Reducing triangular systems of ODEs with rational coefficients, with applications to coupled Regge-Wheeler equations
aa r X i v : . [ m a t h - ph ] J a n Reducing triangular systems of ODEs with rational coefficients, withapplications to coupled Regge-Wheeler equations
Igor Khavkine [email protected]
Institute of Mathematics, Czech Academy of Sciences,ˇZitn´a 25, 115 67 Praha 1, Czech Republic
Abstract
We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE systemof arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linearalgebra problem. This method is then used to solve the problem of conclusively deciding when certain rationalODE systems in upper triangular form can or cannot be reduced to diagonal form by differential operatorswith rational coefficients. As specific examples, we consider systems of coupled Regge-Wheeler equations, whichhave naturally appeared in previous work on vector and tensor perturbations on the Schwarzschild black holespacetime. Our systematic approach reproduces and complements identities that have been previously found bytrial and error methods.
In the recent work [7], we have shown how, after a separation of variables, the radial mode equations of the vectorwave equation (cid:3) v µ = 0 on the Schwarzschild black hole spacetime may be significantly simplified by systematicallydecoupling them into an upper triangular form, where the diagonal components are generalized Regge-Wheeleroperators and only a few of the off-diagonal components are non-vanishing. The original radial mode equationsconstitute a 4 × r ). The existence of such such a simplification, in particular the ability to set to zero most of theoff-diagonal terms in the upper triangular form, follows from specific identities previously discovered by trial anderror in [6], and in part also in [8] (see [7] for a full discussion), and is certainly not obvious a priori . This naturallyraises the questions of how these identities could be recovered in a systematic approach, and whether more couldbe discovered to push the simplifications described above as far as possible. These questions become particularlyrelevant for trying to repeat the same simplifications for the Lichnerowicz equation (cid:3) p µν − R ( µλκν ) p λκ = 0, whichhas a role relative to linearized Einstein equations analogous to that of the vector wave equation relative to Maxwellequations. The relevant identities were also discovered in [6], but only by means of voluminous trial and errorcalculations and without a clear answer to whether they could be further improved. We will revisit this point whenconsidering examples in Section 4.1.The above questions were left open in [7] and are answered in this work. The main systematic tool at ourdisposal is the theory of rational solutions of ordinary differential equations (ODEs) with rational coefficients. Underappropriate mild hypotheses on the equation, the search for such solutions can be reduced to a finite dimensionallinear algebra problem (Theorem 2.4). We summarize this theory in Section 2. The theory of rational solutionsof scalar rational ODEs is fairly well developed (cf. the monograph [3] and the references therein; more precisereferences are given in the text). Our innovation is to synthesize this approach into an economical form, based onwhat we call leading (or trailing ) multipliers , that is directly applicable to our examples of interest, but also moregenerally to ODE systems of arbitrary size and order. In Section 3, we consider the problem of setting to zero anoff-diagonal block in an upper triangular rational ODE system by a transformation with rational coefficients. Thisproblem is first reduced to an operator identity (Equation (15)), which in turn can be solved by converting it into1n inhomogeneous rational ODE system (Theorem 3.1). Finally, in Section 4, we combine the results of Sections 2and 3 to show how the special identities used in [7] and [6] can be recovered with minimal effort, especially whenaided by computer algebra. In particular, we can conclusively decide when simplifications of the kind describedearlier do or do not exist, with several examples given in Section 4.1. Section 5 concludes with a discussion of theresults and an outlook to further work. The main objects under our study will be ordinary differential operators and equations (ODEs) with rationalcoefficients. We will usually denote the independent variable by r . A differential operator e applied to a function u = u ( r ) will be denoted by e [ u ]. Both u and e [ u ] could be vector valued. We do not put any a priori boundson the dimensions or differential order of e . Hence, e can also be seen as a matrix of scalar differential operators.Hence, when e is of order zero, e [ u ] will correspond to multiplication of u by an r -dependent matrix. We will denotethe composition of differential operators by ◦ , so that e ◦ f [ u ] = e [ f [ u ]]. We will restrict our attention only todifferential operators with rational coefficients and in general complex valued.In this section, we will eventually show how to find all the rational solutions u = u ( r ) of a rational ODE e [ u ] = v , by reducing it to a finite dimensional linear algebra problem. Our approach starts with a Laurent seriesrepresentation u = P n u n r n converts the equation e [ u ] = v into a recurrence relation on the coefficients of u . Atdifferent stages, it would be useful to consider Laurent