Spherically symmetric perturbations

Next, the casel=0 exists only in the scalar case, and it corresponds to spherical symme-try. In this case, we do not need to assume the metric perturbationhto be a generalized mode; we shall show thatany spherically symmetric perturbation arises by an infinites-imal change of the black hole mass.

The extension of Birkhoff’s theorem on the classification of solutions of Einstein’s equations with spherical symmetry—namely, the fact that the only such solutions are Schwarzschild spacetimes—to positive cosmological constants was done in a particularly simple manner by Schleich and Witt [126]. One needs to check that their arguments work already at the linear level—a priori there may be solutions of the linearized equation that do not correspond to solutions of the non-linear equation—but this is straightforward, as we show in the remainder of this section.

(¹⁰) Roughly, atµ=0, the Schr¨odinger operator is essentially (µDµ)²+VS, which withx=−logµ equalsD²_x+VS, withVS=O(µ)=O(e^−x) exponentially decaying.

(¹¹) See also the arguments around [78, equations (3.29) and (3.32)] for a discussion in the context of asymptotically hyperbolic spaces—the present problem can be regarded as living on 1-dimensional hyperbolic space, given as the interval (r−, r+)rwith the metricµ⁻²dr², whose Laplacian is (µDr)² as in (7.9)

The proof in [126] proceeds by writing the Lorentzian metric, with the negative of our sign convention, in the form

g=F du²+2X du⊗sdv+Y²/g, (7.10) withF, X and Y independent of the spherical variables, which one may always do by a diffeomorphism; on the linearized level, one can similarly bring a metric perturbation into this form by adding aδ^∗_g term. Note in particular that the Schwarzschild–de Sitter metric is locally inr, but globally otherwise, in this form for an appropriate choice oft_∗ (in terms of the definition (3.6) oft_∗, withcb₀,±≡0) with u=t_∗, v=r, and thenX=±1, Y=v,F=¹₃Λv²+2M/v−1. Notice that, by spherical symmetry, a priorigis of the form

g= ˜f d˜u²+2X d˜e u⊗sd˜v+Z d˜e v²+Ye²/g,

with coefficients independent of the spherical variables, i.e. g simply has an additional Z d˜e v² term; in our near-Schwarzschild–de Sitter regime one may even assume (for con-venience only) thatZeis small; then, the coordinate change isv= ˜v,u=U(˜u,v), and con-˜ versely ˜v=v, ˜u=U(u, v), which gives thee dv² component ˜f(∂_vUe)²+2X∂e _vUe+Z, which ise easily solvable for∂_vUe whether ˜f vanishes (sinceXe is near 1) or not (sinceZeis assumed small, so the discriminant of the quadratic equation is positive). Note that, if δUe de-notes the linearized change inUe(relative toUe=u, corresponding to the trivial coordinate change ˜v=v, ˜u=uneeded in the case of the Schwarzschild–de Sitter metric with Ze=0) andδZedenotes the linearized change inZe(relative toZe=0), then, at Xe=1, we get the equation 2∂_vδUe+δZe=0 for the linearized gauge change; this in particular preserves the property of being au-mode.

The gauge term for the linearized equation around Schwarzschild–de Sitter can be seen even more clearly by considering δ_g^∗_0,±, where g0,±, is the Schwarzschild–de Sitter metric in the above form (7.10), thus with v=r,u=t_∗ with an appropriate choice oft_∗, corresponding to takingcb₀,±≡0 in (3.6). See Figure7.1.

Then, acting on aspherical 1-forms (the only ones forl=0; see (7.6)), written in the basisdu anddr, and with output written in the basisdu⊗du, 2du⊗sdr,dr⊗dr,g, and/ moreover writingµ=µ(r) for thedu²component of Schwarzschild–de Sitter, we compute

δ^∗_g0,±=











∂_u+¹₂µ⁰ −¹₂µµ⁰

1

2∂_r ¹₂∂_u−¹₂µ⁰

0 ∂_r

r −rµ









 ,

i⁺

H⁺ H⁺

Figure 7.1. Level sets ofufor which (u, v=r) gives coordinates near the event horizon, away from the cosmological horizon, in which the Schwarzschild–de Sitter metric takes the form (7.10). Analogous coordinates can be chosen near the cosmological horizon, away from the even horizon.

and thus, with tangent vectors written in terms of the basis∂u,∂r,

δ_g^∗_0,±g0,±=











µ∂u ∂u+¹₂µ⁰

1

2µ∂r+¹₂∂u 1 2∂r

∂r 0

0 r









 ,

where the secondg_0,± on the left is the isomorphism from the tangent to the cotangent bundle. This shows that, given anl=0 symmetric 2-tensor, itsdr²component, say ˙Z dr², can be removed by subtractingδ^∗_g0,±g0,± applied to an appropriate multiple f ∂u of ∂u; one simply solves ˙Z=∂rf. Notice that this gauge change preserves mode expansions.

Einstein’s equations(¹²)Ric(g)−Λg=0 for the metric (7.10) are stated in [126, equa-tions (6)–(9)]; for us the important ones are

−∂vX∂vY+X∂_v²Y = 0, (7.11) X²+Y ∂_vF ∂_vY+F(∂_vY)²−2X∂uY ∂_vY−2XY ∂u∂_vY =−ΛX²Y², (7.12) X∂_uF ∂_vY+2XF ∂_u∂_vY−2F ∂uX∂_vY

−X∂vF ∂_uY+2X∂_uX∂_uY−2X²∂_u²Y = 0.

