Circular Symmetry in Topologically Massive Gravity
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Circular Symmetry in Topologically Massive Gravity
S. Deser and J. Franklin Physics Department, Brandeis University, Waltham, MA 02454 andLauritsen Laboratory, California Institute of Technology, Pasadena, CA 91125 [email protected] Reed College, Portland, OR 97202, USA [email protected]
Abstract
We re-derive, compactly, a TMG decoupling theorem: source-free TMG separates into itsEinstein 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 necessarilynull.
Topologically massive gravity (TMG) [1] is a counterexample to almost all standard lore. The sumof ordinary Einstein ( G µν ) and Cotton-Weyl ( C µν ) sectors in D = 3, it contains very nontrivial bulkexcitations and solutions. Yet the constituent sectors are separately trivial: all Einstein solutionshave locally constant curvature (or are flat if Λ = 0); vanishing Cotton implies conformally flatspace, including of course (A)dS. [While solutions of pure GR always trivially satisfy TMG (andCotton), they are not, in general, its only solutions.] This raises the converse question: underwhat conditions will the combined system necessarily re-dissolve into its (trivial) constituents?Remarkably, a general decoupling criterion exists [2]: presence of a hypersurface-orthogonal Killingvector (HSOK) X µ . The, somewhat abstract, derivation of [2] is based on the “kinematical” lemmathat, for each possible component projection, along and orthogonal to X µ – just one of the respectivecomponents of the Ricci and Cotton tensors vanishes identically. Applied to source-free TMG, thisimplies 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 toits 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 themost familiar and important HSOK, circular ( D = 2) symmetry, but the results are general.The TMG equations with a cosmological term are E µν ≡ √− g ( G µν + Λ g µν ) + m − C µν = κ T µν , µν ≡ ǫ µαβ D α S νβ , S νβ ≡ [ R νβ − / δ νβ 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, ǫ ij ≡ ǫ ij , in C µν ≡ C νµ ,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 x i and ǫ ij x j respectively, and their2-tensor equivalents to ( x i x j , δ ij ) and ǫ k ( i x j ) x k . [We will use this simple notation instead of themore abstract one in terms of X µ = g µφ .] An immediate consequence is that the 2-(pseudo)scalar C , being proportional to ǫ ij , vanishes identically, implying G = 0. The mixed term, E i = a x i + m − b ǫ ij x j = 0 , (2)forces the two functions ( a, b )( r, t ) to vanish separately, as is obvious by projecting (2) with x i or ǫ ik x k : this means G i = 0 = C i . The spatial components, E ij = [ c x i x j + d δ ij ] + m − f ǫ k ( i x j ) x k = 0 , (3)may be projected with the (even) x i x j and δ ij to show that both ( c, d )( r, t ) = 0, hence also f = 0,that is G ij = 0 = C ij . This completes the short proof that all G µν and C µν components ofsource-free TMG vanish separately in presence of HSOK, due to the ǫ ij -“mismatch”.Next, we try to include a (circularly symmetric, of course) source, for example an implodingcircular matter shell, by reinstating T µν in (1). Since T µν is a regular tensor by assumption, theabove steps all apply: the relevant, GR, field equations now include the matter stress-tensor as aright-hand side, while the Cotton sector stays source-free. This would seem to reduce everythingto 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 C µν 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 G µν . From the definition (1), C rφ = ǫ rtφ [ D t S φφ − D φ S φt ] = 0 ,C tφ = ǫ trφ [ D r S φφ − D φ S φr ] = 0 . (4)What constraint, if any, does (4) impose on the Einstein-matter solutions? By circular symmetry,only ( T rr , T , T r ) = 0. Since we have restricted matter to G µν = κ T µν , and none of the cor-responding three G µν components appears in (4), just the scalar curvature parts of S µν survive,and are manifestly required to have vanishing r - and t -derivatives (Λ-terms, being constant, nevercontribute in C µν ). But since this constant R ∼ T µµ , it vanishes for finite sources: we concludethat 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. Thisis then a possibly interesting new class of explicit solutions of TMG. Note incidentally, that theconverse type of source, pure spinning matter proportional to ǫ ij , hence coupled just to C µν , isforbidden: the, now source-free, Einstein sector would require space to be (locally) flat.Some final remarks: First, our demonstration has not used the X µ -parallel/orthogonal com-ponent projection method of [2] explicitly; the familiar shortcuts afforded by circular symmetryclearly sufficed. Of course the two approaches fully agree as to which components vanish identi-2ally. Second, it should be clear that, by simply adapting coordinates, our construction fits anyother HSOK. Third, note that some “HSO”-aspects of X µ were indeed essential: for example,Kerr-like solutions with non-HSO X µ (involving essentially an explicit epsilon factor ∼ ǫ ij x j in themetric) 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 broaddecoupling mechanisms exist.We thanks S Carlip for reminding us of [2]. The work of SD was supported by grants NSFPHY 07-57190 and DOE DE-FG02-164 92ER40701. References [1] S Deser, R Jackiw and S Templeton,
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