A dialogue of multipoles: matched asymptotic expansion for caged black holes
Abstract:
No analytic solution is known to date for a black hole in a compact dimension. We develop an analytic perturbation theory where the small parameter is the size of the black hole relative to the size of the compact dimension. We set up a general procedure for an arbitrary order in the perturbation series based on an asymptotic matched expansion between two coordinate patches: the near horizon zone and the asymptotic zone. The procedure is ordinary perturbation expansion in each zone, where additionally some boundary data comes from the other zone, and so the procedure alternates between the zones. It can be viewed as a dialogue of multipoles where the black hole changes its shape (mass multipoles) in response to the field (multipoles) created by its periodic “mirrors”, and that in turn changes its field and so on. We present the leading correction to the full metric including the first correction to the areatemperature relation, the leading term for black hole eccentricity and the “Archimedes effect”. The next order corrections will appear in a sequel. On the way we determine independently the static perturbations of the Schwarzschild black hole in dimension , where the system of equations can be reduced to “a master equation” — a single ordinary differential equation. The solutions are hypergeometric functions which in some cases reduce to polynomials.
1 Introduction and summary
1.1 Background
In the presence of extra compact dimensions, there exist several phases of black objects, namely massive solutions of General Relativity, depending on the relative size of the object and the relevant length scales in the compact dimensions. For concreteness, we consider a background with a single compact dimension — , where is the total spacetime dimension and (in order to avoid spacetimes with 2 or less extended spatial dimensions where the presence of a massive source is inconsistent with asymptotic flatness).
In this background one expects at least two phases of black object solutions: when the size of the black object is small (compared to the size of the extra dimension) one expects the region near the object to closely resemble a dimensional black hole, while as one increases the mass one expects that at some point the black hole will no longer fit in the compact dimension and a black string, whose horizon winds around the compact dimension will be formed. Thus we distinguish between the black hole and the black string according to their horizon topology which is either spherical — or cylindrical — , respectively. We shall sometimes refer to such a black hole in a compact dimension as a “caged black hole”.
More generally one could consider backgrounds where is any dimensional compact Ricciflat manifold such as the dimensional torus, the K3 surface or a CalabiYau 3fold. For a general more black object phases will exist, but we expect that generically the essential phase transition physics between any two specific phases will be qualitatively similar to the case.
This system raises several deep questions in general relativity [1]: during the realtime phase transition a naked singularity may be encountered which may require intervention from quantum gravity and an amendment to Cosmic Censorship; the transition would certainly produce some sort of a “thunderbolt” [2, 3]; the system exhibits a critical dimension for stability in at least two instances [1, 4]; it is a prototype example for the failure of black hole uniqueness in higher dimensions [5, 6]; a novel kind of topology change is expected to play a central role [1]; and there is an ongoing debate regarding the correct phase diagram, especially whether a stable nonuniform^{1}^{1}1not invariant under translation along the compact direction. string phase exists (for all ) [7].
Considerable work, much of it recent, went into finding the various solutions in this background. A uniform black string is readily described analytically by an arbitrary mass Schwarzschild solution in dimensions with an added spectating coordinate. Gregory and Laflamme (GL,1994) discovered that this solution develops a tachyon below a certain critical mass [8]. Horowitz and Maeda (2000) gave an argument for the existence of stable nonuniform strings [7]. Gubser analytically perturbed the critical GL string to find approximate expressions for nonuniform solutions (which we interpret to be unstable) [9]. DeSmet attempted to find analytic solutions by classifying 5d algebraically special metrics (the same method Kerr used successfully in 4d to obtain the rotating black hole) not finding any novel ones in this background [10]. Harmark and Obers found a clever ansatz that reduces the number of unknown metric functions from 3 to 2, but could not completely solve the equations either [11]. Recently Sorkin generalized the analysis of Gubser from 5d (and 6d which was done by Wiseman [12], fixing some problems with the earlier analysis) to an arbitrary dimension, discovering an interesting critical dimension: for the nonuniform branch emanating from the GL point changes its nature and essentially becomes stable [4]. A critical dimension for a different point in the phase diagram was already predicted in [1].
