IL NUOVO CIMENTO DOI 10.1393/ncb/i2005-10088-1
Vol. 120 B, N. 5
Maggio 2005
Proper Weyl collineations in non-static plane and spherically symmetric space-times(∗ ) G. Shabbir(∗∗ ) Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology Topi Swabi, NWFP, Pakistan (ricevuto il 6 Luglio 2005; revisionato il 22 Agosto 2005; approvato il 23 Agosto 2005)
Summary. — An approach is developed to study proper Weyl collineations in nonstatic plane and spherically symmetric space-times by using the rank of the 6 × 6 Weyl matrix. Studying proper Weyl collineations in each case of the above spacetimes, it is shown that the proper Weyl collineations form an infinite-dimensional vector space. PACS 04.20.-q – Classical general relativity.
1. – Introduction The aim of this paper is to study proper Weyl collineation in non-static plane symmetric and spherically symmetric space-times by using the rank of the 6 × 6 Weyl metric and direct integration techniques. Proper Weyl collineations of some static space-times are given in [1, 2]. Throughout M denotes a (4-dimensional connected, Hausdorff) smooth space-time manifold with Lorentz metric g of signature (−, +, +, +). The usual covariant, partial and Lie derivatives are denoted by a semicolon, a comma and the symbol L, respectively. The curvature tensor associated with gab , through the Levi-Civita connection, is denoted in component form where Rabcd , the Ricci tensor components are Rab = Rcacb , the Weyl tensor components are C abcd , and the Ricci scalar is R = g ab Rab . Round and square brackets denote the usual symmetrization and skew-symmetrization. Let X be a smooth vector field on M , then in any coordinate system on M , one may decompose X in the form (1)
Xa;b =
1 hab + Fab , 2
(∗ ) The author of this paper has agreed to not receive the proofs for correction. (∗∗ ) E-mail:
[email protected] c Societ` a Italiana di Fisica
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where hab = LX gab and Fab (= −Fba ) are symmetric and skew symmetric tensor on M , respectively. If hab = f gab and f (f : M → R) is a real-valued function on M , then X is called a conformal vector field where Fab is called the conformal bivector. The vector field X is called a proper conformal vector field if f is not constant on M. For a conformal bivector Fab one can show that [3] Fab;c = Rabcd X d − 2f;[a gb]c
(2) and
1 fa;b = − Lab;c X c − f Lab + Rc(a Fb)c , 2
(3)
where Lab = Rab − (1/6)Rgab . If X is a conformal vector field on M , then by using (3) one can show that LX Rab = −2fa;b − (f c;c )gab . Further, the conformal vector field X also satisfies [4] LX C abcd = 0
(4) which can be written equivalently as
C abcd;f X f + C abcf X f;d + C abf d X f;c + C af cd X f;b − C fbcd X a;f = 0 . The vector field X satisfying the above equation is called a Weyl collineation (WC). The vector field X is called a proper WC if it is not conformal [5]. The vector field X is called a homothetic vector field if f is constant and a proper homothetic vector field if f = const = 0. If f = 0 on M , then the vector field X is called a Killing vector field. 2. – Main results It has been shown [6,7] that much information on the solutions of (4) can be obtained without integrating it directly. To see this, let p ∈ M and consider the following algebraic classification of the Weyl tensor as a linear map β from the vector space of bivectors to itself; β : Fab → Fcd C cdab , for any bivector Fab at p. The range of the Weyl tensor at p is then the range of β at p and its dimension is the Weyl rank at p. It follows from [6] that the rank of the 6 × 6 Weyl matrix is always even, i.e. 6, 4, 2 or 0. If the rank of the 6 × 6 Weyl matrix is 6 or 4, then every Weyl symmetry is a conformal symmetry [6, 7]. For finding proper WCS, we restrict attention to those cases of rank 2 or less. . 2 1. Proper WCs in non-static plane symmetric space-times. – Consider a non-static plane symmetric space-time in the usual coordinate system (t, x, y, z) (labeled by (x0 , x1 , x2 , x3 ), respectively) with the line element [8] (5)
ds2 = −eA(t,x) dt2 + eB(t,x) dx2 + eC(t,x) (dy 2 + dz 2 ).
