Decoupling of gravitational perturbations

shear. Hence 𝝋 and 𝝋^′ will both satisfy second order decoupled equations if, and only if, 𝜿 = 𝜿^′ = 𝝆 = 𝝆^′ = 0. A natural name for a spacetime admitting such null vector fields seems to be:

Definition 4.2 A spacetime is doubly Kundt if and only if it admits a pair of non- expanding, non-shearing, non-twisting geodesic null vector fields ℓ and 𝑛 with ℓ.𝑛∕= 0.

4.3.2 The Schwarzchild Solution

Consider the special case of the higher-dimensional Schwarzschild solution, which is not Kundt. This solution has 𝝆 = _𝑑−2^𝜌 1 and 𝝉 =0 (a consequence of spherical symmetry).

The latter implies that the terms inFand𝑓 drop out of equation (4.20), leaving us with an equation of the form

(□𝜑)𝑖+ (𝑑−4)

(𝑑−2)²𝜌²𝜑^′_𝑖 = 0, (4.22) where□is a second order differential operator. The second term remains an obstruction to decoupling. For the Schwarzschild solution, the two multiple WANDs have identical properties so we can take the prime of the equation to obtain

(□^′𝜑^′)𝑖+ (𝑑−4)

(𝑑−2)²𝜌^′2𝜑𝑖 = 0, (4.23)

and hence [

□^′ ( 1

𝜌²□𝜑 )]

𝑖− (𝑑−4)²

(𝑑−2)⁴𝜌^′2𝜑𝑖 = 0. (4.24) So in fact𝝋does satisfy a decoupled equation but it is fourth order in derivatives. Note that we had to make use of several special properties of the Schwarzschild solution to obtain this result. It would be interesting to investigate more generally the circumstances under which one can obtain a decoupled equation of higher order for𝝋.

4.4. DECOUPLING OF GRAVITATIONAL PERTURBATIONS 93 The final result will be similar to that of the electromagnetic perturbations; we will find that we can only achieve decoupling when the spacetime is Kundt. We will show that gravitational perturbations of such a Kundt spacetime are described by

(2þ^′þ+k_𝑘k_𝑘+𝜌^′þ−6𝜏_𝑘k_𝑘+ 4Φ− _𝑑−1^2𝑑 Λ) Ω_𝑖𝑗 + 4(

𝜏𝑘k(𝑖∣−𝜏(𝑖∣k𝑘+ Φ^S_(𝑖∣𝑘+ 4Φ^A_(𝑖∣𝑘)

Ω𝑘∣𝑗)+ 2Φ𝑖𝑘𝑗𝑙Ω𝑘𝑙= 0, (4.25) where all quantities exceptΩare evaluated in the background geometry (e.g. Φ denotes Φ⁽⁰⁾ etc.).

In a doubly Kundt spacetime, Ω^′ also will satisfy a decoupled equation, which is given by taking the prime of the above equation.

4.4.2 Derivation of main result

We follow as closely as possible the 4D approach of Stewart & Walker [31]. Many of the equations in this section were checked using the computer algebra package Cadabra [162, 163]. We start by obtaining an equation in which second derivatives act only on Ω_𝑖𝑗. Consider the equations

0 = −k_𝑘Ω_𝑖𝑗 −𝛿_𝑗𝑘k_𝑙Ω_𝑖𝑙+k_𝑗Ω_𝑖𝑘−þ(Ψ_𝑖𝛿_𝑗𝑘+ Ψ_𝑖𝑗𝑘) +𝛿_𝑗𝑘(Φ_𝑙𝑖−2Φ_𝑖𝑙−Φ𝛿_𝑖𝑙)𝜅_𝑙 + (−2Φ𝑖[𝑘∣𝛿𝑗]𝑙+ 2𝛿𝑖𝑙Φ^A_𝑘𝑗 + Φ𝑖𝑙𝑘𝑗)𝜅𝑙+𝛿𝑗𝑘[−Ψ𝑖𝜌−𝜌𝑖𝑙Ψ𝑙−(Ψ𝑚𝑖𝑙+ Ψ𝑖𝑚𝑙)𝜌𝑙𝑚]

