In this paper, a direct rigorous mathematical proof of the Gregory–Laflamme instability for the five-dimensional Schwarzschild black string is presented. Under a choice of ansatz for the perturbation and a gauge choice, the linearized vacuum Einstein equation reduces to an ordinary differential equation (ODE) problem for a single function. In this work, a suitable rescaling and change of variables is applied, which casts the ODE into a Schrödinger eigenvalue equation to which an energy functional is assigned. It is then shown by direct variational methods that the lowest eigenfunction gives rise to an exponentially growing mode solution, which has admissible behavior at the future event horizon and spacelike infinity. After the addition of a pure gauge solution, this gives rise to a regular exponentially growing mode solution of the linearized vacuum Einstein equation in harmonic/transverse-traceless gauge.

The main topic of this paper is the study of the stability problem for the Schwarzschild black string solution to the Einstein vacuum equation in five dimensions. In 1993, the work of Gregory–Laflamme1 gave strong numerical evidence for the presence of an exponentially growing mode instability. This phenomenon has since been known as the Gregory–Laflamme instability. This work has been widely invoked in the physics community to infer instability of many higher dimensional spacetimes, for example, black rings, ultraspinning Myers–Perry black holes, and black Saturns. The interested reader should consult Refs. 2 and 3 and references therein, as well as Ref. 4 and Refs. 5 and 6, which give a general approach to stability problems. The purpose of the present paper is to provide a direct, self-contained, and elementary mathematical proof of the Gregory–Laflamme instability of the 5D Schwarzschild black string.

The most basic solution to the vacuum Einstein equation

Ricg=0
(1.1)

giving rise to the black hole phenomena is the Schwarzschild black hole solution (Schn, gs). It arises dynamically as the maximal Cauchy development of the following initial data: an initial hypersurface Σ0=R×Sn2, a first fundamental form (in isotropic coordinates)

hs=1+M2ρn34n3(dρdρ+ρ2γn2),ρ(0,)R,
(1.2)

and the second fundamental form K = 0, where γn2 is the metric on the unit (n − 2)-sphere Sn2. This spacetime is asymptotically flat and spherically symmetric. The Penrose diagram in Fig. 1 represents the causal structure of (Schn, gs) arising from this initial data, restricted to the future of Σ0. The metric on the exterior EA (see Fig. 1) of the n-dimensional Schwarzschild black hole in traditional Schwarzschild coordinates (t, r, φ1, …, φn−2) takes the form

gs=Dn(r)dtdt+1Dn(r)drdr+r2γn2,Dn(r)=12Mrn3,
(1.3)

where t ∈ [0, ), r(2M)1n3,, and γn2 is the metric on the unit (n − 2)-sphere.

FIG. 1.

The Penrose diagram of the Schwarzschild spacetime (Schn, gs). Here, I+IA+IB+ is future null infinity, i+iA+iB+ and i0iA0iB0 are future timelike infinity and spacelike infinity, respectively, EAJ(IA+)J+(Σ0) is the distinguished exterior region, EBJ(IB+)J+(Σ0) is another exterior region, BSchn\J(I+) is the black hole region, H+=HA+HB+B\int(B) is the future event horizon, and SHA+HB+ is the bifurcation sphere. The wavy line denotes a singular boundary, which is not part of the spacetime (Schn, gs) but towards which the Kretchmann curvature invariant diverges. It is in this sense that (Schn, gs) is singular. Note that every point in this diagram is, in fact, an (n − 2)-sphere.

FIG. 1.

The Penrose diagram of the Schwarzschild spacetime (Schn, gs). Here, I+IA+IB+ is future null infinity, i+iA+iB+ and i0iA0iB0 are future timelike infinity and spacelike infinity, respectively, EAJ(IA+)J+(Σ0) is the distinguished exterior region, EBJ(IB+)J+(Σ0) is another exterior region, BSchn\J(I+) is the black hole region, H+=HA+HB+B\int(B) is the future event horizon, and SHA+HB+ is the bifurcation sphere. The wavy line denotes a singular boundary, which is not part of the spacetime (Schn, gs) but towards which the Kretchmann curvature invariant diverges. It is in this sense that (Schn, gs) is singular. Note that every point in this diagram is, in fact, an (n − 2)-sphere.

Close modal

The Lorentzian manifold that is the main topic of this paper is the Schwarzschild black string spacetime in five dimensions, which is constructed from the 4D Schwarzschild solution (Sch4, gs). Before focusing on this spacetime explicitly, it is of interest to discuss more general spacetimes constructed from the n-dimensional Schwarzschild black hole solution (Schn, gs). Let SR1 denote the circle of radius R, and let Fp{Rp,Rp1×SR1,,R×i=1p1SRi1,i=1pSRi1} with its associated p-dimensional Euclidean metric δp. If one has the n-dimensional Schwarzschild black hole spacetime (Schn, gs) and takes its Cartesian product with Fp, then one realizes the (n + p)-dimensional Schwarzschild black brane (Schn ×Fp, gsδp). This means that the (n + p)-dimensional Schwarzschild black brane (Schn ×Fp, gsδp) is a product manifold made from Ricci-flat manifolds, which is again Ricci-flat and hence satisfies the vacuum Einstein equation (1.1). Note that in contrast to (Schn, gs), the spacetimes (Schn ×Fp, gsδp) are not asymptotically flat but are called “asymptotically Kaluza–Klein.”

The Schwarzschild black brane spacetimes (Schn ×Fp, gsδp) arise dynamically as the maximal Cauchy development of suitably extended Schwarzschild initial data, i.e., (Σ0 ×Fp, hsδp, K = 0). Hence, the above Penrose diagram in Fig. 1 can be reinterpreted as the Penrose diagram for the Schwarzschild black brane, but instead of each point representing a (n − 2)-sphere, it represents a Sn2×Fp. In particular, the notation EA will be used henceforth to denote the distinguished exterior region of (Schn ×Fp, gsδp).

Taking p = 1 gives rise to the (n + 1)-dimensional Schwarzschild black string spacetime Schn×R or alternatively Schn×SR1. The topic of the present paper is the 5D Schwarzschild black string spacetime Sch4×R or alternatively Sch4×SR1. The metric on the exterior EA in standard Schwarzschild coordinates is

gD(r)dtdt+1D(r)drdr+r2γ2+dzdz,D(r)=12Mr,
(1.4)

where t ∈ [0, ), r ∈ (2M, ), and zR or R/2πRZ.

Finally, to analyze the subsequent problem of linear stability on the exterior region EA up to the future event horizon HA+, one requires a chart with coordinate functions that are regular up to this hypersurface HA+\S, where S now denotes the bifurcation surface. A good choice is ingoing Eddington–Finkelstein coordinates defined by

v=t+r*,dr*dr=rn3rn32M,with r*(3M)=3M+2Mlog(M).
(1.5)

The (n + p)-dimensional Schwarzschild black brane metric becomes

gsδ=Dn(r)dvdv+dvdr+drdv+r2γn2+δijdzidzj,Dn(r)=12Mrn3.
(1.6)

For a good introduction to the Gregory–Laflamme instability and the numerical result of Ref. 1, see Ref. 7. A detailed survey of the key work8 related to the present paper is undertaken in Sec. III. A brief history of the problem is presented here:

  1. In 1988, Gregory–Laflamme examined the Schwarzschild black string spacetime and stated that it is stable.9 However, an issue in the analysis arose from working in Schwarzschild coordinates, which lead to incorrect regularity assumptions for the asymptotic solutions.

  2. In 1993, Gregory–Laflamme used numerics to give strong evidence for the existence of a low-frequency instability of the Schwarzschild black string and branes in harmonic gauge.1 

  3. In 1994, Gregory–Laflamme generalized their numerical analysis to show instability of “magnetically-charged dilatonic” black branes10 (see Refs. 10 and 11 for a discussion of these solutions).

  4. In 2000, Gubser–Mitra discussed the Gregory–Laflamme instability for general black branes. They conjectured that a necessary and sufficient condition for stability of the black brane spacetimes is thermodynamic stability of the corresponding black hole.12,13

  5. In 2000, Reall,14 with the aim of addressing the Gubser–Mitra conjecture, explored further the relation between the stability of black branes arising from static, spherically symmetric black holes and thermodynamic stability of those black holes. In particular, the work of Reall argues that there is a direct relation between the “negative mode” of the Euclidean Schwarzschild instanton solution (this mode was initially identified in a paper by Gross, Perry, and Yaffe15) and the threshold of the Gregory–Laflamme instability. This idea was further explored in a work of Reall et al.,16 which extended the idea that “negative modes” of the Euclidean extension of a Myers–Perry black hole (the generalization of the Kerr spacetime to higher dimensions, see Refs. 17 and 2 for details) correspond to the threshold for the onset of a Gregory–Laflamme instability.

  6. In 2006, Hovdebo and Myers8 used a different gauge (which was introduced in Ref. 18) to reproduce the numerics from the original work of Gregory and Laflamme. This gauge choice will be called spherical gauge and will be adopted in the present work. This work discusses the presence of the Gregory–Laflamme instability for the “boosted” Schwarzschild black string and the Emparan–Reall black ring (for a discussion of this solution, see Refs. 2, 19, and 20).

  7. In 2010, Lehner and Pretorius numerically simulated the non-linear evolution of the Gregory–Laflamme instability; see the review21 and references therein.

  8. In 2011, Figueras, Murata, and Reall4 put forward the idea that a local Penrose inequality gives a stability criterion. Furthermore, Ref. 4 showed numerically that this local Penrose inequality was violated for the Schwarzschild black string for a range of frequency parameters, which closely match those found in the original work of Gregory–Laflamme.1 

  9. In 2012, Hollands and Wald5 and, later in 2015, Prabu and Wald6 developed a general method applicable to many linear stability problems, which encompasses the problem of linear stability of the Schwarzschild black string exterior EA. References 5 and 6 are explored in detail in Sec. I E.

The purpose of this paper is to give a direct, self-contained, elementary proof of the Gregory–Laflamme instability for the 5D Schwarzschild black string.

For the statement of the main theorem, one should have in mind the Penrose diagram in Fig. 2 for the 5D Schwarzschild black string spacetime.

FIG. 2.

The Penrose diagram for the 5D Schwarzschild black string illustrating the set up for the linear instability problem. Indicated is a spacelike asymptotically flat hypersurface Σ̃, which extends from spacelike infinity iA0 to intersect the future event horizon HA+ to the future of the bifurcation surface S. Furthermore, F1=R or SR1, B is the black hole region, EA is the exterior region, IA+ is future null infinity, and iA+ is future timelike infinity. The hypersurface Σ can be expressed as Σ = {(t, r*, θ, φ, z) : t = f(r*)} such that f ∼ 1 for r*. An explicit example would be a hypersurface of constant t*, where t* = t + 2M log(r − 2M).

FIG. 2.

The Penrose diagram for the 5D Schwarzschild black string illustrating the set up for the linear instability problem. Indicated is a spacelike asymptotically flat hypersurface Σ̃, which extends from spacelike infinity iA0 to intersect the future event horizon HA+ to the future of the bifurcation surface S. Furthermore, F1=R or SR1, B is the black hole region, EA is the exterior region, IA+ is future null infinity, and iA+ is future timelike infinity. The hypersurface Σ can be expressed as Σ = {(t, r*, θ, φ, z) : t = f(r*)} such that f ∼ 1 for r*. An explicit example would be a hypersurface of constant t*, where t* = t + 2M log(r − 2M).

