Symmetries and gravitational Chern-Simons Lagrangian terms
Loriano Bonora, Maro Cvitan, Predrag Dominis Prester, Silvio Pallua, Ivica Smolic
aa r X i v : . [ h e p - t h ] M a y SISSA 18/2013/FISITZF-2013-02
Symmetries and gravitational Chern-SimonsLagrangian terms
L. Bonora a , M. Cvitan b , P. Dominis Prester c , S. Pallua b , I. Smoli´c b a International School for Advanced Studies (SISSA/ISAS),Via Bonomea 265, 34136 Trieste, Italy b Physics Department, Faculty of Science,University of Zagreb, p.p. 331, HR-10002 Zagreb, Croatia c Department of Physics, University of Rijeka,Radmile Matejˇci´c 2, HR-51000 Rijeka, Croatia
Email: [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract.
We consider some general consequences of adding pure gravitational Chern-Simons termto manifestly diff-covariant theories of gravity. Extending the result of a previous paper we enlargethe class of metrics for which the inclusion of a gCS term in the action does not affect solutions andcorresponding physical quantities. In the case in which such solutions describe black holes (of generalhorizon topology) we show that the black hole entropy is also unchanged. We arrive at these conclusionsby proving three general theorems and studying their consequences. One of the theorems states that thecontribution of the gravitational Chern-Simons to the black hole entropy is invariant under local rescalingof the metric. 1
Introduction
This paper is a follow up of previous papers in which we have analyzed the consequences of adding a purelygravitational Chern-Simons (gCS) term [1] to a manifestly diffeomorphism invariant gravitational actionin (4 k − D = 3dimensions in [2, 3]. Following a proposal by Tachikawa [4], in [5] we analyzed the general consequencesof adding one such gCS term to the action, in particular the appearance of a new contribution to thethermodynamical entropy. In [6] we considered the global geometrical aspects implied by the presence ofa gCS term, both at the level of the action and the entropy, and studied the topological conditions for thewell-definiteness of both. Except in the three-dimensional case, very well studied in the literature, it doesnot seem to be easy to see the effects of a gCS term on observables. In the simplest and more symmetriccases they appear to be null. For this reason in [7] we studied the case of Myers-Perry black hole inseven dimensions. We were able, at least perturbatively, to show that in some sufficiently complicatedconfiguration the effects of the gCS are not identically vanishing.In this paper we would like to enlarge the null effect results of [8], with the purpose of circumscribingas closely as possible the cases in which the addition of a gCS is irrelevant from an observational pointof view. More to the point we are interested in the gravity theories in D = 2 n − n ∈ N )with Lagrangians of the form L = L + λ L gCS (1)where L is some general manifestly diffeomorphism-invariant Lagrangian density and L gCS is the gCSLagrangian density. In (1) λ denotes the gCS coupling constant. It is dimensionless and may be quantized,see [6, 9, 10]. gCS terms have a remarkable set of properties, among which the most notable are: they arenot manifestly diff-covariant, though they preserve diff-covariance in the bulk; they have a topologicalnature which leads to a quantization of their coupling; they are parity-odd and so break parity symmetry;they are conformally covariant, in the sense that under a Weyl rescaling of the metric˜ g µν ( x ) = Ω ( x ) g µν ( x ) (2)Chern-Simons density transforms as (see [1, 11] and Appendix), L gCS [ e Γ ] = L gCS [ Γ ] + d ( . . . ) (3)It is clear that gCS Lagrangian terms have a peculiar role in the set of all possible higher-curvaturegravity terms, which makes them deserve special attention.In the following we shall explicitly refer mainly to irreducible gCS terms, whose Lagrangian densityis given by L gCS [ Γ ] = n Z dt str( Γ R n − t ) (4)Here R t = td Γ + t ΓΓ , Γ is the Levi–Civita connection and str denotes a symmetrized trace, whichis an irreducible invariant symmetric polynomial of the Lie algebra of the SO (1 , D −
1) group, and allproducts are wedge products. A general gCS term is a linear combination of irreducible and reducibleterms, where the form of the latter is obtained from ( D + 1)-dimensional relation d L gCS = tr( R m ) . . . tr( R m k ) , k X j =1 m j = D + 1 , m j ∈ N (5)with k > k = 1 gives irreducible gCS term). For example, in D = 7 aside from the irreducible thereis also a reducible gCS term. ∗ We shall state in what way the obtained results extend to reducible gCSterms.As anticipated above, in this paper we want to improve on the results found in [8]. Our aim isto identify the class of metrics for which the inclusion of a gCS term in (1) does not affect solutions ∗ In string theories compactified to D = 7, they appear in combination when gCS terms are present. L are also solutionscorresponding to (1). In the case in which such solutions describe black holes (of general horizon topology)we shall show that the black hole entropy is also unchanged. As the case n = 2 ( D = 3) has alreadybeen studied in detail in literature (see, e.g., [2, 3, 17, 18, 19]), we focus here on n ≥ D ≥ † Aparticularly important intermediate result is the remark that the terms representing the gCS contributionto the entropy is invariant under local rescaling of the metric.The paper is organized as follows. In section 2 we state and prove three theorems on the vanishingproperties of the generalized Cotton tensor, the Weyl invariance of the gCS entropy and the vanishingof the latter under some general conditions. In section 3 we apply such theorems to various physicalsituations and in section 4 to linearized equations of motion around some highly symmetrical backgrounds.
