# A heuristic for discrete mean values of the derivative of the Riemann zeta
function

Christopher Hughes  Department of Mathematics, University of York, York, YO10
5DD, United Kingdom<EMAIL_ADDRESS>,  Greg Martin  Department
of Mathematics, University of British Columbia, Vancouver, British Columbia,
V6T 1Z2, Canada<EMAIL_ADDRESS>and  Andrew Pearce-Crump  Department of
Mathematics, University of York, York, YO10 5DD, United Kingdom
<EMAIL_ADDRESS>
###### Abstract.

Shanks conjectured that $\zeta^{\prime}(\rho)$, where $\rho$ ranges over non-
trivial zeros of the Riemann zeta function, is real and positive in the mean.
We present a history of this problem, including a generalisation to all
higher-order derivatives $\zeta^{(n)}(s)$, for which the sign of the mean
alternatives between positive for odd $n$ and negative for even $n$.
Furthermore, we give a simple heuristic that provides the leading term
(including its sign) of the asymptotic formula for the average value of
$\zeta^{(n)}(\rho)$.

##  1\. Introduction

Shanks’ conjecture, which dates from 1961, states that

$\zeta^{\prime}(\rho)\text{ is real and positive in the mean}$

as $\rho=\beta+i\gamma$ ranges over non-trivial zeros of the Riemann zeta
function $\zeta(s)$. More recently this assertion has been generalised to
higher derivatives:

$\text{On average, }\zeta^{(n)}(\rho)\text{ is positive if $n$ is odd, but
negative if $n$ is even.}$

The aim of this paper is to show how one can heuristically derive this
Generalised Shanks’ Conjecture, indeed in quantitative form, almost
immediately from an explicit formula known as the Landau–Gonek Theorem.

The Riemann zeta function $\zeta(s)$ has infinitely many non-trivial zeros,
which are numbers $\rho=\beta+i\gamma$ with $0<\rho<1$ that satisfy
$\zeta(\rho)=0$. The locations of these non-trivial zeros, as we have known
for a century and a half (since Riemann), are intimately connected with the
distribution of prime numbers. While impressive calculations were done by hand
over the years (culminating in Titchmarsh and Comrie’s verification [18] that
the first thousand zeros of $\zeta(s)$ lie on the critical line
$\beta=\frac{1}{2}$), the advent of electronic computers in the middle of the
20th century provided a significant boost to computations of the non-trivial
zeros.

Shanks first made his conjecture [17] when he was reviewing Haselgrove’s
tables [10] of numerical values of the Riemann zeta function. He plotted the
graph of $t\mapsto\zeta(\frac{1}{2}+it)$ and noticed that the way this curve
approaches the origin, mainly through the third and fourth quadrants, suggests
the phase of $\zeta^{\prime}(\frac{1}{2}+i\gamma)$ is close to zero in the
mean and thus $\zeta^{\prime}(\frac{1}{2}+i\gamma)$ is positive and real in
the mean. This observation stands in contrast to the general behaviour of the
function $\zeta^{\prime}(\frac{1}{2}+it)$, whose mean value tends quickly to
$0$. We present a different graph to the one Shanks created: Figure 1 shows
the graph of $\zeta^{\prime}(\rho)$ for the first 100,000 zeros of $\zeta(s)$,
which clearly displays symmetry and a bias towards the positive side of the
complex plane.

Figure 1. Scatterplot of $\zeta^{\prime}(\rho)$ for the first 100,000 zeros of
the zeta function

In 1985, Conrey, Ghosh and Gonek [3] were looking for a straightforward proof
of the fact that there are infinitely many simple zeros of $\zeta(s)$. To this
end, they used the Cauchy–Schwarz inequality to write

$\bigg{|}\sum_{0<\gamma\leq
T}\zeta^{\prime}(\rho)\bigg{|}^{2}\leq\sideset{}{{}^{*}}{\sum}_{0<\gamma\leq
T}1\sum_{0<\gamma\leq T}\left|\zeta^{\prime}(\rho)\right|^{2}$

where the star on the middle sum denotes counting only the simple zeros. To
complete the proof they used the asymptotic for the discrete second moment on
the right-hand side, which was proved by Gonek [7] in 1984, and so only needed
to find the leading order behaviour of the sum on the left-hand side. They
finished their proof that there are infinitely many simple zeros of $\zeta(s)$
by showing that

$\sum_{0<\gamma\leq T}\zeta^{\prime}(\rho)=\frac{T}{4\pi}(\log T)^{2}+O(T\log
T),$ (1.1)

which proved Shanks’ conjecture as an additional benefit.

As a side note, this method was not strong enough to show that a positive
proportion of the non-trivial zeros of $\zeta(s)$ are simple (a fact that goes
back to work of Levinson [15]). However, later in 1998 they were able [4] to
prove, by modifying the method to include mollifiers, that at least
$\frac{19}{27}$ of these non-trivial zeros are simple assuming the Riemann
Hypothesis and the Generalised Lindelöf Hypothesis. In 2013, Bui and Heath-
Brown [2] removed the assumption of the Generalised Lindelöf Hypothesis
through careful use of the generalised Vaughan identity.

Before we describe the generalisation of Shanks’ conjecture to higher
derivatives, it is worth describing the ongoing research progress on the
distribution of $\zeta^{\prime}(\rho)$. In 1994 Fujii [5] found explicit lower
order terms for the asymptotic formula (1.1) (he later corrected [6] a slight
error in the lowest-order coefficient when writing a paper where he combined
$\zeta^{\prime}(\rho)$ with the Landau–Gonek Theorem). The full correct
asymptotic is given by

$\sum_{0<\gamma\leq
T}\zeta^{\prime}(\rho)=\frac{T}{4\pi}\biggl{(}\log\frac{T}{2\pi}\biggr{)}^{2}+(-1+C_{0})\frac{T}{2\pi}\log\frac{T}{2\pi}+(1-C_{0}-C_{0}^{2}+3C_{1})\frac{T}{2\pi}+E(T)$

where $C_{0}$ and $C_{1}$ are coefficients in the Laurent expansion of
$\zeta(s)$ about $s=1$. (Unconditional effective bounds for $E(T)$ are known,
and a savings of a power of $T$ can be shown under the Riemann Hypothesis. The
best known unconditional error term can be found in [11]). In 2021 Kobayashi
[13] gave a similar result for Dirichlet $L$-functions.

In 2010 Trudgian [19] gave an alternative proof of Shanks’ conjecture, based
on Shanks’ observation of the connection between $\arg\zeta^{\prime}(\rho)$
and the so-called Gram’s Law. Although care needs to be taken in the
definition of the argument (a priori it is only defined up to a multiple of
$2\pi$), Trudgian was able to show that

$\sum_{0<\gamma\leq
T}\arg\zeta^{\prime}(\rho)\ll_{\varepsilon}T^{\varepsilon}$

for every $\varepsilon>0$.

We are now ready to explore the generalisation of Shanks’ Conjecture to
$\zeta^{(n)}(s)$, the $n$th derivative of $\zeta(s)$, for every positive
integer $n$. This extension, which we call the Generalised Shanks’ Conjecture,
states that $\zeta^{(n)}(\rho)$ is real and positive in the mean if $n$ is
even, while $\zeta^{(n)}(\rho)$ is real and negative in the mean if $n$ is
odd.

|
---|---
Figure 2. Scatterplot of $\zeta^{(n)}(\rho)$ for $n=2$ and $n=3$ for the first
100,000 zeros of the zeta function

In 2011, using ideas similar to those in Conrey, Ghosh and Gonek’s proof of
Shanks’ original conjecture, Kaptan, Karabulut and Yıldırım [12] found the
main term for the summatory function of $\zeta^{(n)}(\rho)$. Specifically,
they showed that

$\sum_{0<\gamma\leq
T}\zeta^{(n)}(\rho)=\frac{(-1)^{n+1}}{n+1}\frac{T}{2\pi}(\log
T)^{n+1}+O\bigl{(}T(\log T)^{n}\bigr{)}$ (1.2)

which clearly implies the Generalised Shanks’ Conjecture as stated above. They
have also proven a similar result for derivatives of Dirichlet $L$-functions,
with the same leading-order asymptotic term. (Note that Shanks’ Conjecture was
first conjectured and later proved, in the usual order of events, but that we
still call the established assertion Shanks’ “Conjecture”. This nomenclature
leads to an unusual situation where the extension is reasonably labeled
“Shanks’ Generalised Conjecture” even though its first appearance in the
literature was as a proven result!)

In 2022 the first and third author [11] investigated the Generalised Shanks’
Conjecture and found lower order terms for the asymptotic formula (1.2). The
result can be stated in the form

$\sum_{0<\gamma\leq
T}\zeta^{(n)}(\rho)=\frac{(-1)^{n+1}}{n+1}\frac{T}{2\pi}\biggl{(}\log\frac{T}{2\pi}\biggr{)}^{n+1}+\frac{T}{2\pi}\mathcal{P}_{n}\biggl{(}\log\frac{T}{2\pi}\biggr{)}+E_{n}(T)$

where $\mathcal{P}_{n}(x)$ is a polynomial of degree $n$ described explicitly
in [11]; as with Fujii’s result, unconditional effective bounds for the error
term $E_{n}(T)$ are known, and a power savings in $T$ can be shown under the
Riemann Hypothesis. Furthermore, the third author [16] has also extended
Fujii’s combination (mentioned earlier) of $\zeta^{\prime}(s)$ and the
Landau–Gonek Theorem to all derivatives of $\zeta(s)$.

All of the results we have described have complicated proofs, and it is
unclear that any of them provide an intuitive explanation for the alternating
signs in the Generalised Shanks’ Conjecture, or indeed of the positive sign in
Shanks’ original conjecture. We now present a simple heuristic that explains
these signs, and indeed recovers (nonrigorously) the leading-order term in the
asymptotic formula (1.2).

##  2\. Heuristic for the Riemann zeta function

We begin with some preliminaries concerning the Riemann zeta function, which
for $\Re(s)>1$ can be written as the convergent Dirichlet series

$\zeta(s)=\sum_{m=1}^{\infty}\frac{1}{m^{s}}.$

Since Dirichlet series converge locally uniformly we may differentiate term by
term; and since the derivative of $\frac{1}{m^{s}}=e^{-s\log m}$ is $({-}\log
m)m^{-s}$, we see that

$\zeta^{(n)}(s)=(-1)^{n}\sum_{m=1}^{\infty}\frac{(\log m)^{n}}{m^{s}}$

for $\Re(s)>1$. Writing $s=\sigma+it$ with $\sigma>1$, a simple application of
Euler–Maclaurin yields

$\zeta^{(n)}(s)=(-1)^{n}\sum_{m\leq t}\frac{(\log
m)^{n}}{m^{s}}+O(t^{-\sigma}(\log t)^{n}).$ (2.1)

We state a corollary of the Landau–Gonek Theorem, found in [9], concerning
sums of the form $X^{\rho}$ over the non-trivial zeros of the Riemann zeta
function. This corollary is the main result that the heuristic relies upon. We
call this result the Landau–Gonek Theorem for the sake of brevity in this
paper, but note it is given in a more general form in the cited papers. It was
proved by Landau [14] for fixed $X$ and made uniform by Gonek [8, 9]. We note
that while our quoted result requires the assumption of the Riemann
hypothesis, the general result does not.

