BRX TH-626 CALT 68-2811 The Bel-Robinson tensor for topologically massive gravity S. Deser Physics Department, Brandeis University, Waltham, MA 02454 and 1 1 Lauritsen Laboratory, California Institute of Technology, Pasadena, CA 91125 0 [email protected] 2 n J. Franklin a J Reed College, Portland, OR 97202 7 1 [email protected] ] c Abstract q - r g We construct, and establish the (covariant) conservation of, a 4-index “super-stress ten- [ sor” for topologically massive gravity (TMG). Separately, we discuss its invalidity in quadratic curvature models and suggest a generalization. 2 v 0 4 The 4-index Bel-Robinson tensor Bγµνρ, quadratic in the Riemann tensor and (covariantly) 2 conserved on Einstein shell, has received much scrutiny in its original D =4 habitat (see references 4 in [1]). There, B is the nearest thing to a covariant gravitational stress-tensor, for example playing . 1 essentially that role in permitting construction of higher (L > 2) loop local counter-terms in 1 0 supergravity [2,3]. It also generalizes to D > 4, at the minor price of losing tracelessness, like its 1 spin 1 model, the Maxwell stress-tensor. : v i In this note, we turn to lower D, asking whether B survives in D = 3 and if so, to what X question is it the answer–in what theory, if any, is it conserved? Since the hallmark of D = 3 is r a the identity of Riemann and Einstein tensors (they are double-duals), it is obvious that B vanishes 1 identicallyonpureEinstein(i.e.,flatspace)shell ,andbecomesthetrivial(andremovable)constant tensor ∼ (Λ2g g +symm) in cosmological GR [4]. This leaves the dynamical hallmark of D = 3, γµ νρ TMG [5], and the new quadratic curvature models [6,7], as the other possible beneficiaries. Our main resultisthat Bbothsurvives dimensionalreduction andis conserved on TMGshell, inaccord with the similar mechanism ensuring the Maxwell tensor’s conservation on topologically massive electrodynamics(TME)shell. Separately,asimpleargumentshowswhyitdoesnotworkforgeneric quadratic curvature actions. One obtains B in D = 3 by inserting the Riemann-Ricci identites (we use de-densitized ǫµνα throughout) Rµανβ ≡ (gµνRαβ +symm)≡ ǫµασG ǫνβρ σρ 1Actually, B can already be made trivial on D=4 GR shell, by addingsuitable terms [8]. 1 into a D = 4 B. The resulting combination is: B = R¯ R¯ +R¯ R¯ −g R¯ R¯β , R¯ ≡ R −1/4g R; (1) γµνρ µν γρ µρ γν µγ νβ ρ µν µν µν the Schouten tensor R¯ also defines the Cotton tensor below. B is manifestly symmetric under (γµ,νρ) pair interchanges (but not totally symmetric here because that depended on special D = 4 identities). Clearly, B vanishes identically for R¯ = 0, and reduces to a constant tensor for the µν cosmological R¯ = Λg extension, a term which may even be removed by suitably adding to the µν µν definition of B there. Turning to TMG, its field equation is [5] Gµν = µ−1Cµν ≡ µ−1ǫµργD R¯ ν (2) ρ γ The Cotton tensor Cµν is identically (covariantly) conserved, symmetric and traceless, so tracing (2) implies R = 0, which simplifies on-shell calculations; µ is a constant with dimension of mass. [Our results will also apply to cosmologically extended TMG [9], much as they do for cosmological GR.] Our question then is whether B of (1) is conserved by virtue of (2). The reason we expect this is the close analogy between TMG and its vector version, TME. The latter model’s abelian version (its non-abelian extension is similar), has (flat space) field equations resembling (2), 1 ∂ Fαβ = µǫαγβF ≡ µ ∗Fα, (3) β γβ 2 while the analog of B is the Maxwell stress tensor T = F βF −1/4g F Fαβ. (4) Mµν µ νβ µν αβ It is indeed conserved on TME shell, as follows: ∂ Tµν = Fµβ∂ Fν = µFµβ ∗F ≡ µǫµαβ ∗F ∗F ≡ 0. (5) ν ν β β α β This success motivates seeking a TMG chain similar to (5), schematically, DB ≡ R (DR−DR) ≡RǫC = µ−1ǫCC ≡? 0; (6) thatis,wearehopingtosetupacurlsoastousethealgebraicidentity D R¯ −D R¯ ≡ ǫ Cµ α βγ γ βα µαγ β as indicated. [There is a major distinction between the two models, however. The Maxwell tensor is also the stress tensor of TME since its Chern-Simons term, being metric-independent, does not contribute. Hence conservation is guaranteed a priori here [5], unlike the very existence, let alone conservation, of a B for TMG.] Taking the divergence of (1) and using (2) indeed yields D Bγµνρ = DγR¯µν −DµR¯νγ R¯ ρ+ DγR¯µρ−DµR¯ργ R¯ ν = µǫσγµ C νC ρ+C ρC ν ≡0 γ γ γ σ γ σ γ (cid:2) (cid:3) (cid:2) (cid:3) (cid:0) (cid:1) (7) where the identity follows by the symmetry under (σγ). This establishes the nontrivial role of B as a “covariant” conserved gravitational tensor for TMG. It may thus find uses here similar to those of the original B in classifying GR solutions. Whether it is relevant to the quantum extensions of these theories is unclear, since D = 3 GR is finite [10] and TMG may be [11]. The other gravitational model of special interest in D = 3 is the “new quadratic curvature” theory. Its L = aR+bR¯2, or even its pure R¯2 variant, does not conserve B. The reason is obvious 2 and applies as well to all quadratic curvature actions in D = 4. The divergence of (any) B behaves 2 as RDR, while the R field equations read DDR+RR = 0, hence they do not tell us anything aboutDR. Sounless RDR vanishes for algebraic reasons, and it does not, thereis nohopealready at linearized, DDR, level, quite apart from the RR terms. A clear example is the R¯2 field equation itself, 3 (cid:3)R¯ + g (cid:3)− D D R+ 2R¯ R¯α −g R¯αβR¯ = 0. (8) µν (cid:18) µν 8 µ ν(cid:19) µα ν µν αβ (cid:16) (cid:17) B-nonconservation also makes physical sense: one would expect the correct candidate (if any) to ′ 2 have the form B = DRDR to reflect the extra derivatives in R actions. In summary, we have obtained a conserved Bel-Robinson tensor for D = 3 TMG, despite TMG’s third derivative order. It is, gratifyingly, the reduction of one originally defined for D = 4 GR, and fits nicely with the Maxwell stress tensor’s conservation in TME. We also noted the unsuitability of B as a conserved tensor in quadratic curvature models, suggesting instead that a modified B′ ∼ DRDR might succeed. SD acknowledges support from NSF PHY 07-57190 and DOE DE-FG02-164 92ER40701 grants. References [1] S. Deser, in “Gravitation and Relativity in General” (ed F. Atrio-Barandela and J. Martins), Wold Publishing (1999), gr-qc/9901007. [2] S. Deser, K.S. Stelle and J.H. Kay, Phys. Rev. Lett., 38 527 (1977). [3] N. Beisert et al, Phys. Lett. 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