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Twistor Space Origins of the Newman-Penrose Map

by Kara Farnsworth, Michael L. Graesser, Gabriel Herczeg

Submission summary

As Contributors: Kara Farnsworth
Arxiv Link: (pdf)
Date submitted: 2022-01-31 16:20
Submitted by: Farnsworth, Kara
Submitted to: SciPost Physics Core
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
  • High-Energy Physics - Phenomenology
Approach: Theoretical


Recently, we introduced the "Newman-Penrose map," a novel correspondence between a certain class of solutions of Einstein's equations and self-dual solutions of the vacuum Maxwell equations, which we showed was closely related to the classical double copy. Here, we give an alternative definition of this correspondence in terms of quantities that are naturally defined on both spacetime and twistor space. The advantage of this reformulation is that it is purely geometrical in nature, being manifestly invariant under both spacetime diffeomorphisms and projective transformations on twistor space. While the original formulation of the map may be more convenient for most explicit calculations, the twistorial formulation we present here may be of greater theoretical utility.

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Reports on this Submission

Anonymous Report 3 on 2022-6-15 (Invited Report)


1.) A novel approach to classical double copy making use of classic results in twistor theory (including the Kerr theorem and spin-raising operators).

2.) Clearly written.


1.) This shares the main weakness of most classical double copy approaches, insofar as it relies on algebraic speciality on the gravitational side (and therefore cannot teach us something new about the space of solutions in GR).


This paper builds on prior work by the authors in defining a 'Newman-Penrose map' which takes Kerr-Schild solutions in general relativity defined using a shear-free null geodesic vector, and maps them to self-dual solutions of the Maxwell equations. In particular, the authors provide a twistor interpretation for this map, demonstrating that various ingredients in the prescription arise naturally in the twistor description of shear-free null geodesic congruences. I think that this is an interesting and novel paper, and deserves to be published in SciPost Physics Core; although prior to publication, the authors should address the issues raised by the other referees.

I also have a (possibly naive) question for the authors, which is as much about their prior work on this topic as it is about this particular paper. Here, the Maxwell field is defined by $A=\hat{k}\Phi$, which results in a complex (in Lorentzian signature), self-dual abelian gauge field. However, a real-valued gauge field can also be obtained by taking $A=\frac{1}{2}\hat{k}\Phi+\frac{1}{2}\hat{\bar{k}}\bar{\Phi}$, and the resulting field strength is a non-chiral solution to the Maxwell equations. (Of course, this isn't true for non-abelian gauge fields, where the SD/ASD parts of the potential will mix in commutators for the field strength, but for Maxwell it is fine.) Furthermore, it seems that this non-chiral Maxwell field coincides with that produced by the 'Kerr-Schild' version of classical double copy. For instance, in the case of Schwarzschild, this will lead to a gauge potential which is real and gauge-equivalent to Coulomb. I haven't checked other examples (e.g., Kerr), but expect that something similar will happen there.

This suggests that the Newman-Penrose map, as stated, is just the chiral half of another map which is really equivalent to the Kerr-Schild double copy, but refined by the shear-free condition. Have the authors considered this, and if so, is there some reason why this is less interesting than their chiral version of the Newman-Penrose map?

Requested changes

1.) The points raised by the other referees, particularly referee 1, should be addressed.

2.) On page 1, the authors say that there is a growing body of evidence that double copy can be "...extended to all orders in perturbation theory [4-11]." This may be nitpicking, but I think that this is an over-optimistic statement: in terms of what is actually known, I think it is fair to say that double copy can be extended up to 5-loops, for the 4-point amplitude. A more accurate statement might be "...extended beyond tree-level in perturbation theory [4-11]."

3.) For completeness, the authors should define the $Q$ and $\varepsilon_0$ appearing in equation (2.8).

  • validity: good
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Anonymous Report 2 on 2022-6-2 (Invited Report)


This paper concerns the application of twistor methods to the study of the “double copy” relating quantities in gauge and gravity theories. The authors reconsider a previously presented proposal for generating self-dual single copies known as the Newman-Penrose map. Gauge fields in this picture are comprised of a certain differential operator acting on a given scalar function. The authors show that both of these ingredients can be given geometric definitions motivated by twistor theory, such that ambiguities in the definitions cancel out in the gauge field. Given the use of twistor methods in other research works concerning the double copy, the present paper is a highly useful addition to the literature, and I recommend publication.

I note that the first referee has a number of comments, which I agree need addressing. But I have a remark that the authors may also wish to comment on. They wonder how their analysis is related to other twistor-related work on the double copy, which relies on the well-known Penrose transform from twistor space to position space. I suspect that the answer lies in their use of the Kerr theorem, which states that shear-free null geodesic congruences (SNGCs) such as those entering Kerr-Schild geometries can be defined by the vanishing of a function of twistor variables. In the Penrose transform approach to the double copy, the contour integral in the Penrose transform picks out the residue of poles in the integrand, which in turn enforces the vanishing of certain twistor functions. So the various approaches seem conceptually related, at least in principle. It would be interesting to know if the authors agree or not with this suggestion, but they should not feel obliged to comment.

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Anonymous Report 1 on 2022-4-19 (Invited Report)


The paper explores the geometrical description in twistor space of the Newman-Penrose map, which is closely related to the classical double copy. The authors discover an interesting geometrical origin of the spin-raising operator $\hat{k}$ which appears in the classical self-dual double copy and in the Newman-Penrose map. While the paper is well-written, some important points should be addressed before I can recommend for publication:
-Under Eq. 2.7 it is mentioned given $\Phi$ specified by 2.6, the solution to Einstein’s equation is completely specified except for its total mass. There are several definitions for a mass in General Relativity, which one is this referring to?
-The building blocks of the Newman-Penrose map are written in terms of spinors, some that naturally arise as components of a null twistor ($\omega^A,\pi_{A’}$) and others that don’t ($l_{AA’}$). Meanwhile, it is mentioned in the abstract, introduction, and discussion that this alternative definition is given in terms of quantities defined on both spacetime and twistor space. The results in Eqs. 4.15 and 4.16 don’t support this claim. They are not written in terms of twistors and include $l_{AA’}$ which is not defined in twistor space. The claim must be clearly proven under these equations or it should be rephrased.
-In Eq. 4.17 an alternative definition of the gauge field in the Newman-Penrose map is considered. It is mentioned that it has no dependence on the phase $\delta$ and that it is equivalent to the definition in 2.9. This is not true unless you ignore the extra constant term. While that term does not contribute to the field strength, it is still present in the gauge field. To make those claims one would have to do a gauge transformation to remove the extra term. These shortcomings should be clarified instead of swept under the carpet.
This paper could be suitable for publication in SciPost Physics Core after the concerns mentioned above have been addressed.

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