Higher order gravities and the Strong Equivalence Principle
IIFT-UAM/CSIC- - arXiv:1705.03495 [gr-qc] May th , Higher order gravitiesandthe Strong Equivalence Principle
Tomás Ortín a Instituto de Física Teórica UAM/CSICC/ Nicolás Cabrera, – , C.U. Cantoblanco, E- Madrid, Spain
Abstract
We show that, in all metric theories of gravity with a general covariantaction, gravity couples to the gravitational energy-momentum tensor in thesame way it couples to the matter energy-momentum tensor order by orderin the weak field approximation around flat spacetime. We discuss therelation of this property to the Strong Equivalence Principle. We also studythe gauge transformation properties of the gravitational energy-momentumtensor. a E-mail:
Tomas.Ortin [at] csic.es a r X i v : . [ g r- q c ] M a y Introduction
Although General Relativity has passed many experimental tests so far, and in spiteof the general problems of physical theories of higher order in derivatives, thereare many reasons to consider possible corrections to the Einstein-Hilbert action con-structed from invariants of higher order in the Riemann curvature tensor: . These are the simplest modifications to General Relativity since they do notrequire the explicit introduction of other “gravitational fields” (such as scalarfields). The modified gravity theories remain dynamical metric theories, auto-matically satisfying the Weak Equivalence Principle (WEP) if matter only couplesminimally to the metric in the action. General covariance ensures the on-shellcovariant divergenceless of the matter energy-momentum tensor. . Most theories of Quantum Gravity (in particular Superstring Theories) predict aninfinite number of higher-order corrections to the Einstein-Hilbert action and theeffective gravitational theories are higher-order gravity theories, but only the low-est order corrections are explicitly known. In this context, some of the illnessesof the theories with terms of higher order in the curvature can be understood asthe result of the truncation of a consistent theory. For Superstring Theories, see, e.g. Refs. [ , , , , , , ] . The AdS/CFT duals of these higher-order gravity theories are much more generalthan those dual to Einstein-Hilbert gravity and have been proven to be very usefulin this context. Some interesting examples can be found in, e.g. , Refs. [ , , , , , , , , ]. . Typically, the higher-order terms modify the behaviour of the gravitational fieldin regions of strong curvature. Thus, these corrections may play important rôlesin inflationary cosmology, in black-hole physics (see, e.g. Refs. [ , , , , , , , , , , , , ]) and in the study of spacetime singularities.In this note we are going to call the theories of gravity resulting from the additionof terms of higher order in the Riemann curvature tensor R or its covariant deriva-tives ∇R , ∇ R , . . . invariant under general coordinate transformations to the Einstein-Hilbert action, that is, theories with actions of the form S [ g ] = χ (cid:90) d d x (cid:113) | g | { R + F ( g , R , ∇R , · · · ) } , χ = π G , ( . ) Most of these problems are related to the linear instability discovered by Ostrogradski in Ref. [ ] [ ]and pedagogically reviewed with its implications in Ref. [ ]. Nevertheless, see also Ref. [ , ]. We are only going to consider metric theories based on a general-covariant action principle. Field redefinitions of the metric involving the curvature can change this property. We will, hence-forth ignore the possibility of making this kind of field redefinitions and we will assume our metric isthe one matter would couple minimally to in the action. here R is the Ricci scalar as higher-order gravities . One of the main obstacles in the study of higher-order gravities is the sheer numberof different combinations of curvature invariants that can be constructed as the orderin curvature and the dimension grow. For this reason, most of the research has focusedon quadratic theories (see, e.g.
