Mode stability for gravitational instantons of type D

We study Ricci-flat perturbations of gravitational instantons of Petrov type D. Analogously to the Lorentzian case, the Weyl curvature scalars of extreme spin weight satisfy a Riemannian version of the separable Teukolsky equation. As a step toward infinitesimal rigidity of the type D Kerr and Taub-bolt families of instantons, we prove mode stability, i.e. that the Teukolsky equation admits no solutions compatible with regularity and asymptotic (local) flatness.


Introduction
A gravitational instanton is a complete and non-compact Ricci-flat Riemannian four-manifold with quadratic curvature decay.There are a number of families of known examples, such as the Riemannian Kerr instanton and the Taub-NUT 1 and Taub-bolt instantons.Furthermore, there are some known general results about gravitational instantons, many of which hold under various symmetry assumptions, such as the existence of a U (1) or U (1) × U (1) isometry group, see e.g.[2,12,7,1].In the compact case, we have the Besse conjecture [6], stating that all compact Ricci-flat manifolds have special holonomy.This is a wide-open conjecture; there are no known examples of compact Ricci-flat four-manifolds with generic holonomy, i.e., holonomy group SO (4).This is in contrast to the non-compact case since there are examples of gravitational instantons with generic holonomy.However, all known examples still satisfy the weaker requirement of hermiticity, and it is, therefore, natural to conjecture that all gravitational instantons are Hermitian, see [2] where this conjecture is stated for the ALF case.A first step toward such a result is given by proving rigidity, i.e., for various known examples of gravitational instantons, showing that there are no other Ricci-flat metrics close to that metric.
In the recent paper [8], it was shown that rigidity holds for the Riemannian Kerr and Taub-bolt families.However, infinitesimal rigidity, i.e., that Ricci-flat linear perturbations of these instantons decaying sufficiently fast at infinity must be perturbations within the respective family, is still open.Due to the fact that rigidity holds and that the relevant families are smooth manifolds, infinitesimal rigidity is equivalent to integrability, i.e., that any such Ricci-flat linear perturbation integrates to a curve of Ricci-flat metrics.
It was shown in [17], that for perturbations of the Lorentzian Kerr metric whose frequency lies in the upper half plane, and satisfying certain boundary conditions, the perturbations of the The Newman-Penrose (NP) formalism [11], commonly used in general relativity, can be adapted to a Riemannian signature (see [3,10]).Let (l, l, m, m) be a tetrad of vector fields with complex coefficients, in which the metric has the form When viewed as first order differential operators, we denote 3 the vector fields l, l, m, m by D, ∆, δ, − δ, respectively.With respect to the tetrad (l, l, m, m), the Levi-Civita connection is represented by 24 spin coefficients, denoted by Greek letters and defined to be the coefficients in the right-hand sides of the equations From the fact that all inner products of the tetrad vectors are constant, it can also be seen that The Levi-Civita connection is thus represented by 6 + 6 independent complex scalars.We also have the Weyl scalars: where W denotes the Weyl curvature tensor.
A tetrad (l, l, m, m) is said4 to be adapted if Ψ 0 = Ψ 1 = Ψ0 = Ψ1 = 0 and Ψ 2 , Ψ2 = 0.It can be shown that a Ricci-flat four-manifold admits an adapted tetrad if and only if it has type D. A proof of this fact in the Lorentzian case can be found in [13,Chapter 7].