+ 2(Ψ[𝑘∣𝛿𝑖𝑙+ Ψ𝑖𝛿[𝑘∣𝑙+ Ψ𝑖[𝑘∣𝑙+ Ψ[𝑘∣𝑖𝑙)𝜌𝑙∣𝑗]+ (Ω𝑖𝑙𝜏_𝑙^′𝛿𝑗𝑘−Ω𝑖𝑘𝜏_𝑗^′+ Ω𝑖𝑗𝜏_𝑘^′) (4.26) and

0 = −2þ^′Ω_𝑖𝑗 +k_𝑘(Ψ_𝑖𝛿_𝑗𝑘+ Ψ_𝑖𝑗𝑘) +(

−Ω_𝑖𝑗𝜌^′+ 2Ω_𝑖𝑘𝜌^′_[𝑗𝑘])

−4(Ψ_(𝑖𝛿_𝑗)𝑘+ Ψ_{(𝑖𝑗)𝑘})𝜏_𝑘

+ Φ_𝑗𝑘𝜌_𝑖𝑘−Φ_𝑘𝑗𝜌_𝑖𝑘+ Φ_𝑖𝑘𝜌_𝑗𝑘−Φ_𝑘𝑖𝜌_𝑘𝑗+ 2Φ_𝑖𝑘𝜌_𝑘𝑗−Φ_𝑖𝑗𝜌+ Φ_{𝑖𝑘𝑗𝑙}𝜌_𝑘𝑙+ Φ𝜌_𝑖𝑗. (4.27) Equation (4.26) is obtained by taking various linear combinations and contractions of the Bianchi equations (B1), while equation (4.27) is constructed from the symmetric part of (B2) and a contraction of (B3). These equations are exact: no decomposition into background and perturbation has been performed at this stage.

Now we consider the linear combination k_𝑘(4.26) +þ(4.27). This contains second derivatives acting onΩand onΨ. However, the point of taking this particular combina- tion is that the second derivatives ofΨ occur in the combination −[þ,k𝑘](Ψ𝑖𝑗𝑘+ Ψ𝑖𝛿𝑗𝑘) and therefore can be eliminated in favour of terms involving one or zero derivatives of Ψusing the formula (C2) for the commutator [þ,k_𝑘].

We can also symmetrize the entire equation on 𝑖𝑗 without losing any useful infor- mation, as the antisymmetric terms do not contain any second derivatives of Ω. This

reduces the equation to

0 = −(2þ^′þ+k_𝑘k_𝑘)Ω_𝑖𝑗 −2[þ,þ^′]Ω_𝑖𝑗 −[k_(𝑖∣,k_𝑘]Ω_𝑘∣𝑗)+þ(𝑇_{𝑖𝑗𝑘𝑙}^Ω 𝜌^′_𝑘𝑙)−k_𝑙(𝑇_{𝑖𝑗𝑘𝑙}^Ω 𝜏_𝑘^′) +[þ,k𝑘]𝑇_𝑖𝑗𝑘^Ψ −4þ(𝑇_𝑖𝑗𝑙^Ψ𝜏𝑙)−2k(𝑖∣(𝑇_{∣𝑗)𝑙𝑘}^Ψ 𝜌𝑘𝑙) + 2k𝑙(𝑇_{(𝑗∣𝑙𝑘}^Ψ 𝜌𝑘∣𝑖))−2k𝑙(𝑇_𝑖𝑗𝑘^Ψ𝜌𝑘𝑙)

+þ(𝑇_{𝑖𝑘𝑗𝑙}^Φ 𝜌𝑘𝑙)−k𝑙(𝑇_{𝑖𝑘𝑗𝑙}^Φ 𝜅𝑘) (4.28)

where

𝑇_{𝑖𝑘𝑗𝑙}^Φ ≡ Φ(𝑖∣𝑘∣𝑗)𝑙+ Φ𝛿(𝑖∣𝑘𝛿∣𝑗)𝑙−Φ^S_𝑖𝑗𝛿𝑘𝑙+ (2Φ(𝑖∣𝑙−Φ𝑙(𝑖∣)𝛿𝑘∣𝑗)+ (2Φ(𝑖∣𝑘−Φ𝑘(𝑖∣)𝛿𝑙∣𝑗), 𝑇_𝑖𝑗𝑘^Ψ ≡ Ψ_{(𝑖𝑗)𝑘}+ Ψ_(𝑖𝛿_𝑗)𝑘,