series of different kinds. In particular, we will mostly dealwith formal series (no requirement of convergence). However, convergence will be automatic if we know in advancethat the series has only finitely many terms or that it comes from the expansion of a rational function. Thus, wemay distinguish unbounded Laurent series C [[ r, r − ]], bounded (from below) Laurent series C [ r − ][[ r ]], bounded fromabove Laurent series C [ r ][[ r − ]] and Laurent polynomials C [ r, r − ]. Of course, we could also consider Laurent seriescentered at some other r = ρ = 0, but for convenience of notation whenever possible we will stick with ρ = 0.For bounded (from below) Laurent series, it is helpful to define leading or trailing orders and coefficients. For u = P n u n r n ∈ C [ r − ][[ r ]], if we can write u = u n r n (1 + O ( r )) u n r n (1 + O ( r ))... , (1)with each O ( r ) ∈ r C [[ r ]], the n , n , . . . are the leading orders of the components u , u , . . . of u , with the exceptionwhen u in i = 0, in which case we set n i = + ∞ . We denote by | ˇ u | the vector where each component of u is replacedby its leading order, and we refer to it as the leading order of u . When n = min i n i < ∞ , u n = 0 and we call it the leading coefficient of u . We define the leading coefficient of 0 to be 0.Similarly, for bounded from above Laurent series u ∈ C [ r ][[ r − ]], if we can write u = u n r n (1 + O ( r − )) u n r n (1 + O ( r − ))... , (2)with each O ( r − ) ∈ r − C [[ r − ]], the n , n , . . . are the trailing orders of the components u , u , . . . of u , with theexception when u in i = 0, in which case we set n i = −∞ . The trailing order | ˆ u | of u is the vector of the trailingorders of the components of u . When n = max i n i > −∞ , u n = 0 and we call it the trailing coefficient of u . Wedefine the trailing coefficient of 0 to be 0.Clearly the leading (trailing) coefficient of a bounded (from above) Laurent series vanishes if and only if thewhole series vanishes. For Laurent polynomials u ∈ C [ r, r − ], both the leading and trailing orders, and coefficients,are well-defined.Consider an ODE e [ u ] = 0 on bounded Laurent series u = P n u n r n ∈ C [ r − ][[ r ]]. What we like to do isturn e [ u ] = 0 into a linear recurrence relation on the coefficients u n of the form E n u n = f n ( u n − , u n − , . . . ) andthen uniquely solve for u n as a function of u n − and lower order coefficients, for almost all n (that is, all butfinitely many). Those finitely many n for which the solution for u n would not be unique would then determinethe dimension of the solution space of the ODE. This approach requires that the coefficients E n in the recurrencerelation be invertible for almost all n . For scalar equations this is an almost trivial requirement, but in matrixequations different components of e may be weighted so differently by powers of r that the coefficient E n comes out2s a singular matrix for infinitely many n . Often this problem can be remedied by applying suitable transformationsto u and to e [ u ].Let S = S ( r ) and T = T ( r ) be matrices with Laurent polynomial components. For future convenience, we alsorequire that the inverses S − and T − also have Laurent polynomial components (or, equivalently, the determinantsof S and T are proportional to single powers of r ). We say that S and T are respectively the source and targetleading multipliers of e when, after expanding all rational coefficients as bounded Laurent series, we have e [ S ( r ) u n r n ] = T ( r )( E n u n r n + r n O ( r )) , (3)with the components of O ( r ) all in r C [[ r ]] and E n an r -independent matrix that is invertible for almost all n . Wecall E n the trailing characteristic matrix of e with respect to the given multipliers. Similarly, we say that S and T are respectively the source and target trailing multipliers of e when, after expanding all rational coefficients asbounded from above Laurent series, we have e [ S ( r ) u n r n ] = T ( r )( E n u n r n + r n O ( r − )) , (4)with the components of O ( r − ) all in r − C [[ r − ]] and E n an r -independent matrix that is invertible for almost all n . We call E n the trailing characteristic matrix of e with respect to the given multipliers.Those integer n ∈ Z such that det E n = 0, which is a polynomial in n , are called (respectively leading or trailing ) (integer) characteristic roots or exponents of e with respect to given multipliers S, T . We denote the set of suchleading characteristic exponents by ˇ σ ( e ) and the set of such trailing characteristic exponents by ˆ σ ( e ), with implicitdependence on the S, T multipliers, of course.We will not dwell on when leading or trailing multipliers exist, but will just assume that they are given for anyparticular problem. Often S and T may be taken to be diagonal, with appropriately chosen powers of r on thediagonal. Otherwise, they could be determined by a recursive procedure similar to that used in the analysis ofregular and irregular singularities for ODEs with meromorphic coefficients [11].Any rational u ∈ C ( r ) will have a (convergent) bounded Laurent series expansion about any point r = ρ .Without loss of generality, let us take ρ = 0. We would like to prove some bounds on the leading order of u at r = 0when it solves e [ u ] = v , with some rational v ∈ C ( r ). For this purpose, it is actually more natural to allow u and v to be bounded Laurent series. Lemma 2.1.
