Asymptotic Expansion of the Equations of Motion for Einstein Gravity with Λ<0

The equations of motion for gravity with a cosmological constant in d+1=5 dimensions are

(Subscript[R, μν])-(1/2)(Subscript[g, μν])(R-2Λ) = 0

with Λ=-d(d-1)/(ℓ^2)=-12/(ℓ^2). We work in coordinates where an asymptotically (Subscript[AdS, 5]) metric takes the form

(ds^2) = ((((ℓ/ρ)))^2)((dρ)^2)+((((ℓ/ρ)))^2)((Subscript[(h^((0))), ab])+ρ (Subscript[(h^((1))), ab])+(ρ^2) (Subscript[(h^((2))), ab])+(ρ^3) (Subscript[(h^((3))), ab])+(ρ^4) (Subscript[(h^((4))), ab])+...)(dx^a)(dx^b)

We' ll determine the (Subscript[(h^((n))), ab]) by solving the various projections of the Einstein equations normal to or parallel to constant-ρ surfaces, with ρ>> ℓ.

Notice that () is a little different than the usual form of the Fefferman-Graham expansion used by Skenderis et al. in, for instance, hep-th/0002230. The coordinate that they call ‘ρ’ is (ρ^2) in our conventions.

We are not making many assumptions about the terms appearing in the expansion. We will derive the following results:

(Subscript[(h^((1))), ab])=(Subscript[(h^((3))), ab])=0

(Subscript[(h^((2))), ab])=-((ℓ^2)/2)((Subscript[(ℛ^((0))), ab])-(1/6)(Subscript[(h^((0))), ab])(ℛ^((0))))

(Subscript[(h^((4)a)), a])=((ℓ^4)/16)((ℛ^((0)ab))(Subscript[(ℛ^((0))), ab])-(2/9)((((ℛ^((0)))))^2))

The covariant divergence (𝒟^a)(Subscript[(h^((4))), ab])

The transverse, traceless part of (Subscript[(h^((4))), ab]) is dynamical.

The only real assumption here is that the "boundary metric" (Subscript[(h^((0))), ab]) is Weyl-flat. This insures that there are no terms in the Fefferman-Graham expansion of the form log ρ. It also implies that first and second derivatives of the Ricci tensor (Subscript[(ℛ^((0))), ab]) and Ricci scalar (ℛ^((0))) satisfy certain identities. This will be important when working with the fourth-order terms in the expansion of the equations of motion.

Preamble : Load Packages

<<xAct`xPert`

Show covariant derivatives in the form (Subscript[D, a])(...), instead of using postfix notation.

$CovDFormat="Prefix"

Define a few functions that collect multiple functions that we frequently apply.

The function ‘org’ contracts metrics in an expression, collects terms by order in the perturbative expansion, and then simplifies/canonicalizes.

org[expr_]:=Collect[ContractMetric[expr],$PerturbationParameter,ToCanonical]

The function ' Simp' takes an expression and contracts metrics, simplifies, and then expands.

Simp[expr_] := expr//ContractMetric//Simplification//Expand

Define Manifold and Variables

Define our manifold M, a 4-dimensional surface of constant ρ.

The command takes the name of the manifold, its dimension, and asks for an array of letters or symbols to be used as indices on tensors. Once declared as indices, they are off-limits and cannot be used as the names of functions, etc.

DefManifold[M,4,{a,b,c,d,e,f}]

Define the “unphysical” metric (Subscript[h, ab]) on M, related to the physical metric by (Subscript[g, ab])= (ρ^(-2))(Subscript[h, ab]).

When defining this metric we specify that it has signature - 1, give it the name "h", indicate that we the metric - compatible covariant derivative is called with a function named "CD", and ask Mathematica to print derivatives using the symbol "D".

DefMetric[-1,h[-a,-b],CD,{";","D"}]

The AdS length scale appears throughout our calculations. We define it as a constant symbol ℓ, so that derivatives and variational operators know to ignore it.

DefConstantSymbol[ℓ]

The variables on M are the metric (Subscript[g, ab]) and the extrinsic curvature (Subscript[K, ab])=(1/2)(Subscript[ℒ, n])(Subscript[g, ab]). But we work with the unphysical metric (Subscript[h, ab]) and a conformally rescaled extrinsic curvature instead. I will write the Einstein equations in terms of a conformally rescaled extrinsic curvature (Subscript[θ, ab]) that is related to the physical extrinsic curvature by (Subscript[K, ab])= (ρ^(-2))(Subscript[θ, ab]). Note that this is not the same as (1/2)(Subscript[ℒ, n])(Subscript[h, ab])!

