Variable Separation and Solutions of Massive Field Equations of Arbitrary Spin in Robertson-Walker Space-Time

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1 Adv. Studies Theor. Phys., Vol. 3, 2009, no. 5, Variable Separation and Solutions of Massive Field Equations of Arbitrary Spin in Robertson-Walker Space-Time Antonio Zecca Dipartimento di Fisica dell Universita Via Celoria, 16 I Milano INFN Sezione di Milano, Italy GNFM, Gruppo Nazionale per la Fisica Matematica, Italy Abstract The arbitrary spin field equations are separated in the Robertson- Walker space-time by the Newmam-Penrose formalism and by using a null tetrad frame previously considered. The eigenvalue problem of the corresponding separated angular equations is solved by looking for polynomial solutions. The radial separated equations are solved in the flat space-time case and the asymptotic behaviour of the solutions explicited. The separated time equations are studied in elementary situations and in the linear and exponential expansion of the universe. The dependence of the time solutions on increasing spin is pointed out. PACS: Pm; K; Jb; Jk Keywords: Spin s field equation; R-W Space-time; Variable Separation; Solutions; Expanding Universe 1 Introduction The general solution of the massive field equation of arbitrary spin seems to be a difficult task even if the equation has received a well established unified formulation in curved space-time [2, 13, 12, 5]. There are indeed solution of the equation in different space-time models and for fixed spin values [6, 7, 14, 15, 16, 19, 20, 21, 22]. They have been generally obtained by variable separation in the line of the separation of Dirac equation in Kerr metric originally obtained by Chandrasekhar [3] by the Newman-Penrose formalism [10].

2 240 A. Zecca In particular, the separability of the field equation of arbitrary spin has been proved, by induction on the spin value, both in the Robertson-Walker and in the Schwarzschild space-time [21, 22]. The corresponding separated equations have also been solved in Robertson-Walker metric [14, 15, 16, 19, 20, 21, 22] for spin 1/2,1,3/2,2. However, in spite of those results it seems to lack, as far as the author knows, a unified solution of the field equation in a given spacetime model. In the following we supply this gap by solving the massive field equation of arbitrary spin in Robertson-Walker space-time. The choice of that space-time has been done on account of its physical interest, being the basis of the Standard Cosmology. Moreover the explicit dependence of the metric on the time has, a priori, an interest also from a mathematical point of view. In the present paper the field equation is formulated in terms of two coupled spinorial equations in a pair of spinor fields with suitable symmetry properties, according to a recent unified formulation [5]. The equations are explicited in terms of the directional derivative and spin coefficients by means of the Newman-Penrose formalism based on a null tetrad frame previously introduced [9]. The equation are separated by an elementary variable separation method. The separated angular equations are integrated. The corresponding eigenvalue problem is solved by looking for polynomial solutions. There results a standard discrete structure of the eigenvalues and a Jacobi polynomial like form of the solutions. The separated radial equations are reduced to independent differential equations and integrated in case of flat space-time. The asymptotic behaviour of the corresponding solutions, that are essentially given by confluent hypergeometrci functions, is explicited. (For spin 1/2 explicit solutions exist also for open and closed space-tme [14, 4, 18]). The separated time dependence results in a pair of coupled differential equations in two functions of the time. Besides elementary situations, the time equations are studied and solved for the linear and the exponential expansion of the universe. The study is developed in an essentially elementary way. The results are coherent with those of previous papers relative to special value of the spin, that can be obtained from the present scheme by simply specializing the value of the spin. 2 Separation of field equation in Robertson- Walker space-time. The spinor formulation of the field equation of spin s in conformally flat spacetime can be written [5] A Ẋ φ AA 1...A n + μ χ A1 A 2...A n Ẋ =0 ŻA χ A1 A 2...A n Ż μ φ AA1 A 2...A n =0, (1)