The limited success of analytical methods created a demand for numerical solutions. The branch of nonuniform black string solutions emanating from the GL point was obtained numerically by Wiseman [12] (see also earlier work in [13] and a postanalysis in [14, 15]) who managed to formulate axiallysymmetric gravitostatics (namely, essentially 2d) in a “relaxation” form (a procedure familiar from electrostatics) while presenting the constraints through “CauchyRiemann — like” relations. Even though there is no definitive answer yet whether these solutions are indeed unstable, the author argued that irrespectively they cannot serve as an endpoint for the decay of the GL string. While all the solutions we mentioned above are static Choptuik et al. performed a demanding numerical time evolution for the decay of the black string, but had to stop before the end state was reached due to an essential limitation of the algorithm used (grid stretching) in the high curvature region which forms [16].
More recently the focus shifted from black strings to black holes. While intuition leads us to expect that a small black hole should exist being indifferent to the existence of a much larger compact dimension, no analytic solution is available to date. In [17] indications for the nature of the phase transition were gained from an analysis of possible timesymmetric initial data. In [18] the closely related problem of black holes in a braneworld was tackled numerically. In [19, 20] it was shown that indeed there are order parameters such that the black hole and black string are at finite values, as was assumed in [1], and moreover [20] announced most of the quantitative results of the current paper. In [21, 22] numerical black hole solutions were presented for the first time in 5d and 6d respectively, giving strong evidence for their existence. Finally, [23] presented a “first order analytic approximation” of small black holes in the framework of the HarmarkObers coordinates [11], and [24] found an analytic approximation for a small black hole on a brane.
1.2 Motivation and basic setup
In this paper we present the first analytic (though perturbative) procedure to obtain solutions for small black holes (BH’s). Let us introduce some notation (see figure 1). We denote by the “cylindrical” coordinates, where is the coordinate along the compact dimension whose period we denote by , and is the radial coordinate in the extended spatial dimensions. The problem is characterized by a single dimensionless parameter, for instance the dimensionless mass where is the dimensional Newton constant and is the mass (measured at infinity), or where is the inverse temperature. In the vicinity of the black hole it is useful to introduce “spherical” coordinates as well. We denote by the Schwarzschild radius of the BH (in the small BH limit), and one has where is a dimensionless constant.
There is reason to expect good analytic control of small black holes even if we do not have a complete analytic solution since we have two good approximation in two different regions which overlap: for the metric is expected to resemble closely a dimensional SchwarzschildTangherlini BH [25], while for the gravitational field is weak and the newtonian approximation holds. Hence or more precisely is our small parameter for the perturbation.
The motivations for this research are first to obtain a theoretical description of this simple system which is important on its own right, and second to gain understanding of the phase transition physics through combination with numerical work. The symbiosis with numerical work comes close to serve as a partial substitute of experiments (which are sorely absent in this field): the numerics are essential for understanding big black holes close to the phase transition where the perturbative expansion is expected to break down, and the analytic control serves to formulate the aims and methods of the numerics. Moreover, the two can be used to test and confirm one another, as was the case for this research and [20, 21]. As it turns out the largest BH’s obtained numerically show only a single multipole mode correction to their spherical horizon [21, 22], and that lends some hope that the analytic expansion would retain some validity for large BH’s as well.
Basic setup.
The first decision to be made it to choose the coordinates and ansatz for the analysis. At first one would hope to use a single coordinate patch for the whole metric. However, [18] showed that in the popular conformal coordinates (where the metric in the plane is in conformal form , see also below (91)) the coordinate size of the horizon is a conformal invariant and hence the coordinate patch necessarily changes with . A similar phenomenon happens in the HarmarkObers coordinates which are a semiinfinite cylinder with a single marked point [] where the coordinate transformation is singular and whose location changes with .
Therefore we choose to work with two coordinate patches (see figure 2): the near zone where the horizon () is fixed and the periodicity of is invisible far away, and the asymptotic zone where is fixed and is invisible. The metric in the two regions must be consistent over the overlap region (which grows indefinitely as ).
