Twisting non-shearing congruences of null geodesics, almost CR structures and Einstein metrics in even dimensions

Abstract

We investigate the geometry of a twisting non-shearing congruence of null geodesics on a conformal manifold of even dimension greater than four and Lorentzian signature. We give a necessary and sufficient condition on the Weyl tensor for the twist to induce an almost Robinson structure, that is, the screen bundle of the congruence is equipped with a bundle complex structure. In this case, the (local) leaf space of the congruence acquires a partially integrable contact almost CR structure of positive definite signature. We give further curvature conditions for the integrability of the almost Robinson structure and the almost CR structure and for the flatness of the latter. We show that under a mild natural assumption on the Weyl tensor, any metric in the conformal class that is a solution to the Einstein field equations determines an almost CR–Einstein structure on the leaf space of the congruence. These metrics depend on three parameters and include the Fefferman–Einstein metric and Taub–NUT–(A)dS metric in the integrable case. In the non-integrable case, we obtain new solutions to the Einstein field equations, which, we show, can be constructed from strictly almost Kähler–Einstein manifolds.

Computation of the curvature

In this section, we compute the curvature tensors of the metric g given in Sect. 5. The use of the open-source software cadabra [51, 52] was particularly helpful for that purpose. Not all components of the curvature tensors are given, but all may be obtained by means of complex conjugation and index manipulation. Parts of the first Bianchi identities are also given to bring out other forms of the components—the remaining, ‘purely CR’, parts being given by equations (4.3). We omit pullback maps for clarity.

1.1 Riemann tensor

$$\begin{aligned} R_{\gamma }{}^{0 0}\,_{\alpha }&= 0 \, , \end{aligned}$$

(A.1)

$$\begin{aligned} R_{{\bar{\gamma }}}{}^{0 0}\,_{\alpha }&= - {\underline{h}}_{\alpha {\bar{\gamma }}} \, , \end{aligned}$$

(A.2)

$$\begin{aligned} R_{0}{}^{0 0}\,_{\alpha }&= {\dot{E}}_{\alpha } - \mathrm {i}E_{\alpha } \, , \end{aligned}$$

(A.3)

$$\begin{aligned} R_{\beta \delta }{}^{0}\,_{\alpha }&= - 2 \mathrm {i}{\underline{{\textsf {N}}}}_{\beta \delta \alpha } \, , \end{aligned}$$

(A.4)

$$\begin{aligned} R_{\beta {\bar{\gamma }}}{}^{0}\,_{\alpha }&= - 2 \mathrm {i}E_{\alpha } {\underline{h}}_{\beta {\bar{\gamma }}} - \mathrm {i}E_{\beta } {\underline{h}}_{\alpha {\bar{\gamma }}} \, , \end{aligned}$$

(A.5)

$$\begin{aligned} R_{\gamma 0}{}^{0}\,_{\alpha }&= - {\underline{\nabla }}_{\gamma } E_{\alpha } + \lambda _{\gamma } {\dot{E}}_{\alpha } + E_{\alpha } E_{\gamma } +E^{\beta } {\underline{{\textsf {N}}}}_{\beta \alpha \gamma } - \frac{1}{2} \mathrm {i}{\underline{A}}_{\alpha \gamma } - \mathrm {i}B_{\gamma \alpha } \, , \end{aligned}$$

(A.6)

$$\begin{aligned} R{}_{{\bar{\beta }} 0}{}^{0}{}_{\alpha }&= - {\underline{\nabla }}_{{\bar{\beta }}} E_{\alpha } + \lambda _{{\bar{\beta }}} {\dot{E}}_{\alpha } + E_{\alpha } E_{{\bar{\beta }}} + \mathrm {i}E_{0} {\underline{h}}_{\alpha {\bar{\beta }}} - \mathrm {i}B_{\alpha {\bar{\beta }}} \, , \end{aligned}$$

(A.7)

$$\begin{aligned} R_{0}{}^{0 0}\,_{0}&= {\dot{E}}_{0} -2 E_{\alpha } E^{\alpha } \, , \end{aligned}$$

(A.8)

$$\begin{aligned} R_{\gamma \delta \alpha \beta }&= 2 {\underline{\nabla }}_{[\gamma |} {\underline{{\textsf {N}}}}_{\alpha \beta |\delta ]} \, , \end{aligned}$$

