$C_{\alpha\beta\gamma\delta}=\Psi_{ABCD}\bar{\epsilon}_{A^{\prime}B^{\prime}}\bar{\epsilon}_{C^{\prime}D^{\prime}}+\bar{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\epsilon_{AB}\epsilon_{CD}\,.$
(197)

The relation between the decompositions (196) and (197) is key to the Weyl
double copy which, given additionally the complex “zeroth copy” scalar $S$
gives a Weyl tensor (197) in terms of a 2-form (196) as (4)

$C^{(\rm
Weyl\,DC)}_{\alpha\beta\gamma\delta}=\frac{1}{S}f_{(AB}f_{CD)}\bar{\epsilon}_{A^{\prime}B^{\prime}}\bar{\epsilon}_{C^{\prime}D^{\prime}}+\frac{1}{S^{*}}\bar{f}_{(A^{\prime}B^{\prime}}\bar{f}_{C^{\prime}D^{\prime})}\epsilon_{AB}\epsilon_{CD}\,.$
(198)

Various curvature and connection components with respect to a null basis have
been conventionally associated to particular notations in what is usually
called the Newman-Penrose formalism. While we will not lean on this heavily,
we next summarize these symbols as used in the main text.

###  A.1 Newman-Penrose formalism

We now introduce our conventions for the Newman-Penrose formalism, which is
mostly an exercise in decomposing spinors into the dyad
$\left\\{o^{A},\iota^{A}\right\\}$ and tensors into the Newman-Penrose frame
$\left\\{k^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\right\\}$. We will make
constant use of (177), (179), (182), and (195).

Let us start with an anti-symmetric tensor field $F_{\alpha\beta}$, as
decomposed into spinors in (196). The Newman-Penrose field strength scalars
$f_{0}$, $f_{1}$ and $f_{2}$ with respect to a dyad
$\left\\{o^{A},\iota^{A}\right\\}$ can be defined in terms of the two-spinor
formulation as

$f_{AB}=f_{2}o_{A}o_{B}-2f_{1}o_{(A}\iota_{B)}+f_{0}\iota_{A}\iota_{B}\,.$
(199)

Using (196), this is expressed in terms of the tensor itself with respect to
the associated null frame
$\left\\{k^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\right\\}$ as

$F_{\alpha\beta}=2f_{2}k_{[\alpha}m_{\beta]}-2f_{1}\left(k_{[\alpha}n_{\beta]}-m_{[\alpha}\bar{m}_{\beta]}\right)+2f_{0}\bar{m}_{[\alpha}n_{\beta]}+cc\,,$
(200)

where we will use “cc” to abbreviate the complex conjugate piece of the
expression. A consequence of (200), that we will regularly use for
calculations, is

$f_{2}=-F_{\alpha\beta}n^{\alpha}\bar{m}^{\beta}\,,\quad
f_{1}=-\frac{1}{2}F_{\alpha\beta}\left(n^{\alpha}k^{\beta}+m^{\alpha}\bar{m}^{\beta}\right)\,,\quad
f_{0}=F_{\alpha\beta}k^{\alpha}m^{\beta}\,.$ (201)

The analogous procedure for the Riemann tensor is somewhat more
(algebraically) complicated but is reviewed in detail in Chapter 4 of
penrose1984spinors . First, introduce the curvature spinors $\Psi_{ABCD}$ and
$\Phi_{ABC^{\prime}D^{\prime}}$ as well as their complex conjugates and the
scalar $\Lambda$ by the decomposition

$\displaystyle-
R_{\alpha\beta\gamma\delta}=\left(\Psi_{ABCD}+\Lambda\left(\epsilon_{AC}\epsilon_{BD}+\epsilon_{AD}\epsilon_{BC}\right)\right)\bar{\epsilon}_{A^{\prime}B^{\prime}}\bar{\epsilon}_{C^{\prime}D^{\prime}}+\Phi_{ABC^{\prime}D^{\prime}}\bar{\epsilon}_{A^{\prime}B^{\prime}}\epsilon_{CD}+cc\,.$
(202)

Now, the Weyl and Ricci scalars are again essentially just components of
$\Psi_{ABCD}$ and $\Phi_{ABC^{\prime}D^{\prime}}$ with respect to the dyad
$\left\\{o^{A},\iota^{A}\right\\}$, similar to (200) for the Maxwell
scalars272727Let us emphasize again that these are identical to the
definitions of penrose1984spinors , given that we introduce an extra minus
sign to correct for the different curvature convention (see (4.11.9) and
(4.11.10)). On the other hand, these differ in sign from say Pound:2021qin ,
where a mostly plus signature is used.

$\displaystyle\Psi_{0}$
$\displaystyle=\Psi_{ABCD}o^{A}o^{B}o^{C}o^{D}=-C_{\alpha\beta\gamma\delta}k^{\alpha}m^{\beta}k^{\gamma}m^{\delta}\,,$
(203) $\displaystyle\Psi_{1}$
$\displaystyle=\Psi_{ABCD}o^{A}o^{B}o^{C}\iota^{D}=-C_{\alpha\beta\gamma\delta}k^{\alpha}n^{\beta}k^{\gamma}m^{\delta}\,,$
$\displaystyle\Psi_{2}$
$\displaystyle=\Psi_{ABCD}o^{A}o^{B}\iota^{C}\iota^{D}=-C_{\alpha\beta\gamma\delta}k^{\alpha}m^{\beta}\bar{m}^{\gamma}n^{\delta}\,,$
$\displaystyle\Psi_{3}$
$\displaystyle=\Psi_{ABCD}o^{A}\iota^{B}\iota^{C}\iota^{D}=-C_{\alpha\beta\gamma\delta}k^{\alpha}n^{\beta}\bar{m}^{\gamma}n^{\delta}\,,$
$\displaystyle\Psi_{4}$
$\displaystyle=\Psi_{ABCD}\iota^{A}\iota^{B}\iota^{C}\iota^{D}=-C_{\alpha\beta\gamma\delta}n^{\alpha}\bar{m}^{\beta}n^{\gamma}\bar{m}^{\delta}\,,$

and

$\displaystyle\Phi_{00}$
$\displaystyle=\Phi_{ABC^{\prime}D^{\prime}}o^{A}o^{B}\bar{o}^{C^{\prime}}\bar{o}^{D^{\prime}}=\frac{1}{2}R_{\mu\nu}k^{\mu}k^{\nu}\,,$
(204) $\displaystyle\Phi_{10}$
$\displaystyle=\Phi_{ABC^{\prime}D^{\prime}}o^{A}\iota^{B}\bar{o}^{C^{\prime}}\bar{o}^{D^{\prime}}=\frac{1}{2}R_{\mu\nu}k^{\mu}\bar{m}^{\nu}\,,$
$\displaystyle\Phi_{20}$
$\displaystyle=\Phi_{ABC^{\prime}D^{\prime}}\iota^{A}\iota^{B}\bar{o}^{C^{\prime}}\bar{o}^{D^{\prime}}=\frac{1}{2}R_{\mu\nu}\bar{m}^{\mu}\bar{m}^{\nu}\,,$
etc.

We do not enumerate all of the $\Phi_{ABC^{\prime}D^{\prime}}$ components as
we do not need most of them explicitly. On the other hand, the component
$\Phi_{00}$ is important as it is the only (potentially) non-vanishing
component of the Ricci spinor in a plane wave spacetime, as discussed around
(34) in the main text. Similarly, the only non-vanishing Weyl scalar, in our
convention, of a type N spacetime is $\Psi_{0}$.

Given that the Weyl curvature $C_{\alpha\beta\gamma\delta}$ is given by

$\displaystyle-
C_{\alpha\beta\gamma\delta}=\Psi_{ABCD}\bar{\epsilon}_{A^{\prime}B^{\prime}}\bar{\epsilon}_{C^{\prime}D^{\prime}}+\bar{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\epsilon_{AB}\epsilon_{CD}\,,$
(205)

let us check some of the equalities in (203). In particular, for the main
text, it is important that the null Fermi coordinate approach to the Penrose
limit (17) Blau:2006ar ; blau2011plane

$H_{ab}(u)=\left(R_{\mu\nu\alpha\beta}E^{\mu}_{a}u^{\nu}E^{\alpha}_{b}u^{\beta}\right)_{|\gamma}\,.$
(206)

is equivalent to the spinor prescription (28) Tod:2019urw

$\Psi(u)=\left(\Psi_{ABCD}\alpha^{A}\alpha^{B}\alpha^{C}\alpha^{D}\right)_{|\gamma}\,,\quad\Phi(u)=\left(\Phi_{ABA^{\prime}B^{\prime}}\alpha^{A}\alpha^{B}\bar{\alpha}^{A^{\prime}}\bar{\alpha}^{B^{\prime}}\right)_{|\gamma}\,,$
(207)

if we identify, from the plane wave, as in (24)