######  Lemma 1.

Under the Riemann Hypothesis, for $T>1,m\in\mathbb{N}$ with $m\geq 2$ and
$\rho$ a non-trivial zero of the Riemann zeta function $\zeta(s)$,

$\sum_{0<\gamma\leq
T}m^{-\rho}=-\frac{T}{2\pi}\frac{\Lambda(m)}{m}+O(\log(2mT)\log\log(3m)),$
(2.2)

where $\Lambda(m)$ is the von Mangoldt function, which therefore only
contributes when $m$ is a prime power.

###  2.1. The heuristic

We now state the short heuristic argument which yields the Generalised Shanks’
Conjecture as stated in Section 1. We ignore error terms for the large part so
the equal signs are not ‘true’ equalities but show the general argument. (But
see Section 2.2.)

######  Remark.

As we were writing this paper, we became aware that the heuristic presented in
this paper can be found in [9] in a different context, where it can be applied
rigorously.

Throughout this section we write $\rho=\beta+i\gamma$ for a non-trivial zero
of $\zeta(s)$.

Using the approximate formula (2.1) for $\zeta(s)$ and the Landau–Gonek
Theorem (2.2), we may write for $n\geq 1$,

$\displaystyle\sum_{0<\gamma\leq T}\zeta^{(n)}(\rho)$
$\displaystyle\approx(-1)^{n}\sum_{0<\gamma\leq T}\sum_{m\leq\gamma}(\log
m)^{n}m^{-\rho}$ (2.3) $\displaystyle=(-1)^{n}\sum_{m\leq T}(\log
m)^{n}\sum_{m<\gamma\leq T}m^{-\rho}$
$\displaystyle\approx(-1)^{n+1}\frac{T}{2\pi}\sum_{m\leq T}\frac{(\log
m)^{n}\Lambda(m)}{m}-(-1)^{n+1}\frac{1}{2\pi}\sum_{m\leq T}(\log
m)^{n}\Lambda(m)$

To finish the heuristic, we need to sum the series in the last line. By
Chebyshev’s Theorem [1],

$C(x)=\sum_{m\leq x}\frac{\Lambda(m)}{m}=\log x+O(1)$

so by partial summation, we have

$\displaystyle\sum_{m\leq T}\frac{(\log m)^{n}\Lambda(m)}{m}$
$\displaystyle=C(T)(\log T)^{n}-n\int_{1}^{T}\frac{C(x)(\log x)^{n-1}}{x}\ dx$
$\displaystyle=\frac{1}{n+1}(\log T)^{n+1}+O\bigl{(}(\log T)^{n}\bigr{)}.$

and similarly

$\sum_{m\leq T}(\log m)^{n}\Lambda(m)=O\bigl{(}T(\log T)^{n}\bigr{)}$

Combining this with our argument above, we have

$\displaystyle\sum_{0<\gamma\leq T}\zeta^{(n)}(\rho)$
$\displaystyle=(-1)^{n+1}\frac{T}{2\pi}\frac{1}{n+1}(\log T)^{n+1}+O(T(\log
T)^{n}),$

as $T\rightarrow\infty$, which is the leading order the asymptotic result for
the Generalised Shanks’ Conjecture.

######  Remark.

Note that this result assumes $n\geq 1$. When $n=0$ the LHS is trivially zero,
whilst the RHS, as written, is not. This is explained by noticing that when
$n\geq 1$ the term when $m=1$ in the sum (2.3) is not present, since
$\log(1)^{n}=0$. However, if $n=0$ this term is present and is not accounted
for in the Landau–Gonek Theorem, which holds for $m\geq 2$ only. The $m=1$
term in that case clearly contributes $N(T)$, which perfectly cancels the
$-\frac{T}{2\pi}\log T+O(T)$ term coming from the calculation given above.

######  Remark.

This argument can be applied _mutatis mutandis_ for Dirichlet $L$-functions,
yielding exactly the same leading-order behaviour as for the Riemann zeta
function. This result was already known [12, 13].

###  2.2. The error terms

In the previous section in the last part of equation (2.3) we ignored the
error terms coming from the Landau–Gonek formula (2.2). Including them one can
see that they contribute

$\ll\sum_{m\leq T}(\log m)^{n}\log\log(3m)\log T\ll T(\log T)^{n+1}\log\log T$

which dominates the main term! However, these are worst-case point-wise
estimates and take no account of any potential cancellation when averaged, so
it is likely that when summed over $m$ the true error is smaller and the main
term is correct. (Indeed that is what is proved, by different methods, in [3,
12] for $n=1$ and $n\geq 2$ respectively).

The point is that whilst this method must remain a heuristic, it is a much
quicker approach to find the main term in the overage of $\zeta^{(n)}(\rho)$
and gives some sense of why the Generalised Shanks’ Conjecture holds true,
that is why the mean of $\zeta^{(n)}(\rho)$ is real and positive / negative.

######  Remark.

In [11] the first and third authors prove that the true value of the mean of
$\zeta^{(n)}(\rho)$ (that is, the main term and all the lower-order terms up
to a power saving) is given by

$\sum_{\begin{subarray}{c}\ell,m\\\ \ell
m\leq\frac{T}{2\pi}\end{subarray}}\Lambda(m)(\log m)^{n}$

which shows that the heuristic in the previous section can only yield the main
term since the last line of equation (2.3) differs from the true value by
$O(T(\log T)^{n})$.

## Acknowledgements

The second author suggested the heuristic in this paper at the PIMS CRG
‘Moments of $L$-functions’ workshop in Prince George, British Columbia to the
third author. The third author would like to thank the group, the LMS and the
University of York for their generous funding.

## References

  * [1] T. M. Apostol, _Introduction to Analytic Number Theory_ , 1st ed., Springer, 1976.
  * [2] H. M. Bui and D. R. Heath-Brown, _On simple zeros of the Riemann zeta-function_ , Bull. Lond. Math. Soc. 45 (2013), 953–961.
  * [3] J. B. Conrey, A. Ghosh, and S. M. Gonek, _Simple zeros of zeta functions_ , Colloque de Théorie Analytique des Nombres “Jean Coquet” (Marseille, 1985), Publ. Math. Orsay, vol. 88, Univ. Paris XI, Orsay, 1988, pp. 77–83.
  * [4] by same author, _Simple zeros of the Riemann zeta-function_ , Proc. Lond. Math. Soc. (3) 76 (1998), 497–522.
  * [5] A. Fujii, _On a conjecture of Shanks_ , Proc. Japan Acad. A 70 (1994), 109–114.
  * [6] by same author, _On the Distribution of Values of the Derivative of the Riemann Zeta Function at Its Zeros. I_ , Proc. Steklov Inst. Math. 276 (2012), 51–76.
  * [7] S. M. Gonek, _Mean values of the Riemann zeta-function and its derivatives_ , Invent. math. 75 (1984), 123–141.
  * [8] by same author, _A formula of Landau and mean values of $\zeta(s)$_, Topics in analytic number theory (Austin, Tex., 1982), Univ. Texas Press, Austin, TX, 1985, pp. 92–97.
  * [9] by same author, _An explicit formula of Landau and its applications to the theory of the zeta-function_ , A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 395–413.
  * [10] C.B. Haselgrove in collaboration with J.C.P. Miller, _Mathematical Tables: Tables of the Riemann zeta function_ Royal Society Mathematical Tables No. 6, Cambridge University Press, (1960).
  * [11] C. Hughes and A. Pearce-Crump, _A discrete mean-value theorem for the higher derivatives of the Riemann zeta function_ , J. Number Theory 241 (2022), 142–164.
  * [12] D. Kaptan, Y. Karabut, and C. Yildirim, _Some mean value theorems for the Riemann zeta-function and Dirichlet $L$-functions._, Comment. Math. Univ. St. Pauli 60 (2011), 83–87.
  * [13] H. Kobayashi, _On a certain sum of the derivatives of Dirichlet $L$-functions_, arXiv:2003.05629, 2021.
  * [14] E. Landau, _Uber die Nullstellen der Zetafunction_ , Math Annalen 71 (1911), 548–564.
  * [15] N. Levinson, _More than one-third of zeros of Riemann’s zeta-function are on $\sigma=\frac{1}{2}$_, Adv. in Math. 13 (1974) 383–436.
  * [16] A. Pearce-Crump, _A further generalisation of sums of higher derivatives of the Riemann zeta function_ , arXiv:2111.02756, 2021.
  * [17] D. Shanks, _Review of ‘Tables of the Riemann zeta function’ by C.B. Haselgrove in collaboration with J.C.P. Miller_ , Math. Comp. (1961), 84–86.
  * [18] E. C. Titchmarsh, _The Zeros of the Riemann Zeta-Function_ , Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, The Royal Society, 157 (1936): 261–-263.
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# Label uncertainty-guided multi-stream model for disease screening

###### Abstract

The annotation of disease severity for medical image datasets often relies on
collaborative decisions from multiple human graders. The intra-observer
variability derived from individual differences always persists in this
process, yet the influence is often underestimated. In this paper, we cast the
intra-observer variability as an uncertainty problem and incorporate the label
uncertainty information as guidance into the disease screening model to
improve the final decision. The main idea is dividing the images into simple
and hard cases by uncertainty information, and then developing a multi-stream
network to deal with different cases separately. Particularly, for hard cases,
we strengthen the network’s capacity in capturing the correct disease features
and resisting the interference of uncertainty. Experiments on a fundus image-
based glaucoma screening case study show that the proposed model outperforms
several baselines, especially in screening hard cases.

Index Terms—  Label uncertainty, disease screening.

##  1 Introduction

Deep learning (DL) models for image-based disease screening heavily rely on
large-scale datasets consisting of ($x_{i},y_{i}$) pairs, where $x_{i}$ is the
image data instance and $y_{i}$ is the corresponding label of disease
severity. Practically, the label annotation is performed by multiple human
experts (e.g., doctors or trained graders) in a collaborative pattern, where
the consistent decision from the majority is often regarded as the “ground
truth” of $y_{i}$ [1, 2, 3, 4]. Given individual differences in e.g., domain
expertise, judgement and bias, intra-observer variability always persists in
the annotation process [5, 6]. However, defining ground truth by majority
consistency is somehow oversimplified, underestimating the influence of intra-
observer variability and may even mislead the ground truth. For instance, two
images, one is labelled as “positive” by $100\%$ of graders while the other
one by $51\%$ of graders, will have the same final annotation as “positive”,
despite the grading difficulties being intuitively unequal.

In fact, the label uncertainty resulted from intra-observer variability
contains important prior information, which implies how difficult an image is
in identifying its disease severity. Previous studies exploited such
information for quantifying imaging quality [7], or identifying hard patient
cases that may require a medical second opinion [8]. Different from previous
viewpoints, we believe that label uncertainty can provide prior guidance to
improve DL models’ decisions, and should be carefully considered in the
initial model design. We have two observations in a fundus image-based
glaucomatous optic neuropathy (GON) screening task:

O1: A DL model generally performs worse on samples with higher label
uncertainty. Fig.1.a shows the performances of a screening model in different
uncertainty score (computed by Eq.1) groups. The average group accuracy
degrades with the uncertainty score increases.

O2: Disease severity distribution is potentially correlated with label
uncertainty. Fig.1.b shows the distributions of GON severity in images with
different uncertainty scores. The label uncertainty of suspect GON samples are
generally higher than those of unlikely and certain GON samples.

Fig. 1: (a) A DL model’s screening accuracy in different label uncertainty
groups; (b) GON severity distributions on different label uncertainty. Both
x-axes are label uncertainty scores.