Such a procedure is known in General Relativity as a “matched asymptotic expansion” — the metric is solved for in each asymptotic region and certain quantities are determined by matching the metrics over the overlap (for some recent examples see [26, 27, 28], in mathematical physics this idea goes back as far as Laplace who used it to find the shape of a drop of liquid on a surface — see [29] and references therein for a historical review). However, this is probably the first time such a procedure is used to find a static black hole solution.
We start by defining the domain for each zone and the zeroth order solution. In the near zone, whose domain is we have a dimensional Schwarzschild black hole metric, with fixed and the perturbation is in orders of . In the asymptotic zone, on the other hand, the domain is , the zeroth order solution is simply the flat “cylinder” with the origin omitted, and the perturbation parameter is .
Our objective is to describe the perturbation process (to any order) and apply it. The paper is organized as follows. In section 2 we describe the equations in the asymptotic zone and especially the newtonian potential. In section 3 we determine the linearized corrections to the Schwarzschild black hole in the near zone. In section 4 we describe the general perturbation procedure for this system and in 5 we present quantitative results on the leading corrections to the zeroth order metric. In the appendices we review some information on Heun’s equation and the Hypergeometric equation, review the definition of the surface gravity and give some details on vector harmonics in 5d (on ). We now turn to the summary.
1.3 Summary
The asymptotic zone.
In section 2 we describe the asymptotic zone. The first correction to the zeroth order metrics described above is readily computed — it is the newtonian approximation in the asymptotic region (in standard harmonic gauge), where the newtonian potential (6) is obtained through the method of mirror images (considering the infinite sequence of sources in the covering space at for any integer ). This term is proportional to and hence belongs to order in the asymptotic zone. Actually, the whole postnewtonian procedure is relevant and we review it, though in this paper all we need is the lowest order (newtonian) approximation.
Black hole perturbations.
The next correction to consider is the leading correction to the Schwarzschild solution. It is the analogue of the newtonian approximation only here the unperturbed background is curved, and it describes the response of the geometry to the mirror sources far away. Despite the analogy with the newtonian approximation the implementation is involved and is described in section 3.
In 4d this computation was carried out by Regge and Wheeler [30] who solved for all the linear perturbations, not only the static ones. Recently Ishibashi and Kodama succeeded to generalize their result to an arbitrary dimension, and to include a cosmological constant as well [31, 32, 33]. Our treatment is independent and we compare the two approaches below after describing our own.
The computation involves a few steps. We start by writing down the most general static perturbation to the metric. The spherical symmetry guarantees that at linear order perturbations in different representations of the rotation group will not mix, and we find by counting degrees of freedom that it suffices to consider “scalar harmonics” — representations which are the symmetric product of the vector representation, or equivalently, metric perturbations which are determined by scalar functions on the sphere. It turns out that the spherical symmetry also suggests a natural gauge which we term “no derivatives gauge” and completely fixes the reparameterization invariance, leaving us with 3 undetermined metric functions (fields).
Writing down the equations of motion and separating the angular variables we find that a Ricci flatness condition in the angular directions yields an algebraic relation among the radial functions (39) which is similar to a trace condition and allows us to eliminate one of the fields. After substitution one can express one of the remaining fields in terms of the other and its first and second derivatives. Performing the second substitution we are left with a second order ordinary differential equation (ODE), rather than a third or fourth order one would initially expect. So finally one has a single second order ODE in the radial direction, for one metric function (and for each spherical harmonic mode) from which the whole metric may be recovered. This is the so called “master equation” which after a change of variables simplifies further to become (55). It would be nice to have a deeper understanding why these reductions were to be expected.
The master equation belongs to the Heun class of Fuchsian equations, where Fuchsian means that the equation has only regularsingular points on the complex sphere which includes infinity, and Heun means that there are exactly 4 such points. Unlike the Hypergeometric case of 3 regular singularities there is no general solution to the Heun equation, though several methods are available. In this case however, it turns out that the solutions can be written in terms of a hypergeometric function, and it would be nice to understand why that had to be the case. Interestingly, we observe that in some of the relevant cases these hypergeometric functions simplify further to polynomials, and in particular in 5d all relevant solutions are polynomials (solutions which are of even multipole number and are regular at the horizon).