𝑇_{𝑖𝑗𝑘𝑙}^Ω ≡ −Ω𝑖𝑗𝛿𝑘𝑙+ Ω(𝑖∣𝑙𝛿𝑘∣𝑗)−Ω(𝑖∣𝑘𝛿𝑙∣𝑗). (4.29) Note that these quantities satisfy the following relations:

𝑇_{𝑖𝑗𝑘𝑙}^Φ =𝑇_{(𝑖∣𝑗∣𝑘)𝑙}^Φ =𝑇_{𝑖(𝑗∣𝑘∣𝑙)}^Φ , 𝑇_{𝑖𝑗𝑖𝑙}^Φ = 0 and 𝑇_{𝑖𝑗𝑘𝑗}^Φ =−(𝑑−2)Φ^S_𝑖𝑘+ Φ𝛿𝑖𝑘, (4.30) 𝑇_𝑖𝑗𝑘^Ψ =𝑇_{(𝑖𝑗)𝑘}^Ψ , 𝑇_𝑖𝑖𝑘^Ψ = 0 and 𝑇_𝑖𝑗𝑖^Ψ = ¹₂𝑑Ψ𝑗 (4.31) 𝑇_{𝑖𝑗𝑘𝑙}^Ω =𝑇_(𝑖𝑗^Ω_)𝑘𝑙, 𝑇_{𝑖𝑗(𝑘𝑙)}^Ω =−Ω𝑖𝑗𝛿𝑘𝑙 𝑇_{𝑖𝑖𝑘𝑙}^Ω = 0 and 𝑇_{𝑖𝑗𝑘𝑘}^Ω =−(𝑑−2)Ω𝑖𝑗. (4.32) In this notation, the parts of (4.26) and (4.27) symmetric on 𝑖𝑗 become

þ𝑇_𝑖𝑗𝑘^Ψ −k𝑙𝑇_{𝑖𝑗𝑙𝑘}^Ω =−𝑇_{𝑖𝑗𝑙𝑘}^Ω 𝜏_𝑙^′+ 2𝑇_{(𝑖∣𝑘𝑙}^Ψ 𝜌𝑙∣𝑗)−2𝑇_𝑖𝑗𝑙^Ψ𝜌𝑙𝑘−2𝑇_{𝑙(𝑖∣𝑚}^Ψ 𝜌𝑚𝑙𝛿𝑘∣𝑗)−𝑇_{𝑖𝑘𝑗𝑙}^Φ 𝜅𝑙 (4.33) and

−k𝑘𝑇_𝑖𝑗𝑘^Ψ + 2þ^′Ω𝑖𝑘 =𝑇_{𝑖𝑘𝑗𝑙}^Φ 𝜌𝑗𝑙−4𝑇_𝑖𝑗𝑘^Ψ𝜏𝑘+𝑇_{𝑖𝑗𝑘𝑙}^Ω 𝜌^′_𝑘𝑙. (4.34) Next we perform the following steps:

1. Use the commutator (C2) to eliminate the terms [þ,k𝑘]𝑇_𝑖𝑗𝑘^Ψ from (4.28) (note that this introduces a new kind of term, of the schematic form 𝜿þ^′Ψ).

2. Expand out the brackets using the Leibniz rule for GHP derivatives.

3. Eliminate the term þ𝑇_𝑖𝑗𝑘^Ψ using equation (4.33).

4. Use the NP equations (NP1), (NP2) and (NP4)^′ to eliminate terms in whichþacts on 𝝆, 𝝉 and 𝝆^′ respectively.

5. Take a linear combination of the Bianchi equations (B2,B3,B4) to get an equation þ𝑇_{𝑖𝑘𝑗𝑙}^Φ = þ^′𝑇_{𝑖𝑗𝑘𝑙}^Ω +k_(𝑖∣Ψ_{𝑙∣𝑗)𝑘}−k_𝑙Ψ_{(𝑖𝑗)𝑘}−𝛿_(𝑖∣𝑘𝛿_∣𝑗)𝑙k_𝑚Ψ_𝑚+𝛿_𝑘𝑙k_(𝑖Ψ_𝑗)

+(−2k𝑙Ψ(𝑖∣+k(𝑖∣Ψ𝑙)𝛿𝑘∣𝑗)+ (−2k𝑘Ψ(𝑖∣+k(𝑖∣Ψ𝑘)𝛿𝑙∣𝑗)+. . . , (4.35) where the ellipsis indicates terms that involve no derivatives. Use this to eliminate þ𝑇_{𝑖𝑘𝑗𝑙}^Φ from (4.28).