(A.9)

$$\begin{aligned} R_{\gamma {\bar{\delta }} \alpha \beta }&= 2 \mathrm {i}B_{\alpha \beta } {\underline{h}}_{\gamma {\bar{\delta }}} - 2 \mathrm {i}B_{\gamma [\alpha } {\underline{h}}_{\beta ] {\bar{\delta }}} - {\underline{\nabla }}_{{\bar{\delta }}} {\underline{{\textsf {N}}}}_{\alpha \beta \gamma } + \mathrm {i}{\underline{A}}_{\gamma [\alpha } {\underline{h}}_{\beta ] {\bar{\delta }}} \, , \end{aligned}$$

(A.10)

$$\begin{aligned} R_{\gamma {\bar{\delta }} {\bar{\beta }} \alpha }&= {\underline{R}}_{\gamma {\bar{\delta }} \alpha {\bar{\beta }}} - 2 \mathrm {i}B_{\alpha {\bar{\beta }}} {\underline{h}}_{\gamma {\bar{\delta }}} - 2 \mathrm {i}B_{\gamma {\bar{\delta }}} {\underline{h}}_{\alpha {\bar{\beta }}} - \mathrm {i}B_{\gamma {\bar{\beta }} } {\underline{h}}_{\alpha {\bar{\delta }}} - \mathrm {i}B_{\alpha {\bar{\delta }}} {\underline{h}}_{\gamma {\bar{\beta }}} + {\underline{{\textsf {N}}}}_{\epsilon \alpha \gamma } {\underline{{\textsf {N}}}}{}^{\epsilon }{}_{{\bar{\beta }} {\bar{\delta }}} \, , \end{aligned}$$

(A.11)

$$\begin{aligned} R_{0}{}^{0}{}_{\alpha 0}&= \frac{1}{2} {\underline{\nabla }}_{0} E_{\alpha } - \frac{1}{2}\lambda _{0} {\dot{E}}_{\alpha }+ E^{\beta } B_{\alpha \beta } +E_{\beta } B_{\alpha }\,^{\beta } + \mathrm {i}C_{\alpha } - {\dot{C}}_{\alpha } \, , \end{aligned}$$

(A.12)

$$\begin{aligned} R_{\gamma 0 \alpha \beta }&= {\underline{\nabla }}_{\gamma } B_{\alpha \beta } - \lambda _{\gamma } {\dot{B}}_{\alpha \beta } + 2 B_{[\alpha }\,^{\delta } {\underline{{\textsf {N}}}}_{\beta ] \delta \gamma } +E_{[\alpha } {\underline{A}}_{\beta ] \gamma } + 2 E_{[\alpha } B_{\beta ] \gamma } - \frac{1}{2} {\underline{\nabla }}_{0} {\underline{{\textsf {N}}}}_{\alpha \beta \gamma } \, , \end{aligned}$$

(A.13)

$$\begin{aligned} R_{\beta {\bar{\gamma }} \alpha 0}&= - {\underline{\nabla }}_{{\bar{\gamma }}} B_{\alpha \beta } + \lambda _{{\bar{\gamma }}} {\dot{B}}_{\alpha \beta } + {\underline{\nabla }}_{\beta } B_{\alpha {\bar{\gamma }}} - \lambda _{\beta } {\dot{B}}_{\alpha {\bar{\gamma }}} -2 E_{\alpha } B_{\beta {\bar{\gamma }}} +E_{\beta } B_{\alpha {\bar{\gamma }}} -E_{{\bar{\gamma }}}B_{\alpha \beta } \nonumber \\&\qquad + 2 \mathrm {i}C_{\alpha } {\underline{h}}_{\beta {\bar{\gamma }}} - \frac{1}{2}E_{{\bar{\gamma }}} {\underline{A}}_{\alpha \beta } + B_{{\bar{\gamma }}}{}^{\delta } {\underline{{\textsf {N}}}}_{\delta \alpha \beta } - \frac{1}{2} {\underline{\nabla }}_{{\bar{\gamma }}} {\underline{A}}_{\alpha \beta } - \frac{1}{2}{\underline{A}}_{{\bar{\gamma }}}{}^{\delta } {\underline{{\textsf {N}}}}_{\delta \alpha \beta } \, , \end{aligned}$$