Let e [ u ] = 0 be an ODE with rational coefficients, leading multipliers S, T , and leading characteristicmatrix E n . Let u, v ∈ C [ r − ][[ r ]] with leading orders m = min i | ˇ u i | , n = min i | ˇ v i | (the values m = ∞ or n = ∞ areboth permissible). If e [ Su ] = T v , then either (a) m = n and (provided n < ∞ ) v n = E n u n or (b) m is a leadingcharacteristic exponent of e , E m u m = 0 , and m < n . In other words min { n } ∪ ˇ σ ( e ) ≤ m. (5)This result and proof are analogous to those presented in § S, T and the way theylead to the leading (trailing) characteristic matrix E n . Proof. If m = ∞ , this means that u = 0. Then also v = 0 and n = ∞ , meaning that (a) holds. For the rest wewill assume that m < ∞ , meaning that u has the non-vanishing leading coefficient u m = 0. If E m u m = 0, then thedefining property (3) of leading multipliers S and T directly implies part (a), that is n = m . On the other hand, if E m u m = 0 and since by definition u m = 0, the leading order of u must be a characteristic exponent of e , m ∈ ˇ σ ( e ).Using again (3), we also find m < n .Part (a) implies n ≤ m , while part (b) implies min ˇ σ ( e ) ≤ m and m < n . Since at least one of (a) or (b) alwaysholds, the lower bound (5) on m is always true.All the same arguments apply to Laurent expansions about r = ∞ , though after making use of the transformation r /r . For convenience, we state the corresponding result without the need to invoke this transformation. Lemma 2.2.
Let e [ u ] = 0 be an ODE with rational coefficients, trailing multipliers S, T , and trailing characteristicmatrix E n . Let u, v ∈ C [ r ][[ r − ]] with trailing orders m = max i | ˆ u i | , m = max i | ˆ v i | (the values m = −∞ or n = −∞ are both permissible). If e [ Su ] = T v , then either (a) m = n and (provided n > −∞ ) v n = E n u n or (b) m is atrailing characteristic exponent of e and E m u m = 0 . In other words m ≤ max { n } ∪ ˆ σ ( e ) . (6)3ow we know how to bound the order of the pole of a rational solution u at any particular value of r = ρ ∈ C .For the following class of ODEs, we can also identify all the potential locations of the poles of u . Lemma 2.3.
Let e [ u ] = 0 be an ODE of differential order p with rational coefficients, for which there exists aninvertible matrix P = P ( r ) with rational coefficients such that P e [ u ] = d p dr p u + ˜ e [ u ] , where ˜ e is of differential orderat most p − . For rational u, v ∈ C ( r ) , if e [ u ] = v , then u ( r ) is smooth (i.e., has no pole) at all but finitely manypoints of C . The only possible exceptions are r = ρ , with ρ one of the poles of P v or of the coefficients of ˜ e .Proof. By our hypotheses, we can put the equation e [ u ] = v into the equivalent form d p dr p u + ˜ e [ u ] = P v, (7)where
P v is rational and ˜ e [ u ] has rational coefficients. Consider a point r = ρ ∈ C , that is not pole of P v or of thecoefficients of ˜ e [ u ]. There are obviously only finitely many such excluded points. If u has a pole of type ( r − ρ ) − k ,then d p dr p u has a pole of type ( r − ρ ) − k − p , while P v and ˜ e [ u ] will only have poles of lower order. But this meansthat such a u cannot be a solution of our equation. Hence, any rational solution u ∈ C ( r ) can have poles only inthe already mentioned excluded set.Given a rational ODE e [ u ] = 0, when considering Laurent expansions at r = ρ , let us denote the correspondingleading multipliers by S ρ , T ρ , which are by definition rational and have poles only at r = ρ (and r = ∞ , of course).For a rational u ∈ C ( r ), if we know that its poles are restricted to a finite set of points in C and we have a boundon the degree of the pole at each of these points, then we can find a rational matrix R = R ( r ) such that Ru has nopoles in C . Theorem 2.4.
Let e [ u ] = v be an ODE with rational coefficients and rational v ∈ C ( r ) , satisfying the hypothesesof Lemma 2.3. Suppose also that we have the leading multipliers S ρ , T ρ of e at each of the finitely many exceptionalpoints r = ρ ∈ C identified in Lemma 2.3. Then, there exists a rational matrix R = R ( r ) such that, for any rational u ∈ C ( r ) satisfying e [ u ] = v , there is a Laurent polynomial ˜ u ∈ C [ r, r − ] satisfying u = R ˜ u . We call such a matrix R a universal multiplier for the rational inhomogeneous ODE e [ u ] = v . A universalmultiplier certainly need not be unique. The existence of universal multipliers for scalar equations is discussedin [3, § Proof.