We started working out this problem before we were aware of the results of Kodama and Ishibashi [31, 32, 33] and we continued independently even after learning about these papers in order to avoid the formalism of gauge invariant perturbation theory, and the various changes of variables which are employed there. We were able to do so and actually found a somewhat different master (Heun) equation. Yet the final reduction of our master Heun equation to a hypergeometric one was motivated by those papers.
The matching procedure.
One of the main results of this paper is the construction of a perturbation method for the metric (in both patches) which may be carried in principle to an arbitrarily high order in the small parameter. The method is described in section 4. A crucial step is to identify a dimensionful expansion parameter on each patch: in the asymptotic zone and in the near region. As in any perturbative expansion, at each order one needs to solve a nonhomogenous linear equation — the linear equation being the same as the one which appears at first order and the nonhomogeneous source term being constructed from lower order metric functions. The precise form of the source term depends on the higher order gauge choice which we do not specify, but will not change the method we describe. The solution to this equation is determined up to a solution of the homogeneous equation. This indeterminacy for each zone on its own reflects the freedom of adding external field multipoles — in the asymptotic zone they are situated at the origin while in the near zone they are at infinity. These external multipoles must be determined by matching with the other zone, a procedure which requires to identify (after matching the gauge) the leading terms in the metric on both zones. We call this process “a dialogue of multipoles”.
A priori it is not obvious that the required terms from the other zone are already available at the right time, namely that the method is wellposed (that there are sufficient boundary conditions). Hence it is interesting to study at any given order in a specific zone which orders must be already available for matching from the other one, and thereby describe the pattern of the dialogue — the orders at which one should alternate between the zones. This pattern can be determined by a simple dimensional analysis of the multipole coefficients as we describe in the text, and indeed we find the system to be wellposed. Interestingly, the dialogue pattern which emanates is dimension dependent: 5d is special in that one scales a single order in the perturbation ladder on each zone and then alternates to the other zone; for one needs to climb several steps before going to the other zone, and in the limit one gets infinitely many constants already from matching with the newtonian potential alone.
Quantitative matching results.
In section 5 we apply the general procedure to the leading order and match the newtonian potential at the asymptotic zone to get the leading correction to the Schwarzschild solution in the near zone. To this purpose it is essential to have available certain matching constants which can be read from the explicit solutions we derived for the linear perturbations of Schwarzschild. The next order correction is currently under study [34].
¿From the metric which we obtain one may extract certain “measurables”:

The leading correction to the mass — temperature relation is given in (77). At this order the BH is still spherical but there is a correction to this relation since the small black hole does not asymptote in the near zone to flat space with zero potential, but rather there is a nonzero potential shift due to the images.

The leading (quadrupole) departure from a spherical horizon — measured by the “eccentricity” is given in (90). The result of the deformation is to make the black hole longer along the axis compared to the axis as in figure 9 and can be understood from the shape of small (newtonian) equipotential lines around (see figure 3).

The coefficient of the “interpolar distance” is given in (102). By “interpolar” distance we mean the proper distance from the “north pole” of the black hole around the compact circle and up to the “south pole”(see figure 11). Actually the black hole tends to “make room” for itself, in the sense that the interpolar distance added to the black hole size in conformal coordinates is always larger than , the size of the compact dimension. This can be restated as the observation that such black holes seem to always have a positive scalar charge as seen from infinity similar to the ordinary positive mass theorem (where the scalar is the one which arises from the dimensional reduction of the metric component — the size of the extra dimension). In 5d the effect is the strongest, where to leading order in the small parameter the interpolar distance does not decrease at all. We term that “a black hole Archimedes effect” since the black hole repels or expands an amount of space equal to its size in 5d (and less in higher dimensions).