The insights stemming from the observations is that DL screening models should
be improved on the samples with higher label uncertainty with identifying
“purer” disease features in the presence of label uncertainty. To this end, we
first model label uncertainty as scalar scores using the empirical
distribution of intra-observer variability. The images are accordingly divided
into simple and hard cases. Then a multi-stream disease screening model is
proposed, comprising two main streams, one for simple case screening (SC-Net)
and one for hard case screening (HC-Net), and an auxiliary stream (US-Net) for
extracting uncertainty-associated features and predicting uncertainty scores.
The label uncertainty information is elaborately embedded into the HC-Net’s
learning process in various ways to offer guidance for classifying hard cases.
Specifically, in the training phase, the variability-based encoding and
uncertainty-guided joint loss are used to enforce the network to capture
correct disease features in the presence of label uncertainty and disentangle
the uncertainty-associated features in the latent space. In the inference
phase, the uncertainty-guided adaptive threshold is applied to unseen samples
to improve the model’s decision on hard cases.

##  2 Methodology

###  2.1 Modeling label uncertainty

Given an image $x_{i}$, we let $y_{i}^{(1)},y_{i}^{(2)},...,y_{i}^{(M)}$ be
the assigned labels from $M$ individual graders. In practice, the minimum of
$M$ is $3$ to gain a significant voting. Letting $c_{1},...,c_{K}$ be $K$
classes of disease severity, the empirical distribution of intra-observer
variability is
$\hat{\mathbf{p}}_{\mathbf{i}}=\left[\hat{p}_{i}^{(1)},\ldots,\hat{p}_{i}^{(K)}\right]$,
with $\hat{p}_{i}^{(k)}=\frac{1}{M}\sum_{m}\mathcal{H}({y_{i}^{(m)}}=c_{k})$
where $\mathcal{H}(\cdot)=1$ when the inside condition stands, otherwise $0$
[8]. Then the label uncertainty of $x_{i}$ can be defined by entropy which
indicates the stability of a system:

$u_{i}=-\sum_{k}\hat{p}_{i}^{(k)}\log\hat{p}_{i}^{(k)}.$ (1)

$u_{i}$ reaches its peak when human disagreements are equally distributed on
$K$ classes, indicating the most unstable state.

Note that label uncertainty differs from model uncertainty, also known as
epistemic uncertainty. The latter measures the model’s confidence in making a
decision. Those probabilistic approximation-based method for estimating model
uncertainty, such as Monte Carlo Dropout (MC-Dropout) [9], cannot be directly
applied to label uncertainty.

Fig. 2: Overview of the training phase and the inference phase of the
uncertainty-guided multi-stream screening model.

###  2.2 Multi-stream screening model

Fig. 2 shows an overview of the training phase (left) and the inference phase
(right) of the multi-stream screening model. The three sub-streams (US-Net,
SC-Net and HC-Net) employ the same deep neural network (DNN) structure as
backbones. Notably, the proposed model is independent with DNN structures and
compatible with various DNN backbones.

####  2.2.1 US-Net stream

The US-Net is an auxiliary stream for disentangling uncertainty-associated
features and predicting uncertainty scores for unseen samples (see Section
2.2.2). It is optimized with a mean squared error (MSE) loss (Fig.2: the
purple learning flow). Given the training data pair $(x_{i},u_{i})$, the MSE
loss is:

$\mathcal{L}_{MSE}=\frac{1}{N}\sum_{i}\left(u_{i}-\tilde{u}_{i}\right)^{2},$
(2)

where $\tilde{u}_{i}$ is the predicted uncertainty score.

####  2.2.2 SC-Net and HC-Net streams

In the training phase, the image samples are divided into simple cases and
hard cases via a predefined threshold on the empirical uncertainty score $u$.
Then the two cases are fed into the SC-Net and the HC-Net for learning disease
severity representation, respectively. The learning of disease severity can be
formulated as a multi-classification problem.

Regarding simple cases, the underlying disease severity features can be well
captured by a vanilla DNN. Since human graders easily achieve a high
consistency for simple cases, we follow previous studies [1, 2, 3, 4]using the
majority-voting result $\bar{y}$ as the training ground truth. For the
training pair $(x_{i},\bar{y}_{i})$, the optimization is performed with a
cross-entropy loss (shown as the green learning flow in Fig. 2):

$\mathcal{L}_{CE}=-\frac{1}{N}\sum_{i}\sum_{k}\bar{p}_{i}^{(k)}\log\left(\tilde{p}_{i}^{(k)}\right),\bar{p}_{i}^{(k)}\in\bar{\mathbf{p}}_{\mathbf{i}},\tilde{p}_{i}^{(k)}\in\tilde{\mathbf{p}}_{\mathbf{i}}$
(3)

where $\bar{\mathbf{p}}_{\mathbf{i}}$ denotes the one-hot encoding vector of
$\bar{y}_{i}$ and $\tilde{\mathbf{p}}_{\mathbf{i}}$ is the class probability
yielded by the softmax layer of DNN.

Compared to simple cases, hard cases are more difficult for both human and
vanilla DNNs to recognize the disease severity correctly. Therefore, directly
applying the majority-voting-based ground truth and the SC-Net for hard case
learning is impractical. Taking advantage of label uncertainty information, we
design the HC-Net with several specific strategies to address the problem. The
red arrow in Fig. 2 illustrates the learning flow of HC-Net.

Variability-based encoding. Rethinking the above one-hot encoding method for a
multi-classification problem, where the target class is assigned with a scalar
$1$ while other classes are $0$, a strong assumption is that all samples
belonging to the same class contain equal amount of ground truth information
in the label space, irrespective of their intrinsic difficulties in disease
screening being different. This could lead to a model bias in identifying the
correct disease features for hard cases. Instead, we propose the variability-
based encoding method which applies the empirical distribution of intro-
observer variability
$\hat{\mathbf{p}}_{\mathbf{i}}=\left[\hat{p}_{i}^{(1)},\ldots,\hat{p}_{i}^{(K)}\right]$
as the ground-truth. The label uncertainty information retained in the encoded
vector can help ensuring the model to capture disease features properly in the
presence of uncertainty. Particularly, when all graders arrive at the same
decision (i.e., no uncertainty exists), $\hat{\mathbf{p}}_{\mathbf{i}}$ equals
to the one-hot vector $\bar{\mathbf{p}}_{\mathbf{i}}$.

Uncertainty-guided focal loss. Inspired by the focal loss [10], we propose the
uncertainty-guided focal loss to replace the cross-entropy loss, which can
promote the HC-Net to pay more attentions to hard cases during training:

$\mathcal{L}_{UGF}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{K}\left[\left(1-\tilde{p}_{i}^{(k)}\right)^{g(u_{i})}\cdot\hat{p}_{i}^{(k)}\log\left(\tilde{p}_{i}^{(k)}\right)\right],\\\
$ (4)

where $\left(1-\tilde{p}_{i}^{(k)}\right)^{g(u_{i})}$ is imposed to adjust the
model’s focus according to the difficulty of given samples, i.e., attaching
more importance to the samples with lower prediction confidences and higher
uncertainty scores; $g(u_{i})=\gamma\cdot u_{i}$ is an adjusting function
relying on $u_{i}$ with a constant weight $\gamma$.

Uncertainty feature decoupling loss. According to the observations in Fig. 1,
the potential correlation between uncertainty and disease severity may bias
the screen model’s decision. Therefore, we propose the uncertainty feature
decoupling loss to disentangle disease features $f_{di}$ from uncertainty
features $f_{un}$ in the latent space.

$\mathcal{L}_{UFD}=-\frac{1}{N}\sum_{i}\max(0,h(u_{i})-\mathcal{D}(f_{di}(x_{i}),f_{un}(x_{i}))),$
(5)

where $f_{di}(x_{i})$ and $f_{un}(x_{i})$ are the flattened feature vectors of
image $x_{i}$ output by the last convolutional layer of the HC-Net and the
pre-trained US-Net, respectively; $h(u_{i})=\min(\alpha\cdot u_{i},1)$ imposes
a $u_{i}$-guided dynamic margin to ensure a lower bound of the two features’
distance, where the case is harder, the margin is larger. $\mathcal{D}(\cdot)$
is the Pearson distance [11].

Consequently, the final loss for optimizing the HC-Net is:

$\mathcal{L}_{joint}=\mathcal{L}_{UGF}+\mathcal{L}_{UFD}.$ (6)

Uncertainty-guided adaptive threshold. The practical population-scale disease
screening models are normally expected to make a binary decision in the
inference phase to support clinical recommendations, e.g., “non-referable”
versus “referable” cases [1, 2, 3, 4]. A common way is applying a fixed
threshold (e.g. $0.5$) to $\tilde{\mathbf{p}}_{i}$ to identify the best trade-
off between sensitivity and specificity of a classification model [12, 13,
14]. The fixed threshold could degrade the flexibility facing cases with
varying uncertainty. Instead, we design an uncertainty-guided adaptive
threshold for this process.

Given an unseen sample, first its uncertainty score $\tilde{u}$ is predicted
by the US-Net to decide whether the sample should be allocated to SC-Net or
HC-Net. For HC-Net screening, an adaptive threshold $\tau$ is applied to the
inference probability of the negative class:

$\tau=1-\frac{K-1}{K}(1-\tilde{u})^{\beta},$ (7)

where $K$ is the number of GON severity classes and $\beta$ is an adjustable
weight; $\tau$ rises when $\tilde{u}$ increases, meaning that for samples with
larger uncertainty, the model is allowed to be more inclined to classify them
as “referable”.

##  3 Experiments

###  3.1 Experimental settings

Datasets. The evaluations were performed on the fundus image-based GON
screening dataset _LabelMe_ [3]. Images were collected from various clinical
settings in China. Twenty-one ophthalmologists participated in the grading
process. Each image was assigned to different graders sequentially until three
consistent individual decisions were achieved. The five-class grading criteria
was applied [3], including unlikely, suspect and certain GON, and poor
quality/location. Images with poor quality/location were excluded and the
final dataset consists of $47012$ images. The dataset was randomly divided
into training, validation and testing sets, as detailed in Table 1. Note that
in all subsets, the mean uncertainty scores (Eq. 1) of suspect GON samples are
significantly higher than the other two classes, in line with our previous
observation O2. According to the observation in Fig.1.b, we empirically set
the predefined threshold of $u$ as $0.25$. The bottom two rows show the
details of simple cases and hard cases divided by this threshold. Fig.3 shows
some examples with different GON severity levels and uncertainty scores from
this dataset.

Fig. 3: Image examples with different GON severity levels and empirical uncertainty scores from the GON dataset. Table 1: The details of the GON dataset. MUS: mean uncertainty score. | Training set | Validation set | Testing set
---|---|---|---
| N | MUS | N | MUS | N | MUS
Unlikely | 27525 | 0.068 | 6147 | 0.068 | 1511 | 0.067
Suspect | 2338 | 0.410 | 518 | 0.423 | 94 | 0.412
Certain | 6947 | 0.162 | 1537 | 0.158 | 395 | 0.151
Simple cases | 28050 | 0.0003 | 6243 | 0.0003 | 1550 | 0.0003
Hard cases | 8760 | 0.452 | 1959 | 0.447 | 450 | 0.442

Settings. All images were central-cropped and downsized to $299*299$ with
pixel values rescaled to the range of $[0,1]$. We employed Xception [15] as
the backbone DNN for all sub-streams. The SC-Net and US-Net were initialized
with random weights. The HC-Net were initialized with the pre-trained SC-Net
weights in a transfer learning manner [4], due to that the shallow layers
learn generic low-level domain features that can be shared in similar tasks to
accelerate training. The batch size was $220$ and the epoch number was $100$.
The Adam optimizer [16] with an initial learning rate of $0.01$ plus an
adaptively-decay scheduler was adopted. The hyper-parameters
$\\{\gamma,\alpha,\beta\\}$ were empirically set as $\\{4,1.4,2\\}$ (the
decision process is detailed in the supplementary).