A rational u ∈ C ( r ) has only finitely many poles, and at each of those poles it has a bounded Laurent seriesexpansion. By invoking Lemma 2.3 we can constrain the poles of u to a finite set of points. Then, by invokingLemma 2.1, for each of those points, say r = ρ , we can find a finite lower bound ˇ n ρ for the leading Laurent orderof S − ρ u at r = ρ . Recall that one of the defining properties of S ρ is that both it and S − ρ only have poles at r = ρ (and of course at r = ∞ ). This means that ˜ u = Q ρ S − ρ ( r − ρ ) − ˇ n ρ u , where the product is taken over the potentialpole locations (possibly excluding ρ = 0), is still rational but no longer has any poles in r ∈ C , with the possibleexception of r = 0. But that can only be if ˜ u ∈ C [ r, r − ] is a Laurent polynomial. Therefore, we can take R ( r ) = Y ρ S ρ ( r )( r − ρ ) ˇ n ρ (8)as the desired universal multiplier. Since any of the ˇ n ρ can be decreased without breaking this result, we have manypossible choices for R . Corollary 2.5.
Let e and v be as in Theorem 2.4, with universal multiplier R . In addition, suppose that we havethe leading multipliers S , T at r = 0 and the trailing multipliers S ∞ , T ∞ at r = ∞ for ˜ e = e ◦ R . Then theequation e [ u ] = v for u can be reduced to a finite dimensional linear system, and hence its solution space is finitedimensional.Proof. By invoking Theorem 2.4, solving e [ u ] = v for u ∈ C ( r ) is equivalent to solving ˜ e [˜ u ] = v for ˜ u ∈ C [ r, r − ],with u = R ˜ u and ˜ e [˜ u ] = e [ Ru ]. Invoking Lemma 2.1 we can find a finite lower bound on the leading order of S − ˜ u and hence of ˜ u , which we will call ˇ n . Invoking Lemma 2.2 we can find a finite upper bound on the leading order of S − ∞ ˜ u and hence of ˜ u , which we will call ˆ n . Therefore, we can parametrize all solutions as Laurent polynomials˜ u = ˆ n X n =ˇ n u n r n , (9)4hich has ˆ n − ˇ n + 1 < ∞ undetermined coefficients. Plugging this parametrization into the equation ˜ e [˜ u ] = v ,putting both sides over a common denominator, and comparing coefficients reduces the original problem to a finitedimensional linear system of equations. The dimension of the solution space of this system is of course finite, and(crudely) bounded by the number of coefficients in (9).Of course, once an equation has been reduced to an explicit finite dimensional linear system, it can be solvedon a computer, even symbolically. In this section, we are interested in the following question. Given an ODE system in block upper triangular form,is it possible to find an equivalent ODE system where the off-diagonal block has been set to zero, hence in diagonalform? If possible, we call this a reduction to block diagonal form and say that the original system can be reduced .A refined version of the question is whether a rational ODE system can be reduced while remaining rational.The first thing we need to clarify is the notion of equivalence. Roughly speaking, two ODE systems should beequivalent when there is an isomorphism between their solution spaces. A further practical requirement is that thisisomorphism be given, in either direction, by differential operators. After all, transformations given by differentialoperators tend to be easier to write down in terms of explicit formulas, while also allowing rather precise controlover some properties of the coefficients of the ODE systems, like rationality or upper triangular form. Also, anequivalence should make explicit the transformation of one ODE system into the other one, again by a differentialoperator.We formalize these ideas as follows. Given two ODE systems, e [ u ] = 0 and ¯ e [¯ u ] = 0, an equivalence betweenthem consists of pairs of differential operators k, g and ¯ k, ¯ g obeying the operator identities, for any u, ¯ u and v, ¯ v ,¯ e [ k [ u ]] = g [ e [ u ]] , ¯ k [ k [ u ]] = u, ¯ g [ g [ v ]] = v, (10) e [¯ k [¯ u ]] = ¯ g [¯ e [¯ u ]] , k [¯ k [¯ u ]] = ¯ u, g [¯ g [¯ v ]] = ¯ v. (11)Graphically, if we represent each differential operator by an arrow and appropriate function spaces by • ’s, theseidentities mean that the squares in the following diagram are commutative and that the horizontal arrows composeto identity in either direction: • •• • e k ¯ e ¯ kg ¯ g . (12)Basically, these identities imply that for a solution u of e [ u ] = 0, ¯ u = k [ u ] is a solution of ¯ e [¯ u ] = 0, and vice versa,where the barred and unbarred transformation operators are mutually