Non-linear perturbations on spherical backgrounds (Formalism and ...

ackground connection, or from (20) if we want it in terms of the <strong>perturbations</strong>of the metric. As there is no metric contraction involved in the passagefrom the Riemman to the Ricci tensor, it is clear that∆ n [R µρ ] = ∆ n [R µνρ ν ]. (22)The resulting expression is symmetric in µ and ρ owing to the identityH αβ {k} h αβρ;µ = H αβ {k} h αβµ;ρ , (23)valid for any symmetric tensor H αβ such that H αβ {k} h β µ is also symmetricin α and µ.When considering the Ricci scalar, one has to take into account that thereis a metric contraction,R = g αβ R αβ , (24)which will have a non-zero contribution when the perturbation operator isapplied. Using formula (17), it is easy to compute the general perturbativeterm:( )n∑ n∆ n [R] = ∆ k [g αβ ]∆ n−k [Rkαβ ]. (25)k=0These formulas allow us to obtain the general perturbative term of theEinstein tensorG µν = R µν − 1 2 g µνg αβ R αβ , (26)where now formula (17) has to be applied twice, as the second term of theequation is the product of three objects. It is straightforward to rearrangelabels and write∆ n [G µν ] = ∆ n [R µν ]− 1 2n∑n∑k=0 j=kn!{n − j}h µν ∆ j−k [g αβ ]∆ k [R αβ ].k!(j − k)!(n − j)!(27)As explained in the introduction, the full Einstein equations are equivalentto an infinite hierarchy of <strong>linear</strong> differential equations. In order to make thisexplicit, it suffices to take the Einstein equations and substitute the tensorsby their expansions in power series of ε:G µν = 8πT µν ⇔ ∆ n [G µν ] = 8π {n} T µν for all n. (28)We can separate the nth order perturbation of the Einstein tensor into a partthat is <strong>linear</strong> in the nth perturbation of the metric and a source that will be

composed of products of lower order <strong>perturbations</strong>. Then equation (28) canbe written, for every n, in a schematic formL[ {n} h µν ] + {n} S µν [ {1} h, ..., {n − 1} h] = 8π {n} T µν , (29)where {n} S is the source term and L is a <strong>linear</strong> differential operator that onlydepends on the background geometry. If we define [8]{n}h µν ≡ {n} h µν − 1 2 g µν {n} h α α , (30)in general the operator can be written asL[ {n} h µν ] = 1 2 (g µν {n} h αβ R αβ − {n} h µν R + {n} h αµ;ν α + {n} h αν;µα(31)− g µν{n}h αβ ;αβ − {n} h µν;α α ).There is a special case which deserves particular attention. If the backgroundis a vacuum (T µν = 0), the background Ricci tensor will cancel outreducing the expression (31) toL[ {n} h µν ] = 1 2 ( {n} h αµ;ν α + {n} h αν;µ α − g µν{n}h αβ ;αβ − {n} h µν;α α ). (32)Now, we can interchange the order of the covariant derivatives in the firsttwo terms and impose the Lorenz gauge,{n}h µα;α = 0, (33)obtaining in this way the vacuum propagation equation,{n}h µν;α α + 2R αµβν{n}h αβ − 2 {n} S µν [ {1} h, ..., {n − 1} h] = −16π {n} T µν . (34)This is a wave equation for the nth order perturbation of the metric. Theinteraction between the wave with the background is encoded in the secondterms, whereas the self-interaction of the wave is given by the source {n} S µν .In order to close the system of equations (28), the matter evolution equationsare necessary. These are equations for the matter fields ˜Φ(ε) and mustalso be <strong>linear</strong>ized.It is worth emphasizing that the combinatorial formulas for the <strong>perturbations</strong>of the curvature tensors at a general order that we have deduced in

