Black hole nonmodal linear stability: the Schwarzschild (A)dS cases
The nonmodal linear stability of the Schwarzschild black hole established in Phys. Rev. Lett. 112 (2014) 191101 is generalized to the case of a nonnegative cosmological constant . Two gauge invariant combinations of perturbed scalars made out of the Weyl tensor and its first covariant derivative are found such that the map with domain the set of equivalent classes under gauge transformations of solutions of the linearized Einstein’s equation, is invertible. The way to reconstruct a representative of in terms of is given. It is proved that, for an arbitrary perturbation consistent with the background asymptote, and are bounded in the the outer static region. At large times, the perturbation decays leaving a linearized Kerr black hole around the Schwarzschild or Schwarschild de Sitter background solution. For negative cosmological constant it is shown that there are choices of boundary conditions at the time-like boundary under which the Schwarzschild anti de Sitter black hole is unstable. The root of Chandrasekhar’s duality relating odd and even modes is exhibited, and some technicalities related to this duality and omitted in the original proof of the case are explained in detail.
- I Introduction
- II Tensor fields on a spherically symmetric space-time
III The Linearized Einstein equation: odd sector
- III.1 Gauge transformations and gauge invariants
- III.2 Solution of the linearized Einstein equation
- III.3 Measurable effects of the perturbation on the geometry
- III.4 Nonmodal linear stability of the black holes
IV The Linearized Einstein equation: even sector
- IV.1 Gauge transformations and gauge invariants
- IV.2 The linearized Einstein equation
- IV.3 Solution of the linearized Einstein equation
- IV.4 The ubiquitous Regge-Wheeler equation
- IV.5 Chandrasekhar’s duality
- IV.6 Measurable effects of the perturbation on the geometry
- IV.7 Non-modal linear stability of the black holes
- V Discussion
- VI Acknowledgments
One of the most salient open problems in classical General Relativity (GR) is proving the stability of the outer region of the stationary electro-vacuum black holes in the Kerr-Newman family. A complete proof of stability in the context of the non linear GR equations has only been given for Minkowski spacetime Christodoulou:1993uv ; the stability problem of more complex solutions of Einstein’s equation is usually approached by analyzing the behavior of linear test fields satisfying appropriate boundary conditions in order to establish if unbounded solutions are allowed. Scalar test fields provide a first clue, whereas gravitational waves, that is, metric perturbations propagating on the background spacetime , give a more realistic approach to the problem. For vacuum spacetimes, assuming a cosmological constant , the metric perturbation satisfies the linearized Einstein equation (LEE)
obtained by assuming that on a fixed four dimensional manifold (the spacetime) there is a mono-parametric family of solutions of the vacuum Einstein field equation
around the “unperturbed background” , and Taylor expanding (2) at . For tensor fields that depend functionally on the metric we use a dot to denote the “perturbed field”, which is the field obtained by taking the first derivative with respect to at , e.g., for the Riemann tensor,
We make an exception for the metric field itself by adopting the standard notation and, as usually done in linear perturbation theory, defining , and . Note that this convention implies that
where is an arbitrary vector field; these amount to the first order in change of the metric under the pullback by the flow generated by the vector field . Any two solutions and such that
are related by this diffeomorphism and thus physically equivalent, this being the gauge freedom of linearized gravity. If a tensor field is a functional of the metric , then is a linear functional of and, under the gauge transformation (5) we find that
In particular, if is a scalar field, is gauge invariant iff for every
vector field, that is,
if is constant in the background.
Curvature related scalar fields (CS for short, not to be confused with the tetrad components of the Weyl tensor in the Newman-Penrose formalism), are scalar fields obtained by a full contraction of a polynomial in the Riemann tensor and its covariant derivatives, the metric and the volume form. Although these fields partially characterize the metric, it is well known that the information they carry is limited, an extreme example of this fact being the vanishing scalar invariants spacetimes, which have a nonzero Riemann tensor and yet every CS vanishes Pravda:2002us . This fact may suggest that the perturbation of CS under a given solution of the LEE provide very limited information on . It was shown in Dotti:2013uxa , however, that for a Schwarzschild black hole background, there are two gauge invariant combinations of perturbed CSs that fully characterize the gauge equivalence class of the corresponding solution of the LEE. More precisely, let be the Weyl tensor, the volume form, and consider the CSs
The background values of these fields, that is, their value for the Schwarzschild solution, are , and , where is the mass and the areal radius, then the fields
made out of the first order variations of these CS, are gauge invariant. It was shown in Dotti:2013uxa that the linear map
with domain the equivalence classes of smooth solutions of the LEE preserving the asymptotic flatness, is injective.
