PLB34434S0370-2693(19)30096-610.1016/j.physletb.2019.02.005The Author(s)TheoryAre all identically conserved geometric tensors metric variations of actions? A status reportS.Deserabdeser@brandeis.eduY.Pangc⁎Yi.Pang@maths.ox.ac.ukaCalifornia Institute of Technology, Pasadena, CA 91125, USACalifornia Institute of TechnologyPasadenaCA91125USAbBrandeis University, Waltham, MA 02454, USABrandeis UniversityWalthamMA02454USAcMathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UKMathematical InstituteUniversity of OxfordWoodstock RoadOxfordOX2 6GGUK⁎Corresponding author.Editor: M. CvetičAbstractNoether's theorem, that local gauge variations of gauge invariant actions are identically conserved (more tautologically, that gauge variations of gauge invariants vanish) was established a century ago. Its converse, in the geometric context: are all identically conserved local symmetric tensors variations of some coordinate invariant action? remains unsolved to this day. We survey its present state and discuss some of our concrete attempts at a solution, including a significant improvement. For notational simplicity, details are primarily given in D=2, but we discuss generic D as well.KeywordsGeneral relativityConservation laws1IntroductionNoether's theorem is a textbook truism that the field equations of gauge theories–Maxwell, Yang–Mills, Einstein et al.–obey conservation, “Bianchi”, identities as a consequence of their Lagrangian origins: The actions being invariant, their local gauge variations vanish. But the latter are just the divergences of the action's field variations: It suffices for models to be Lagrangian for them to obey gauge identities. But is it also necessary–are all identically conserved currents derived from actions? This converse hypothesis is almost as old as Noether's and remains unsolved–for the gravitational case–despite its simple form and intuitive appeal. Over the last few decades, only limited success has been achieved. For instance, when the tensor has at most two metric derivatives, ∂2g, it is Lagrangian [1]; at ∂3g order, [2] proved the Lagrangian nature in definite signature spaces. That assumption was lifted in [3] in D=3, while [4] gave the general ∂3g proof in all D. For a detailed history, see, e.g., [5]. To our knowledge, there is no result beyond ∂3g until our present ∂6g one. This is not merely a formal conjecture, but has direct physical consequences: Non-Lagrangian terms have recently been proposed as alternative geometrical models. But the physics requires them to be separately conserved: Since coordinate invariant matter actions' stress-tensors are identically conserved (on matter shell), irrespective of their couplings, if any, to gravity, the proposed field equations,(1)Gμν(g)+Eμν(g)=Tμν(matt;g) imply that the non-Lagrangian gravitational addition Eμν must be identically conserved, since both the Lagrangian gravity part Gμν (including Gμν=0) and–as we saw11A recent suggestion [6] that a matter Lagrangian is not needed to specify matter systems, but only conservation of the stress-tensors, can be understood in this light as being entirely equivalent to the standard lore: A correct stress tensor is always the metric variation of an action, and is conserved IFF the matter field equations are invoked. –Tμν(matt) both are. Hence counterexamples to the necessity hypothesis, if they existed, would be of physical interest and conversely their absence would remove a sea of models. We shall first review the vector gauge theories, where there are manifold counter-examples to the conjecture, before coming to the gravitational story. Concentrating on the most elementary geometrical systems, those in D=2 where only the scalar curvature enters, we will discuss some differential and integral approaches to exhibit the nature of some of the obstacles involved as well as all-order versus perturbative attempts; in the former case we have succeeded in reaching several derivative order improvements over past results. Higher-dimensional similarities and differences will also be discussed. For completeness, we emphasize that we are only interested in local