Evaluation metrics. For the multi-classification of GON severity, we computed
F1-score for each single severity class and the classification accuracy for
the overall performance. For the final binary GON screening (i.e., referable
vs. non-referable), sensitivity ($SE$), specificity ($SP$) and area under the
ROC curve ($AUC$) were evaluated.

###  3.2 Results

Ablation experiments. The ablation experiments were performed in two groups -
the whole testing set and only the hard testing cases, respectively. The
candidate models include a base model without any strategies (equivalent to
the SC-Net) and models with gradually-added strategies. The uncertainty-guided
adaptive threshold is excluded here since it is only applicable to the final
binary screening setting. Table 2 shows the $F_{1}$ scores of individual
classes and the overall classification accuracy. For all classes in both
groups, the $F_{1}$ scores and the overall accuracy generally increases with
adding more strategies to the base model. The final model combining three
strategies outperforms all other models in both groups, and the overall
improvement is much more significant in the hard case group
($72.89\%\rightarrow 84.22\%$), indicating the effectiveness of the proposed
strategies.

Table 2: Results of the ablation experiments. M1: Base model; M2: M1 \+ variability-based encoding; M3: M2 \+ uncertainty-guided focal loss; M4: M3 \+ uncertainty feature decoupling loss. The highest value is in bold. | The whole dataset | Hard cases only
---|---|---
| Unlikely | Suspect | Certain | Overall | Unlikely | Suspect | Certain | Overall
M1 | 95.84 | 30.99 | 88.83 | 92.11 | 80.89 | 31.40 | 76.60 | 72.89
M2 | 95.83 | 43.04 | 88.70 | 92.28 | 83.13 | 47.41 | 77.42 | 76.01
M3 | 96.82 | 55.90 | 90.80 | 93.94 | 85.59 | 58.33 | 79.42 | 79.33
M4 | 97.08 | 62.35 | 91.56 | 94.49 | 88.70 | 67.97 | 85.61 | 84.22
Table 3: Binary screening performances of the proposed multi-stream model (with three different backbones as shown in brackets) and two baselines. The highest value is in bold. | The whole dataset | Hard cases only
---|---|---
| $SE$ | $SP$ | $AUC$ | $SE$ | $SP$ | $AUC$
BCNet [3] | 87.59 | 92.96 | 95.54 | 80.23 | 85.83 | 87.80
DENet [17] | 92.51 | 95.96 | 96.79 | 84.23 | 88.10 | 90.84
CaliNet [18] | 90.11 | 91.27 | 93.74 | 81.01 | 86.22 | 89.04
Ours (Inception-V3) | 90.60 | 94.76 | 96.13 | 85.21 | 85.22 | 93.56
Ours (ResNet50) | 92.59 | 96.36 | 98.21 | 89.44 | 88.13 | 95.30
Ours (Xception) | 91.65 | 97.68 | 98.90 | 89.12 | 89.57 | 95.79

Screening performance. We tested the final binary GON screening performance of
the multi-stream model with all strategies (also including the uncertainty-
guided adaptive threshold). According to the screening criteria [3], unlikely
GON is regarded as “non-referable” while the other two classes are
“referable”. To show our model’s compatibility with different DNN structures,
we employed three DNNs as backbone, including Inceptivon-V3 [19], ResNet-50
[20] and Xception [15]. We also compared our method with two state-of-the-art
models for large-scale fundus image-based GON screening: an Inception-V3-based
binary classification network (BCNet) [3] and a disc-aware ensemble network
(DENet) [17], as well as a label uncertainty-based model calibration method
(CaliNet) [18]. Table 3 shows the results in two groups in terms of $SE,SP$
and $AUC$ scores. The models with three different backbones can all achieve
$AUC$ scores larger than $96.00\%$ in the whole dataset group and $93.50\%$ in
the hard case group, showcasing its satisfied DNN-compatibility. And their
performances are generally better or comparable than the two baselines in both
groups. Likewise, the improvement in the hard case group is much more
significant, e.g., the $AUC$ score is raised from $90.84\%$ from DENet, which
is the best result of baselines, to $95.79\%$ from our method with the
Xception backbone.

###  3.3 Conclusion

In this paper, we investigated how to leverage the label uncertainty existing
in medical image annotation as prior guidance to meliorate disease screening
models’ decisions. We developed a multi-stream model for the cases with
different uncertainty levels, where multiple uncertainty-guided strategies
were incorporated specifically for improvement on cases with high label
uncertainty. The evaluations conducted in a GON screening case study showed
the effectiveness of our method. Our method can benefit general DL models
developed on medical image dataset annotated by multiple graders.

Conflicts of Interest. This work was supported in part by a grant from the
Fundamental Research Funds of the State Key Laboratory of Ophthalmology
(303060202400362), National Natural Science Foundation of China (grant numbers
81420108008, 81271037, 82101171).

Compliance with Ethical Standards. This study was performed in line with the
principles of the Declaration of Helsinki. Approval was granted by the he
institutional review board of Zhongshan Ophthalmic Center (2017KYPJ049). The
review board determined that informed consent was not necessary in this study
due to the retrospective nature and fully anonymised use of images.

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Double Kerr-Schild spacetimes and the Newman-Penrose map

Kara Farnsworth1, Michael L. Graesser2, Gabriel Herczeg3

1 Départment de Physique Théorique, Université de Genève, CH-1211 Genève,
Switzerland

2 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA

3 Brown Theoretical Physics Center, Department of Physics, Brown University,
Providence, RI 02912, USA

###### Abstract

The Newman-Penrose map, which is closely related to the classical double copy,
associates certain exact solutions of Einstein’s equations with self-dual
solutions of the vacuum Maxwell equations. Here we initiate an extension of
the Newman-Penrose map to a broader class of spacetimes. As an example, we
apply principles from the Newman-Penrose map to associate a self-dual gauge
field to the Kerr-Taub-NUT-(A)dS spacetime and we show that the result agrees
with previously studied examples of classical double copies. The corresponding
field strength exhibits a discrete electric-magnetic duality that is distinct
from its (Hodge star) self-dual property.

###### Contents

  1. 1 Introduction
  2. 2 Newman-Penrose map for single Kerr-Schild spacetimes
  3. 3 Newman-Penrose map for double Kerr-Schild spacetimes
  4. 4 Kerr-Taub-NUT-(A)dS metric
    1. 4.1 Scalar functions
    2. 4.2 Self-dual gauge field
    3. 4.3 Electric-magnetic duality
  5. 5 Discussion

##  1 Introduction

The double copy is a family of correspondences between gravitational and gauge
theories with origins in string theory [1] that can be summarized
schematically as

$\displaystyle\textrm{gravity}=\textrm{gauge}\otimes\textrm{gauge}.$

Years after the initial discovery that closed string amplitudes can be
factorized into a product of open string amplitudes, a new manifestation of
the double copy was discovered in the form of color-kinematics duality, where
a kinematic identity analogous to the Jacobi identity for color factors was
introduced to argue that the Kawai-Lewellen-Tye (KLT) relations of [1] are
equivalent to a “diagram-by-diagram numerator ‘squaring’ relation with gauge
theory” [2]. This relationship greatly simplifies calculations in gravity, and
has provided useful insight into gauge theory calculations. It has been proven
at tree level for Yang-Mills theory and Einstein gravity amplitudes [3], and
in various limits and cases with extra symmetry at higher loop level [4, 5, 6,
7, 8, 9], however a general exact equivalence at all orders has yet to be
found. For a recent review on the subject of the double copy, see [10].

In an effort to understand how general this double copy relationship is, a
proliferation of similar approaches has been initiated in recent years,
particularly relating _exact_ non-perturbative classical solutions of gauge
theory and gravity [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62,
63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81,
82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92]. The first classical “Kerr-Schild”
double copy was introduced in [11], where stationary, vacuum solutions of
Einstein’s equations of Kerr-Schild type were mapped to solutions of the
vacuum Maxwell’s equations. Since then, many other classical double copies
have been proposed, including one called the Newman-Penrose map, which we
introduced in [12] and elaborated upon in [13] using the language of twistors.

The Newman-Penrose map associates a self-dual solution of the vacuum Maxwell
equations with a solution of the Einstein equations in Kerr-Schild form.
Unlike other versions of the double copy, including the Kerr-Schild double
copy, this map generically applies to gravitational solutions that are not
necessarily stationary or vacuum. When other versions of the double copy and
the Newman-Penrose map are both defined, they agree in all known examples,
although a general proof of their equivalence has not been conclusively
established.

In this note, we apply concepts from the Newman-Penrose map in an exploratory
manner to a family of spacetimes for which the Newman-Penrose map is not well
defined a priori. In particular, we consider the Kerr-Taub-NUT-(A)dS family,
which is Petrov type D, and which can be given a double Kerr-Schild form after
analytic continuation to Kleinian signature [93]. A straightforward
application of the Newman-Penrose map to each of the null, geodesic and shear-
free Kerr-Schild vectors leads to a complex self-dual gauge field whose real
part agrees with previous double copy studies of the same metric. The
resulting field strength is shown to exhibit an electric-magnetic duality,
that acts to relate the real (imaginary) parts of the electric and magnetic
fields to each other, and consequently is a symmetry of the solution that is
distinct from its (Hodge star) self-dual property.

##  2 Newman-Penrose map for single Kerr-Schild spacetimes

In this section we give a concise overview of the Newman-Penrose map [13, 12]
for single Kerr-Schild spacetimes in a general metric signature. We focus on
the basic ingredients of the map that must be generalized in order to
accommodate more complicated spacetimes.

The Newman-Penrose map consists of two main components:

  * •

$\Phi$: a scalar function, harmonic with respect to the background metric,
which arises in a particular choice of null tetrad that is suitable for a
generic111By ‘generic,’ we mean that a tetrad of the form (3) exists whenever
the Kerr-Schild vector is shear-free, geodesic and expanding. This includes
nearly all known vacuum Kerr-Schild spacetimes with the exception of the PP
wave spacetimes which are Petrov type N and have a non-expanding congruence.
Kerr-Schild spacetime.

  * •

$\hat{k}$: a differential spin-raising operator.

From these elements we can construct the gauge field

$\displaystyle A=\hat{k}\Phi.$ (1)

When $\Phi$ is harmonic with respect to the flat background metric, the field
strength $F=dA$ defined by the gauge field defined in (1) is self-dual, and as
a consequence automatically satisfies the vacuum Maxwell equations in flat
spacetime.