inverse. Finally, when dealing with ODEsystems with rational coefficients, we require the coefficients of the operators k, g and ¯ k, ¯ g to be rational as well.The above notion of equivalence is actually somewhat more rigid than absolutely necessary, but it will be sufficientfor our purposes. A discussion of a somewhat looser notion of equivalence can be found in [7], with references todeeper literature on this topic. Below, we use this notion of equivalence to discuss reduction of triangular ODEsystems. A similar discussion can already be found in [7, Sec.2.3].An ODE system of the form (cid:20) e ∆0 e (cid:21) (cid:20) u u (cid:21) = (cid:20) (cid:21) (13)is said to be (block) upper triangular , or (block) diagonal if ∆ = 0. We will always assume that this system hasrational coefficients. We presume also that the equations e [ u ] = 0 and e [ u ] = 0 are ODE systems of unspecifieddimensions and differential orders, but such that they can be solved for the highest derivatives, as in the hypothesesof Lemma 2.3.A reduction of the upper triangular ODE system (13) is an equivalence given by the following pair of commutative5iagrams • •• • e ∆0 e id δ e e id ε , • •• • e e id − δ e ∆0 e id − ε , (14)where the corresponding horizontal arrows are clearly mutual inverses. Of course, we require the differential oper-ators δ and ε to have rational coefficients. By direct calculation, we can check that the above diagrams commuteif and only if δ and ε satisfy the operator identity e ◦ δ = ∆ + ε ◦ e . (15)Note that solutions of (15) are certainly not unique. For instance, for any δ, ε solution pair, ( δ + α ◦ e ) , ( ε + e ◦ α )is another solution, with arbitrary α , since e ◦ ( α ◦ e ) = ( e ◦ α ) ◦ e . In addition, having a solution pair δ, ε for a given ∆, automatically gives us the solution pairs ( δ − α ) , ( ε + β ) for ∆ replaced with ∆ + e ◦ α + β ◦ e .When e [ u ] = 0 and e [ u ] = 0 can be solved for their highest derivatives, we can use the above freedom to reduceequation (15), with δ, ε and ∆ of potentially high differential orders, to the same equation, but with the differentialorders of δ, ε and ∆ bounded by the orders of e and e . Theorem 3.1.
Suppose that the rational ODE systems e [ u ] = 0 and e [ u ] = 0 of differential orders p and p ,respectively. Suppose also that they can be solved for the highest order derivatives, that is, for i = 0 , there existrational invertible matrices P i such that P i e i [ u ] = d pi dr pi u + ˜ e i [ u ] , where ˜ e i is of differential order < p i . (a) Theknowledge of δ and ∆ in (15) is sufficient to reconstruct ε uniquely. (b) For given ∆ and a δ of fixed differentialorder, the existence of an ε satisfying (15) is equivalent to a rational ODE system on the coefficients of δ . (c) If ∆ if of differential order < p + p , then (15) has a solution if and only if it has a solution where δ is of differentialorder < p and ε is of differential order < p .Proof. We first make the standard observation is that, under our hypotheses on e i ( i = 0 , f i [ u i ], we can find a unique differential operators g i and ˜ f i such that f i = g i + ˜ f i ◦ e i , with g i of differentialorder < p i . This is easy to prove by noting that we cannot decrease the differential order of e i by pre-composing itwith a non-zero differential operator and then recursively rewriting the highest order derivatives in f i , say d pi + q dr pi + q u i ,as − d q dr q ˜ e i [ u ] + d q dr q P i e i [ u i ]. Obviously, both g i and ˜ f i also have rational coefficients and, in fact, their coefficientsare linear rational differential operators applied to the coefficients of f i .To prove part (a), note that the identity e ◦ δ − ∆ = 0 + ε ◦ e , combined with our initial observation, impliesthat ε is uniquely fixed once we know ∆ and δ .To prove part (b), consider the decomposition e ◦ δ − ∆ = ˜∆ + ε ◦ e , with ˜∆ of differential order < p , whichby our initial observation always exists and is unique. Thus, δ, ε and ∆ satisfy (15) if and only if the coefficientsof ˜∆ are all zero. But construction, the coefficients of ˜∆ are linear rational differential operators acting on thecoefficients of δ and ∆.To prove part (c), we first apply our initial observation to get the decomposition δ = ˜ δ + ε ◦ e , where thedifferential order of ˜ δ is < p . Then e ◦ ˜ δ = ∆ + ˜ ε ◦ e , with ˜ ε = ε − ε . The differential orders of e ◦ ˜ δ and ∆ areboth < p + p , hence by comparison we can conclude that