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The non-zero independent components of the Weyl tensor are B eA 2 eB ˙ 2 ˙ + ¨ + e (A˙ B˙ − A˙ C˙ + C˙ B) (A + 2A ) − (B + 2B) 12 12 12 1 eA eB ¨ + (C B − A C − A B ) − C + C ≡ F1, 12 6 6 C eC−B+A 2 eC ˙ 2 ˙ − ¨ − e (A˙ B˙ − A˙ C˙ + C˙ B) (A + 2A ) + (B + 2B) =− 24 24 24 eC−B+A eC−B+A eC ¨ − (C B − A C − A B ) + C − C ≡ F2, 24 12 12 ≡ F2, C−A+B eC 2 eC−A+B ˙ 2 ˙ + ¨ +e (A + 2A ) − (B + 2B) (A˙ B˙ − A˙ C˙ + C˙ B) = 24 24 24 eC eC eC−A+B ¨ + (C B − A C − A B ) − C + C ≡ F3, 24 12 12 ≡ F3, 2C−A e2C−B 2 e2C−A ˙ 2 ˙ − ¨ −e (A + 2A ) + (B + 2B) (A˙ B˙ − A˙ C˙ + C˙ B) =− 12 12 24 e2C−B e2C−B e2C−A ¨ − (C B − A C − A B ) + C − C ≡ F4, 12 6 6
C0101 =
C0202
(6) C0303 C1212
C1313 C2323
where prime and dot denote the derivative with respect to x and t, respectively. The Weyl tensor of M can be described by components Cabcd written in the well-known way [9] Cabcd = diag (F 1, F 2, F 2, F 3, F 3, F 4). We restrict attention to those cases of rank 2 or less, since by theorem [6] no proper WCS can exist when the rank of the 6 × 6 Weyl matrix is > 2. We thus obtain the following cases: ¨ = 0, (M1) Rank = 0, A, C ∈ R and B˙ 2 + 2B 2
(M2) Rank = 0, B, C ∈ R and A + 2A = 0, (M3) Rank = 0, C ∈ R, A = A(t) and B = B(x), (M4) Rank = 0, A, B, C ∈ R. In all the above cases the rank of the 6 × 6 Weyl matrix is zero, i.e. the space-time is conformally flat. It turns out that any vector field is a WC. Clearly, they form an infinite-dimensional vector space. . 2 2. Proper WCs in non static spherically symmetric space-times. – Consider a nonstatic spherically symmetric space-time in the usual coordinate system (t, r, θ, φ) (labeled by (x0 , x1 , x2 , x3 ), respectively) with the line element [8] (7)
ds2 = −eA(t,r) dt2 + eB(t,r) dr2 + r2 (dθ2 + sin2 θ dφ2 ) .
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The non-zero independent components of the Weyl tensor are A eA 2 eB ˙ 2 ¨ + e (B − A ) + (A + 2A ) − (B + 2B) 12 12 6r eB eA eB ˙ ˙ B AB − A B ≡ E1 , + 2 (1 − e ) + 3r 12 12 A−B r2 eA−B 2 r2 ¨ + re (A + 2A ) + (B˙ 2 + 2B) (B − A ) − =− 24 24 12 r2 eA−B eA−B r2 (1 − eB ) − A˙ B˙ + A B ≡ E2 , − 6 24 24 = sin2 θ E2 ≡ E3 , r2 2 r2 eB−A ˙ 2 ¨ + r (B − A ) + (A + 2A ) − (B + 2B) = 24 24 12 2 B−A 2 1 r e r + (1 − eB ) + A˙ B˙ − A B ≡ E4 , 6 24 24 = sin2 θ E4 ≡ E5 , = −2r2 sin2 θ e−B E4 ≡ E6 ,
C0101 =
C0202
(8)
C0303 C1212
C1313 C2323
where prime and dot denote the derivative with respect to r and t, respectively. The Weyl tensor of M can be described by components Cabcd written in the well-known way [9] Cabcd = diag(E1, E2, E3, E4, E5, E6) . We are interested in those cases when the rank of the 6 × 6 Weyl matrix is less than or equal to two, since we know from theorem [6] that when the rank of the 6 × 6 Weyl matrix is greater than two there exist no proper WCs. For rank less than or equal to two, one may set four components of the Weyl matrix in (8) to be zero. Thus there exists only one possibility: (M5) Rank = 0, A ∈ R and B = 0. In all the above case the rank of the 6 × 6 Weyl matrix is zero, i.e. the space-time is conformally flat. It turns out that any vector field is a WC. The WCs in all the above case form an infinite-dimensional vector space. 3. – Conclusions In this paper a study of proper Weyl collineations in non-static plane symmetric and spherically symmetric space-times is given by using the rank of the 6 × 6 Weyl matrix and the theorem given in [6, 7]. Studying proper Weyl collineations in each of the above space-times, we have shown that when the above space-times admit the proper Weyl collineations, they form an infinite-dimensional vector space. ∗ ∗ ∗ The author would like to thank ICTP, Trieste, Italy for hospitality during his stay at the Centre, where a part of this work was done.
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