Adv. Studies Theor. Phys., Vol. 7, 2013, no. 21, 1043 - 1049 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2013.3892
On Scalar Curvature of Toric Selfdual Four-Manifolds Rio N. Wijaya2 , Fiki T. Akbar2 and Bobby E. Gunara1,2 1
Indonesian Center for Theoretical and Mathematical Physics (ICTMP) and Theoretical Physics Laboratory Theoretical High Energy Physics and Instrumentation Research Group Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung Jl. Ganesha no. 10 Bandung, Indonesia, 40132
[email protected] ft
[email protected] bobby@fi.itb.ac.id
c 2013 Rio N. Wijaya, Fiki T. Akbar and Bobby E. Gunara. This is an open Copyright access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract In this paper, we prove the nonexistence of constant scalar curvature of four dimensional selfdual Joyce manifolds. Since the manifold admits a torus symmetry, our work is greatly reduced to the case of two dimensional upper-half planes.
1
Introduction and Main Result
Selfduality properties have been widely considered for physicist in the context of field theories. For example, in four dimensional Yang-Mills gauge theories, the selfduality of field strength corresponds to the instanton solutions which minimize the Yang-Mills action. While, in four dimensional gravitational field theories, the selfduality of Riemannian curvatures is related to the so called gravitational instantons firstly introduced by G. Gibbons and S. Hawking [1, 2]. Their construction shows that this instanton is an example of hyper-K¨ahler
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Rio N. Wijaya, Fiki T. Akbar and Bobby E. Gunara
manifolds in four dimensions which are invariant under a circle action. Our interest here is to study a four dimensional manifold whose Weyl curvature is self dual. This class of manifolds has been explicitly constructed by D. Joyce [3] which is a torus vibration over two dimensional upper-half planes. In other words, this manifold has a torus symmetry. This property transforms the selfduality into a set of coupled partial differential equations on two dimensional upper-half planes. In this paper, we prove the following statement: Theorem 1. Let M be a four dimensional Riemannian manifold endowed with self dual Joyce metric. Then, there is no exist spaces of constant Ricci scalar curvature. Some remarks are in order. Our work here is purely to study Joyce form of four dimensional toric self dual spaces (2.3) without employing any scaling. These spaces are conformal to N × T 2 , where T 2 is the torus fibration over two dimensional upper-half plane N, whose metric is in Gowdy form. Another line of work related to this paper such as [4] in which, it has been shown in a special case that one can explicitly construct a family of self dual Einstein spaces which are conformal to Joyce metric (2.3). The structure of this paper can be mentioned as follows. In section 2, we shortly review some material about the self dual metric with torus symmetry in four dimensions. Then, we put the proof of Theorem 1 in section 3. Finally, we give a simple solution of Joyce self dual equations using separation variables method in section 4.
2
4d Selfdual Metric with Torus Symmetry
In this section, we give a short review of toric self dual four-spaces constructed by D. Joyce [3]. We only write some materials which are useful for our analysis in this paper. For an excellent review, interested reader can further consult, for example, [4, 5]. In four dimensions, we can decompose the Weyl tensor into two parts W = W+ ⊕ W− since the rotation group SO(4) is locally isomorphic to SU(2) ⊗ SU(2). W+ and W− are self dual and anti-self dual parts, respectively. Since the Weyl tensor is invariant under conformal transformation, g˜ → Ω2 g, then one can define a family of conformal scaling metric [g]. The self dual condition for g of [g] is W− = 0 which implies that [g] has a self dual structure. Let M be a four dimensional Riemannian manifold whose topology is M = N × T 2 . In local coordinates, (ρ, η, θ, ϕ), the metric on M is in Gowdy form
On scalar curvature of toric selfdual four-manifolds
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given by
g = dρ2 + dη 2 +
(ρA0 dθ − ρB0 dϕ)2 + (ρA1 dθ − ρB1 dϕ)2 , (A0 B1 − A1 B0 )2
(2.1)
with A0 , A1 , B0 , B1 depend on ρ and η. By scaling the above metric (2.1) with scale factor Ω2 =
(A0 B1 − A1 B0 ) , ρ2
(2.2)
we obtain the Joyce form of the metric with torus symmetry
gJ =
(A0 dθ − B0 dϕ)2 + (A1 dθ − B1 dϕ)2 (A0 B1 − A1 B0 ) 2 2 dρ + + dη . ρ2 (A0 B1 − A1 B0 ) (2.3)
The selfduality of gJ , i.e. W− = 0 further implies that all functions A0 , A1 , B0 , B1 satisfy a set of coupled linear partial differential equations, A0 , ρ = 0, B0 = , ρ = 0,
(A0 )ρ + (A1 )η = (A0 )η − (A1 )ρ (B0 )ρ + (B1 )η (B0 )η − (B1 )ρ
(2.4)
where ( )ρ ≡ ∂ρ and ( )η ≡ ∂η .
