect is inferred from the fact that in the linearized theory the Lagrangian of the field appears with the sign opposite to that for linearized Einstein gravity [16]. This fact is interpreted in classical theory as related to the occurrence of excitations with negative energy for the field, and in consequence as a signal of instability of the theory. However, in this paper we show (an incomplete proof was given previously in [13]) that the ground state solution (vacuum) is classically stable at the linear level. This does not prove the stability of the vacuum state whenever nonlinear terms are taken into account: we stress, however, that all the above mentioned features usually advocated as signals of instability, are equally derived within the linear approximation. The problem of energy is more subtle: in HJF the variational energy–momentum tensor is evidently unphysical, while in EF the Lagrangian is highly nonlinear (not even polynomial), and we will show in sect. 7 that the linear approximation tells rather little about the energy density of the exact theory. Hence, the massive spin–two field generated by NLG theories is still worth investigating in the framework of classical Lagrangian field theory.
Most previous works [12, 13, 14] were centered on the particle spectrum of the theory and dealt only with the action integrals, while less attention was paid to the field equations and the structure of the theory. The field equations in HJF were given in [13], but these authors regarded the field equations in EF as extremely involved and thus intractable, they only studied the case where the spin–two field is assumed to be proportional to the metric tensor and thus can be described by a scalar function (its trace). In consequence, the dynamical consistency of the theory has always been taken for granted since the fourth–order equations in JF are consistent.
In the present paper we systematically investigate the nonlinear spin–two field generated by a NLG theory with a quadratic Lagrangian (1) in the framework of classical field theory, employing a Legendre transformation. As is well known, a spin–two field may be mathematically represented by tensor variables of different rank and symmetry properties [17, 2]. Any NLG theory generates in a natural way a representation of the field in terms of a symmetric, second rank tensor , and this representation will be employed in this work.
Although we study a concrete Lagrangian (this is motivated in sects. 2 and 3) we employ no tricks adjusted to it. The paper is self–contained and provides an (almost) full exposition of the subject. The first part describes the nonlinear spin–two and spinzero fields in HJF. The particle content in this frame is well known. The main new results shown here are:

the field equations for the metric and the other two fields for any spacetime dimension ;

the fact that dimension is distinguished in that the scalar field is decoupled and can be easily removed from the theory (we do so in the rest of the paper since we are interested in the description of the spin–two field);

the fact (sect. 3) that the resulting equations of motion for the spin–two field do not generate further constraints besides the five ensuring the purely spin–2 character of the field;

the internal consistency of the theory in HJF, ensured by strong Noether conservation laws;

the linear stability of the unique ground state solution representing flat spacetime and vanishing spin–two field, assessed by the fact that small perturbations form plane waves with constant amplitudes.
In sect. 4 we show that the two possible ways to obtain the massless limit of the spin–two field described in the previous section yield the same result, so the massless limit is well–defined and leads to the propagation equations for gravitational perturbations in a Ricci–flat spacetime.
The main thrust of the paper is its second part (sect. 5 to 9), where we investigate the equations of motion in Einstein frame. Here most results are new.

A generic presentation of the Lagrangian theory for the nonlinear spin–two field in Einstein frame is given in sect. 5: in order to better exhibit the structure of the theory in this frame, we consider a Lagrangian being an arbitrary function of the Ricci tensor of the original metric in JF. We explicitly give the equations of motion for the massive field in the generic case, an expression (highly nonlinear) for the full energy–momentum tensor of the field and four differential constraints imposed on the field by its dynamics.

This generic theory is then specialized in sect. 6 to the case of the particular Lagrangian of eq. (33), which in HJF ensured that the scalar field drops out. The previous, generic equations produce in this case a fifth, algebraic constraint, which together with those already found ensures that also in EF the massive field has five degrees of freedom and is purely spin–two.

