aa r X i v : . [ g r- q c ] M a r ADM analysis and massive gravity
Alexey Golovnev
Saint-Petersburg State University, high energy physics department,Ulyanovskaya ul., d. 1; 198504 Saint-Petersburg, Petrodvoretz; Russia [email protected], [email protected]
Abstract
This is a contribution to the Proceedings of the 7th Mathematical Physics Meeting: SummerSchool and Conference on Modern Mathematical Physics, held in Belgrade 09 – 19 September 2012.We give an easily accessible introduction to the ADM decomposition of the curvature components.After that we review the basic problems associated with attempts of constructing a viable massivegravity theory. And finally, we present the metric formulations of ghost-free massive gravity models,and comment on existence problem of the matrix square root.
General relativity is a very successful theory. We are not aware of any experimental contradictions to itspredictions whenever we are able to test it directly. However, at the largest scales, it fails unless we arewilling to introduce huge amounts of Dark Matter and Dark Energy. The Dark Matter must be of someyet unknown mysterious nature. And the Dark Energy might be a mere cosmological constant, but ofan overwhelmingly unnatural value. At the end of the day, it pushes us towards reconsidering the basicfoundations of the theory of gravitational interactions.Modifying the General Relativity, even if just by giving the graviton a mass, is not an easy game toplay [1]. And it is only very recently that we have got a decent hope [2, 3] of obtaining a viable modelof massive gravity. In this contribution we review some non-perturbative aspects of this theory from theviewpoint of the Hamiltonian ADM analysis.
In the ADM approach, we use the following (3 + 1)-decomposition of the metric [4] ds = − (cid:0) N − N i N i (cid:1) dt + 2 N i dx i dt + γ ij dx i dx j (1)where N and N i are commonly called lapse and shifts respectively, and the spatial indices are raisedand lowered by the spatial metric γ . Equivalently, g = − (cid:0) N − N i N i (cid:1) , g i = N i and g ij = γ ij . Andit is fairly straightforward to check that the inverse metric components are g = − N , g i = N i N and g ij = γ ij − N i N j N . Comparing this result with the Cramer’s rule for g we get the useful relation fordeterminants, √− g = N √ γ .Let us also introduce a unit normal vector n µ ≡ (cid:16) N , − N i N (cid:17) which corresponds to the one-form n µ ≡ (cid:16) − N, −→ (cid:17) nullifying the tangent vectors. Generically, these objects are not covariantly constant,and one can define the extrinsic curvatures of the spatial slices t = const : K ij ≡ − ▽ i n j = Γ µij n µ = − N Γ ij (2)with covariant derivatives ▽ µ n ν ≡ ∂ µ n ν − Γ αµν n α and Christoffel symbolsΓ αµν = g αβ Γ βµν = 12 g αβ ( ∂ µ g βν + ∂ ν g βµ − ∂ β g µν )of the metric g . 1e easily get Γ ijk = (3) Γ ijk where (3) Γ denotes thel Christoffel symbols for the metric γ , and Γ ij = ( ∂ i N j + ∂ j N i − ˙ γ ij ) which gives K ij = − N Γ ij = 12 N (3) ▽ i N j + (3) ▽ j N i − ˙ γ ij ! (3)for the extrinsic curvatures (2) where the three-dimensional covariant derivatives are taken with respectto γ . Incidentally, we note that g − = (cid:18) γ − (cid:19) − n × n T . (4)In order to analyse a gravitational theory, we need to compute the curvature components R µναβ = ∂ α Γ µβν − ∂ β Γ µαν + Γ µαρ Γ ρβν − Γ µβρ Γ ραν (5)in terms of the ADM metric decomposition (1). For the Levi-Civita connection we have the symmetryproperties R αβµν = − R βαµν = − R αβνµ = R µναβ , and therefore need to know only three types of compo-nents R ijkl , R ijk and R i j . The calculations are rarely presented in an explicit and detailed form. Onecan either use the geometric meaning of the curvature tensor and play with parallelly transporting normaland tangent vectors, or one can follow more direct derivations in a somewhat involved algebraic disguisein mathematical textbooks. We want to present a straightforward, almost brute force, computation. Ourmain