Close modal

Definition 1.1
(Mode Solution). A solution of the linearized vacuum Einstein equation
gcdcdhab+abh2bchac+2Racbdhcd=0
(1.7)
on the exteriorEAof the Schwarzschild black stringSch4×Rof the form
hαβ=eμt+ikzHαβ(r,θ)
(1.8)
withμ,kR and (t, r, θ, φ, z) standard Schwarzschild coordinates will be called a mode solution of (1.7).

A way of establishing the linear instability of an asymptotically flat black hole is exhibiting a mode solution of the linearized Einstein equation (1.7), which is smooth up to and including the future event horizon and decays toward spacelike infinity and such that μ > 0.

Theorem 1.1
(Gregory–Laflamme Instability). For all|k|[320M,820M], there exists a non-trivial mode solutionh of the form (1.8) to the linearized vacuum Einstein equation (1.7) on the exteriorEAof the Schwarzschild black string backgroundSch4×Rwithμ>14010M>0and
Hαβ(r,θ)=Htt(r)Htr(r)000Htr(r)Hrr(r)00000Hθθ(r)00000Hθθ(r)sin2θ000000.
(1.9)
The solutionhextends regularly toHA+and decays exponentially towardsiA0and can thus be viewed as arising from regular initial data on a hypersurface Σ extending from the future event horizonHA+ to iA0. In particular,h|Σandh|Σare smooth on Σ. Moreover, the solutionhis not pure gauge and can, in fact, be chosen such that the harmonic/transverse-traceless gauge conditions
ahab=0,gabhab=0
(1.10)
are satisfied.

SupposeR > 4M, then one can chooseksuch that there exists an integern[3R20M,8R20M], and therefore,hinduces a smooth solution on the exteriorEAof the Schwarzschild black stringSch4×SR1. Moreover, the initial data for such a mode solution on the exteriorEAofSch4×SR1have finite energy.

Hence, the exteriorEAof the Schwarzschild black stringSch4×RorSch4×SR1forR > 4Mis linearly unstable as a solution of the vacuum Einstein equation (1.7), and the instability can be realized as a mode instability in harmonic/transverse-traceless gauge (1.10), which is not pure gauge.

Remark.

One can construct a gauge invariant quantity, thetztz-component of the linearized Weyl tensorW(1), which is non-vanishing for a non-trivial mode solutionh withk ≠ 0 andμ ≠ 0 and exhibits exponential growth int whenμ > 0. This allows one to show that the mode solution constructed in Theorem 1.1 is not pure gauge. Hence, one expects that the above mode solution persists in any “good” gauge, not just (1.10).

Remark.

The reader should note that the lower bound on the frequency parameterk should not be interpreted as ruling out the existence of unstable modes with arbitrarily long wavelengths. The lower bound onk in Theorem 1.1 results from the use of a test function in the variational argument (see Proposition 4.5 in Sec. IV C). The numerics of Gregory–Laflamme and Hovdebo–Myers1,8both provide evidence that there are unstable modes fork arbitrarily small.

It may seem natural to directly consider the problem in harmonic gauge since the equation of study (1.7) reduces to a tensorial wave equation

gcdcdhab+2Racbdhcd=0.
(1.11)

The above equation (1.11) results from the linearization of the gauge reduced non-linear vacuum Einstein equation (1.1), which is strongly hyperbolic and therefore well-posed. Equation (1.11) reduces to a system of ordinary differential equations (ODEs) under the mode solution ansatz (1.8) with (1.9). This system can be reduced to a single ODE of the form

d2udr2+Pμ,k(r)dudr+Qμ,k(r)u=μ2D(r)2u,D=12Mr,
(1.12)

where u = Htt, Htr, Hrr, or Hθθ, and Pμ,k(r) and Qμ,k(r) depend on μ, k, and r. However, if one insists on this decoupling, one introduces a regular singular point in the range r ∈ (0, ). For certain ranges of μ and k, this value occurs on the exterior EA, i.e., the regular singular point occurs in r ∈ (2M, ). In particular, this regular singularity occurs on the exterior for the numerical values of k and μ for which Gregory–Laflamme identified instability. In the original works of Gregory and Laflamme, the decoupled ODE for Htr was studied (see Refs. 1, 7, and 9)

It turns out that in looking for an instability, one can make a different gauge choice called spherical gauge. As shown in Sec. III, the linearized vacuum Einstein equation (1.7) for a mode solution (1.8) in spherical gauge can be reduced to a second-order ODE of the form (1.12), where, in contrast to harmonic/transverse-traceless gauge, Pμ,k(r) = Pk(r) and Qμ,k(r) = Qk(r) depend only on k and r. Hence, the existence of solution to ODE (1.12) becomes a simple eigenvalue problem for μ. Spherical gauge was originally introduced in Ref. 18 and has another advantage over harmonic/transverse-traceless gauge, which is that all r ∈ (2M, ) are ordinary points of ODE (1.12). Hence, the spherical gauge choice also avoids the issues of a regular singularity at some r ∈ (2M, ). However, in contrast to harmonic gauge, for this gauge choice, well-posedness is unclear. If one were trying to prove stability, then exhibiting a well-posed gauge would be key since well-posedness of the equations is essential for understanding general solutions. For instability, it turns out that it is sufficient to exhibit a mode solution of the non-gauge reduced Eq. (1.7), which is not pure gauge. One expects then that such a mode solution will persist in all “good” gauges, of which harmonic gauge is an example. The discussion of pure gauge mode solutions in spherical gauge in Sec. III C provides a proof that if k ≠ 0 and μ ≠ 0, then a mode solution in spherical gauge is not pure gauge. This can be shown directly or from the computation of a gauge invariant quantity, namely, the tztz-component of the linearized Weyl tensor, W(1). Furthermore, it is shown that if a non-trivial mode solution in spherical gauge grows exponentially in t, then W(1)tztz is non-zero and grows exponentially t.

An issue with spherical gauge is that mode solutions in the spherical gauge do not, in general, extend smoothly to the future event horizon HA+, even when they represent physically admissible solutions. However, as shown in Sec. III D, one can detect what are the admissible boundary conditions at the future event horizon in spherical gauge by adding a pure gauge perturbation to the metric perturbation to try and construct a solution that indeed extends smoothly to HA+. In fact, the pure gauge perturbation found is precisely one that transforms the metric perturbation to harmonic/transverse-traceless gauge (1.10). Hence, after also identifying the admissible boundary conditions at spacelike infinity iA0 in Sec. III D, proving the existence of an unstable mode solution to the linearized vacuum Einstein equation (1.7) that is not pure gauge is reduced to showing the existence of a solution to ODE (1.12) with μ > 0 and k ≠ 0, which satisfies the admissible boundary conditions that are identified in this work.

In this paper, ODE problem (1.12) is approached from a direct variational point of view in Sec. IV. To run a direct variational argument, the solution u of ODE (1.12) is rescaled and change of coordinates is applied. It is shown in Sec. IV A that Eq. (1.12) can be cast into a Schrödinger form

Δr*u+Vk(r*)u=μ2u,r*=r+2Mlog(r2M),
(1.13)

with Vk independent of μ. ODE (1.13) can be interpreted as an eigenvalue problem for −μ2; finding an eigenfunction, in a suitable space, with a negative eigenvalue will correspond to an instability. As shown in Sec. IV B, this involves assigning the following energy functional to the Schrödinger operator on the left-hand side of (1.13):

E(u)r*u,r*uL2(R)+Vku,uL2(R).
(1.14)

Using a suitably chosen test function, one can show that the infimum over functions in H1(R) of this functional is negative for a range of k. One then needs to argue that this infimum is attained as an eigenvalue by showing that this functional is lower semicontinuous and that the minimizer is non-trivial. The corresponding eigenfunction is then a weak solution in H1(R) to ODE (1.13) with μ > 0 for a range of kR\{0}. Elementary one-dimensional elliptic regularity implies that the solution is indeed smooth away from the future event horizon, HA+, and therefore corresponds to a classical solution of the problem (1.13). Finally, the solution can be shown to satisfy the admissible boundary conditions by the condition that the solution lies in H1(R).

This paper is organized in the following manner. The remainder of the present section contains additional background on the Gregory–Laflamme instability. In Sec. II, linear perturbation theory is reviewed and the linearized Einstein equation (1.7) is derived. In Sec. III, the analysis in spherical gauge is presented. The decoupled ODE (1.12) resulting from the linearized Einstein equation (1.7) is derived, and it is established that the problem can be reduced to the existence of a solution to the decoupled ODE with μ > 0 and k ≠ 0 satisfying admissible boundary conditions. In Sec. IV, the proof of the existence of such a solution is presented via the direct variational method.

 Appendix A contains a list of the Riemann tensor components and the Christoffel symbols for the Schwarzschild black string spacetime Sch4×R or Sch4×SR1.  Appendix B collects results on singularities in second order ODE relevant for the discussion of the boundary conditions for the decoupled ODE (1.12).  Appendix C provides a method of transforming a second order ODE into a Schrödinger equation.  Appendix D collects some useful results from analysis that are needed in the Proof of Theorem 1.1.  Appendix E compliments Theorem 1.1 with some stability results.

The reader should note that there are two papers5,6 concerning a very general class of spacetimes, which are of relevence to the stability problem for the Schwarzschild black string. In particular, it follows from Refs. 5 and 6 that there exists a linear perturbation of the Schwarzschild black string spacetime, which is not pure gauge and grows exponentially in the Schwarzschild t-coordinate. The following describes the results of these works.

In 2012, a paper of Hollands and Wald5 gave a criterion for linear stability of stationary, axisymmetric, vacuum black holes and black branes in D ≥ 4 spacetime dimensions under axisymmetric perturbations. They define a quantity called the “canonical energy” E of the perturbation, which is an integral over an initial Cauchy surface of an expression quadratic in the perturbation. It can be related to thermodynamic quantities by

E=δ2MBΩBδ2JBκ8πδ2A,
(1.15)

where M and JB are the ADM mass and ADM angular momenta in the Bth plane and A is the cross-sectional area of the horizon. Note that the right-hand side of (1.15) refers to the second variation of thermodynamic quantities. It is remarkable that the combination E of these second variations is, in fact, determined by linear perturbations.

Reference 5 considers initial data for a perturbation of either a stationary, axisymmetric black hole or black brane with the following properties: (i) the linearized Hamiltonian and momentum constraints are satisfied, (ii) that δM = 0 = δJA and that the ADM momentum vanishes, and (iii) specific gauge conditions and finiteness/regularity conditions at the future horizon and infinity are satisfied. In what follows, initial data satisfying (i)–(iii) will be referred to as admissible. Hollands and Wald showed that if E0 for all admissible initial data, then one has mode stability. The work also establishes that if there exist admissible initial data such that E<0, then there exist admissible initial data for a perturbation, which cannot approach a stationary perturbation at late times, i.e., one has failure of asymptotic stability.

For the Schwarzschild black hole, one can take initial data, which corresponds simply to a change of the mass parameter MM + α, and therefore, by Eq. (1.15) and since the cross-sectional area of the horizon is given by A = 16π(M + α)2, it follows that E<0. This is the “thermodynamic instability” of the Schwarzschild black hole. However, the initial data for a change of mass perturbation is manifestly not admissible (the family of Schwarzschild black holes is, after all, dynamically stable).