$H_{ab}=-\begin{pmatrix}\frac{1}{2}\left(\Psi(u)+2\Phi(u)+\bar{\Psi}(u)\right)&\frac{i}{2}\left(\Psi(u)-\bar{\Psi}(u)\right)\\\
\frac{i}{2}\left(\Psi(u)-\bar{\Psi}(u)\right)&\frac{1}{2}\left(-\Psi(u)+2\Phi(u)-\bar{\Psi}(u)\right)\end{pmatrix}\,,$
(208)

with, on the plane wave ((33) and (34))

$\Phi(u)=\Phi^{(\gamma)}_{00}\,,\quad\Psi(u)=\Psi^{(\gamma)}_{0}\,,$ (209)

and, as in (29)

$u^{\delta}=\alpha^{D}\bar{\alpha}^{D^{\prime}}\,,\quad
M^{\delta}=-\frac{1}{\sqrt{2}}\left(E^{\delta}_{1}-iE^{\delta}_{2}\right)=\alpha^{D}\bar{\beta}^{D^{\prime}}\,.$
(210)

First observe that

$\displaystyle\Psi^{(\gamma)}_{0}=-C_{\alpha\beta\delta\gamma}M^{\alpha}u^{\beta}M^{\delta}u^{\gamma}$
$\displaystyle=\left(\Psi_{ABCD}\bar{\epsilon}_{A^{\prime}B^{\prime}}\bar{\epsilon}_{C^{\prime}D^{\prime}}+cc\right)\alpha^{A}\alpha^{B}\alpha^{C}\alpha^{D}\bar{\beta}^{A^{\prime}}\bar{\alpha}^{B^{\prime}}\bar{\beta}^{C^{\prime}}\bar{\alpha}^{D^{\prime}}$
(211)
$\displaystyle=\left(\Psi_{ABCD}\alpha^{A}\alpha^{B}\alpha^{C}\alpha^{D}\right)\,,$

and

$\displaystyle\Phi^{(\gamma)}_{00}=\frac{1}{2}R_{\alpha\beta}u^{\alpha}u^{\beta}$
$\displaystyle=\frac{1}{2}\left(\Phi_{ABC^{\prime}D^{\prime}}\bar{\epsilon}_{A^{\prime}B^{\prime}}\epsilon_{CD}+cc\right)\alpha^{B}\bar{\alpha}^{B^{\prime}}\alpha^{C}\bar{\alpha}^{C^{\prime}}\epsilon^{AD}\bar{\epsilon}^{A^{\prime}D^{\prime}}$
(212)
$\displaystyle=\Phi_{ABC^{\prime}D^{\prime}}\alpha^{B}\alpha^{A}\bar{\alpha}^{C^{\prime}}\bar{\alpha}^{D^{\prime}}\,.$

(211) and (206) imply in particular the traceless components of (208)

$\displaystyle
C_{\alpha\beta\delta\gamma}E^{\alpha}_{1}u^{\beta}E^{\delta}_{1}u^{\gamma}$
$\displaystyle=-\frac{1}{2}\left(\Psi^{(\gamma)}_{0}+\bar{\Psi}^{(\gamma)}_{0}\right)$
(213) $\displaystyle
C_{\alpha\beta\delta\gamma}E^{\alpha}_{1}u^{\beta}E^{\delta}_{2}u^{\gamma}$
$\displaystyle=\frac{1}{2i}\left(\Psi^{(\gamma)}_{0}-\bar{\Psi}^{(\gamma)}_{0}\right)$
$\displaystyle
C_{\alpha\beta\delta\gamma}E^{\alpha}_{2}u^{\beta}E^{\delta}_{2}u^{\gamma}$
$\displaystyle=\frac{1}{2}\left(\Psi^{(\gamma)}_{0}+\bar{\Psi}^{(\gamma)}_{0}\right)\,,$

while the trace follow from (212) and (206) by

$\displaystyle
H_{11}+H_{22}=R_{\alpha\beta\delta\gamma}u^{\beta}u^{\gamma}\delta^{ab}E_{a}^{\alpha}E_{b}^{\delta}$
$\displaystyle=R_{\alpha\beta\delta\gamma}u^{\beta}u^{\gamma}(-g^{\alpha\delta})$
(214) $\displaystyle=-2\Phi^{(\gamma)}_{00}\,.$

Note that it was our choice to build the dyad
$\left\\{\alpha^{A},\beta^{B}\right\\}$ with

$\epsilon_{AB}\alpha^{A}\beta^{B}=1\,,$ (215)

on the null geodesic together with (207) and (203) that led us to use the
convention that type N spacetimes only have non-vanishing $\Psi_{0}$.

To take covariant derivatives of spinors in a dyad basis, we need the spin
coefficients

$\gamma_{AA^{\prime}C}{}^{B}=\epsilon_{D}{}^{B}\nabla_{AA^{\prime}}\epsilon_{C}{}^{D}\,,$
(216)

such that in particular

$\displaystyle\nabla_{\mu}\kappa^{A}$
$\displaystyle=\nabla_{\mu}\left(\kappa^{o}o^{A}+\kappa^{\iota}\iota^{A}\right)$
(217) $\displaystyle=\left(\partial_{\mu}\kappa^{o}+\gamma_{\mu
o}{}^{o}\kappa^{o}+\gamma_{\mu\iota}{}^{o}\kappa^{\iota}\right)o^{A}+\left(\partial_{\mu}\kappa^{\iota}+\gamma_{\mu
o}{}^{\iota}\kappa^{o}+\gamma_{\mu\iota}{}^{\iota}\kappa^{\iota}\right)\iota^{A}\,.$

One way to compute (216) is using the associated Newman-Penrose frame.
Introducing first the Newman-Penrose spin coefficient symbols of
penrose1984spinors ; Penrose:1986ca , which give names to the individual
components,

$\displaystyle\kappa$
$\displaystyle=m^{\alpha}k^{\mu}\nabla_{\mu}k_{\alpha}\,,\quad\epsilon$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(n^{\alpha}k^{\mu}\nabla_{\mu}k_{\alpha}+m^{\alpha}k^{\mu}\nabla_{\mu}\bar{m}_{\alpha}\right)\,,\quad$
$\displaystyle\gamma^{\prime}$
$\displaystyle=\frac{1}{2}\left(k^{\alpha}k^{\mu}\nabla_{\mu}n_{\alpha}+\bar{m}^{\alpha}k^{\mu}\nabla_{\mu}m_{\alpha}\right)\,,$
$\displaystyle\rho$
$\displaystyle=m^{\alpha}\bar{m}^{\mu}\nabla_{\mu}k_{\alpha}\,,\quad\alpha$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(n^{\alpha}\bar{m}^{\mu}\nabla_{\mu}k_{\alpha}+m^{\alpha}\bar{m}^{\mu}\nabla_{\mu}\bar{m}_{\alpha}\right)\,,\quad$
$\displaystyle\beta^{\prime}$
$\displaystyle=\frac{1}{2}\left(k^{\alpha}\bar{m}^{\mu}\nabla_{\mu}n_{\alpha}+\bar{m}^{\alpha}\bar{m}^{\mu}\nabla_{\mu}m_{\alpha}\right)\,,$
$\displaystyle\sigma$
$\displaystyle=m^{\alpha}m^{\mu}\nabla_{\mu}k_{\alpha}\,,\quad\beta$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(n^{\alpha}m^{\mu}\nabla_{\mu}k_{\alpha}+m^{\alpha}m^{\mu}\nabla_{\mu}\bar{m}_{\alpha}\right)\,,\quad$
$\displaystyle\alpha^{\prime}$
$\displaystyle=\frac{1}{2}\left(k^{\alpha}m^{\mu}\nabla_{\mu}n_{\alpha}+\bar{m}^{\alpha}m^{\mu}\nabla_{\mu}m_{\alpha}\right)\,,$
$\displaystyle\tau$
$\displaystyle=m^{\alpha}n^{\mu}\nabla_{\mu}k_{\alpha}\,,\quad\gamma$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(n^{\alpha}n^{\mu}\nabla_{\mu}k_{\alpha}+m^{\alpha}n^{\mu}\nabla_{\mu}\bar{m}_{\alpha}\right)\,,\quad$
$\displaystyle\epsilon^{\prime}$
$\displaystyle=\frac{1}{2}\left(k^{\alpha}n^{\mu}\nabla_{\mu}n_{\alpha}+\bar{m}^{\alpha}n^{\mu}\nabla_{\mu}m_{\alpha}\right)\,,$
$\displaystyle\tau^{\prime}$
$\displaystyle=\bar{m}^{\alpha}k^{\mu}\nabla_{\mu}n_{\alpha}\,,\quad\sigma^{\prime}$
$\displaystyle=$
$\displaystyle\bar{m}^{\alpha}\bar{m}^{\mu}\nabla_{\mu}n_{\alpha}\,,\quad\rho^{\prime}=\bar{m}^{\alpha}m^{\mu}\nabla_{\mu}n_{\alpha}\,,\quad$
$\displaystyle\kappa^{\prime}$
$\displaystyle=\bar{m}^{\alpha}n^{\mu}\nabla_{\mu}n_{\alpha}\,,$ (218)

