The trace formulas yield the inverse metric formula

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1 The trace formulas yield the inverse metric formula arxiv:math-ph/ v4 3 May 2000 Ronaldo Rodrigues Silva Centro Brasileiro de Pesquisas Físicas Rua Dr Xavier Sigaud 150, , Rio de Janeiro, RJ, Brazil Abstract It is a well-known fact that the first and last non-trivial coefficients of the characteristic polynomial of a linear operator are respectively its trace and its determinant This work shows how to compute recursively all the coefficients as polynomial functions in the traces of the successive powers of the operator With the aid of Cayley-Hamilton s theorem the trace formulas provide a rational formula for the resolvent kernel and an operator-valued null identity for each finite dimension of the underlying vector space The 4-dimensional resolvent formula allows an algebraic solution of the inverse metric problem in general relativity PACS: 0210, 0230T, 0420 I Introduction The eigenvalue problem arises in a variety of different branches of mathematical physics For instance, it is well-known that quantum systems reach stationary states that are the eigenvectors of a suitable linear operator defined throughout a complex vector space representing the physical quantum states In the infinite dimensional case real and complex analytical methods have been developed to compute the eigenvalues of a linear operator We mention, for instance, global variational methods, which consist of investigating stationary levels of a suitable energy functional, and local perturbative methods by means of complex analytic continuation On the other hand, if the underlying vector space is finite dimensional the eigenvalue problem becomes an algebraically well-posed problem by means of the characteristic polynomial It is exactly this feature that justifies the algebraic approach in this work Following this general device, the algebraic environment is set by the complex number field After working out a few ronaldor@catcbpfbr 1

2 tools, the most important of these being Newton s identities, and setting very fundamental concepts and basic results in complex linear algebra, we succeed in achieving the mathematical core of this work: a recursive algorithm to compute the coefficients of the characteristic polynomial as algebraic functions in the traces of the successive powers of the linear operator With the aid of Cayley-Hamilton s theorem the trace formulas already obtained provide an operator-valued null identity for each finite dimension of the underlying vector space As a by-product it sheds light on the algebraic structure of the associative algebra of complex linear operators The computational skill just attained is enough to yield an operator-valued polynomial with rational coefficients for the finite-dimensional resolvent kernel, which improves a known result by revealing its rational dependence with respect to the spectral variable, as well as with respect to the linear operator In the context of general relativity, the 4-dimensional characteristic formula endows a polynomial expression for the volume scalar density Furthermore, the 4-dimensional resolvent formula yields a tensor-valued third-degree polynomial for the inverse metric, thus avoiding the computacional and formal drawbacks of the Neumann series II Newton s identities The fundamental theorem of algebra [1] together with Euclid s division algorithm imply algebraic closureness [2] of the complex number field; this basic result is encompassed without proof by Theorem 0 If pz) = z n +D 1 z n 1 + +D n 1 z +D n is a polynomial with complex coefficients D k then there exist complex numbers λ 1, λ 2,, λ n, called roots of p, such that pz) = z λ 1 )z λ 2 ) z λ n ) As a consequence of the identity principle, the following relations between coefficients and roots hold: D 1 = λ 1 +λ 2 + +λ n, +D 2 = λ 1 λ 2 + +λ n 1 λ n, ) k D k = k-products of λ s, ) n D n = λ 1 λ 2 λ n 1) The right-hand side of 1) defines the elementary symmetric functions [3] in the variables λ 1, λ 2,, λ n Next to them, the most important symmetric 2

