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Twistor spaces for real four-dimensional Lorentzian manifolds. (English) Zbl 0801.53047
In this paper we construct twistor spaces of real 4-dimensional Lorentzian manifolds under a suitable curvature condition. For a Lorentzian manifold, \(*^ 2 = -id\) for Hodge’s star-operator \(*\) and the space of 2-forms has no canonical real decomposition. The set of all Lorentz orthogonal complex structures on \(\mathbb{R}^ 4_ 1\) is isomorphic to \(SO_ +(3,1)/U(1)\) and of 5-dimensions. It is different from the Riemannian case. As the underlying space of the twistor space of a 4- dimensional Lorentzian manifold \(M\), we take the space \(P\) of all future- pointing null directions on \(M\). This fits in with the original ideal of Penrose and it will have physical applications. The space \(P\) is a fibre bundle over \(M\) with fibre \(S^ 2\). By choosing a future-pointing timelike vector field \(T\) on \(M\), we can naturally define an almost complex structure \(J = J_ T\) on \(P\). The point \(x\) and a timelike unit tangent vector \(T\) at \(x\) form a model of the instantaneous observer in the Lorentzian world \(M\). The vector indicates the direction of the observer’s individual time. The subspace in \(M\) Lorentz orthogonal to this vector is a model of the 3-dimensional physical space of the instantaneous observer. The physical spaces of two different instantaneous observers are different, even if the observers are located at the same point of \(M\). The result is to state the integrability condition of the almost complex structure \(J\) on \(P\) by the vanishing of certain parts of an irreducible \(SO(3)\)-decomposition of the curvature tensor \(R\) of the 4-dimensional Lorentzian manifold \(M\). The proof is given by transfering the integrability condition to that of the linear frame bundle. Conformal flatness does not imply the integrability of the twistor space. But any Robertson-Walker space-time gives a global example of integrable twistor space. It is conformally flat and the flow of the perfect fluid on it defines an integrable twistor space. A relation between the twistor space and the normal CR-structures of the tangent sphere bundles of 3-dimensional manifolds is stated. A relation of the integrability of the twistor space of the real 4-dimensional Riemannian manifold made by changing a real time to the imaginary time is also stated.

MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C80 Applications of global differential geometry to the sciences
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