then

$\displaystyle\gamma_{oo^{\prime}o}{}^{o}$
$\displaystyle=\epsilon\,,\quad\gamma_{oo^{\prime}o}{}^{\iota}$
$\displaystyle=$ $\displaystyle-\kappa\,,\quad\gamma_{oo^{\prime}\iota}{}^{o}$
$\displaystyle=$
$\displaystyle-\tau^{\prime}\,,\quad\gamma_{oo^{\prime}\iota}{}^{\iota}$
$\displaystyle=$ $\displaystyle\gamma^{\prime}$ $\displaystyle\gamma_{\iota
o^{\prime}o}{}^{o}$ $\displaystyle=\alpha\,,\quad\gamma_{\iota
o^{\prime}o}{}^{\iota}$ $\displaystyle=$
$\displaystyle-\rho\,,\quad\gamma_{\iota o^{\prime}\iota}{}^{o}$
$\displaystyle=$ $\displaystyle-\sigma^{\prime}\,,\quad\gamma_{\iota
o^{\prime}\iota}{}^{\iota}$ $\displaystyle=$ $\displaystyle\beta^{\prime}$
$\displaystyle\gamma_{o\iota^{\prime}o}{}^{o}$
$\displaystyle=\beta\,,\quad\gamma_{o\iota^{\prime}o}{}^{\iota}$
$\displaystyle=$
$\displaystyle-\sigma\,,\quad\gamma_{o\iota^{\prime}\iota}{}^{o}$
$\displaystyle=$
$\displaystyle-\rho^{\prime}\,,\quad\gamma_{o\iota^{\prime}\iota}{}^{\iota}$
$\displaystyle=$ $\displaystyle\alpha^{\prime}$
$\displaystyle\gamma_{\iota\iota^{\prime}o}{}^{o}$
$\displaystyle=\gamma\,,\quad\gamma_{\iota\iota^{\prime}o}{}^{\iota}$
$\displaystyle=$
$\displaystyle-\tau\,,\quad\gamma_{\iota\iota^{\prime}\iota}{}^{o}$
$\displaystyle=$
$\displaystyle-\kappa^{\prime}\,,\quad\gamma_{\iota\iota^{\prime}\iota}{}^{\iota}$
$\displaystyle=$ $\displaystyle\epsilon^{\prime}\,.$ (219)

Although we will not generally need the expressions (219) and symbols (218) in
the main text, they are useful performing explicit calculations for the
examples. Moreover, the expressions above quoted from penrose1984spinors ;
Penrose:1986ca  retain the possibility to work with an unnormalized dyad,
while we always impose (177). Therefore

$\epsilon=-\gamma^{\prime}\,,\quad\alpha=-\beta^{\prime}\,,\quad\beta=-\alpha^{\prime}\,,\quad\gamma=-\epsilon^{\prime}\,.$
(220)

##  Appendix B Examples: further details

In this appendix, we gather some technical details that are used in Section 4.
We start with two, closely related, common problems one encounters in all the
examples. These end up being essentially a combination of simple algebraic
problems and first order linear, ordinary differential equations. However, the
solutions will generically still become rather intractable symbolically and
may not have a very satisfying analytic form.

The first of these problems is to express

$u^{AA^{\prime}}=\alpha^{A}\bar{\alpha}^{A^{\prime}}\,.$ (221)

With respect to a null basis $\left\\{K,N,M,\bar{M}\right\\}$ and the
associated dyad $\left\\{o^{A},\iota^{A}\right\\}$, such that in particular
$\alpha^{A}=\alpha^{o}o^{A}+\alpha^{\iota}\iota^{A}$

$\begin{pmatrix}u^{\mu}N_{\mu}&-u^{\mu}\bar{M}_{\mu}\\\
-u^{\mu}M_{\mu}&u^{\mu}K_{\mu}\end{pmatrix}=\begin{pmatrix}\alpha^{o}\bar{\alpha}^{o}&\alpha^{o}\bar{\alpha}^{\iota}\\\
\alpha^{\iota}\bar{\alpha}^{o}&\alpha^{\iota}\bar{\alpha}^{\iota}\end{pmatrix}=\begin{pmatrix}\alpha^{o}\\\
\alpha^{\iota}\end{pmatrix}\begin{pmatrix}\bar{\alpha}^{o}&\bar{\alpha}^{\iota}\end{pmatrix}\,.$
(222)

The absolute values of the spinor components and a relative phase are
immediate. For instance

$\alpha^{A}=e^{iS}\left(|\alpha^{o}|o^{A}+e^{i\delta}|\alpha^{\iota}|\iota^{A}\right)$
(223)

where282828We assume $u^{\mu}$ is generic; $u^{\nu}N_{\mu}$,
$u^{\mu}K_{\mu}\neq 0$. If this does not hold the problem simplifies even
further.

$|\alpha^{o}|^{2}=u^{\mu}N_{\mu}\,,\quad|\alpha^{\iota}|^{2}=u^{\mu}K_{\mu}\,,\quad
e^{i\delta}=-\frac{u^{\mu}M_{\mu}}{\sqrt{|u^{\mu}N_{\mu}||u^{\mu}K_{\mu}|}}\,.$
(224)

In addition to $\alpha^{A}$, we often want to find a $\beta^{A}$ such that

$\epsilon_{AB}\alpha^{A}\beta^{B}=1\,.$ (225)

For an $\alpha^{A}$ as in (223), such a $\beta^{A}$ can be given by

$\beta^{A}=\frac{e^{-i\delta-
iS}}{2|\alpha^{\iota}||\alpha^{o}|}\left(|\alpha^{o}|o^{A}-e^{i\delta}|\alpha^{\iota}|\iota^{A}\right)\,.$
(226)

A fundamental ambiguity will always remain in the choice of $e^{iS}$, in the
sense that it is simply not fixed by only specifying $u^{\mu}$. However, we
will want to impose that

$u^{\mu}\nabla_{\mu}\alpha^{A}=0\,.$ (227)

Therefore this ambiguity cannot take an arbitrary functional form, as would be
allowed by (221). Let $\tilde{\alpha}^{A}$ and $\tilde{\beta}^{A}$ be two-
spinors constructed as in (223) and (226) with $S=0$. From (227), the
appropriate choice of $S$ can be constructed by solving

$u^{\mu}\nabla_{\mu}S=i\epsilon_{AB}\left(u^{\mu}\nabla_{\mu}\tilde{\alpha}^{A}\right)\tilde{\beta}^{B}\,.$
(228)

Note that by (221), (228) can be written in terms of the spin-coefficient
$\epsilon$ (see (218)) with respect to the dyad
$\left\\{\tilde{\alpha},\tilde{\beta}\right\\}$ (we therefore denote it as
$\tilde{\epsilon}$) as

$u^{\mu}\nabla_{\mu}S-i\tilde{\epsilon}=0\,.$ (229)

The second reoccurring algebraic problem is to find basis vectors
$E_{i}{}^{a}$ which square to a given (real) symmetric positive definite
matrix $C_{ij}$

$C_{ij}=E_{i}{}^{a}\delta_{ab}E_{j}{}^{b}=\sum_{a}E_{i}{}^{a}E_{j}{}^{a}={\boldsymbol{E}}\cdot\left({\boldsymbol{E}}\right)^{T}\,.$
(230)

One natural approach would be to first diagonalize $C_{ij}$, solve the trivial
version of the above, and subsequently transform back. Slightly more
efficiently, one can simply make an ansatz for $E_{i}{}^{a}$ to be triangular
and solve the resulting set of equations from (230) step-by-step. In other
words, one can construct $E_{i}{}^{a}$ from a type of Gram-Schmidt
orthogonalization procedure.

As with the previous problem however, the complication for our purposes
actually arises because in addition to (230), we need $E_{i}{}^{a}$ to be
parallel propagated which, as discussed in the main text, results in the
condition (14)

$\dot{E}^{i}{}_{a}E_{ib}=\dot{E}^{i}{}_{b}E_{ia}\,.$ (231)

Given a solution to $\tilde{E}_{i}{}^{a}$ which does not solve the above, the
problem is to find a time-dependent orthogonal transformation such that

$E_{j}{}^{a}=O_{j}{}^{i}\tilde{E}_{i}{}^{a}\,.$ (232)

does satisfy (231).

In a four dimensional spacetime, there are only two transverse dimensions and
one can rather easily parameterize explicitly

${\boldsymbol{O}}=\begin{pmatrix}\cos{\theta}&\sin{\theta}\\\
-\sin{\theta}&\cos{\theta}\end{pmatrix}\,,$ (233)

and solve the ordinary differential equation for $\theta$ that will ensure
(231). We will not work this out explicitly here but in practice this ends up
being identical to the first problem we discussed, as once we have found a
parallel propagated dyad $\left\\{\alpha^{A},\beta^{A}\right\\}$, we can
readily construct a parallel propagated frame.