the differential order of ˜ ε is < p . Let f ( r ) = 1 − Mr , f ′ ( r ) = f ( r ) r , f ( r ) = 2 Mr . (16)Define the (generalized) spin- s Regge-Wheeler operator with mass parameter M , angular momentum quantumnumber l and frequency ω by D s φ = ∂ r f ∂ r φ − r [ B l + (1 − s ) f ] φ + ω f φ, (17)where B l = l ( l + 1), with l = 0 , , , . . . . We will assume that ω = 0 and that s is a non-negative integer. Of course,for any s , D s has rational coefficients and satisfies the hypotheses of Lemma 2.3.6onsider the upper triangular rational ODE system (cid:20) D s ∆0 D s (cid:21) (cid:20) u u (cid:21) = (cid:20) (cid:21) , (18)where we suppose that ∆ is of differential order at most 1. As discussed in Section 3, this system is reducible todiagonal by an equivalence (Section 3) with rational coefficients if and only if the following version of Equation 15is satisfied: D s ◦ δ = ∆ + ε ◦ D s . (19)By Theorem 3.1, without loss of generality, we can consider this problem restricted to the following class of operators:∆ = 1 r (∆ r∂ r + ∆ ) , (20) δ = δ r∂ r + δ , (21) ε = δ r∂ r + [2 ∂ r ( rδ ) − f f δ + δ ] , (22)where δ i , ∆ i , for i = 0 ,
1, are all rational functions. Plugging this parametrization into (19) and comparing coeffi-cients, we find the equivalent ODE system e (cid:20) δ δ (cid:21) := (cid:20) f f (cid:21) r ∂ r (cid:20) δ δ (cid:21) + " f − ω r f + 2[ B l + f (1 − s )]2 f f − f r∂ r (cid:20) δ δ (cid:21) + " f ( s − s ) − ω r ( f − f ) f − f f [ B l + 1 − s ]0 f ( s − s + f ) δ δ (cid:21) = (cid:20) ∆ ∆ (cid:21) (23)for δ , δ , with ∆ , ∆ as inhomogeneous sources.Next, we will apply the analysis of Section 2 to check the conditions under which the system (23) has rationalsolutions for δ , δ , with given ∆ , ∆ .It is easy to see that the only singular points of the equation (23) are r = 0 , M, ∞ (recall that f (2 M ) = 0). Inthis work, we will not consider ∆ , ∆ with poles at other values of r , which means that these points are the onlypossible locations of the poles of δ , δ .For each of the singular points, we have the following multipliers ( S and T ), characteristic matrices ( E n ) andcharacteristic exponents ( σ ( e )). • r = 0: ˇ σ ( e ) = {± s ± s } , with det E n = (2 M ) ( n + s + s )( n + s − s )( n − s + s )( n − s − s ) and S = (cid:20) (cid:21) , T = (cid:20) r − r − (cid:21) , E n = − M (cid:20) n − n − s + s − n (1 − s )2 n n + 2 n − s + s (cid:21) . (24) • r = 2 M : ˇ σ M ( e ) = {− } , with det E n = ( n + 1) [( n + 1) + 16 M ω ] and S = " ( r − M )2 M ( r − M ) M , T = (cid:20) M ( r − M ) (cid:21) , E n = (cid:20) ( n + 1) − M ω ( n + 1)2( n + 1) ( n + 1) (cid:21) . (25) • r = ∞ : ˆ σ ∞ ( e ) = {− , } , with det E n = 4 ω n ( n + 1) and S = (cid:20) (cid:21) , T = (cid:20) r
00 1 (cid:21) , E n = (cid:20) − ω ( n + 1)2 n n ( n + 1) (cid:21) . (26)Suppose that A ρ = T − ρ [ ∆ ∆ ] has leading order ˇ m = min i | ˇ A i | , and trailing order ˆ m = max i | ˆ A i ∞ | . At r = 2 M ,we do not need the specific order, but just some lower bound m ≤ min i | ˇ A i M | r =2 M for some integer m ≤ −
1. Wechoose a bound of this form because then the identitymin { m } ∪ ˇ σ M ( e ) = min { m, − } = m (27)7etermines that the Laurent series expansion of the solution δ = [ δ δ ] must belong to δ ∈ " ( r − M )2 M ( r − M ) M ( r − M ) min { m }∪ ˇ σ M ( e ) C [[( r − M )]] = (cid:20) ( r − M )2 M (cid:21) ( r − M ) m +1 C [[ r ]] . (28)Since this is the only condition to be satisfied for poles other than at r = 0 , ∞ , without loss of generality, we cantake R = f m +1 = ( r − M M ) m +1 as a convenient universal multiplier. So that, according to Theorem 2.4, any rationalsolution of (23) must satisfy δ ∈ R C [ r, r − ] = f m +1 C [ r, r − ] . (29)Finally, we can parametrize any such solution with the following bounded order Laurent polynomial: δ = f m +1 ˆ n X n =ˇ n d n r n , where ( ˆ n = max { ˆ m } ∪ σ ∞ ( e ) = max { ˆ m, } , ˇ n = min { ˇ m } ∪ σ ( e ) = min { ˇ m, − s − s } . (30) We finish with a few explicit examples. Sometimes, in specific examples, a δ, ǫ solution for a given ∆ can be foundby trial and error. However, when unguided, such a process can be quite laborious. And at the end, if no solutionwas found, one cannot automatically conclude that a solution does not exist. Using the method presented above,when the trial and error method becomes too laborious, it can be automated using a computer algebra system.Moreover, our method can also furnish a proof