Most of these results were already announced in [20] and here we add the determination of the interpolar distance and the generalization of the eccentricity for . They were numerically confirmed in 5d [21] as well as in 6d [35, 36] and other dimensions [36]. Recently a paper [23] has appeared deriving the leading order form of the metric within the framework of the HarmarkObers coordinates [11], and as such overlaps with the results announced in [20] and proven here. The overlap includes the corrections to the temperature and area, while [23] obtains also the corrections to the mass and tension, and this paper derives the eccentricity and “Archimedes effect”. Moreover, here we go beyond and demonstrate a method for an arbitrary number of successive approximations.
2 The asymptotic zone
In this section we write the static Einstein equations in a form that will be convenient for iterative expansion in a small parameter around the flat Minkowsky spacetime. This type of expansion is known as “postnewtonian expansion” (see [29]) and we follow here the usual conventions for this type of expansion. The leading order in the expansion is the newtonian approximation. We repeat here the calculation of the newtonian approximation for the caged black hole which appeared in many places (see for example [11, 23]). The following orders in the expansion are called “postnewtonian” and their calculation will appear in [34].
For the postnewtonian expansion it is convenient to write the Ricci tensor in the following form [37]
(1) 
where and are the Christoffel symbols of the first and the second kind, respectively, and in addition one defines
Next one chooses the harmonic (or de Donder ^{2}^{2}2The first introduction of this gauge appeared in [38].) gauge by the requirement that
(2) 
where we denote by the determinant of the metric . In this gauge, the last term in the expression of the Ricci tensor above vanishes. This choice of gauge is very convenient for expansion in the asymptotic zone. Finally one attempts to solve Einstein’s equations.
The first step in this iterative procedure is to look at the linearized equations valid for weakly gravitating regions, namely making the newtonian approximation. The metric is taken to be
(3) 
The harmonic gauge equation (2) takes the more famous form (for example in the treatment of gravitational waves)
One defines
(4) 
in terms of which the linearized field equations become
(5) 
where is the flat space D’alambertian.
In our case, working on the covering space implies an infinite array of newtonian sources in the direction and the only nonzero components of the energy momentum tensor is
where denote the extended spatial coordinates. The method of images can be used to solve the equation for
(6)  
where is the newtonian potential, conventionally normalized such that its flux through a surface enclosing a mass is , is given by [39]
(7) 
and
is the area of a unit . A more formal alternative to obtain the prefactor of the newtonian potential in equation (6) is through matching with the Schwarzschild metric in the near zone.
We see that the first correction to the metric in the asymptotic region is of order . We will see later that we must choose the perturbation parameter in this region to be rather than , namely^{3}^{3}3or more precisely to the power .
(8) 
and hence we see that the leading correction comes at order , namely
(9) 
Transforming back to using the inverse of (4)
(10) 
yields the expression for the metric perturbation in terms of the newtonian potential (6)
(11) 
where the Latin indices stand for the spatial components.
In the 5d case one can express the newtonian potential as
(12) 
In figure 3 we give the equipotential surfaces of the newtonian potential in 5d, and they look qualitatively the same in any dimension .
We close this section with two comments. First, at higher orders in the perturbation procedure the form of the equations is dominated by the linearized equations and is given by
where is the order under study and are source terms which are quadratic, at least, in lower order metric components and their derivatives. The second remark is that in this section all the expressions for the metric components were in the harmonic gauge. In the next sections we use the Schwarzschild gauge in the near zone. To avoid cluttering the notation we will not introduce always a different letter for every gauge — in subsection 4.1 we give different notation for in the two gauges ( in the Schwarzschild coordinates and in the harmonic gauge) but later we omit the difference in the notation. However, it is important to remember the difference in the gauge between the two zones.
3 Black hole perturbations
As explained in the introduction the zeroth order in the near (horizon) zone is the dimensional SchwarzschildTangherlini metric [25]
where , is related to via (7) and
is the metric on .