4.4. DECOUPLING OF GRAVITATIONAL PERTURBATIONS 95 6. Use a combination of (B5) and (B7) to show that

k𝑙𝑇_{𝑖𝑗𝑘𝑙}^Φ = 3þ^′𝑇_𝑖𝑗𝑘^Ψ + 3𝑇_{𝑖𝑘𝑗𝑙}^Φ 𝜏𝑙+. . . (4.36) where the ellipsis denotes first order terms not involving any derivatives. Use this to eliminate k_𝑙𝑇_{𝑖𝑗𝑘𝑙}^Φ from (4.28).

The resulting equation is very long so we shall not write it out in full. It has the schematic form

(þ^′þ+k.k+ [þ,þ^′] + [k,k] +𝝆^′þ+𝝆þ^′+𝝉k+𝝉^′k+𝝉 𝝉^′ +𝝆𝝆^′+Φ) Ω +𝜿þ^′Ψ+𝝆kΨ+ (𝝉 𝝆+𝝉^′𝜌+𝜿𝜌^′+þ^′𝜿+k𝝆)Ψ

+ (𝝉 𝜿+𝝉^′𝜿+𝝆²)Φ+ (𝜿𝝆)Ψ^′ = 0 (4.37) Here, we neglect terms that are of quadratic order or higher when we decompose quan- tities into a background piece and a perturbation. Recall thatΩ and Ψ are first order quantities. Note that the only terms containing derivatives of Weyl components other than Ωare of the schematic form 𝜿þ^′Ψ and 𝝆kΨ. For decoupling to occur, these must vanish for any possible perturbation. We shall now examine the circumstances under which we can eliminate these terms.

The detailed form of the 𝜿þ^′Ψ terms is

4𝜅_𝑘þ^′(Ψ_{(𝑖𝑗)𝑘}+ Ψ_(𝑖𝛿_𝑗)𝑘) (4.38)

If𝜿⁽⁰⁾ ∕=0 then there is nothing we can do to eliminate these terms. The only Bianchi equation containingþ^′Ψis (B5), and using this again would reintroduce the 1-derivative terms that we have eliminated above. Hence the only way for these terms to drop out is for𝜿to vanish in the background. Hence𝜿⁽⁰⁾ =0is a necessary condition for decoupling.

Henceforth we assume𝜿is a first-order quantity, in which case the above terms become second order terms and can be neglected.

Recall that 𝜿⁽⁰⁾ = 0 is equivalent to the statement that ℓ is geodesic in the back- ground. By Theorem 3.2, this places no further restrictions on the spacetimes that can be analysed.

Having set 𝜿⁽⁰⁾ =0, the only remaining terms involving derivatives of Weyl compo- nents other than Ωare of the form 𝝆kΨ. The detailed form of these terms is:

4𝜌(𝑖∣𝑘k𝑘Ψ∣𝑗)+𝜌𝑘𝑙[

2k𝑙Ψ(𝑖𝑗)𝑘+k(𝑖∣Ψ𝑙∣𝑗)𝑘−k(𝑖Ψ𝑗)𝑘𝑙+ 2k𝑘Ψ(𝑖𝑗)𝑙−k(𝑖∣Ψ𝑘∣𝑗)𝑙] +𝜌_𝑘(𝑖∣[

−k_𝑙Ψ_{∣𝑗)𝑙𝑘}−k_𝑙Ψ_{𝑙∣𝑗)𝑘}−k_∣𝑗)Ψ_𝑘+ 2k_𝑘Ψ_∣𝑗)]

(4.39) For decoupling we need to eliminate these terms in favour of terms in which derivatives act only on Ω.