that the scalars , beyond measuring the “amount of distortion” caused by the perturbation,
encode all the relevant information on the perturbation class . A way to construct a class representative from
the fields is indicated in Dotti:2013uxa .
The stability concept introduced in Dotti:2013uxa is based on i) the existence of the CSs for which (9) is injective on smooth solutions of the LEE that preserve the asymptotic flatness, and ii) the proof that for this class of solutions of the LEE the scalar fields are bounded. More precisely, it is proved in Dotti:2013uxa that in the outer region
where are constants that can be obtained from the perturbation field datum at a Cauchy slice. Given that
the scalars contain all the gauge invariant information on the perturbation, the fact that they remain bounded
as the perturbation evolves through spacetime gives a meaningful notion of linear stability.
This concept of stability, that we call nonmodal, should be compared with prior linear gravity stability notions for the Schwarzschild black hole. To this end we review the results in a short list of papers that were crucial in the development of this subject. It is important to stress that they all use the spherical symmetry of the Schwarzschild spacetime to decompose a metric perturbation
into even () and odd ()
modes. Here refers to
the eigenspace of the Laplace-Beltrami operator acting on real scalar fields on corresponding to the eigenvalue , is an index labeling a particular
basis of this dimensional space, and the parity accounts for the way transforms when pull-backed
by the antipodal map on (for details refer to Section II.1).
The first work on the linear stability of the Schwarzschild black hole is T. Regge and J. Wheeler 1957 paper Regge:1957td , where the decomposition (11) was proposed and the modes where recognized to be non-dynamical. At the time the very notion of black hole was unclear (the term “black hole” was coined by J. Wheeler some ten years later), and Kerr’s solution had not yet been discovered. This is probably why, although the even piece of the perturbation was readily identified as a mass shift in Regge:1957td , the odd modes, which corresponds to perturbing along a Kerr family with , was misunderstood (see the paragraph between equations (37) and (38) in Regge:1957td ) and the opportunity of producing a “slowly rotating” black hole at a time when there was no clue about a rotating black hole solution was missed 111I thank Reinaldo Gleiser for this observation.. The decomposition (11) in Regge:1957td was done in Schwarzschild coordinates in what came to be called the Regge-Wheeler (RW, for short) gauge. The LEE in the even and odd sectors were shown to decouple, and the dynamical odd sector of the LEE reduced to an infinite set of two dimensional wave equations, individually know as the RW equation:
where (adding a cosmological constant for future reference) , , and
The even sector LEE, a much more intricate system of equations, was disentangled by F. Zerilli in his 1970 paper Zerilli:1970se and shown to be equivalent to the wave equations
For the RW and Zerilli potentials first appeared in guven . The solution of the LEE in the RW gauge is then given in the form
where is a bilinear differential operator acting on and the
spherical harmonics . Note that, since focus is on non-stationary modes, only enter this formulation.
The non-stationary solution space of the LEE is thus parametrized by the infinite set of scalar fields
that enter the series (11) through (14).
Every notion of gravitational linear stability of the Schwarzschild black hole prior to Dotti:2013uxa was concerned with the behavior of the potentials of isolated modes (we call this “modal linear stability”). In particular:
In Regge:1957td it was shown that separable solutions that do not diverge as require , ruling out exponentially growing solutions in the odd sector.
In Zerilli:1970se it was shown that separable solutions that do not diverge as require , ruling out exponentially growing solutions in the even sector.
In Price:1971fb it was shown that, for large and fixed , decays as (an effect known as “Price tails”).
In wald , the conserved energy
was used to rule out uniform exponential growth in time.
