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Solutions of the Rarita–Schwinger equation in Einstein spaces
20 May 2002
Physics Letters A 297 (2002) 359–362 www.elsevier.com/locate/pla
Solutions of the Rarita–Schwinger equation in Einstein spaces A. Szereszewski, J. Tafel ∗ Institute of Theoretical Physics, University of Warsaw, Ho˙za 69, 00-681 Warsaw, Poland Received 19 July 2001; accepted 18 March 2002 Communicated by A.P. Fordy
Abstract We find special solutions of the Rarita–Schwinger equation in spacetimes admitting shear free congruences of null geodesics. 2002 Elsevier Science B.V. All rights reserved. PACS: 4.40 Keywords: Rarita–Schwinger; Einstein; Algebraically special
1. Introduction
ΨAB C˙ which is symmetric in the indices A and B. The field equations are
Spin- 23 fields [1,2] are interesting because of their role in supergravity [3,4], twistor constructions [5,6] and possible significance for the integrability of the Einstein equations [7,8]. Very few explicit solutions of the Rarita–Schwinger equations in curved spacetimes are known [9,10] (see also [11] and references therein). In this Letter we present a construction of such solutions in the case of algebraically special metrics satisfying the vacuum Einstein equations with cosmological constant. Examples include fields in the Schwarzschild space, pp wave and conformally flat Einstein spaces (Minkowski, de-Sitter, anti-de-Sitter). We consider the massless Rarita–Schwinger equation in a 4-dimensional spacetime with metric g of signature + − − −. In the Weyl spinor approach [12] the Rarita–Schwinger field is represented by a spinor
∇AB˙ Ψ AC D = 0,
* Corresponding author.
E-mail address: [email protected] (J. Tafel).
˙
(1)
where ∇ is the covariant derivative related to the LeviCivita connection ω. Integrability conditions of (1) are trivially satisfied if the Ricci tensor of g satisfies [13] Rµν = Λgµν .
(2)
be an orthonormal basis dual to eµ (µ = Let 0, 1, 2, 3). We assume the following relations between the Lorentz indices and the Weyl indices θµ
˙
˙
v AB = v µ σµ AB , sA = AB s B ,
˙
vµ = vAB˙ σµ AB , s A = − AB sB .
(3)
Here v is a vector, s is a spinor, AB is the completely antisymmetric tensor (12 = 1) and 1 1 0 1 0 1 σ0 = √ , σ1 = √ , 2 0 1 2 1 0 1 0 −i 1 1 0 , σ3 = √ . (4) σ2 = √ 2 i 0 2 0 −1
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 2 8 3 - 9
360
A. Szereszewski, J. Tafel / Physics Letters A 297 (2002) 359–362
The covariant derivatives in Eq. (1) contain components of the spinor connection γ A B and its conjugate ˙ γ¯ A B˙ . These connections are defined by the Levi-Civita connection ωABC ˙ D˙ = γAC B˙ D˙ + γ¯B˙ D˙ AC .
(5)
Let θ µ be a null basis such that √ 3 √ 0 2 θ = θ 0 − θ 3, 2 θ = θ 0 + θ 3, √ 2 √ 1 1 2 2 θ = θ + iθ , 2 θ = θ 1 − iθ 2 .
(6)
Then g = 2θ 0 θ 3 − 2θ 1 θ 2 , and v
AB˙
=
v 3 v 1
v 2 v 0
(7)
,
vAB˙ =
v 3 v 1
v 2 v 0
.
(8)
The spinor connection is given by γ 1 1 = −γ 2 2 = γ 1 2 = ω01 ,
1 , ω03 − ω12 2 γ 2 1 = ω32 ,
(9)
is the Levi-Civita connection related to θ µ . where ωµν
2. Decomposable fields in algebraically special spacetimes Assume that the Rarita–Schwinger field Ψ is decomposable in the following way ˙
˙
Ψ AB C = s A s B q C . It follows from (1) that spinor
(10) sA
has to satisfy
s A s C ∇AB˙ sC = 0.