The paper is based on three results that we state in the form of theorems. The first concern the effects ofa gCS term on the generalized Cotton tensor. The second the invariance of the gCS entropy contributionunder Weyl rescalings of the metric. Thanks to these result the third states that metrics such as thosein the first theorem do not contribute to the gCS entropy.
Adding a gCS term in the Lagrangian brings about additional terms in the equations of motion. It wasshown in [11] that the equation for the metric tensor g αβ acquires an additional term C αβ , which, for theirreducible gCS term (4), is of the form C αβ = − ǫ µ ··· µ n − ( α ∇ ρ (cid:16) R β ) σ µ µ R σ σ µ µ · · · R σ n − σ n − µ n − µ n − R σ n − ρµ n − µ n − (cid:17) (6)Under the Weyl rescaling of metric (2) the tensor C αβ transforms as C αβ [˜ g ] = Ω − (2 D +2) C αβ [ g ] (7)Aside from being conformally covariant, the tensor C αβ is also traceless and covariantly conserved andso may be considered as a generalization of the Cotton tensor to D = 4 k − k > SO ( D −
1) isometrysubgroup) [8, 10]. Here we want to show that there is a much broader class of metrics for which tensor C αβ vanishes. This is guaranteed by the following theorem. Theorem 1.
Assume that the metric of D -dimensional spacetime ( M, g µν ) can be cast, in some region O ⊂ M , in the following form, ds = g µν ( x ) dx µ dx ν = D ( x ) (cid:0) A ( z ) g ab ( y ) dy a dy b + B ( y ) h ij ( z ) dz i dz j (cid:1) , (8)where local coordinates on O are split into x µ = ( y a , z i ), µ ∈ { , . . . , D } , a ∈ { , . . . , p } , and i ∈ { , . . . , q } (so that p + q = D ). The functions B ( y ), g ab ( y ) and A ( z ), h ij ( z ) depend only on the { y a } and { z i } coordinates, respectively. If p ≥ q ≥ x ∈ O C µν ( x ) = 0 (9) † In string theory n = 2 gravitational CS terms play an important and unique role in some black hole analyses (see, e.g.,[18, 20, 21, 22, 23]). roof. Due to property (7), equality (9) is preserved under Weyl rescalings (2). By taking Ω = (
DAB ) − the metric (8) may be put in the direct product form d ˜ s = ˜ g µν dx µ dx ν = g ab ( y ) dy a dy b + h ij ( z ) dz i dz j , (10)so we only have to prove that the theorem hold for the metrics of the type (10). This greatly simplifiesour job because both Riemann tensor and its covariant derivative are completely block-diagonal, and asa consequence also the tensor ∇ ρ (cid:16) R βσ µ µ R σ σ µ µ · · · R σ n − σ n − µ n − µ n − R σ n − ρµ n − µ n − (cid:17) (11)present in the definition of C µν (6) is. This means that the components of the tensor in (11) arenonvanishing only when all the indices are either from the y -subspace or the z -subspace. Because thereare D − p > q > C µν is also zero. (cid:4) For reducible gCS terms the Theorem 1 gets modified, allowing other possibilities (aside p or q equalto 0 or 1) in which one may have C αβ = 0 for geometries of the type (8). Their contribution to C αβ is asum of terms which are of the form [8] ǫ µ ··· µ D − ( α (cid:0) tr( R m ) · · · tr( R m k − ) (cid:1) µ ··· µ D +1 − mk ∇ ρ (cid:16) R β ) σ µ D +2 − mk µ D +3 − mk · · · R σ n − ρµ D − µ D − (cid:17) (12)Following the same logic as above it is easy to conclude that for the reducible gCS term, defined implicitlyby (5), exceptions to Theorem 1 may appear when there is a subset { m j , . . . , m j l } of the set of exponents { m j , j = 1 , . . . , k } such that 2 l X r =1 m j r + σ = p or q (13)for σ = 0 or 1. When p or q satisfy (13) it is possible that C αβ = 0. For example, in D = 7 there is aunique reducible gCS term, which has k = 2 and m = m = 2, so (13) gives no restrictions on p and q (all values from 0 to 7 are allowed), so in this case Theorem 1 by itself has no content. However, if any ofthe submetrics, g ab or h ij , is maximally symmetric and p > q >
