A 4-dimensional reference linearised Kerr family K

The other class of interesting special solutions which we shall identify corresponds to the 4-dimensional family that arises by linearising 1-parameter representations of Kerr (which of course solves the non-linear equations (75)) around Schwarzschild in an ap-propriate coordinate system. We will present such a family here, giving first in§6.2.1a 1-dimensional linearised Schwarzschild family, and then in§6.2.2a 3-dimensional family corresponding to Kerr with fixed massM.

6.2.1. Linearised Schwarzschild solutions

We begin by reminding the reader that, in view of the pure gauge solutions identified in

§6.1, there is no unique way of identifying a 1-parameter family of linearised Schwarzschild solutions. Thisuniqueness up to pure gauge solutions is reflected in the choice of double null coordinates in which one linearises the 1-parameter Schwarzschild family. A par-ticularly simple such choice is given by writing the 1-parameter Schwarzschild family in rescaled null coordinates

g_M= 4M²

−4

1−1 x

dˆu dˆv+x²dσ₂

, (169)

with x defined via the relation (x−1)e^x=e^{v−ˆˆ u}. Note that, setting r=2M x, u=2Muˆ andv=2Mvˆproduces the metric in standard Eddington– Finkelstein coordinatesuand v. Since the x in (169) does not depend on M at all, the linearisation of (169) in the parameterM is immediate.

One obtains thus a proof of the following proposition (which can alternatively be proven by directly verifying that the system of linearised gravity is satisfied).

Proposition 6.2.1. For every m∈R, the following is a (spherically symmetric) solution of the system of gravitational perturbations (131)–(157)in M:

(1)

ˆ/ g=

(1)

χb=

(1)

χb=⁽¹⁾α=⁽¹⁾α= 0,

(1)

b=⁽¹⁾η=⁽¹⁾η=

(1)

β=

(1)

β= 0,

(1)

(Ω trχ) = Ω⁻²

(1)

(Ω trχ) =⁽¹⁾ω=⁽¹⁾ω=σ= 0, and

2Ω⁻¹

(1)

Ω =−m, tr_/g (1)

/g=−2m, ⁽¹⁾%=−2M r³ ·m,

(1)

K=m r².

We refer to the above 1-parameter family as the reference linearised Schwarzschild solu-tions.

As mentioned, the above proposition exhibits the family of linearised Schwarzschild solutions in a particular gauge.

Remark6.4. With respect to standard Eddington–Finkelstein coordinates (u, v), the Schwarzschild family is given by

gM=−4

1−2M rM

du dv+(rM)²dσ2, (170) with rM defined via (rM/2M−1)e^rM/2M=e^(v−u)/2M. If one linearises (170) with re-spect to the parameterM fixing the (u, v)-differential structure, one obtains the sum of the family of Lemma6.2.1 and the pure-gauge transformation generated by f1=u/2M and f2=v/2M (and f3=f4=j3=j4=0) in (158), the reason being that the coordinate transformation relating (u, v) and (ˆu,v) mentioned above depends onˆ M itself.

6.2.2. Linearised Kerr solutions leaving the mass unchanged

Recall from the discussion in§2.1.2 that the Kerr family can globally be brought into the double null form (115) in its exterior. This was achieved in [59] (see also [18]). One can linearise a 1-parameter representation of the metric in this form with respect to the angular momentum parameter a=εa, to obtain what we shall call the (reference) linearised Kerr solution below. Alternatively one can take a shortcut and start from the Kerr metric expressed in standard Boyer–Lindquist coordinates ignoring all terms quadratic or higher ina:

gKerr=−

1−2M r

dt²+ dr²

1−2M /r+r²(dθ²+sin²θ dφ²)−4M a

r sin²θ dφ dt+O(a²).

One can now introduce the standard Eddington–Finkelstein coordinates (u, v) for the Schwarzschild part and do a coordinate transformationφ7!φ+f˜ (v−u) for an appropriate functionf to bring the metric into the form (115) to first order inεand still read off the metric perturbation. Either of these procedures leads to the (m=0 case of the) following proposition.

Proposition 6.2.2. Let Y_m^`=1, for m=−1,0,1, denote the spherical harmonics (108). For any a∈R, the following is a smooth solution of the system of gravitational perturbations (131)–(157)on M. The non-vanishing metric coefficients are

(1)

b^A= (b^Kerr,m)^A=4Ma

r /ε^AB∂_BY_m^`=1. (171) The non-vanishing Ricci coefficients are

(1)η^A= (η^Kerr,m)^A=3Ma

r² /ε^AB∂BY_m^`=1 and ⁽¹⁾η=η^Kerr,m=−η^Kerr,m. (172) The non-vanishing curvature components are

(1)

β =β^Kerr,m=Ω

rη^Kerr,m,

(1)

b=β^Kerr,m=−β^Kerr,m, ⁽¹⁾σ=σ^Kerr,m= 6

r⁴aM·Y_m<sup>`=1</sup>. We will refer to this3-parameter family spanned by the above solutions (m=−1,0,1) as the reference`=1 linearised Kerr solutions.