3 functions are the sums of like powers: T 1 = λ 1 +λ 2 + +λ n, T 2 = λ 1 2 +λ λ n 2, T k = λ 1 k +λ 2 k + +λ n k, D k and T k, besides being symmetric, are homogeneous functions of degree k in the variables λ 1, λ 2,, λ n We shall derive a set of recursive relations connecting D s and T s by which the λ s are eliminated from 1) and 2) To pursuit this goal we need two lemmas The first one is an improved version of the remainder theorem Proposition 1 If pz) = z n + D 1 z n D n 1 z + D n is a polynomial with complex coefficients D k and λ is a complex number then pz) pλ) z λ = z n 1 +λ+d 1 )z n 2 +λ 2 +D 1 λ+d 2 )z n 3 + +λ n 1 +D 1 λ n 2 + +D n 1 ) Proof: By induction on the degree For n=1, pz) pλ) = z +D 1 ) λ+ D 1 ) = z λ For generic n, pz) pλ) = = z n +D 1 z n 1 + +D n 1 z +D n ) λ n +D 1 λ n 1 + +D n 1 λ+d n ) = zz n 1 +D 1 z n 2 + +D n 1 ) λλ n 1 +D 1 λ n 2 + +D n 1 ) = zz n 1 +D 1 z n 2 + +D n 1 ) λ n 1 +D 1 λ n 2 + +D n 1 ))+ +z λ)λ n 1 +D 1 λ n 2 + +D n 1 ) From the inductive hypothesis the last expression reads zz λ)z n 2 +λ+d 1 )z n 3 + +λ n 2 +D 1 λ n 3 + +D n 2 ))+ +z λ)λ n 1 +D 1 λ n 2 + +D n 1 ) = z λ)z n 1 +λ+d 1 )z n 2 + +λ n 2 +D 1 λ n 3 + +D n 2 )z+ +λ n 1 +D 1 λ n 2 + +D n 1 )) The second lemma is a contribution of differential calculus to algebra Proposition 2 If λ 1, λ 2,, λ n are the complex roots of the nth-degree polynomial p and p is its derivative polynomial, then p z) = n k=1 pz) z λ k Proof: From pz) = d 0 z λ 1 )z λ 2 ) z λ n ), it follows p z) = d 0 z λ 2 )z λ 3 ) z λ n )+d 0 z λ 1 )z λ 3 ) z λ n )+ +d 0 z λ 1 )z λ 2 ) z λ n 1 ) = pz) z λ + pz) 1 z λ + + pz) 2 z λ n 3 2)

4 Thus, we are able to get Theorem 1 Newton s formulas [4]) If λ 1, λ 2,, λ n arethe complexroots of the polynomial pz) = z n +D 1 z n 1 +D 2 z n 2 + +D n 1 z+d n with complex coefficients D k and T k = λ 1 k +λ 2 k + +λ n k then T k +D 1 T k 1 + +D k 1 T 1 +kd k = 0, k = 1, 2,, n Proof: Propositions 1 and 2 yield nz n 1 +n 1)D 1 z n 2 +n 2)D 2 z n 3 + +D n 1 = = n { z n 1 +λ k +D 1 )z n 2 +λ 2 k +D 1λ k +D 2 )z n 3 + k=1 } +λk n 1 +D 1 λk n 2 + +D n 2 λ k +D n 1 ) = nz n 1 +T 1 +nd 1 )z n 2 +T 2 +D 1 T 1 +nd 2 )z n 3 + +T n 1 +D 1 T n 2 + +D n 2 T 1 +nd n 1 ) Equating coefficients we obtain n 1)D 1 = T 1 +nd 1, n 2)D 2 = T 2 +D 1 T 1 +nd 2, D n 1 = T n 1 +D 1 T n 2 + +D n 2 T 1 +nd n 1, from which follow the n 1) first relations to be shown Since each λ k is a root of p it verifies 0 = λ n k + D 1λk n D n 1 λ k + D n The addition of these relations yields 0 = n k=1 { λ n k +D 1 λ n 1 k + +D n 1 λ k +D n } = Tn +D 1 T n 1 + +D n 1 T 1 +nd n, which is the remaining formula to be proved III The trace formulas The characteristic polynomial [5] of a complex linear operator T in a finite dimensional complex vector space \V is defined by pz) = detzi T), where I is the identity operator in \V The degree of p as a polynomial in the complex variable z equals the dimension of \V as a complex vector space; its roots are called the characteristic values of T and the set of characteristic values is called the spectrum of T These definitions and properties lead to 4