Certain combinations of terms of the formkΨcan be eliminated using Bianchi equa- tions. In order to understand precisely what kinds of terms can be so eliminated, we can decompose k_𝑖Ψ_𝑗𝑘𝑙 into parts that transform irreducibly under 𝑆𝑂(𝑑−2). If we do the same for the Bianchi equations at our disposal (or combinations of them such as (4.27)) then we will see which irreducible parts of kΨcan be eliminated from the above equation.

Decomposing into tracefree and trace parts gives, for 𝑑 >4:

k𝑖Ψ𝑗𝑘𝑙 =𝑉𝑖𝑗𝑘𝑙+ 2𝛿𝑖[𝑘∣𝑊𝑗∣𝑙]+𝛿𝑖𝑗𝑋𝑘𝑙+ 2𝛿𝑗[𝑘∣𝑌𝑖∣𝑙]+ 2𝛿𝑖[𝑘∣𝛿𝑗∣𝑙]𝑍, (4.40) where𝑉_{𝑖𝑗𝑘𝑙} is traceless and satisfies 𝑉_{𝑖[𝑗𝑘𝑙]}=𝑉_{𝑖𝑗(𝑘𝑙)} = 0. The other terms are given by

𝑊_[𝑖𝑗] = 1

2𝑋_𝑖𝑗 = 1 𝑑(𝑑−4)

(−(𝑑−3)k_𝑘Ψ_{[𝑖𝑗]𝑘}+k_[𝑖Ψ_𝑗])

, (4.41)

𝑌_[𝑖𝑗] = 1 𝑑(𝑑−4)

(3k_𝑘Ψ_{[𝑖𝑗]𝑘}−(𝑑−1)k_[𝑖Ψ_𝑗])

, (4.42)

𝑊(𝑖𝑗)= 1 (𝑑−2)(𝑑−4)

(−(𝑑−3)k𝑘Ψ(𝑖𝑗)𝑘+k(𝑖Ψ𝑗)−k𝑘Ψ𝑘𝛿𝑖𝑗)

, (4.43)

𝑌_(𝑖𝑗)= 1 (𝑑−2)(𝑑−4)

(k_𝑘Ψ_{(𝑖𝑗)𝑘}−(𝑑−3)k_(𝑖Ψ_𝑗)+k_𝑘Ψ_𝑘𝛿_𝑖𝑗)

, (4.44)

𝑍 = 1

(𝑑−2)(𝑑−3)k𝑘Ψ𝑘. (4.45)

Note that𝑊_(𝑖𝑗) and 𝑌_(𝑖𝑗) are traceless and 𝑋_(𝑖𝑗)= 0.

The traceless part 𝑉𝑖𝑗𝑘𝑙 can be decomposed further into parts that transform irre- ducibly under𝑆𝑂(𝑑−2). The relevant irreducible representations correspond to Young tableaux with 4 boxes. However, it turns out that we will not need to discuss these. As well as these quantities, we have two independent quantities transforming as , namely 𝑊_[𝑖𝑗] and 𝑌_[𝑖𝑗], two quantities transforming as , namely 𝑊_(𝑖𝑗) and 𝑌_(𝑖𝑗), and a singlet 𝑍.

Consider first the singlet𝑍. The contribution of this to equation (4.39) is 4(𝑑−3)𝜎_𝑖𝑗𝑍 = 4

𝑑−2𝜎_𝑖𝑗k_𝑘Ψ_𝑘, (4.46) where the shear 𝝈 is the traceless symmetric part of 𝝆. In order to achieve decoupling, we would need to add to (4.39) a combination of Bianchi components containing a singlet term that cancelled this, and did not introduce any 1-derivative terms (e.g. þΦ terms) that we have already eliminated. However, there is no such combination. For example, the singlet drops out of equation (4.27). Therefore, the only way to eliminate the singlet term from our equation, as required for decoupling, is to set 𝝈⁽⁰⁾ = 0. This is the

4.4. DECOUPLING OF GRAVITATIONAL PERTURBATIONS 97 condition that, in the background geometry, the shear of the multiple WAND ℓ must vanish. Henceforth we assume that this is the case.