Also in wald , a point wise bound on the RW and Zerilli potentials was found in the form
where the constants are given in terms of the initial data
To understand the limitations of these results it is important to keep in mind that
the are an infinite set of potentials defined on the orbit space , whose
first and second order derivatives
enter the terms in the series (11) through (14), together with first and second derivatives of
the spherical harmonics.
Two extra derivatives are required to calculate the perturbed Riemann tensor, as a first step to measure the
effects of the perturbation on the curvature. Thus, the relation of the potentials
to geometrically meaningful
quantities is remote, and the usefulness
of the bounds (16) to measure the strength of the perturbation is far from obvious.
The motivation of the nonmodal approach can be better understood if we put in perspective the progress made in wald , equation (16). The fact that the two dimensional wave equations (RWE) and (ZE) can be solved by separating variables () had previous works on linear stability focus on showing that must be real, a limited notion of stability that does not even rule out, e.g., linear growth wald . To obtain the type of bounds (16) one must “undo” the separation of variables and reconsider the original equations (RWE) and (ZE), as this allows to work out results that are valid for generic, non separable solutions. The idea behind the non modal stability concept introduced in this paper, is to go one step further and “undo” the separation of the variables that antecedes the reduction of the LEE to (RWE) and (ZE), in order to get bounds for truly four dimensional quantities. When trying to make this idea more precise, one is immediately faced to the problem of which four dimensional quantity one should look at. If this quantity is to measure the strength of the perturbation, it should be a field related to the curvature change, and it should also be a scalar field, since there is no natural norm for a tensor field in a Lorentzian background. If this scalar field does not obey a four dimensional wave equation (or some partial differential equation), we do not have a clear tool to place bounds on it. Thus, we are naturally led to the question of the existence of scalar fields made from perturbed CSs that, as a consequence of the LEE, obey some wave-like equation. This is the problem we address in this paper. The name nonmodal stability is borrowed from fluid mechanics, where the limitations of the normal mode analysis were realized some thirty years ago in experiments involving wall bounded shear flows schmid . In that problem, the linearized Navier-Stokes operator is non normal, so their eigenfunctions are non orthogonal and, as a consequence, even if they individually decay as , a condition that assures large stability, there may be important transient growths schmid . Take, e.g., the following simple example (from schmid , section 2.3 and figure 2) of a system of two degrees of freedoms: obeying the equation , with a matrix with (non orthogonal!) eigenvectors , say,
Consider the case . Note that, although , that is, normal modes decay exponentially, if
reaches a maximum norm at a finite time before decaying to zero.
For the odd sector of the LEE, a four dimensional approach relating the metric perturbation with a scalar potential defined on (instead of ) was found in Dotti:2013uxa , where it was noticed that the sum over of (14) simplifies to
where is a field assembled using spherical harmonics and the RW potentials:
The odd sector LEE equations (RWE) for combined to the spherical harmonic equations for the spherical harmonics , turn out to be equivalent to what we call the four dimensional Regge-Wheeler equation which, adding a cosmological constant, reads
Note however that is no more than the collection of ’s, so its connection to geometrically relevant
fields is loose.
Much more important is the fact, also proved in Dotti:2013uxa for , that
the LEE implies that the field also satisfies the 4DRWE, as this is what allows us to place a point wise bound on .
The even sector of the LEE is more difficult to approach. Is the simplicity of the RW potential (12), with the obvious term, what suggested considering the field (19). The way appears in (13), instead, is a clear indication that there is no natural 4D interpretation of (ZE). Is a map exchanging solutions of the RWE and ZE equations, found by Chandrasekhar ch2 , what ultimately allows us to also reduce the even non stationary LEE equations to (4DRWE). As a consequence, the entire set of non stationary LEE reduces to two fields satisfying equation (4DRWE), as stated in Theorem 4 below.
The purpose of this paper is twofold: (i) to extend the results in Dotti:2013uxa
to Schwarzschild black holes in cosmological backgrounds, and (ii) to explain in detail
a number of technicalities omitted in Dotti:2013uxa due to the space limitations imposed by the letter format.
For we give a proof of non-modal stability.
We leave aside the treatment of stability of the Schwarzschild anti de Sitter (SAdS) black hole, since the issue of non global hiperbolicity and
ambiguous dynamics due to the conformal timelike boundary takes us away from
the core of the subjects addressed here.