A unique, linearly stable ground state solution is then found (without any simplifying assumptions); clearly it corresponds via the Legendre transformation to the ground state in HJF. The spin–two field is then redefined to make it vanish in this state (and in vacuum in general). It turns out that the consistency and hyperbolicity problems in this frame, investigated in sect. 7, are harder that in HJF and should be studied perturbatively; we study them in the linearized theory. The Lagrangian is computed in the lowest order (quadratic) approximation around the ground state solution to show the ghostlike character of the spin–two field. A detailed comparison is made with the theory of the linear spin–two field, and the fact that the energy–momentum tensor of the nonlinear theory is not approximated by the energy–momentum tensor derived from the Wentzel Lagrangian is fully explained.

Finally, contrary to the common belief that the full (nonlinear) system of equations of motion in EF are intractably involved, in sect. 8 we give a simple but nontrivial solution to them.
Conclusions are in sect. 9, and Lagrange field equations and the energy–momentum tensor in EF for the redefined spin–two field as well as some other useful formulae are contained in Appendix.
2 Equations of motion for the gravitational
multiplet in
Helmholtz–Jordan frame
We will investigate dynamical structure and particle content of a nonlinear gravity (NLG) theory using Legendre transformation method [9, 10, 11]. The starting point is a –dimensional manifold , (later the dimensionality will be fixed to ) endowed with a Lorentzian metric . The inverse contravariant metric tensor will be denoted by and ; we introduce this nonstandard notation for further purposes. One need not view as a physical spacetime metric, actually whether or its ”canonically conjugate” momentum is the measurable quantity determining all spacetime distances in physical world should be determined only after a careful examination of the physical content of the theory, rather than prescribed a priori. Formally plays both the role of a metric tensor on and is a kind of unifying field which will be decomposed in a multiplet of fields with definite spins; pure gravity is described in terms of the fields with the metric being a component of the multiplet. In general dynamics for is generated by a nonlinear Lagrangian density where and is the Riemann tensor for ; is any smooth (not necessarily analytic) scalar function. Except for Hilbert–Einstein and Euler–Poincaré topological invariant densities the resulting variational Lagrange equations are of fourth order. The Legendre transformation technique allows one to deal with fully generic Lagrangians; from the physical standpoint, however, there is no need to investigate complicate or generic Lagrangians. Firstly, in the bosonic sector of low energy field theory limit of string effective action one gets in the lowest approximation the Hilbert–Einstein Lagrangian plus terms quadratic in the curvature tensor. Secondly, to obtain an explicit form of field equations and to deal with them effectively one needs to invert the appropriate Legendre transformation and in a generic case this amounts to solving nonlinear matrix equations. Hindawi, Ovrut and Waldram [18] have given arguments that a generic NLG theory has eight degrees of freedom and the same particle spectrum as in the quadratic Lagrangian (1) below, the only known physical difference lies in the fact that in the generic case one expects multiple nontrival (i.e. different from flat spacetime) ground state solutions. This result can be also derived from the observation that after the Legendre transformation the kinetic terms in the resulting (Helmholtz) Lagrangian are universal, and only the potential terms keep the trace of the original nonlinear Lagrangian. If the latter is a polynomial of order higher than two in the curvature tensor, the Legendre map is only locally invertible and this leads to multivalued potentials, generating a ground state solution in each “branch”; yet the form of the potential could produce additional dynamical contraints, affecting the number of degrees of freedom, only in non–generic cases. The physically relevant Lagrangians in field theory depend quadratically on generalized velocities and then conjugate momenta are linear functions of the velocities. For both conceptual and practical purposes it is then sufficient to envisage a quadratic Lagrangian
(1) 
In principle one should also include the term
(in
four dimensions it can be eliminated via Gauss–Bonnet theorem), but the
presence of Weyl tensor causes troubles: although formally the Legendre
transformation formalism works well there are problems with providing
appropriate propagation equations for the conjugate momentum and with
physical interpretation (particle content) of the field. We therefore
suppress Weyl tensor in the Lagrangian. The Lagrangian cannot be purely
quadratic: it is known from the case of restricted NLG theories (Lagrangian
depends solely on the curvature scalar, ) that the
linear term is essential [15] and we will see that
the same holds for Lagrangians explicitly depending on Ricci tensor
. The coefficients and have dimension
; contrary to some claims in the literature there are no
grounds to presume that they are of order
unless the Lagrangian (1) arises from a more fundamental theory (e.g.
string theory) where is explicitly present. Otherwise in a pure
gravity theory the only fundamental constants are and ; then
and need not be new fundamental constants, they are rather related
to masses of the gravitational multiplet fields. Here we assume that the
NLG theory with the Lagrangian (1) is an independent one, i.e. it inherits
no features or relationships from a possible more fundamental theory.