this section are extremely useful for computational purposes. Essentially, theproblem of <strong>perturbations</strong> is reduced to that of listing the sorted partitionsof a given number, which can be done very fast in any computer algebrasystem. This fact simply reflects the recursive differential origin of the perturbationprocess. The formulas in this section have been implemented inthe free package xPert that will be explained in Section 5.3 Spherical background3.1 Gerlach and Sengupta notationWe will only consider spherically symmetric backgrounds and use the notationintroduced by GS [2]. We take the four-dimensional background manifoldM and write it as the product M 2 ×S 2 , where M 2 is a two-dimensionalLorentzian manifold with boundary (this boundary will be associated withthe center of symmetry) and S 2 the two-sphere. We will use Greek indicesfor the four dimensional manifold and choose a coordinate system adaptedto this decomposition x µ ≡ (x A , x a ). Capital Latin indices denote basis indiceson the manifold M 2 , x A ≡ (x 0 , x 1 ), and lowercase Latin letters indicateindices on the two sphere, x a ≡ (x 2 , x 3 ).Any spherically symmetric metric and stress-energy tensor can be respectivelywritten as [2]g µν (x λ )dx µ dx ν = g AB (x D )dx A dx B + r 2 (x D ) γ ab (x d )dx a dx b ,t µν (x λ )dx µ dx ν = t AB (x D )dx A dx B + 1 2 r2 (x D ) Q(x D ) γ ab (x d )dx a dx b ,where g AB is the metric of the manifold M 2 , γ ab is the round metric of thesphere and r is a scalar defined on M 2 . The associated covariant derivativeof the round metric will be denoted by a semicolon (γ ab:c = 0).3.2 Spherical tensor harmonicsThe so-called RWZ tensor harmonics [3, 4] provide bases for expandingscalars, vectors and symmetric rank-two tensors on the sphere. These harmonicshave a well-defined parity and are tangent to the sphere. They areclassified into two polarity families: under a parity transformation the polar

and axial harmonics change sign as (−1) l and (−1) l+1 , respectively, wherethe integer l is the corresponding harmonic label.In order to face high-order perturbation theory, we generalize the RWZharmonics to any rank defining the polar harmonic Zl m a 1 ...a s≡ (Ylm :a 1 ...a s) STand the axial harmonic Xl m a 1 ...a s≡ ε b (a1 Zl m ba 2 ...a s), where Ylm is the scalarharmonic and the superscript ST stands for the symmetric and trace-freepart. In this way, a complete basis for tensors of rank s is composed byZ m l a 1 ...a s, X m l a 1 ...a sand independent <strong>linear</strong> combinations of products betweenγ ab and ε ab with the basis of rank (s − 2).It is also neccesary to obtain a closed formula that expresses the productbetween any pair of RWZ generalized tensor harmonics as <strong>linear</strong> combinationof harmonics with some Clebsch-Gordan-like coefficients [1].4 <strong>Non</strong>-spherical <strong>perturbations</strong>We expand the <strong>perturbations</strong> of the metric using the RWZ basis of harmonics:{n}h µν ≡ ∑ {n}Hl,m( m l AB Zlm {n}H m l A Zl m b + {n} h m l A Xl m )b{n}Sym. Kl m r 2 γ ab Zl m + {n} G m l r 2 Zl m ab + {n} h m l Xl m .(35)abA similar decomposition is done for the matter <strong>perturbations</strong> {n} T µν . Butthese <strong>perturbations</strong> have some gauge (nonphysical) degrees of freedom. Thegauge freedom of the nth-order perturbation of a background tensor is encodedin n vector fields [5]. We have demostrated that choosing vanishingharmonics coefficients {n} H m l A,{n}G m l and {n} h m l fixes the gauge at any order.For the case n = 1 this gauge was introduced by Regge and Wheeler [3].Another way of removing these nonphysical degrees of freedom is by usinggauge-invariant quantities. These invariants were introduced by Moncrief [6]for Schwarzschild at <strong>linear</strong> order. GS [2] generalized them for any sphericallysymmetric background and, taking <strong>linear</strong> combinations of the matter andmetric harmonic coefficients (35), they constructed seven polar ({ {1} Ψ}) andthree axial ({ {1} ψ}) matter invariants, as well as six metric invariants, givenby the polar objects { {1} K m l AB, {1} K m l } and the axial vector {1} κ m l A. We havedesigned [7] an iterative method to generalize these objects up to any order,under certain restrictions on the harmonic labels.At second-order, we have explicitly calculated [7] the invariants that arisefrom the same <strong>linear</strong> combinations that appear at first order, but supplementedwith some first-order quadratic terms. The second-order Einstein

metric g[−µ, −ν]. There is a command Perturbation[expr,n] which is ableto give the nth-order <strong>perturbations</strong> of any expression expr. It plays the roleof ∆, so it is defined with the properties of a derivative,Perturbation[x +y,1] = Perturbation[x,1] +Perturbation[y,1], (38)Perturbation[xy,1] = Perturbation[x,1] y +x Perturbation[y,1]. (39)We define its action on thenth perturbation of the metric, and, as a particularcase, on the metric itself,Perturbation[h[LI[n], −µ, −ν],1] = h[LI[n +1], −µ, −ν], (40)Perturbation[g[−µ, −ν],1] = h[LI[1], −µ, −ν], (41)and, because of the relation (7), we also definePerturbation[g[µ, ν],1] = −g[µ, α] h[LI[1], −α, −β] g[β, ν]. (42)In this way, the internal procedure to calculate Perturbation[expr,n] isthe following:• expr is written in terms of the metric g[−µ, −ν], its inverse g[µ, ν] andthe nth order perturbation of the metric h[LI[n], −µ, −ν]. This step isimmediate because the package xTensor knows the definition of all thecurvature tensors in terms of the metric and its inverse.• Perturbation[expr,n] is expanded via Perturbation[Perturbation[expr,1],n −1]. From now on we only will deal with the most internalhead Perturbation.• Perturbation acts on expr making use of <strong>linear</strong>ity (38) and the Leibnitzrule (39).• Definitions (40-42) are applied.• If anyPerturbation head remains, the process goes back to the secondstep of this chain.There is another command namedGeneralPerturbation[expr,n] whichimplements the general expansion formulas for the the inverse metric (10),the Christoffel symbols (15), the Riemann tensor (20), and the Ricci tensorand scalar. This general expansions are much faster than the recursive