5 Proposition 3 The trace of a complex linear operator T equals the sum of its characteristic values counted with multiplicities Proof: detzi T) = z λ 1 )z λ 2 ) z λ n ) By the matrix representation T i j of T we cast the left-hand side as z n T 1 1 +T T n n)z n 1 + terms of order n 2 In the right-hand side we have z n λ 1 +λ 2 + +λ n )z n 1 + terms of order n 2 Equating coefficients we obtain T 1 1 +T T n n = λ 1 +λ 2 + +λ n The roots of unity in the complex field yield the following factorization lemma in the associative algebra of complex linear operators Proposition 4 If T and W are commuting complex linear operators and θ = e i2π/k = cos 2π k +i sin 2π k with k a positive integer, then T k W k = T W)T θw)t θ 2 W) T θ k 1 W) Proof: As T and W commute we collect terms as T W)T θw)t θ 2 W) T θ k 1 W) = = T k +d 1 WT k 1 +d 2 W 2 T k 2 + +d k 1 W k 1 T+d k W k, where the complex numbers d 1, d 2,, d k are the coefficients of the polynomial z 1)z θ)z θ 2 ) z θ k 1 ) = z k 1, since θ j ) k = e i2π/k) j ) k = 1 for each j = 0, 1,, k 1 Therefore d 1 = d 2 = = d k 1 = 0 and d k = 1 From the multiplicative and homogeneity properties of determinants we get Proposition 5 If λ 1, λ 2,, λ n are the characteristic values of the complex linear operator T then λ 1 k, λ 2 k,, λ n k are the characteristic values of the linear operator T k Proof: For the particular case where W = wi, proposition 4 yields det T k w k I ) = dett wi)dett θwi) det T θ k 1 wi ) = λ 1 w) λ n w)λ 1 θw) λ n θw) λ 1 θ k 1 w) λ n θ k 1 w) = λ 1 w)λ 1 θw) λ 1 θ k 1 w) λ n w)λ n θw) λ n θ k 1 w) = λ 1 k w k ) λ n k w k ) Thus det zi T k) = z λ 1 k ) z λ n k ) Collecting the former results we are ready to achieve the recursive formulas enclosed by 5

6 Theorem 2 If pz) = z n + D 1 z n 1 + +D n 1 z + D n is the characteristic polynomial of the complex linear operator T and T k = tracet k ) then T k +D 1 T k 1 + +D k 1 T 1 +kd k = 0, k = 1, 2,, n Proof: It is a straightforward consequence of theorem 1 together with propositions 3 and 5 The key content of theorem 2 shall be stood out by the trace formulas statement: the coefficients of the characteristic polynomial of a linear operator can be recursively computed as polynomial functions in the traces of its successive powers The trace formulas D k = D k T 1, T 2,, T k 1, T k ) for the coefficients of the characteristic polynomial are listed below for k up to 8 D 1 = T 1, D 2 = 1 2 T T 2, D 3 = 1 6 T T 1T T 3, D 4 = 1 24 T T 1 2 T T 1T T T 4, D 5 = 1 T T T 2 1T T 3 1T 8 1T T 4 1T 4 + 1T 6 2T 3 1T 5 5, D 6 = 1 T T T T T T T 2 2 1T T 4 1T 6 1T 2 T 3 + 1T 5 1T 5 1 T T 8 2T T T 6 6, D 7 = 1 T T T 2 1 T T 3 1 T T T T T T 2 T 3 1 T T T 48 1T T 8 1T 2 T 4 1 T 18 1T T 6 1T 6 1 T T T 10 2T T 12 3T 4 1T 7 7, D 8 = 1 T T T T T T T T T 4 1 T T 2 T T T 5 1 T T T T 2 T T T 3 1 T T T 24 1T 2 2 T 3 1 T 10 1T 2 T 5 1 T 12 1T 3 T T 7 1T T T T 4 1 T 36 2T T 12 2T T 15 3T T T 8 8 The characteristic formulas for dimensions up to four are written down: det zi T) = z T 1, det zi T) = z 2 T 1 z T 1 2 T 2 ), det zi T) = z 3 T 1 z T 1 2 T 2 )z 1 6 T 1 3 3T 1 T 2 +2T 3 ), 4) det zi T) = z 4 T 1 z T 1 2 T 2 )z T 1 3 3T 1 T 2 +2T 3 )z T 1 4 6T 1 2 T 2 +8T 1 T 3 +3T 2 2 6T 4 ) 3) 6

7 IV Null identities The polynomial pz) = d 0 z n + d 1 z n d n 1 z + d n with complex coefficients d k is said to annihilate the complex linear operator T if pt) = d 0 T n +d 1 T n 1 + +d n 1 T+d n I = O, the identically null operator The knowledge of the principal ideal [6] of polynomials that annihilate a linear operator T is essential to attain computational skill in the associative algebra generated by T To ensure this goal we state without proof one of the fundamental results in linear algebra Theorem 3 Cayley-Hamilton [7]) The characteristic polynomial of a linear operator annihilates it Joining Cayley-Hamilton s theorem with trace formulas statement we conclude that for each finite dimension of the underlying vector space there is a fundamental null identity in the associative algebra of linear operators For instance, in dimensions up to four, the characteristic formulas 4) yield the following null identities: T T 1 I = O, T 2 T 1 T+ 1 2 T 1 2 T 2 )I = O, T 3 T 1 T T 1 2 T 2 )T 1 6 T 1 3 3T 1 T 2 +2T 3 )I = O, T 4 T 1 T T 1 2 T 2 )T T 1 3 3T 1 T 2 +2T 3 )T T 1 4 6T 1 2 T 2 +8T 1 T 3 +3T 2 2 6T 4 )I = O V The finite-dimensional resolvent kernel The resolvent of a complex linear operator T is the operator-valued function Rdefined by Rz) = zi T) 1 It isawell-known fact that Risanoperatorvalued analytic function outside the spectrum of T We shall refine such a result showing that in the finite dimensional case R is an operator-valued rational function completely tied by a finite set of complex-valued rational functions, whose coefficients are exactly the trace formulas In this vein we need an improved version of the remainder theorem in the associative algebra of complex linear operators, whose content is the extension of proposition 1 to operator-valued polynomials Proposition 6 If pw) = w n +D 1 w n 1 + +D n 1 w +D n is a polynomial with complex coefficients D k and T and W are commuting complex linear 7