###  B.1 The Schwarzschild black hole

In this section, for completeness, we provide some additional details related
to the Schwarzschild example in Section 4.1. First, using the frame (70) and
definitions (218), the spin coefficients are given by

$\displaystyle\kappa$ $\displaystyle=0\,,\quad\sigma$ $\displaystyle=$
$\displaystyle 0\,,\quad$ $\displaystyle\rho$
$\displaystyle=\frac{1}{\sqrt{2}r}\,,\quad$ $\displaystyle\tau$
$\displaystyle=0\,,$ $\displaystyle\kappa^{\prime}$
$\displaystyle=0\,,\quad\sigma^{\prime}$ $\displaystyle=$ $\displaystyle
0\,,\quad$ $\displaystyle\rho^{\prime}$
$\displaystyle=-\frac{1}{\sqrt{2}r}(1-\frac{2M}{r})\,,\quad$
$\displaystyle\tau^{\prime}$ $\displaystyle=0\,,$ $\displaystyle\epsilon$
$\displaystyle=0\,,\quad\epsilon^{\prime}$ $\displaystyle=$
$\displaystyle\frac{M}{\sqrt{2}r^{2}}\,,\quad$ $\displaystyle\beta$
$\displaystyle=-\frac{\cot{\theta}}{2\sqrt{2}}\,,\quad$
$\displaystyle\beta^{\prime}$
$\displaystyle=-\frac{\cot{\theta}}{2\sqrt{2}}\,,$ (234)

For comparisons, using instead the flat frame, given also in (70), on the flat
background in Kerr-Schild coordinates, we of course just find (234) with $M\to
0$.

The flat space spin coefficients are just telling us that the null directions
in flat space generate radially propagating null congruences. Compared to
(234), the main difference is that the expansion of the outgoing congruence is
reduced by the black hole. In the Geroch-Held-Penrose (or compacted spin
coefficient) formalism geroch1973space , $\beta$ and $\beta^{\prime}$ act as
gauge fields for rotations in the transverse planes, $m^{\mu}\to
e^{i\varphi(x)}m^{\mu}$ for some function $\varphi(x)$. In this spherically
symmetric Schwarzchild example these are related to the induced connection on
the transverse spheres. $\epsilon$ and $\epsilon^{\prime}$ play a similar role
for the timelike surface spanned by the time and radial coordinates, $\hat{t}$
and $r$, and the transformations $k^{\mu}\to e^{\varphi(x)}k^{\mu}$ and
$n^{\mu}\to e^{-\varphi(x)}n^{\mu}$. In the flat case, this surface is just a
flat plane, so $\epsilon$ and $\epsilon^{\prime}$ vanish, but the plane is
curved by the black hole inducing also a non-zero value for
$\epsilon^{\prime}$. The fact that, contrary to $\rho^{\prime}$ and
$\epsilon^{\prime}$, $\rho$ and $\epsilon$ are unchanged between the black
hole and the flat space is a consequence of the Kerr-Schild form of the metric
and the frame.

In order to check the solution (75) to the valence-2 Killing spinor equations
(74), let us write out these equations explicitly for the ansatz

$\chi_{AB}=2\chi_{01}(r)o_{(A}\iota_{B)}\,.$ (235)

When we evaluate the Killing spinor equations (74) on (235), setting the
vanishing spin coefficients in (234) to zero, we find

$\displaystyle\left(\frac{d}{dr}-\sqrt{2}\rho\right)\chi_{01}=0\,,\quad\left(\left(1-\frac{2M}{r}\right)\frac{d}{dr}+\sqrt{2}\rho^{\prime}\right)\chi_{01}=0\,.$
(236)

A solution to (74) is thus given by $\chi_{01}(r)\propto r$.

Next, we provide a first step to check that the two-spinor $\alpha^{A}$, given
in (80), satisfies the parallel transport equation (79) along the null
geodesic $\gamma$ with tangent
$u^{\alpha}=\alpha^{A}\bar{\alpha}^{A^{\prime}}$, given in Kerr-Schild
coordinates in (77) and (78). Simply writing out the parallel transport
equation while using that the components of $\alpha^{A}$ have no explicit
dependence on $\hat{t}$ and $\phi$, and setting the vanishing spin
coefficients in (234) to zero already, we find,

$\displaystyle-\iota_{A}\alpha^{B}\bar{\alpha}^{B^{\prime}}\nabla_{BB^{\prime}}\alpha^{A}$
$\displaystyle=\left[-\frac{1}{\sqrt{2}r}|\alpha^{o}|^{2}\frac{\partial\alpha^{o}}{\partial
r}+|\alpha^{\iota}|^{2}\left(\frac{1}{\sqrt{2}}\left(1-\frac{2M}{r}\right)\frac{\partial\alpha^{o}}{\partial
r}-\epsilon^{\prime}\alpha^{o}\right)\right]$ (237)
$\displaystyle+\left[\alpha^{o}\bar{\alpha}^{\iota^{\prime}}\left(-\frac{1}{\sqrt{2}r}\frac{\partial\alpha^{o}}{\partial\theta}-\rho^{\prime}\alpha^{\iota}+\beta\alpha^{o}\right)+\alpha^{\iota}\bar{\alpha}^{o^{\prime}}\left(-\frac{1}{\sqrt{2}r}\frac{\partial\alpha^{o}}{\partial\theta}-\beta^{\prime}\alpha^{o}\right)\right]\,,$
$\displaystyle
o_{A}\alpha^{B}\bar{\alpha}^{B^{\prime}}\nabla_{BB^{\prime}}\alpha^{A}$
$\displaystyle=\left[-\frac{1}{\sqrt{2}r}|\alpha^{o}|^{2}\frac{\partial\alpha^{\iota}}{\partial
r}+|\alpha^{\iota}|^{2}\left(\frac{1}{\sqrt{2}}\left(1-\frac{2M}{r}\right)\frac{\partial\alpha^{\iota}}{\partial
r}+\epsilon^{\prime}\alpha^{\iota}\right)\right]$
$\displaystyle+\left[\alpha^{o}\bar{\alpha}^{\iota^{\prime}}\left(-\frac{1}{\sqrt{2}r}\frac{\partial\alpha^{\iota}}{\partial\theta}-\beta\alpha^{\iota}\right)+\alpha^{\iota}\bar{\alpha}^{o^{\prime}}\left(-\frac{1}{\sqrt{2}r}\frac{\partial\alpha^{\iota}}{\partial\theta}-\beta^{\prime}\alpha^{\iota}-\rho\alpha^{o}\right)\right]\,.$

At this point we can just insert (80) and grind through the algebra, which we
have done. Alternatively, we could pretend for a moment that we did not have
or know of the Killing spinor $\chi_{AB}$ from which we could derive (80) by
demanding (81) is constant. Then, we would instead use the ansatz (224)
together with (223) and solve for the unkown overall phase $S$. In that case,
there is still a non-trivial partial differential equation to be solved, in
stark contrast with the simple algebraic problem that results from demanding
$\chi_{AB}\alpha^{A}\alpha^{B}$ is constant.

Finally, let us indicate how to derive the adapted coordinates (93), although
see also blau2011plane . The general strategy is to first choose a
hypersurface in spacetime which parameterizes the initial conditions for the
null geodesics of fixed momenta $\hat{E}$, $L$, $L_{\phi}=0$. A coordinate
system on this initial condition hypersurface together with the affine time
parameterizing the evolution along the geodesics is then almost in adapted
form. To get a coordinate system into adapted form, we use that the action as
one of the hypersurface coordinates. The reason this use of the action as a
coordinate yields an adapted coordinate system is that the induced coordinate
one-form of the action is the momentum of the congruence while the metrically
dual vector field is the tangent vector field along the geodesic congruence,
which by construction is the null coordinate vector field $\partial_{U}$ of
the affine parameter $U$. Therefore $g_{UV}=1$ and $g_{UU}=0$. Moreover, again
by construction, the other transverse coordinate vector fields keep $V$ fixed
such that $g_{Ui}=0$. As a result, “the gauge conditions” for the adapted
coordinates (9) are satisfied.