Refs. [ , , , , , , , ]) and in the search fortheories with special properties [ , , , , , , , ]. Typically, one looks fortheories which only propagate, in maximally symmetric vacua, a massless, transverse,graviton or, at the very least, theories that do not propagate the ghost-like spin- modewhich generically appears in higher-order gravities together with scalar modes [ , , ]. Actually, some of the higher-order gravities (the f ( R ) [ , , ] or f ( Lovelock ) [ , ] theories, for example) can be reformulated as scalar-tensor gravities. In thesealternative representations there are issues related to the use of the Jordan or Einsteinframes and to the qualification of the scalar-mediated interactions, which are usuallydifferent for different kinds of matter, as gravitational or non-gravitational.Some of these issues can be avoided by sticking to the original, higher-order, met-ric representation. However, if the theory does propagate other modes besides themassless graviton, one can expect to see some effects associated to them in that repre-sentation as well. One of such effects would be a violation of the Strong EquivalencePrinciple (SEP) as shown in Ref. [ ]. The following paragraph in Section . . ofRef. [ ] summarizes very well our current understanding: after arguing that onlymetric theories of gravity have a chance of satisfying the SEP it is stated that Empirically it has been found that almost every metric theory other than GRintroduces auxiliary gravitational fields, either dynamical or prior geometric, andthus predicts violations of SEP at some level (here we ignore quantum-theory in-spired modifications to GR involving “R ” terms). The reasons why quantum-theory inspired modifications to GR involving “R ” terms ,which are precisely the subject of this note, should be kept out of the discussion arenot clear to us. As we have just mentioned, some of them admit a scalar-tensor repre-sentation which suggests violations of SEP. On the other hand, there are higher-ordertheories such as the Einsteinian cubic gravity of Ref. [ ] which only propagate a trans-verse, traceless, massless spin- mode. Do they violate the SEP? Is it possible to makea general statement concerning the SEP in higher-order gravities?The difference between the Einstein and the Strong Equivalence Principles (EEP andSEP, resp.) is that the second extends the former to situations in which the presenceof gravitational fields is significant. The main effects of gravitational fields in a localsystem in free fall are . Contributions to the binding energy of macroscopic bodies or Keplerian systems. Observe that purely higher-order gravities (without Einstein-Hilbert term) and Palatini ( f ( R ) or else)theories are excluded from our considerations. For a review of the latter see, e.g. Ref. [ ]. The existence of other representations for a generic higher-order gravity is, by no means, guaranteed. A general formulation of the different Equivalence Principles can be found in, e.g. [ ]. . Determination, through the curvature, of the boundary conditions for locallyMinkowskian metrics.The second effect, which falls out of the scope of this note, is responsible for mostof the violations of the SEP in higher-order gravities since, via the curvature of a ex-ternal gravitational field, free-falling systems can suffer spacetime position-dependenteffects. Often they are associated to the emergence of an effective spacetime position-dependent Newton constant, although this concept is far from having a unique andclear definition.The consequences of the first effect depend on the nature of the coupling of gravityto gravity: the SEP states that it must be identical to the coupling of gravity to otherforms of matter/energy so that self-gravitation