8 operators then pw) pt) = W T) [ T n 1 +W+D 1 I)T n 2 +W 2 +D 1 W+D 2 I)T n W n 1 +D 1 W n 2 + +D n 1 I)I ] Proof: By induction on the degree For n = 1, pw) pt) = W+D 1 I) T+D 1 I) = W T For generic n, pw) pt) = = WW n 1 +D 1 W n 2 + +D n 1 I)+D n I+ TT n 1 +D 1 T n 2 + +D n 1 I) D n I = W T)W n 1 +D 1 W n 2 + +D n 1 I)+ [ +T W n 1 +D 1 W n 2 + +D n 1 I)+ ] T n 1 +D 1 T n 2 + +D n 1 I) From the inductive hypothesis the last expression reads W T)W n 1 +D 1 W n 2 + +D n 1 I) +TW T) [ T n 2 +W+D 1 I)T n 3 + +W n 2 +D 1 W n 3 + +D n 2 I)I ] = W T) [ T n 1 +W+D 1 I)T n 2 + +W n 2 +D 1 W n D n 2 I)T+W n 1 +D 1 W n 2 + +D n 1 I)I ] We are ready to achieve a rational formula for the resolvent kernel Theorem 4 If pw) = w n +D 1 w n 1 + +D n 1 w+d n is the characteristic polynomial of the complex linear operator T and z is any complex number not belonging to the spectrum of T then zi T) 1 = pz) 1 Tn 1 + z +D 1 pz) Tn 2 + z2 +D 1 z +D 2 T pz) n zn 1 +D 1 z n 2 + +D n 1 I pz) Proof: It is enough to consider Cayley-Hamilton s theorem for the linear operator T and set W = zi in proposition 6; notice that in such case W k + D 1 W k 1 + +D k I = z k +D 1 z k 1 + +D k )I for each k = 1, 2,, n The joint content of theorem 4 and the trace formulas statement shall be pointedoutas: for each finite dimension of the underlying vector space there is a fundamental rational formula for the resolvent kernel of a 8

9 linear operator For instance, in dimensions up to four, the trace formulas 3) provide the following resolvent formulas: zi T) 1 = z T 1 ) 1 I, zi T) 1 = z 2 T 1 z T [ 1 2 T 2 )) T+z T1 )I ], zi T) 1 = z 3 T 1 z T 1 2 T 2 )z 1 ) 1 [ 6 T 1 3 3T 1 T 2 +2T 3 ) T 2 + +z T 1 )T+ z 2 T 1 z + 1 ) ] 2 T 1 2 T 2 ) I, zi T) 1 = z 4 T 1 z T 1 2 T 2 )z T 1 3 3T 1 T 2 +2T 3 )z+ + 1 ) 1 [ 24 T 1 4 6T 2 1 T 2 +8T 1 T 3 +3T 2 2 6T 4 ) T 3 + +z T 1 )T 2 + z 2 T 1 z + 1 ) 2 T 1 2 T 2 ) T+ z 3 + T 1 z T 1 2 T 2 )z 1 ) ] 6 T 1 3 3T 1 T 2 +2T 3 ) I 5) VI Applications to general relativity In general relativity the fundamental physical object is an effective geometry, mathematically represented by a second-rank covariant tensor field g, defined throughout a suitable 4-dimensional manifold In a local coordinate system x = x α ) the metric tensor can be written as g = g µν dx µ dx ν Investigations on general relativity are frequently carried out under the assumption that there exists some background geometry o g= o g µν dx µ dx ν The metric properties to be assumed on o g vary depending on the gravitational scenario In order to perform calculations it suffices to set o g a Ricci-flat metric However, to interpret the results as physically meaningful it is generally agreed that one should require o g to be a flat metric Even this case is sometimes thought to be too broad, as some authors claim to set harmonic coordinates [8] or even cartesian coordinates [9] We do not take into account here any suplementary conditions on the background geometry o g The effective and background geometries are related by a tensor field h = h µν dx µ dx ν by a connecting equation, the generally accepted form of which being g = o g+h The scalar density gdx 4 associated with g requires us to compute detg) = det o g +h) This can be achieved by means of 9