In this section we find the linear static perturbations for the dimensional Schwarzschild solution. Regge and Wheeler [30] derived the linear equations that describe small perturbations to the four dimensional Schwarzschild black hole, and here we generalize their method for the static case. These perturbations can be interpreted as deviations of the black hole from spherical symmetry due to remote masses. In the case of a compact dimension, we are interested in the influence of the black hole (images) on itself. Therefore we assume that the symmetry in the spherical coordinates is still preserved and the deformation of the black hole takes place only in the plane (which in “cylindrical” coordinates will be part of the plane). We denote in this section the dimensional Schwarzschild metric by and the perturbation metric by . Therefore, is a function only of and .
The linearized vacuum Einstein equations can be brought to the simplified form [40]
(13) 
where , is the covariant derivative with respect to the background metric and stands for the symmetric part of a tensor .
3.1 Spherical harmonics on
Our goal is to simplify the equations by reducing them to a system of ordinary differential equations. Following Regge and Wheeler we start by expanding the solution, , into generalized spherical harmonics on the sphere . Each component of is transformed under local coordinate changes of like a scalar, a vector or a tensor. Hence, we decompose into 3 types of spherical harmonics: scalar, vector and tensor harmonics. The scalar components are: , and . The vectors are: and . The tensor is formed of the block
where we denote by symmetric components. Counting degrees of freedom we find that we should have 3 elements in the basis of the scalar harmonics, in the basis of the vector harmonics and in the basis of the tensor harmonics. Together we obtain all the components of .
Since we are interested in expansion to harmonics which are static and have symmetry the nonvanishing components of the tensor are:
(14) 
where in addition, the symmetry implies that the only independent angular components on the diagonal are and . The rest of the angular components are obtained through the relations
Note that there are only 5 independent components of , namely , , , , , comprising of 2 scalars, 1 vector and 2 tensors, and these numbers are independent of the dimension. Accordingly, we will find 5 linear ordinary differential equations (the equations of motion).
3.1.1 Scalar harmonics
By considering the flat dimensional Laplace equation we can get both the scalar spherical harmonics on , and the leading radial profile of the multipoles in the nearly flat region . Since we assume symmetry we consider a Laplace equation for a function which depends only on one angular variable . Thus, the Laplace equation for a function on would be
(15) 
Separation of variables gives us two separated equations for each eigenvalue which denotes also the number of the multipole in the expansion
is the angular function which is associated with each one of the multipoles. The extra zero index in the angular function stands to remind us that had we not had the symmetry, the angular functions, which depends on , would have additional indices (additional “quantum numbers”). (For an example see the 5d case later in this subsection). The equation for is
(16) 
where are the eigenvalues. This equation can be brought to the form of a Legendre equation [41] in dimensions using the substitution (then the solutions of the equation are called Legendre polynomials of the variable in dimensions). The solutions can be expressed by a Rodriguez formula (see [41])
(17) 
where the prefactor with the Gamma functions fixes the usual normalization of the Legendre polynomials.
The functions give us the radial part of the expansion. is obtained as the solution of the eigenvalue equation
(18) 
Therefore
(19) 
where and are constants. For fixed and the first term is the multipole of a mass distribution at and the second term is the multipole of a mass distribution at infinity. Note that is the first multipole — the monopole, is the dipole and so on.
Since the linear equations for (13) are invariant we may separate the angular variables, and since the perturbed metric depends only on , we may expand the scalar components and in spherical harmonics as follows
(20) 
where the radial functions satisfy some differential equations to be discussed later.
5d scalar spherical harmonics.
As a concrete simple example, where we can give explicit formulae for the scalar harmonics, let us consider the 5d case. Gerlach and Sengupta used this type of decomposition in 5d for the RobertsonWalker spacetime [42]. Let us denote the scalar spherical harmonics in 5 dimensions^{4}^{4}4Note that in 5 dimensions we use different notation for the indices; we keep the index for the usual spherical harmonics . Thus, we will use instead of in the general case. by . They can be separated into a product of two types of functions
where are the usual spherical harmonics on and are the “Fock” harmonics (see [42, 43]), which are given by
Since we require symmetry for the twosphere, we take in the spherical harmonics. Thus we arrive to (16) in the 5d case (with the index instead of )
(21) 
whose solutions are Chebyshev polynomials of the second kind
3.1.2 Vector and tensor harmonics
Given a family of scalar harmonics one can form a family of “scalar derived” vector harmonics simply by taking its gradient. There are other, more involved, vector harmonics as well, but we shall see now that the “scalar derived” family suffices for our purposes. For a concrete example of a basis of vector harmonics in 5d (on ) see appendix C.