1, one has ‡ C αβ = 0 and so in thiscase the original statement of the Theorem applies also to all reducible gCS terms. This will be relevantwhen we discuss applications in Sec. 3The obvious consequence of the above theorem is that if we can find coordinates around every pointof spacetime in which the metric, which is a solution to the equations of motion obtained from someLagrangian L , is of the form (8), then this metric will also be a solution in the theory defined by theLagrangian (1). In other words, adding gCS Lagrangian terms does not affect solutions which are of theform specified by the theorem.The theorem covers many classes of metrics frequently discussed in the literature. In Sec. 3 we shallmention a few examples of particular interest. If the metric describes a black hole one is also interested in its thermodynamical behavior, and in particularin the black hole entropy. It was shown in [4] that the irreducible gCS Lagrangian term (4) brings anadditional term in the black hole entropy formula, which must be added to Wald’s formula [12] obtainedfrom the L part of the total Lagrangian (1), given by [5] S gCS [ g ] = 4 πn Z B ω ( d ω ) n − , (14) ‡ This follows because maximally symmetric spaces satisfy (25). B is the ( D − ω is a 1-form tobe identified with the SO (1 ,
1) (or U (1) in Euclidean signature) connection on the normal bundle of B .Here we want to discuss some general properties of the gCS entropy term (14). Theorem 2.
The gCS entropy term (14) is invariant under Weyl rescalings (2) of the spacetime metric.
Proof.
In [5] we showed that ω can be written as ω µ = − q νµ n ρ ∇ ν ℓ ρ (15)where q µν is the induced metric on B , and ℓ µ and n µ is a pair of two future directed null vector fields,normal to the black horizon and arbitrary up to a normalization ℓ µ n µ = −
1. For the Weyl rescaledmetric (2) we can take the null vectors to be˜ ℓ µ = √ Ω ℓ µ , ˜ n µ = √ Ω n µ (16)By using ˜ q µν = q µν and a well-known relation (e.g., see Appendix D of [13]) e ∇ ν ℓ ρ = ∇ ν ℓ ρ + C ραν ℓ α , C ραν = δ ρ ( α ∇ ν ) ln Ω − g αν g ρβ ∇ β ln Ω (17)a straightforward calculation gives e ω µ = ω µ . (18)Using this in (14) we obtain S gCS [˜ g ] = S gCS [ g ] (19)which proves the theorem. (cid:4) The third theorem describes some general consequences of the first two.
Theorem 3.
If a metric g µν ( x ) describing a black hole is of the form (8), with q ≥ z are tangential to the bifurcation surface of the horizon, then the gCS entropy term (14)evaluated on such metric vanishes S gCS [ g ] = 0 . (20) Proof.
First we make a Weyl rescaling (2) with Ω = (
DAB ) − to obtain the metric ˜ g µν in the directproduct form (10). Theorem 2 says that the gCS entropy term is invariant under such transformation,so we can use ˜ g µν to evaluate it. Let us focus on the components of ω µ in “ z -directions”, i.e., for µ = i .Due to the block diagonality of the metric we have e ω i = − ˜ q ji ˜ n a e ∇ j ˜ ℓ a (21)where we have used the fact that metric ˜ g µν has direct product form and that ˜ n and ˜ ℓ are defined purelyin the y -block. Moreover, the direct product form of the metric implies that the covariant derivative e ∇ j is defined purely in the z -block. From this follows that e ∇ j ˜ ℓ ρ = 0 and so e ω i = 0. As indices from the z -block must appear when performing the integration in (14), it directly follows that S gCS [˜ g ] = 0 , (22)which completes the proof of the theorem (cid:4) Using the results from [5] (see Eq. (4.26)) and conformal properties of tr( R k ) [1, 11] it directly followsthat the theorems 2 and 3 are valid also for the reducible gCS terms.5 Applications
The Theorem 1 from Sec. 2.1 covers many classes of metrics appearing in different contexts. Here wediscuss a few situations frequently occurring in the literature.