The work of Hollands and Wald5 also shows an additional result relevant specifically to the problem of stability of black branes. Suppose that there exist initial data for a perturbation of the ADM parameters of a black hole such that E<0. Reference 5 shows that starting from such a perturbation of the black hole, one can infer the existence of admissible initial data, which depend on a parameter l, for a perturbation (which is not pure gauge) of the associated black brane such that again E<0. One should note that this argument does not give an explicit bound on l. This criterion formalized a conjecture by Gubser–Mitra that a necessary and sufficient condition for stability of the black brane spacetimes is thermodynamic stability of the corresponding black hole.12,13 Since the change of mass perturbation of the Schwarzschild black hole produces E<0, this argument implies that the Schwarzschild black string fails to be asymptotically stable.

Remark.

The reader should note that the Hollands and Wald paper5 also showed that a necessary and sufficient condition for stability, with respect to axisymmetric perturbations, is that a “local Penrose inequality” is satisfied. The idea that a local Penrose inequality gives a stability criterion was originally discussed in the work of Figueras, Murata, and Reall,4 which gave strong evidence in favor of sufficiency of this condition for stability. Furthermore, Ref. 4 showed numerically that this local Penrose inequality was violated for the Schwarzschild black string for a range of frequency parameters that closely match those found in the original work of Gregory–Laflamme.1 

The failure of asymptotic stability does not in itself imply that perturbations grow. However, the results of Ref. 5 were strengthened in 2015 by Prabhu and Wald.6 They showed, using some spectral theory, that if there exist admissible initial data for a perturbation such that E<0 for a black brane, then there exists initially well-behaved perturbations that are not pure gauge and that grow exponentially in time. Having established that there exist admissible initial data for a perturbation such that E<0 for the Schwarzschild black string in Ref. 5, the existence of a linear perturbation, which is not pure gauge and has exponential growth, follows.

The present work differs from the above as it gives a direct, self-contained, elementary proof of the Gregory–Laflamme instability following the original formulation of Refs. 1 and 7–9, which is completely explicit. In particular, it gives an exponentially growing mode solution with an explicit growth rate of the form defined by equations (1.8) and (1.9) in harmonic/transverse-traceless gauge, which is not pure gauge.

Remark.

It would also be of interest to see if Theorem 1.1 in the form stated could be inferred from the canonical energy method of Hollands, Wald, and Prabu5,6 in an explicit way bypassing some of the functional calculus applied there. In particular, it would be interesting to explore the possible relation between the variational theory applied to E and that applied here (see Sec. IV B).

This paper brings together what is known about the Gregory–Laflamme instability as well as providing a direct elementary mathematically rigorous proof of its existence without the use of numerics and with an explicit bound on μ and k. Note that while only the 5D Schwarzschild black string was considered here, the result of instability readily extends to higher dimensions with the replacement of kz in the exponential factor with ∑ikizi.

Further directions of work could be to study the non-linear problem, the extension to Kerr4×S1 or Kerr4×R, the extension to charged black branes of the work,10 and the extension to black rings or ultraspinning Myers–Perry black holes.

1. Motivation for the study of higher dimensions

The study of higher dimensions merits a few words of motivation since, from a physical standpoint, only 3 + 1 are perceived classically. First, from a purely mathematical perspective, it is of interest to see how general relativity differs in higher dimensions from the 4D case. This throws light on how general Lorentzian manifolds obeying the vacuum Einstein equation (1.1) behave. Second, the physics community is very interested in higher-dimensional gravity from the point of view of string theory. Understanding how general relativity behaves in higher dimensions is therefore of relevance to the low energy limit of string theory.2 

2. Some differences in higher dimensions

In higher dimensions, many results from 4D general relativity no longer hold. As shown by Hawking, in 4D, the cross sections of the event horizon of an asymptotically flat stationary black hole spacetime must be topologically S2 (under the dominant energy condition).22 In higher dimensions, it is possible to construct explicit examples of black hole spacetimes with non-spherical cross-sectional horizon topology. For example, the black ring solution with horizon topology is S2×S1.19 In higher dimensions, there also exists a generalized Kerr solution known as the Myers–Perry black hole,17 which has cross-sectional horizon topology S3. Hawking’s theorem has been generalized to higher dimensions,23 which shows that the horizon topology must be of positive scalar curvature. In 5D, under the assumptions of stationarity, asymptotic flatness, two commuting axisymmetries and “rod structure” black holes are unique, and further the horizon topology is either S3, S1×S2, or lens space.24 

In 4D, it is conjectured that maximal developments of “generic” asymptotically flat initial data sets can asymptotically be described by a finite number of Kerr black holes. This “final state conjecture” cannot generalize immediately since there exist at least two distinct families of black hole solutions that can have the same mass and angular momentum: the Myers–Perry black hole and the black ring. Moreover, there exist distinct black ring solutions with the same mass and angular momentum.2,20 The final state conjecture may need to be modified to include the property of stability.

3. Related works

A few other works are of relevance to this discussion. The review paper2 and book chapter20 discuss the black ring solution19 in great detail. This relates to the work presented here since the Gregory–Laflamme instability is often heuristically invoked when discussing higher-dimensional black hole solutions. In particular, if the black ring of study has a large radius and is sufficiently thin, then it “looks like” a Schwarzschild black string and therefore would be susceptible to the Gregory–Laflamme instability. There have been heuristic and numerical results to give evidence to this claim.8,25 Finally, note that in 2018, Ref. 26 produced the first mathematically rigorous result on the stability problem for the black ring spacetime.

This section provides a derivation and review of the linearized vacuum Einstein equation (1.7) around a general spacetime background metric (M, g) satisfying the vacuum Einstein equation (1.1).

Consider a Lorentzian manifold (M, g) with metric satisfying the vacuum Einstein equation

Ricg=0.
(2.1)

In this section, a “perturbation” of the spacetime metric will be discussed. This will be represented by a new metric of the form g + ϵh with ϵ > 0. h here is a symmetric bilinear form on the fibers of TM. In the following, a series of results on how various quantities change to O(ϵ) (the linear level) are derived. This will result in an expression for the Ricci tensor under such a perturbation to linear order.

Remark.

An important point to note is that indices are raised and lowered here with respect tog.

Proposition 2.1
(change in the Ricci Tensor). Consider a Lorentzian manifold (M, g). Suppose the metricg̃ab=gab+ϵhabis a Lorentzian metric. Then, the Ricci tensor,(Ricg̃)ab, ofg̃abtoO(ϵ) is
(Ricg̃)ab=(Ricg)abϵ12ΔLhab,
(2.2)
where ΔLdenotes the Lichnerowicz operator given by
ΔLhab=gcdcdhab+2Racbdhcd2(Ricg)cahbc2achbc+abh,
(2.3)
andh = gabhab.

Proof.

Direct computation.

If one assumes that g satisfies the vacuum Einstein equation (1.1) and g + ϵh satisfies the vacuum Einstein equation (1.1) to O(ϵ), then it follows from Proposition 2.1 that h must satisfy

gcdcdhab+abh2bchac+2Racbdhcd=0
(2.4)

to O(ϵ). In what follows, Eq. (2.4) will be called the linearized vacuum Einstein equation. This will be the main equation of interest, with g being the Schwarzschild black string metric,

gD(r)dtdt+1D(r)drdr+r2dθdθ+sin2θdφdφ+dzdz,D(r)=12Mr.
(2.5)

The vacuum Einstein equation (1.1) is a system of second order quasilinear partial differential equations of the pair (M, g), which are invariant under the diffeomorphisms of M. This means that for given initial data, the vacuum Einstein equation (1.1) only determines a spacetime unique up to diffeomorphism, i.e., if there exists a diffeomorphism Φ : MM, then (M, g) and (M, Φ*(g)) are equivalent solutions of the vacuum Einstein equation (1.1). For constructing spacetimes, one often imposes conditions on local coordinates called a gauge choice. For linearized theory, this can be formulated as follows.

Consider a Lorentzian manifold (M,g̃g+ϵh) with ϵ > 0. Let {Φτ} be a one-parameter family of diffeomorphisms generated by a vector field X and define ξτXTM. Then, from the definition of the Lie derivative, one has

(Φτ)*(g̃)=g̃+Lξg+O(ϵ2)
(2.6)

if one treats τ=O(ϵ). Hence, in the context of linearized theory, one considers two solutions to the linearized vacuum Einstein equation (2.4), h1 and h2, as equivalent if

h2=h1+Lξg(h2)ab=(h1)ab+2aξb
(2.7)

for some vector field ξTM.

Definition 2.1
(Pure Gauge Solution). Let (M, g) be a vacuum spacetime. A solution h to the linearized vacuum Einstein equation (2.4) will be called pure gauge if there exists a vector fieldξTMsuch that
hab=2aξb.
(2.8)
The notationhpgwill be used to denote a pure gauge solution to the linearized vacuum Einstein equation (2.4).

Showing that a solution h to the linearized vacuum Einstein equation (2.4) is not pure gauge is tantamount to showing that h is not equivalent to the trivial solution. It is thus essential that the solution constructed in this paper not be pure gauge. The following propositions establish that the tztz-component of the linearized Weyl tensor W(1) is invariant under gauge transformation. This means that if W(1) is non-zero for a solution h to the linearized vacuum Einstein equation (2.4), then h cannot be pure gauge.

Proposition 2.2
(Change to the Weyl Tensor). Let (M, g) be a vacuum spacetime. Suppose the metricg̃ab=gab+ϵhabis a Lorentzian metric such thathsatisfies the linearized vacuum Einstein equation (2.4). Then the Weyl tensor,W̃abcd, ofg̃abtoO(ϵ) is
W̃abcd=Wabcd+ϵW(1)abcd,
(2.9)
where
W(1)abcd=cbhad+dahbc+12RebcdhaeReacdheb.
(2.10)
Henceforth,W(1)will be referred to as the linearized Weyl tensor.

Proof.

Direct computation.

Proposition 2.3.

For the 5DSchwarszchild black string,W(1)tztzevaluated on a pure gauge solution vanishes.

Proof.
Let W(1)pg denote the linearized Weyl tensor evaluated on a pure gauge solution hpg. Recall that a pure gauge solution hpg can always be written as hpg=Lξg for some vector field ξTM. Using Proposition 2.2, one has that
(W(1)pg)abcd=cbaξd+dabξc+12RebcdaξeReacdeξb+c|b|dξa+d|a|cξb+12RebcdeξaReacdbξe.
(2.11)
By repeated use of the Ricci identity with the first and second Bianchi identities, one can compute that
(W(1)pg)abcd=(aR)ebcdξe+(bR)eadcξe+Readcbξe+Rebcdaξe+Redabcξe+Recbadξe.
(2.12)
From  Appendix A, one has Rμαβz=0, Rμαzβ=0, Rμzαβ=0, and Γzβα=0. Furthermore, the black string metric (1.4) is independent of t and z. Hence,
(W(1)pg)tztz=0.
(2.13)

In this section, a mode solution, h, of the linearized vacuum Einstein equation (2.4) on the exterior EA of the Schwarzschild black string spacetime Sch4×R or Sch4×SR1 is considered. One makes the additional assumption that this mode solution preserves the spherical symmetry of Sch4. Hence, in particular, the solution can be expressed in (t, r, θ, φ, z) coordinates as

hαβ=eμt+ikzHtt(r)Htr(r)00Htz(r)Htr(r)Hrr(r)00Hrz(r)00Hθθ(r)00000Hθθ(r)sin2θ0Htz(r)Hrz(r)00Hzz(r),
(3.1)

where α, β ∈{t, r, θ, φ, z}. Moreover, in search of instability, the most interesting case for the present work is μ > 0.