There are many ways in which we could choose an initial condition
hypersurface. One natural choice would be to take a slice of fixed Kerr-Schild
time $\hat{t}$. On the other hand, it is usually simplest if the choice of
hypersurface respects as many isometries as possible Papadopoulos:2020qik .
That means that it is in particular more convenient to parameterize the
congruence on an initial condition hypersurface that is invariant under
(spacetime) time translations. For Schwarzschild, the setup is simple enough
that the choice makes little difference but it is for this isometry reason
that we instead choose a constant radius $r=r_{\rm ref}$ hypersurface to
parametrize the initial positions of the null geodesic congruence. With that
choice the Kerr-Schild coordinates satisfy

$\hat{t}|_{U=0}=\hat{t}_{0}\,,\quad r|_{U=0}=r_{\rm
ref}\,,\quad\theta|_{U=0}=\theta_{0}\,,\quad\phi|_{U=0}=\phi_{0}\,.$ (238)

By our choice $L_{\phi}=0$, the initial azimuthal angle is preserved such that
even in general $\phi=\phi_{0}$. In addition, the radial null geodesic motion
at fixed energy and angular momentum doesn’t depend on any of the other
coordinates. Therefore, using the radial geodesic equation (83), we can find
$r(U)$ by inverting

$U=\int^{r(U)}_{r_{\rm
ref}}\frac{d\rho}{R^{\prime}(\rho)-\frac{2M}{\rho}\left(\hat{E}+R^{\prime}(\rho)\right)}\,.$
(239)

On the other hand, the solutions to the geodesic equations for $\theta$ and
$\hat{t}$ do of course depend on $\theta_{0}$ and $\hat{t}_{0}$ in addition to
radial motion, but simply as integration constants such that, from

$u^{\theta}=\frac{d\theta}{dU}=\frac{L}{r(U)^{2}}\,,\quad
u^{\hat{t}}=\frac{d\hat{t}}{dU}=\hat{E}+\frac{2M}{r(U)}\left(\hat{E}+R^{\prime}(r)\right)\
\,,$ (240)

with $R^{\prime}(r)|_{r=r(U)}$, we find

$\displaystyle\theta(U,\hat{t}_{0})$
$\displaystyle=\theta_{0}(V,\hat{t}_{0})+L\int^{r(U)}_{r_{\rm
ref}}d\rho\,\frac{1}{\rho^{2}}\frac{1}{R^{\prime}(\rho)-\frac{2M}{\rho}\left(\hat{E}+R^{\prime}(\rho)\right)}\,,$
(241) $\displaystyle\hat{t}(U,\hat{t}_{0})$
$\displaystyle=\hat{t}_{0}+\int^{r(U)}_{r_{\rm
ref}}d\rho\,\frac{\hat{E}+\frac{2M}{\rho}\left(\hat{E}+R^{\prime}(\rho)\right)}{R^{\prime}(\rho)-\frac{2M}{\rho}\left(\hat{E}+R^{\prime}(\rho)\right)}\,.$

Next, the action associated to the null geodesic congruence, from (77), is

$V=\hat{E}\hat{t}-(R(r)-R(r_{\rm ref}))-L\theta\,,$ (242)

which, when evaluated on (239) and (241) gives

$V=\hat{E}\hat{t}_{0}-L\theta_{0}\,.$ (243)

Specifically, as expected, $V$ does not depend on $U$ and can therefore be
used as a coordinate for initial condition hypersurface. To see very
explicitly how this happens, note that using (78)

$-L^{2}+\rho^{2}\hat{E}^{2}-\rho^{2}(R^{\prime}(\rho))^{2}+2M\rho\left(\hat{E}+R^{\prime}(\rho)\right)R^{\prime}(\rho)+2M\rho\hat{E}\left(\hat{E}+R^{\prime}(\rho)\right)=0\,.$
(244)

Our choice of adapted coordinates is to use $V$ instead of $\theta_{0}$

$\theta_{0}(V,\hat{t}_{0})=\frac{1}{L}\left(\hat{E}\hat{t}_{0}-V\right)\,.$
(245)

In summary, (239), (241), and (245) together are presented as (93) in the main
text.

###  B.2 The Kerr black hole

In this section, for completeness, we provide some additional details related
to the Kerr example in Section 4.2. First, using the frame (120) and
definitions (218), the spin coefficients are given by

$\displaystyle\kappa$ $\displaystyle=0\,,\quad$ $\displaystyle\sigma$
$\displaystyle=0\,,$ $\displaystyle\rho$
$\displaystyle=\frac{1}{r-iy}\,,\quad$ $\displaystyle\tau$
$\displaystyle=-i\frac{(r+iy)\sqrt{a^{2}-y^{2}}}{\sqrt{2}(r^{2}+y^{2})^{3/2}}\,,$
$\displaystyle\kappa^{\prime}$ $\displaystyle=0\,,\quad$
$\displaystyle\sigma^{\prime}$ $\displaystyle=0\,,$
$\displaystyle\rho^{\prime}$
$\displaystyle=-\frac{\Delta_{r}}{\sqrt{2}(r-iy)(r^{2}+y^{2})}\,,\quad$
$\displaystyle\tau^{\prime}$
$\displaystyle=-i\frac{(r+iy)\sqrt{a^{2}-y^{2}}}{\sqrt{2}(r^{2}+y^{2})^{3/2}}\,,$
$\displaystyle\epsilon$ $\displaystyle=\frac{iy}{2(r^{2}+y^{2})}\,,\quad$
$\displaystyle\epsilon^{\prime}$
$\displaystyle=\frac{r^{3}-(r+iy)\Delta_{r}+2(r-M)y^{2}-ra^{2}}{4(r^{2}+y^{2})^{2}}\,,$
(246) $\displaystyle\beta$
$\displaystyle=-i\frac{(a^{2}-y^{2})(r+iy)+i(r^{2}-a^{2})y}{2\sqrt{2}(r^{2}+y^{2})^{3/2}\sqrt{a^{2}-y^{2}}}\,,\quad$
$\displaystyle\beta^{\prime}$
$\displaystyle=-i\frac{(a^{2}-y^{2})(r+iy)+i(r^{2}+a^{2})y}{2\sqrt{2}(r^{2}+y^{2})^{3/2}\sqrt{a^{2}-y^{2}}}\,.$
(247)

The principal null congruences are still geodesic and twist-free but now
additionally have some twist. The timelike surface-element292929The presence
of the twists of the principal null congruences and the analogous quantities
in $\tau$, $\tau^{\prime}$ prevent having actual embeddings. In fact, the
absence of these orthogonal embedded 2-surfaces is arguably the most relevant,
qualitative difference from Schwarzschild. spanned by the null direction now
is not flat, even if $M=0$ while its transverse spatial counterpart (the
2-sphere for Schwarzschild) still does not depend on the mass $M$, so it is
identical for flat space with the corresponding Kerr-Schild frames, but now
also depends on $r$. Notable, embeddable 2-surfaces orthonormal or along the
principal null-frame directions are the equator at $y^{2}=a^{2}$ and the
horizon at $\Delta_{r}=0$.

The equations (236) for Schwarzschild, with the slightly more general ansatz,

$\chi_{AB}=2\chi_{01}(r,y)o_{(A}\iota_{B)}\,,$ (248)

and allowing for the additional, non-vanishing spin coefficients in (247)
compared to (234), generalizes to

$\displaystyle\left(\frac{d}{dr}-\rho\right)\chi_{01}$
$\displaystyle=0\,,\quad\left(\sqrt{\frac{a^{2}-y^{2}}{r^{2}+y^{2}}}\frac{d}{dy}+\sqrt{2}\tau^{\prime}\right)\chi_{01}=0\,,$
(249)
$\displaystyle\left(\frac{\Delta_{r}}{r^{2}+y^{2}}\frac{d}{dr}-2\rho^{\prime}\right)\chi_{01}$
$\displaystyle=0\,,\quad\left(\sqrt{\frac{a^{2}-y^{2}}{r^{2}+y^{2}}}\frac{d}{d\theta}+\sqrt{2}\tau\right)\chi_{01}=0\,.$

As a result, $\chi_{01}\propto r-iy=\zeta$ as desired.

Next, we will provide a first step in checking that the tangent two-spinor
$\alpha^{A}$, given in (127), is indeed parallel transported along the null
geodesic with tangent given in (124).