(A.14)

$$\begin{aligned} R_{\beta 0 \alpha 0}&= - \frac{1}{2} {\underline{\nabla }}_{0} B_{\alpha \beta } + \frac{1}{2} \lambda _{0} {\dot{B}}_{\alpha \beta } + {\underline{\nabla }}_{\beta } C_{\alpha } - \lambda _{\beta } {\dot{C}}_{\alpha } -B_{\alpha }\,^{\gamma } {\underline{A}}_{\beta \gamma } -E_{\alpha } C_{\beta } +B_{\alpha \gamma } B_{\beta }\,^{\gamma } \nonumber \\&\qquad - C^{\gamma } {\underline{{\textsf {N}}}}_{\gamma \alpha \beta } + B_{\beta \gamma } B_{\alpha }\,^{\gamma } +E_{\beta } C_{\alpha } - \frac{1}{2}E_{0} {\underline{A}}_{\alpha \beta } -E_{0} B_{\alpha \beta } - \frac{1}{4} {\underline{\nabla }}_{0} {\underline{A}}_{\alpha \beta } \, , \end{aligned}$$

(A.15)

$$\begin{aligned} R_{{\bar{\beta }} 0 \alpha 0}&= - \frac{1}{2} {\underline{\nabla }}_{0} B_{\alpha {\bar{\beta }}} + \frac{1}{2} \lambda _{0} {\dot{B}}_{\alpha {\bar{\beta }}} + {\underline{\nabla }}_{{\bar{\beta }}} C_{\alpha } - \lambda _{{\bar{\beta }}} {\dot{C}}_{\alpha } -B_{\alpha \gamma } {\underline{A}}_{{\bar{\beta }}}{}^{\gamma } \nonumber \\&\qquad -E_{\alpha } C^{\beta } + B_{\alpha \gamma } B_{{\bar{\beta }} }{}^{\gamma } -B_{\alpha }\,^{\gamma } B_{\gamma {\bar{\beta }}} +E_{{\bar{\beta }}} C_{\alpha } -E_{0} B_{\alpha {\bar{\beta }}} - \frac{1}{4}{\underline{A}}_{\alpha \gamma } {\underline{A}}_{{\bar{\beta }} }{}^{\gamma } \, . \end{aligned}$$

(A.16)

1.2 The Ricci tensor

$$\begin{aligned} {\mathrm {Ric}}^{0 0}&= 2m \, , \end{aligned}$$

(A.17)

$$\begin{aligned} {\mathrm {Ric}}_{\alpha }{}^{0}&= {\dot{E}}\,_{\alpha } -4 \mathrm {i}E_{\alpha }\, , \end{aligned}$$

(A.18)

$$\begin{aligned} {\mathrm {Ric}}_{\alpha \beta }&=2 {\underline{\nabla }}_{(\alpha }{E_{\beta )}} - 2 \lambda _{(\alpha } {\dot{E}}\,_{\beta )} - 2 E_{\alpha } E_{\beta } + \mathrm {i}m {\underline{A}}_{\alpha \beta } + 2 {\underline{\nabla }}^{\gamma }{{\underline{{\textsf {N}}}}_{\gamma (\alpha \beta )}} - 2 {\underline{{\textsf {N}}}}_{\gamma (\alpha \beta )}E^{\gamma } \, , \end{aligned}$$

(A.19)

$$\begin{aligned} {\mathrm {Ric}}_{\alpha {\bar{\beta }}}&= {\underline{\nabla }}_{\alpha }{E_{\bar{\beta }}} + {\underline{\nabla }}_{\bar{\beta }}{E_{\alpha }} - \lambda _{\alpha } {\dot{E}}_{\bar{\beta }} - \lambda _{\bar{\beta }}{\dot{E}}_{\alpha } -2E_{\alpha } E_{\bar{\beta }} \nonumber \\&\qquad - 4 \mathrm {i}B_{\alpha {\bar{\beta }}} + {\underline{{\mathrm {Ric}}}}_{\alpha {\bar{\beta }}} - {\underline{{\textsf {N}}}}_{\alpha \delta \gamma } {\underline{{\textsf {N}}}}_{\bar{\beta }}\,^{\delta \gamma } \, , \end{aligned}$$

