I correct some recent misunderstandings about, and amplify some details of,
an old explicit non-geometrical derivation of GR.Comment: Final, amplified, published version; GRG (2009

D. Boulware

H. Dam van

I. Kogan

L. Butcher

L. Faddeev

M. Porrati

S. Deser

S. Wong

Stanley Deser

T. Padmanabhan

V. Zakharov

English

arXiv

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Gravity from self-interaction redux

I correct some recent misunderstandings about, and amplify some details of, an old explicit non-geometrical derivation of GR

Deser, Stanley

Caltech Authors - Main

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BRX TH-614
CALT 68-2764
Circular Symmetry in Topologically Massive Gravity
S. Deser1 and J. Franklin2
1Physics Department, Brandeis University, Waltham, MA 02454 and
Lauritsen Laboratory, California Institute of Technology, Pasadena, CA 91125
deser@brandeis.edu
2 Reed College, Portland, OR 97202, USA
jfrankli@reed.edu
Abstract
We re-derive, compactly, a TMG decoupling theorem: source-free TMG separates into its
Einstein and Cotton sectors for spaces with a hypersurface-orthogonal Killing vector, here con-
cretely for circular symmetry. We can then generalize it to include matter, which is necessarily
null.
1 Introduction
Topologically massive gravity (TMG) [1] is a counterexample to almost all standard lore. The sum
of ordinary Einstein (Gµν) and Cotton-Weyl (Cµν) sectors in D = 3, it contains very nontrivial bulk
excitations and solutions. Yet the constituent sectors are separately trivial: all Einstein solutions
have locally constant curvature (or are flat if Λ = 0); vanishing Cotton implies conformally flat
space, including of course (A)dS. [While solutions of pure GR always trivially satisfy TMG (and
Cotton), they are not, in general, its only solutions.] This raises the converse question: under
what conditions will the combined system necessarily re-dissolve into its (trivial) constituents?
Remarkably, a general decoupling criterion exists [2]: presence of a hypersurface-orthogonal Killing
vector (HSOK) Xµ. The, somewhat abstract, derivation of [2] is based on the “kinematical” lemma
that, for each possible component projection, along and orthogonal toXµ – just one of the respective
components of the Ricci and Cotton tensors vanishes identically. Applied to source-free TMG, this
implies the separate vanishing of the two sectors’ tensors, reducing the solutions to those of GR
– no “true” TMG extensions exist. Our aim here is the twofold one of tracing this decoupling to
its cause – a “mismatch” between Einstein and Cotton tensors, thereby providing a short, simple,
proof – then to analyze its applicability in the presence of matter. For concreteness, we use the
most familiar and important HSOK, circular (D = 2) symmetry, but the results are general.
The TMG equations with a cosmological term are
Eµν ≡ √−g (Gµν + Λ gµν) +m−1 Cµν = κT µν ,
1
Cµν ≡ µαβ Dα Sνβ , Sνβ ≡ [Rνβ − 1/4 δνβ R]. (1)
For simplicity of notation (only), the Λ-term is understood implicitly in G below. Also, we set
T µν = 0 to start with. The key to decoupling is the Levi-Civita tensor, 0ij ≡ ij , in Cµν ≡ Cνµ,
along with the elementary fact that, in circularly symmetric (but not necessarily time-independent)
spaces, all 2-vectors and their axial versions are proportional to xi and ij xj respectively, and their
2-tensor equivalents to (xi xj, δij) and k(i xj) xk. [We will use this simple notation instead of the
more abstract one in terms of Xµ = gµφ.] An immediate consequence is that the 2-(pseudo)scalar
C00, being proportional to ij, vanishes identically, implying G00 = 0. The mixed term,
E0i = a xi +m−1 b ij xj = 0, (2)
forces the two functions (a, b)(r, t) to vanish separately, as is obvious by projecting (2) with xi or
ik xk: this means G0i = 0 = C0i. The spatial components,
Eij = [c xi xj + d δij ] +m−1 f k(i xj) xk = 0, (3)
may be projected with the (even) xi xj and δij to show that both (c, d)(r, t) = 0, hence also f = 0,
that is Gij = 0 = Cij. This completes the short proof that all Gµν and C
µν components of
source-free TMG vanish separately in presence of HSOK, due to the ij-“mismatch”.
Next, we try to include a (circularly symmetric, of course) source, for example an imploding
circular matter shell, by reinstating T µν in (1). Since T µν is a regular tensor by assumption, the
above steps all apply: the relevant, GR, field equations now include the matter stress-tensor as a
right-hand side, while the Cotton sector stays source-free. This would seem to reduce everything
to GR (now with a source) again; however, the Cotton sector, even if free, constrains its solutions.
To study this, we use the “kinematical” lemma of [2]: the only two non-identically vanishing com-
ponents of Cµν are those with one C-index along X (here φ), and its other either of the orthogonal
(r, t); these are also the only identically vanishing components of Gµν . From the definition (1),
Crφ = rtφ [Dt S
φ
φ −Dφ Sφt ] = 0,
Ctφ = trφ [Dr S
φ
φ −Dφ Sφr ] = 0. (4)
What constraint, if any, does (4) impose on the Einstein-matter solutions? By circular symmetry,
only (Trr, T00, T0r) 6= 0. Since we have restricted matter to Gµν = κTµν , and none of the cor-
responding three Gµν components appears in (4), just the scalar curvature parts of Sµν survive,
and are manifestly required to have vanishing r- and t-derivatives (Λ-terms, being constant, never
contribute in Cµν). But since this constant R ∼ T µµ , it vanishes for finite sources: we conclude
that decoupling is permitted (only) in presence of null matter. This is not surprising physically,
being driven by the remaining field equation, vanishing of the source-free Cotton-Weyl tensor. This
is then a possibly interesting new class of explicit solutions of TMG. Note incidentally, that the
converse type of source, pure spinning matter proportional to ij , hence coupled just to Cµν , is
forbidden: the, now source-free, Einstein sector would require space to be (locally) flat.
Some final remarks: First, our demonstration has not used the Xµ-parallel/orthogonal com-
ponent projection method of [2] explicitly; the familiar shortcuts afforded by circular symmetry
clearly sufficed. Of course the two approaches fully agree as to which components vanish identi-
2
cally. Second, it should be clear that, by simply adapting coordinates, our construction fits any
other HSOK. Third, note that some “HSO”-aspects of Xµ were indeed essential: for example,
Kerr-like solutions with non-HSO Xµ (involving essentially an explicit epsilon factor ∼ ij xj in the
metric) do not decouple. The basic, metric tensor, variables must possess the HSOK symmetry;
the only “pseudo-”source is the explicit epsilon in Cotton. Given the latter’s identical traceless-
ness, conformal HSOK might conceivably also suffice, but it seems unlikely that any other broad
decoupling mechanisms exist.
We thanks S Carlip for reminding us of [2]. The work of SD was supported by grants NSF
PHY 07-57190 and DOE DE-FG02-164 92ER40701.
References
[1] S Deser, R Jackiw and S Templeton, Phys. Rev. Lett., 48 975 (1982); Ann. Phys. 140 372
(1982); S Deser, in Quantum Theory of Gravity, ed. S Christensen (A Hilger Ltd, London
1984).
[2] A Aliev and Y Nutku, Class. Quant. Grav. 13 L29 (1996), gr-qc/9812089.
3


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