(11) ˙
Eq. (11) implies that the null vector field k AB = ˙ s A s B defines a shear-free congruence of null geodesics (SGN congruence) [12]. Due to the Goldberg– Sachs theorem [14] the Weyl tensor of the metric must be degenerated. Given such a metric and the corresponding s A the Rarita–Schwinger equation (1) reduce to the following equation for spinor q s A ∇AB˙ qC˙ = − ∇AB˙ s A + pB˙ qC˙ , (12) where pB˙ is defined by the relation s A ∇AB˙ s C = pB˙ s C .
(13)
Integrability conditions of Eq. (12) are satisfied for any q iff AB C˙ D˙ s A s B = 0,
(14)
where is the spinor corresponding to the traceless part of Rµν . Eq. (14) constitute a part of the Einstein equations which can be exactly solved for algebraically special metrics [14]. Our approach to the Rarita–Schwinger equation in algebraically special spacetimes is probably simpler (but less general) than that using the Teukolsky type equations [11]. In the following we give examples of solutions of the form (10) in the case of the Schwarzschild metric, pp waves and (Section 3) conformally flat metrics. In the Eddington–Finkelstein coordinates the Schwarzschild metric reads g = 1 − 2Mr −1 du2 + 2 du dr − r 2 dθ 2 + sin2 θ dφ 2 (15) and the SGN congruence (one of two) is defined by the null vector field k = ∂r . As the null basis (6) we take 1 θ 3 = dr + 1 − 2Mr −1 du, (16) θ 0 = du, 2 √ −1 1 ¯ dξ, θ = 2 r 1 + ξξ √ −1 2 θ = 2 r 1 + ξ ξ¯ (17) d ξ¯ , where ξ = ctg(θ/2) exp(iφ) is the complex stereographic coordinate on S2 . The spinor s corresponding to the congruence is given by s A = δ1A . Solving Eq. (12) yields the following expressions for nonvanishing components of the Rarita–Schwinger field Ψ 1/2 ˙ ˙ Ψ 111 = q 1 = r −1 1 + ξ ξ¯ (18) a(u, ξ ), ˙
˙
Ψ 112 = q 2 √ 3/2 ˙ = 2 ξ −1 q 1 + r −1 1 + ξ ξ¯ b(u, ξ ).
(19)
Here a and b are arbitrary functions of u and ξ . For any choice of them Ψ is singular at some ξ . For pp wave the metric tensor can be written in the form g = 2 du (dv + H du) − 2 dξ d ξ¯ ,
(20)
where H = Re h(u, ξ ). The SGN congruence is represented by the vector field k = ∂v . In the null basis θ 0 = du,
θ 3 = dv + H du,
A. Szereszewski, J. Tafel / Physics Letters A 297 (2002) 359–362
θ 2 = d ξ¯
θ 1 = dξ,
(21)
the Rarita–Schwinger field of the form (10) has ˙ ˙ nonvanishing components Ψ 11A = q A (u, ξ ), which are arbitrary functions of u and ξ .
Splitting Eq. (12) into the symmetric and antisymmet˙ C) ˙ yields ric part (with respect to indices B, A 3/2 = 0, ∂A(B˙ qC) (30) ˙ s Ω ˙ ∂AB˙ q B s A Ω 7/2 = 0.
3. Solutions in conformally flat spacetimes
(22)
where ηµν = diag(1, −1, −1, −1). The Einstein equations with cosmological constant yield Ω
−1
= axµ x + bµ x + c, µ
µ
(23)
where xµ = ηµν x ν and a, bµ , c are constants. Due to a freedom of coordinate transformation it is sufficient to consider Ω −1 = c 1 + xµ x µ , (24) 4 where = 0 and c = 1 (Minkowski space), = −1 (de Sitter space) or = 1 (anti-de-Sitter space) [15]. We work in the orthonormal basis θ µ = Ω dx µ . According to the Kerr theorem [12] any analytic SGN congruence in a conformally flat spacetime is defined by the spinor s A = (1, λ),
(25)
where λ satisfies the equation ˙ F λ, sA x AB = 0.