This section contains the analysis of the ODEs resulting from the linearized Einstein vacuum equation (2.4) for a mode solution of the form (3.1) when it is expressed in spherical gauge.

Definition 3.1
(Spherical Gauge). A mode solutionhof the linearized vacuum Einstein equation (2.4) on the exteriorEAof the Schwarzschild black string spacetimeSch4×Ris said to be in spherical gauge if it is of the form
hμν=eμt+ikzHt(r)μHv(r)000μHv(r)Hr(r)00ikHv(r)00000000000ikHv(r)00Hz(r).
(3.2)
For the Schwarzschild black string spacetimeSch4×SR1, one makes the same definition with the additional assumption thatkRZ.

Remark.

The terminology “spherical gauge” is motivated by the fact that a mode solution of this form preserves the area of the spheres of the original spacetime.

First, it is shown in Sec. III A that one can impose the gauge consistently at the level of modes, i.e., if there is a mode solution of the form (3.1), with μ ≠ 0 and either k ≠ 0 or dHtzdrHrz=0, then there is a mode solution of the form (3.2) differing from the original one by a pure gauge solution. In the case where Htz = 0, Hrz = 0, and Hzz = 0, this consistency condition is already implicit in Refs. 18 and 8. In Sec. III B, the original decoupling of the ODEs resulting from the linearized vacuum Einstein equation (2.4) and the spherical gauge ansatz (3.2) is reproduced from Ref. 8. This decoupling results in a single ODE for the component Hz(r) in Eq. (3.2). It is then shown, in Sec. III C, that if k ≠ 0 and μ ≠ 0, then mode solutions in spherical gauge (3.2) are not pure gauge. This is proved by examining the tztz-component of the linearized Weyl tensor W(1) associated with a mode solution in spherical gauge, which is gauge invariant by Proposition 2.3. In this section, it is also proved that if a non-trivial mode solution in spherical gauge has μ > 0 (i.e., it grows exponentially in t) and k ≠ 0, then W(1)tztz is non-zero and also grows exponentially. By the gauge invariance of W(1)tztz, this behavior will persist in all gauges. Next, in Sec. III D, the admissible boundary conditions for the solution at the future event horizon HA+ and finiteness conditions at spacelike infinity iA0 are identified. Note this issue is subtle since, in general, both “basis” elements for a mode solution h of the form (3.2) are, in fact, singular at the future event horizon HA+ in this gauge. By adding a pure gauge perturbation, the admissible boundary conditions for the solution h in the form (3.2) can be identified. Moreover, this pure gauge solution can be chosen such that, after adding it, the harmonic/transverse-traceless gauge (1.10) conditions are satisfied. Finally, in Sec. III E, the problem of constructing a linear mode instability of the form (3.1) is reduced to show that there exists a solution to the decoupled ODE for Hz(r), with μ > 0 and k ≠ 0, that satisfies the admissible boundary conditions at the future event horizon HA+ and spacelike infinity iA0 (see Proposition 3.8).

In Ref. 8, it is stated that any mode solution of the form in Eq. (3.1) with Htz = 0, Hrz = 0, and Hzz = 0 can be brought to the spherical gauge form (3.2) by the addition of a pure gauge solution. Slightly more generally, one, in fact, has the following proposition:

Proposition 3.1

(Consistency of the Spherical Gauge). Consider a mode solutionhto the linearized Einstein vacuum equation (2.4) on the exteriorEAof the Schwarzschild black string spacetimeSch4×RorSch4×SR1of the form (3.1) withμ ≠ 0. Further suppose that eitherk ≠ 0 orddrHtzμHrz=0. Then, there exists a pure gauge solutionhpgsuch thath + hpgis of form (3.2). It is in this sense that the spherical gauge (3.2) can be consistently imposed on the exteriorEAof the Schwarzschild black stringSch4×RorSch4×SR1.

Proof.

From Sec. II B, a pure gauge solution is given by hpg=2aξb for a vector field ξ. Hence, h̃ab=hab+2aξb is the new mode solution. Consider a diffeomorphism generating vector field of the form ξ = eμt+ikz(ζt(r), ζr(r), 0, 0, ζz(r)).

If k ≠ 0, one can take
ζt(r)=ir(r2M)2MkrHtz(r)μHrz(r)+r(r2M)2MHtr(r)rμ2MHθθ(r),ζr(r)=Hθθ(r)2(r2M),ζz(r)=Htz(r)+ikζt(r)μ
(3.3)
and immediately verify that h̃ is of the form (3.2).
If ddrHtzμHrz=0, then one can take
ζt(r)=r(r2M)2MHtr(r)rμ2MHθθ(r),ζr(r)=Hθθ(r)2(r2M),ζz(r)=Htz(r)+ikζt(r)μ
(3.4)
and immediately verify that h̃ is of the form (3.2).

Under a spherical gauge ansatz (3.2) with μ ≠ 0 and k ≠ 0, the linearized vacuum Einstein equation (2.4) reduces to a system of coupled ODEs for the components Ht, Hv, Hr, and Hz. This system of ODEs can be decoupled to the single ODE for hHz,

d2hdr2(r)+Pk(r)dhdr(r)+Qk(r)μ2r2(r2M)2h(r)=0,
(3.5)

with

Pk(r)12Mr(k2r3+2M)5r+1r2M,
(3.6)
Qk(r)6Mr2(r2M)rk2r2M12M2r2(r2M)(k2r3+2M).
(3.7)

The following proposition establishes this decoupling of the linearized vacuum Einstein equation (2.4) to ODE (3.5) and the construction of a mode solution h in spherical gauge (3.2) from a solution h to ODE (3.5).

Proposition 3.2.

Given a mode solutionhin spherical gauge (3.2) withμ ≠ 0 andk ≠ 0 on the exteriorEAof the Schwarzschild black stringSch4×R or Sch4×SR1, ODE (3.5) is satisfied byhzz. Conversely, given aC2((2M, )) solutionh(r)to ODE (3.5) withk ≠ 0 andμ ≠ 0, one can construct a mode solutionhin spherical gauge (3.2) to the linearized vacuuum Einstein equation (2.4) on the exterior EAof the Schwarzschild black stringSch4×R. IfkRZ, thenhinduces a mode solution onSch4×SR1.

Proof.
Let h be a mode solution in spherical gauge (3.2) with μR and kR satisfying the linearized vacuum Einstein equation (2.4) on the exterior EA of the Schwarzschild black string Sch4×RorSch4×SR1. Equivalently, the following system of ODE has to be satisfied:
μkHr=2Mμkr(r2M)Hv,
(3.8)
μk2Hv=μ2dHzdrμ(r2M)Hrr2μMHz2r(r2M),
(3.9)
kdHtdr=kMHtr(r2M)k(r2M)(2r3M)Hrr3+2μ2kHv,
(3.10)
Ht=(r2M)(r(k2μ2)2Mk2)MHv+(r2M)2(r+M)Mr2Hr,
(3.11)
+(r2M)32MrdHrdr(r2M)22MdHzdr+r(r2M)2MdHtdr,d2Hzdr2=k2Hr+r2(r2M)2μ2Hzk2Ht+2(rM)r(r2M)2k2HvdHzdr+2k2dHvdr,
(3.12)
d2Hzdr2=2M(2r3M)r(r2M)3Ht6M2(μ2+k2)r4+2Mr(k2r22)r3(r2M)Hr,
(3.13)
2M(2Mk2+r(μ2k2))r(r2M)2Hv+2r3Mr2dHrdr2μ2r+4Mk22k2rr2MdHvdrMr(r2M)dHzdrM(r2M)2dHtdr+rr2Md2Htdr2,
d2Htdr2=k2r42Mk2r32M2r2(r2M)2Htμ2+2M2r4Hrrμ2r2MHz+4μ2r2+4M2k22Mr(3μ2+k2)r2(r2M)Hv2r5Mr(r2M)dHtdrM(r2M)r3dHrdr+2μ2dHvdr+Mr2dHzdr.
(3.14)
Now, if μ ≠ 0 and k ≠ 0, then from Eqs. (3.8) and (3.9), one can find Hv in terms of Hz and dHzdr. This can then be used in Eq. (3.10) to give an equation for dHtdr in terms of Ht, Hz, and dHzdr. All of these expressions can be used to express Ht in terms of Hz, dHzdr, and d2Hzdr2 via Eq. (3.11). The resulting equations are
Hr(r)=M2r(r2M)2(k2r2+2M)Hz(r)+Mr2(r2M)(k2r2+2M)dHzdr,
(3.15)
Hv(r)=Mr22(r2M)(k2r2+2M)Hz(r)+r32(k2r2+2M)dHzdr,
(3.16)
Ht(r)=2M2(r3M)+Mk2r3(2r5M)k4r6(r2M)r(k2r3+2M)2Hz,
(3.17)
2(r2M)(M(r4M)+(2r5M)k2r3)(k2r3+2M)2dHzdr+r(r2M)2k2r3+2Md2Hzdr2.
Finally, one can use the above expressions to obtain a decoupled ODE for hHz, namely,
d2hdr2(r)+Pk(r)dhdr(r)+Qk(r)μ2r2(r2M)2h(r)=0,
(3.18)
with
Pk(r)12Mr(k2r3+2M)5r+1r2M,
(3.19)
Qk(r)6Mr2(r2M)rk2r2M12M2r2(r2M)(k2r3+2M).
(3.20)
Conversely, given any C2((2M, )) solution h(r) to ODE (3.5) with k ≠ 0 and μ ≠ 0, one can define Hz(r)=h(r). Since k ≠ 0, one can use Eqs. (3.15)–(3.17) to construct Ht(r), Hr(r), and Hv(r). These then define the components of a mode solution h in spherical gauge (3.2). Explicitly,
h=eμt+ikzHt(r)μHv(r)000μHv(r)Hr(r)00ikHv(r)00000000000ikHv(r)00Hz(r).
(3.21)
If ODE (3.5) is satisfied and (3.15)–(3.17) define Hr, Hv, and Ht, then Eqs. (3.8)–(3.14) are also satisfied. Therefore, a mode solution h constructed in this manner solves the linearized vacuum Einstein equation (2.4) on the exterior EA of the Schwarzschild black string Sch4×R. If kRZ, then this construction also gives a mode solution h, which solves the linearized vacuum Einstein equation (2.4) on the exterior EA of the Schwarzschild black string Sch4×SR1.