$\displaystyle-\iota_{A}\alpha^{B}\bar{\alpha}^{B^{\prime}}\nabla_{BB^{\prime}}\alpha^{A}=\left[|\alpha^{o}|^{2}\left(-\frac{\partial\alpha^{o}}{\partial
r}-\tau^{\prime}\alpha^{\iota}+\epsilon\alpha^{o}\right)+|\alpha^{\iota}|^{2}\left(\frac{\Delta_{r}}{2(r^{2}+y^{2})}\frac{\partial\alpha^{o}}{\partial
r}-\epsilon^{\prime}\alpha^{o}\right)\right]$ (250)
$\displaystyle+\left[\alpha^{o}\bar{\alpha}^{\iota^{\prime}}\left(-\sqrt{\frac{a^{2}-y^{2}}{2(r^{2}+y^{2})}}\frac{\partial\alpha^{o}}{\partial
y}-\rho^{\prime}\alpha^{\iota}+\beta\alpha^{o}\right)+\alpha^{\iota}\bar{\alpha}^{o^{\prime}}\left(-\sqrt{\frac{a^{2}-y^{2}}{2(r^{2}+y^{2})}}\frac{\partial\alpha^{o}}{\partial
y}-\beta^{\prime}\alpha^{o}\right)\right]\,,$ $\displaystyle
o_{A}\alpha^{B}\bar{\alpha}^{B^{\prime}}\nabla_{BB^{\prime}}\alpha^{A}=\left[-|\alpha^{o}|^{2}\left(\frac{\partial\alpha^{\iota}}{\partial
r}+\epsilon\alpha^{\iota}\right)+|\alpha^{\iota}|^{2}\left(\frac{\Delta_{r}}{2(r^{2}+y^{2})}\frac{\partial\alpha^{\iota}}{\partial
r}+\epsilon^{\prime}\alpha^{\iota}-\tau\alpha^{o}\right)\right]$
$\displaystyle+\left[\alpha^{o}\bar{\alpha}^{\iota^{\prime}}\left(-\sqrt{\frac{a^{2}-y^{2}}{2(r^{2}+y^{2})}}\frac{\partial\alpha^{\iota}}{\partial
y}-\beta\alpha^{\iota}\right)+\alpha^{\iota}\bar{\alpha}^{o^{\prime}}\left(-\sqrt{\frac{a^{2}-y^{2}}{2(r^{2}+y^{2})}}\frac{\partial\alpha^{\iota}}{\partial
y}+\beta^{\prime}\alpha^{\iota}-\rho\alpha^{o}\right)\right]\,.$

At this point, it serves no purpose to continue to write down the calculation,
which is uninformative. However, we have verified that the equations (250) are
satisfied by (127).

###  B.3 Vacuum Petrov type D spacetimes

In this section, we provide some additional details related to the vacuum
Petrov type D spacetimes example in Section 4.3. From the build-up towards
this general case, it is clear that for the general structure of the double
copy and the Penrose limit, the main ingredient that we need is the existence
of the Killing spinor $\chi_{AB}$. On the other hand, in practice, one would
still need to be more explicitly about, say, the Kerr-Schild form and
geodesics. We will therefore present here for completeness the (uncharged)
Plebanski-Demianski line element Kinnersley:1969zza ; Plebanski:1976gy , which
is the most general Petrov type D solution of the vacuum Einstein equations.
However, see Griffiths:2005qp  for more details.

The (vacuum) Plebanski-Demianski line element, with vanishing cosmological
constant, in complexified coordinates $\\{u,v,p,q\\}$ is given by

$(1-pq)^{2}ds^{2}=-2i(du+q^{2}dv)dp+2(du-p^{2}dv)dq-\frac{P(p)}{p^{2}+q^{2}}(du+q^{2}dv)^{2}+\frac{Q(q)}{p^{2}+q^{2}}(du-p^{2}dv)^{2}\,,$
(251)

where the the auxiliary polynomials $P(p)$ and $Q(q)$ are given by

$\displaystyle P(p)$ $\displaystyle=\Gamma(1-p^{4})+2Np-\Xi p^{2}+2Mp^{3}\,,$
(252) $\displaystyle Q(q)$ $\displaystyle=\Gamma(1-q^{4})-2Mq+\Xi
q^{2}-2Nq^{3}\,,$

with free parameters $M$, $N$, $\Gamma$, and $\Xi$. We present the metric
immediately in the complex form (251), which is of the double Kerr-Schild form
that was used in Luna:2018dpt  to connect to the single copy flat space.

The double Kerr-Schild form of (251) requires us to slightly modify parts of
the discussion in the main text. First, instead of (2), the metric is of the
form

$g_{\mu\nu}=\eta_{\mu\nu}+\phi_{k}k_{\mu}k_{\nu}+\phi_{m}m_{\nu}m_{\mu}\,.$
(253)

for null, geodesic vector fields $k$ and $m$, which are moreover orthogonal to
each other303030The fact that $k_{\mu}m^{\mu}=0$ is our motivation to call the
second null vector $m$ instead of $l$. Moreover, the requirement to have a
double Kerr-Schild metric, which involves say $m$, explains why it was
necessary to complexify the metric., as well as scalar functions $\phi_{k}$
and $\phi_{m}$. The mutual orthogonality of $m$ and $k$ implies first of all
that one can unambiguously raise and lower their indices with either the flat
or the curved metric. Moreover, the extension of the Kerr-Schild shift (52) is
given by

$\displaystyle n^{(g)}_{\mu}$
$\displaystyle=n^{(\eta)}+\frac{\phi_{k}}{2}k^{(\eta)}_{\mu}\,,\quad
k^{(g)}_{\mu}=k^{(\eta)}_{\mu}$ (254) $\displaystyle\tilde{m}^{(g)}_{\mu}$
$\displaystyle=\tilde{m}^{(\eta)}+\frac{\phi_{m}}{2}m^{(\eta)}_{\mu}\,,\quad
m^{(g)}_{\mu}=m^{(\eta)}_{\mu}\,,$

where, as before

$\eta_{\mu\nu}=2k^{(\eta)}_{(\mu}n^{(\eta)}_{\nu)}-2m^{(\eta)}_{(\mu}\tilde{m}^{(\eta)}_{\nu)}\,,\quad
g_{\mu\nu}=2k^{(g)}_{(\mu}n^{(g)}_{\nu)}-2m^{(g)}_{(\mu}\tilde{m}^{(g)}_{\nu)}\,,$
(255)

but we use $\tilde{m}$ instead of $\bar{m}$ to emphasize that these are not
necessarily complex conjugate for the metric in its complex form. Note also
that the vector fields $k$ and $m$ are identical on both flat and curved
spacetime, so again unambiguously $k_{\mu}=k_{\mu}^{(\eta)}=k_{\mu}^{(g)}$ and
$m_{\mu}=m_{\mu}^{(\eta)}=m_{\mu}^{(g)}$. Similarly, the shift in Infeld-van
der Waerden symbols (54) is readily extended to

$\sigma^{(g)}{}{}^{AA^{\prime}}_{\mu}=\sigma^{(\eta)}{}{}^{AA^{\prime}}_{\mu}+\frac{\phi_{k}}{2}k_{\mu}\sigma^{(\eta)}{}{}^{AA^{\prime}}_{\nu}k^{\nu}+\frac{\phi_{m}}{2}m_{\mu}\sigma^{(\eta)}{}{}^{AA^{\prime}}_{\nu}m^{\nu}\,.$
(256)

A principal null frame for the metric (251), that takes the form (254), is
given by

$\displaystyle k^{(g)}_{\mu}dx^{\mu}$
$\displaystyle=k^{(\eta)}_{\mu}dx^{\mu}\,,$ $\displaystyle
n^{(g)}_{\mu}dx^{\mu}$
$\displaystyle=n^{(\eta)}_{\mu}dx^{\mu}-\frac{q(M+Nq^{2})}{(1-pq)^{2}(p^{2}+q^{2})}k^{(\eta)}_{\mu}dx^{\mu}\,,$
$\displaystyle m^{(g)}_{\mu}dx^{\mu}$
$\displaystyle=m^{(\eta)}_{\mu}dx^{\mu}\,,$
$\displaystyle\tilde{m}^{(g)}_{\mu}dx^{\mu}$
$\displaystyle=\tilde{m}^{(\eta)}_{\mu}dx^{\mu}+\frac{p(N+Mp^{2})}{(1-pq)^{2}(p^{2}+q^{2})}m^{(\eta)}_{\mu}dx^{\mu}\,.$
(257)

with

$\displaystyle k^{(\eta)}_{\mu}dx^{\mu}$ $\displaystyle=du-p^{2}dv\,,$
$\displaystyle n^{(\eta)}_{\mu}dx^{\mu}$
$\displaystyle=\frac{\Gamma(1-q^{4})+\Xi
q^{2}}{2(1-pq)^{2}(p^{2}+q^{2})}du-p^{2}\frac{\Gamma(1-q^{4})+\Xi
q^{2}}{2(1-pq)^{2}(p^{2}+q^{2})}dv+\frac{dq}{(1-pq)^{2}}\,,$ $\displaystyle
m^{(\eta)}_{\mu}dx^{\mu}$ $\displaystyle=du+q^{2}dv\,,$
$\displaystyle\tilde{m}^{(\eta)}_{\mu}dx^{\mu}$
$\displaystyle=\frac{\Gamma(1-p^{4})+\Xi
p^{2}}{2(1-pq)^{2}(p^{2}+q^{2})}du+q^{2}\frac{\Gamma(1-p^{4})-\Xi
p^{2}}{2(1-pq)^{2}(p^{2}+q^{2})}dv+\frac{idp}{(1-pq)^{2}}\,.$ (258)

The scalar functions $\phi_{k}$ and $\phi_{m}$, in (253) and (254), are thus
given by

$\phi_{k}=-\frac{q(M+Nq^{2})}{(1-pq)^{2}(p^{2}+q^{2})}\,,\quad\phi_{m}=\frac{p(N+Mp^{2})}{(1-pq)^{2}(p^{2}+q^{2})}\,.$
(259)