We show however that there is (at least) one choice of
Robin boundary condition at the time-like boundary for which
there is an instability, and we explicitly exhibit this instability and its effect on the background geometry.
To the best of our knowledge, this has not been informed before.
A systematic study of the gravitational linear stability of SAdS black holes under different boundary conditions
is to be found in bernardo .
We have found that defined in (8) are appropriate variables to study the most general gravitational linear perturbations of Schwarzschild (A)dS black holes. The pieces of these fields encode the relevant information on the stationary modes, which are perturbations within the Kerr family (parametrized by mass and the angular momenta components), whereas the terms encode the dynamics. More precisely:
contains no term, time independent terms proportional to the first order angular momenta components , and a time dependent piece obeying the 4DRW equation.
contains no term, a time independent piece proportional to a mass shift , and a time dependent piece which, for , can be written in terms of fields obeying the 4DRW equation.
Once the appropriate set of perturbation fields is given, and their relation to the 4DRW equation established for ,
we may adapt to the 4DRW equation the techniques used to prove boundedness of solutions of the scalar wave equation,
in order to analyze the behavior of the fields. As an example, the result of Kay and Wald Kay:1987ax was
used in Dotti:2013uxa to prove (10) in the Schwarzschild case, and is adapted here to prove that (10) holds
also for positive .
We can go further and
take advantage of the growing literature on decay of solutions of the scalar wave equation on S(A)dS backgrounds, as many of these results
are expected to hold also for (4DRWE). Specific time decay results for the 4DRW equation, somewhat expected
from Price’s result Price:1971fb , can be found in Blue:2003si (see also the recent preprint Dafermos:2016uzj ).
Putting together the bijection (9), the above description of the stationary () and dynamic ()
pieces of , and the time decay results,
the following picture emerges for a perturbed Schwarzschild (SdS) black hole: a generic perturbation contains a mass shift, infinitesimal angular momenta
and dynamical degrees of freedom; at large times the dynamical degrees of freedom decay and what is left
is a linearized Kerr (Kerr dS) black hole around the background Schwarzschild (SdS) solution.
Through the paper, calculations are carried leaving unspecified whenever possible, and specializing when necessary. Among the many current treatments of linear perturbations of spherically symmetric spacetimes, we have made heavy use of the excellent paper Chaverra:2012bh , which we found particularly well suited to our approach.
Ii Tensor fields on a spherically symmetric space-time
A spherically symmetric space-time is a warped product of a Lorentzian two-manifold with the unit sphere , for which we will use the standard angular coordinates :
The form of the metric (20) implies that inherits the isometry group of
as an isometry subgroup. Here
are the proper rotations and is the antipodal map .
Since acts transitively on , we find that , this is why
is called the ()
Equation (20) exhibits our index conventions, which we have adopted form ref Chaverra:2012bh : lower case Latin indexes are used for tensors on , upper case Latin indexes for tensors on , and Greek indexes for space-time tensors. We will furthermore assume that
Tensor fields introduced
with a lower index (say ) and then shown with an upper index are assumed
to have been acted upon with the unit metric inverse ,
and similarly for upper indexes moving down. and are the covariant derivative, volume form (any chosen orientation) and metric inverse
for ; and are the covariant derivative and volume form
on the unit sphere, for which we assume the standard orientation .
As an example, in terms of the differential operators and , the Laplacian on scalar fields reads
ii.1 Covector and symmetric tensor fields
The Einstein field equation, as well as its linearized version around a particular solution, is expressed as an equality among symmetric tensor fields.
The metric perturbation may be subjected to gauge transformations of the form (5).
This is why we are interested in the decomposition on a spherically symmetric spacetime of covector fields
(such as )
and symmetric rank two tensor fields such as .
We will assume all tensor fields on are smooth. As a consequence their components will be square integrable on and can be expanded using a real orthonormal basis of scalar spherical harmonics
where numbers an orthonormal basis of the -dimensional eigenspace with eigenvalue of the Laplace-Beltrami operator on scalar functions (). An explicit choice for the subspaces is and
We denote the subspace of and
A covector field on
contains the covector which, according to Proposition 2.1 in Ishibashi:2004wx , can be uniquely decomposed as where . This last condition implies that is dual to the differential of an scalar, . It then follows that, introducing and for later convenience,
where the odd piece of is
and its even piece is
For a given covector field , the scalar fields are unique up to an -constant, thus they are unique if we require that they belong to
a condition that we will assume. The symmetric tensor field
contains two covector fields
and a symmetric tensor field . Note that
and are covector fields on parametrized on .