(7.13) (Equation (7.11) is the dv² component of the Einstein equation, equation (7.12) is the spherical part, simplified using (7.11), and lastly(¹³) (7.13) is the du² component, sim-plified by plugging in the expression for Λ from (7.12).)

Now, the linearized version of (7.11), linearized around Schwarzschild–de Sitter, so X=1,Y=v, with dotted variables denoting the linearization, is

−∂_vX˙+∂²_vY˙ = 0, (7.14)

(¹²) Recall that the sign conventions in [126], which we are using presently, are the opposite of what we use in the rest of the paper.

(¹³) There seem to be two typos in [126, equation (9)]: a missing factor F in the second term, and the differentiations in the fifth term are with respect tourather thanv. This does not affect the argument in [126] however, since this equation is only used once one has arranged Y=v andX=±1, hence∂uX=∂vX≡0 and∂u∂vY≡0.

so

X˙−∂vY˙ =ξ(u),

withξindependent of v. With this in mind, it is convenient to further arrange that ing one has Y=v, as one always may do by a diffeomorphism, and thus infinitesimally at Schwarzschild–de Sitter by a δ_g^∗_0,± term. Indeed, we are assuming thatY is near v by virtue of considering deformations of Schwarzschild–de Sitter, soY can be used as a co-ordinate in place ofv, and thus the inverse function theorem is applicable at least locally.

Thusv=V(u, Y) and the form of the metric (with nodv²term) is preserved. Arranging this directly on the linearized level can in fact be done globally (except inr, since our Schwarzschild–de Sitter metric is only local inr): one can remove the ˙Y component (ap-pearing as a coefficient of/g in the linearized metric) by subtractingδ_g^∗_0,±g0,±(2r⁻¹Y ∂˙ r), which preserves mode expansions. Having arranged ˙Y=0, we conclude that ˙X=ξ(u).

In fact, we can use this additional information to further simplify the metric by a simple change of variables to fix the coefficient of du⊗_sdv as 2. Indeed, the du⊗_sdv term is 2X du⊗_sdv=2ξ(u)du⊗_sdv, so changing u appropriately, namely by doing a change of variables ˜u=U(u) withe Ue⁰=ξ, arranges this. In the linearized version, we note that solving∂uf=2 ˙X with f independent of r, and subtractingδ_g^∗_0,±g0,±(f ∂u) from the symmetric 2-tensor removes itsdu⊗sdrcomponent as well. Note that this uses strongly that ˙X=ξ(u): this is what ensures that the∂rderivative inδ_g^∗_0,±g0,±does not give adr² component. Again, notice that this gauge change keeps theu-modes unchanged except the 0-mode, in which case a linear inuterm is generated.

But now, with ˙X=0 and ˙Y=0, the linearization of (7.12) is

v∂vF˙+ ˙F= 0. (7.15)

This says∂_v(vF˙)=0, and hence

F˙=M(u)v⁻¹. Finally, (7.13) becomes

∂_uF˙= 0, (7.16)

and thus M(u)=M is independent of u. Comparing with (3.4), this is exactly the infinitesimal Schwarzschild–de Sitter deformation corresponding to changing the mass.

Thus, one concludes that locally in r, but globally in t_∗, the only solution, without fixing a frequencyσin t_∗, i.e. not working on the Fourier-transformed (int_∗) picture, is linearized Schwarzschild–de Sitter—which is a zero-mode—up to gauge terms, which are mode gauge terms, with possibly linear growth int_∗, as described above.

Now, this argument applies separately in two regions of the formr−−δ <r<r2 and r1<r<r++δ, withr± as in (6.1), wherer−<r1<r2<r+. Thus, in the overlapr1<r<r2, we can write the linearized solution as

˙

g=δ^∗_gθ˙₁+ ˙g_0,−=δ_g^∗θ˙₂+ ˙g_0,+,

where ˙g0,± are the linearized Schwarzschild–de Sitter solutions in the two regions, and where we are using the global Schwarzschild–de Sitter metricg, so theθ±differ, from the ones constructed above by pull-back, by the diffeomorphism that puts g into the form considered above, with vanishingdr² term; notice that this again preserves theu-mode expansions. This, in particular, implies that the ˙g0,± have the same mass parameter M, thus they are equal, that is, ˙g0,±= ˙g0. Then one concludes that δ_g^∗( ˙θ−−θ˙+)=0, i.e.

θ˙−−θ˙+ is the 1-form corresponding to a local Killing vector field which is spherically symmetric. This is necessarily a constant multiple of∂t=∂t∗=∂u, and is thus globally defined. Adding this multiple of the 1-form g(∂t) to ˙θ+ to get ˙θ⁰₊ we then have ˙θ−= ˙θ₊⁰ in the overlap, and thus the 1-form ˙θ, equal to ˙θ− and ˙θ⁰₊in the domains of definition of these two 1-forms, is a well-defined global 1-form. This proves thel=0 case of UEMS.