Using the frame (257), we find

$\displaystyle\kappa$ $\displaystyle=0\,,\quad$ $\displaystyle\sigma$
$\displaystyle=0\,,$ $\displaystyle\rho$
$\displaystyle=-\frac{(i+p^{2})(1-pq)}{p+iq}\,,\quad$ $\displaystyle\tau$
$\displaystyle=\frac{(-i+q^{2})(1-pq)}{p+iq}\,,$
$\displaystyle\kappa^{\prime}$ $\displaystyle=0\,,\quad$
$\displaystyle\sigma^{\prime}$ $\displaystyle=0\,,$
$\displaystyle\rho^{\prime}$
$\displaystyle=-\frac{(i+p^{2})(2Mq+2Nq^{3}-\Gamma(1-q^{4})-\Xi
q^{2})}{2(p-iq)(p+iq)^{2}(1-pq)}\,,$ (260) $\displaystyle\tau^{\prime}$
$\displaystyle=-\frac{(-i+q^{2})(2Np+2Np^{3}+\Gamma(1-p^{4})-\Xi
p^{2})}{2(p-iq)(p+iq)^{2}(1-pq)}\,,$ (261)

where for brevity we omit the GHP gauge fields $\epsilon$,
$\epsilon^{\prime}$, $\beta$, and $\beta^{\prime}$. In addition, as expected
for a Petrov type D spacetime

$\Psi_{2}=i\frac{(M-iN)(1-pq)^{3}}{(p+iq)^{3}}\,,\quad\quad\Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0\,.$
(262)

We should stress here that the above spin coefficients and Weyl scalars are
potentially incomplete as written. Contrary to the previous examples, $m$ and
$\tilde{m}$ are not complex conjugates. Therefore, there are additional
scalars that are not necessarily related to the ones presented above. An
interesting example is that $\tilde{\tau}\neq\bar{\tau}$. On the other hand,
we do find

$\tilde{\Psi}_{2}=\bar{\Psi}_{2}=-i\frac{(M+iN)(1-pq)^{3}}{(p-iq)^{3}}\,.$
(263)

which is the quantity of relevance for the discussion of the Weyl double copy
in the main text. As in the other Petrov type D examples, the Weyl spinor is
given by

$\Psi_{ABCD}=6i\frac{(M-iN)(1-pq)^{3}}{(p+iq)^{3}}o_{(A}o_{B}\iota_{C}\iota_{D)}\,,$
(264)

or

$\Psi_{ABCD}=-\frac{\tilde{M}^{5}}{\zeta^{5}}3\chi_{(AB}\chi_{CD)}\,,\quad\chi_{AB}=\sqrt{2}\frac{\zeta}{\tilde{M}^{2}}o_{(A}\iota_{B)}\,.$
(265)

with

$\tilde{M}=M-iN\,,\quad\zeta=\frac{q-ip}{1-pq}\,.$ (266)

The double Kerr-Schild prescription of Luna:2018dpt , compatible with the Weyl
double copy, is given by313131One can (and would generically) change the
constant charges $N^{(\eta)}$ and $M^{(\eta)}$ to which $N$ and $M$ are
mapped, see Luna:2018dpt .

$A=\frac{2M^{(\eta)}q}{p^{2}+q^{2}}k+\frac{2N^{(\eta)}p}{p^{2}+q^{2}}m\,.$
(267)

However, as was also pointed out by Luna:2018dpt ,

$\phi_{k}\not\propto\frac{2M^{(\eta)}q}{p^{2}+q^{2}}\,,\quad\phi_{m}\not\propto\frac{2N^{(\eta)}p}{p^{2}+q^{2}}\,.$
(268)

Moreover, the prefactors of $k$ and $m$ in (267) do not satisfy the flat space
wave equation. However, a rescaling of the frames323232The discussion of
rescaling the frame vectors can also be usefully phrased in terms of the GHP-
weight of the Kerr-Schild single and zeroth copy, which indeed have definite,
non-zero weight. The requirement that the flat space field equations are
satisfied generically fixes the GHP frame up to a constant.

$\left(l,n,m,\tilde{m}\right)\to\left(\frac{l}{1-pq},\left(1-pq\right)n,\frac{m}{1-pq},\left(1-pq\right)\tilde{m}\right)\,,$
(269)

together with

$\phi_{k}\to\left(1-pq\right)^{2}\phi_{k}\,,\quad\phi_{m}\to\left(1-pq\right)^{2}\phi_{m}\,,$
(270)

leaves invariant the form of the flat Kerr-Schild metric as well as of
$\Psi_{2}$ but it changes (267) into

$A=\frac{2M^{(\eta)}q\left(1-pq\right)}{p^{2}+q^{2}}k+\frac{2N^{(\eta)}p\left(1-pq\right)}{p^{2}+q^{2}}m\,.$
(271)

while

$\phi_{k}=-\frac{q(M+Nq^{2})}{p^{2}+q^{2}}\,,\quad\phi_{m}=\frac{p(N+Mp^{2})}{p^{2}+q^{2}}\,.$
(272)

Obviously, $\phi_{k}$ and $\phi_{m}$ in this frame still do not correspond to
the prefactors of $k$ and $m$ in (271). On the other hand, those prefactors,
denoted for convenience and to distinguish from (272) as

$\phi^{(KS)}_{k}=\frac{2M^{(\eta)}q\left(1-pq\right)}{p^{2}+q^{2}}\,,\quad\phi^{(KS)}_{m}=\frac{2N^{(\eta)}p\left(1-pq\right)}{p^{2}+q^{2}}\,.$
(273)

now at least also satisfy the flat space wave equation.

Even if (273) and (272) are not quite the same, we would like to understand
the relation to the Weyl zeroth copy

$S=-2\frac{(\tilde{M}^{(\eta)})^{2}}{\tilde{M}}\frac{1-pq}{q-ip}\,,$ (274)

which of course also satisfies the flat space wave equation. In the earlier,
ordinary Kerr-Schild backgrounds, the relation was that the Kerr-Schild zeroth
copy is, up to a constant, the real part of the Weyl zeroth copy, see (5).
Here, taking for simplicity $\tilde{M}^{(\eta)}=\tilde{M}$, that gives

$S+S^{*}=-2\frac{1-pq}{p^{2}+q^{2}}(N^{(\eta)}p+M^{(\eta)}q)=\phi^{(KS)}_{k}+\phi^{(KS)}_{m}\,.$
(275)