Using Proposition 2.2 in Ishibashi:2004wx and the fact that there are no transverse traceless symmetric tensor fields on Higuchi:1986wu , we find that the symmetric tensor in (31) can be uniquely decomposed into three terms:
Introducing , and we arrive at (c.f., Chaverra:2012bh , Section IV.A)
A generic smooth metric perturbation admits the following decomposition:
where the odd piece of is
and the even piece is
The proof of the following lemma follows from straightforward calculations:
From now on we will assume the required conditions for uniqueness of and .
The linearized Ricci tensor admits a decomposition analogous to (31)-(36).
Given that scalar fields, divergence free covector fields (which are all of the form ) and transverse traceless symmetric tensors
on span inequivalent representations, and that the linear map
is invariant, this map cannot mix odd and even sectors Ishibashi:2004wx . This implies that
is a linear functional of only, and similarly
depends only on .
Note from (25) that
, is a basis of Killing vector fields on generating rotations around orthogonal axis, normalized such that the maximum length of their orbits is . The square angular momentum operator is the sum of the squares of the Lie derivatives along these Killing vector fields:
This operator commutes with the maps and (and similarly in the even sector). As a consequence, the piece of depends only on the piece of (see equation (38)):
These perturbations correspond to infinitesimal rotation, i.e., to deformations towards a stationary Kerr solution.
The mode is defined in a way analogous to (42), i.e., keeping a single term in the spherical harmonic expansion of the even fields . It is important to note that
The different behavior under parity is the signature that distinguishes odd from even modes.
ii.2 The Schwarzschild (A)dS solution
The Schwarzschild / Schwarzschild (anti) de-Sitter (S(A)dS) is the only spherically symmetric solution of the vacuum Einstein equation with a cosmological constant :
the Ricci tensor of the Lorentzian metric . The manifold is , the metric (c.f. equation (20))
is the mass of the solution, in (47) gives the Minkowski (), de Sitter () and anti de Sitter
() spacetimes. in (47) corresponds to a Schwarzschild black hole if , a
Schwarzschild de Sitter (SdS) black hole if and ,
and SAdS black hole if .
The Killing vector field
is timelike in the open sets defined by and spacelike in the open sets defined by . The null hypersurfaces of constant
positive where are the horizons, they cover the curvature singularity at .
In any open set where we may define a “tortoise” radial coordinate by
and a coordinate through
The metric in static coordinates is
ii.2.1 Horizons and the static region
The Schwarzschild () and the SAdS () black holes have a single horizon at , satisfying
in terms of which
These black holes have a non-static region I defined by and a static region II defined by .
The SdS black holes are those for which ( has single real root when , and this root is negative). For SdS black holes has one negative () and two positive roots , , with as , and Lake
There is a non-static region I defined by adjacent to a static region II (), and a further
non-static region III defined by .
This paper focuses on the stability of the static region II of Schwarzschild and S(A)dS black holes.
ii.2.2 The bifurcation sphere at
Let be the function
is smooth in I II (i.e., for SdS, for Schwarzschild and SAdS), since is smooth with a simple zero at
and no zeros in I and II.
Consider the following solution of (48)
The most general solution is obtained by adding (possibly different) constants to the left and right of , however, for as in (55), the function
is smooth in I II, and monotonically growing, so it has an inverse . Introduce in addition to above, then
Note that and , therefore (47) is equivalent to
and . We may now take two extra copies of I and II (call these and ) and define
The black hole metric (47) in is again given by equation (59),
except that now .
It can be checked that (59) is smooth on times the open region of the plane defined
by ( for SdS). This region contains two copies of I and two copies of II.
Since is a function of the product , the metric (59) has the discrete symmetry
The invariant set is a sphere of radius , called bifurcation sphere.
Integrating (48) in region II we find that, for , after choosing an integration constant,
for the integration constant can be chosen such that