DefTensor[θ[-a,-b],M,Symmetric[{-a,-b}]]

Because of the method we use to implement the expansions, we also need to define a new tensor for the ρ derivative of the extrinsic curvature

DefTensor[ℒθ[-a,-b],M,Symmetric[{-a,-b}]]

Define perturbations, which will be terms in the Fefferman-Graham expansion. Because of the way xPert works, we need to define perturbations for (Subscript[h, ab]), as well as (Subscript[θ, ab]) and any other quantity that involves ρ derivatives (like (Subscript[ℒ, n])(Subscript[θ, ab])).  These quantities are of course all related, so we will establish replacement rules that express the perturbations of (Subscript[θ, ab]) and (Subscript[ℒ, n])(Subscript[θ, ab]) in terms of perturbations of (Subscript[h, ab]).

DefMetricPerturbation[h,δh,ρ]

PrintAs[δh] ^= "h";

DefTensorPerturbation[δθ[LI[order],-a,-b],θ[-a,-b],M]

PrintAs[δθ] ^= "θ";

DefTensorPerturbation[δℒθ[LI[order],-a,-b],ℒθ[-a,-b],M]

PrintAs[δℒθ] ^= "ℒθ";

Perturbations are usually printed like this :

Perturbation[h[-a,-b],1]

This looks a little different than the usual notation, so we redefine the way perturbations are printed to more closely resemble the standard notation for terms in the Fefferman-Graham expansion:

Unprotect[IndexForm];
IndexForm[LI[x_]]:=ColorString[StringJoin["(",ToString[x],")"],RGBColor[0,0,1]];
Protect[IndexForm];

IndexForm[LI[n]]

Perturbation[ h[-a,-b] ]

The zeroth order perturbation is just the unperturbed quantity.

Perturbation[h[-a,-b] ,0]

The variational operator δ satisfies a Leibniz rule. One consequence is that perturbations carry a factor (1/(n!))at (n^th) order.

Perturbed[h[-a,-b],4]

Perturbed[θ[-a,-b],4]

Perturbed[ℒθ[-a,-b],4]

Now we need to set rules that relate the perturbations of (Subscript[θ, ab]) and (Subscript[ℒ, n])(Subscript[θ, ab]) to the perturbations of (Subscript[h, ab]).

First work out (Subscript[θ, ab]) in terms of (Subscript[h, ab]) and its derivatives. The spacelike unit vector normal to M is (n^μ)=(ρ/ℓ)(Subscript[(δ^μ), ρ]) in our coordinate system, and (Subscript[K, ab])=(1/2)(Subscript[ℒ, n])(Subscript[h, ab]), so:

(Subscript[K, ab])=(1/2)(ρ/ℓ)(∂_ρ)(Subscript[g, ab])=(1/2)(ρ/ℓ)(∂_ρ)((ρ^(-2))(Subscript[h, ab])) = (ρ^(-2))((1/2)(ρ/ℓ)(∂_ρ)(Subscript[h, ab])-(1/ℓ)(Subscript[h, ab])) = (ρ^(-2))(Subscript[θ, ab])
⟶ (Subscript[θ, ab])=(1/2)(ρ/ℓ)(∂_ρ)(Subscript[h, ab])-(1/ℓ)(Subscript[h, ab])

Next, relate terms in the asymptotic expansions of (Subscript[θ, ab]) and (Subscript[h, ab]) out to fourth order:

Collect[(1/2)(ρ/ℓ)D[Perturbed[h[-a,-b],4],ρ]-(1/ℓ)Perturbed[h[-a,-b],4]//ContractMetric,ρ,Simplification]

Perturbed[θ[-a,-b],4]

Compare the terms in the two expansions and define the replacement rules.

θRule={Table[IndexSolve[(Coefficient[ρ(Perturbed[θ[-a,-b],4]-((ρ/(2ℓ)))D[Perturbed[h[-a,-b],4],ρ]+(1/ℓ)Perturbed[h[-a,-b],4]),(ρ^(K+1))]//Simp)==0,δθ[LI[K],-a,-b], ContractMetrics->True],{K,0,4}]}//Flatten

Make sure this works by checking perturbations of θ = (h^ab)(Subscript[θ, ab])=K = (g^ab)(Subscript[K, ab]). xPert knows how to expand (h^ab)(Subscript[θ, ab]):

Collect[(Perturbed[h[a,b]θ[-a,-b],4]/.θRule)//ExpandPerturbation//ContractMetric,ρ,Simplification]

Now compare to the expansion of (g^ab)(Subscript[K, ab]), working out each factor to out to (4^th) order, combining them, and throwing away terms with powers of (ρ^n), n≥ 5.