As was mentioned in the Introduction, in the Legendre transformation one
replaces the higher derivatives of the field
by additional fields.
In this section we assume that the original field keeps the role of the physical
spacetime metric, and the self–gravitating spin–two field originates from the
“conjugate momenta” to .
We recall that for a second–order Lagrangian such as (1) one should properly choose the quantities to be taken as generalized velocities to define, via a Legendre map, conjugate momenta [11]. One cannot, for instance, use the partial derivatives as generalized velocities since for covariant Lagrangians, e.g. (1), the Legendre map cannot be inverted: the Hessian, being the determinant of a matrix, vanishes,
(2) 
General covariance indicates which linear combinations of can be used as the velocities, i.e. with respect to which combinations the Lagrangian is regular (the Hessian does not vanish). Clearly this is Ricci tensor . The explicit use of generally covariant quantities in this approach is supported by the Wald’s theorem [19, 20] that only a generally covariant theory may be a consistent theory of a spin–two field. Following [11], in order to decompose into fields with definite spins, one makes Legendre transformations of the Lagrangian (1) with respect to the two irreducible components of : its trace and the traceless part . In terms of , , the Lagrangian reads
(3) 
one assumes and . One then defines a scalar and a tensor canonical momentum via corresponding Legendre transformations:
(4) 
it is convenient to identify with rather than with alone. From (3):
(5) 
hence fields and are dimensionless and is traceless, . The new triplet of field variables {} defines the Helmholtz–Jordan Frame (HJF).
Equations of motion for this frame arise as variational Lagrange equations from Helmholtz Lagrangian [22, 9, 10, 11]. First one constructs the Hamiltonian
(6) 
expressed in terms of and the canonical momenta, it reads
(7) 
here ; all indices are raised and lowered with the aid of and . Next one evaluates Helmholtz Lagrangian defined as
(8) 
where the derivatives and are set equal to the canonical momenta and respectively, while the ”velocities” and explicitly depend on first and second derivatives of . In classical mechanics for a first order Lagrangian one has
(9) 
i.e. is a scalar function on the tangent bundle to the cotangent bundle to the configuration space; does not depend on . Similarly, in a field theory is independent of partial derivatives of canonical momenta. In classical mechnics the action gives rise, when varied with respect to , to the equation , while varied with respect to generates
(10) 
the latter equation is equivalent to
(11) 
Thus simultaneously generates both Hamilton and Lagrange equations of motion. In the case of NLG theories one is interested in replacing the fourth order Lagrange equations by the equivalent second order Hamilton ones. For the Lagrangian (1) reads
(12) 
One sees that the linear Hilbert–Einstein Lagrangian for the metric field is recovered. This means that Hamilton equations for are not just second order ones of any kind but exactly Einstein field equations^{3}^{3}3We use units , the signature is . We use all the conventions of [22]. . The nonminimal coupling interaction terms and will cause that will depend on second derivatives of and and will contain Ricci tensor. Since Hilbert–Einstein Lagrangian for the metric field in general relativity is , then , where is the Lagrangian for the non–geometric components of gravity, and . The energy–momentum tensor is then
Explicitly the equations read
(14)  
here denotes the covariant derivative with respect to and . As remarked above more resembles the stress tensor for the conformally invariant scalar field [23, 24] than that for ordinary matter. The equations of motion for and are purely algebraic and clearly coincide with (5),
(15) 
These equations can be recast in the form of Einstein ones,
(16) 
Comparison of eqs. (14) and (16) shows that for solutions there exists a simple linear expression for the stress tensor:
(17) 
This relationship allows one to derive differential propagation equations for and . Before doing it we simplify the expression (14) for with the aid of (16) by replacing by and and making use of the Bianchi identity for . The latter provides a first order constraint on and ,
(18) 
the constraint is already solved with respect to . Upon inserting (16) into the r.h.s. of (14) one gets
(19)  
Equating the trace computed from (17) to the trace of (19) and applying (18) one arrives at a quasilinear equation of motion for ,
(20) 
The field is self–interacting and is coupled to , the term acts as a source for . Eq. (20) is a Klein–Gordon equation with a potential and an external source. The mass of is
(21) 
and may be of both signs depending on the parameters.
To derive a propagation equation for one replaces derivatives of in (19) by derivatives of with the help of (18). Then depends on via terms and ; the latter is eliminated with the aid of eq. (20) and the reappearing term is again removed employing (18). The resulting expression for , which contains terms and , is set equal to the r.h.s. of eq. (17), then the terms cancel each other and finally one arrives at the following equation of motion for ,
(22) 
The triplet of gravitational fields is described by a coupled system
of eqs. (16), (20) and (2) and the constraint (18). The equations (20)
and (2) are quasilinear and contain interaction and self–interaction
terms which cannot be removed for dimensions .
Since for generic dimension the dynamics of the fields and cannot be decoupled, one can obtain some information on the individual behaviour of each field by considering particular solutions in which only one of the two fields is excited.