procedures because one only needs to calculate all the sorted partitions ofa certain integer number and this can be done very fast in any computeralgebra system.As an example, we present here the calculation and canonicalization ofthe fifth order perturbation of the Riemman tensor:DefManifoldM,1, 2, 3, 4,Μ,Ν,Ρ,Σ,Λ,Α,Β,Η,Τ,ΩDefMetric1, gΜ,Ν, CD,";", ""DefMetricPerturbationg, h, ΕGeneralPerturbationRiemannCD4Μ,Ν,Α,Β, 5 ToCanonical Short8.54854 Second5 h 3ΗΛ 1 1h ΝΛ;Α h ;Β ΜΗ 1554 h ΝΗ;Α2 h2ΗΛ h ΝΛ;Α4 1h Η;Β Μ 15 h 1ΗΛ h 2ΡΗ1Ρ2 1h ;Β ΜΗ 5 h 1ΗΛ 3h ΝΛ;Α h ΜΗ1 ;Β 1 1h ΝΡ;Α h ;Β 1ΜΛ 30 h ΗΡ h 1ΗΛ h 1ΡΣ 1 1h ΝΣ;Α h ;Β ΜΛ 2 1h ΝΡ;Α h ;Β ΜΛ 992 30 h 1ΒΗ 1h ΗΛ h 1ΛΡ 1 1h ΝΡ;Σ h ;Σ ΜΑ 15 h 1ΗΛ h Η30 h 1ΒΗ 1h ΗΛ h 1ΛΡ 1 1h ΡΣ;Μ h ;Σ ΝΑ 30 h 1ΒΗ 1h ΗΛ h 1ΛΡ 1 1h ΜΣ;Ρ h ;Σ ΝΑ 30 h 1ΒΗ 1h ΗΛ h 1ΛΡ 1 1 ;Σh ΜΡ;Σ h ΝΑIn fact, this expression is only a shortened representation of the resultbecause means that there are another 992 terms that have not beendisplayed.In order to give an idea of the high efficiency of this package, the followingfigure displays the canonicalization times for the <strong>perturbations</strong> of the Riemanntensor with a small PC. The horizontal axis represents the perturbativeorder whereas the vertical axis describes in a logarithmic scale the total numberof terms of the perturbation of the Riemann tensor, which appears inblue, and the time spent to calculate and canonicalize the expression, printedin red.10000Hour100Minute1Second2 4 6 8 10

5.2 HarmonicsThe package Harmonics implements different kinds of tensor sphericalharmonics [10]. For a description of the conventions used in this package,see reference [1]. The commands PureOrbital[j, l, m][a,b,...] andPureSpin[j, ±1, m][a,b,...] provide, respectively, any pure-orbital orpure-spin harmonics, both in abstract form or giving their components inany coordinate or noncoordinate basis. The generalized RWZ harmonicsZ[LI[l],LI[m],a,b, ...] and X[LI[l],LI[m],a,b, ...] have also been defined, incorporatingall their symmetries and properties [1]. The product formula hasalso been included by a rule named productRule.As an example of the use of this package, we present here the expansionof the product between two tensorial spherical harmonics into a <strong>linear</strong> combinationof harmonics, in spherical coordinates (θ, φ). The harmonic labels land m appear in black, while we show in red the coordinate indices with thenotation (2, 3) ≡ (θ, φ).DefManifoldS2,2, 3,a, b, c, d, e, f, wDefMetric1, gammaa,b, cd,":", "D", PrintAs "Γ"RicciScalarcd :⩵ 2DefChartspherical, S2,Θ,Φ, BasisColorRedXLI7, LI2,2, spherical,3, sphericalZLI4, LI1,2, spherical,2, spherical,3, spherical72X 3 412 Z 223%. productRule Simplify0.736047 Second14 CscΘ2,7,2 E 3,4,1;42,7,2X 41 2,7,23 E 3,4,1;6SinΘ X 2232X 61 2,7,23 E 3,4,1;82,7,2X 81 2,7,23 E 3,4,1;10X 101 2,7,23 4 E 3,4,1;551SinΘ X 3 2232 714 E 3 913,4,1;7 4 E 3,4,1;9 SinΘ X 3 1112232 4 E 3,4,1;11 SinΘ X 3 2232 2,7,2E 3,4,1;3 Z 31 2,7,23 E 3,4,1;5 Z 51 2,7,23 E 3,4,1;7 Z 71 2,7,23 E 3,4,1;9 Z 91 2,7,23 E 3,4,1;11 Z 111 3 2,7,2614 E 3,4,1;6 SinΘ Z 3 2,7,2812232 4 E 3,4,1;8 SinΘ Z 3 2,7,22232 4 E 3,4,1;10 SinΘ Z 2232%. coeffERule HarmonicComponent0.64404 Second2,7,2101 3 1 8505 7 131072 Π Φ 338 1351 Cos2 Θ 990 Cos4 Θ 1001 Cos6 Θ Sin2 Θ 2