Next consider the traceless symmetric tensors 𝑊_(𝑖𝑗) and 𝑌_(𝑖𝑗) that arise in the above decomposition ofk_𝑖Ψ_𝑗𝑘𝑙. The contribution of these to (4.39) is:

−5𝜌(𝑊(𝑖𝑗)+𝑌(𝑖𝑗))+¹₂(𝑑−10)(

𝑊(𝑖𝑘)𝜔𝑗𝑘+𝑊(𝑗𝑘)𝜔𝑖𝑘)

−³₂(𝑑−2)(

𝑌(𝑖𝑘)𝜔𝑗𝑘+𝑌(𝑗𝑘)𝜔𝑖𝑘)

, (4.47) where𝜔_𝑖𝑗 ≡𝜌_[𝑖𝑗].

Now consider again the Bianchi equations. The only combination of equations involv- ing 𝑊(𝑖𝑗) and 𝑌(𝑖𝑗) that does not introduce any 1-derivative terms that we have already eliminated is (4.34), which gives an expression for

k𝑘𝑇_𝑖𝑗𝑘^Ψ ≡ −(𝑑−2)(

𝑊_(𝑖𝑗)+𝑌_(𝑖𝑗))

. (4.48)

We can use this to eliminate, say,𝑌(𝑖𝑗)from (4.47), via the expression𝑌(𝑖𝑗) =−𝑊(𝑖𝑗)+. . ., where the ellipsis denotes terms in which derivatives act only onΩ. Equation (4.47) then reduces to

2(𝑑−4)(

𝑊(𝑖𝑘)𝜔𝑗𝑘+𝑊(𝑗𝑘)𝜔𝑖𝑘)

+. . . . (4.49)

Since we have no independent equation that will allow us to eliminate𝑊(𝑖𝑗), we conclude that in order for thekΨterms to decouple we must have𝝎 =0in the background, i.e., the multiple WANDℓ must be shearfreeand rotation free. Note the factor of 𝑑−4: for 𝑑= 4, vanishing rotation isnot necessary for decoupling.⁴

Having set 𝝈⁽⁰⁾ =𝝎⁽⁰⁾ =0, we find that that the 1-derivative terms (4.39) reduce to 5𝜌

𝑑−2

(k_𝑘Ψ_{(𝑖𝑗)𝑘}+k_(𝑖Ψ_𝑗))

= 5𝜌

𝑑−2k_𝑘𝑇_𝑖𝑗𝑘^Ψ, (4.50) where𝜌≡𝜌𝑖𝑖. These terms can be eliminated from (4.37) with equation (4.34).

In the resulting equation, we now use (NP3) to argue thatk𝑖𝜌is a first order quantity.

It appears only when multiplied byΨ, so such terms are second order and can be dropped.

The only Weyl components that are now acted on by derivatives areΩ, and the equation has been reduced to the schematic form

(þ^′þ+k.k+[þ,þ^′]+[k,k]+𝜌^′þ+𝜌þ^′+𝝉k+𝝉^′k+𝜌𝜌^′+𝝉 𝝉^′+Φ)Ω+𝜌²Φ+𝝉𝜌Ψ= 0 (4.51) At this point, we can also simplify the form of the terms involving Ω, by using the commutators (C1,C3) to eliminate the terms of the form [þ,þ^′]Ωand [k,k]Ωrespectively, in favour of terms that involve at most first derivatives of Ω.

4For𝑑= 4, Ψ_𝑖𝑗𝑘=−2𝛿_𝑖[𝑗Ψ_𝑘], so the irreducible parts ofk_𝑖Ψ_𝑗𝑘𝑙are just the trace, tracefree symmetric and antisymmetric parts of k𝑖Ψ𝑗. Considering the trace gives 𝝈 = 0 as for 𝑑 > 4. The tracefree symmetric part can be eliminated with (4.27). The antisymmetric part simply drops out of (4.39), using the fact that all 2×2 antisymmetric matrices commute.