The original formulation of the Newman-Penrose map applies to single Kerr-
Schild spacetimes with a flat background

$\displaystyle g_{\mu\nu}=g^{(0)}_{\mu\nu}+V\ell_{\mu}\ell_{\nu},$ (2)

where $\ell_{\mu}$ is taken to be null with respect to both the full
($g_{\mu\nu}$) and background ($g^{(0)}_{\mu\nu}$) metrics. In Lorentzian
signature, if the stress tensor satisfies $T_{\mu\nu}\ell^{\mu}\ell^{\nu}=0$
(which includes vacuum solutions of Einstein’s equations with or without a
cosmological constant), then $\ell^{\mu}$ is also geodesic with respect to
both the full and background metrics (see theorem (32.1) of [94]). Moreover,
when the vacuum Einstein equations are satisfied, it can be shown that
$g_{\mu\nu}$ is algebraically special and $\ell^{\mu}$ is a repeated principle
null direction of the Weyl tensor [95], so the Goldberg Sachs theorem ensures
$\ell^{\mu}$ is also shear-free [96]. Here, we wish to consider more general
spacetimes, which may have an arbitrary metric signature, or even be complex,
and which may be sourced by more general stress tensors. However, we will
nevertheless assume that $\ell^{\mu}$ is geodesic and shear-free. Any such
spacetime admits a null tetrad of the form

$\displaystyle\begin{split}\ell&=du+\tilde{\Phi}d\zeta+\Phi
d\tilde{\zeta}+\Phi\tilde{\Phi}dv\,,\\\ n&=dv+\tfrac{1}{2}V\ell,\\\
m&=-(\tilde{\Phi}dv+d\tilde{\zeta})\\\ \tilde{m}&=-(\Phi
dv+d\zeta)\end{split}$ (3)

where $\ell=\ell_{\mu}dx^{\mu}$, etc. $u,v,\zeta,\tilde{\zeta}$ are null
coordinates222In Kleinian signature (which will be the focus of the paper)
these null coordinates are defined in terms of Cartesian and twisted
spheroidal coordinates by
$\displaystyle\begin{split}u&=\tfrac{1}{\sqrt{2}}(t-z)=\frac{1}{\sqrt{2}}(t-r\cos\theta)\\\
v&=\tfrac{1}{\sqrt{2}}(t+z)=\frac{1}{\sqrt{2}}(t+r\cos\theta)\\\
\quad\zeta&=\tfrac{1}{\sqrt{2}}(x+y)=\frac{1}{\sqrt{2}}(r+a)e^{\varphi}\sin\theta\\\
\tilde{\zeta}&=\tfrac{1}{\sqrt{2}}(x-y)=\frac{1}{\sqrt{2}}(r-a)e^{-\varphi}\sin\theta.\end{split}$
(4) , and $V$, $\Phi$ and $\tilde{\Phi}$ are scalar functions. In Lorentzian
signature, the tilded and untilded quantities are related by complex
conjugation, however in general signature they are independent.

The full metric can then be written

$\displaystyle g_{\mu\nu}$
$\displaystyle=n_{\mu}\ell_{\nu}+\ell_{\mu}n_{\nu}-m_{\mu}\tilde{m}_{\nu}-\tilde{m}_{\mu}m_{\nu}$
(5) $\displaystyle=g^{(0)}_{\mu\nu}+V\ell_{\mu}\ell_{\nu}$ (6)

with $g^{(0)}_{\mu\nu}dx^{\mu}dx^{\nu}=2(dudv-d\zeta d\tilde{\zeta})$. For
vacuum spacetimes, $V$ is determined by $\Phi$ and $\tilde{\Phi}$ and the mass
$M$, so $\Phi$ and $\tilde{\Phi}$ completely determine a vacuum solution up to
the constant $M$.

The shear-free and geodesic conditions imply that $\Phi$ satisfies the
following non-linear partial differential equations:

$\left(\Phi\partial_{\zeta}-\partial_{v}\right)\Phi=0,\qquad\big{(}\Phi\partial_{u}-\partial_{\tilde{\zeta}}\big{)}\Phi=0.$
(7)

Importantly, these equations also imply that $\Phi$ is harmonic with respect
to the flat background metric

$\displaystyle\Box_{0}\Phi=(\partial_{u}\partial_{v}-\partial_{\zeta}\partial_{\tilde{\zeta}})\Phi=0.$
(8)

This is also the (linearized) equation of motion for the zeroth copy biadjoint
scalar field. Pushing this intuition further, we can construct a gauge field
by acting on $\Phi$ with the spin-raising operator

$\hat{k}=-\frac{Q}{2\pi\epsilon_{0}}(dv\,\partial_{\zeta}+d\tilde{\zeta}\,\partial_{u}).$
(9)

The gauge field defined by

$\displaystyle A_{\textnormal{NP}}=\hat{k}\Phi$ (10)

automatically satisfies the self-dual vacuum Maxwell equations when $\Phi$ is
harmonic with respect to the flat background metric. It is worth noting that
the operator $\hat{k}$ used above in defining the Newman-Penrose map also
appears in the definition of the self-dual Kerr-Schild double copy originally
presented in [11] and further developed in subsequent works. However, here
$\Phi$ satisfies the non-linear PDEs (7), but we do not impose the Plebański
equation. Consequently, $h_{\mu\nu}=\hat{k}_{\mu}\hat{k}_{\nu}\Phi$ need not
correspond to a self-dual solution of Einstein’s equations. With this
construction, we can define a map from any single Kerr-Schild metric with an
expanding, shear-free, null geodesic congruence to a self-dual solution of the
flat space vacuum Maxwell equations. This includes all vacuum Kerr-Schild
spacetimes with an expanding repeated principle direction. However, unlike the
Kerr-Schild double copy, the map also applies to non-vacuum and non-stationary
spacetimes obeying these conditions. While there is no general proof of the
equivalence of the Newman-Penrose map and the Kerr-Schild double copy, they
have been shown to be equivalent where both formalisms apply [12].

##  3 Newman-Penrose map for double Kerr-Schild spacetimes

To explore the generality of this framework, we attempt to apply the map to
both double Kerr-Schild spacetimes and solutions in non-flat (maximally
symmetric) backgrounds:

$\displaystyle g_{\mu\nu}=\bar{g}_{\mu\nu}+\phi
k_{\mu}k_{\nu}+\psi\ell_{\mu}\ell_{\nu}.$ (11)

Here $\bar{g}_{\mu\nu}$ is maximally symmetric, and we will write it in a
manifestly conformally flat form for ease of comparison with the single Kerr-
Schild case. This metric is in double Kerr-Schild form, so we have two scalar
fields $\phi$ and $\psi$, as well as two vectors $k_{\mu}$ and $\ell_{\mu}$
which are assumed to each be null, geodesic, and shear-free with respect to
the background metric, from which it follows that they are null, shear-free
and geodesic with respect to the full metric. The vectors $k_{\mu}$ and
$\ell_{\mu}$ are also assumed to be orthogonal, which cannot be accomplished
for real metrics in Lorentzian signature. Hence, our discussion of double
Kerr-Schild metrics will be primarily in Kleinian signature unless otherwise
noted, although it applies equally well to complex spacetimes where metric
signature is not a well-defined notion.

In Kleinian signature, the background metric (which can be de Sitter (dS),
anti-de Sitter (AdS), or flat) can be written in null coordinates as [97]

$\displaystyle\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{2}{\Delta^{2}}\left(dUdV-
dXdY\right)$ (12)

with $\Delta=1-\frac{\lambda}{2}(UV-XY)$ and the sign of $\lambda$
corresponding to dS (positive), AdS (negative) or zero (flat). In this
signature, the coordinates $X$ and $Y$ are real and are not complex conjugates
of one another, but they become conjugate upon analytic continuation back to
Lorentzian signature.

In these coordinates, our null vectors can be normalized so that they take the
form

$\displaystyle k$
$\displaystyle=dU+\tilde{\Phi}_{k}\,dX+\Phi_{k}\,dY+\Phi_{k}\,\tilde{\Phi}_{k}\,dV$
(13) $\displaystyle\ell$
$\displaystyle=dU+\tilde{\Phi}_{\ell}\,dX+\Phi_{\ell}\,dY+\Phi_{\ell}\,\tilde{\Phi}_{\ell}\,dV.$
(14)

In analogy with the single Kerr-Schild case, the (real) scalar functions
$\Phi_{k}$ and $\Phi_{\ell}$ will be used to construct the gauge fields
associated with the vectors $k$ and $\ell$.

We now also have two spin-raising operators

$\displaystyle\hat{k}_{e}=\frac{-Q_{e}}{2\pi\epsilon_{0}}\left(dV\partial_{X}+dY\partial_{U}\right),\quad{}\hat{k}_{m}=\frac{-Q_{m}}{2\pi\mu_{0}}\left(dV\partial_{X}+dY\partial_{U}\right).$
(15)

The operator $\hat{k}_{e}$ is identical to $\hat{k}$ for single Kerr-Schild
metrics and the operator $\hat{k}_{m}$ is the magnetic analog. The total gauge
field obtained from this double Kerr-Schild metric via the Newman-Penrose map
is then the linear combination

$\displaystyle A_{\textnormal{NP}}$
$\displaystyle=\hat{k}_{e}\Phi_{\ell}+\hat{k}_{m}\Phi_{k}.$ (16)

As was the case for single Kerr-Schild metrics, the self-dual gauge field (16)
satisfies the vacuum Maxwell equations by construction, provided both
$\Phi_{k}$ and $\Phi_{\ell}$ are harmonic with respect to the flat metric.
Note that since the wave operator is not conformally invariant, $\Phi_{k}$ and
$\Phi_{\ell}$ are _not_ solutions of the wave equation on an (A)dS background.
However, since the vacuum Maxwell equations _are_ conformally invariant, the
gauge field (16) is a solution on both the flat and maximally symmetric
backgrounds. Generalizing this formalism to double Kerr-Schild metrics and
maximally symmetric backgrounds then amounts to understanding when this
harmonic condition holds.

Although we are not able to prove that the functions $\Phi_{\ell}$ and
$\Phi_{k}$ are harmonic in general double Kerr-Schild spacetimes, for the
Kerr-Taub-NUT-(A)dS family of solutions, the Newman-Penrose map outlined here
can be constructed explicitly in the affirmative. This is the subject of the
proceeding section, and here we state the main results. The function
$\Phi_{\ell}$ is shown to satisfy the following non-linear PDEs

$\displaystyle\begin{split}\left(\Phi_{\ell}\partial_{X}-\partial_{V}\right)\Phi_{\ell}=0,\quad{}\left(\Phi_{\ell}\partial_{U}-\partial_{Y}\right)\Phi_{\ell}=0\end{split}$
(17)

which are the same as the non-linear PDEs (7) but for the null coordinates of
the curved spacetime. This result is not surprising, but also non-trivial, at
least for these coordinates. It is not surprising because the (A)dS background
is conformally flat, and the geodesic and shear-free properties of $k$ and
$\ell$ are conformally invariant. Hence, both essential properties of $k$ and
$\ell$ required to define the Newman-Penrose map are preserved in passing to a
maximally symmetric background. On the other hand, the explicit form for
$\Phi_{\ell}$ has a complicated dependence on both the cosmological constant
and null coordinates, such that satisfying (17) is not trivial.

It follows from (17) that $\Phi_{\ell}$ is harmonic with respect to the flat
metric, and consequently the field strength obtained from the gauge potential
$A_{\ell}=\hat{k}\Phi_{\ell}$ is automatically self-dual and satisfies the
vacuum Maxwell equations in both flat and conformally flat spacetimes. The
real part of the gauge field $\Re(A_{\ell})$ is shown to be gauge-equivalent
to the original double copy prescription for obtaining a gauge field from the
Kerr-Schild form of the metric.

As shown below, a similar story holds for $\tilde{\Phi}_{\ell}$,
$\tilde{\Phi}_{k}$, and $\Phi_{k}$ \- they satisfy similar non-linear PDEs and
are therefore each harmonic with respect to the flat background metric. For
double Kerr-Schild spacetimes, a priori there could exist multiple Newman-
Penrose maps using different combinations of $\Phi_{\ell}$,
$\tilde{\Phi}_{\ell}$, $\tilde{\Phi}_{k}$, and $\Phi_{k}$, however for this
family of spacetimes these four functions are sufficiently related to one
other that the possible different self-dual gauge fields obtained from acting
with $\hat{k}_{e}$ and $\hat{k}_{m}$ are equivalent.