The Perturbation Equations
When referring to a perturbation ġ of a metric g, we are referring to a linear perturbation of g, i.e. a symmetric two-tensor ġ.When g is Ricci-flat, we say that ġ is a Ricci-flat perturbation if it is Ricci-flat to first order, i.e. if ġ ∈ ker((D Ric) g ).In general, for a quantity depending on the metric g, we let a dot above the quantity denote its derivative in the direction ġ.Then ġ is a Ricci-flat perturbation if and only if Ṙic = 0.
Ricci-flat perturbations of the Lorentzian Kerr metric have been studied extensively, and in [15], Teukolsky derived a well-known equation for the perturbation of the Weyl scalars of extreme spin weight, for such perturbations of the metric.The following theorem gives a Riemannian analog of that perturbation equation.Theorem 3. Consider a Ricci-flat perturbation ġ of a Ricci-flat type D metric g.For an adapted tetrad, the perturbation Ψ0 satisfies the equation (12) and the perturbation Ψ0 satisfies the equation Proof.Since we have an adapted tetrad, Ψ 0 = Ψ 1 = Ψ0 = Ψ1 = 0, and using (89) and (91) along with their tilded versions, we also have κ = κ = σ = σ = 0.The linearized versions of (91) and (89) become and respectively.Operating on (14) with D and on (15) with δ, subtracting the resulting equations and using the commutation relation (76), we get We eliminate the first term on the right: where and By using (80), ( 85) and (88), we see that A 1 = 0.For an adapted tetrad, (90) and (92) become Therefore, by the Leibniz rule, where we used the linearization of (84) in the last step, showing that (12) holds.The proof of ( 13) is similar, referring to the tilded versions of the NP equations instead.

The Riemannian Kerr Instanton
In Boyer-Lindquist coordinates (t, r, θ, φ), the Riemannian Kerr family of metrics is given by the expression Here, M > 0 and a ∈ R are the parameters of the family, ∆ = ∆(r) = r 2 − 2M r − a 2 and Σ = r 2 − a 2 cos 2 θ, and the coordinates have the ranges r > r + , 0 < θ < π, where r ± = M ± √ M 2 + a 2 are the roots of ∆.Like its Lorentzian counterpart, this metric is Ricci-flat.Now define new coordinates ( t, r, θ, φ) by where and Ω = a 2Mr+ .Then r is a smooth function of r2 , and (25) gives Letting (r, t) be polar coordinates on R 2 and letting (θ, φ) be spherical coordinates on S 2 , it follows that g extends to a complete metric on R 2 × S 2 , provided that we identify t and φ with period 2π independently.Note that this is equivalent to performing the identifications (t, φ)

