We construct, and establish the (covariant) conservation of, a 4-index ‘super stress tensor’ for topologically massive gravity. Separately, we discuss its invalidity in quadratic curvature models and suggest a generalization

Deser, S.

Franklin, J.

Caltech Authors - Main

The Bel-Robinson tensor for topologically massive gravity

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BRX TH-626
CALT 68-2811
The Bel-Robinson tensor for topologically massive gravity
S. Deser
Physics Department, Brandeis University, Waltham, MA 02454 and
Lauritsen Laboratory, California Institute of Technology, Pasadena, CA 91125
deser@brandeis.edu
J. Franklin
Reed College, Portland, OR 97202
jfrankli@reed.edu
Abstract
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.
The 4-index Bel-Robinson tensor Bγµνρ, quadratic in the Riemann tensor and (covariantly)
conserved on Einstein shell, has received much scrutiny in its original D = 4 habitat (see references
in [1]). There, B is the nearest thing to a covariant gravitational stress-tensor, for example playing
essentially that role in permitting construction of higher (L > 2) loop local counter-terms in
supergravity [2,3]. It also generalizes to D > 4, at the minor price of losing tracelessness, like its
spin 1 model, the Maxwell stress-tensor.
In this note, we turn to lower D, asking whether B survives in D = 3 and if so, to what
question is it the answer–in what theory, if any, is it conserved? Since the hallmark of D = 3 is
the identity of Riemann and Einstein tensors (they are double-duals), it is obvious that B vanishes
identically on pure Einstein (i.e., flat space) shell1, and becomes the trivial (and removable) constant
tensor ∼ (Λ2gγµ gνρ+symm) in cosmological GR [4]. This leaves the dynamical hallmark of D = 3,
TMG [5], and the new quadratic curvature models [6,7], as the other possible beneficiaries. Our
main result is that B both survives dimensional reduction and is conserved on TMG shell, in accord
with the similar mechanism ensuring the Maxwell tensor’s conservation on topologically massive
electrodynamics (TME) shell. Separately, a simple argument shows why it does not work for generic
quadratic curvature actions.
One obtains B in D = 3 by inserting the Riemann-Ricci identites (we use de-densitized ǫµνα
throughout)
Rµανβ ≡ (gµνRαβ + symm) ≡ ǫµασ Gσρ ǫ
νβρ
1Actually, B can already be made trivial on D = 4 GR shell, by adding suitable terms [8].
1
into a D = 4 B. The resulting combination is:
Bγµνρ = R¯µν R¯γρ + R¯µρ R¯γν − gµγ R¯νβ R¯
β
ρ , R¯µν ≡ Rµν − 1/4 gµν R; (1)
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¯µν = 0, and reduces to a constant tensor for the
cosmological R¯µν = Λ gµν extension, a term which may even be removed by suitably adding to the
definition of B there. Turning to TMG, its field equation is [5]
Gµν = µ−1Cµν ≡ µ−1 ǫµργ Dρ R¯
ν
γ (2)
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),
∂β F
αβ =
1
2
µ ǫαγβFγβ ≡ µ
∗Fα, (3)
while the analog of B is the Maxwell stress tensor
TMµν = F
β
µ Fνβ − 1/4 gµν Fαβ F
αβ. (4)
It is indeed conserved on TME shell, as follows:
∂ν T
µν = Fµβ ∂ν F
ν
β = µF
µβ ∗Fβ ≡ µ ǫ
µαβ ∗Fα
∗Fβ ≡ 0. (5)
This success motivates seeking a TMG chain similar to (5), schematically,
DB ≡ R (DR−DR) ≡ R ǫC = µ−1ǫC C
?
≡ 0; (6)
that is, we are hoping to set up a curl so as to use the algebraic identity DαR¯βγ−Dγ R¯βα ≡ ǫµαγ C
µ
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
Dγ B
γµνρ =
[
Dγ R¯µν −Dµ R¯νγ
]
R¯ ργ +
[
Dγ R¯µρ −Dµ R¯ργ
]
R¯ νγ = µ ǫ
σγµ
(
C νσ C
ρ
γ +C
ρ
σ C
ν
γ
)
≡ 0
(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+ b R¯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
as RDR, while the R2 field equations read DDR + RR = 0, hence they do not tell us anything
about DR. So unless RDR vanishes for algebraic reasons, and it does not, there is no hope already
at linearized, DDR, level, quite apart from the RR terms. A clear example is the R¯2 field equation
itself,
 R¯µν +
(
gµν −
3
8
DµDν
)
R+
(
2 R¯µα R¯
α
ν − gµν R¯
αβ R¯αβ
)
= 0. (8)
B-nonconservation also makes physical sense: one would expect the correct candidate (if any) to
have the form B′ = DRDR to reflect the extra derivatives in R2 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. B, 694 265 (2010), hep-th/1009.1643.
[4] S. Deser and R. Jackiw, Ann. Phys. 153 405 (1984).
[5] S.Deser, R.Jackiw and S.Templeton, Ann. Phys. 140 372 (1982); Phys. Rev. Lett., 48 975
(1982).
[6] E. A. Bergshoeff, O. Hohm and P. K. Townsend, Phys. Rev. Lett. 102 201301(2009), hep-
th/0901.1766.
[7] S. Deser, Phys. Rev. Lett. 103 101302 (2009), hep-th/0904.4473.
[8] R.K. Sachs, Z. Phys. 157, 462 (1960).
[9] S. Deser, in “Quantum Theory of Gravity” (ed S.M. Christensen), Hilger London (1984).
[10] E. Witten, Nucl. Phys. B., 311 46 (1988); S.Deser, J.McCarthy and Z.Yang, Phys. Lett. B,
222 61 (1989).
[11] S. Deser and Z. Yang, Class. Quant. Grav., 7 1603 (1990).
3


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The Bel-Robinson tensor for topologically massive gravity

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