##  4 Kerr-Taub-NUT-(A)dS metric

A particular solution that admits a double Kerr-Schild form containing many
well known solutions as various limits is the Kerr-Taub-NUT-(A)dS metric. This
solution can be written in a double Kerr-Schild form that is real-valued in
$(2,2)$ signature after analytic continuation.

In Plebanski coordinates it is given as [93]

$\displaystyle
ds^{2}=d\bar{s}^{2}-\frac{2Mq}{q^{2}-p^{2}}\left[d\tilde{\tau}+p^{2}d\tilde{\sigma}\right]^{2}-\frac{2Np}{q^{2}-p^{2}}\left[d\tilde{\tau}+q^{2}d\tilde{\sigma}\right]^{2}.$
(18)

Here $M$ is the Schwarzschild mass, $N$ is the NUT charge, and

$\displaystyle d\bar{s}^{2}=$
$\displaystyle\frac{\bar{\Delta}_{p}}{q^{2}-p^{2}}\left[d\tilde{\tau}+q^{2}d\tilde{\sigma}\right]^{2}+\frac{\bar{\Delta}_{q}}{q^{2}-p^{2}}\left[d\tilde{\tau}+p^{2}d\tilde{\sigma}\right]^{2}+2\left[d\tilde{\tau}+q^{2}d\tilde{\sigma}\right]dp+2\left[d\tilde{\tau}+p^{2}d\tilde{\sigma}\right]dq$
(19)

is the background (A)dS metric. Here $\bar{\Delta}_{p}=\gamma-\epsilon
p^{2}+\lambda p^{4}$ and $\bar{\Delta}_{q}=-\gamma+\epsilon q^{2}-\lambda
q^{4}$ with $\lambda$ related to the scalar curvature via $R=12\lambda$, and
$\gamma$ and $\epsilon$ will be related to the rotational parameter $a$.

In these coordinates the two orthogonal Kerr-Schild vectors take especially
simple forms,

$\displaystyle\ell$ $\displaystyle=d\tilde{\tau}+p^{2}d\tilde{\sigma}$ (20)
$\displaystyle k$ $\displaystyle=d\tilde{\tau}+q^{2}d\tilde{\sigma}$ (21)

which hereafter are referred to as the “mass” and “NUT” vectors, respectively.

To guide our intuition, we first transform this to twisted spheroidal
coordinates using

$\displaystyle\begin{split}dq&=dr\\\ dp&=d(a\cos\theta)\\\
d\tilde{\tau}&=\frac{1}{(1-\lambda a^{2})}\left(dt-\frac{1}{(1-\lambda
r^{2})}dr+ad\varphi\right)-\frac{a\cos^{2}\theta}{(1-\lambda
a^{2}\cos^{2}\theta)\sin\theta}d\theta\\\ d\tilde{\sigma}&=\frac{1}{(1-\lambda
a^{2})}\left(-\lambda dt+\frac{\lambda}{(1-\lambda
r^{2})}dr-\frac{1}{a}d\varphi\right)+\frac{1}{(1-\lambda
a^{2}\cos^{2}\theta)a\sin\theta}d\theta.\end{split}$ (22)

With the choices $\gamma=a^{2}$ and $\epsilon=1+\lambda a^{2}$ the background
metric becomes

$\displaystyle\begin{split}d\bar{s}^{2}=&\ \frac{(1-\lambda r^{2})(1-\lambda
a^{2}\cos^{2}\theta)}{(1-\lambda a^{2})}dt^{2}-\frac{(1-\lambda
a^{2}\cos^{2}\theta)}{(1-\lambda r^{2})(1-\lambda a^{2})}dr^{2}\\\
&-\frac{(r^{2}-a^{2}\cos^{2}\theta)}{(1-\lambda
a^{2}\cos^{2}\theta)}d\theta^{2}+\frac{(r^{2}-a^{2})\sin^{2}\theta}{(1-\lambda
a^{2})}d\varphi^{2}+2\frac{a\sin^{2}\theta}{(1-\lambda a^{2})}d\varphi
dr.\end{split}$ (23)

After analytic continuation to Lorentzian signature ($a\rightarrow-ia,\
\varphi\rightarrow i\varphi$), this metric reduces to known results in
particular limits: taking $\lambda\rightarrow 0$ gives the Minkowski metric in
twisted spheroidal coordinates used in [12], and taking $a\rightarrow 0$ gives
the (A)dS metric in the familiar static spheroidal coordinates. The parameter
$a$ is associated with the angular momentum of the black hole, as in the case
of the Kerr metric. In these coordinates, the mass vector $\ell$ and NUT
vector $k$ become

$\displaystyle\ell$ $\displaystyle=\frac{(1-\lambda
a^{2}\cos^{2}\theta)}{(1-\lambda a^{2})}dt-\frac{(1-\lambda
a^{2}\cos^{2}\theta)}{(1-\lambda r^{2})(1-\lambda
a^{2})}dr+\frac{a\sin^{2}\theta}{(1-\lambda a^{2})}d\varphi$ (24)
$\displaystyle k$ $\displaystyle=\frac{(1-\lambda r^{2})}{(1-\lambda
a^{2})}dt-\frac{1}{(1-\lambda
a^{2})}dr+\frac{(r^{2}-a^{2}\cos^{2}\theta)}{a\sin\theta(1-\lambda
a^{2}\cos^{2}\theta)}d\theta-\frac{(r^{2}-a^{2})}{a(1-\lambda
a^{2})}d\varphi.$ (25)

Once again we see that in the limit $\lambda\rightarrow 0$, $\ell$ matches
with the Schwarzschild ($a=0$) or Kerr ($a\neq 0$) vectors used in [12] after
analytic continuation. For the vector $k$, the $a\rightarrow 0$ limit is
singular, however the functions $\Phi_{k}$ and $\tilde{\Phi}_{k}$ are not, as
shown below.

###  4.1 Scalar functions

In order to extract the scalar functions analogous to $\Phi$ in the single
Kerr-Schild case, we perform a second coordinate transformation

$\displaystyle\begin{split}U&=\frac{\Delta}{\sqrt{2}}\bigg{[}\frac{1}{\sqrt{\lambda}}\frac{\sqrt{1-\lambda
r^{2}}}{\sqrt{1-\lambda a^{2}}}\sqrt{1-\lambda
a^{2}\cos^{2}\theta}\sinh(\sqrt{\lambda}t)-r\cos\theta\bigg{]}\\\
V&=\frac{\Delta}{\sqrt{2}}\bigg{[}\frac{1}{\sqrt{\lambda}}\frac{\sqrt{1-\lambda
r^{2}}}{\sqrt{1-\lambda a^{2}}}\sqrt{1-\lambda
a^{2}\cos^{2}\theta}\sinh(\sqrt{\lambda}t)+r\cos\theta\bigg{]}\\\
X&=\frac{\Delta}{\sqrt{2}}\frac{1}{\sqrt{(1-\lambda
a^{2})}}(r-a)e^{\varphi}\sin\theta\\\
Y&=\frac{\Delta}{\sqrt{2}}\frac{1}{\sqrt{(1-\lambda
a^{2})}}(r+a)e^{-\varphi}\sin\theta\end{split}$ (26)

with $\Delta=1-\frac{\lambda}{2}(UV-XY)$ and $r$ defined by the equation

$\displaystyle\frac{(V-U)^{2}}{2r^{2}}+\frac{2XY}{r^{2}-a^{2}}(1-\lambda
a^{2})=\Delta^{2}.$ (27)

The limit $\lambda\rightarrow 0$ recovers the flat spacetime transformation to
null coordinates (4). In these null coordinates, the maximally symmetric
background metric becomes

$\displaystyle d\bar{s}^{2}=\frac{2}{\Delta^{2}}\left(dUdV-dXdY\right)$ (28)

which is conformally flat and reduces to the flat space metric in null
coordinates for $\lambda\rightarrow 0$.

The mass and NUT vectors written in these coordinates are

$\displaystyle\ell=$ $\displaystyle\
\mathcal{N}_{\ell}\left[dU+\tilde{\Phi}_{\ell}dX+\Phi_{\ell}dY+\Phi_{\ell}\tilde{\Phi}_{\ell}dV\right]$
(29) $\displaystyle k=$ $\displaystyle\
\mathcal{N}_{k}\left[dU+\tilde{\Phi}_{k}dX+\Phi_{k}dY+\Phi_{k}\tilde{\Phi}_{k}dV\right]$
(30)

with normalization factors

$\displaystyle\mathcal{N}_{\ell}$
$\displaystyle=\frac{(1+\cos\theta)}{2\Delta(1-\lambda
r^{2})}\left[\sqrt{2}-\lambda
rV-\frac{\lambda\Delta}{\sqrt{2}}(1-\cos\theta)\frac{(r^{2}-a^{2})}{1-\lambda
a^{2}}\right]$ (31) $\displaystyle\mathcal{N}_{k}$
$\displaystyle=\frac{(r+a)}{2a\Delta(1-\lambda
a^{2}\cos^{2}\theta)}\left(\sqrt{2}-\lambda
aV\cos\theta-\frac{\lambda\Delta}{\sqrt{2}}(1-\cos^{2}\theta)\frac{a(r-a)}{(1-\lambda
a^{2})}\right)$ (32)

and scalar functions

$\displaystyle\Phi_{\ell}$
$\displaystyle=\frac{\left[\frac{1}{\sqrt{2}}\lambda(r-a)(U+V+\sqrt{2}\Delta
r)-2(1-\lambda ar)\right]}{2\mathcal{N}_{\ell}\Delta^{2}(r-a)(1-\lambda
r^{2})}X$ (33) $\displaystyle\tilde{\Phi}_{\ell}$
$\displaystyle=\frac{\left[\frac{1}{\sqrt{2}}\lambda(r+a)(U+V+\sqrt{2}\Delta
r)-2(1+\lambda ar)\right]}{2\mathcal{N}_{\ell}\Delta^{2}(r+a)(1-\lambda
r^{2})}Y$ (34)

$\displaystyle\Phi_{k}$ $\displaystyle=\frac{\left[\frac{1}{\sqrt{2}}\lambda
a(1-\cos\theta)\left(U+V+\sqrt{2}\Delta a\cos\theta\right)+2(1-\lambda
a^{2}\cos\theta)\right]}{2\mathcal{N}_{k}\Delta^{2}a(1-\lambda
a^{2}\cos^{2}\theta)(1-\cos\theta)}X$ (35) $\displaystyle\tilde{\Phi}_{k}$
$\displaystyle=\frac{\left[\frac{1}{\sqrt{2}}\lambda
a(1+\cos\theta)\left(U+V+\sqrt{2}\Delta a\cos\theta\right)-2(1+\lambda
a^{2}\cos\theta)\right]}{2\mathcal{N}_{k}\Delta^{2}a(1-\lambda
a^{2}\cos^{2}\theta)(1+\cos\theta)}Y.$ (36)

It can be shown by a direct computation that $\Phi_{\ell}$ satisfies the non-
linear PDEs (17).