10 Proposition 7 If g = o g +h, H = o g 1 h, H k = trace H k) and the underlying manifold is 4-dimensional then detg) det o g) = 1+H H 1 2 H 2 )+ 1 6 H 1 3 3H 1 H 2 +2H 3 ) H 1 4 6H 1 2 H 2 +8H 1 H 3 +3H 2 2 6H 4 ) Proof: From g = o g+h = o g I+H) it follows that detg) = det o g)deti+h) Now it is enough to consider the 4-dimensional characteristic formula in 4) with z = 1 and T = H; notice that T k = ) k H k implies T k = ) k H k The Levi-Civita connection associated with g requires us to compute g 1 = og +h ) 1 = I+H) 1 o g 1 The known explicit form is given by the Neumann series [10] I+H) 1 = I H+H 2 H 3 + In local coordinates it reads g µν = δ µ β g o µα h αβ + g o µα o h αλ g λǫ h ǫβ g o µα o h αλ g λǫ o og h ǫρ g ρσ h σβ + ) βν, where g µν and o g µν are well-defined by g µα g αν = δ µ ν = o g µβ o g βν Besides convergence requirements on the above series, we stress that such expression leads to technical difficulties when developing the Lagrangian variational formalism These problems were already dealt with in the literature, the proposed solution being to modify the above form of the connecting equation [11] We show how to overcome such drawbacks by means of Proposition 8 If g = o g +h, H = o g 1 h, H k = trace H k) and the underlying manifold is 4-dimensional then g 1 = 1+H H 1 2 H 2 )+ 1 6 H 1 3 3H 1 H 2 +2H 3 ) H 1 4 6H 1 2 H 2 +8H 1 H 3 +3H 2 2 6H 4 ) ) 1 [ H H 1 )H 2 1+H H 1 2 H 2 ) ) H H H 1 2 H 2 ) H 1 3 3H 1 H 2 +2H 3 ) ) I] og 1 Proof: From g = o g +h = o g I+H) it follows that g 1 = I+H) 1 o g 1 Now it is enough to consider the 4-dimensional resolvent formula in 5) with z = 1 and T = H; notice that T k = ) k H k implies T k = ) k H k 10

11 Acknowledgements The author thanks Dr Renato Klippert Barcellos for having collaborated with him in this work, for valuable discussions, and for the implementation of the computer algebra algorithm The author owes special thanks to Dr Nelson Pinto Neto and Dr Vitorio Alberto De Lorenci for encouragement, and for having drawn the attention of him to the relevance of the subject of the present work The author is also indebted to Carla da Fonseca-Bernetti for helpful corrections of the manuscript This work was financially supported by CNPq References [1] J V Uspensky, Theory of Equations p 293, McGraw-Hill Book Company, NY 1948) [2] S Lang, Algebra rev printing p 169, Addison Wesley Publishing Company, London 1971) [3] B L van der Warden, Algebra Vol I 5 th ed p 99, Springer-Verlag, NY 1991) [4] See Uspensky, op cit p 260 [5] S Mac Lane and G Birkhoff, Algebra 4 th printing, p 310, The MacMillan Company, London 1970); see also Lang, op cit p 399 [6] See Mac Lane and Birkhoff, op cit p 152 and Lang, op cit p 58 [7] See Mac Lane and Birkhoff, op cit p 316 and Lang, op cit p 400 [8] V Fock, The Theory of Space, Time and Gravitation, 2 nd rev ed p 369, Pergamon Press, Oxford 1966) [9] L D Landau and E M Lifshitz, Course of Theoretical Physics V 2: The Classical Theory of Fields, 4 th rev english ed p 330, Butterworth- Heinemann Ltd, Oxford 1975) [10] M Reed and B Simon, Methods of Modern Mathematical Physics I Functional Analysis, p 191, Academic Press, NY 1972) [11] L P Grishchuk, A N Petrov and A D Popova, Commun Math Phys 94, ) 11