If the spacetime allows a foliation with maximally symmetric d -dimensional subspaces (with d ≥ D ( x ) = A ( z ) = 1 and q = d (see, e.g., Sec. 13.5 of [14]). There are at least three frequent contexts where such metrics appear:1. Cosmology – If the metric describes a cosmological model of the Universe (in some extra-dimensionalset-up) then the SO (3) isometry subgroup, following from the isotropy of 3-dimensional “physical”space, implies that the spacetime may be foliated by 2-spheres (in this case we have d = 2).2. Stationary rotating black holes – If a stationary rotating black hole has k angular momenta vanishingwith k ≥
2, this typically implies that the isometry group has an SO (2 k ) factor. Then there existsa foliation in (2 k − d = 2 k −
1. In this case Theorems 1 and 3 apply. In the k = 1case it may naively seem that Theorem 3 implies the contribution of the gCS entropy term againto be vanishing; however this is not so. We have explicitly shown in [7] on a particular examplein D = 7 (which gives Myers-Perry solution [15] for λ = 0) that the effect of the gCS term in theequations of motion in such a case is such that it forces the metric to depart from the form (8). Sowe cannot apply Theorem 3 to the full black hole solution.3. Flat p -branes – Geometries of flat p -branes are of the form (8) with q = p + 1, so for p ≥ p -branes, then also Theorem 3 applies. In many extra-dimensional scenarios appearing more or less frequently in the literature, either in theform of braneworlds or Kaluza-Klein (KK) compactifications, the metric of the vacuum is of the form(8). In realistic scenarios p = 4, thus it follows that q = D − p ≥ D ≥ q ≥
2. Thus a gCS Lagrangian term does notaffect solutions of this form.Let us explain the situation with a simple example. We consider some diff-covariant theory in D = 11and add to it an n = 6 gCS Lagrangian term (4). Then we proceed to the standard KK reduction to D = 4. If in the vacuum all the seven KK gauge fields coming from the metric vanish, then the vacuummetric is of the form (8), with m = 4 and q = 7. If we excite the vacuum by switching on (among other) k gauge fields, then the metric will still be of the form (8), but now with q = 7 − k . This implies thatfor k ≤ It is quite obvious that gCS Lagrangian terms do not contribute to the linearized equations of motion(EOM) around a flat Minkowski background metric. We now show that this also holds for more generalbackgrounds, including (A)dS metrics. 6 heorem 4. In D >
Proof.
Linearized EOM’s around background metrics g (0) µν are obtained by writing the metric as g µν = g (0) µν + h µν (23)and expanding the equations of motion around g (0) µν while keeping only the terms which are at mostfirst-order in h µν . We now show that inserting (23) in the C µν term (6) and expanding in h µν , does notproduce terms of zeroth and first order whenever the background metric is maximally symmetric. Firstnote that Theorem 1 applies to all maximally symmetric metrics, so there is no contribution at zerothorder C (0) µν = C µν [ g (0) ] = 0 (24)Let us turn next to first order terms. We use the following properties of maximally symmetric metrics ∇ (0) µ R (0) νρσκ = 0 , (cid:16) R (cid:17) α β ≡ R α (0) γ ∧ R γ (0) β = 0 (25)where R is the tensor-valued 2-form curvature defined by( R αβ ) µν = R αβµν . (26)The second equation in (25) follows from R (0) µνρσ = ± ℓ − (cid:16) g (0) µρ g (0) νσ − g (0) µσ g (0) νρ (cid:17) (27)and from maximally symmetric metrics being diagonal. It is obvious from the form of (6) that relations(25) guarantee that there are no first order terms in h µν when n ≥ D ≥ D = 7 ( n = 4) thereis one suspicious term C αβ (1) = − ǫ µ ··· µ ( α (0) R β )(0) σ µ µ ∇ (0) ρ R σ (1) σ µ µ R σ ρ (0) µ µ (28)coming from the irreducible gCS term, which is not obviously vanishing. However, by using Eqs. (7.5.7)–(7.5.8) and (3.2.12) from [13] one can put (28) in a form in which the second equation in (25) and (27)again force it to vanish. If the background is a direct product of maximally symmetric spaces, the prooffollows in the same way. The only difference is that in (25) and (27) there is a different radius ℓ i forevery maximally symmetric subspace i , but this does not affect the proof. Note that the theorem is validoff-shell, i.e. regardless of whether the background metric satisfies the EOM or not. (cid:4) There are many important situations where linearization enters. Let us mention three of them andemphasize direct consequences of Theorem 4: (1) Perturbative degrees of freedom around flat and (A)dSspaces in
D > g (0) µν - gCS Lagrangianterms do not affect the stability analysis of maximally symmetric spaces (or their products) in D >