These four functions are not independent, but are all related to each other by
a sequence of discrete transformations. Although not obvious from the form
written above, the functions $\tilde{\Phi}_{\ell}$ and $\tilde{\Phi}_{k}$ are
identical. By inspection, we also have

$\displaystyle\begin{split}\tilde{\Phi}_{\ell}(a,U,V,X,Y)&=\Phi_{\ell}(-a,U,V,Y,X),\\\
\tilde{\Phi}_{k}(a,U,V,X,Y)&=-\Phi_{k}(a,-U,-V,Y,X),\end{split}$ (37)

or in spheroidal coordinates

$\displaystyle\begin{split}\tilde{\Phi}_{\ell}(a,t,r,\cos\theta,\varphi)&=\Phi_{\ell}(-a,t,r,\cos\theta,-\varphi),\\\
\tilde{\Phi}_{k}(a,t,r,\cos\theta,\varphi)&=\Phi_{k}(a,-t,-r,\cos\theta,-\varphi).\end{split}$
(38)

The functions $\Phi_{\ell}$ and $\Phi_{k}$ which we will use for the Newman-
Penrose map are then related by

$\displaystyle\Phi_{\ell}(a,t,r,\cos\theta,\varphi)=\Phi_{k}(-a,-t,-r,\cos\theta,\varphi).$
(39)

It then follows that $\Phi_{k}$ satisfies the same non-linear PDEs (17) as
$\Phi_{\ell}$, while $\tilde{\Phi}_{\ell}=\tilde{\Phi}_{k}$ satisfies the non-
linear PDEs obtained from (17) by interchanging $X$ with $Y$. Each are
therefore harmonic with respect to the flat metric, and for these double Kerr-
Schild spacetimes the Newman-Penrose map can be defined. This means that the
gauge field constructed from these scalar functions via the Newman-Penrose map
will automatically be a self-dual solution of the vacuum Maxwell equations in
both flat and conformally flat backgrounds. The relation (39) also implies
that the two parts of the gauge field are related via

$\displaystyle A_{\ell}(Q_{e},a,t,r,\cos\theta,\varphi)$
$\displaystyle=A_{k}(Q_{m},-a,-t,-r,\cos\theta,\varphi)$ (40)

with $A_{\ell}=\hat{k}_{e}\Phi_{\ell}$ and $A_{k}=\hat{k}_{m}\Phi_{k}$.

###  4.2 Self-dual gauge field

To obtain the self-dual gauge field from this metric via the Newman-Penrose
map, we act the operator $\hat{k}$ (with appropriate charges) defined in (15)
on the functions $\Phi_{k}$ and $\Phi_{\ell}$ from the previous section, i.e.

$\displaystyle A_{\textnormal{NP}}$
$\displaystyle=\hat{k}_{e}\hat{\Phi}_{\ell}+\hat{k}_{m}\hat{\Phi}_{k}.$ (41)

We will set $\epsilon_{0}=\mu_{0}=1$ from now on for ease of notation. In
practice, we compute $A_{\ell}$ and use the relationship (40) to find $A_{k}$.

In order to compare with known results, we then analytically continue the
gauge field to Lorentzian signature using $a\rightarrow-ia,\
\varphi\rightarrow i\varphi,\ Q_{m}\rightarrow iQ_{m}$. Our total gauge field
in Lorentzian signature is

$\displaystyle\begin{split}A_{\textnormal{NP}}=&\
\frac{(Q_{e}+iQ_{m})}{4\pi(r+ia\cos\theta)}\left[\left(1+\lambda
iar\cos\theta\right)dt+dr+(r\cos\theta+ia)id\varphi\right]\\\
&+A_{\textnormal{NP, gauge}}\end{split}$ (42)

with the pure gauge term

$\displaystyle\begin{split}A_{\textnormal{NP, gauge}}=&\
\frac{(Q_{e}+iQ_{m})}{2\pi}\frac{\sqrt{2}}{4r\Omega}\frac{(1-\lambda
r^{2})}{(1+\lambda a^{2})}\bigg{(}2(1+\lambda a^{2})+i\lambda\Delta
a(r+ia)(1-\cos\theta)\bigg{)}\\\ &\ \ \ \ \times\left[dt-\frac{1}{(1-\lambda
r^{2})}dr-\frac{ia}{(1+\lambda a^{2}\cos^{2}\theta)}d(\cos\theta)\right]\\\
&-\frac{(Q_{e}+iQ_{m})}{4\pi r}\left[dt-ia\frac{1}{(1+\lambda
a^{2}\cos^{2}\theta)}d(\cos\theta)\right]\\\
&-\frac{(Q_{e}+iQ_{m})}{2\pi}\frac{1}{2}id\varphi-\frac{(Q_{e}+iQ_{m})}{2\pi}\frac{d\Delta}{(2-\Delta)}\\\
&-\frac{(Q_{e}+iQ_{m})}{2\pi}\left[\frac{\sinh(\sqrt{\lambda}t)}{\cosh(\sqrt{\lambda}t)}\sqrt{\lambda}dt\right]+\frac{(Q_{e}+iQ_{m})}{2\pi}\frac{1}{2r}\left[\frac{(1+\lambda
r^{2})}{(1-\lambda r^{2})}dr\right]\\\
&+\frac{(Q_{e}+iQ_{m})}{2\pi}\left[\frac{(1-\lambda
a^{2}\cos\theta)}{2(1+\cos\theta)(1+\lambda
a^{2}\cos^{2}\theta)}d(\cos\theta)\right]\end{split}$ (43)

and

$\displaystyle\Omega$
$\displaystyle=\frac{1}{\sqrt{2}}\bigg{[}2-\sqrt{\lambda}r(2-\Delta)\frac{\sinh(\sqrt{\lambda}t)}{\cosh(\sqrt{\lambda}t)}-\lambda\Delta
r^{2}-\lambda\Delta a^{2}(1-\cos\theta)\frac{(1-\lambda r^{2})}{(1+\lambda
a^{2})}\bigg{]}.$ (44)

Finally, looking separately at the real and imaginary parts, we get

$\displaystyle\begin{split}\Re(A_{\textnormal{NP}})=&\
\frac{Q_{e}r}{4\pi(r^{2}+a^{2}\cos^{2}\theta)}\left[\left(1+\lambda
a^{2}\cos^{2}\theta\right)dt+dr-a\sin^{2}\theta d\varphi\right]\\\
&+\frac{Q_{m}a\cos\theta}{4\pi(r^{2}+a^{2}\cos^{2}\theta)}\left[(1-\lambda
r^{2})dt+dr-\frac{(r^{2}+a^{2})}{a}d\varphi\right]\end{split}$ (45)

and

$\displaystyle\begin{split}\Im(A_{\textnormal{NP}})=&\
\frac{Q_{m}r}{4\pi(r^{2}+a^{2}\cos^{2}\theta)}\left[(1+\lambda
a^{2}\cos^{2}\theta)dt+dr-a\sin^{2}\theta d\varphi\right]\\\
&-\frac{Q_{e}a\cos\theta}{4\pi(r^{2}+a^{2}\cos^{2}\theta)}\left[(1-\lambda
r^{2})dt+dr-\frac{(r^{2}+a^{2})}{a}d\varphi\right].\end{split}$ (46)

For comparison, the Kerr-Schild double copy of a metric in double Kerr-Schild
form (11) is defined as [14]

$\displaystyle A_{\textnormal{KS}}=\psi\ell+\phi k.$ (47)

For the Kerr-Taub-NUT-(A)dS metric in $(2,2)$ signature (18), we can perform a
slightly modified version of the coordinate transformation (22)
($t\rightarrow-t$) which does not change the background metric

$\displaystyle\begin{split}dq&=dr\\\ dp&=d(a\cos\theta)\\\
d\tilde{\tau}&=\frac{1}{(1-\lambda a^{2})}\left(-dt-\frac{1}{(1-\lambda
r^{2})}dr+ad\varphi\right)-\frac{a\cos^{2}\theta}{(1-\lambda
a^{2}\cos^{2}\theta)\sin\theta}d\theta\\\ d\tilde{\sigma}&=\frac{1}{(1-\lambda
a^{2})}\left(\lambda dt+\frac{\lambda}{(1-\lambda
r^{2})}dr-\frac{1}{a}d\varphi\right)+\frac{1}{(1-\lambda
a^{2}\cos^{2}\theta)a\sin\theta}d\theta.\end{split}$ (48)

In these coordinates the two scalar functions $\phi$ and $\psi$ are

$\displaystyle\psi$
$\displaystyle=\frac{2Mq}{q^{2}-p^{2}}=\frac{2Mr}{(r^{2}-a^{2}\cos^{2}\theta)}$
(49) $\displaystyle\phi$
$\displaystyle=\frac{2Np}{q^{2}-p^{2}}=\frac{2Na\cos\theta}{(r^{2}-a^{2}\cos^{2}\theta)},$
(50)

and the vectors are

$\displaystyle\ell$ $\displaystyle=\frac{(1-\lambda
a^{2}\cos^{2}\theta)}{(1-\lambda a^{2})}dt+\frac{(1-\lambda
a^{2}\cos^{2}\theta)}{(1-\lambda r^{2})(1-\lambda
a^{2})}dr-\frac{a\sin^{2}\theta}{(1-\lambda a^{2})}d\varphi$ (51)
$\displaystyle k$ $\displaystyle=\frac{(1-\lambda r^{2})}{(1-\lambda
a^{2})}dt+\frac{1}{(1-\lambda
a^{2})}dr-\frac{(r^{2}-a^{2}\cos^{2}\theta)}{a\sin\theta(1-\lambda
a^{2}\cos^{2}\theta)}d\theta-\frac{(r^{2}-a^{2})}{a(1-\lambda
a^{2})}d\varphi.$ (52)

After analytic continuation to Lorentzian signature ($a\rightarrow-ia,\
\varphi\rightarrow i\varphi,\ N\rightarrow iN$), the Kerr-Schild gauge field
for this metric is then (mapping $M\rightarrow Q_{e}/8\pi$ and $N\rightarrow
Q_{m}/8\pi$)

$\displaystyle\begin{split}A_{\textnormal{KS}}=&\
\frac{Q_{e}r}{4\pi(r^{2}+a^{2}\cos^{2}\theta)(1+\lambda
a^{2})}\bigg{[}(1+\lambda a^{2}\cos^{2}\theta)dt+dr-a\sin^{2}\theta
d\varphi\bigg{]}\\\
&+\frac{Q_{m}a\cos\theta}{4\pi(r^{2}+a^{2}\cos^{2}\theta)(1+\lambda
a^{2})}\bigg{[}(1-\lambda
r^{2})dt+dr-\frac{(r^{2}+a^{2})}{a}d\varphi\bigg{]}\\\ &+A_{\textnormal{KS,
gauge}}\end{split}$ (53)

with

$\displaystyle A_{\textnormal{KS, gauge}}$
$\displaystyle=\frac{Q_{e}}{4\pi(1+\lambda a^{2})}\bigg{[}\frac{\lambda
r}{(1-\lambda
r^{2})}dr\bigg{]}-\frac{iQ_{m}}{4\pi}\bigg{[}\frac{\cos\theta}{\sin\theta(1+\lambda
a^{2}\cos^{2}\theta)}d\theta\bigg{]}.$ (54)

This agrees exactly with the real part of the Newman-Penrose map (45) up to an
overall constant factor of $\frac{1}{(1+\lambda a^{2})}$ and pure gauge terms.