Let . Then eq. (2) holds identically while the constraint (18) implies and eq. (20) reduces to a quadratic equation, . One solves it separately for and for .

. There are two solutions, and . From eq. (16) they correspond to and i.e. Einstein space respectively. Therefore there exist two distinct ground state solutions: Minkowski space for and –dimensional de Sitter space or anti–de Sitter for depending on the sign of .

For the equation has only one solution corresponding to and there is a unique ground state solution (Minkowski metric) and .

2.1 The four–dimensional case
Dimension four is clearly distinguished by the scalar field eq. (20). Setting , one finds two relevant properties:

both the self–interaction and the source in eq. (20) vanish and satisfies the linear Klein–Gordon equation
(24) for the mass is real,
(25) We shall see below that this allows one to decouple completely the propagation equations for the fields and ;

furthermore, for eq. (24) admits only one solution, , so for this particular choice of coefficients the scalar field disappears from the theory: we shall exploit this fact in the next section to concentrate our investigation only on the spin–two field.
We stress that for vanishing of the coefficient of in eq. (20) does not imply that that the scalar drops out. In fact, the equation gives rise to an algebraic relationship between the scalar and tensor fields,
(26) 
which replaces a propagation equation for .
To prove the statements above, one starts from eq. (2), and for one eliminates from the interaction term applying eq. (24): then is replaced by with the help of (18). Then the equations of motion for read
(27) 
The linear equation (24) for and quasilinear eq. (2.1) for
are decoupled. Equations (2.1) are linearly dependent since
the trace (w.r.t. ) vanishes identically, hence there are
9 algebraically independent equations (they satisfy also differential
identities, see below).
For the special solution the differential constraint reads . Then eq. (2.1) reduces to
(28) 
and the mass of the field is the same as in , i.e. . The masses of the massive components of the gravitational triplet, and , agree with the values found in the linear approximation to the quadratic Lagrangian for spin–0 and spin–2 fields by [16, 25] and [26]. The non–tachyon condition is then and [7].
This condition shows that the term in the Lagrangian (1) is essential. In fact, if this term is absent and the Lagrangian reads , then (for dimensionality four)
(30)  
and
(31) 
The two fields have masses and and
one of them is necessarily a tachyon. It is worth noting that in the case
of restricted NLG theories, i.e. , the
term is also essential in the Taylor expansion of the Lagrangian: for
Minkowski space is stable as the ground state solution of the
theory, while for it is classically unstable and for the
solution may be unstable or stable [15].
Finally we show that all nine algebraically independent equations (2.1) are hyperbolic propagation equations for and they contain no constraint equations. To this end one replaces by with the aid of (18) and one arrives at the following equations:
(32) 
In this form the hyperbolicity of all the equations is evident.
3 The spin–2 field in the gravitational doublet in Helmholtz–Jordan frame
We have seen in the previous section that the generic quadratic Lagrangian (1), subject only to the non–tachyon condition and , describes a triplet of gravitational fields, HJF= , where the nongeometric fields in the triplet represent (on the quantum–mechanical level) interacting particles with positive masses. It has long been known that the theory (1) has 8 degrees of freedom [25, 27, 28, 29]. As it was first found in the linear approximation [16] and then in the exact theory [13, 12], these degrees of freedom are carried by a massless spin–2 field (graviton, 2 degrees of freedom), a massive spin–2 field (5 d.o.f.) and a massive scalar field.