3
Proof of Theorem 1
Now, we ready to prove Theorem 1 using the above data in the preceding section. Applying the selfduality condition (2.4) and differentiating it with
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Rio N. Wijaya, Fiki T. Akbar and Bobby E. Gunara
respect to ρ and η, one can simplify the components of Ricci tensor into
R11 = R21 = R22 = R33 =
R43 =
R44 =
1 −B1 A0ρ − B0 A1ρ + A1 B0ρ + A0 B1ρ − 2 , 2ρ(A0 B1 − A1 B0 ) 2ρ B0 A0ρ − B1 A1ρ − A0 B0ρ + A1 B1ρ , 2ρ(A0 B1 − A1 B0 ) 1 B1 A0ρ + B0 A1ρ − A1 B0ρ − A0 B1ρ − 2 , 2ρ(A0 B1 − A1 B0 ) ρ ρ A1 B0ρ (A21 − 3A20 ) + A0 B1ρ (A20 − 3A21 ) 3 2(A0 B1 − A1 B0 ) +(A0 B0 − A1 B1 )(A1 A0ρ − A0 A1ρ ) +(A0 B1 + A1 B0 )(A0 A0ρ + A1 A1ρ ) , ρ 2(A B − A1 B0 )3 0 1 (A0 B0 − A1 B1 )(A1 B0ρ − A0 B1ρ − B1 A0ρ + B0 A1ρ ) +(A0 B1 + A1 B0 )(A0 B0ρ + A1 B1ρ − B0 A0ρ − B1 A1ρ ) , −ρ B1 A0ρ (B12 − 3B02 ) + B0 A1ρ (B02 − 3B12 ) (3.1) 2(A0 B1 − A1 B0 )3 +(A0 B0 − A1 B1 )(B1 B0ρ − B0 B1ρ ) +(A0 B1 + A1 B0 )(B0 B0ρ + B1 B1ρ ,
whose Ricci scalar is simply
R=
3 . 2(A1 B0 − A0 B1 )
(3.2)
Thus, if Joyce metric (2.3), namely gJ , belongs to spaces of constant Ricci scalar, then it must be A0 B1 − A1 B0 = λ ,
(3.3)
where λ is a nonzero constant. We have two possible solutions. First, one can set that all functions A0 , B0 , A1 , B1 are constants, so the selfduality condition (2.4) demands that all of them are trivial which are not the solution. Second, suppose there exists some functions E, F, G, H which depend only on ρ and η
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On scalar curvature of toric selfdual four-manifolds
such that λ1/2 E EF − GH λ1/2 G = √ EF − GH λ1/2 F = √ EF − GH λ1/2 H = √ EF − GH
A0 = √
,
A1
,
B0 B1
, .
(3.4)
But this case is conformal to the Joyce metric (2.3) with scale factor Ω2 = (A0 B1 − A1 B0 )−1 ,
(3.5)
and this is also not the solution. Thus, the selfdual Joyce’s metric has no solutions of constant Ricci scalar which proves Theorem 1.