currents constructed only from metric and its local derivatives, as these are meant to be primary models. There are of course non-local quantum contributions to effective actions (from anomalies inter alia) but these are all action-generated anyway. We also underline the irrelevance of (action-defined) matter's Tμν details. The complementary question of non-action matter sources' gravity problems is treated in [7]. Given the simplicity and plausibility of the hypothesis, we cannot help but feel some obvious proof is being overlooked; perhaps this résumé will attract one!2VectorsA sufficiently general set of field equations, first in the abelian, D=4 Maxwell, case, is(2)Mν=∂μ[X(F2,F˜F)Fμν]=0, where F˜μν is the (D=4) dual of Fμν and we have used only its two simplest, algebraic, invariants in the arbitrary function X. The divergence identities ∂νMν=0 are manifest from the antisymmetry of F contracted with the symmetric ∂μ∂ν, irrespective of X. However, not all such M are Aμ variation of a Lagrangian: they must obey the usual Helmholz integrability conditions, which set stringent limits on the X. So here identical conservation does NOT require an action. Perhaps surprisingly this is not some purely linear, abelian property, but holds also for non-abelian fields: there, we replace ∂μ by the usual covariant color derivatives Dμ whose commutator is now the non-abelian field strength, [Dμ,Dν]∼Fμν. Yet the generalization of (2) remains transverse owing to the antisymmetry of the structure constants: fabcFbμνFμνc=0 (the arguments of X are now the (color-singlet) traces of F2 and F˜F). Again, only the algebraic factor: antisymmetry, is relevant.3GravityWe now come to our problem: the origin of identically conserved geometric tensors. The formalism is enormously simplified by working first in D=2, where all essentials are already present, index proliferation is at a minimum and the issues are manifest. Only the scalar curvature R and its covariant derivatives, ∇nR, (since Rμν=12gμνR), and explicit metrics contracting indices are present. Our convention is(3)R=gμνRμν=gμνRαμαν=gμν(∂αΓαμν−∂νΓααμΓααβΓβμν−ΓανβΓβμα). Its variation is(4)δR(x)δgμν(y)=[12gμνR+(gμν∇2−∇μ∇ν)]δ(2)(x−y). Note that the ∇∇ part of δR is the covariant version of the flat space transverse projector Oμν=[ημν∂2−∂μ∂ν], but it is of course no longer transverse; there are none in curved space. Indeed this is the 2D version of the flat superpotentials Vμν=∂α∂βH[μα][νβ], where H has the algebraic symmetries of the Riemann tensor, so V is identically conserved. In D=2, H degenerates into εμαενβS where S is a scalar, namely into the Oμν above. First, a reminder of why invariant action-based tensors are conserved here (non-invariant actions' variations are of course not even tensors). The variation of(5)A=∫d2xL(gμν;∇nR),n≥0 is(6)δAδgμν(x)|total=δAδgμν(x)|Rconst+∫d2yδR(y)δgμν(x)δAδR(y)|gconst, and of course the Noether identity ∇νδAδgμν|total=0 holds because A is invariant under arbitrary coordinate variations, δgμν=∇(μξν). Note that both terms in (6) are “normal” tensors, as against “projector” ones, OμνS–this point is critical to our problem, so we explain it. (Ex-)projectors are of course tensors, but strange ones whose divergences are NOT in general total derivatives: despite the notation, ∇ν(OμνS) is of the form S∂R (or R∂S, depending on choice); that is manifestly NOT always the divergence of any regular, NON-OS, tensor–for example if S=(∂R)2. The Lagrangian case is the one where OS is normal, because it also can be written as δR/δg, so for S=δA−gδR|g we recover (6).The above illustrates sufficiency; Now for necessity: are there NON-Lagrangian identically conserved Xμν(gμν;∇nR)? In the vector cases, we saw that such (vector) terms existed because one merely algebraically contracted antisymmetric with symmetric indices, unlike the differential nature of the present problem. The lowest-level cases are easy: if Xμν is R-independent, it must be proportional to gμν, namely to a cosmological action L=−g. Likewise, X=X(g;R) obviously comes from an L=−gf(R). This is no longer so obvious when X does depend on derivatives