We are interested in the dynamics and physical properties of the massive spin–2 field. In this context, the scalar is undesirable and its presence only makes the system of the equations of motion more involved. One can get rid of the unwanted scalar by a proper choice of the coefficients in the original Lagrangian. As mentioned previously, for eq. (24) has only one trivial solution^{4}^{4}4That the scalar degree of freedom disappears in this case was previously found in [16, 12]. . We therefore restrict our further study to the special Lagrangian
(33) 
and denote assuming . As it was noticed in [16, 13] this Lagrangian can be neatly expressed in terms of Weyl tensor,
(34) 
where , the Gauss–Bonnet term, is
a total divergence in four dimensions.
One formally repeats the operations of the previous section and replaces the expressions (15) to (19) by
(35) 
(36) 
and
(37)  
respectively. The scalar field is still present in these equations. However the trace of eq. (37) is while the trace of eq. (36) yields . Then and the scalar drops out from the theory. Although the scalar field vanishes one cannot, however, remove it from the Lagrangian in HJF unless one imposes the constraint already on the level of the initial Lagrangian in HF. It is more convenient to deal with Lagrangians containing no Lagrange multipliers and therefore the auxiliary nondynamic (i.e. having no physical degrees of freedom) scalar remains in the Helmholtz Lagrangian (12) which now reads
(38) 
The system of field equations for the gravitational doublet HJF=, having together seven degrees of freedom, consists of Einstein field equations,
(39) 
and quasilinear hyperbolic propagation equations for ,
(40) 
It should be stressed that, as is clearly seen from the method of deriving eqs. (20) and (2), the eqs. (39) and (40) are not simply the variational equations and with the substitutions and .
The field satisfies 5 constraints,
and . Notice
that the field equations give rise to no further constraints. In fact,
the trace of (40) and divergence of (39) vanish identically if the
constraints hold. A possible source of a further constraint is
divergence of eq. (40). It may be shown by a direct calculation that
if the equations (39) and (40) hold throughout the spacetime and if
the five constraints are satisfied everywhere, then divergence of eq. (40) vanishes identically. This confirms the previous result
[16, 25, 26, 12, 13, 7] that this field has spin two without
any admixture of lower spin fields.
We now investigate the internal consistency of the theory based on
eqs. (39) and (40). It is well known [17, 30, 29]
that a linear spin–two field , massive or massless, has
inconsistent dynamics in the presence of gravitation since in a curved
nonempty spacetime the field loses the degrees of freedom it had in
flat spacetime and the five conditions which ensured there its purely
spin–2 character, , are replaced by four differential constraints
imposed on the field and Ricci tensor^{5}^{5}5For linear fields with
spins higher than 2 it was long believed
[31] that there was no easy way to have physical fields on anything
but Minkowski space; only recently a progress has been made for
massive fields [32].. Here we are dealing with the nonlinear
spin–2 field
and one expects that this field is consistent. This
expectation follows from the dynamical equivalence of the Helmholtz
Lagrangian (38) to the purely metric Lagrangian (33) of
the fourth–order theory and the latter one (as well as any other NLG
theory with a Lagrangian being any smooth scalar function of the
curvature tensor) is known to be consistent. However the spin–2 field
theory in HJF should be a selfcontained one and one should show its
consistency without invoking its equivalent fourth–order version.
We first notice a difference in the structures of the theories for the
linear and nonlinear spin–two fields. For the linear field
in Minkowski space one first derives (quite involved)
Lagrange field equations and then either by employing the gauge
invariance (for the massless field) or by taking the trace and
divergence of the field equations for the massive one, one derives the
five constraints . The method does not work in a generic curved
spacetime [17]. For the nonlinear field the
tracelessness is a direct consequence of the tracelessness of