4
Separable Solutions of Selfduality Condition
In this section, we construct the separable solutions of selfduality condition given by equations (2.4). Since the set of equations for Ai and Bi are similar up to a constant, then we only consider the first and the second equations in (2.4). By differentiating the first and the second equations in (2.4) with respect to ρ and respect to η respectively, we have (A0 )ρ A0 − 2 , ρ ρ = (A1 )ρη .
(A0 )ρρ + (A1 )ηρ = (A0 )ηη
(4.1) (4.2)
Since the partial derivative is commute, then by equating (A1 )ηρ = (A1 )ρη , we have differential equation only for A0 , (A0 )ρρ + (A0 )ηη =
(A0 )ρ A0 − 2 . ρ ρ
(4.3)
To solve (4.3), we apply separation variables method by assuming that A0 has the form A0 (ρ, η) = P (ρ)N(η) .
(4.4)
Inserting into (4.3), we have ¨ 1 N P 1 P − + 2 =− , P ρP ρ N
(4.5)
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Rio N. Wijaya, Fiki T. Akbar and Bobby E. Gunara
where and ˙ denote the derivative with respect to ρ and η respectively. Since the left side and the right side depend on different variables, then each side must be equal to constant, P −
P P + 2 = kP, ρ ρ ¨ = kN, N
where k is a real constant and first, we assume k = 0. The solutions of these equations are √ √ P (ρ) = ρ C1 J0 iρ k + C2 Y0 −iρ k , √ √ N(η) = C3 cos η k + C4 sin η k ,
(4.6) (4.7)
(4.8) (4.9)
where C1 , C2 , C3 , and C4 are a constants and J0 and Y0 are the Bessel function of the first kind and the second kind respectively. Thus, we have solution for A0 , √ √ √ √ C3 cos η k + C4 sin η k . A0 (ρ, η) = ρ C1 J0 iρ k + C2 Y0 −iρ k (4.10) Using the the second equation in (2.4), we have solution for A1 , √ √ √ √ C4 cos η k − C3 sin η k . A1 (ρ, η) = ρ C1 I1 ρ k + iC2 Y1 −iρ k (4.11) For the case k = 0, equations (4.6) and (4.7) become P −
P P + 2 = 0, ρ ρ ¨ = 0. N
(4.12) (4.13)
The solutions for these equations are P (ρ) = ρ (C1 + C2 ln(ρ)) , N(η) = C3 η + C4 .
(4.14) (4.15)
Then we have solution for A0 , A0 (ρ, η) = ρ (C1 + C2 ln(ρ)) (C3 η + C4 ) .
(4.16)
Using the first and the second equations in (2.4), we have solution for A1 , A1 (ρ, η) = C3
ρ2 (2C1 − C2 + 2C2 ln(ρ)) . 4
(4.17)
On scalar curvature of toric selfdual four-manifolds
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Acknowledgement We thank to M. Fakhrul Rozi and Agus Suroso for useful discussion in early stage of developments of this paper. This paper is funded by Riset KK ITB 2013 No. 122.6/AL-J/DIPA/PN/SPK/2013 and Riset Desentralisasi DIKTIITB 2013 No. 122.69/AL-J/DIPA/PN/SPK/2013.
References [1] S. W. Hawking, Gravitational Instantons, Phys. Lett. A60 (1977) 81. [2] G. W. Gibbons and S. W. Hawking, Gravitational Multi - Instantons, Phys. Lett. B78 (1978) 430. [3] D. Joyce, Explicit Construction of Self-Dual Manifolds, Duke Math. J. 77 (1995) 519-552. [4] D. Calderbank and H. Pedersen, Selfdual Einstein Metric with Torus Symmetry, J. Differential Geom. 60 (2002) No. 3, 485-521. [5] O. Santillan and A. Zorin, Toric Hyperkahler Manifolds with Quaternionic Kahler Bases and Supergravity Solutions, Commun. Math. Phys. 255 (2005) 33. Received: August 21, 2013
