Slowly rotating Kerr–de Sitter black holes

Given Schwarzschild–de Sitter parameters b0=(M,0,0)∈B and the smooth manifold (3.9) with time functiont_∗∈C^∞(M), we now proceed to define the Kerr–de Sitter fam-ily of metrics, depending on the parameters b∈B, as a smooth family gb of stationary Lorentzian metrics onMforb close tob0.

Given the angular momentuma=|a|of a black hole of massM(spinning around the axisa/|a|∈R³fora6=0), the Kerr–de Sitter metric with parametersb=(M,a) inBoyer–

Lindquist coordinates(t, r, φ, θ)∈R×Ib×S¹φ×(0, π), withIb⊂Ran interval defined below, takes the form

gb=−%²_b dr²

˜ µ_b +dθ²

b

− bsin²θ

(1+λ_b)²%²_b(a dt−(r²+a²)dφ)² + µ˜b

(1+λ_b)²%²_b(dt−a(sin²θ)dφ)²,

(3.12)

where

˜

µb(r) = (r²+a²) 1−¹₃Λr²

−2Mr, %²_b=r²+a²cos²θ, λb=¹₃Λa², b= 1+λbcos²θ.

(3.13) For a=0, we have ˜µb(r)=r²µb(r), with µb defined in (3.4). For a6=0, the spherical coordinates (φ, θ) are chosen such that the vector a/|a|∈S² is defined by θ=0, and the vector field∂φ generates counterclockwise rotation around a/|a|, with R³ carrying the standard orientation. Thus, for a=(0,0, a), a>0, the coordinates (φ, θ) are the standard spherical coordinates ofS²,!R³. We note that, using these standard spherical coordinates, the expression forg_{(M,(0,0,−a))}is given by (3.12) withareplaced by−a: this simply means that reflecting the angular momentum vector across the origin is equivalent to reversing the direction of rotation.

Lemma 3.3. Let r_b0,−<r_b0,+denote the unique positive roots of µ˜_b0. Then,for b in an open neighborhood b₀∈U_B⊂B, the largest two positive roots

r_b,−< r_b,+

of µ˜b depend smoothly on b∈UB. In particular, for b near b0,we have

|rb,±−rb₀,±|< εM, and so r_b,±∈I, with I defined in (3.9).

Proof. This follows from the simplicity of the roots rb₀,± of ˜µb₀ and the implicit function theorem.

The interval Ib in which the radial variable r of the Boyer–Lindquist coordinate system takes values is then

Ib= (rb,−, rb,+).

The coordinate singularity of (3.12) is removed by a change of variables

t_∗=t−Fb(r), φ_∗=φ−Φb(r), (3.14) whereFb,Φb are smooth functions on (rb,−, rb,+) such that

F_b⁰(r) =±

(1+λ_b)(r²+a²)

˜ µb

+c_b,±

, Φ⁰_b(r) =±

(1+λ_b)a

˜ µb

+˜c_b,±

(3.15) nearr=rb,±, withcb,± and ˜cb,± smooth up torb,±. Here, forb=b0, we take

cb₀,−∈ C^∞((rI,−, rb₀,+)) and cb₀,+∈ C^∞((rb₀,−, rI,+)),

withr_I,± defined in (3.9), to be equal to any fixed choice for the Schwarzschild–de Sitter space (M, g_b0), e.g. the one in Lemma3.1.

Lemma 3.4. Fix radii r1 and r2 with rb₀,−+εM<r1<r2<rb₀,+−εM. Then, there exists a neighborhood b0∈UB⊂B such that the following holds:

(1) There exist smooth functions

U_B×(r_I,−, r₂)3(b, r)7−!c_b,−(r), U_B×(r₁, r_I,+)3(b, r)7−!c_b,+(r),

which are equal to the given c_b0,± for b=b₀,such that the two functions

±

(1+λ_b)(r²+a²)

˜ µb

+c_b,±

agree on (r₁, r₂).

(2) There exist functions

UB×(rI,−, r2)3(b, r)7−!c˜b,−(r), UB×(r1, rI,+)3(b, r)7−!c˜b,+(r), with a⁻¹c˜b,± smooth, and c˜b₀,±≡0,such that the two functions

±

(1+λb)a

˜

µ_b +˜cb,±

agree on (r1, r2).

Proof. We can takecb,−≡cb₀,− on (rI,−, r2); then, for a cutoffχ∈C^∞(R), withχ≡1 on [r1, r2] andχ≡0 on [rb₀,+−εM, rI,+), we put

cb,+=−

2(1+λb)(r²+a²)

˜

µ_b +cb₀,−

χ+cb₀,+(1−χ).