Collect[(ρ^2)Perturbed[h[a,b],4](ρ/(2ℓ))D[(1/(ρ^2))Perturbed[h[-a,-b],4],ρ]//ExpandPerturbation//ContractMetric,ρ,Simplification]/. { (ρ^5)-> 0, (ρ^6)-> 0,(ρ^7)-> 0,(ρ^8)-> 0}

They are the same, so it works.

%%-%//Simplification

Do the same for (Subscript[ℒ, n])(Subscript[θ, ab]):

(Subscript[ℒ, n])(Subscript[θ, ab])=(ρ/ℓ)(∂_ρ)(Subscript[θ, ab])=(ρ/ℓ)(∂_ρ)((Subscript[(θ^((0))), ab])+(Overscript[(∑_(n=1)), n=∞])(1/(n!))(ρ^n)(Subscript[((δθ)^((n))), ab]))=(ρ/ℓ)(Overscript[(∑_(n=1)), n=∞])(1/((n-1)!))(ρ^(n-1))(Subscript[((δθ)^((n))), ab])=(1/ℓ)(Overscript[(∑_(n=1)), n=∞])(1/((n-1)!))(ρ^n)(Subscript[((δθ)^((n))), ab])=(Overscript[(∑_(n=0)), n=∞])(1/(n!))(ρ^n)(Subscript[((δℒθ)^((n))), ab])

Collect[(ρ/ℓ)D[Perturbed[θ[-a,-b],4],ρ],ρ,Simplification]

Perturbed[ℒθ[-a,-b],4]

Compare the two expressions and define replacement rules.

ℒθRule={Table[IndexSolve[(Coefficient[ρ(Perturbed[ℒθ[-a,-b],4]-((ρ/ℓ))D[Perturbed[θ[-a,-b],4],ρ]),(ρ^(K+1))]//Simp)==0,δℒθ[LI[K],-a,-b], ContractMetrics->True],{K,0,4}]}//Flatten

Perturbed[ℒθ[-a,-b],4]/.ℒθRule

%-Collect[(ρ/ℓ)D[Perturbed[θ[-a,-b],4],ρ],ρ,Simplification]//Expand

Join both sets of rules. Applying this rule to a given expression will replace (Subscript[θ, ab]) and (Subscript[ℒ, n])(Subscript[θ, ab]) (with indices in any position) by the appropriate quantities in terms of (Subscript[h, ab]) and its perturbations.

PertRules = {θRule,ℒθRule}//Flatten

So, we will write all equations in terms of (Subscript[h, ab]), (Subscript[θ, ab]), and (Subscript[ℒ, n])(Subscript[θ, ab]), perturb the equations to the desired order, and use “PertRules” to rewrite everything in terms of the (Subscript[(h^((n))), ab]) appearing in the Fefferman-Graham expansion of the metric.

The “Momentum Constraint”

Contract one index of (Subscript[G, μν]) with the normal vector (n^μ), and project the other index tangent to M:

((Subscript[P, b])^ν)((n^μ)(Subscript[G, μν]))= (D^a)(Subscript[θ, ab])-(Subscript[D, b])θ = 0

This is the analog of the momentum constraint in the ADM formalism.

(Perturbed[h[a,c]CD[-c][θ[-a,-b]]-CD[-b][h[a,c]θ[-a,-c]],4])//ExpandPerturbation;

Apply the “PertRules” to rewrite everything in terms of (Subscript[h, ab]), and contract any metrics that show up after replacement.

%/.PertRules;

Now collect terms order - by - order and simplify them.

MomentumEqn = Collect[% ,ρ,Simp];

Since we already applied the rules that relate (Subscript[θ, ab]) to (Subscript[h, ab]), the zeroth order term in the equation is satisfied:

MomentumEqn /. ρ-> 0

The (1^st) order term:

Coefficient[MomentumEqn,ρ]

The (2^nd) order term:

Coefficient[MomentumEqn,(ρ^2)]

Notice that this contains factors of ((δh)^((2))) and (((((δh)^((1)))))^2). As we go to higher order the equations will of course become much longer and more complex.

The (3^rd) order term:

Coefficient[MomentumEqn,(ρ^3)]

The (4^th) order term:

Coefficient[MomentumEqn,(ρ^4)]

The “Hamiltonian Constraint”

Contract both indices of (Subscript[G, μν]) with the normal vector normal to M:

(n^μ)(n^ν)(Subscript[G, μν]) = (1/2)(θ^2)-(1/2)(θ^ab)(Subscript[θ, ab])-(1/2)(ρ^2)ℛ[h]-(6/(ℓ^2))= 0

This is the analog of the Hamiltonian constraint in the ADM formalism.