The Separated Perturbation Equations in Coordinates
We shall be interested in a particular choice of complex null tetrad (l, l, m, m), called the Carter tetrad, defined by Note that |l| g = |m| g = 1.The spin coefficients for the Carter tetrad are given explicitly in Section B.1.For this tetrad, we have and all other Weyl scalars vanish.In particular, this is an adapted tetrad.
We shall now analyze the perturbation equations in the Carter tetrad.The relevant properties of the equations are given in the following four lemmas.Lemma 1.For the Carter tetrad, the perturbation equation ( 12) is equivalent to the equation6 LΦ = 0, where Φ = Ψ Furthermore, if Φ is a solution to this equation coming from a perturbation of the metric, then we can write where m runs over Z, ω runs over Ω + κZ and for each choice of m, ω, Λ, the function R = R m,ω,Λ solves the equation RR = 0.The function S = S m,ω,Λ is the unique solution to the boundary value problem SS = 0, S ′ (0) = S ′ (π) = 0, where and Here, S is normalized with respect to the L 2 product with measure sin θ dθ, and the separation constant Λ runs over the (countable set of) values for which such an S exists.The same statement holds for the perturbation equation (13), if Φ is replaced by Φ, where Proof.The fact that ( 12) is equivalent to LΦ = 0 follows from a direct computation, using the expressions for the spin coefficients in Section B.1.Now note that the boundary value problem SS = 0, S ′ (0) = S ′ (π) = 0 is a Sturm-Liouville problem.Thus, there exists an orthonormal L 2 basis of functions {S m,ω,Λ } Λ solving it, and furthermore, we can perform a Fourier series decompositions in the coordinates (t, φ).From these considerations, we can write (32), where The fact that R = R m,ω,Λ satisfies RR = 0 now follows directly from (37), along with the fact that LΦ = 0.For the statement involving Φ, the proof is entirely analogous.
Lemma 2. The equation RR = 0 is an ordinary differential equation in a complex variable r, which has regular singular points at r = r ± .The point r = ∞ is an irregular singular point of rank 1, except when ω = 0, in which case it is a regular singular point.Thus, the equation RR = 0 is a confluent Heun equation (see [14,Section 3]) when ω = 0, and a hypergeometric equation (see [14,Section 2]) when ω = 0.The characteristic exponents at r = r + are and those at r = r − are When ω = 0, we have Λ ≥ 0, and the characteristic exponents at r = ∞ are When ω = 0, the equation RR = 0 admits normal solutions (see [9,Section 3.2]), near r = ∞, of the asymptotic form R ∼ e ±rω r −1±2(Mω−1) . (41) Proof.The fact that r = r ± are regular singular points follows directly from the fact that ∆ = (r − r + )(r − r − ), the statement about the type and rank of the singular point at r = ∞ follows directly from the discussion in [9, Section 3.1], and the expressions for the characteristic exponents can be seen from the discussion in [14, Section 1.1 where Following [9, Section 3.2], the equation RR = 0 therefore has normal solutions of the asymptotic form R ∼ e ±rω r −1±2(Mω−1) .(44) Lemma 3.For a solution to the equation RR = 0 coming from a (globally smooth) perturbation of the Kerr metric, the corresponding characteristic exponent at r = r + is Proof.Since the set corresponding to r = r + is compact, and by assumption, the perturbation Ẇ of the Weyl tensor is continuous, Ẇ has bounded norm in a neighborhood of this set.Consequently, since l and m have norm 1, it follows that Ψ0 = − Ẇ (l, m, l, m) is bounded near r = r + .Since Ψ −2/3 2 = O(r 2 ), it follows that Φ, and therefore R, is bounded near r = r + .The statement now follows immediately.
A perturbation ġ of the Kerr metric g is said to be asymptotically flat (AF) if | ġ| g = O(r −1 ), with corresponding decay on derivatives, i.e. |∇ k ġ| g = O(r −1−k ) for every positive integer k.Here, r is the radial coordinate of Kerr defined earlier, and the norm and covariant derivative are taken with respect to g. Lemma 4. Let R be a solution to the equation RR = 0 coming from an asymptotically flat perturbation of the Kerr metric.When ω = 0, none of the characteristic exponents at r = ∞ are compatible with the asymptotic flatness assumption.When ω = 0, exactly one of the asymptotic normal solutions is compatible with this assumption, namely R ∼ e −r|ω| r −1−2(Mω−1) sgn(ω) . (46) Proof.By the assumption of asymptotic flatness, we have W = O(r −3 ), which means that Φ, and therefore R, decays as r −1 as r → ∞.In particular, we must have lim r→∞ R(r) = 0, and the result now follows immediately.

Mode Stability
Equipped with the lemmas of the previous subsection, we are now in a position to prove Theorem 1.
Proof of Theorem 1.For r > r + and −1 < x < 1, note that Here, the strict negativity follows from that of the first term, which holds because r + > |a|.By an integration of parts, we have where the last equality follows from (47) and the normalization of S. Multiplying (33) by R and integrating, the first term being integrated by parts, we get We claim that the first term vanishes; to see this, we consider the endpoints separately.Near r = ∞, we have ∆ ∼ r 2 , while R and its derivative decays exponentially.Thus, the term in square brackets In the same way as for the previous coordinate system, this shows that g extends to a smooth metric on another copy of C 2 .Note that the identifications made to ensure regularity in the coordinate system ( t, r, θ, φ) also ensure regularity in the coordinate system ( t, r, θ, φ).When defined on the union of these copies of C 2 , this metric is complete.
Computing the transition map between the two coordinate systems, we see that they are related by ( t, r, θ, φ) = ( t − φ, r, 4  θ , φ).In other words, the two copies of C 2 are glued together according to the map Topologically, this is the same thing as gluing two such copies along the map (z ), or equivalently, gluing together two copies of D 2 × C along the map We now claim that the manifold is diffeomorphic to CP 2 minus a point.To see this, consider two of the projective coordinate charts for CP 2 , (U 0 , φ 0 ) and (U 1 , φ 1 ), where and follows that the latter is topologically equivalent to two copies of C 2 , glued together along the transition map Again, topologically this is the same thing as gluing together two copies of D 2 × C along the map (57).This shows that the manifold is homeomorphic to CP 2 minus a point.To show that these are diffeomorphic, we can replace the closed disk with an open disk of radius slightly larger than 1, gluing the two spaces together along a thin open strip around S 1 .The gluing map will then be isotopic to the corresponding transition map in CP 2 .