Remark.
Ifk = 0 andμ ≠ 0, then one can add an additional pure gauge solutionhpgto a mode solutionhin spherical gauge (3.2) such thath + hpgis also in spherical gauge (3.2) withHt(r) ≡ 0. The relevant choice of pure gauge solution is given by(hpg)ab=2aξbwith
ξ=eμtHt(r)2μ,0,0,0,0.
(3.22)

A mode solutionhin spherical gauge withHt(r) ≡ 0 satisfying the linearized vacuum Einstein equation (2.4) on the exteriorEAof the Schwarzschild black string is then again equivalent to the system of ODE (3.8)–(3.14) [withk = 0 andHt ≡ 0] being satisfied. Equations (3.8) and (3.10) are automatically satisfied byk = 0. Eq. (3.12) automatically gives the decoupled equation (3.5) forHz. Then, Eq. (3.9) can be solved forHrin terms ofHzanddHzdr. This gives the relation in Eq. (3.15) forHrwithk = 0. Equation (3.11) can be used to solve forHvin terms ofHzanddHzdr. At this point, the equations (3.13) and (3.14) are automatically satisfied. Therefore, again a solution to ODE (3.5) induces a mode solution in spherical gauge with Ht = 0.

This section contains two proofs that if k ≠ 0 and μ ≠ 0, then a non-trivial mode solution h of the form (3.2) cannot be a pure gauge solution. One can prove this directly via the following proposition:

Proposition 3.3.

Supposek ≠ 0 andμ ≠ 0. A non-trivial mode solutionhin spherical gauge (3.2) of the linearized vacuum Einstein equation (2.4) on the exteriorEAof the Schwarzschild black stringSch4×RorSch4×SR1cannot be pure gauge.

Proof.
If h is pure gauge, it must be possible to write hab=2aξb for some vector field ξ. Therefore, one finds
hzz=Hz(r)eμt+ikz2zξz=Hz(r)eμt+ikz,
(3.23)
hzθ=0θξz+zξθ=0.
(3.24)
Applying z to Eq. (3.24), using that partial derivatives commute and that, from Eq. (3.23), zξz clearly does not depend on θ gives
z2ξθ=0.
(3.25)
Next, hθθ = 0 implies
θξθΓθθrξr=0.
(3.26)
From  Appendix A, Γθθr=(r2M). Hence, taking two derivatives of (3.26) in the z direction and using z2ξθ=0 give
Γθθrz2ξr=(r2M)z2ξr=0.
(3.27)
Therefore, z2ξr=0 on EA.
From the hrr component, one has,
2rξr2Γrrrξr=2rξr+2Mr(r2M)ξr=Hreμt+ikz,
(3.28)
where one uses Γrrr=Mr(r2M) from  Appendix A. Taking the second z derivative of Eq. (3.28) and using z2ξr=0 on EA give
k2Hr=0onEA.
(3.29)
Since k ≠ 0, this implies Hr ≡ 0 on the exterior EA. Since k ≠ 0 and μ ≠ 0, Eq. (3.8) implies that if Hr = 0 on EA, then Hv ≡ 0 on EA. Using the hzr component, one finds
zξr+rξz=ikHveμt+ikz=0r(zξz)=0dHzdr=0onEA,
(3.30)
where one uses the identity z2ξr=0 on EA in the first implication and that zξz = Hz(r)eμt+ikz in the second implication. The linearized vacuum Einstein equation (2.4) under this ansatz [Eq. (3.9)] then implies Hz ≡ 0 on EA, and therefore, from equations (3.10) and (3.11), Ht ≡ 0 on EA. Hence, h ≡ 0 on EA.

Perhaps more satisfactorily, one can establish that if h is a non-trival mode solution in spherical gauge (3.2) with k ≠ 0 and μ ≠ 0, then the tztz-component of the linearized Weyl tensor W(1) is non-vanishing. Moreover, if h has μ > 0, then W(1)tztz grows exponentially. Since W(1)tztz is gauge invariant, this behavior persists in all gauges. More precisely, one has the following proposition:

Proposition 3.4.

Supposek ≠ 0, μ ≠ 0, andhis a non-trivial mode solution in spherical gauge (3.2) of the linearized vacuum Einstein equation (2.4) on the exteriorEAof the Schwarzschild black stringSch4×RorSch4×SR1. Then,W(1)tztzis non-vanishing andhis not pure gauge. Moreover, ifμ > 0 thenW(1)tztzalso grows exponentially.

Proof.
By Proposition 2.3, W(1)tztz is gauge invariant. Hence, if W(1)tztz is non-zero when evaluated on a non-trivial mode solution h in spherical gauge (3.2), h cannot be pure gauge. Using Proposition 2.2 gives the following expression for W(1)tztz:
W(1)tztz=12eμt+ikzk2Ht(r)2Mk2(r2M)r3Hv(r)+M(r2M)r3dHzdr(r)μ2Hz(r).
(3.31)
If k ≠ 0 and μ ≠ 0, one can use Eqs. (3.15)-(3.17) and ODE (3.5) to simplify this to
W(1)tztz=eμt+ikzM(r2M)(k2r3(3r7M)2M2)r3(k2r3+2M)2dHzdr(r)+M(k4r3(r2M)+k2(μ2r4Mr+2M2)+2Mμ2r)r(k2r3+2M)2Hz(r).
(3.32)
Suppose W(1)tztz0 identically, then
dHzdr(r)=r2(Mr(2k4r2+k22μ2)k2(μ2+k2)r42M2k2)(r2M)(3k2r47Mk2r32M2)Hz(r).
(3.33)
Substituting this into ODE (3.5) gives that either
k4r3(r2M)2+Mr(4r9M)μ2+r5μ4+k2(r2M)(2r4μ22Mr+5M2)=0
(3.34)
for all r ∈ (2M, ) or Hz(r) ≡ 0. If μ ≠ 0 and k ≠ 0, then the polynomial in Eq. (3.34) has at most five roots in r ∈ (2M, ). Therefore, if W(1)tztz=0, then Hz(r) = 0, which is a contradiction. Moreover, since W(1)tztz0, it is clear from Eq. (3.32) that if μ > 0, then W(1)tztz grows exponentially.

One can construct two sets of distinguished solutions to ODE (3.5) associated with the “end points” of the interval (2M, ). Note that by Definition B.1 from  Appendix B, r = 2M is a regular singularity, as 2M is not an ordinary point and

(r2M)Pk(r) and (r2M)2Qk(r)μ2r2(r2M)2
(3.35)

are analytic near r = 2M. By Definition B.3, ODE (3.5) has an irregular singularity at infinity since there exist convergent series expansions

Pk(r)=n=0pnzn and Qk(r)μ2r2(r2M)2=n=0qnzn
(3.36)

in a neighborhood of infinity with p0 = 0, p1 = −4, q0 = −(k2μ2), and q1 = −2M(k2 + 2μ2). The asymptotic analysis of the ODEs around these points is examined in Secs. III D 1 and III D 2. This analysis of ODE (3.5) near r = 2M and r = will lead to the identification of the admissible boundary conditions for a mode solution h in spherical gauge (3.2) of the linearized Einstein vacuum equation (2.4).

1. The future event horizon HA+

The goal of this section is to identify the admissible boundary conditions for a solution h to ODE (3.5) near r = 2M. This requires one to understand the behavior near r = 2M of the mode solution h in spherical gauge (3.2) of the linearized vacuum Einstein equation (2.4), which results (through the construction in Proposition 3.2) from h.

Associated with the future event horizon HA+, there exists a basis h2M,± for solutions to ODE (3.5). From h2M,±, one can examine the behavior near r = 2M of any mode solution h in spherical gauge (3.2) with μ ≠ 0 and k ≠ 0 through Proposition 3.2. A mode solution h in spherical gauge (3.2) with μ > 0 and k ≠ 0 constructed from h2M, never smoothly extends to the future event horizon. A mode solution h in spherical gauge (3.2) with μ > 0 and k ≠ 0 constructed from h2M,+ also does not smoothly extend to the future event horizon unless μ satisfies particular conditions. However, if h is a mode solution in spherical gauge (3.2) with μ > 0 and k ≠ 0 constructed from h2M,+, then after the addition of a pure gauge solution hpg, it turns that out one can smoothly extend h + hpg to the future event horizon. Moreover, it will be shown that h + hpg satisfies the harmonic/transverse-traceless gauge (1.10) conditions. This will be the content of Proposition 3.5.

First, some preliminaries: The coefficients of ODE (3.5) extend meromorphically to r = 2M and behave asymptotically as

Pk(r)=1r2M+O(1)Qk(r)μ2r2(r2M)2=4M2μ2(r2M)2+O1r2M.
(3.37)

Hence, one may write ODE (3.5) as

d2hdr2+1r2M+O(1)dhdr4M2μ2(r2M)2+O1r2Mh=0.
(3.38)

From  Appendix B, the indicial equation associated with the ODE (3.38) is

I(α)=α24M2μ2,
(3.39)

which has roots

α±±2Mμ.
(3.40)

If α+α=4MμZ, then one can deduce from Theorem B.1 the asymptotic basis for solutions near r = 2M. If α+α=4MμZ, then the relevant result for the asymptotic basis of solutions is Theorem B.2. Combining the results of Theorems B.1 and B.2, one has the following basis for solutions for μ > 0:

h2M,+(r)(r2M)2Mμn=0an+(r2M)n,
(3.41)
h2M,(r)an(r2M)n2Mμ+CNh2M,+(r)ln(r2M)if4Mμ=NZ>0(r2M)2Mμan(r2M)notherwise,
(3.42)

where the coefficients an+, an, and the anomalous term CN can be calculated recursively (see Theorems B.1 and B.2). A general solution to ODE (3.5) will be of the form

h(r)=k1h2M,+(r)+k2h2M,(r),
(3.43)

with k1,k2R.

If 4 is not an integer or 4 is an integer and CN = 0, then the asymptotic basis for solutions for μ > 0 reduces to

h2M,+(r)=(r2M)2Mμn=0an+(r2M)n,
(3.44)
h2M,(r)=(r2M)2Mμn=0an(r2M)n.
(3.45)

In Eqs. (3.44) and (3.45), the first order coefficients of the basis can be calculated to be

a1±=±μ(20M2k21)+4M(μ2k2+4M2μ2k2+2M2k4)(1±4Mμ)(4M2k2+1).
(3.46)

The main result of this section is the following:

Proposition 3.5.

Supposeμ > 0, k ≠ 0, and let hbe a solution to ODE (3.5). Lethbe the mode solution on the exteriorEAof the Schwarzschild black stringSch4×Rconstructed fromHz=hin Proposition 3.2. Then, there exists a pure gauge solution hpgsuch thath + hpgextends to a smooth solution of the linearized vacuum Einstein equation (2.4) at the future event horizonHA+ if k2 = 0, wherek2is defined in Eq. (3.43). Moreover,h + hpgcan be chosen to satisfy the harmonic/transverse-traceless gauge (1.10) conditions.

Remark.
To determine admissible boundary conditions of h at r = 2M, it is essential that one works in coordinates that extend regularly across this hypersurface. A good choice is ingoing Eddington–Finkelstein coordinates (v, r, θ, φ, z) defined by
v=t+r*(r),r*(r)=r+2Mlog|r2M|.
(3.47)
Also note that for the boundary conditions to be admissible, one needs to consider all components of the mode solution h constructed fromhvia Proposition 3.2. These remarks will be implemented in the Proof of Proposition 3.5.

Before proving the statement of Proposition 3.5, it is useful to prove the following lemma:

Lemma 3.6.
Lethbe a mode solution of the linearized vacuum Einstein equation (2.4) of the form
hαβ=eμt+ikzHtt(r)Htr(r)000Htr(r)Hrr(r)00000Hθθ(r)00000Hθθ(r)sin2θ000000.
(3.48)
Then,hsatisfies the following harmonic/transverse-traceless gauge conditions:
ahab=0,gabhab=0
(3.49)
ifk ≠ 0.