3 Spin s field in R-W space-time 241 with n =0, 1, 2,.. and where μ = im 0 / 2, m 0 the mass of the particles of the field; n is related to the spin value s by s =(n +1)/2 and the spinor fields are assumed to have the symmetry properties φ AA1...A n = φ (AA1...A n) and χ A1 A 2...A n Ẋ = χ (A 1 A 2...A n)ẋ. In the following we will study the solution of eq. (1) in the Robertson-Walker space-time whose line element is given by ds 2 = dt 2 R(t) 2[ dr 2 1 ar + 2 r2 (dθ 2 + sin θ 2 dϕ 2 ) ], a =0, ±1 (2) and that is conformally flat [12] as it can be directly checked by the vanishing of the Weyl spinor (e.g. [17]). On account of the symmetry properties of the spinor fields it is useful to set: φ h φ AA1 A 2...A n A + A A n = h, h =0, 1, 2,...,n+1 χ jẋ χ A 1 A 2...A n Ẋ A 1 + A A n = j, j =0, 1,...,n (3) The eq. (1) can be separated by variable separation by preliminary developing the scheme in the Newman-Penrose formalism. To that end we consider the null tetrad frame e μ a, (a =1, 2, 3, 4, μ= t, r, θ, ϕ) whose corresponding directional derivatives and non zero spin coefficients are given by [9] D = oȯ = e μ 1 μ = 1 2 ( t + 1 ar 2 R r ) Δ= = 1 1 eμ 2 μ = 1 2 ( t 1 ar 2 R r ) δ = 0 1 = e μ 3 μ = 1 rr ( 2 θ + i csc θ ϕ ) δ = 1 0 = e μ 4 μ = 1 rr ( 2 θ i csc θ ϕ ) (4) ( ρ = 1 Ṙ + 1 ar 2 2 R rr ( μ = 1 Ṙ 1 ar 2 2 R rr ), ɛ = γ = Ṙ 2 2 R ), α = β = cot θ 2rR 2 (Ṙ = dr/dt). By expliciting the spinorial derivatives [12] in eq. (1) one then obtains in correspondence to the first eq. (1) [D (n +1)ρ +(1 n)ɛ]φ 1 [δ +(n +1)β]φ 0 = μ χ 0 0 [D n + ρ +(3 n)ɛ]φ 2 [δ +(n 1)β]φ 1 = μ χ 1 0 [D (n 1)ρ +(5 n)ɛ]φ 3 [δ +(n 3)β]φ 2 = μ χ (5) [D ρ +(n +1)ɛ]φ n [δ +(1 n)β]φ n 1 = μ χ n 0 [Δ + μ +(n +1)ɛ]φ 0 [δ +(1 n)β]φ 1 = μ χ 0 1 [Δ + 2μ +(n 1)ɛ]φ 1 [δ +(3 n)β]φ 2 = μ χ 1 1 [Δ + 3μ +(n 3)ɛ]φ 2 [δ +(5 n)β]φ 3 = μ χ [Δ + (n +1)μ +(1 n)ɛ]φ n [δ +(n +1)β]φ n+1 = μ χ n 1