The time spent by the computer to make the computation between eachinput and output is also shown above.5.3 xPertGSThe package xPertGS needs the previous two packages and has threesections:• Spherical background. The reduced manifold M 2 is defined with itsmetric g[-A,-B] (and corresponding derivative CD[-A]). The scalarfield r[] is also defined. The manifold M 4 is constructed as a productof M 2 and S 2 ; all tensors on M 4 can be block-decomposed in theirrespective parts on those submanifolds.• First order. First-order <strong>perturbations</strong> are defined and their gauge invariantsconstructed. The evolution equations for these invariants arealso computed. The energy-momentum conservation equations are calculatedand, making use of the first-order Einstein equations, it ischecked that they are trivially zero. The total execution time of thispart is of the order of two minutes in a PC.• Second order. The scheme is the same as at first order. Second-order<strong>perturbations</strong> are defined and their equations computed and manipulatedin order to simplify them as much as possible. The energymomentumconservation equations are also computed and the Bianchiidentity verified. The total execution time of this part is of the orderof two hours.6 ConclusionsIn this work, we have summarized the formalism that has been presentedrecently in Refs. [1] and [7] in order to deal with high-order <strong>perturbations</strong> ofspherical backgrounds in a gauge-invariant way. We have focused on somespecific features of our formalism to make them more transparent.On the one hand we have given the detailed derivation of the closedformulas for the nth order perturbation of every curvature tensor of interest.In particular, restricting the background to a vacuum, we have deduced thepropagation equation of the nth order perturbation of the metric. These

ecursive formulas are important from a computational point of view becausethey allow to reduce the problem of calculating the nth order perturbationof any curvature tensor essentially to that of listing all the sorted partitionsof the number n.On the other hand, we have explained both the internal structure and theexternal use of the Mathematica packages which we have developed during ourinvestigation. Each of the packages is designed for one part of the problem.More precisely, xPert deals with the <strong>perturbations</strong> of all the tensors thatappear in our analysis without imposing any symmetry in the background.Harmonics implements some structures defined in a two-sphere, in particularit incorporates different definitions of spherical tensor harmonics that havebeen used in several applications. Finally, xPertGS combines the last twopackages in order to compute second-order <strong>perturbations</strong> on a sphericallysymmetric background.AcknowledgmentsD.B. acknowledges financial support from the FPI program of the RegionalGovernment of Madrid. J.M.M.-G. acknowledges the financial aid providedby the I3P framework of CSIC and the European Social Fund. This workwas supported by the Spanish MEC Project No. FIS2005-05736-C03-02.References[1] D. Brizuela, J. M. Martín-García and G. A. Mena Marugán, Phys. Rev.D 74 044039 (2006).[2] U. H. Gerlach and U. K. Sengupta, Phys. Rev. D 19 2268 (1979).[3] T. Regge and J. A. Wheeler, Phys. Rev. 108 1063 (1957).[4] F. J. Zerilli, Phys. Rev. Lett. 24 737 (1970).[5] S. Sonego and M. Bruni, Commun. Math. Phys. 193 209 (1998).[6] V. Moncrief, Annals of Physics 88 323 (1973).[7] D. Brizuela, J. M. Martín-García and G. A. Mena Marugán, Phys. Rev.D 76 024004 (2007).

[8] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman,New York, 1973).[9] J. M. Martín-García, “xTensor, a free abstract tensor manipulator forMathematica”, http://metric.iem.csic.es/Martin-Garcia/xAct/.[10] K. S. Thorne, Rev. Mod. Phys. 52 299 (1980).