Abstract
Einstein complex spacetimes admitting null Killing or null homothetic Killing vectors are studied. Such vectors define totally null and geodesic 2-surfaces called the null strings or twistor surfaces. Geometric properties of these null strings are discussed. It is shown, that spaces considered are hyperheavenly spaces ( $$\mathcal {HH}$$ -spaces) or, if one of the parts of the Weyl tensor vanishes, heavenly spaces ( $$\mathcal {H}$$ -spaces). The explicit complex metrics admitting null Killing vectors are found. Some Lorentzian and ultrahyperbolic slices of these metrics are discussed.
Highlights
The idea of using the complex numbers in analysis of the spacetime is almost so old, as theory of relativity. [Einstein himself used the imaginary time coordinate in special theory of relativity]
The main aim of our work is to find all complex hyperheavenly and heavenly metrics admitting null homothetic and isometric Killing symmetry
5.4 Spaces of the type [III,N,−]n ⊗ [N,−]e. In this case we deal with the hyperheavenly spaces of types [III,N]n ⊗ [N]e with nonexpanding congruence of self-dual null strings defined by the null Killing vector; the congruence of anti-self-dual null strings is still expanding
Summary
The idea of using the complex numbers in analysis of the spacetime is almost so old, as theory of relativity. [Einstein himself used the imaginary time coordinate in special theory of relativity]. A few works devoted to Killing symmetries in heavenly and hyperheavenly spaces appeared [27,28,29,30] These papers generalized the previous ideas of Plebanski, Finley III and Sonnleitner [16,17,18]. The main aim of our work is to find all complex hyperheavenly and heavenly metrics admitting null homothetic and isometric Killing symmetry Such metrics appear to be important in (++−−) real geometries. We develop this idea and examine all possible Lorentzian slices of the complex spacetimes admitting the null Killing vector It is well known [23] that if a complex spacetime admits any real Lorentzian slice both self-dual and anti-self-dual part of the Weyl tensor must be of the same Petrov–Penrose type.
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