point is that it can also become rather simple if cleverly done. It is reasonable to calculate the connection components first. Anyway, they might be needed for thematter part of the action. We will thoroughly trade ˙ γ for the extrinsic curvatures using (3). In thefollowing we need: Γ ij = Γ i j = − N K ij + (3) ▽ j N i , (6)Γ ijk = (3) Γ ijk , (7)Γ = 1 N (cid:16) ˙ N + N i ∂ i N − N i N j K ij (cid:17) , (8)Γ i = Γ i = 1 N (cid:0) ∂ i N − N j K ij (cid:1) , (9)Γ i j = Γ ij = − N i ∂ j NN − N (cid:18) γ ik − N i N k N (cid:19) K kj + (3) ▽ j N i , (10)Γ ij = − N K ij , (11)Γ ijk = (3) Γ ijk + N i N K jk . (12) Using the formulae (6) – (12), we readily obtain the curvature tensor (5). First, R ijkl = g iρ ∂ k Γ ρlj − g iρ ∂ l Γ ρkj + Γ ikρ Γ ρlj − Γ ilρ Γ ρkj = − N i ∂ k (cid:18) N K jl (cid:19) + γ im ∂ k (cid:18) (3) Γ mjl + N m N K jl (cid:19) − N K jl − N K ik + (3) ▽ k N i ! + (3) Γ ikm (cid:18) (3) Γ mlj + N m N K lj (cid:19) − ( k ↔ l )= (3) R ijkl + K ik K jl − K il K jk (13)Then it would be easier to calculate n µ R µiαj = n µ R µiαj = 1 N R iαj − N k N R kiαj (14)2nstead of R iαj . Indeed, we get n µ R µijk = − N (cid:0) ∂ j Γ ki + Γ jρ Γ ρki (cid:1) + N (cid:0) ∂ k Γ ji + Γ kρ Γ ρji (cid:1) = ∂ j K ki + (3) Γ mki K jm − ( j ↔ k )= (3) ▽ j K ki − (3) ▽ k K ji (15)Finally, after just a little bit of very simple algebra, we find n µ R µi j = ˙ K ij + (3) ▽ i (3) ▽ j N + N K ik K kj − (3) ▽ j (cid:0) K ik N k (cid:1) − K kj (3) ▽ i N k . Using (15) and (14) we trivially transform it to a more symmetric form n µ n ν R µiνj = 1 N ˙ K ij + (3) ▽ i (3) ▽ j N + N K ik K kj − L ie −→ N K ij ! (16)where L ie −→ N K ij ≡ N k ∂ k K ij + K ik ∂ j N k + K jk ∂ i N k , and the partial derivatives can be substituted by thecovariant ones. The formulae (13) – (16) represent the full Riemann tensor for the Levi-Civita connection. Now theRicci tensor R µν ≡ R αµαν components R ij , n µ R µi and n µ n ν R µν can be easily obtained. We skip thispoint and find the Ricci scalar R ≡ g µν R µν directly. Using (4) and symmetry properties of the Riemanntensor, and the obvious relation γ ij ˙ K ij = ∂ K ii + K ij ˙ γ ij together with (3) we get R = g µν g αβ R µανβ = γ ik γ jl R ijkl − n µ n ν γ ij R µiνj = (3) R + K ii K jj − K ij K ij − N γ ij ˙ K ij + 4 N K ij (3) ▽ j N i + 2 N j N (3) ▽ j K ii − N (3) △ N = (3) R + K ij K ij + K ii K jj − N ˙ K ii + 2 N j N (3) ▽ j K ii − N (3) △ N (17)As a final step, let us put the Einstein-Hilbert density √− gR = √ γN R into a convenient form. Weuse the relation ∂ √ γ = √ γ γ ij ˙ γ ij and exclude ˙ γ by virtue of (3): √− gR = √ γN (cid:18) (3) R + K ij K ij − K ii K jj (cid:19) − √− g ▽ µ (cid:0) K ii n µ (cid:1) − √ γ (3) △ N where − √− g ▽ µ (cid:0) K ii n µ (cid:1) ≡ − ∂ (cid:0) √ γK ii (cid:1) + 2 √ γ (3) ▽ j (cid:0) K ii N j (cid:1) .Neglecting total time derivative and covariant divergence terms, we obtain the Einsten-Hilbert actionin the ADM formalism: S = Z dtd x √ γN (cid:18) (3) R + K ij K ij − K ii K jj (cid:19) . (18)The action (18) would be enough for us as the kinetic part of the massive gravity models. However, as wehave computed all the curvature components, these results can be used for more complicated modifiedgravity models, too. Note also that the curvature tensor in theories with non-metric connection can beviewed as an (exact at second order) variation of the Riemannian one with respect to the connectionbeing varied from its Levi-Civita value by non-metricity and contortion tensors. Now we come to massive deformations of general relativity. Note that only the six γ ij variables aredynamical in the action (18). The lapse and shifts play the role of Lagrange multipliers which enforcefour first-class constraints corresponding to the diffeomorphism invariance of the theory. The total numberof physical degrees of freedom is two. A generic mass term would break the gauge invariance and producea model with six independent degrees of freedom. However, a massive spin-two field should posess onlyfive of them. Therefore, we have an extra scalar. 3hat this scalar might be problematic, can be seen already at the level of quadratic perturbationsaround Minkowski space. For linearised theory, the Einstein-Hilbert action acquires the form of −
