We attempt to generalize the familiar covariantly conserved Bel-Robinson
tensor B_{mnab} ~ R R of GR and its recent topologically massive third
derivative order counterpart B ~ RDR, to quadratic curvature actions. Two very
different models of current interest are examined: fourth order D=3 "new
massive", and second order D>4 Lanczos-Lovelock, gravity. On dimensional
grounds, the candidates here become B ~ DRDR+RRR. For the D=3 model, there
indeed exist conserved B ~ dRdR in the linearized limit. However, despite a
plethora of available cubic terms, B cannot be extended to the full theory. The
D>4 models are not even linearizable about flat space, since their field
equations are quadratic in curvature; they also have no viable B, a fact that
persists even if one includes cosmological or Einstein terms to allow
linearization about the resulting dS vacua. These results are an unexpected, if
hardly unique, example of linearization instability.Comment: published versio

Deser S

J Franklin

S Deser

English

arXiv

We attempt to generalize the familiar covariantly conserved Bel–Robinson tensor B_(μναβ) ~ RR of GR and its recent topologically massive third derivative order counterpart B ~ RDR to quadratic curvature actions. Two very different models of current interest are examined: fourth-order D = 3 'new massive gravity' and second-order D > 4 Lanczos–Lovelock. On dimensional grounds, the candidates here become B ~ DRDR + RRR. For the D = 3 model, there indeed exist conserved B ~ ∂R∂R in the linearized limit. However, despite a plethora of available cubic terms, B cannot be extended to the full theory. The D > 4 models are not even linearizable about flat space, since their field equations are quadratic in curvature; they also have no viable B, a fact that persists even if one includes cosmological or Einstein terms to allow linearization about the resulting dS vacua. These results are an unexpected, if hardly unique, example of linearization instability

Deser, S.

Franklin, J.