does not modify the free fall motion ofmassive bodies. (cid:48) In metric theories, thus, the SEP demands that the metric couples tothe gravitational energy-momentum tensor in the same way it couples to the energy-momentum tensor of any other kind of matter. This property only makes sense in theweak-field limit in which a special-relativistic gravitational energy-momentum tensorcan be (non-uniquely) defined, but it is a property that any metric theory of gravitycan be tested for unambiguously. This is what we are going to do for higher-ordergravities.Let us start by formulating more precisely this property.If we expand the gravitational field around the Minkowski spacetime in the weakfield approximation g µν = η µν + χ h µν then the SEP requires the equations of motion ofthe gravitational field to second order in h µν to have the form D ( ) µν = χ ( t ( ) µν + T µν matter ) , ( . )where D ( ) µν is a wave operator that acts linearly on h µν but which is of arbitrary The necessity of the self-coupling of the relativistic gravitational field arises naturally in the con-struction of a classical interacting special-relativistic theory of gravity, as reviewed in Ref. [ ]. TheLorentz invariance of the S matrix of the quantized theory requires the graviton field to couple to thetotal energy-momentum tensor [ , ], which , in the long wavelength limit must be that derived fromGeneral Relativity in the weak-field limit [ ]. At the same time, the consistency of the self-couplingrequires the introduction of an infinite number of corrections as first noticed by Gupta [ , ] whodevised the possible construction of the full theory by demanding consistency of the self-coupling of thegravitational field at all orders (the so-called “Gupta program”). (Obviously, the meaning of consistency is one of the keys in this problem.) It has been argued that the resulting theory is equivalent to GeneralRelativity [ , , , ] although this conclusion seems to depend strongly on the starting point, asshown in Ref. [ ] where a different theory, equivalent to unimodular gravity, was found. Although thetheories that we are considering have an Einstein-Hilbert term and, therefore, a Fierz-Pauli term in theweak-field limit (which seems to lead unavoidably to General Relativity) we are not going to use explic-itly this fact in what follows and our results will be valid for more general classes of general covarianttheories. The violation of this part of the SEP would give rise to the Nordtvedt effect [ , ]. Section . Here we follow the notation and conventions of Ref. [ ]. The consequences of having differentcouplings for the gravitational and matter energy-momentum tensors ( χ (cid:48) and χ ) at the linear level werefirst studied by Kraichnan in Ref. [ ]. rder in derivatives, computed from the quadratic (zeroth-order) term in the action, t ( ) µν is zeroth-order gravitational energy-momentum tensor in Minkowski spacetimecomputed from the same term in the action, quadratic in h µν but also of arbitraryorder in derivatives and T µν matter is the matter energy-momentum tensor in Minkowskispacetime computed according to Rosenfeld’s prescription [ ]. The consistency of the above equation will be the main ingredient of our discussion.Observe that, if it can be derived from the lowest-order terms of an action − δ S δ h µν = D ( ) µν − χ t ( ) µν + . . . , S = S ( ) + χ S ( ) + χ S ( ) + . . . ( . )as we have assumed, the wave operator originates in the variation of the zeroth-orderterm D ( ) µν = − δ S ( ) δ h µν , ( . )while the gravitational energy-momentum tensor must originate in the first-order termin the action t ( ) µν = δ S ( ) δ h µν . ( . )The variation of S ( ) will always give some term quadratic in h µν , but the SEPdemands