(26)
(F is a function of three complex variables. The case with s A = (1, 0) can be obtained from (25) with λ = 0 by a simple SL(2, C) transformation.) Due to (26) the spinor s satisfies s A ∂AB˙ s C = 0,
(27)
where ∂AB˙ denotes the partial derivative with respect ˙ to x AB . It follows from (27) that operators s A ∂AB˙ commute and ∂AB˙ λ,x s A = 0, (28) where λ,x = cµ ∂µ λ,
cµ = const.
(31)
Assume that λ = const. Then λ,x = 0 for some cµ . Under the substitution
Consider a conformally flat metric g = Ω 2 ηµν dx µ dx ν ,
361
(29)
Ω 3/2qC˙ = λ,x qC ˙
(32)
Eqs. (30) and (31) transform into s A ∂A(B˙ qC) ˙ =0
(33)
and ˙ s A ∂AB˙ Ω 2 q B = 0.
(34)
The general solution of (33) reads ˙
˙
˙
q C = f C + hx 1C ,
(35)
where ˙
s A ∂AB˙ f C = s A ∂AB˙ h = 0.
(36)
Substituting (35) into (34) yields a definition of h in terms of other variables. In the case (24) it reads ˙ h = s A xAB˙ f B . 2
(37)
If λ = const then formulas (32)–(37) are still valid provided one set λ,x = 1.
(38)
Summarizing, in Minkowski, de-Sitter and antide-Sitter spacetimes one can relate with every SGN congruence (given by (25) and (26)) a solution of the Rarita–Schwinger equation of the form (10), where ˙ ˙ ˙ q C = Ω −3/2 λ,x f C + hx 1C ,
(39)
λ,x is given by (29) or (38), h is given by (37) and ˙ ˙ f C are functions of λ and sA x AB (note that only two of these variables are independent due to (26)). The fields corresponding to different congruences (with nonequivalent functions F ) can be added, yielding in this way nondecomposable solutions.
362
A. Szereszewski, J. Tafel / Physics Letters A 297 (2002) 359–362
Acknowledgements This work was partially supported by the Polish Committee for Scientific Research (grant 2 P03b 060 17).
References [1] M. Fierz, W. Pauli, Proc. R. Soc. A 155 (1939) 447. [2] W. Rarita, J. Schwinger, Phys. Rev. 60 (1941) 61. [3] D.Z. Freedman, P. van Nieuwanhuizen, S. Ferrara, Phys. Rev. 13 (1976) 3214. [4] S. Deser, B. Zumino, Phys. Lett. B 62 (1976) 335. [5] R. Penrose, in: A. Zichichi, N. de Sabbata, N. Sanchez (Eds.), Gravitation and Modern Cosmology, Plenum, New York, 1991.
[6] L.J. Mason, R. Penrose, Twistor Newsletter 37 (1994) 1. [7] L.J. Mason, J.P. Nicolas, Proc. London Math. Soc. 79 (1999) 694. [8] A. Szereszewski, J. Tafel, Class. Quantum Grav. 18 (2001) L129. [9] P.C. Aichelburg, T. Dereli, Phys. Rev. 18 (1978) 1754. [10] G.F. Torres del Castillo, Phys. Rev. D 42 (1990) 4082. [11] G.F. Torres del Castillo, J. Math. Phys. 30 (1989) 2114. [12] R. Penrose, W. Rindler, Spinors and Space–Time, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, 1984 and 1986. [13] H.A. Buchdahl, Nuovo Cimento 10 (1959) 96. [14] D. Kramer, H. Stephani, M. MacCallum, E. Herlt, Exact Solutions of Einstein’s Field Equations, VEB Deutcher Verlag, Berlin, 1980. [15] S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Spacetime, Cambridge Univ. Press, Cambridge, 1973.