Caltech Authors - Main

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BRX TH-635
CALT 68-2825
No Bel-Robinson Tensor for Quadratic Curvature Theories
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 attempt to generalize the familiar covariantly conserved Bel-Robinson tensor Bµναβ ∼
RR of GR and its recent topologically massive third derivative order counterpart B ∼ RDR,
to quadratic curvature actions. Two very different models of current interest are examined:
fourth order D = 3 “new massive”, and second order D > 4 Lanczos-Lovelock, gravity. On
dimensional grounds, the candidates here become B ∼ DRDR + RRR. For the D = 3 model,
there indeed exist conserved B ∼ ∂R∂R in the linearized limit. However, despite a plethora
of available cubic terms, B cannot be extended to the full theory. The D > 4 models are not
even linearizable about flat space, since their field equations are quadratic in curvature; they
also have no viable B, a fact that persists even if one includes cosmological or Einstein terms
to allow linearization about the resulting dS vacua. These results are an unexpected, if hardly
unique, example of linearization instability.
1 Introduction
The Bel-Robinson tensor Bµναβ was first discovered in ordinary Einstein gravity (GR) at D = 4,
in a search for a gravitational counterpart of the usual matter stress-tensor Tµν . Since there can
be neither local tensorial gravitational candidates of second derivative order (because quantities
∼ ∂gµν ∂gαβ , being frame-dependent, can be made to vanish at any point), nor any non-covariantly
conserved ones, the successful candidate was indeed quadratic in the tensorial “field strengths” –
curvatures B ∼ RR, and covariantly conserved, in analogy with the quadratic Maxwell Tµν ∼ F F .
Subsequently, conserved B were found for arbitrary dimension and matter analogs have also been
constructed (see, e.g., [1] for earlier references).
Very recently, we showed that a conserved B ∼ RDR could be defined for topologically
massive gravity [2]. Given this theory’s third derivative order, we speculated there on extending B
1
to other gravitational systems, in particular to the currently popular quadratic curvature models.
The present note reports a negative outcome in two active, very different theories. For the fourth
derivative D = 3 new massive gravity (NMG) [3], without an Einstein term [4] for simplicity,
there is a B ∼ ∂R∂R at linearized level but it cannot be extended to the full theory, despite
an enormous number of possible cubic correction terms. The Lanczos-Lovelock (LL) [5] models’
quadratic curvature field equations, ∼ RR = 0, obviously cannot even be linearized about flat
space and turn out to have no conserved B either. Adding cosmological or Einstein terms to
permit linearization (to effective cosmological GR) about the resulting dS vacua only allows the
usual linear B ∼ RR of GR, but no nonlinear extension.
2 Fourth Order Models
Since the machinery is considerably more complicated here than in GR, we analyze the (purely
quadratic part of) NMG [3]:
I =
∫
d3 x
√−g (R¯µν R¯µν − R¯2) , R¯µν ≡ Rµν − 1
4
gµν R. (1)
[The Cotton tensor of TMG is the curl of the Schouten tensor R¯µν .] The resulting field equations,
R¯µν −DµDν R¯− 4 R¯ σµ R¯νσ + R¯ R¯µν + gµν R¯2 = 0 (2)
of course obey Bianchi identities since (1) is an invariant. The field equations begin as DDR¯ ∼ 0,
so B cannot just imitate the RR of GR, but must instead start as B ∼ DR¯DR¯, along with possible
cubic, R¯R¯R¯, terms since they have the same dimension as DR¯DR¯. Schematically then, we expect
that
B ∼ DR¯DR¯+ R¯R¯R¯. (3)
[One can also consider similar terms ∼ D(R¯DR¯) or R¯DDR¯, but they are already irrelevant at
linear level.] We begin with the linearized (about flat space) truncation, ∂∂R¯ = 0, of (2); the
corresponding linearized B is simply given by
Bµναβ = R¯
σ
αβ,
(
R¯σν,µ − R¯µν,σ
)
. (4)
Its conservation,
∂µBµναβ = 0 (5)
is verified using the field equations, the Bianchi identity, and antisymmetry of the parenthesis in
(4). Conservation also holds for all permutations of B’s (ναβ) indices; for completeness, we note
also the identically conserved, if irrelevant, bµναβ = D
γ H[γµ]ναβ, H antisymmetric in γ ↔ µ.
Thus encouraged, we turn to candidates (3) at nonlinear level, where the now covariant
derivatives Dµ no longer commute. At first sight, there are so many available independent terms
cubic in R¯µν and gµν R¯, that success seems guaranteed; however, it turns out to be unattainable.
The simplest procedure is to take the covariant divergence of the initial B ∼ DR¯ (DR¯−DR¯) ansatz
and try to compensate for the resulting cubic ∼ R¯R¯DR¯ terms by adding suitable R¯R¯R¯ to B. The
2
procedure is straightforward, using the D = 3 properties
Rµνρσ = gµρ R¯νσ + gνσ R¯µρ − gνρ R¯µσ − gµσ R¯νρ,
[Dα,Dβ ]V
γ =
[
δγβ R¯σα − δγα R¯σβ + gσα R¯γβ − gσβ R¯γα