A completely analogous construction works for ˜c_b,±: we can take ˜c_b,−≡0 and put

˜

c_b,+=−2(1+λ_b)a

˜ µb

χ.

Clearly, the functionsa⁻¹˜c_b,± depend smoothly onb.

This lemma ensures that the definitions on F_b⁰ and Φ⁰_b in the two regions in (3.15) coincide, hence makingF_b and Φ_b well-defined up to an additive constant. Using

aF_b⁰−(r²+a²)Φ⁰_b=±(ac_b,±−(r²+a²)˜c_b,±)

and

F_b⁰−a(sin²θ)Φ⁰_b=±

(1+λ_b)%²_b

˜ µb

+cb,±−a(sin²θ)˜cb,±

forr>r1(+ sign) or r<r2 (−sign), one now computes gb=− bsin²θ

(1+λb)²%²_b(a(dt_∗±cb,±dr)−(r²+a²)(dφ_∗±˜cb,±dr))² + µ˜b

(1+λb)%²_b(dt∗±cb,±dr−a(sin²θ)(dφ∗±˜cb,±dr))²

± 2 1+λb

(dt∗±cb,±dr−a(sin²θ)(dφ∗±˜cb,±dr))dr−%²_b b

dθ²,

(3.16)

which now extends smoothly to (and across)r_b,±. Since one can compute the volume form to be(⁴)|dg_b|=(1+λ_b)⁻²%²_b(sinθ)dt_∗dr dφ_∗dθ, the metric g_b in the coordinates used in (3.16) is a non-degenerate Lorentzian metric, apart from the singularity of the spherical coordinates at θ=0, π, which we proceed to discuss: first, we compute the dual metric to be

%²_bGb=−˜µb(∂r∓cb,±∂t∗∓˜cb,±∂φ∗)²

±2a(1+λb)(∂r∓cb,±∂t∗∓˜cb,±∂φ∗)∂φ∗

±2(1+λb)(r²+a²)(∂r∓cb,±∂t∗∓˜cb,±∂φ∗)∂t∗

−(1+λ_b)²

bsin²θ(a(sin²θ)∂t∗+∂φ∗)²−b∂_θ².

(3.17)

Smooth coordinates on S² near the polesθ=0, π are x=sinθcosφ_∗ and y=sinθsinφ_∗, and for the change of variables ζ dφ_∗+η dθ=λ dx+ν dy, one finds sin²θ=x²+y² and ζ=νx−λy, that is,

∂φ_∗=y∂x−x∂y,

and thus the smoothness of%²_bG_bnear the poles follows from writing−btimes the term coming from the last line of (3.17) as

(1+λb)²

sin²θ ∂_φ²_∗+²b∂_θ²= (1+λb)²((sin⁻²θ)∂²_φ∗+∂_θ²)+(b²−(1+λb)²)∂²_θ. Indeed, the first summand is smooth at the poles, since (sin⁻²θ)∂²_φ

∗+∂_θ²=G/ is the dual metric of the round metric onS²in spherical coordinates; and we can rewrite the second summand as

−(2+λ_b(1+cos²θ))λ_b(sin²θ)∂_θ² (3.18)

(⁴) For these calculations, a convenient frame ofT Mis

v1=∂r∓c_b,±∂t∗∓˜c_b,±∂_φ∗, v2=a(sin²θ)∂t∗+∂_φ∗, v3=∂t∗ and v4=∂_θ.

and observe that (sin²θ)∂_θ²=(1−x²−y²)(x∂x+y∂y)²is smooth at (x, y)=0 as well. Since the volume form is given by

|dgb|= (1+λb)⁻²(r²+a²(1−x²−y²))(1−x²−y²)^−1/2dt_∗dr dx dy,

and thus smooth at the poles, we conclude that gb indeed extends smoothly and non-degenerately to the poles.

Using the map

(t_∗, r, φ_∗, θ)7−!(t_∗, rsinθcosφ_∗, rsinθsinφ_∗, rcosθ)∈Rt∗×X⊂M,

withX⊂R³as in (3.10), we can thus push the metricgbforward to a smooth, stationary, non-degenerate Lorentzian metric, which we continue to denote bygb, onM, andgb₀ is equal (pointwise!) to the extended Schwarzschild–de Sitter metric defined in§3.1. The Boyer–Lindquist coordinate patch of the Kerr–de Sitter black hole with parametersb∈B is the subset{rb,−<r<rb,+}⊂M.

Since the choice of spherical coordinates does not depend smoothly onanear a=0, the smooth dependence ofgb, as a family of metrics on M, on b is not automatic; we thus prove the following result.

Proposition 3.5. Let the neighborhood U_B⊂B of b₀ be as in Lemma 3.3. Then, the smooth Lorentzian metric g_b on M depends smoothly on b∈B.