3. (3) Determination of asymptotic charges, in particular mass and angular momenta, which in themethod described in Section 7.6 of [14] (asymptotically flat configuration) or [16] (asymptotically AdSconfigurations) are calculated from linearized EOM’s - in
D > C µν term does notcontribute directly to charges, so that the only possible contribution of a gCS Lagrangian term is indirectthrough changing the asymptotic behavior of the metric (in the spatial infinity). In [7] we have shown,on an explicit example of stationary rotating black hole in D = 7, that the gCS term does not changethe relevant asymptotic behavior of the metric and, as a consequence, relations for mass and angular7omentum are perturbatively unchanged when a gCS Lagrangian term is introduced. § It remains tobe shown how general this result is, and in particular whether it is valid also for asymptotically (A)dSconfigurations.It should be emphasized that the n = 2 ( D = 3) case, which is excluded in Theorem 4, is indeedexceptional. It was shown in [2, 3] that in D = 3 the gCS term contributes to the linearized EOM, in a waywhich may make graviton massive (like in Topologically Massive Gravity), and/or generates additionalterms in the expressions for mass and angular momentum [24, 25]. As discussed above, Theorem 4guarantees that nothing like this happens in D > This paper is another step in our endeavor to understand the consequences and nature of adding a purelygravitational Chern-Simons (gCS) term to an otherwise ordinary gravitational action. With the exceptionof the D=3 case the effects of such addition are rather elusive, and are present only for configurationswith rather modest space-time symmetries. Conversely the problem of circumscribing metric solutionsto the equations of motion that are left unchanged by the same addition is also very elusive. Similarlyfar from obvious is the related question of whether the effects of gCS terms are of topological nature ornot. We believe the best course in this situation is to try to enlarge as far as possible the class of caseswhere the effect of a gCS term are null. This is what we have done in this paper. We have proved threetheorems that allow us to conclude for a rather large class of metrics that the corresponding physics isnot affected by the addition of a gCS term. It include cosmological, black hole and p-brane solutions. Wehave also shown that these theorems are helpful for a much larger class of problems in which linearizedgravity is involved.
Acknowledgements
One of us (L.B.) would like to thank the Theoretical Physics Department, University of Zagreb, forhospitality and financial support during his visits there. The work of L.B. was supported in part by theMIUR-PRIN contract 2009-KHZKRX.. M.C., P.D.P., S.P. and I.S. would like to acknowledge supportby the Croatian Ministry of Science, Education and Sport under the contract no. 119-0982930-1016.
AppendixA A simple proof of the Cotton tensor’s conformal covariance
There is a simple way to derive the property (7), based on theorem by Chern and Simons [1]. It isinstructive to review it because it highlights the global issues underlying the proof. Let LM be the framebundle over the manifold M (with structure group GL ( D )) and let φ be a connection on this bundle, φ t = t φ and Φ , Φ t the respective curvatures. Let P denote an invariant polynomial of GL ( D ) and let uswrite, as usual, the transgression T P ( φ ) = n Z dt P ( φ , Φ t , . . . , Φ t ) (29) § This was explicitly shown up to first order in gCS coupling λ , but, based on the structure of the gCS contribution tothe EOM, we conjectured that the result is valid to all orders in λ . Theorem (CS). If g and ˜ g are conformally related, (2), and φ , Φ and ˜ φ , ˜ Φ are their respectiveconnections and curvatures, we have T P ( ˜ φ ) = T P ( φ ) + d Θ ncf In the case φ is a Riemannian connection, T P ( φ ) reduces to Υ CS ( Γ ), see [8], that is to gCS Lagrangianterm. Now, the Cotton tensor is defined via the relation δ Υ CS (Γ) = C µν δg µν ǫ + d Θ cov + d Θ nc (30)where ǫ = √− gd D x . Taking the analogous variation for Υ CS (Γ) δ Υ CS (˜Γ) = ˜ C µν δ ˜ g µν ˜ ǫ + d ˜ Θ cov + d ˜ Θ nc (31)where δ ˜ g = Ω δg , comparing the two and taking into account the transformation properties of the volumeelement, one gets immediately (7).In deriving the equations of motion any exact term in the previous formulas is discarded, but in thederivation of the entropy formula by means of the phase space formalism also Θ cov and Θ nc play a role.Therefore the term Θ ncf has to be taken into account when comparing conformally related metrics. As aconsequence it is not a priori obvious that the entropy formula for a gCS term is conformally invariant.But this turns out to be the case. References [1] S. Chern and J. Simons,
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