Physics Letters A 297 (2002) 359–362 www.elsevier.com/locate/pla
Solutions of the Rarita–Schwinger equation in Einstein spaces A. Szereszewski, J. Tafel ∗ Institute of Theoretical Physics, University of Warsaw, Ho˙za 69, 00-681 Warsaw, Poland Received 19 July 2001; accepted 18 March 2002 Communicated by A.P. Fordy
Abstract We find special solutions of the Rarita–Schwinger equation in spacetimes admitting shear free congruences of null geodesics. 2002 Elsevier Science B.V. All rights reserved. PACS: 4.40 Keywords: Rarita–Schwinger; Einstein; Algebraically special
1. Introduction
ΨAB C˙ which is symmetric in the indices A and B. The field equations are
Spin- 23 fields [1,2] are interesting because of their role in supergravity [3,4], twistor constructions [5,6] and possible significance for the integrability of the Einstein equations [7,8]. Very few explicit solutions of the Rarita–Schwinger equations in curved spacetimes are known [9,10] (see also [11] and references therein). In this Letter we present a construction of such solutions in the case of algebraically special metrics satisfying the vacuum Einstein equations with cosmological constant. Examples include fields in the Schwarzschild space, pp wave and conformally flat Einstein spaces (Minkowski, de-Sitter, anti-de-Sitter). We consider the massless Rarita–Schwinger equation in a 4-dimensional spacetime with metric g of signature + − − −. In the Weyl spinor approach [12] the Rarita–Schwinger field is represented by a spinor
∇AB˙ Ψ AC D = 0,
* Corresponding author.
E-mail address: [email protected] (J. Tafel).
˙
(1)
where ∇ is the covariant derivative related to the LeviCivita connection ω. Integrability conditions of (1) are trivially satisfied if the Ricci tensor of g satisfies [13] Rµν = Λgµν .
(2)
be an orthonormal basis dual to eµ (µ = Let 0, 1, 2, 3). We assume the following relations between the Lorentz indices and the Weyl indices θµ
˙
˙
v AB = v µ σµ AB , sA = AB s B ,
˙
vµ = vAB˙ σµ AB , s A = − AB sB .
(3)
Here v is a vector, s is a spinor, AB is the completely antisymmetric tensor (12 = 1) and 1 1 0 1 0 1 σ0 = √ , σ1 = √ , 2 0 1 2 1 0 1 0 −i 1 1 0 , σ3 = √ . (4) σ2 = √ 2 i 0 2 0 −1
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 2 8 3 - 9
360
A. Szereszewski, J. Tafel / Physics Letters A 297 (2002) 359–362
The covariant derivatives in Eq. (1) contain components of the spinor connection γ A B and its conjugate ˙ γ¯ A B˙ . These connections are defined by the Levi-Civita connection ωABC ˙ D˙ = γAC B˙ D˙ + γ¯B˙ D˙ AC .
(5)
Let θ µ be a null basis such that √ 3 √ 0 2 θ = θ 0 − θ 3, 2 θ = θ 0 + θ 3, √ 2 √ 1 1 2 2 θ = θ + iθ , 2 θ = θ 1 − iθ 2 .
(6)
Then g = 2θ 0 θ 3 − 2θ 1 θ 2 , and v
AB˙
=
v 3 v 1
v 2 v 0
(7)
,
vAB˙ =
v 3 v 1
v 2 v 0
.
(8)
The spinor connection is given by γ 1 1 = −γ 2 2 = γ 1 2 = ω01 ,
1 , ω03 − ω12 2 γ 2 1 = ω32 ,
(9)
is the Levi-Civita connection related to θ µ . where ωµν
2. Decomposable fields in algebraically special spacetimes Assume that the Rarita–Schwinger field Ψ is decomposable in the following way ˙
˙
Ψ AB C = s A s B q C . It follows from (1) that spinor
(10) sA
has to satisfy
s A s C ∇AB˙ sC = 0.