that in some situation no solution exists.Below, we find it helpful to use the notation ˇ O ( r p ) to denote the leading order of a Laurent series at r = 0 andˆ O ( r q ) to denote the trailing order of a Laurent series at r = ∞ .1. The equation D ◦ δ = f r + ε ◦ D , (31)where (cid:20) ∆ ∆ (cid:21) = (cid:20) f (cid:21) = (cid:20) ˇ O ( r − )0 (cid:21) = (cid:20) ˆ O ( r − )0 (cid:21) , (32)gives rise to the normalized sources | ˇ A | = (cid:20) ∞ (cid:21) , | ˇ A M | = (cid:20) ∞ (cid:21) , | ˆ A ∞ | = (cid:20) − −∞ (cid:21) . (33)Since A M = T − M ∆ has no pole at r = 2 M , we can choose the universal multiplier to be R = 1. A and A ∞ give the orders ˇ m = 0 and ˆ m = − n = min { , − − } = − n = max {− , } . From this information, we can conclude that there exists the unique solution δ = − , ε = − . (34)2. The equation D ◦ δ = f r + ε ◦ D (35)where ∆ , ∆ and the corresponding normalized sources are the same as in Example 2. Again, since A M = T − M ∆ has no pole at r = 2 M , we can choose the universal multiplier to be R = 1. A and A ∞ givethe orders ˇ m = 0 and ˆ m = − n = min { , − − } = 0 andˆ n = max {− , } = 0. From this information, we can conclude that there exists no solution for δ, ε .3. The equation D ◦ δ = − f r (cid:18) B l + f (cid:19) + ε ◦ D (36)where (cid:20) ∆ ∆ (cid:21) = " − f (cid:16) B l + f (cid:17) = (cid:20) ˇ O ( r − )0 (cid:21) = (cid:20) ˆ O ( r − )0 (cid:21) (37)8ives rise to the normalized sources | ˇ A | = (cid:20) − ∞ (cid:21) , | ˇ A M | = (cid:20) ∞ (cid:21) , | ˆ A ∞ | = (cid:20) − −∞ (cid:21) . (38)Since A M = T − M ∆ has no pole at r = 2 M , we can choose the universal multiplier to be R = 1. A and A ∞ give the orders ˇ m = − m = − n = min {− , − − } = − n = max {− , } = 0. From this information, we can conclude that there exists no solution for δ, ε .4. The equation D ◦ δ = ∆ + ε ◦ D , (39)with∆ =24 if r ω − if (6 f f + 6 B l f + A l ) rω∂ r − i (cid:18) A l + 2( B l −
3) + ( A l − B l )(1 + 2 B l ) + 2( A l + 6 B l ) f − A l B l f − f (cid:19) ω + f f B l ( − f + 8 f f − B l + 16 f B l + A l ) irω ∂ r + if B l ( A l ( B l − f ) + 12 f (1 − (2 + B l ) f + f )) r ω (40) (cid:20) ∆ ∆ (cid:21) = (cid:20) ˇ O ( r − )ˇ O ( r − ) (cid:21) = (cid:20) ˆ O ( r )ˆ O ( r ) (cid:21) (41)gives rise to the normalized sources | ˇ A | = (cid:20) − − (cid:21) , | ˇ A M | = (cid:20) (cid:21) , | ˆ A ∞ | = (cid:20) (cid:21) . (42)Since A M = T − M ∆ has no pole at r = 2 M , we can choose the universal multiplier to be R = 1. A and A ∞ give the orders ˇ m = − m = 2 and hence the Laurent polynomial bounds ˇ n = min {− , − − } = − n = max { , } = 2. From this information, we can conclude that there exists a unique solution δ = − if f r ω∂ r − i (6 f f + 12 B l f + A l ) r ω + if B l (4 A l − − f + 24 f f + 3 B l + f l ( l − r∂ r ω + i ( A l + 2 A l (9 + 5 B l ) f − B l f ( − f − B l ))4 ω , (43)with ε given by (22). The above result was obtained and checked with computer algebra.5. The equation D ◦ δ = ∆ + ε ◦ D , (44)with ∆ = − if f rω∂ r − i A l ω + 6 f f (3 f − B l irω ∂ r − − if B l (18 f f − f B l + A l ) r ω (45) (cid:20) ∆ ∆ (cid:21) = (cid:20) ˇ O ( r − )ˇ O ( r − ) (cid:21) = (cid:20) ˆ O ( r )ˆ O ( r ) (cid:21) (46)gives rise to the normalized sources | ˇ A | = (cid:20) − − (cid:21) , | ˇ A M | = (cid:20) (cid:21) , | ˆ A ∞ | = (cid:20) (cid:21) . (47)Since A M = T − M ∆ has no pole at r = 2 M , we can choose the universal multiplier to be R = 1. A and A ∞ give the orders ˇ m = − m = 1 and hence the Laurent polynomial bounds ˇ n = min {− , − − } = − n = max { , } = 1. From this information, we can conclude that there exists a unique solution δ = − if r ω + 2 if ( B l − f )( B l − f + f ) r∂ r ω + i ( A l B l + 6 A l B l ( B l + 3) f − B l − f − f )3 ω , (48)with ε given by (22). The above result was obtained and checked with computer algebra.9he solution from Example 1 can be generalized to D s s − s ) = f r + 1( s − s ) D s , (49)which actually works for any complex values of s , s , except for s = ± s . We obtained this parametric solutionby trial and error, while trying to understand and generalize some identities from [6]. However, simply havingthis formula does not tell us whether it is the unique solution. Applying our systematic approach, we can checkuniqueness as we did in Example 1 for s = 0 and s = 1, but only for specific values of s , s at a time. The reasonis that the lower bound ˇ n on the Laurent polynomial order of δ in (30) is influenced by