14 ( ∂ α h µν )( ∂ α h µν ) + 12 ( ∂ α h µν )( ∂ ν h µα ) −
12 ( ∂ α h αµ )( ∂ µ h ββ ) + 14 ( ∂ µ h αα )( ∂ µ h ββ )which can be studied in the standard parametrisation of h = 2 φ , h i = ∂ i b + s i with ∂ i s i = 0, and h ij = 2 ψδ ij + 2 ∂ ij σ + ∂ i v j + ∂ j v i + h ( T T ) ij with ∂ i v i = 0, ∂ i h ( T T ) ij = 0 and h ( T T ) ii = 0: −
14 ( ∂ α h ( T T ) ij )( ∂ α h ( T T ) ij ) + 12 ( ∂ j ( ˙ v i − s i )) − ψ + 2( ∂ i ψ ) + 4 ψ △ (cid:16) φ − ˙ b + ¨ σ (cid:17) where we have neglected all total derivative terms. One can see that there are two gauge invariant variables h ( T T ) ij in helicity-two (transverse traceless) sector, two gauge invariant variables ˙ v i − s i in helicity-onesector, and two gauge invariant variables, ψ and φ − ˙ b + ¨ σ , in helicity-zero sector. Of course, these are justthe standard gauge invariant variables of the cosmological perturbation theory in the limit of vanishingHubble constant. Only the helicity-two modes are physical, the others are constrained. And the latter isextremely good because otherwise we would have got severe problems with the wrong-sign kinetric termof ψ . Unfortunately, we do indeed get them once the gauge invariance is broken by, say, a mass term.In general, we can think of two types of a mass term, h µν h µν and h µµ h νν . Both of them contain h and h i fields non-linearly (quadratically) which implies that the corresponding Einstein equations ceaseto provide constraints for the spatial sector. However, there is one particular combination of them, m (cid:0) h µν h µν − h µµ h νν (cid:1) discovered by Fierz and Pauli [5], from which h drops. The resulting constraint kills the unwanted sixthdegree of freedom and gives a ghost-free massive gravity in the linear approximation.Unfortunately, even in the vanishing mass limit, the linear theory anyway contradicts observations [6]due to the scalar graviton which couples to dust modifying the effective gravitational constant, but not toradiation keeping the bending of light intact. It was later argued by Vainshtein [7] that non-linear effectswill take over at small scales and restore the GR limit. But, almost at the same time, Boulware andDeser have shown [8] that the sixth degree of freedom comes back at the non-linear level reintroducingthe ghost mode. And therefore, a stable theory of massive gravity is probably not possible at all. Recently, a potentially ghost-free non-linear construction of massive gravity was found [2] in the decou-pling limit ( m → M P l → ∞ with m M P l fixed) by carefully getting rid off the sixth mode at everyorder of perturbation theory starting from the Fierz-Pauli term. The model can be resummed [3, 9] to adeceptively simple potential V = 2 m (cid:16) Tr p g − f − (cid:17) (19)around any background metric f µν . Up to the − h ≡ g − f contribution to the action, this is the first symmetric polynomial of eigenvalues of the square-root matrix p g − f . (Interestingly, it is noted in [10] that this model was known to Wess and Zumino as early asin 1970.) The other two suitable potentials are the second (cid:16) Tr p g − f (cid:17) − Tr (cid:16)p g − f (cid:17) and the third (cid:16) Tr p g − f (cid:17) − (cid:16) Tr p g − f (cid:17) Tr (cid:16)p g − f (cid:17) + 2Tr (cid:16)p g − f (cid:17) symmetric polynomials. In order to count the degrees of freedom non-perturbatively, one has to perform the Hamiltonian analysisof the action (18) with the potential term −√ γN V given by (19) in the case of the minimal dRGT model.We find the momenta π ij ≡ ∂ L ∂ ˙ γ ij = √ γ (cid:0) K kk γ ij − K ij (cid:1) , the primary constraints π N = π N i = 0, and theHamiltonian H = − Z d x √ γ N (cid:18) ( ) R + 1 γ (cid:18) (cid:16) π jj (cid:17) − π ik π ik (cid:19) − V (cid:19) + 2 N i ( ) ▽ k π ik ! . Hassan and Rosen have found [11, 12] a redefinition of shifts N i = ( δ ij + N D ij ( γ, n )) n j such that, for the new shifts n i , the square-root matrix takes the form p g − f = 1 N √ − n k n k (cid:18) n i − n j − n i n j (cid:19) + (cid:18) X ij ( γ, n ) (cid:19) (20)which means that the lapse assumes the role of the Lagrange multiplier and imposes a non-trivial con-straint on the spatial sector. One can check that the same is true for the higher ghost-free potentialsbecause the first matrix in (20) preserves its form under exponentiation to an integer positive power.A careful investigation shows [13] that the subsequent consistency conditions provide the second spatialsector constraint to form a non-degenerate pair of second class constraints on γ ij , and finally, an equa-tion for determining the lapse. And the same is true [14] for bimetric versions (with an independentEinstein-Hilbert term for f µν ), too. One can also approach the problem in a different manner. We can introduce a matrix Φ of auxiliaryfields with a constraint that Φ = N g − f which amounts to N V = 2 m Φ µµ + κ µν (cid:0) Φ να Φ αµ − N g να f αµ (cid:1) and, varying with respect to Φ, integrate out the κ -s: N V = m (cid:16) Φ µµ + (cid:0) Φ − (cid:1) µν N g να f αµ (cid:17) . In this method it is not necessary to calculate the matrix square root explicitly, and the procedure isstandard and elementary. Unfortunately, the calculations are cumbersome. But one can easily find [15]a direction in the space of must-be unphysical variables ( N , N i , Φ) along which there is no restrictionat the level of secondary constraints which proves that the number of degrees of freedom is less than six.Note that, in this construction, the equations of motion automatically demand that a real square-rootmatrix p g − f does exist. Non-perturbative existence of the real square-root is one of the very interesting issues about the dRGTmode [16]. In the cited paper the reader can find the necessary and sufficient condition for it: if there isa real negative eigenvalue of the g − f matrix, then it must be accompanied by even numbers of identicalJordan blocks. However, let us note that if g − f has a negative eigenvalue − λ , then the correspondingeigenvector lies in the kernel of λg + f matrix. In other words, there is a degenerate linear combination of g and f with positive coefficients. Even though it is anyway undesirable to couple matter to a non-triviallinear combination of metrics, one might consider imposing a condition of non-degeneracy of all positivelinear combinations. On the other hand, in full quantum gravity it might appear to be not so muchnon-sensical to consider complex potentials.We would point out that there is also a uniqueness problem for the squre root. Even the 2 × (cid:18) (cid:18) − − − (cid:19)(cid:19) = (cid:18) (cid:19) . Note thatthis one has zero trace which shifts the vacuum energy with respect to the trivial square root. It mightplay an important role in a fully quantum regime. 5 n conclusion , let us mention that there are lots of other problems to take care of. Some of remainingfive modes may become ghosts in certain regimes, and there can be strong coupling issues in some phys-ically relevant solutions. And there are interesting problems concerning multimetric theories. Moreover,we have to accept some form of acausality in our physical world since the superliminal propagation wasshown to be quite generic in the models of dRGT type [10]. However, it is very remarkable that, in thefirst place, we do have a reasonable candidate for a viable massive, or bimetric, gravity model. Acknowledgements.
The Author is supported by Russian Foundation for Basic Research Grant No.12-02-31214. It is also of a great pleasure to thank the orginisers of the 7th Mathematical Physics Meeting:Summer School and Conference on Modern Mathematical Physics for the opportunity to participate inthis wonderful scientific event.
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