Note that the above family may be parameterised by the`=1-modes of the curvature component⁽¹⁾σ.(²²)

Proof. To ease notation, we suppress the superscriptmfor the proof. We first note thatdiv/ b^Kerr=0 andD/^?₂b^Kerr=0, as well as

∂_u(b^Kerr)^A=∂_u 4M a

r³ γ^ACε_C^B∂_BY_m^`=1

=12M aΩ²

r² /ε^AB∂_BY^`=1= 2Ω²(η^Kerr−η^Kerr)^A, where we recall that/g=r²γand thatε_A^B=/ε_A^Bdoes not depend onr. Hence, (131)–(133) all hold.

Since also div/ η^Kerr=div/ η^Kerr=0 andD/^?₂η^Kerr=D/^?₂η^Kerr=0, all null structure equa-tions (135)–(147) hold trivially except (142) and (146). Note that the Codazzi equaequa-tions (145) hold by definition of

(1)

β and

(1)

β in terms of ⁽¹⁾η and⁽¹⁾η. To verify (146), we compute curl/ η^Kerr=/ε^BA∂Bη_A^Kerr=3M a

r² /ε^BA∂B/ε_A^C∂CY_m^`=1

=−3M a

r⁴ ∆_S2Y_m^`=1= 6

r⁴aM·Y_m^`=1=σ^Kerr,

(²²) The scalar⁽¹⁾σ has no`=0 mode, as it satisfies the equation⁽¹⁾σ=curl⁽¹⁾η.

and similarly forη^Kerr. To verify (142), we compute (Ω∇/₄r²η^Kerr)A=∂v(r²η^Kerr_A )−Ω²

r (r²η_A^Kerr) =−Ωβ^Kerr_A , (Ω∇/₃r²η^Kerr)A=∂u(r²η^Kerr_A )+Ω²

r (r²η^Kerr_A ) = Ωβ^Kerr_A .

We finally turn to verifying the Bianchi equations (148)–(157). We first note that the ones for⁽¹⁾α and⁽¹⁾α, as well as those for ⁽¹⁾%, are trivially satisfied. Also,

(Ω∇/₄(r⁴Ω⁻¹β^Kerr))A= (Ω∇/₄(r³η^Kerr))A= 0, (Ω∇/₃(r⁴Ω⁻¹β^Kerr))A= (Ω∇/₃(r³η^Kerr))A= 0,

verifying (149) and (156). It remains to check that (150) and (153)–(155) are satisfied.

For the⁽¹⁾σ equations, we note Ω∇/₄(r³σ^Kerr) =−6

r²aMΩ²Y_m^`=1=−r²σ^KerrΩ²=−Ωr³curl/ β^Kerr,

and similarly for equation (154). We finally verify (150), noting that (155) is verified analogously:

Ω∇/₃(r²Ωβ^Kerr)A= Ω∇/₃(rΩ²η^Kerr)A

=∂u

3M a

r Ω²ε_A^B∂BY_m^`=1

+Ω² r

3M a

r Ω²ε_A^B∂BY_m^`=1

=6M a r² Ω²

1−2M

r

ε_A^B∂_BY_m^`=1−3M a

r³ 2MΩ²ε_A^B∂_BY_m^`=1

=r²Ω²ε_A^B∂Bσ^Kerr−6M

r Ω²η^Kerr_A .

The reader might wonder why the family in Proposition6.2.2is a 3-parameter family of solutions, while the full Kerr metric with fixed mass is a 1-parameter family. This can be explained as follows. When writing down the Kerr metric one fixes an axis of symmetry. Rotations of this axis in space correspond to the same Kerr metric expressed in different coordinates. At the linear level, if we linearised the metric at anon-trivial (a6=0) member of the Kerr family, this would manifest itself in the existence of non-trivial pure gauge solutions corresponding to a rotation of the axis. In contrast, here we are linearising with respect to thespherically symmetric Schwarzschild metric. The associated pure gauge solutions of rotating the axis are then trivial in view of the isometry group of the round sphere. Hence, we must see three “basis” Kerr metrics which cannot be connected by a pure gauge transformation. Note the aforementioned trivial pure gauge solutions are seen as (q₁=0, q₂=Y_m¹) generating the trivial solution in Lemma 6.1.3.

Finally, let us combine the 1-dimensional space of reference linearised Schwarzschild solutions and the 3-dimensional space of reference`=1 linearised Kerrs in the following definition.

Definition 6.1. Letm, s₋₁,s0 ands1 be four real parameters. We call the sum of the solution of Proposition6.2.1with parametermand the solution of Proposition6.2.2 satisfying σ^Kerr=P

msmY_m^`=1 the reference linearised Kerr solution with parameters (m, s−1, s0, s1), and denote it byKm,s_i, or simplyK.

7. The Teukolsky and Regge–Wheeler equations and the gauge invariant

Dans le document The linear stability of the Schwarzschild solution to gravitational perturbations (Page 71-75)