The terms of the formΦ𝜌are simplified by noting that equation (4.27), evaluated in the background geometry implies that

𝜌⁽⁰⁾Φ⁽⁰⁾_𝑖𝑗 = 1

𝑑−2𝜌⁽⁰⁾Φ⁽⁰⁾𝛿_𝑖𝑗. (4.52) Equation (4.51) now reduces to something sufficiently simple to write out explicitly:

(2þ^′þ+k𝑘k𝑘+𝜌^′þ+ ^𝑑+6_𝑑−2𝜌þ^′ +_𝑑−2² 𝜌𝜌^′−6𝜏𝑘k𝑘+ 4Φ−_𝑑−1^2𝑑 Λ) Ω𝑖𝑗

+(

4𝜏𝑘k(𝑖∣−4𝜏(𝑖∣k𝑘+_𝑑−2² 𝜌(𝜌^′_𝑘(𝑖∣−𝜌^′_(𝑖∣𝑘) + 4Φ^S_(𝑖∣𝑘+ 16Φ^A_(𝑖∣𝑘)

Ω𝑘∣𝑗)+ 2Φ𝑖𝑘𝑗𝑙Ω𝑘𝑙

+ 2𝜌² 𝑑−2

(Φ^S_𝑖𝑗 − _𝑑−2¹ Φ𝛿_𝑖𝑗)

+ 2𝜌𝜏_𝑘(

Ψ_{(𝑖𝑗)𝑘}−Ψ_(𝑖𝛿_𝑗)𝑘+_𝑑−2² 𝛿_𝑖𝑗Ψ_𝑘)

= 0. (4.53) This equation is the analogue of equation (4.20) for the Maxwell field. Note that (4.52) implies that𝜌(Φ^S_𝑖𝑗 −_𝑑−2¹ Φ𝛿𝑖𝑗) is a first order quantity.

To achieve decoupling we have to eliminate the terms not involving Ω𝑖𝑗, i.e., those on the final line of this equation. For 𝑑 = 4, this is automatic since the particular combination of Φ terms appearing in this equation vanishes identically (i.e. Φ^S_𝑖𝑗 = ¹₂Φ𝛿_𝑖𝑗 if 𝑑 = 4), as does the particular combination of Ψ terms. For 𝑑 > 4, the only way of eliminating the Φterms above is to set 𝜌⁽⁰⁾ = 0, i.e., take 𝜌 to be first order. All terms on the final line above are then of higher order and can be neglected.

Hence we see that, for 𝑑 >4, decoupling requires that

𝜿⁽⁰⁾ =0 =𝝆⁽⁰⁾, (4.54)

that is the multiple WAND must be geodesic and free of expansion, rotation and shear.

The existence of such a vector field implies, by definition, that the spacetime is Kundt.

This is a necessary condition for decoupling; it is also sufficient since we now have an equation in which the only perturbed Weyl components that appear areΩ.

The resulting decoupled equation is:

(2þ^′þ+k𝑘k𝑘+𝜌^′þ−6𝜏𝑘k𝑘+ 4Φ− _𝑑−1^2𝑑 Λ) Ω𝑖𝑗

+ 4(

𝜏_𝑘k_(𝑖∣−𝜏_(𝑖∣k_𝑘+ Φ^S_(𝑖∣𝑘+ 4Φ^A_(𝑖∣𝑘)

Ω_𝑘∣𝑗)+ 2Φ_{𝑖𝑘𝑗𝑙}Ω_𝑘𝑙= 0. (4.55) We remind the reader that Ωis a first order quantity, so quantities multiplying Ω(e.g.

Φ,𝝉) must be evaluated in the background geometry.

4.4.3 Comment on the expanding case

Just as we did for Maxwell perturbations, it is interesting to consider what happens if ℓ is geodesic with vanishing rotation and shear, but non-vanishing expansion (i.e. the spacetime is Robinson-Trautman). Under these circumstances, we have equation (4.53),

4.4. DECOUPLING OF GRAVITATIONAL PERTURBATIONS 99 a perturbation equation for a gauge invariant quantity Ω. However, it contains two terms that obstruct the decoupling of the equation. It is interesting to ask how these terms are consistent with gauge invariance. The answer is supplied by:

Lemma 4.3 Letℓbe an expanding, non-twisting, non-shearing geodesic multiple WAND for an Einstein spacetime of dimension 𝑑 >4. Then

(Φ^S_𝑖𝑗 − _𝑑−2¹ Φ𝛿_𝑖𝑗)₍₁₎

(4.56) is a gauge invariant quantity. If 𝝉⁽⁰⁾ ∕=0, then

Ψ⁽¹⁾_𝑖 and 𝜏_𝑘⁽⁰⁾Ψ⁽¹⁾_𝑖𝑗𝑘 (4.57) also are gauge invariant quantities.