(11) ˙
Eq. (11) implies that the null vector field k AB = ˙ s A s B defines a shear-free congruence of null geodesics (SGN congruence) [12]. Due to the Goldberg– Sachs theorem [14] the Weyl tensor of the metric must be degenerated. Given such a metric and the corresponding s A the Rarita–Schwinger equation (1) reduce to the following equation for spinor q s A ∇AB˙ qC˙ = − ∇AB˙ s A + pB˙ qC˙ , (12) where pB˙ is defined by the relation s A ∇AB˙ s C = pB˙ s C .
(13)
Integrability conditions of Eq. (12) are satisfied for any q iff AB C˙ D˙ s A s B = 0,
(14)
where is the spinor corresponding to the traceless part of Rµν . Eq. (14) constitute a part of the Einstein equations which can be exactly solved for algebraically special metrics [14]. Our approach to the Rarita–Schwinger equation in algebraically special spacetimes is probably simpler (but less general) than that using the Teukolsky type equations [11]. In the following we give examples of solutions of the form (10) in the case of the Schwarzschild metric, pp waves and (Section 3) conformally flat metrics. In the Eddington–Finkelstein coordinates the Schwarzschild metric reads g = 1 − 2Mr −1 du2 + 2 du dr − r 2 dθ 2 + sin2 θ dφ 2 (15) and the SGN congruence (one of two) is defined by the null vector field k = ∂r . As the null basis (6) we take 1 θ 3 = dr + 1 − 2Mr −1 du, (16) θ 0 = du, 2 √ −1 1 ¯ dξ, θ = 2 r 1 + ξξ √ −1 2 θ = 2 r 1 + ξ ξ¯ (17) d ξ¯ , where ξ = ctg(θ/2) exp(iφ) is the complex stereographic coordinate on S2 . The spinor s corresponding to the congruence is given by s A = δ1A . Solving Eq. (12) yields the following expressions for nonvanishing components of the Rarita–Schwinger field Ψ 1/2 ˙ ˙ Ψ 111 = q 1 = r −1 1 + ξ ξ¯ (18) a(u, ξ ), ˙
˙
Ψ 112 = q 2 √ 3/2 ˙ = 2 ξ −1 q 1 + r −1 1 + ξ ξ¯ b(u, ξ ).
(19)
Here a and b are arbitrary functions of u and ξ . For any choice of them Ψ is singular at some ξ . For pp wave the metric tensor can be written in the form g = 2 du (dv + H du) − 2 dξ d ξ¯ ,
(20)
where H = Re h(u, ξ ). The SGN congruence is represented by the vector field k = ∂v . In the null basis θ 0 = du,
θ 3 = dv + H du,
A. Szereszewski, J. Tafel / Physics Letters A 297 (2002) 359–362
θ 2 = d ξ¯
θ 1 = dξ,
(21)
the Rarita–Schwinger field of the form (10) has ˙ ˙ nonvanishing components Ψ 11A = q A (u, ξ ), which are arbitrary functions of u and ξ .
Splitting Eq. (12) into the symmetric and antisymmet˙ C) ˙ yields ric part (with respect to indices B, A 3/2 = 0, ∂A(B˙ qC) (30) ˙ s Ω ˙ ∂AB˙ q B s A Ω 7/2 = 0.
3. Solutions in conformally flat spacetimes
(22)
where ηµν = diag(1, −1, −1, −1). The Einstein equations with cosmological constant yield Ω
−1
= axµ x + bµ x + c, µ
µ
(23)
where xµ = ηµν x ν and a, bµ , c are constants. Due to a freedom of coordinate transformation it is sufficient to consider Ω −1 = c 1 + xµ x µ , (24) 4 where = 0 and c = 1 (Minkowski space), = −1 (de Sitter space) or = 1 (anti-de-Sitter space) [15]. We work in the orthonormal basis θ µ = Ω dx µ . According to the Kerr theorem [12] any analytic SGN congruence in a conformally flat spacetime is defined by the spinor s A = (1, λ),
(25)
where λ satisfies the equation ˙ F λ, sA x AB = 0.