]
V σ.
(6)
We find
DµBµναβ =
[
R¯ρµ R¯ρβ
(
R¯µν;α − R¯αν;µ
)
+
1
2
R¯αβ;σ
(
R¯σρ R¯ρν − R¯σν R¯
)]
+ (α↔ β) , (7)
to which could be added the above (αβν) permutations. The problem is now to find, among all
(R¯ R¯ R¯)µναβ , those whose µ-divergence cancels (7), or the sum of permutations. Note that (7)
contains both µ- and α , β-derivatives, and that the latter must arise from cubic terms of the form
∼ gµα(R¯ R¯ R¯)νβ , and possibly gµα gνβ (R¯ R¯ R¯); these are also plentiful. This systematic approach
failed: any compensating term also created a new problematic one. We then undertook the brute
force approach, involving the sum of all dimensionally possible candidate terms, ∼ DR¯DR¯ +
R¯DDR¯ + R¯R¯R¯; to be sure, any DR¯DR¯ combinations that fail the linearized conservation test
could be excluded a priori.
We sketch the procedure and counts, but omit the uninstructive, laborious, details1. Start
with the thirty nine possible terms of the form R¯R¯R¯, incuding g R¯R¯R¯ terms, together with the 75
D(R¯DR¯) candidates, giving a total of m = 114 potential terms. Take the divergence of each of
these, enforcing the on-shell, Bianchi and three-dimensional simplifications. That gives, for each
term, a right-hand side that is a linear combination of (DDR¯)DR¯, R¯R¯DR¯ and R¯DDDR¯. There
are n = 231 unique terms of this form in these sums. We can view the action of Dµ as a linear
operator taking the m starting ingredients to a space of n outputs: assign to each candidate term
a basis vector {ei}mi=1, and to each term appearing in the sums Dµ ei, the basis vector {Ei}ni=1, the
matrix D ∈ Rn×m just maps ek to its sum: D ek =
∑n
i=1 αiEi. We constructed this matrix using a
symbolic algebra package, and found that D has no null space, and so no conserved Bel-Robinson
tensor exists.
3 Second Order, LL, Theories
At the other end of the quadratic curvature spectrum from the D = 3 model (1) lie the D > 4 LL
theories, whose field equations remain of second derivative order, but do not admit any linearized
expansion about flat space. The Lagrangians are, in vielbein formulation,
L = ǫabcdf... ǫABCDF...RabAB RcdCD efF . . . (8)
The field equations simply consist of removing one vielbein factor,
EfF ≡ (ǫǫRRe)fF = 0, (9)
1These may be found in the notebook at: http://people.reed.edu/∼jfrankli/NullSpace.nb.
3
since the curvatures’ variations vanish by the cyclic identities. Lowercase indices are local, capitals
are world. Diffeomorphism invariance of the action ensures that DF E
fF vanishes identically, by
the cyclic identities. Indeed, this is even true in D > 5: if we open any second pair of indices (gG),
there results an identically conserved, hence uninteresting, Hfg...FG.... We cannot even start with
a linearized B, since the field equations (9) are intrinsically nonlinear. That there is no true B is
rather clear: for example, attaching a curvature to H above loses conservation, which no additional
cubics can help, while no DRDR term is relevant just because it is linearizable. Extending LL to
include a cosmological or Einstein term does allow for linearization about a nonflat, dS, vacuum:
a dS metric ansatz reduces (9) to the equation ∼ aΛ2 = bΛ in either choice. There is always
a flat, Λ = 0, but also now a dS, Λ = b/a, vacuum2. About the latter, the system linearizes to
cosmological Einstein, which of course enjoys the same conserved B ∼ RR as Λ = 0 GR. However,
just like its predecessor in NMG, this B evaporates as soon as we go to the full theory, since there
is still no way to take into account the nonlinear RR part of (9).
4 Conclusion
We conclude that both NMG and Einstein+LL violate Fermi’s rule: if the lowest order works, the
conjecture is correct, while pure LL doesn’t even have a lowest (here linearized) order. Clearly
there is no invariance here like that [1] underlying B-conservation in GR.
SD acknowledges support from NSF PHY-1064302 and DOE DE-FG02-164 92ER40701 grants.
References
[1] S. Deser, in “Gravitation and Relativity in General” (ed F. Atrio-Barandela and J. Martins),
World Publishing (1999), gr-qc/9901007; S. Deser and Z. Yang, Class. Quant. Grav. 7 1491
(1990).
[2] S. Deser and J. Franklin, Class. Quant. Grav. 28 032002 (2011), arXiv:1011.4240.
[3] E. A. Bergshoeff, O. Hohm and P. K. Townsend, Phys. Rev. Lett. 102 201301 (2009), hep-
th/0901.1766.
[4] S. Deser, Phys. Rev. Lett. 103 101302 (2009), hep-th/0904.4473.
[5] C. Lanczos, Ann. Math. 39 842 (1938); D. Lovelock, J. Math. Phys. 12 498 (1971).
[6] D.G. Boulware and S. Deser, Phys. Rev. Lett. 55 2656 (1985).
2While the appearance of a second, dS, vacuum in R+LL – indeed in all R+R2 models but (D = 4) R+Weyl2 –
is reasonably clear [6], it may seem counter-intuitive that, unlike GR, quadratic models with cosmological term allow
both flat and dS vacua.
4


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