Proof.
First, it is instructive to write out explicit expressions for ∇chab and ∇cdhab in coordinates. These are the following:
γhαβ=γhαβΓγαλhλβΓγβλhαλ,
(3.50)
γδhαβ=γ(δhαβΓδαλhλβΓδβλhαλ)Γγδμ(μhαβΓμαλhλβΓμβλhαλ),
(3.51)
Γγαμ(δhμβΓδμλhλβΓδβλhμλ)Γγβμ(δhαμΓδαλhλμΓδμλhαλ).
If one takes ansatz (3.48) and α = z in Eq. (3.51), then since h = 0 for all β ∈{t, r, θ, φ, z} and, from  Appendix A, Γzβλ=0 for all β, λ ∈ {t, r, θ, φ, z},
γδhαβ=0(α=z).
(3.52)
Hence,
gγδγδhαβ=0(α=z),
(3.53)
gδβγδhαβ=0(α=z).
(3.54)
Consider the linearized vacuum Einstein equation (2.4) in coordinates
gγδγδhαβ+αβhαγhβγβγhαγ+2Rαγβδhγδ=0.
(3.55)
Since from Eqs. (3.53) and (3.54) and from  Appendix A, Rzβγδ = 0, it follows that the linearized vacuum Einstein equation in local coordinates with α = z and under ansatz (3.48) reduces to
z(βhγhβγ)=0.
(3.56)
Furthermore, ∇z = z, so using the explicit z-dependence of ansatz (3.48), Eq. (3.56) reduces to
k(βhγhβγ)=0.
(3.57)
Since k ≠ 0, the harmonic gauge condition
βhγhβγ=0
(3.58)
is satisfied. If β = z, then using Eq. (3.50) and ∇z = z, Eq. (3.58) reduces to
zh=kh=0h=0
(3.59)
since k ≠ 0. Substituting (3.59) into Eq. (3.58) gives the transverse condition
γhβγ=0.
(3.60)

Proof of Proposition 3.5.
Consider Hz2M,±h2M,± where h2M,± are given by Eqs. (3.44) and (3.45) with first order coefficients (3.46). Taking k2 = 0 is equivalent to examining the basis element Hz2M,+. Since μ > 0 and k ≠ 0, one can use Proposition 3.2 to construct the components Ht, Hr, and Hv associated to Hz2M,±. Substituting the basis into Eqs. (3.15)–(3.17), one finds
Hr2M,±=(r2M)2±2MμM2(±4Mμ1)1+4M2k2+M(4M2(2μ2+k2)±6Mμ1)2(1+4M2k2)(r2M)+O((r2M)2),
(3.61)
Ht2M,±=(r2M)±2Mμ(1+4Mμ)(4Mμ1)4(1+4M2k2)+3+4M2(8μ2k2)±2Mμ(8M2(2μ2+k2)11)8M(1+4M2k2)(r2M)+O((r2M)2),
(3.62)
Hv2M,±=(r2M)1+2MμM2(±4Mμ1)1+4M2k2+M(2M2(2μ2+k2)1±5Mμ)1+4M2k2(r2M)+O((r2M)2).
(3.63)
Consider a pure gauge solution hpg=2aξb generated by the following vector field
ξ=eμt+ikzμHz(r)2k2,2k2Hv(r)dHzdr(r)2k2,0,0,iHz(r)2k,
(3.64)
where Hv is defined via Eq. (3.16). This gives a new solution to the linearized vacuum Einstein equation (2.4),
h̃μν=hμν+2μξν=eμt+ikzH̃tt(r)H̃tr(r)000H̃tr(r)H̃rr(r)00000H̃θθ(r)00000H̃θθ(r)sin2θ000000,
(3.65)
with the following expressions for the matrix components:
H̃tt(r)=c1(r)Hz(r)+c2(r)dHzdr(r),
(3.66)
H̃θθ(r)=c3(r)Hz(r)+c4(r)dHzdr(r),
(3.67)
H̃rr(r)=r2(r2M)2H̃tt(r)2r(r2M)H̃θθ(r),
(3.68)
H̃tr(r)=2Mμk2(2M+r3k2)dHzdr(r)Mr(r2M)Hz(r),
(3.69)
where
c1(r)6M2(r2M)r(k2r3+2M)22M(r2M)r(k2r3+2M)+μ2r3k2r3+2Mμ2k2,c2(r)M(r2M)k2r3M(r2M)k2r3+2M6M(4M24Mr+r2)(k2r3+2M)2,c3(r)Mr2k2r3+2M,c4(r)r3(r2M)k2r3+2Mr2Mk2.
(3.70)
Note that Eqs. (3.5) and (3.15)–(3.17) have been used to derive Eqs. (3.66)–(3.69). By Lemma 3.6, this new mode solution (3.65) satisfies the harmonic/transverse-traceless gauge,
gμνh̃μν=0,μh̃μν=0.
(3.71)
As remarked above, to determine admissible boundary conditions of h at r = 2M, it is essential that one works in coordinates that extend regularly across this hypersurface. Moreover, to identify the boundary conditions to be admissible, one needs to consider all components of the mode solution h constructed from h via Proposition 3.2. The following formulas give the transformation to ingoing Eddington–Finkelstein coordinates for the components of the mode solution h defined in Eq. (3.65):
H̃vv=tv2H̃tt,H̃vr=tvrrH̃tr+tvtrH̃tt=H̃trrr2MH̃tt,H̃rr=tr2H̃tt+trrrH̃tr+rr2H̃rr=r2(r2M)2H̃ttrr2MH̃tr+H̃rr,
(3.72)
where one uses t = vr*(r) with r*(r) = r + 2M log|r − 2M|. Explicitly, Eq. (3.72) can be computed to be
H̃vv=2M(2Mμ2r+k2(μ2r4Mr+2M2)+k4r3(r2M))r(k3r3+2Mk)2Hz(r)
(3.73)
2M(r2M)(k2r3(3r7M)2M2)r3(k3r3+2Mk)2dHzdr(r),H̃vr=μ(μr2+M)rk2(r2M)μ2r4+Mμr22M(r2M)(r2M)(k2r3+2M)6M2(k2r3+2M)2Hz
(3.74)
+6Mr(r2M)(k2r3+2M)2+r(μr2+M)k2r3+2Mμr2+Mk2r2dHzdr,H̃rr=2r(Mμr2+μ2r4M(r2M))(r2M)2(k2r3+2M)+12M2r(k2r3+2M)2(r2M)2μ(μr+M)k2(r2M)2Hz
(3.75)
+2(μr2+rM)k2r(r2M)12Mr2(k2r3+2M)22r2(μr2+rM)(r2M)(k2r3+2M)Hz,H̃θθ=Mr2k2r3+2MHz(r)2M(r2M)k4r3+2Mk2dHzdr(r),
(3.76)
where ODE (3.5) with h=Hz has been used. To determine the behavior of these new metric perturbation components close to the future event horizon HA+, one must substitute Hz2M,±(r)h2M,±(r) from Eqs. (3.41) and (3.42). Substituting Hz2M,±(r)h2M,±(r) from Eqs. (3.44) and (3.45) into these expressions gives leading order behavior close to the future event horizon HA+ determined by the relations
H̃vv2M,±=fvv(r)(r2M)±2Mμ,
(3.77)
H̃vr2M,±=(μμ)(1+4Mμ)2k2(1+4M2k2)(r2M)1+fvr(r)(r2M)±2Mμ,
(3.78)
H̃rr2M,±=2(11)Mμ(1+4Mμ)k2(1+4M2k2)(r2M)2+k±(r2M)1+frr(r)(r2M)±2Mμ,
(3.79)
H̃θθ2M,±=fθθ(r)(r2M)±2Mμ,
(3.80)
with fvv, fvr, frr, fθθ being smooth functions of r ∈ [2M, ), which are non-vanishing at 2M, k+ = 0, and k being a non-zero constant depending on k, M and μ. Therefore, multiplying H̃vv2M,+, H̃vr2M,+, H̃rr2M,+, and H̃θθ2M,+ by eμt = eμveμr(r − 2M)−2 gives
eμt+ikzH̃vv2M,+=fvv(r)eμvμr+ikz,
(3.81)
eμt+ikzH̃vr2M,+=fvr(r)eμvμr+ikz,
(3.82)
eμt+ikzH̃rr2M,+=frr(r)eμvμr+ikz,
(3.83)
eμt+ikzH̃θθ2M,+=fθθ(r)eμvμr+ikz,
(3.84)
which can indeed be smoothly extended to the future event horizon HA+.

Remark.
The form of the pure gauge solution defined by Eq. (3.64) can be derived as follows: From Lemma 3.6, a mode solutionh̃of the form (3.48) satisfies the harmonic/transverse-traceless (1.10) gauge conditions. Take a mode solution h in spherical gauge (3.2) add the pure gauge solutionhpg=2aξbfor some vector field
ξ=eμt+ikzζ,
(3.85)
whereζis a vector field which depends only onr. From a direct calculation ofh + hpg, one can see that to obtain a solutionh̃of the form (3.48),ζmust be given by Eq. (3.64).

Remark.
To explicitly see the singular behavior of the mode solutionh±in spherical gauge (3.2) withμ > 0 andk ≠ 0 associated, via Proposition 3.2, to eitherh2M,±, consider directly transforming to ingoing Eddington–Finkelstein coordinates. This transformation gives the following basis elements:
Hrr2M,±=tr2Ht2M,±+2trμHv2M,±+Hr2M,±(r),
(3.86)
Hvv2M,±=Ht2M,±(r),
(3.87)
Hvr2M,±=trHt2M,±(r)+μHv2M,±(r),
(3.88)
Hzz2M,±=Hz2M,±(r),
(3.89)
whereHv2M,±, Ht2M,±, andHr2M,±are the basis for solutions forHv, Ht, andHrconstructed from Proposition (3.2). These relevant expressions can be found from Eqs. (3.15)–(3.17).

First, if 4is a positive integer and the coefficientCNdoes not vanish, then by Eq. (3.89), the basis elementHzz2M,(r)=Hz2M,=h2M,has an essential logarithmic divergence and is therefore always singular at the future event horizonHA+.