4 242 A. Zecca In correspondence to the second eq. (1) one has [D ρ +(1 n)ɛ]χ 0 1 [δ +(1 n)β]χ 0 0 = μ φ 0 [D ρ +(3 n)ɛ]χ 1 1 [δ +(3 n)β]χ μχ 0 0 = μ φ 1 [D ρ +(5 n)ɛ]χ [δ +(5 n)β]χ μχ 1 0 = μ φ 2... [D ρ +(n +1)ɛ]χ n 1 [δ +(1+n)β]χ n 0 + nμχ n 1 0 = μ φ n [Δ + μ +(n +1)ɛ]χ 0 0 [δ +(n +1)β]χ 0 1 nρχ 1 1 = μ φ 1 (6) [Δ + μ +(n 1)ɛ]χ 1 0 [δ +(n 1)β]χ (n 1)ρχ = μ φ 2 [Δ + μ +(n 3)ɛ]χ 2 0 [δ +(n 3)β]χ 2 1 (n 2)ρχ 3 1 = μ φ 3... [Δ + μ +(3 n)ɛ]χ n 1 0 [δ +(3 n)β]χ ρχ n 1 1 n 1 = μ φ n [Δ + μ +(1 n)ɛ]χ n 0 [δ +(1 n)β]χ n 1 = μ φ n+1 By noting that the spin coefficients do not depend on the variable ϕ, itis possible to separate the equations (5), (6) by an elementary variable separation by setting φ j (t, r, θ, ϕ) =α(t)φ j (r)s j (θ) exp(imϕ), j =0, 1,...,n+1 χ h 0(t, r, θ, ϕ) =A(t)φ h+1 (r)s h+1 (θ) exp(imϕ), (7) χ h 1 (t, r, θ, ϕ) = A(t)φ h(r)s h (θ) exp(imϕ), h =0, 1,...,n It is convenient to assume m =0, ±1, ±2, ±3,... By using these expressions into eqs. (5), (6) one obtains the separated angular equations: L s j S j = λ j S j+1 L + 1 s+j S j+1 = λ 2s+j S j, j =0, 1,...,n (8) where it has been set L ± d = θ csc θ + d cot θ and λ i, i =0, 1, 2,.., n are the integration separation constants. These equations come from the separation of eq. (5). The separation of eq. (6) gives again the eqs. (8) after a suitable identification of the corresponding separation constants with the λ i s.

5 Spin s field in R-W space-time 243 The separated radial equations are instead given by the coupled equations: ikφ j+1 = 1 ar 2 φ j+1 + 2s j r 1 ar2 φ j+1 λ j φ r j, ikφ j = 1 ar 2 φ j j+1 r 1 ar2 φ j λ 2s+j φ r j+1, j =0, 1, 2,...,n (9) that are consistently obtained by identifying all the separation constants relative to the separation of the r, t variables with the one only constant k. One can choose k R. Finally, for what concerns the separated time dependence, one is left with αr +(s +1)αṘ im 0 AR = ikα ȦR +(2 s) A Ṙ im oαr = ika (10) that depends only on the value of the spin and on the motion of the cosmological background. 3 Solution of the angular and radial equations From eq. (8) one has that the angular function S j satisfies simultaneously the eigenvalue equations: L s j+1 L + j s S j = λ j λ 2s+j 1 S j L + 1+j s L s j S j = λ j λ 2s+j S j, j =0, 1,...,n (11) The problem of compatibility can be solved by noting that L + 1+j sl s j L s j+1 L + j s = 2j 2s j =0, 1,...,n (12) This implies for the λ j s the relation λ j λ 2s+j = λ j 1 λ 2s+j 1 = λ 0 λ 2s + j 2 jn j =0, 1,...,n (13)

6 244 A. Zecca In the following we will set λ 2 = λ 0 λ 2s. By expliciting the differential operators L ± d one obtains for S j the differential equation S j + cot θs j + { + λ 2 j 2 + jn m2 sin θ (j s)[1 + 2m cos θ +(s j 1) cos θ2 ] sin θ 2 } Sj = 0 (14) By putting S j =(ξ 1) m+s j 2 (ξ +1) m s+j 2 f j (ξ), ξ = cos θ, into eq. (14) and then ξ = 2x 1 in the resulting equation, one obtains for f j (x) the hypergeometric equation [1] x(1 x)f j +[j+m+1 s 2x(m+1)]f j [m(m+1) s(s 1) λ2 ]f j = 0 (15) If m 0 and one looks for polynomial solutions of eq. (14), then the acceptable solutions correspond to λ 2 = l(l +1) s(s 1), l = m, m +1,m+2,... (16) and can be written in terms of Jacobi polynomials [1]: P (m+j s,m j+s) l m ( cos θ). Therefore, apart an irrelevant factor, S j =(1 cos θ) m+s j 2 (cos θ +1) m s+j 2 P (m+j s,m j+s) l m (cos θ), m 0 (17) If m<0 it suffices to replace m by m and ξ by ξ (see eq. (14)). Therefore S j = (1 + cos θ) m +s j 2 (1 cos θ) m s+j 2 P ( m +j s, m j+s) l m (cos θ), (18) with l = m, m +1,... Note that, on account of the orthogonality properties of the Jacoby polynomials, the angular solutions (17), (18) times exp(imϕ), are already, for fixed j, s, a complete orthogonal set in L 2 (Ω). For what concerns the separated radial equations, the system (8) implies, by substitution, the differential equations r(1 ar 2 )φ j +[2s +2 (2s +3)ar 2 ]φ j +