Here, the smoothness of the familygb is equivalent to the statement that the map B×Rt∗×X3(b, t_∗, p)7−!gb(t_∗, p),

with gb(t_∗, p) on the right the matrix of gb at the given point in the global coordinate system (t_∗, p)∈M, is a smooth mapUB×Rt∗×X!R^4×4.

Proof. Given (M,a)∈B witha=|a|6=0, let us denote the spherical coordinate sys-tem onXwith north poleθ=0 ata/|a|by (φ_b,∗, θ_b), so the push-forwards of the functions in (3.13) toM are simply obtained by replacing θ byθ_b. Then, if (r, φ_b,∗, θ_b) are the polar coordinates of a pointp∈X, we have

r=|p|, acosθ_b=

a, p

|p|

, a²sin²θ_b=|a|²−a²cos²θ_b,

where| · |andh ·,· idenote the Euclidean norm and inner product onX⊂R³, respectively.

Since 0∈X, this shows that/ r,acosθ_b anda²sin²θ_b, and hence the push-forwards of ˜µ_b,

%_b,λ_bandbare smooth (inb) families of smooth functions onM, as arec_b,±anda⁻¹˜c_b,±

(which only depend onr), on their respective domains of definition, by Lemma3.4.

We can now prove the smooth dependence of the dual metric Gb onb: in light of the expression (3.17) and the discussion around (3.18), all we need to show is that the vector fields∂t∗, ∂r, a∂φb,∗ and a(sinθb)∂θ_b depend smoothly on a, in particular near a where they are defined to be identically zero. (Note that the 2-tensor in (3.18) is a smooth multiple of a²(sin²θb)∂_θ²

b.) Indeed, this proves that Gb is a smooth family of smooth sections of S²T M, and we already checked the non-degeneracy of Gb at the poles, where the spherical coordinates are singular.

For ∂_t∗ and for the radial vector field ∂_r=|p|⁻¹p∂_p, which do not depend onb, the smoothness is clear. Further, we have

a∂φ_b,∗=∇_a×p at p∈X,

i.e. differentiation in the direction of the vectora×p. Indeed, ifa=a~e3:=(0,0, a), both sides equala(x∂y−y∂x) onR³x,y,z, and ifa∈R³is any given vector andR∈SO(3) is a ro-tation withRa=a~e3,a=|a|, then (R_∗(a∂φb,∗))|p=(a∂φ_a~e3,∗)|_R(p), which we just observed to be equal to∇_a~e3×R(p)=R∗∇a×p.

In a similar vein, one sees that

a(sinθb)∂θ_b=|p|⁻¹∇_p×(p×a) atp∈X,

and the latter expression is clearly smooth ina, finishing the proof of the proposition.

Remark 3.6. If one were interested in analyzing the non-linear stability of Kerr–

de Sitter spacetimes for general parameters—i.e. dropping the assumption of small an-gular momentum—one notes that the construction described in this section can be per-formed in the neighborhood of any Kerr–de Sitter spacetime which is non-degenerate in the sense that the two largest roots of ˜µb are simple and positive.

Given the smooth family of Kerr–de Sitter metricsgbonMdefined in the previous section, we can define its linearization around anyg_b,b∈UB.

Definition 3.7. Forb∈U_Bandb⁰∈T_bB, we define the elementg_b⁰(b⁰) of the linearized Kerr–de Sitter family, linearized aroundg_b, by

g⁰_b(b⁰) = d dsg_b+sb⁰

_s=0

.

Notice here that U_B⊂B⊂R⁴ is an open subset of R⁴, and thus we can identify T_bB=R⁴. The linearization of the Kerr–de Sitter family aroundg_b is the 4-dimensional vector spaceg⁰_b(T_bB)≡{g⁰_b(b⁰):b⁰∈T_bB}.

Remark 3.8. Define db to be the number of parameters needed to describe a lin-earized Kerr–de Sitter metric modulo Lie derivatives (i.e. the number of ‘physical degrees of freedom’); that is,

Γb= g⁰_b(TbB) (ranδ_g^∗

b)∩g⁰_b(TbB), db= dim Γb. Then one can show that

d_b=

4, ifa=0, 2, ifa6=0.

See also the related discussion at the end of [37, §6.2.2]. The reason for db=4 for Schwarzschild–de Sitter parametersbis that slowly rotating Kerr–de Sitter metrics with rotation axes which are far apart are related by a rotation by a large angle. This is one of the reasons to use the redundant (for non-zero angular momenta) parametrization (3.1) of the Kerr–de Sitter family.

Dans le document of the Kerr–de Sitter family of black holes (Page 34-40)