IfCN = 0 or 4is not a positive integer, then the basis elementsHz2M,±=h2M,±are given by Eqs. (3.44) and (3.45) with first order coefficients (3.46). Substituting the basis into Eqs. (3.15)–(3.17) for the other metric perturbation component, one finds
Hr2M,±=(r2M)2±2MμM2(±4Mμ1)1+4M2k2+M(4M2(2μ2+k2)±6Mμ1)2(1+4M2k2)(r2M)+O((r2M)2),
(3.90)
Ht2M,±=(r2M)±2Mμ(1+4Mμ)(4Mμ1)4(1+4M2k2)+3+4M2(8μ2k2)±2Mμ(8M2(2μ2+k2)11)8M(1+4M2k2)(r2M)+O((r2M)2),
(3.91)
Hv2M,±=(r2M)1+2MμM2(±4Mμ1)1+4M2k2+M(2M2(2μ2+k2)1±5Mμ)1+4M2k2(r2M)+O((r2M)2)
(3.92)
Transforming to ingoing Eddington–Finkelstein coordinates gives
Hrr2M,±=(r2M)2±2Mμ2M2(12Mμ(11))(4Mμ1)1+4M2k2
(3.93)
+2M2μ(34)+2(97)Mμ(11)4M2(2μ2+k2)1+4M2k2(r2M)+O((r2M)2),Hvv2M,±=(r2M)±2Mμ(1+4Mμ)(4Mμ1)4(1+4M2k2)+O(r2M),
(3.94)
Hvr2M,±=(r2M)1±2MμM(2Mμ(12)1)(±4Mμ1)2(1+4M2k2)+O(r2M),
(3.95)
Hzz2M,±=(r2M)±2Mμ(1+O(r2M)).
(3.96)

Note that the full mode solution h constructed from Proposition 3.2 involves a factor of eμt = eμveμr(r − 2M)−2, so after multiplication by this exponential factor, one can see that the basis elements Hμν2M, are always singular, i.e., a solution with k2 ≠ 0 is always singular at the future event horizon. The components eμtHvv2M,+ and eμtHz2M,+ are unconditionally smooth. However, in general, the components eμtHrr2M,+ and eμtHvr2M,+ remain singular at the future event horizon HA+ unless 4 = 1 or 2+2MμN{0} or −2 + 2 > 2. [In  Appendix E, it is shown that for existence of a solution h with μ > 0, which has k2 = 0 and is finite at infinity (see Sec. III D 2), then μ<316M32<14M.] Hence, neither basis perturbation h± in spherical gauge (3.2) extends, in general, smoothly across the future event horizon HA+.

2. Spacelike infinity iA0

The goal of this section is to identify the admissible boundary conditions for a solution h to ODE (3.5) as r. This requires one to understand the behavior as r of the mode solution h in spherical gauge (3.2) of the linearized vacuum Einstein equation (2.4), which results (through the construction in Proposition 3.2) from h.

In this section, a basis for solution h,± associated with r is constructed. This basis h,± captures the asymptotic behavior of any solution to ODE (3.5) as r. In particular, as r, h,+ grows exponentially and h, decays exponentially. It will be shown that after the addition of the pure gauge solution hpg defined in equations (3.64) and (3.65), h + hpg is a mode solution in harmonic/transverse-traceless gauge (1.10) to the linearized Einstein vacuum equation, which is a linear combination of solutions that grow or decay exponentially as r. The admissible boundary condition will be that the solution should decay exponentially, from which it will follows that h=ah,.

One should note that the functions Pk(r) and Qk(r)μ2r2(r2M)2 admit convergent series expansions in a neighborhood of r = ,

Pk(r)=n=0pnrn,Qk(r)=n=0qnrn,
(3.97)

with p0 = 0, p1 = −4, q0 = −(k2μ2), and q1 = −2M(k2 + 2μ2). Therefore, r = is an irregular singular point of ODE (3.5) according to the discussion of  Appendix B. Eqs. (B18) and (B19) from  Appendix B give

λ±=±μ2+k2,μ±=2±M(2μ2+k2)μ2+k2.
(3.98)

From Theorem B.3, there exists a unique basis for solutions h,±(r) to ODE (3.5) satisfying

h,±=e±μ2+k2rr2±M(2μ2+k2)μ2+k2+Oe±μ2+k2rr1±M(2μ2+k2)μ2+k2.
(3.99)

Therefore, a general solution will be of the form

h=c1h,++c2h,,
(3.100)

with c1,c2R.

Proposition 3.7.

Let h be a solution to ODE (3.5). Let h be the mode solution to the linearized vacuum Einstein equation (2.4) in spherical gauge (3.2) associated with the solution h, and let hpg be the pure gauge solution defined by Eqs. (3.64) and (3.65) such that h + hpg satisfies the harmonic/transverse-traceless gauge (1.10) conditions. Then, the solution h + hpg to ODE (3.5) decays exponentially towards spacelike infinity iA0 if c1 = 0, where c1 is defined by Eq. (3.100).

Proof.
Defining Hz,±(r)h,±(r) and using Eqs. (3.66) and (3.67), one can construct the corresponding basis for solutions as H̃tt, H̃tr, H̃rr, and H̃θθ from Proposition 3.2. Note that Eqs. (3.66) and (3.67) define the components of the mode solution h + hpg to the linearized vacuum Einstein equation (2.4), which satisfies harmonic/transverse-traceless gauge (1.10). Asymptotically, H̃tt, H̃tr, H̃rr, and H̃θθ have the following behavior:
Htt,±=e±μ2+k2rr1±M(2μ2+k2)μ2+k2+Oe±μ2+k2rr2±M(2μ2+k2)μ2+k2,
(3.101)
Htr,±=e±μ2+k2rr1±M(2μ2+k2)μ2+k2+Oe±μ2+k2rr2±M(2μ2+k2)μ2+k2,
(3.102)
Hrr,±=e±μ2+k2rr1±M(2μ2+k2)μ2+k2+Oe±μ2+k2rr2±M(2μ2+k2)μ2+k2,
(3.103)
Hθθ,±=e±μ2+k2rr1±M(2μ2+k2)μ2+k2+Oe±μ2+k2rr±M(2μ2+k2)μ2+k2.
(3.104)
It is clear from these expressions that if c1 = 0, then the mode solution h + hpg decays exponentially as r.

This section summarizes Propositions 3.2–3.5 and 3.7 to give a full description of the permissible asymptotic behavior of a mode solution h in spherical gauge (3.2), which is not pure gauge. This provides a reduction of Theorem 1.1 to proving that there exists a solution h to ODE (3.5), which has μ > 0, k ≠ 0, and obeys the admissible boundary conditions: k2 = 0 and c1 = 0.

Proposition 3.8.
Letμ > 0 andkRwithk ≠ 0. Leth2M,±be the basis for the space of solutions to ODE (3.5) as defined in Eqs. (3.41) and (3.42) andh,±be the basis for the space of solutions to ODE (3.5) as defined in Eq. (3.99). In particular, to any solutionhof ODE (3.5), one can ascribe four numbers k1,k2,c1,c2Rdefined by
h(r)=k1h2M,+(r)+k2h2M,(r),
(3.105)
h(r)=c1h,+(r)+c2h,(r).
(3.106)
Lethbe the mode solution in spherical gauge (3.2) to the linearized vacuum Einstein Eq. (2.4) on the exteriorEAof Schwarzschild black stringSch4×Rassociated with hvia Proposition 3.2. Lethpgbe the pure gauge solution as defined in Eqs. (3.64) and (3.65). Then,h + hpgdecays exponentially towards spacelike infinityiA0and is smooth at the future event horizonHA+ if k2 = 0 andc1 = 0. Moreover,h + hpgsatisfies the harmonic/transverse-traceless gauge conditions (1.10) and cannot be a pure gauge solution.

Under the additional assumption thatkRZ, the mode solutionhdefined above can be interpreted as a mode solution to the linearized vacuum Einstein equation (2.4) on the exteriorEAof the Schwarzschild black string Sch4×SR1. Hence, ifkRZ, the above statement applies to the exteriorEAofSch4×SR1.

Section IV (see Proposition 4.1) will prove the existence of a solution h to ODE (3.5) satisfying the properties of Proposition 3.8. In particular, for all |k|[320M,820M], a solution h to ODE (3.5) with μ>14010M>0, k2 = 0, and c1 = 0 is constructed. If R > 4M, then there exists an integer n[3R20M,8R20M]. Hence, one can choose k such that the constructed h gives rise to a mode solution on Sch4×SR1. Moreover, on Sch4×SR1, h will manifestly have finite energy in the sense that h|ΣH1 and t*h|ΣL2 are finite. (Note that on Sch4×R, h will not have finite energy due to the periodic behavior in z on R.) Thus, Theorem 1.1 follows from Propositions 3.8 and 4.1.

By Proposition 3.8, the Proof of Theorem 1.1 has now been reduced to exhibiting a solution h to (3.5) with μ > 0, k ≠ 0, k2 = 0, and c1 = 0. This section establishes the required proposition thus completing the proof.

Proposition 4.1.

For all |k|[320M,820M], there exists aC((2M, )) solution hto ODE (3.5) with μ > 0, and in the language of Proposition 3.8,k2 = 0 andc1 = 0.

In order to exhibit such a solution h to ODE (3.5), it is convenient to rescale the solution and change coordinates in ODE (3.5) so as to recast as a Schrödinger equation for a function u. This transformation is given in Sec. IV A. In Sec. IV B, an energy functional is assigned to the resulting Schrödinger operator. With the use of a test function (constructed in Sec. IV C), a direct variational argument can be run to establish that for |k|[320M,820M], there exists a weak solution uH1(R) with uH1(R)=1 such that μ > 0. The Proof of Proposition 4.1 concludes by showing that the solution u is indeed smooth for r ∈ (2M, ) and satisfies the conditions of Proposition 3.8, i.e., k2 = 0 and c1 = 0.

To reduce the number of parameters in ODE (3.5), one can eliminate the mass parameter with xr2M, μ̂2Mμ, and k̂2Mk to find

d2hdx2(x)+pk̂(x)dhdx+qk̂(x)μ̂2x2(x1)2h(x)=0,
(4.1)

with

pk̂(x)=1x15x+6x(k̂2x3+1),
(4.2)
qk̂(x)=3x2(x1)k̂2xx13x2(x1)(1+k̂2x3).
(4.3)

Following Proposition C.1 from  Appendix C, one can now transform Eq. (4.1) into the regularized Schrödinger form by introducing a weight function h(x)=w(x)h̃(x) and changing coordinates to x*=r*2M=x+log|x1|. This will produce a Schrödinger operator with a potential, which decays to zero at the future event horizon and tends to the constant k̂2 at spatial infinity. From Proposition C.1, the weight function must satisfy the ODE,

dwdx+(12k2x3)x(1+k2x3)w=0.
(4.4)

The desired solution for the weight function is

w(x)=(1+k̂2x3)x.
(4.5)

ODE (4.1) becomes

d2h̃dx*2(x*)+V(x*)h̃(x*)=μ̂2h̃(x*),
(4.6)

where V:RR can be found from Eq. (C10) to be

V(x*)=k̂2(x1)x+(6x11)(x1)x4+18(x1)2x4(1+k̂2x3)26(4x5)(x1)x4(1+k̂2x3),x(1,),
(4.7)

where x is understood as an implicit function of x*.

As a trivial consequence of Proposition 3.8 in Sec. III D on asymptotics of the solution to ODE (3.5), one has the following proposition for the asymptotics of the Schrödinger equation (4.6).

Proposition 4.2.
Assumeμ̂>0. To any solutionh̃to the Schrödinger equation (4.6), one can ascribe four numbersk̃1,k̃2,c̃1,c̃2Rdefined by
h̃(x*)=k̃1h̃2M,+(x*)+k̃2h̃2M,(x*)asx*,
(4.8)
h̃(x*)=c̃1h̃,+(x*)+c̃2h̃,(x*)asx*,
(4.9)
with
h̃2M,±h2M,±w,
(4.10)
h̃,±h,±w.
(4.11)
The conditions thatc̃1=0andk̃2=0are equivalent to, in the language of Proposition 3.8,c1 = 0 andk2 = 0.

Remark.
In the case 4is not a positive integer or 4is a positive integer andCN = 0, the leading order terms of these basis elements are
h̃2M,±=(x1)±μ̂11+k̂2+O(x1),
(4.12)
h̃,±=e±μ̂2+k̂2xx±(2μ̂2+k̂2)2μ̂2+k̂21k̂2+O1x.
(4.13)

This section establishes a variational argument which, will be used to infer the existence of a negative eigenvalue to the Schrödinger operator in Eq. (4.6).