The Separated Perturbation Equations in Coordinates
As for Kerr, we are interested in a particular choice of complex null tetrad (l, l, m, m), in this case given by The spin coefficients for this tetrad are given explicitly in Section B.2.For this tetrad, we have and the rest of the Weyl scalars vanish.Thus, this is an adapted tetrad, and we see that the Taub-bolt metric is of type D.
The following four lemmas give the relevant properties of the perturbation equations (12) (13) for our analysis.The proofs are entirely analogous to those in Section 3.1 and are therefore omitted.Lemma 5.For the tetrad given in (59) and (60), the perturbation equation ( 12) is equivalent to the equation LΦ = 0, where Φ = Ψ Furthermore, if Φ is a solution to this equation coming from a perturbation of the metric, then we can write where m runs over 1 2 Z, and ω runs over m + Z, and for each choice of m, ω, Λ, the function R = R m,ω,Λ solves the equation RR = 0, and the function S = S m,ω,Λ solves the boundary value problem SS = 0, S ′ (0) = S ′ (π) = 0, where and The separation constant Λ runs over the (countable set of) values for which such an S exists, all of which are non-negative.The same statement holds if Φ is replaced by Φ = Ψ−2/3 2 Ψ0 , the operator L is replaced by L, and the operator R replaced by R, defined in the same way but using a potential Ũ in place of U . Here, and Lemma 6.The equation RR = 0 is an ordinary differential equation in a complex variable r, which has regular singular points at r = 2N and r = N/2.The point r = ∞ is an irregular singular point of rank 1, except when ω = 0, in which case it is a regular singular point.Thus, the equation RR = 0 is a confluent Heun equation (see [14,Section 3]) when ω = 0, and a hypergeometric equation (see [14,Section 2]) when ω = 0.The characteristic exponents at r = 2N are and those at r = N/2 are When ω = 0, the characteristic exponents at r = ∞ are When ω = 0, the equation RR = 0 admits normal solutions (see [9,Section 3.2]), near r = ∞, of the asymptotic form R ∼ e ±rω/2N r −1±(5ω/4−2) .(73) Lemma 7.For a solution to the equation RR = 0 coming from a (globally smooth) perturbation of the Taub-bolt metric, the corresponding characteristic exponent at r = 2N is |ω − 1|.
A perturbation ġ of the Taub-bolt metric is said to be asymptotically locally flat (ALF) if it decays as O(r −1 ), with corresponding decay on derivatives, just like the definition of AF perturbations of the Kerr metric given in Section 3.1.

Lemma 8.
Let R be a solution to the equation RR = 0 coming from an asymptotically locally flat perturbation of the Taub-bolt metric.When ω = 0, none of the characteristic exponents at r = ∞ are compatible with the assumption of asymptotic local flatness.When ω = 0, exactly one of the asymptotic normal solutions is compatible with this assumption, namely R ∼ e −r|ω|/2N r −1−(5ω/4−2) sgn(ω) . (74) Corresponding lemmas regarding the asymptotics of the equation RR = 0 also hold.We omit them since they are entirely analogous.

Mode Stability
Proof of Theorem 2. In this case, we directly see that U (r) < 0, and integrating the equation RR = 0 by parts like in the proof of Theorem 1, we see that R vanishes identically.The case involving the equation RR = 0 is entirely analogous.

A Newman-Penrose Equations
Given an equation expressed in terms of the spin coefficients, Weyl scalars and tetrad derivative operators, we can apply the tilde operation, given by formally replacing any such quantity x by the tilded quantity x.Here, we adopt the convention that x = x, D = D and ∆ = ∆.The result is a new, a priori independent equation, the tilded version of the original equation.
We have the Newman-Penrose commutation relations, given by the four equations