(26)
(F is a function of three complex variables. The case with s A = (1, 0) can be obtained from (25) with λ = 0 by a simple SL(2, C) transformation.) Due to (26) the spinor s satisfies s A ∂AB˙ s C = 0,
(27)
where ∂AB˙ denotes the partial derivative with respect ˙ to x AB . It follows from (27) that operators s A ∂AB˙ commute and ∂AB˙ λ,x s A = 0, (28) where λ,x = cµ ∂µ λ,
cµ = const.
(31)
Assume that λ = const. Then λ,x = 0 for some cµ . Under the substitution
Consider a conformally flat metric g = Ω 2 ηµν dx µ dx ν ,
361
(29)
Ω 3/2qC˙ = λ,x qC ˙
(32)
Eqs. (30) and (31) transform into s A ∂A(B˙ qC) ˙ =0
(33)
and ˙ s A ∂AB˙ Ω 2 q B = 0.
(34)
The general solution of (33) reads ˙
˙
˙
q C = f C + hx 1C ,
(35)
where ˙
s A ∂AB˙ f C = s A ∂AB˙ h = 0.
(36)
Substituting (35) into (34) yields a definition of h in terms of other variables. In the case (24) it reads ˙ h = s A xAB˙ f B . 2
(37)
If λ = const then formulas (32)–(37) are still valid provided one set λ,x = 1.
(38)
Summarizing, in Minkowski, de-Sitter and antide-Sitter spacetimes one can relate with every SGN congruence (given by (25) and (26)) a solution of the Rarita–Schwinger equation of the form (10), where ˙ ˙ ˙ q C = Ω −3/2 λ,x f C + hx 1C ,
(39)
λ,x is given by (29) or (38), h is given by (37) and ˙ ˙ f C are functions of λ and sA x AB (note that only two of these variables are independent due to (26)). The fields corresponding to different congruences (with nonequivalent functions F ) can be added, yielding in this way nondecomposable solutions.
362
A. Szereszewski, J. Tafel / Physics Letters A 297 (2002) 359–362
Acknowledgements This work was partially supported by the Polish Committee for Scientific Research (grant 2 P03b 060 17).
References [1] M. Fierz, W. Pauli, Proc. R. Soc. A 155 (1939) 447. [2] W. Rarita, J. Schwinger, Phys. Rev. 60 (1941) 61. [3] D.Z. Freedman, P. van Nieuwanhuizen, S. Ferrara, Phys. Rev. 13 (1976) 3214. [4] S. Deser, B. Zumino, Phys. Lett. B 62 (1976) 335. [5] R. Penrose, in: A. Zichichi, N. de Sabbata, N. Sanchez (Eds.), Gravitation and Modern Cosmology, Plenum, New York, 1991.
[6] L.J. Mason, R. Penrose, Twistor Newsletter 37 (1994) 1. [7] L.J. Mason, J.P. Nicolas, Proc. London Math. Soc. 79 (1999) 694. [8] A. Szereszewski, J. Tafel, Class. Quantum Grav. 18 (2001) L129. [9] P.C. Aichelburg, T. Dereli, Phys. Rev. 18 (1978) 1754. [10] G.F. Torres del Castillo, Phys. Rev. D 42 (1990) 4082. [11] G.F. Torres del Castillo, J. Math. Phys. 30 (1989) 2114. [12] R. Penrose, W. Rindler, Spinors and Space–Time, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, 1984 and 1986. [13] H.A. Buchdahl, Nuovo Cimento 10 (1959) 96. [14] D. Kramer, H. Stephani, M. MacCallum, E. Herlt, Exact Solutions of Einstein’s Field Equations, VEB Deutcher Verlag, Berlin, 1980. [15] S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Spacetime, Cambridge Univ. Press, Cambridge, 1973.