and of the field eq. (15), which now reads (in
terms of ) rather than of )
(41) 
in the Helmholtz Lagrangian (38) it is not assumed that has vanishing trace. After eliminating the scalar field one gets and the other four constraints and the algebraic equation for reduces to
(42) 
while the Einstein field equations for are
(43) 
where is given by eq. (37) for . Hence the constraints ensuring that has 5 degrees of freedom hold whenever the field equations hold. Clearly the system (42)–(43) is equivalent to the system (39)–(40) but the former is more convenient for studying the consistency of the equations. To this end one employs the coordinate invariance of the action integral [30]. Under an infinitesimal coordinate transformation , , the metric and the spin2 field vary as and
(44) 
here L is the Lie derivative. The action integral
(45) 
is generally covariant, therefore it is invariant under the transformation
(46) 
where and . Thus the coordinate invariance implies a strong Noether conservation law
(47) 
Now assume that the field equations and hold.
Then also holds and the identity reduces to
. This is a consistency condition for Einstein
field equations (43). Divergence of should vanish due to
the system of field equations and constraints without giving rise to
further independent constraints.
By a direct calculation one proves the following proposition:
if the field equations and (42) and the five constraints
hold
throughout the spacetime,
then the stress tensor given by eq. (37) is divergenceless,
.
This shows that the system (42)–(43) is consistent.
Owing to the tracelessness of there is only one ground state solution for the system (39)–(40) [13, 18] (i.e. the spacetime is maximally symmetric and is covariantly constant). This is Minkowski space, and . This state is linearly stable since small metric perturbations and excitations of around can be expanded into plane waves with a constant wave vector , and a constant wave amplitude satisfying .
4 Massless spin–two field in HJF
The massive spin–2 field has a finite range with the
length scale . According to the principle of physical
continuity [29] as the mass tends to zero, the longrange force
mediated by should have a smooth limit and in this limit
it should coincide with the strictly infiniterange theory. We
therefore consider two cases: first the exactly massless theory
resulting from the Helmholtz Lagrangian for and then the field
equations of the previous section in the limit .
After setting in eq. (38) it is convenient to express the resulting Lagrangian in terms of Einstein tensor and the trace of the spin–2 field, . One gets
(48) 
Owing to the Bianchi identity this Lagrangian is invariant under the gauge transformation
(49) 
with arbitrary vector ^{6}^{6}6Another possible
decomposition of in this frame into fields with
definite spin, as is done in [13], yields a different gauge
transformation not affecting the scalar field.. The
scalar
is gauge invariant. We notice that the scalar
cannot be removed already on the level of the Lagrangian
since it would break the gauge invariance. Moreover the term in
is essential to obtain appropriate field equations. Hence in this
approach the scalar cannot be eliminated by using the first principles.
One can only set
in a specific gauge.
The field equations are
which imply . It is clear that unlike in the massive case now one cannot recover the original fourthorder Lagrangian (1) since the two fields are here unrelated to Ricci tensor. The fields and do not interact with the metric which acts as an empty–spacetime background metric. This suggests in turn that the two components of the gravitational triplet are test fields on the metric background, e.g. some excitations, and one may expect that they satisfy linear equations of motion. Variation of with respect to the metric yields Einstein field equations which are reduced in comparison to eq. (14),
(50) 
the last equality follows from . The scalar field does not appear here in the combination as in the Helmholtz Lagrangian since upon varying with respect to the metric one has