Due to the symmetries the vector has a single component (14)
(22) 
where runs over the angular coordinates only. Denoting we have
(23) 
Therefore, after expanding into spherical harmonics and substituting in (23) we can write the expansion of into radial functions as
(24) 
A similar argument holds for the tensor components which are essentially , (14). Again there are exactly two “scalar derived” tensor harmonics
(25) 
and
(26) 
where is the metric on the sphere , is the covariant derivative on , and the second term is proportional to the trace of the first. So we have two tensor harmonics to decompose into, which is exactly the number we need, and indeed one can verify that these two families always suffice.
Hence we can decompose the tensor part into radial functions ,
(27) 
The discussion above led us to the conclusion that the only elements of both the vector and tensor basis which survived under our symmetry requirements are the “scalar derived” ones. These are in the “scalar type” representation under the rotation group, where by “scalar type” we mean representations whose Dynkin indices are for some , namely those in the times traceless symmetric product of the vector representation. More generally, Kodama and Sasaki [44] pointed out that one can classify the different basis elements into three groups with respect to their different representations under the rotation isometry group: scalar, vector and tensor ‘‘type’’.^{5}^{5}5The vector and tensor types are defined in analogy with the scalar type: a vector type representation has Dynkin indices , namely the traceless product of the 2nd rank antisymmetric representation with the times traceless symmetric product of the vector representation, and a tensor type is . This classification into representations of the isometry group should not be confused with the classification that we used above with respect to local coordinate transformations of . So, according to this classification, (24) is a vector of “scalar type” under the rotation group and (27) is a tensor of “scalar type” as well. Since we deal with linear equations two different representations cannot mix. Thus even if there were any representations of nonscalar “type” in the decomposition of the perturbed metric, they would not appear in the equations for the scalar type radial functions .
3.2 The choice of gauge
We have now 5 radial fields (,..,) defined in (20), (24), (27) for each mode of the expansion. We can reduce further the number of fields to 3 using the gauge freedom. The gauge transformations of linearized general relativity about a solution are of the form
where is an arbitrary vector field — the generator of an infinitesimal transformation. The most general generator consistent with the symmetries is
(28) 
It is natural to eliminate the “scalar derived” functions and in each mode, putting in a diagonal form. We term this the “no derivative gauge”. Thus,
The required gauge implies that the functions and should satisfy the following conditions
(29) 
Moreover, since is arbitrary function of we can redefine it using the transformation . These equations for and , although including differential operators, are actually algebraic and have a single solution (for each mode) without any additional gauge freedom, i.e., the gauge is completely fixed. Applying the gauge transformation to any component of (,…,) yields the same equations as in the case of using (16).
3.3 The field equations
3.3.1 The master equation
After gauge fixing the perturbed metric diagonalizes and we are left with 3 metric functions , which are functions of
(30)  
It turns out that Einstein’s equations simplify if we use instead of . Thus we define , , where as usual . The ansatz in the new variables reads
(31)  
We substitute this ansatz into the linearized Einstein equations, (13) and using (16) we obtain, as expected, five equations
(32) 
(33) 
(34) 
(35) 
(36) 
where we define
(37)  
(38)  
Three of the equations are second order in the derivatives. These are the evolution equations. One of the equations is first order () — the constraint equation. For the expression in the brackets in (36) is nonzero and we get the following algebraic relation
(39) 
The case is degenerate and we will discuss it separately.
Now the variables can be separated in (35) and it becomes a second order ordinary differential equation
(40)  
Using the algebraic relation to eliminate and its derivative from the other equations, equations (32) and (33) become
(41) 
where we use the abbreviation
¿From the last two equations we can express in terms of and its derivatives
(42) 
Now using the remaining equations ((34)–(35)), we arrive to a second order linear differential equation for where we introduce , a dimensionless variable
(43) 