Proposition 4.3.
LetW:RRand define
E0infvH1(R)vL2(R)=1E(v)v,vL2(R)+Wv,vL2(R).
(4.14)
Suppose that W = p + qwithqC0(R)such that
lim|x|q(x)=0
(4.15)
andp(x)L(R)positive. IfE0 < 0, then there existsuH1(R)such thatuL2(R)=1 and E(u) = E0.

Proof.
By the definition of the infimum, there exists a minimizing sequence (um)mH1(R) and umL2=1 such that
limnE(un)=E0.
(4.16)
Now, un are bounded in H1(R) by the following argument. There exists an MN such that for all mM,
E(um)E0+1.
(4.17)
Hence, for mM,
um,umL2(R)E0+1+supxR|p(x)|+supxR|q(x)|.
(4.18)
Hence, umH1(R) is controlled. Now, using Theorem D.1 from  Appendix D, there exists a subsequence (umn)n such that umnu in H1(R).
Consider
E(um)=R|um|2+p(x)|um|2+q(x)|um|2dx.
(4.19)
Since the Dirichlet energy is lower semicontinuous, only the latter two terms under the integral (4.19) need to be examined more carefully. The middle term in integral (4.19) is simply a weighted L2 integral, so lower semicontinuity is established via
unuLp22=unu,unuLp2=unLq222un,uLp2+uLp22.
(4.20)
Hence,
uLp22unLp222u,unuLp2.
(4.21)
Hence, by weak convergence,
uLp22lim infnunLp22.
(4.22)
Proposition D.2 from  Appendix D establishes that the multiplication operator Mq : uqu is compact from H1(R) to L2(R). Hence, by the characterization of compactness through weak convergence (Theorem D.1 from  Appendix D), qumqu in L2(R). Therefore,
qu,uL2=limmqum,umL2=lim infmqum,umL2.
(4.23)
Hence, the last term under integral (4.19) is also lower semicontinuous. Therefore,
E(u)lim infnE(un)=E0.
(4.24)
Since the infimum is negative, the minimizer is non-trivial. One needs to show that there is no loss of mass, i.e., uL2=1. Note uL2lim infnunL2=1. Hence, suppose uL2<1 and define ũ=uuL2 so ũL2=1, then
E(ũ)=E0uL2(R)2E0
(4.25)
since uL21. Hence, one would obtain a contradiction to the infimum if uL2<1.

Corollary 4.4.
LetW = VwithVas defined in Eq. (4.7), then
E(v)v,vL2(R)+Vv,vL2(R)E0infvH1(R)vL2(R)=1E(v)
(4.26)
satisfies the assumptions of Proposition 4.6.

Proof.
The function V:RR can be written as V = p + q with p and q as follows. Define
p(x*)k̂2x1x,
(4.27)
q(x*)(6x11)(x1)x4+18(x1)2x4(1+k̂2x3)26(4x5)(x1)x4(1+k̂2x3),
(4.28)
where in these expressions x considered as an implicit function of x*. Since x ∈ (1, ), it follows that p(x*) > 0 for all x*R. Moreover,
supx*R|p(x*)|=1.
(4.29)
Therefore, pL(R). Note that the function q satisfies lim|x*|q(x*)=0. Hence, the assumptions of Proposition 4.3 hold.

ODE (4.6) is now in a form where a direct variational argument can be used to prove that there exists an eigenfunction of the Schrödinger operator associated with the left-hand side of ODE (4.6) with a negative eigenvalue, i.e., μ̂2<0. The following proposition constructs a suitable test function such that it is in the correct function space, H1(R), and, for all |k̂|[310,810], implies that the infimum of the energy functional in Eq. (4.26) is negative. (As will be apparent, the negativity is inferred via complicated but purely algebraic calculations.)

Proposition 4.5.

DefineuT(x*)x(1+|k̂|2x3)(x1)1ne4|k̂|(x1), wherexis an implicit function ofx*, nis a finite non-zero natural number,k̂R\{0}and defineEandE0as in Eq. (4.26) of Corollary 4.4. Then,uTH1(R)and forn = 100 and|k̂|[310,810], E0E(uT)uTL2(R)2<14000.

Proof.
Let kN{0} and define the following functions:
fj(x)xj1(x1)2n1e8|k̂|(x1).
(4.30)
The H1(R)-norm of uT can be expressed as
uTH1(R)2=1x1xduTdx2xx1dx+1|uT|2xx1dx,
(4.31)
where on the right-hand side the change of variables from x*R to x ∈ (1, ) has been made. To calculate uTL2(R), it is useful to write it as a linear combination of the functions fk in Eq. (4.30). Explicitly, one can show that
|uT|2xx1=f4(x)+2|k̂|2f7(x)+|k̂|4f10(x).
(4.32)
Similarly, one can show that
x1xduTdx2xx1=j=111cjfj1(x),
(4.33)
with
c1=1,c2=22n8|k̂|,c3=1+1n2+2n+16|k̂|+8|k̂|n+16|k̂|2,c4=8|k̂|8|k̂|n24|k̂|2,c5=10|k̂|2n40|k̂|3,c6=8|k̂|2+2|k̂|2n2+10|k̂|2n+80|k̂|3+16|k̂|3n+32|k̂|4,c7=40|k̂|316|k̂|3n48|k̂|4,c8=8|k̂|4n32|k̂|5,c9=16|k̂|4+|k̂|4n2+8|k̂|4n+64|k̂|5+8|k̂|5n+16|k̂|6,c10=32|k̂|58|k̂|5n32|k̂|6,c11=16|k̂|6.
(4.34)
One can express E(uT) with the change of variables from x* to x as
E(uT)=1x1xduTdx2+V(uT)2xx1dx.
(4.35)
The integrand can be written as
x1xduTdx2+V(uT)2xx1=j=111ajfj1(x),
(4.36)
with
a1=0,a2=2+n+8n|k̂|n,a3=1+1n2+16|k̂|+16|k̂|2+2+8|k̂|n,a4=|k̂|8(1+n)n+33|k̂|,a5=(21n10)|k̂|2n40|k̂|3,a6=|k̂|22(1+5n2n2)n2+16(1+5n)|k̂|n+32|k̂|2,a7=|k̂|38(2+5n)n+39|k̂|,a8=|k̂|4(8+15n)n+32|k̂|,a9=|k̂|41+8n+22n2n2+8(1+8n)n+16|k̂|2,a10=|k̂|58(1+4n)n+33|k̂|,a11=17|k̂|6.
(4.37)
Therefore, if one can compute the integrals
Ij1fj(x)dx
(4.38)
for k = 0, …, 10, then one can compute uTL2(R), duTdx*L2(R), and E(uT).
Defining a change variables in the integrals (4.38) by t = x − 1, integrals (4.38) become
Ij=0(t+1)j1t2n1e8|k̂|t.
(4.39)
Note that the confluent hypergeometric function of the second kind U(a, b; z) can be defined as
U(a,b;z)1Γ(a)0(t+1)ba1ta1ezt
(4.40)
for a,b,zC with Re(a) > 0 and Re(z) > 0, where Γ(a) is the Euler Gamma function, which can be defined through the integral
Γ(a)=0ta1etdt
(4.41)
for aC with Re(a) > 0. For a reference, see chapter 9 of Ref. 27. Therefore, setting a=2n, b=k+2n, and z=8|k̂| gives
Ij=Γ2nU2n,j+2n;8|k̂|.
(4.42)
The function U(a, b; z) satisfies the following recurrence properties (see Chap. 9 of Ref. 27 and Chap. 16 of Ref. 28):
U(0,b;z)=1,
(4.43)
U(a,b;z)z1bU(1+ab,2b;z)=0,
(4.44)
U(a,b;z)aU(a+1,b;z)U(a,b1;z)=0,
(4.45)
(ba1)U(a,b1;z)+(1bz)U(a,b;z)+zU(a,b+1;z)=0.
(4.46)
Setting a=2n, b=1+2n, and z=8|k̂| in Eq. (4.44) and using Eq. (4.43) allow one to calculate I1. Setting a=2n, b=2+2n, and z=8|k̂| in Eq. (4.45) and using I1 and Eq. (4.43) allow one to calculate I2. Setting a=2n, b=j+2n, and z=8|k̂| in Eq. (4.46) and using Ij−1, …, I1 and Eq. (4.43) allow one to calculate Ij. Finally, one can show that I0 < by the following argument. One can see from the definition of Ij in Eq. (4.39) that
I0=11x(x1)(x1)2ne8|k̂|(x1)dx.
(4.47)
Now, since e8|k̂|(x1)<1 on x ∈ (1, ) and (x1)2n1x<12(x1) for n ≥ 1 on x ∈ (2, ),
I0121x(x1)(x1)2n+122(x1)e8|k̂|(x1)<.
(4.48)
Using the recurrence properties in Eqs. (4.43)–(4.46) and estimate (4.48) allows one to explicitly show that uTH1(R)< for n ≥ 1, k̂R\{0}, i.e., uTH1(R). Moreover, one can calculate E(uT)uTL2(R). Explicitly, E(uT)uTL2(R) is given by
E(uT)uTL2(R)2=|k̂|2i=19pi(n)|k̂|i1j=110qj(n)|k̂|i1,
(4.49)
with
p1(n)16+416n+5576n2+36176n3+123809n4+234794n5+244459n6+128034n7+25560n8,p2(n)32n(16+336n+3296n2+15572n3+29107n4+21238n5+4361n6366n7),p3(n)128n2(56+924n+6130n2+20133n3+11972n43365n5466n6),p4(n)1024n3(56+700n+2750n2+6041n31715n418n5),p5(n)2048n4(140+1260n+2225n2+3443n31758n4),p6(n)32768n5(28+168n+43n2+111n3),p7(n)917504n6(2+7n3n2),p8(n)1048576n7(2+3n),p9(n)1048576n8,q1(n)116+288n+2184n2+9072n3+22449n4+33642n5+29531n6+13698n7+2520n8,q2(n)4n(144+2016n+12104n2+39120n3+71801n4+73494n5+38171n6+7590n7),q3(n)128n2(72+756n+3534n2+8535n3+11180n4+7137n5+1642n6),q4(n)1536n3(56+420n+1510n2+2535n3+2351n4+706n5),q5(n)2048n4(252+1260n+3485n2+3495n3+2554n4),q6(n)8192n5(252+756n+1653n2+669n3+512n4),q7(n)393216n6(14+21n+39n2),q8(n)524288n7(18+9n+16n2),q9(n)9437184n8,q10(n)4194304n9.
(4.50)
Taking n = 100, one can check, via Sturm’s algorithm,29 that the polynomial
p(n,|k̂|)i=19pi(n)|k̂|i1
(4.51)
has two distinct real roots in |k̂|(0,1). Evaluating p(100,|k̂|) at |k̂|=0, |k̂|=310, |k̂|=810, and |k̂|=1 yields
p(100,0)>0,p100,310<0,p100,810<0andp(100,1)>0.
(4.52)
Hence, E(uT)uTL2(R)2 must be negative for all |k̂|[310,810]. Taking the derivative of E(uT)uTL2(R)2 with respect to |k̂| yields another rational function of