that this term is precisely the energy-momentum tensor of the gravitationalfield corresponding to S ( ) .The rest of this note is devoted to proving that this property, which requires avery precise relation between S ( ) and S ( ) , holds in all higher-order gravities. Wewill start by recalling some well- and less-well-known facts about special-relativisticenergy-momentum tensors of theories of higher-order in derivatives in Section . InSection we will study the implications of general covariance concerning the gravita-tional energy-momentum tensor in the context of perturbative gravity and in Section we will use them to prove Eq. ( . ). Section contains our conclusions and a discussionof how the SEP may be satisfied when terms of higher order in h µν are considered. That is: coupling matter minimally to the metric g µν and computing T µν matter = δ S matter δ g µν (cid:12)(cid:12)(cid:12)(cid:12) g µν = η µν . ( . )In the context of the weak-field expansion this is the energy-momentum tensor that occurs naturallycoupled to h µν . We ignore the matter fields from now onwards. Energy-momentum tensors
Let us consider a d -dimensional field theory with action S = (cid:90) d d x L , ( . )where the Lagrangian L is a function of the field φ and its derivatives ∂ µ φ , ∂ µ ∂ ν φ , ∂ µ ∂ ν ∂ ρ φ , . . . up to an arbitrary order.Imposing adequate conditions for the boundary values of the field and a numberof its derivatives, demanding the extremization of the above action leads to the Euler-Lagrange equations δ S δφ = ∂ L ∂φ − ∂ µ ∂ L ∂∂ µ φ + ∂ µ ∂ ν ∂ L ∂∂ µ ∂ ν φ − ∂ µ ∂ ν ∂ ρ ∂ L ∂∂ µ ∂ ν ∂ ρ φ + . . . ( . )According to the first Noether theorem, the invariance of the above action underconstant displacements of the coordinates δ x µ , with δφ = δ x µ ∂ µ φ leads to the relation ∂ µ t can µν = δ S δφ ∂ ν φ , ( . )where t can µν ≡ η µν L − ∂ L ∂∂ µ φ ∂ ν φ − ∂ L ∂∂ µ ∂ α φ ∂ ν ∂ α φ + ∂ α (cid:18) ∂ L ∂∂ µ ∂ α φ (cid:19) ∂ ν φ + . . . ( . )is the canonical energy-momentum tensor. The above relation implies its on-shell con-servation ( i.e. when δ S / δφ = ∂ µ t ( ) can µν = δ S ( ) δ h ρσ ∂ ν h ρσ . ( . )Noether currents and, in particular, the canonical energy-momentum tensor, are notunambiguously defined: one can add to them terms proportional to the equations ofmotion, that will vanish on shell, and superpotential terms of the form ∂ ρ Ψ ρµν , with Ψ ρµν = Ψ [ ρµ ] ν . ( . )The positive side of this ambiguity is that it can be used to make gauge-invariant orsymmetrize the canonical energy-momentum tensor. Belinfante [ ] found a system-atic procedure to symmetrize the canonical energy-momentum tensor of fundamentalfields that gives the same result as the Rosenfeld prescription for many fields, but notfor the gravitational field h µν unless one adds terms proportional to the equations ofmotion [ ]. In the General Relativity case, it was shown in Refs. [ , ] that thegravitational energy-momentum tensor which is singled-out by the theory (and satis-fies the consistency condition Eq. ( . )) is completely determined by gauge invariance nd it is related to the canonical one by a superpotential and on-shell-vanishing terms.It was also shown that using other energy-momentum tensors (as done by Thirringin Ref. [ ], for instance) leads to the wrong value for the secular shift of Mercury’sperihelion. The same principle should determine the gravitational energy-momentumtensor in higher-order gravities. On the other hand, given a conserved -index ten-sor, quadratic in the gravitational field, we will identify it with the energy-momentumtensor if it differs from the canonical one by superpotential and on-shell-vanishingterms.Let us know examine in more details how gauge invariance acts in this context. Perturbative gravity and gauge invariance
