Two-dimensional twistor manifolds and Teukolsky operators

Abstract

The Teukolsky equations are currently the leading approach for analysing stability of linear massless fields propagating in rotating black holes. It has recently been shown that the geometry of these equations can be understood in terms of a connection constructed from the conformal and complex structure of Petrov type D spaces. Since the study of linear massless fields by a combination of conformal, complex and spinor methods is a distinctive feature of twistor theory, and since versions of the twistor equation have recently been shown to appear in the Teukolsky equations, this raises the question of whether there are deeper twistor structures underlying this geometry. In this work we show that all these geometric structures can be understood naturally by considering a 2-dimensional twistor manifold, whereas in twistor theory the standard (projective) twistor space is 3-dimensional.

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Notes

  1. 1.

    This subsection is related to the ‘usual’ twistor space, i.e. not to its ‘dual’ version, which is the one that we use in the rest of the paper.

  2. 2.

    If we fix \(\zeta ^{A'}\) and vary \(\lambda ^A\) instead, the resulting 2-plane is an ‘\(\alpha \)-plane’, and (projective) twistor space \(\mathbb {PT}\) is the space of \(\alpha \)-planes.

  3. 3.

    There are important subtleties that we are omitting here, namely the fact that it is not actually the whole \(\mathbb {PT}^{*}\) which enters (2.7) but the region with \(\lambda _{A}\ne 0\); we do not need to discuss this for the purposes of our presentation.

  4. 4.

    Note that, consistently, equation (3.1) is conformally invariant if \(\xi ^A\) has well-defined conformal weight.

  5. 5.

    Our main reference for concepts and definitions regarding conformal geometry is [43].

  6. 6.

    Recall that the spin structure of a conformal manifold is a well-defined concept, see e.g. [37, Section 5.6] and also Note 8 to Chapter 9 in [32].

  7. 7.

    Note that the spin group \(\mathrm{SL}(2,\mathbb {C})\) can be decomposed as \(\mathrm{SL}(2,\mathbb {C})\cong \mathbb {C}^{\times }\times \mathbb {C}^{+}\times \mathbb {C}^{+}\), where \(\mathbb {C}^{\times }\) is the ‘GHP part’ and the two factors of \(\mathbb {C}^{+}\) correspond to null rotations around the spinors of the frame.

  8. 8.

    Note that (3.10) is a complex map, whereas the usual notion of an almost-complex structure requires it to be real. However, as shown in Theorem VIII.3 in [25], a Lorentzian manifold (which is ultimately the most interesting case for our purposes) cannot admit a (real) almost-Hermitian structure, so we are forced to consider this complex-valued almost-Hermitian structure (in [25] this is referred to as a ‘modified’ Hermitian structure). We will give an interpretation of (3.10) in Sect. 4.2 below.

  9. 9.

    I am grateful to J. L. Jaramillo for suggesting looking into this.

  10. 10.

    Note that, roughly speaking, a fibre of \(\langle \xi ^A\rangle \) corresponds to a single point in a fibre of \(\mathbb {P}\mathcal {S}^A\).

  11. 11.

    Of course, the ‘if and only if’ part of the linearized version of the Goldberg-Sachs theorem is not valid, as shown in [19].

  12. 12.

    In what follows, for a quantity \(T(\varepsilon )\) we use the notation \(\mathring{T}\equiv T(0)\) and \({\dot{T}}\equiv \frac{d}{d\varepsilon }|_{\varepsilon =0}T(\varepsilon )\).

  13. 13.

    I am very grateful to M. Dunajski and L. Mason for discussions about this and for suggesting references.

  14. 14.

    I thank M. Dunajski for bringing this reference to my attention.

  15. 15.

    The equation \(\nabla _{A'}{}^{(A}X^{BC)}=0\) is conformally invariant, so \(\nabla _{AA'}\) here is any Levi-Civita connection in the conformal class.

  16. 16.

    I am grateful to L. Mason for this suggestion.

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Acknowledgements

It is a pleasure to thank Steffen Aksteiner, Lars Andersson, Thomas Bäckdahl, Igor Khavkine and Lionel Mason for very helpful discussions, that took place at the Institut Mittag-Leffler (Djursholm, Sweden) in the fall 2019. I am also very thankful to Tim Adamo, Maciej Dunajski and George Sparling for comments about this work during the conference “Twistors meet Loops in Marseille”, held at CIRM (France) in September 2019; in particular I want to thank M. Dunajski for several illuminating conversations in this conference and also during a visit to Cambridge University in November 2019. The hospitality and support of all the institutions mentioned above are also gratefully acknowledged. Finally I thank Gustavo Dotti, José Luis Jaramillo, Oscar Reula and Juan Valiente Kroon for supportive comments on this work and on a previous version of this manuscript. This work is partially supported by a postdoctoral fellowship from CONICET (Argentina).

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The current version of this paper is based upon work supported by the Swedish Research Council under Grant No. 2016-06596 while the author was in residence at Institut Mittag-Leffler in Djursholm, Sweden, during the fall 2019.

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Araneda, B. Two-dimensional twistor manifolds and Teukolsky operators. Lett Math Phys (2020). https://doi.org/10.1007/s11005-020-01307-8

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