of R. We must fall back on the projector basis of flat space conservation for inspiration. As we saw above, if the R-dependence is such that a scalar S is of the form δA−gδR|g, then ∫d2y−gδR(y)/δgμν(x)S(y) is the R-variation of an action and the total conserved current is its sum with δA−gδgμν(x)|R. The inspiration is of course (4), showing that the flat Oμν must be extended to the curved one, plus the (natural) gR-term. We can now state the general problem in its tersest form, at least in the present approach. Are there NON-Lagrangian solutions of the local equation ∇ν(OμνS+Zμν)=0, where Z is a “normal” tensor, S a scalar and O the δR/δgμν of (4)? So far the only way a compensating “normal” Z can exist is for OS to have a normal divergence as discussed above. Although we have not succeeded in settling the question, it seems so intuitively simple that these lines may inspire a resolution. In higher D, there are a few novel wrinkles, such as the existence of 4-index Oμνρλ from the variations of the–identically conserved–Einstein tensor, multiplied by a 2-tensor Sρλ and of course the complications of dependence on the index-rich (covariant derivatives of) Ricci and Riemann tensors. These are all examples of the general superpotential ∂α∂βH[μα][νβ] mentioned earlier. Then there are Chern–Simons like operators in odd D, and finally for D>4 the Lanczos–Lovelock [1,8]22[8] merely noted the quadratic curvature topological invariants in D=4, namely Gauss–Bonnet and its axial counterpart ∫d4xR˜R, while [1] showed that the G–B action becomes dynamical for D>4 and listed all such extensions. actions' variations have no contributions from their curvature dependence, but rather entirely from their explicit metric dependence, in complete contrast with D=2, where the latter is trivial.Let us now look (back in D=2) at the problem, first in a perturbative way. [For space reasons, we will be very terse about the still unsolved approaches.]. We seek an identically conserved tensor Xμν(7)Xμν=(∇μ∇ν−gμν□)S−12SRgμν+Zμν, whose vanishing divergence means that(8)∇μZμν=12S∂νR, an equation that resembles that of a scalar–tensor model with R an independent scalar. In a weak field expansion about flat space,(9)gμν≃ημν+ϵhμν,ϵ≪1, the leading term in (8) becomes(10)∂μZμν(L)=12S(L)∂νR(L), in an obvious notation; all covariant derivatives are here replaced by partials. The right hand side of (10) is annihilated by Euler–Lagrangian operator (since it kills all total derivatives), is a necessary, but not sufficient, condition for conservation. While we have not been able to solve the resulting condition iteratively in general, we have at least succeeded in pushing the known results [4] several orders higher in derivatives of R, namely to ∂6 in the metric, as we briefly sketch. When Xμν depends only on the first six derivatives of the metric, we can in fact construct a non-perturbative proof by solving (8) directly. In this case, S depends on at most the second derivative of the R so the most general Zμν must take the form(11)Zμν=A(R,T)(∇μ∇ν−gμν□)R+B(R,T)∇μR∇νR+gμνC(R,T),T≡(∂R)2. Notice that the ∇2R terms in Zμν must appear in the combination OμνR or ∇μZμν would depend on ∇3R. Now demanding(12)∇μZμν=12S(R,∇R,∇2R)∂νR yields(13)B=−∂A∂R−2∂C∂T. In deriving (13), we have used the D=2 identities for any scalar quantity ϕ(14)2ϕμνϕνλ∇λϕ−2ϕμν∇νϕ□ϕ=∇μϕ[(□ϕ)2−ϕνλϕνλ],ϕμν≡∇μ∇νϕ,gμνϕλρ∇λϕ∇ρϕ−2ϕλ(μ∇ν)ϕ∇λϕ=(∂ϕ)2(gμν□−∇μ∇ν)ϕ−□ϕ∇μϕ∇νϕ. One can show that the particular Zμν satisfying (13) results from varying the following action with respect to gμν with R fixed(15)A=∫d2xg(A∇μlogT∇μR−2C). Therefore, general covariance implies gS=δAδR|g which contains at most second derivative of R. By the reasoning given in previous paragraphs, Xμν constructed in (7) comprises the general divergence free symmetric tensor depending on at most ∂6g. This procedure can of course continue to higher orders with the encounter of complicated new Schouten-type identities at each order. We did not proceed further because our main goal is an all-order proof which seems to be beyond the limit of the current approach. However, equation (8) does