7 Spin s field in R-W space-time { r [ k 2 a(j + 1)(2s +1 j) ] + +2ik(s j) 1 ar 2 + 2s λ2 } φj = 0 (19) r Note that the substitution j n +1 j (= 2s j) changes the eq. (19) into its complex conjugate. Therefore the solutions of the different equations are related by φ n+1 j = φ j. The solution of eq. (19) is difficult for a = ±1. Instead, in the flat space-time case, the equation can be reported, by setting φ j = r l s exp(ikr) f j (r) and then ξ = 2ikr in the resulting equation, to the confluent hypergeometric equation ξf j +(2l +2 ξ)f j (s + l +1 j)f j =0 (a = 0) (20) Therefore the radial solutions for a = 0 are [1] φ j (r) = r l s exp(ikr)φ(s + l +1 j;2l +2; 2ikr) (a = 0) (21) By using properties of the confluent hypergeometric function one can give the asymptotic behaviour of the radial solutions: φ j r 0 r l s, j φ j r 1 r j+1 exp( ikr), j < s φ j r φ 2s j 1 r 2s+1 j exp(ikr), j > s (22) φ s r 1 cos[kr π (l + 1)], j r s+1 2 = s the last result being valid for bosons only. The solution of eq. (19) has been determined also in the open and closed space-time case for s = 1/2 (e.g. [14, 4, 18]). For arbitrary higher values of the spin the solution of the equation is still an open problem, as fa as the author knows.

8 246 A. Zecca 4 Time evolution of spin s field equation in expanding universe. The time equation can be easily decoupled by substitution for a general time evolution of the cosmological background. Besides the static cosmological backgroundd and the massless field case, the equations will be integrated in two expanding situations of the standard cosmology of physical interest. Let first R(t) =const. = R 0. The eqs. (10) give then for both A and α oscillating solutions: α(t) = α 0 e ±iωt, ω = m k 2 /R 2 0 A(t) = α 0 m 0 R 0 [ ± ωr 0 + k]e ±iωt (23) For m 0 = 0 the time equations decouple and give α(t) A(t) = α(0) [ R(0) R(t) ] s+1 exp ( ik t 0 = A(0) [ R(0) R(t) ] 2 s exp ( ik t 0 ) dt R(t ) ) dt R(t ) (24) The result is coherent with the massless case studied in [16]. Note that A(t) rapidly increases with s for expanding universe. Suppose now R(t) = Ht, H constant. This is the situation that occurs in a fluid dominated Friedman equation [8] whose pressure and density are related by p = ρ/3. From eqs. (10) one has then the equation for A(t): Ä + 3 t Ȧ + [ s(2 s)+ k2 +2i k (1 s) ] H 2 H + m 2 t 2 0 A = 0 (25) By setting A = t β exp(im 0 t)z(t), β= 1±(s 1+i k H ) and then ξ = 2im 0t, one is left with the confluent hypergeometric equation ξz +(2β +3 ξ)z 5 2 Z = 0 (26)