Higher-order gravities are covariant under general coordinate transformations δ ξ S = χδ ξ = δ ( ) ξ + χδ ( ) ξ + χ δ ( ) ξ + . . . ( . )The variation of the gravitational fields h µν , only has zeroth- and first-order terms:zeroth- and first-order transformations δ ξ h µν = (cid:16) δ ( ) ξ + χδ ( ) ξ (cid:17) h µν , ( . )with δ ( ) ξ h µν = ∂ ( µ ξ ν ) , δ ( ) ξ h µν = £ ξ h µν = ξ ρ ∂ ρ h µν + ∂ ( µ ξ ρ h ν ) ρ . ( . )The parameter ξ µ is completely arbitrary and it does not satisfy any constraints re-stricting the general covariance of the original theory. This condition excludes theunimodular theories which are invariant under the above transformations for diver-genceless parameters ∂ µ ξ µ = ].These transformations relate terms of consecutive orders in the action: δ ( ) ξ S ( ) = . ) δ ( ) ξ S ( n ) + δ ( ) ξ S ( n − ) = n ≤ . ) It should also determine the higher-order corrections for Fierz-Pauli that eventually lead to GeneralRelativity. In the search for theories with consistent self-coupling of the gravitational field h µν (Gupta’sprogram) mentioned in footnote , if one just tries to couple the gravitational field to its own energy-momentum tensor, the ambiguity in the definition of the latter turns the problem into the problem ofwhich is the “right” energy-momentum tensor. Even if one can show that there is a prescription that inthe end gives General Relativity, as in Ref. [ ], one needs some physical principle to justify it and itsuniqueness. Gauge invariance plays here this rôle. o our purposes, each of these relations can be seen as an independent gauge sym-metry, and there is a Noether (also called gauge, or Bianchi) identity associated to eachof these local symmetries via Noether’s second theorem. Using the explicit form ofthe transformations of the gravitational field one arrives, after integration by parts andelimination of total derivatives, to the first two Noether identities ∂ µ δ S ( ) δ h µν = . ) ∂ µ δ S ( ) δ h µν = − ∂ µ (cid:32) δ S ( ) δ h µρ h ρν (cid:33) + δ S ( ) δ h ρσ ∂ ν h ρσ . ( . )Together, if Eq. ( . ) is satisfied, they are consistent with the on-shell conservationof the gravitational energy-momentum tensor to lowest order in χ : if we take thedivergence of both sides of the equation of motion ( . ) and use the first identity, weget χ∂ µ t ( ) µν = . )The second identity says that χ∂ µ t ( ) µν = − χ∂ µ (cid:32) δ S ( ) δ h µρ h ρν (cid:33) + χ δ S ( ) δ h ρσ ∂ ν h ρσ , ( . )and, using again the equations of motion χ∂ µ t ( ) µν = O ( χ ) , on shell. ( . )Apart from this consistency check, let us stress that the above identities Eqs. ( . ),( . )hold off-shell for any higher-order gravity. The energy-momentum tensor of higher-order gravity
We are now ready to prove that Eq. ( . ) holds; that is: that 2 δ S ( ) / δ h µν can be iden-tified with one of the special-relativistic energy-momentum tensors that one can asso-ciate to S ( ) , such as the canonical energy-momentum tensor t ( ) can µν plus some on-shell-vanishing terms (to lowest order in χ !) and a superpotential term.Using in the Noether identity Eq. ( . ) the property Eq. ( . ) we can rewrite it asfollows: The restricted gauge invariance of linearized unimodular gravity leads to different Noether identi-ties [ , ]. The classical equations of motion of unimodular gravity [ ] are those of General Relativitywith a cosmological constant and, presumably, this theory must enjoy the same property we are provinghere, although it must arise in a more complicated way. µ (cid:32) δ S ( ) δ h µν (cid:33) = ∂ µ (cid:32) t ( ) can µν − δ S ( ) δ h µρ h ρν (cid:33) , ( . )from which it follows that2 δ S ( ) δ h µν = t ( ) can µν − δ S ( ) δ h µρ h ρν + ∂ ρ Ψ ( ) ρµν , with Ψ ( ) ρµν = − Ψ ( ) µρν . ( . )Given that the second term in the r.h.s. vanishes on-shell at the order in χ weare working at, the l.h.s. of the above equation can be identified with a special-relativistic gravitational energy-momentum tensor associated to the zeroth-order action S ( ) , proving Eq. ( . ). Extension to higher orders