provide a link between scalar–tensor models and the divergenceless symmetric D=2 tensor, the latter being four derivative orders higher than the former in their respective fundamental fields.A different approach to the problem would be to establish that Xμν obeys the integrability condition(16)δgXμν(x)δgρσ(y)=δgXρσ(y)δgμν(x), namely, δgXμν(x)δgρσ(y) is a formally self-adjoint differential operator comprised of the Riemann tensor and its covariant derivatives. The integral form of (16) can be expressed as(17)∫M(δ2(gXμν)δ1gμν−δ1(gXμν)δ2gμν)=0, for arbitrary variations δ1g and δ2g. To approach our goal (17), first define the functional(18)AX(Y):=∫MgXμνYμν, in which the tensor Yμν has finite support on M. Conservation of Xμν implies this functional vanishes when Y is the Lie derivative of the metric with respect to a compactly supported vector field:(19)AX(Lξg):=2∫MgXμν∇μξν=0. Here L denotes the Lie derivative and we have used that Lξgμν=∇μξν+∇νξμ. Hence, the variation of A(Lξg) also vanishes so that(20)δ1AX(Lξg)=∫M(δ1(gXμν)Lξgμν+gXμνLξδ1gμν)=0. The functional AX(Y) is diffeomorphism-invariant, so a variation δ2AX(δ1g) with δ2δ1g=Lξ(δ1g) also vanishes. This gives(21)∫M(δ2(gXμν)δ1gμν+gXμνLξ(δ1g))=0. The difference of the above two displays(22)∫M(δ2(gXμν)δ1gμν−δ1(gXμν)δ2gμν)|δ2g=Lξg=0. Were δ2g not restricted to variations of the form Lξg, this would complete the proof. However, Xμν has three components in D=2, of which only two are affected by (22). We have unfortunately been unable to complete this “integral” approach either.4CommentsWe have reviewed and summarized the current standing of a century-old conjecture-validity of the converse of Noether's theorem: are all identically conserved geometrical 2-tensors the metric variations of some invariant action? This intuitively attractive proposition has proved remarkably recalcitrant to date, although we have managed to push the proof to sixth derivative order in the metric. A number of quite different approaches have been pursued and we have summarized them by concentrating on the simplest curved space dimension, D=2, where the problem is most clearly stated without the obscuring higher D index proliferation. A proof (or indeed disproof) in D=2 all but guarantees the same for all D. There are important physical consequences of this seemingly formal question to real physics: Of the many attempts to go beyond GR, addition of non-Lagrangian terms on the “left hand side” of the field equations requires them to be identically conserved, since both Gμν and the (Lagrangian-based) matter stress tensors on their mass shell are. This would close the floodgates to a wide range of speculation. [Conversely, in the unlikely event that there are such tensors, a whole new field would open up!] In string theory, one always obtains DX=0 equation for the target space fields from the world-sheet BRST invariance. So if our conjecture is true, it also implies that all stringy gravity models are Lagrangian.We have used locality as a physical demand. If that is lifted, it is trivial to provide counter-examples, albeit non-symmetric ones, such as Xμν=(∇μ□−1∇ν−gμν)S (conserved on one index). Finally, we have not investigated the recently proposed [9,10] amusing D=3 models whose X-divergences only vanish on-shell.AcknowledgementsWe thank Andrew Waldron for his useful suggestions in the early stages of this endless saga. The work of S.D. is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632. The work of Y.P. is supported by a Newton International Fellowship NF170385 of the UK Royal Society.References[1]D.LovelockThe Einstein tensor and its generalizationsJ. Math. Phys.121971498[2]G.W.HorndeskiDivergence-free third order tensorial concomitants of a pseudo-Riemannian metricTensor2919752129[3]D.LovelockDivergence-free third order concomitants of the metric tensor in three dimensionsTopics in Differential Geometry (in Memory of Evan Tom Davies)1976Academic PressNew York8798[4]Ian M.Anderson Juha PohjanpeltoVariational principles for natural divergence-free tensors in metric field theoriesJ. Geom. 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