(A.20)

$$\begin{aligned} {\mathrm {Ric}}_{0}{}^{0}&= {\underline{\nabla }}_{\alpha }{E^{\alpha }} + {\underline{\nabla }}^{\alpha }{E_{\alpha }} - \lambda _{\alpha } {\dot{E}}^{\alpha } - \lambda ^{\alpha } {\dot{E}}_{\alpha } - 4E_{\alpha } E^{\alpha }+2 \mathrm {i}B_{\alpha }\,^{\alpha }+{\dot{E}}\,_{0} \, , \end{aligned}$$

(A.21)

$$\begin{aligned} {\mathrm {Ric}}_{0 \beta }&= {\underline{\nabla }}_{\beta }{E_{0}} - \frac{1}{2}{\underline{\nabla }}_{0}{E_{\beta }} - \lambda _{\beta } {\dot{E}}_{0} + \frac{1}{2}\lambda _{0} {\dot{E}}\,_{\beta } -\frac{1}{2}E^{\alpha } {\underline{A}}_{\beta \alpha } - 2 E^{\alpha } B_{\alpha \beta } + 2 E_{\alpha } B_{\beta }\,^{\alpha } \nonumber \\&\qquad -\mathrm {i}C_{\beta } + {\underline{\nabla }}^{\alpha }{B_{\alpha \beta }} - \lambda ^{\alpha } {\dot{B}}\,_{\alpha \beta } - {\underline{\nabla }}_{\alpha }{B_{\beta }\,^{\alpha }} + \lambda _{\alpha }{\dot{B}}_{\beta }\,^{\alpha } \nonumber \\&\qquad +\frac{1}{2} B^{\alpha \gamma } {\underline{{\textsf {N}}}}_{\alpha \gamma \beta } + \frac{1}{2} {\underline{\nabla }}^{\alpha }{{\underline{A}}_{\beta \alpha }} +\frac{1}{2}{\underline{A}}^{\alpha \gamma } {\underline{{\textsf {N}}}}_{\beta \alpha \gamma } \, , \end{aligned}$$

(A.22)

$$\begin{aligned} {\mathrm {Ric}}_{0 0}&= {\underline{\nabla }}_{\alpha } {C^{\alpha }} + {\underline{\nabla }}^{\alpha }{C_{\alpha }} - \lambda _{\alpha } {\dot{C}}^{\alpha } - \lambda ^{\alpha }{\dot{C}}_{\alpha } - \frac{1}{2} {\underline{A}}_{\alpha \beta } {\underline{A}}^{\alpha \beta }+2B_{\alpha \beta } B^{\alpha \beta }-2B_{\alpha }\,^{\beta } B_{\beta }\,^{\alpha }\, . \end{aligned}$$

(A.23)

1.3 The Ricci scalar

$$\begin{aligned} {\mathrm {Sc}}= & {} 4 {\underline{\nabla }}_{\alpha }{E^{\alpha }} +4 {\underline{\nabla }}^{\alpha }{E_{\alpha }} - 4 \lambda _{\alpha } {\dot{E}}^{\alpha } -4\lambda ^{\alpha } {\dot{E}}\,_{\alpha }-12E_{\alpha } E^{\alpha } \nonumber \\&+2{\dot{E}}\,_{0} -4 \mathrm {i}B_{\alpha }\,^{\alpha } +2 {\underline{{\mathrm {Sc}}}} -2 {\underline{{\textsf {N}}}}_{\alpha \beta \gamma } {\underline{{\textsf {N}}}}^{\alpha \beta \gamma } \, . \end{aligned}$$

(A.24)

1.4 The first Bianchi identities

$$\begin{aligned} 0&= {\dot{B}}_{\alpha \beta } + 2 {\underline{\nabla }}_{[\alpha } {E_{\beta ]}} - 2 \lambda _{[\alpha } {\dot{E}}_{\beta ]} - {\underline{{\textsf {N}}}}_{\alpha \beta \gamma } E^{\gamma } \, , \end{aligned}$$