The Schwarzschild black hole in arbitrary dimension is an example of a spacetime ad- mitting such a multiple WAND (although in this case, 𝝉⁽⁰⁾ = 0). In four dimensions, (4.56) vanishes identically in all spacetimes, while the quantities (4.57) are not gauge invariant.

Proof: From equation (4.52) we have

Φ⁽⁰⁾_𝑖𝑗 = _𝑑−2¹ Φ⁽⁰⁾𝛿𝑖𝑗 (4.58) in any such spacetime. Hence we see immediately that (4.56) is invariant under infinites- imal coordinate transformations, and also under infinitesimal spins. Furthermore, Ref.

[158] showed that all such spacetimes are of algebraic Type D so we can choose our basis so that all Weyl tensor components with non-zero boost weight vanish. Under an infinitesimal null rotation aboutℓ, equation (2.58) implies that, to first order in𝑧𝑖,

Φ^S_𝑖𝑗 7→Φ^S_𝑖𝑗+𝑧(𝑖Ψ𝑗)−𝑧𝑘Ψ(𝑖𝑗)𝑘, (4.59) but Ψ is a first order quantity and hence Φ^S(1)_𝑖𝑗 and Φ⁽¹⁾ are both invariant in a Type D background. An identical argument applies to null rotations about𝑛, and hence (4.56) is a gauge invariant quantity.

For an algebraically special spacetime, Ψ_𝑖𝑗𝑘 and Ψ_𝑖 both vanish in the background, and so, to first order, they are invariant under infinitesimal spins and infinitesimal coor- dinate transformations. They are also invariant under infinitesimal null rotations about ℓ, as these can only introduce terms involving Ω𝑖𝑗 which also vanishes in the background.

We now consider the effect of an infinitesimal null rotation about 𝑛. Taking the prime of (2.61) implies that, to linear order,

Ψ𝑖𝑗𝑘7→Ψ𝑖𝑗𝑘+_𝑑−2² Φ⁽⁰⁾𝛿𝑖[𝑗𝑧𝑘]+𝑧𝑙Φ⁽⁰⁾_{𝑙𝑖𝑗𝑘} (4.60)

and

Ψ_𝑗 7→Ψ_𝑗 − ^𝑑−1_𝑑−2Φ⁽⁰⁾𝑧_𝑗, (4.61) where we have used (4.58). We will show that the quantities (4.57) are invariant under this transformation if𝜏_𝑖⁽⁰⁾ ∕= 0.

Take a double trace of the Bianchi equation (B7) for the background spacetimes.

This implies that (𝑑−4)k_𝑘Φ⁽⁰⁾ = 0, and hence, for𝑑 >4, k_𝑘Φ⁽⁰⁾ = 0. The trace of (B5) gives

k𝑗Φ⁽⁰⁾= ^𝑑−1_𝑑−3𝜏_𝑗⁽⁰⁾Φ⁽⁰⁾, (4.62) and hence Φ⁽⁰⁾ = 0 if 𝜏_𝑖⁽⁰⁾ ∕= 0. From (4.58) we then have Φ⁽⁰⁾_𝑖𝑗 = 0. Putting these results back into (B5) implies that Φ⁽⁰⁾_{𝑖𝑗𝑘𝑙}𝜏_𝑙⁽⁰⁾ = 0. Inserting these results into (4.60,4.61) implies that, although Ψ𝑖𝑗𝑘 is not invariant under infinitesimal null rotations about 𝑛, both 𝜏𝑘Ψ𝑖𝑗𝑘 and Ψ𝑖 are invariant, and hence both of these are new gauge invariant quantities, provided that 𝑑 >4 and𝜏_𝑖⁽⁰⁾ ∕= 0. □

In document New approaches to higher-dimensional general relativity (página 104-112)