9 Spin s field in R-W space-time 247 Therefore [1] the solution for A(t) is given by A(t) =t β exp(im 0 t)φ ( 5 ;1± 2 2(s 1+i k ); 2im H 0t ) while α(t) follows from this solution and from eqs. (10). Note that for s 1, β s. Therefore, for large t and s, α(t) A(t) t s 5 2 exp(im 0 t). As a second example consider the expansion R = exp(ht), H constant. This expanding law describes an inflationary phase or a vacuum dominated expansion of the Standard Cosmology [8, 11]. By setting τ = exp( Ht), the system (10) implies for A(τ) the differential equation: A 2 τ A + [ m s(2 s) 2ik(1 s) + + k2 ] A = 0 (27) H 2 τ 2 Hτ H 2 As in the previous case, by setting A(τ) =τ β exp( i k H t)z(τ), β = 1 2H [3H ± 9H 2 m s(s 2)] one obtains from (27) ξz +[2β 2 ξ]z (s 2+β)Z =0, ξ =2i k H τ (28) A solution for A is then in terms of confluent hypergeometric function [1] A(t) =τ β exp( i k H τ)φ(s 2+2β;2β 2; 2i k τ),τ = exp( Ht) (29) H Note that, for large t, A(t) exp( βht). If then one chooses β s/(2h) for s 1, one has A(t) α(t) exp( s 2 t). References [1] W. Abramowits, and I.E. Stegun, Handbook of Mathematical Functions Dover. New York, [2] H.A. Buchdahl, On the compatibility of relativistic wave equations for particle of higher spin in the presence of gravitational fields Nuovo Cimento 10, (1958)

10 248 A. Zecca [3] S. Chandrasekhar, The Mathematical Theory of Black Holes. Oxford University press. London, [4] X.B. Huang, Exact solution of the Dirac equation in Robertson-Walker space-time. arxiv: gr-qc/ [5] R. Illge, Massive fields of arbirary spin in curved space-time Comm. Math. Phys. 158, 433 (1993). [6] E.K. Kalnins, and M. Miller, Complete sets of functions for perturbaions of Robertson-Walker cosmologies and spin 1 equations in Robertson-Walker space-time Jour. Math. Phys. 32, 698 (1991) [7] E.G. Kalnins, M. Miller and G.C.Williams, Recent advances in the use of separation of variables methods in general relativity Phil. Trans. R. Soc. Lond. A340, 337 (1992). [8] E.W Kolb, and M.S. Turner, The Early Universe. Addison-Wesley publishing Company. New York, [9] E. Montaldi and A. Zecca, Neutrino wave equation in the Robertson- Walker Geometry. Int. Jour. Theor. Phys. 33, 1053 (1994) [10] E. T. Newman, R. Penrose, An appoach to gravitational radiation by a method od spin coefficients Jour. Math. Phys. 3,566(1962). [11] P.J.E.Peebles, Principles of Physical Cosmology. Princeton University Press. Princeton, New Jersey, [12] R. Penrose and W. Rindler, Spinors and space-time. Cambridge University Press. Cambridge, Vol. I, II.

11 Spin s field in R-W space-time 249 [13] V. Wünsch, Cauchy s problem and Huygens principle for relativistic higher spin wave equation in arbitrary curved space-time Gen. Rel. Grav. 17, 15 (1983) [14] A. Zecca, The Dirac equation in the Robertson-Walker space-time J. Math. phys. 37, 874 (1996). [15] A. Zecca, Separation of the massless spin 1 equation in Robertson-Walker space-time Int. J. Theor. Phys. 35, 323 (1996). [16] A. Zecca, Separation of the massless field equation for arbitrary spin in the Robertson-Walker space-time J. Math. phys. 37, 3539 (1996) [17] A. Zecca, Weyl spinor and solutions of massless free field equations. Int. J. Theor. Phys. 39, 377 (2000) [18] A. Zecca, Solution of the Dirac equation in expanding universes Int. J. Theor. Phys. 45, 47(2006) [19] A. Zecca, Proca fields interpretation of spin 1 equation in Robertson- Walker space-time. Gen. Rel. Grav. 38, 837 (2006) [20] A. Zecca, Separation of massive field equation of arbitrary spin in Robertson-Walker space-time. Nuovo Cimento B 121, 167(2006) [21] A. Zecca, Massive field equations of arbitrary spin in Schwarzschild geometry: separation induced by spin 3/2 case. Int. J. Theor. Phys. 45, 2208 (2006) [22] A. Zecca, Spin 2 Field Equation in Expanding Universe. Int. Jour. Theor. Phys., DOI /s z

12 250 A. Zecca Received: November, 2008