(A.25)

$$\begin{aligned} 0&= {\dot{B}}_{\alpha {\bar{\beta }}} + {\underline{\nabla }}_{\alpha }{E_{{\bar{\beta }}}} - {\underline{\nabla }}_{{\bar{\beta }}}{E_{\alpha }} - \lambda _{\alpha } {\dot{E}}_{{\bar{\beta }}} + \lambda _{{\bar{\beta }}} {\dot{E}}_{\alpha } + \mathrm {i}E_{0} {\underline{h}}_{\alpha {\bar{\beta }}} \, , \end{aligned}$$

(A.26)

$$\begin{aligned} 0&= {\dot{C}}_{\alpha } + {\underline{\nabla }}_{\alpha }{E_{0}} - {\underline{\nabla }}_{0}{E_{\alpha }} - \lambda _{\alpha } {\dot{E}}_{0} + \lambda _{0}{\dot{E}}_{\alpha } - {\underline{A}}_{\beta \alpha } E^{\beta } \, , \end{aligned}$$

(A.27)

$$\begin{aligned} 0&= {\underline{\nabla }}_{[\alpha }{B_{\beta \gamma ]}} - \lambda _{[\alpha } {\dot{B}}_{\beta \gamma ]} + B_{[\alpha }\,^{\delta } {\underline{{\textsf {N}}}}_{\beta \gamma ] \delta } +2E_{[\alpha } B_{\beta \gamma ]} \, , \end{aligned}$$

(A.28)

$$\begin{aligned} 0&= 2 {\underline{\nabla }}_{[\alpha }{B_{\beta ] {\bar{\gamma }}}} +{\underline{\nabla }}_{{\bar{\gamma }}}{B_{\alpha \beta }} - 2 \lambda _{[\alpha }{\dot{B}}_{\beta ] {\bar{\gamma }}} - \lambda _{{\bar{\gamma }}} {\dot{B}}_{\alpha \beta } \nonumber \\&\qquad \qquad + 4E_{[\alpha } B_{\beta ] {\bar{\gamma }}} + 2E_{{\bar{\gamma }}} B_{\alpha \beta } -2 \mathrm {i}C_{[\alpha } {\underline{h}}_{\beta ] {\bar{\gamma }}} + {\underline{{\textsf {N}}}}_{\alpha \beta \delta } B_{{\bar{\gamma }}}{}^{\delta } \, , \end{aligned}$$

(A.29)

$$\begin{aligned} 0&= {\underline{\nabla }}_{[\alpha }{C_{\beta ]}} - \lambda _{[\alpha } {\dot{C}}_{\beta ]} + \frac{1}{2} {\underline{\nabla }}_{0}{B_{\alpha \beta }} - \frac{1}{2} \lambda _{0} {\dot{B}}_{\alpha \beta } \nonumber \\&\qquad + 2E_{[\alpha } C_{\beta ]} +E_{0} B_{\alpha \beta } + B_{[\alpha }\,^{\gamma } {\underline{A}}_{\beta ] \gamma } -\frac{1}{2} {\underline{{\textsf {N}}}}_{\alpha \beta \gamma } C^{\gamma } \, , \end{aligned}$$

(A.30)

$$\begin{aligned} 0&= {\underline{\nabla }}_{\alpha } {C_{{\bar{\beta }}}} - {\underline{\nabla }}_{{\bar{\beta }}}{C_{\alpha }} - \lambda _{\alpha } {\dot{C}}_{{\bar{\beta }}} + \lambda _{{\bar{\beta }}}{\dot{C}}_{\alpha } + {\underline{\nabla }}_{0}{B_{\alpha {\bar{\beta }}}} - \lambda _{0}{\dot{B}}_{\alpha {\bar{\beta }}} \nonumber \\&\qquad +2E_{\alpha } C_{{\bar{\beta }}} -2E_{{\bar{\beta }}} C_{\alpha } +2E_{0} B_{\alpha {\bar{\beta }}} - B_{{\bar{\beta }}}{}^{\gamma } {\underline{A}}_{\gamma \alpha } - {\underline{A}}_{{\bar{\beta }}}{}^{\gamma } B_{ \gamma \alpha } \, . \end{aligned}$$

(A.31)