In the $\lambda\rightarrow 0$ limit the metric is the Kerr-Taub-NUT solution,
and the gauge field becomes

$\displaystyle\begin{split}\Re(A_{\textnormal{NP}})\bigg{|}_{\lambda\rightarrow
0}=&\
\frac{Q_{e}r}{4\pi(r^{2}+a^{2}\cos^{2}\theta)}\left[dt+dr-a\sin^{2}\theta
d\varphi\right]\\\
&+\frac{Q_{m}a\cos\theta}{4\pi(r^{2}+a^{2}\cos^{2}\theta)}\left[dt+dr-\frac{(r^{2}+a^{2})}{a}d\varphi\right]\end{split}$
(55)

$\displaystyle\begin{split}\Im(A_{\textnormal{NP}})\bigg{|}_{\lambda\rightarrow
0}=&\
\frac{Q_{m}r}{4\pi(r^{2}+a^{2}\cos^{2}\theta)}\left[dt+dr-a\sin^{2}\theta
d\varphi\right]\\\
&-\frac{Q_{e}a\cos\theta}{4\pi(r^{2}+a^{2}\cos^{2}\theta)}\left[dt+dr-\frac{(r^{2}+a^{2})}{a}d\varphi\right].\end{split}$
(56)

The electric part of this solution in this limit is exactly equivalent to the
one found in [12] (known as $\sqrt{\textnormal{Kerr}}$) for the Kerr metric,
and the addition of the NUT charge simply adds the magnetic dual part. The
electric part of this gauge field was shown in [11, 29] to be sourced by an
axisymmetric electric charge distribution rotating at a uniform rate about the
$z$-axis (where $z=r\cos\theta$). We expect the magnetic part of this field to
be similarly sourced by an axisymmetric magnetic charge distribution rotating
at a uniform rate about the $z$-axis.

In the $a\rightarrow 0$ limit, the gravitational solution is the Taub-
NUT-(A)dS metric. The gauge solution can be written (with the
addition/subtraction of pure gauge terms) as

$\displaystyle\Re(A_{\textnormal{NP}})\bigg{|}_{a\rightarrow 0}=$
$\displaystyle\ \frac{Q_{e}}{4\pi
r}dt+\frac{Q_{m}}{4\pi}(1-\cos\theta)d\varphi$ (57)
$\displaystyle\Im(A_{\textnormal{NP}})\bigg{|}_{a\rightarrow 0}=$
$\displaystyle\ \frac{Q_{m}}{4\pi
r}dt-\frac{Q_{e}}{4\pi}(1-\cos\theta)d\varphi.$ (58)

The real part of this solution is a dyon (a Coulomb charge plus a magnetic
monopole). This is equal to the gauge solution found via the Kerr-Schild
double copy in [14] in flat space. The electric part of this solution also
agrees with the Schwarzschild-(A)dS solution (up to pure gauge terms) found
via the Kerr-Schild double copy in [20, 21].

In this limit, the $\lambda$ dependence of the solution completely drops out.
Although a priori in the $a\rightarrow 0$ limit the $\Phi$’s appear to depend
on $\lambda$ through $\Delta$, this dependence is completely eliminated upon
expressing $\Phi$’s in terms of null coordinates ($U,V,X,Y$) only. For the
$\Phi$’s depend on $r$ and $\Delta$ only in the combination $\Delta r$, and it
is only this combination that appears in the quadratic equation relating $r$
to the null coordinates.

In other words,

$\displaystyle\Phi(a=0,\lambda,t,r,\cos\theta,\varphi)$
$\displaystyle=\Phi(a=0,\lambda=0,U,V,X,Y).$ (59)

is independent of $\lambda$. Again, this result isn’t too surprising given
that $\Phi$ describes a shear-free null geodesic congruence, physical
properties that are conformally invariant.

###  4.3 Electric-magnetic duality

The field strength $F_{\textnormal{NP}}=dA_{\textnormal{NP}}$ is of the form

$\displaystyle F_{\textnormal{NP}}=(Q_{e}+iQ_{m})F$ (60)

where both $F_{\textnormal{NP}}$ and $F$ are self-dual, with
$F_{\textnormal{NP}}=i\star_{0}F_{\textnormal{NP}}$ and $F=i\star_{0}F$. This
solution exhibits a discrete electric-magnetic duality that acts on the
charges, separate from its Hodge star self-dual property. From the self-dual
property of $F$ it follows that

$\displaystyle F=f+i\star_{0}f$ (61)

for some real two-form $f$, and also that

$\displaystyle
F_{\textnormal{NP}}=Q_{e}f-Q_{m}\star_{0}f+i(Q_{m}f+Q_{e}\star_{0}f).$ (62)

This result implies a relation between the complex electric $E$ and magnetic
$B$ fields (vector notation suppressed) of $F_{\textnormal{NP}}$ and the real-
valued electric $E_{f}$ and magnetic $B_{f}$ components of $f$, namely

$\displaystyle\begin{split}\Re{(E)}&=Q_{e}E_{f}-Q_{m}B_{f}\\\
\Re{(B)}&=Q_{e}B_{f}+Q_{m}E_{f}\\\ \Im{(E)}&=Q_{m}E_{f}+Q_{e}B_{f}\\\
\Im{(B)}&=Q_{m}B_{f}-Q_{e}E_{f}\end{split}$ (63)

where $\star E=B$ and $\star B=-E$ has been used. Comparing these equations
gives $\Im{(E)}=\Re{(B)}$ and $\Im{(B)}=-\Re{(E)}$, which taken together is
none other than the statement that $F_{\textnormal{NP}}$ is a self-dual
solution, i.e., $E=iB$. Note however that $\Re{(E)}$ and $-\Re{(B)}$, which
are not related by the Hodge star operation, are obtained from $\Re{(B)}$ and
$\Re{(E)}$, respectively, by $(Q_{e},Q_{m})\rightarrow(-Q_{m},Q_{e})$, which
is a counter-clockwise $\mathit{U}(1)$ rotation on the electric and magnetic
charge by an angle $\theta=\pi/2$.

Given such a self-dual $F_{\textnormal{NP}}$, one can take
$\Re{(F_{\textnormal{NP}})}$ alone (or $\Im{(F_{\textnormal{NP}})}$), forget
the other, and obtain a non-self dual solution to the source-free Maxwell’s
equation. The resulting field strength will also exhibit electric-magnetic
duality.

For gravitational solutions on a flat background with vanishing rotational
parameter $a$, the object on the gauge field side describes a dyon which has
manifest electric-magnetic duality. Here we find this electric-magnetic
duality of the solution carries over to maximally symmetric spacetimes,
including those having non-vanishing rotational parameter. The gravitational
analog of electric-magnetic duality has been explored in a general setting
using the Kerr-Taub-NUT spacetime as a case study in [98].

##  5 Discussion

In this paper we applied the intuition of the Newman-Penrose map to more
general spacetimes, including those in double Kerr-Schild form and on
maximally symmetric backgrounds. Although we do not have a rigorous
formulation of the Newman-Penrose map for general double Kerr-Schild metrics,
for a particular example metric we are able to reproduce the gauge field from
the Kerr-Schild and Weyl double copies, as shown in table 1.

Gravity solution | Gauge solution | KS/Weyl DC | NP Map
---|---|---|---
Schwarzschild | Coulomb | [11] | [12]
Kerr | rotating disc of charge | [11] | [12]
Photon Rocket | Lienard-Weichert potential | [15] | [12]
Taub-NUT | dyon | [14] | this work
Kerr-Taub-NUT | rotating dyon | [14, 35] | this work
Schwarzschild-(A)dS | Coulomb in (A)dS | [20, 21] | this work
Kerr-(A)dS | rotating disc in (A)dS | [21] | this work
Kerr-Taub-NUT-(A)dS | aligned fields in (A)dS | [78] | this work
Table 1: Table enumerating example metrics and their gauge fields obtained via
either the Kerr-Schild/Weyl double copy or the Newman-Penrose map.

In particular, we studied the Kerr-Taub-NUT-(A)dS metric which can be put into
double Kerr-Schild form after analytic continuation to Kleinian signature. In
analogy with the Newman-Penrose map for single Kerr-Schild spacetimes with a
flat background, we obtained scalar functions $\Phi_{\ell}$ and $\Phi_{k}$ for
each of the null vectors by identifying null coordinates that make the
conformal flatness and Lorentz invariance of the maximally symmetric
background metric manifest. These scalar functions are shown to satisfy the
non-linear PDEs sufficient for them to be harmonic in flat spacetime (but not
conformally flat spacetime).

Acting with our spin-raising operator gives a gauge field, which, because of
the harmonic conditions on the $\Phi_{\ell}$ and $\Phi_{k}$, satisfies the
vacuum Maxwell equations in _both_ flat and conformally flat spacetime. We
then compare our results with those of the Kerr-Schild and Weyl double copies
and show they match up to pure gauge terms. We also find that the field
strength obtained from the Newman-Penrose map gauge field displays a discrete
electric-magnetic duality.

The fact that $\Phi_{\ell}$ satisfies the non-linear PDEs implies it be can be
obtained from the zero set of a holomorphic function $F$ (the “Kerr function”)
of twistor variables, one of the consequences of Kerr’s Theorem [99, 100, 101,
102]. What this function is in general for the Kerr-Taub-NUT-(A)dS family of
solutions is presently unknown. In the $a=0,\lambda\neq 0$ limit it is seen to
be identical to the function $F_{Sch}$ of the Schwarzschild spacetime, and for
$\lambda=0$ it is a quadratic function of $\Phi_{\ell}$ and invariants of the
non-linear PDEs (and obtainable from $F_{Sch}$ upon applying the Newman-Janis
trick). Knowing that the Kerr function is holomorphic in twistor variables is
insufficient information to fully fix it. For more general metrics of the
single Kerr-Schild metric form, the Kerr function is obtained by integrating
the Einstein equations [103] (see also,[104]). Finding the Kerr function in
general for double Kerr-Schild metrics, or even for the subset of the Kerr-
Taub-NUT-(A)dS family of solutions, presumably involves similar steps.

Our results suggest that the Newman-Penrose map might be generalizable to all
double Kerr-Schild spacetimes admitting an integrable distribution of totally
null vectors. For flat backgrounds, these can all be solved exactly [105]. The
tetrad can be generalized to accommodate double Kerr-Schild spacetimes, but
this only leads to a subset of the non-linear PDEs being satisfied, so there
remains a conceptual obstruction to defining the Newman-Penrose map for
general double Kerr-Schild spacetimes.

These results provide further evidence that the Newman-Penrose map is a
manifestation of the double copy, with the potential to generically study non-
vacuum and non-stationary solutions. In future work we hope to understand the
relationship between the Newman-Penrose map and the Kerr-Schild double copy,
which a priori appear very different, yet agree in all known examples. This
promises to provide insight into the structure of the double copy more
generally, and the gravitational and gauge theories to which it applies. The
apparently close relationships between the Kerr-Schild and Weyl double copies
and the Newman-Penrose map also serve to highlight a shortcoming that all
three constructions share—namely that they map exact gravitational solutions
to _abelian_ gauge fields, while the presumably more fundamental double copy
for amplitudes necessarily relates graviton amplitudes with _gluon_
amplitudes. Generalizing the Newman-Penrose map to the non-abelian case could
potentially illuminate the structures underlying both classical and quantum
mechanical double copies.

## Acknowledgements

The work of KF is supported by the Swiss National Science Foundation under
grant no. 200021-205016. KF also acknowledges support from Simons Foundation
Award Number 658908. The work of MG is supported by the LDRD program at Los
Alamos National Laboratory and by the U.S. Department of Energy, Office of
High Energy Physics, under Contract No. DE-AC52-06NA25396. The Los Alamos
preprint number for this manuscript is LA-UR-23-26863.

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