Perturbed[(1/2)θ[a,-a]θ[b,-b]-(1/2)θ[a,b]θ[-a,-b],4]-(1/2)(ρ^2)Perturbed[RicciScalarCD[],2]-(6/(ℓ^2))//ExpandPerturbation;

Note that the term involving ℛ[h] carries an overall factor of (ρ^2), so we only need to perturb it to second order if we want the terms in the equation out to 4th order.

%/.PertRules;

HamiltonianEqn = Collect[%,ρ,Simp];

The (0^th) order term:

HamiltonianEqn/.ρ-> 0

The (1^st) order term:

Coefficient[HamiltonianEqn,ρ]

The (2^nd) order term:

Coefficient[HamiltonianEqn,(ρ^2)]

The (3^rd) order term:

Coefficient[HamiltonianEqn,(ρ^3)]

The (4^th) order term:

Coefficient[HamiltonianEqn,(ρ^4)]

Fully Projected Einstein Equation

Project both indices of the Einstein equations tangent to M:

(ρ^2)((Subscript[P, a])^μ)((Subscript[P, b])^ν)((Subscript[G, μν])) = (ρ^2)(Subscript[𝒢, ab])[h]-(Subscript[ℒ, n])(Subscript[θ, ab])+(Subscript[h, ab])(h^cd)(Subscript[ℒ, n])(Subscript[θ, cd])+(2/ℓ)(Subscript[θ, ab])-(2/ℓ)(Subscript[h, ab])θ-θ (Subscript[θ, ab])+2((Subscript[θ, a])^c)(Subscript[θ, bc])+(1/2)(Subscript[h, ab])(θ^2)-(3/2)(Subscript[h, ab])(θ^cd)(Subscript[θ, cd])-(6/(ℓ^2))(Subscript[h, ab]) = 0

(ρ^2)Perturbed[EinsteinCD[-a,-b],2]+Perturbed[-ℒθ[-a,-b]+h[-a,-b]h[c,d]ℒθ[-c,-d]+ (2/ℓ)θ[-a,-b]-(2/ℓ)h[-a,-b]θ[c,-c]-θ[c,-c]θ[-a,-b]+2θ[-a,c]θ[-b,-c]+(1/2)h[-a,-b]θ[c,-c]θ[d,-d]-(3/2)h[-a,-b]θ[c,d]θ[-c,-d]-(6/(ℓ^2))h[-a,-b],4]//ExpandPerturbation;

%//.PertRules;

ProjectedEinsteinEqn = Collect[%,ρ,Simp];

The (0^th) order term:

ProjectedEinsteinEqn/.ρ-> 0

The (1^st) order term:

Coefficient[ProjectedEinsteinEqn,ρ]

The (2^nd) order term:

Coefficient[ProjectedEinsteinEqn,(ρ^2)]

The (3^rd) order term:

Coefficient[ProjectedEinsteinEqn,(ρ^3)]

The (4^th) order term:

Coefficient[ProjectedEinsteinEqn,(ρ^4)]

Solve the Equations

First Order

Here are the first - order terms in the equations:

{Coefficient[HamiltonianEqn,ρ],Coefficient[MomentumEqn,ρ],Coefficient[ProjectedEinsteinEqn,ρ]}//Simplification

The solution is obviously (Subscript[((δh)^((1))), ab])=0.

AsympSoln = MakeRule[{δh[LI[1],-a,-b],0}];

Second Order

Now look at the second-order equations, simplified by substituting the solution to the first-order equations. First the Hamiltonian constraint:

(Coefficient[HamiltonianEqn,(ρ^2)])/. AsympSoln

Add the solution to this equation to our solutions from the first-order equations:

AsympSoln=Append[AsympSoln,MakeRule[{δh[LI[2],a,-a],-((ℓ^2)/3)RicciScalarCD[]}]]//Flatten;

Now the Projected Einstein equation, with the solution to the first-order EOM and the second-order Hamiltonian constraint applied:

Coefficient[ProjectedEinsteinEqn,(ρ^2)]/.AsympSoln

Replace the Einstein tensor with Riccis and simplify:

%//EinsteinToRicci//Simplification

Add this to the list of solutions:

AsympSoln=Append[AsympSoln,MakeRule[{δh[LI[2],-a,-b],(ℓ^2)(-RicciCD[-a,-b]+(1/6)h[-a,-b]RicciScalarCD[])}]]//Flatten;

Now make sure the momentum constraint is satisfied at this order:

Coefficient[MomentumEqn,(ρ^2)]/.AsympSoln

Check that this vanishes by first converting Riccis to Einstein tensors (since xTensor knows to apply the contracted Bianchi identity in that case) and simplify:

%//RicciToEinstein//Simp

Third Order

The third order equations are trivial (like at first order) once the solutions at first and second order are applied:

Coefficient[HamiltonianEqn,(ρ^3)]/. AsympSoln

AsympSoln=Append[AsympSoln,MakeRule[{δh[LI[3],a,-a],0}]]//Flatten;

Coefficient[ProjectedEinsteinEqn,(ρ^3)]/.AsympSoln

AsympSoln=Append[AsympSoln,MakeRule[{δh[LI[3],-a,-b],0}]]//Flatten;

Fourth Order

The momentum constraint fixes the trace of the fourth order term:

(Coefficient[HamiltonianEqn,(ρ^4)]/.AsympSoln)//Simp

%//RicciToEinstein//EinsteinToRicci//Simp

Solve[% == 0, δh[LI[4],a,-a]]//Expand

AsympSoln=Append[AsympSoln,MakeRule[{δh[LI[4],a,-a],(3/2) (ℓ^4) ({{R[D], ({{ ,  }, {a, b}})}}) ({{R[D], ({{a, b}, { ,  }})}})-(((ℓ^4) ((R[D])^2))/3)}]]//Flatten;

All factors of the uncontracted (Subscript[((δh)^((4))), ab]) cancel at this order in the projected Einstein equation.

Coefficient[ProjectedEinsteinEqn,(ρ^4)]/.AsympSoln//Simp

All that is left is a complicated expression involving Ricci tensors, Ricci scalars, and their derivatives.

terms4=%;

Does this expression vanish? It should, but showing this is non-trivial. We’ve assumed that the boundary metric is Weyl-flat -- otherwise there would be log terms in the Fefferman-Graham expansion that contribute to the equations at this order. So we need to define a few rules that implement identities for Weyl-flat metrics. First, the Weyl tensor vanishes. Second, since derivatives of Weyl must also vanish we get some identities that relate one and two derivatives action on the Ricci tensor and scalar. (See, for example, equations (12) and (13) of these notes, with the Weyl tensor set to zero.)

Be careful! Trying to apply the rules defined below with “//.” instead of “/.” can cause an infinite loop.

WeylFlatRules={MakeRule[{CD[a][RicciCD[-a,-b]],(1/2)CD[-b][RicciScalarCD[]]}],MakeRule[{WeylCD[a,b,c,d],0}],MakeRule[{CD[-c][CD[c][RicciCD[-a,-b]]],(1/6)h[-a,-b]CD[-c][CD[c][RicciScalarCD[]]]+(1/3)CD[-a][CD[-b][RicciScalarCD[]]]+2RicciCD[-a,c]RicciCD[-b,-c]-(2/3)RicciScalarCD[]RicciCD[-a,-b]-(1/2)h[-a,-b](RicciCD[c,d]RicciCD[-c,-d]-(1/3)((RicciScalarCD[])^2))}],MakeRule[{CD[b]@RicciCD[a,c],CD[a]@RicciCD[b,c]+(1/6)h[a,c]CD[b]@RicciScalarCD[]-(1/6)h[b,c]CD[a]@RicciScalarCD[]}]}//Flatten;

To show the fourth-order terms in the projected Einstein equation vanish, we need to “massage” the expression until the terms are in a form we can simplify. First, commute covariant derivatives in a few terms to get factors of (D^c)(Subscript[R, ac]):

CommuteCovDs[CommuteCovDs[terms4,CD,{-b,-c}],CD,{-a,-c}]

Now replace factors of the Riemann tensor with factors of the Weyl tensor...

%//RiemannToWeyl//Simp

... and apply our rules for Weyl-flat boundary metric:

%/.WeylFlatRules//Simp

So, to summarize: the fourth-order term in the projected Einstein equation vanishes when the metric (Subscript[(h^((0))), ab]) is Weyl-flat. The equation does not determine (Subscript[(h^((4))), ab]). This is the first term in the asymptotic expansion of the metric to contain dynamical information.

Finally, what about the momentum constraint? It determines the divergence (D^b)(Subscript[(h^((4))), ab]):

(Coefficient[MomentumEqn,(ρ^4)]/.AsympSoln)//Simp

Apply the rules for Weyl-flat boundary metric:

%/.WeylFlatRules//Simp

Solve[%==0, CD[-a][δh[LI[4],-b,a]]][[1,1,2]]//Simp

((%)/(4!))//Simp

This isn' t the exact form you see quoted in the literature -- the terms can be simplified a bit more using other identities -- but it is equivalent.