min σ ( e ) = − s − s (atleast for non-negative integer values of the s i ), which depend on the s i .Note also that the above formula is singular for s = ± s and no longer tells us anything about the existenceof solutions in those cases. On the other hand, our systematic approach can check that indeed no solution exists,again on a case by case basis, as we did for s = 0 in Example 2.In Equation (88) of [7], we had managed to reduce a 3 × harmonic or Lorenz gauge-fixed versionof Maxwell’s equations) on the background of a Schwarzschild black hole. That work was strongly inspired by [6],which achieved a similar decoupling for the Lichnerowicz equation (which could be interpreted as the harmonic or de Donder gauge-fixed version of linearized Einstein’s equations) also on Schwarzschild. The methods and resultsresult achieved in [6] are unfortunately somewhat obscure and implicit. We aim to clarify those results usingthe systematic methods that we have outlined in [7] and in the present work. For instance, the existence of thesolutions from Examples 4 and 5 is equivalent to Equations (3.49–51) from [6], which were instrumental to theirmain decoupling results, but apparently obtained by trail and error, without a clear guide to how they could bereproduced independently. Fortunately, Examples 4 and 5 show that our systematic approach can rediscover theseformulas in a straight forward way using computer algebra. The main goal of this work was to conclusively decide when it is or is not possible to reduce an upper triangularrational ODE system like (18) to diagonal form by a transformation like (14) with rational coefficients, where onthe diagonal we have generalized Regge-Wheeler operators. This question was left open in our previous work [7].In Section 3, we showed how to reduce this question to the existence of a rational solution to an auxiliary rationalODE system. In Section 2 we showed that, under mild hypotheses, the existence of a rational solution of a rationalODE system can be reduced to a finite dimensional linear algebra problem. Hence, such a question can alwaysbe conclusively decided, at least on a case by case basis. In Section 4.1, we gave several examples illustrating ourmethods. These examples reproduce, in a systematic way, some identities previously discovered by voluminous trialand error calculations in [6].These identities were used in [7] to significantly simplify, after a separation of variables, the coupled radial modeequations of the vector wave equation on Schwarzschild spacetime. Our Example 3 shows that this simplificationcannot be further improved. The vector wave equation plays a role relative to the Maxwell equation that is analogousto the Lichnerowicz equation relative to the linearized Einstein equations. In a future work, we will further buildon the results of [6] to apply to the Lichnerowicz equation the same simplifications as were applied to the vectorwave equation in [7]. The methods developed in this work, will help decide how much these simplifications could beimproved. Of course, it will also be very interesting to see how much the simplifications studied jointly in [7] and thecurrent work will translate from the (non-rotating) Schwarzschild black hole to the significantly more complicatedcase of the (rotating) Kerr black hole.An interesting generalization of the question of the existence of rational solutions to the rational ODE e [ u ] = v is the characterization of the image of e when applied to arbitrary rational arguments. An equivalent question isthe characterization of the rational cokernel of e . Then, even if no rational solution to e [ u ] = v exists, preciselyidentifying the equivalence class of v in the cokernel of e might allow us to choose a representative from theequivalence class of v that is simplest, with respect to some reasonable criteria. Such questions also have connections10ith the theory of D -modules with rational coefficients [10, Ch.2], [9, Sec.10.5], which is an algebraic formalism forstudying linear differential equations, especially those with polynomial or rational coefficients. These topics mayalso be explored in future work. Acknowledgments
Research of the author was partially supported by the GA ˇCR project 18-07776S and RVO: 67985840. The authoralso thanks Francesco Bussola for help with converting Equations (3.49–51) of [6] into the form given in Examples 4and 5.
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