Can we extend this O ( h ) result to higher orders?The n th Noether identity is ∂ µ δ S ( n ) δ h µν = − ∂ µ (cid:32) δ S ( n − ) δ h µρ h ρν (cid:33) + δ S ( n − ) δ h ρσ ∂ ν h ρσ , ( . )and using again Eq. ( . ) and the same reasoning as in the n = δ S ( n ) δ h µν = t ( n − ) can µν − δ S ( n − ) δ h µρ h ρν + ∂ ρ Ψ ( n − ) ρµν , with Ψ ( n − ) ρµν = − Ψ ( n − ) µρν .( . )where t ( n − ) can µν is the contribution to the gravitational special-relativistic canonical energy-momentum tensor coming from the S ( n − ) . Adding all these relations to order n mul-tiplied by the corresponding power of χ we get2 (cid:32) χ δ S ( ) δ h µν + χ δ S ( ) δ h µν + · · · + χ n δ S ( n ) δ h µν (cid:33) = χ t ( ) can µν + χ t ( ) can µν + · · · + χ n t ( n − ) can µν − χ (cid:32) δ S ( ) δ h µν + χ δ S ( ) δ h µν + · · · + χ n − δ S ( n − ) δ h µν (cid:33) h ρν + ∂ ρ (cid:16) χ Ψ ( ) ρµν + χ Ψ ( ) ρµν + · · · + + χ n Ψ ( n − ) ρµν (cid:17) . ( . ) Equivalently, if we use the equations of motion, that term would be of O ( h ) . he n th-order equations of motion say that δ S ( ) δ h µν + χ δ S ( ) δ h µν + · · · + χ n − δ S ( n − ) δ h µν = − χ n δ S ( n ) δ h µν ( . )and, therefore, the term in the second line of the r.h.s. of Eq. ( . ) vanishes up to thisorder on-shell and the term in the l.h.s. can be identified with a special-relativisticenergy-momentum tensor for the gravitational field. Gauge transformations of the energy-momentum ten-sor
The lowest-order energy-momentum tensor of the spin- field, t ( ) GR µν is not expected tobe invariant under the δ ( ) ξ gauge transformations. Its gauge transformation rule mustbe completely determined by the gauge-invariance of the action and the same mustbe true for t ( ) µν and its higher order in h corrections in higher-order theories. In orderto find this gauge transformation rule we proceed as follows: we first compute thegauge trasformations Eqs. ( . ) and ( . ) of the action δ ξ S [ h ] order by order in χ . Byassumption δ ξ S [ h ] = h µν , δδ ξ S [ h ] =
0, taking into account that δδ ( ) ξ h µν δ h αβ = δδ ( ) ξ h µν δ h αβ (cid:54) =
0, sothat, integrating by parts (cid:90) d d x δ S ( n ) δ h µν δδ ( ) ξ h µν δ h αβ = (cid:90) d d x (cid:40) − ∂ ρ (cid:32) ξ ρ δ S ( n ) δ h αβ (cid:33) + ∂ ( α ξ ρ δ S ( n ) δ h β ) ρ (cid:41) . ( . )Next, we interchange the variations of the S ( n ) , arriving to δδ ξ S [ h ] = (cid:90) d d x (cid:40) δ ( ) ξ δ S ( ) δ h αβ + χ (cid:34) δ ( ) ξ δ S ( ) δ h αβ + δ ( ) ξ δ S ( ) δ h αβ − ∂ ρ (cid:32) ξ ρ δ S ( ) δ h αβ (cid:33) + ∂ ( α ξ ρ δ S ( ) δ h β ) ρ (cid:35) + · · · (cid:41) δ h αβ = . )Finally, the standard arguments lead to the identities δ ( ) ξ δ S ( n ) δ h αβ = − δ ( ) ξ δ S ( n − ) δ h αβ + ∂ ρ (cid:32) ξ ρ δ S ( n − ) δ h αβ (cid:33) − ∂ ( α ξ ρ δ S ( n − ) δ h β ) ρ , ∀ n ≥ . )Obviously, the sums of the terms n = · · · N and those of the terms n = · · · N − or GR at lowest order, these identities imply δ ( ) ξ D ( ) µν = . ) δ ( ) ξ t ( ) µν GR = δ ( ) ξ D ( ) µν + ∂ ρ (cid:16) ξ ρ D ( ) µν (cid:17) − ∂ ( µ | ξ ρ D ( ) | ν ) ρ , ( . )which can be checked to hold using the Fierz-Pauli equation of motion and the explicitexpression for t ( ) µν GR in Eq. ( . ) of Ref. [ ]. The equation D ( ) µν = χ t ( ) µν GR , ( . )is only invariant under ( δ ( ) ξ + χδ ( ) ξ ) h µν and only up to terms proportional to D ( ) µν and its derivatives which, on-shell, are of order O ( χ ) . Discussion