## References

  * (1) Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, _The Duality Between Color and Kinematics and its Applications_ , 1909.01358.
  * (2) Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, _The SAGEX review on scattering amplitudes Chapter 2: An invitation to color-kinematics duality and the double copy_ , _J. Phys. A_ 55 (2022) 443003 [2203.13013].
  * (3) T. Adamo, J.J.M. Carrasco, M. Carrillo-González, M. Chiodaroli, H. Elvang, H. Johansson et al., _Snowmass White Paper: the Double Copy and its Applications_ , in _Snowmass 2021_ , 4, 2022 [2204.06547].
  * (4) Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, _Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order_ , _Phys. Rev. Lett._ 122 (2019) 201603 [1901.04424].
  * (5) Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, _Black Hole Binary Dynamics from the Double Copy and Effective Theory_ , _JHEP_ 10 (2019) 206 [1908.01493].
  * (6) Z. Bern, J. Parra-Martinez, R. Roiban, E. Sawyer and C.-H. Shen, _Leading Nonlinear Tidal Effects and Scattering Amplitudes_ , _JHEP_ 05 (2021) 188 [2010.08559].
  * (7) Z. Bern, J. Parra-Martinez, R. Roiban, M.S. Ruf, C.-H. Shen, M.P. Solon et al., _Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at O(G4)_ , _Phys. Rev. Lett._ 128 (2022) 161103 [2112.10750].
  * (8) Z. Bern, J. Parra-Martinez, R. Roiban, M.S. Ruf, C.-H. Shen, M.P. Solon et al., _Scattering amplitudes and conservative dynamics at the fourth post-Minkowskian order_ , _PoS_ LL2022 (2022) 051.
  * (9) H. Kawai, D.C. Lewellen and S.H.H. Tye, _A Relation Between Tree Amplitudes of Closed and Open Strings_ , _Nucl. Phys. B_ 269 (1986) 1.
  * (10) T. Adamo, E. Casali, L. Mason and S. Nekovar, _Scattering on plane waves and the double copy_ , _Class. Quant. Grav._ 35 (2018) 015004 [1706.08925].
  * (11) T. Adamo, G. Bogna, L. Mason and A. Sharma, _Scattering on self-dual Taub-NUT_ , 2309.03834.
  * (12) A. Sivaramakrishnan, _Towards color-kinematics duality in generic spacetimes_ , _JHEP_ 04 (2022) 036 [2110.15356].
  * (13) A. Herderschee, R. Roiban and F. Teng, _On the differential representation and color-kinematics duality of AdS boundary correlators_ , _JHEP_ 05 (2022) 026 [2201.05067].
  * (14) C. Cheung, J. Parra-Martinez and A. Sivaramakrishnan, _On-shell correlators and color-kinematics duality in curved symmetric spacetimes_ , _JHEP_ 05 (2022) 027 [2201.05147].
  * (15) A. Ilderton and W. Lindved, _Toward double copy on arbitrary backgrounds_ , 2405.10016.
  * (16) T. Adamo, _Lectures on twistor theory_ , _PoS_ Modave2017 (2018) 003 [1712.02196].
  * (17) R. Monteiro, D. O’Connell and C.D. White, _Black holes and the double copy_ , _JHEP_ 12 (2014) 056 [1410.0239].
  * (18) A. Luna, R. Monteiro, I. Nicholson, D. O’Connell and C.D. White, _The double copy: Bremsstrahlung and accelerating black holes_ , _JHEP_ 06 (2016) 023 [1603.05737].
  * (19) A. Luna, R. Monteiro, D. O’Connell and C.D. White, _The classical double copy for Taub–NUT spacetime_ , _Phys. Lett. B_ 750 (2015) 272 [1507.01869].
  * (20) K. Kim, K. Lee, R. Monteiro, I. Nicholson and D. Peinador Veiga, _The Classical Double Copy of a Point Charge_ , _JHEP_ 02 (2020) 046 [1912.02177].
  * (21) A. Luna, S. Nagy and C. White, _The convolutional double copy: a case study with a point_ , _JHEP_ 09 (2020) 062 [2004.11254].
  * (22) A. Luna, R. Monteiro, I. Nicholson and D. O’Connell, _Type D Spacetimes and the Weyl Double Copy_ , _Class. Quant. Grav._ 36 (2019) 065003 [1810.08183].
  * (23) H. Godazgar, M. Godazgar, R. Monteiro, D. Peinador Veiga and C.N. Pope, _Asymptotic Weyl double copy_ , _JHEP_ 11 (2021) 126 [2109.07866].
  * (24) D.S. Berman, E. Chacón, A. Luna and C.D. White, _The self-dual classical double copy, and the Eguchi-Hanson instanton_ , _JHEP_ 01 (2019) 107 [1809.04063].
  * (25) K. Lee, _Kerr-Schild Double Field Theory and Classical Double Copy_ , _JHEP_ 10 (2018) 027 [1807.08443].
  * (26) W. Cho and K. Lee, _Heterotic Kerr-Schild Double Field Theory and Classical Double Copy_ , _JHEP_ 07 (2019) 030 [1904.11650].
  * (27) A. Cristofoli, _Gravitational shock waves and scattering amplitudes_ , _JHEP_ 11 (2020) 160 [2006.08283].
  * (28) D.A. Easson, T. Manton and A. Svesko, _Sources in the Weyl Double Copy_ , _Phys. Rev. Lett._ 127 (2021) 271101 [2110.02293].
  * (29) D.A. Easson, T. Manton and A. Svesko, _Einstein-Maxwell theory and the Weyl double copy_ , _Phys. Rev. D_ 107 (2023) 044063 [2210.16339].
  * (30) D.A. Easson, C. Keeler and T. Manton, _Classical double copy of nonsingular black holes_ , _Phys. Rev. D_ 102 (2020) 086015 [2007.16186].
  * (31) K. Mkrtchyan and M. Svazas, _Solutions in Nonlinear Electrodynamics and their double copy regular black holes_ , _JHEP_ 09 (2022) 012 [2205.14187].
  * (32) A. Ilderton, _Screw-symmetric gravitational waves: a double copy of the vortex_ , _Phys. Lett. B_ 782 (2018) 22 [1804.07290].
  * (33) J. Barrientos, A. Cisterna, M. Hassaine and J. Oliva, _Revisiting Buchdahl transformations: New static and rotating black holes in vacuum, double copy, and hairy extensions_ , 2404.12194.
  * (34) M. Ortaggio, V. Pravda and A. Pravdova, _Kerr-Schild double copy for Kundt spacetimes of any dimension_ , _JHEP_ 2024 (2024) 069 [2312.00706].
  * (35) T. Adamo, A. Cristofoli and P. Tourkine, _The ultrarelativistic limit of Kerr_ , _JHEP_ 02 (2023) 107 [2209.05730].
  * (36) C. Keeler, T. Manton and N. Monga, _From Navier-Stokes to Maxwell via Einstein_ , _JHEP_ 08 (2020) 147 [2005.04242].
  * (37) C. Keeler and N. Monga, _On type-II Spacetimes and the Double Copy for Fluids Metrics_ , 2404.03195.
  * (38) H. Godazgar, M. Godazgar, R. Monteiro, D. Peinador Veiga and C.N. Pope, _Weyl Double Copy for Gravitational Waves_ , _Phys. Rev. Lett._ 126 (2021) 101103 [2010.02925].
  * (39) C.D. White, _Twistorial Foundation for the Classical Double Copy_ , _Phys. Rev. Lett._ 126 (2021) 061602 [2012.02479].
  * (40) E. Chacón, S. Nagy and C.D. White, _The Weyl double copy from twistor space_ , _JHEP_ 05 (2021) 2239 [2103.16441].
  * (41) E. Chacón, S. Nagy and C.D. White, _Alternative formulations of the twistor double copy_ , _JHEP_ 03 (2022) 180 [2112.06764].
  * (42) S. Han, _Weyl double copy and massless free-fields in curved spacetimes_ , _Class. Quant. Grav._ 39 (2022) 225009 [2204.01907].
  * (43) R. Alawadhi, D.S. Berman, B. Spence and D. Peinador Veiga, _S-duality and the double copy_ , _JHEP_ 03 (2020) 059 [1911.06797].
  * (44) R. Alawadhi, D.S. Berman and B. Spence, _Weyl doubling_ , _JHEP_ 09 (2020) 127 [2007.03264].
  * (45) G. Alkac, M.K. Gumus, O. Kasikci, M.A. Olpak and M. Tek, _Regularized Weyl double copy_ , _Phys. Rev. D_ 109 (2024) 084047 [2310.06048].
  * (46) K. Armstrong-Williams and C.D. White, _A spinorial double copy for $\mathcal{N}$ = 0 supergravity_, _JHEP_ 05 (2023) 047 [2303.04631].
  * (47) S. Chawla and C. Keeler, _Aligned fields double copy to Kerr-NUT-(A)dS_ , _JHEP_ 04 (2023) 005 [2209.09275].
  * (48) A. Luna, N. Moynihan and C.D. White, _Why is the Weyl double copy local in position space?_ , _JHEP_ 12 (2022) 046 [2208.08548].
  * (49) E.T. Newman and R. Penrose, _New conservation laws for zero rest-mass fields in asymptotically flat space-time_ , _Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences_ 305 (1968) 175.
  * (50) T. Adamo and U. Kol, _Classical double copy at null infinity_ , _Class. Quant. Grav._ 39 (2022) 105007 [2109.07832].
  * (51) R. Penrose, _Any space-time has a plane wave as a limit_ , in _Differential geometry and relativity_ , pp. 271–275, Springer (1976).
  * (52) R. Penrose and W. Rindler, _Spinors and space-time: Volume 1, Two-spinor calculus and relativistic fields_ , vol. 1, Cambridge University Press (1984).
  * (53) R. Penrose and W. Rindler, _SPINORS AND SPACE-TIME. VOL. 2: SPINOR AND TWISTOR METHODS IN SPACE-TIME GEOMETRY_ , Cambridge Monographs on Mathematical Physics, Cambridge University Press (4, 1988), 10.1017/CBO9780511524486.
  * (54) D.A. Kosower, R. Monteiro and D. O’Connell, _The SAGEX review on scattering amplitudes Chapter 14: Classical gravity from scattering amplitudes_ , _J. Phys. A_ 55 (2022) 443015 [2203.13025].
  * (55) M. Blau, _Plane waves and penrose limits_ , _Lecture Notes for the ICTP School on Mathematics in String and Field Theory (June 2-13 2003)_ (2011) .
  * (56) S.W. Hawking and G.F.R. Ellis, _The Large Scale Structure of Space-Time_ , Cambridge Monographs on Mathematical Physics, Cambridge University Press (2, 2023), 10.1017/9781009253161.
  * (57) R. Penrose, _Techniques in differential topology in relativity_ , SIAM (1972).
  * (58) N. Rosen, _Plane polarized waves in the general theory of relativity_ , _Phys. Z. Sowjetunion_ 12 (1937) 366.
  * (59) I. Robinson, _Report to the eddington group_ , 1956.
  * (60) R. Penrose, _A remarkable property of plane waves in general relativity_ , _Reviews of Modern Physics_ 37 (1965) 215.
  * (61) H. Brinkmann, _Einstein spaces which are mapped conformally on each other_ , _Mathematische Annalen_ 94 (1925) 119.
  * (62) M. Blau, D. Frank and S. Weiss, _Fermi coordinates and Penrose limits_ , _Class. Quant. Grav._ 23 (2006) 3993 [hep-th/0603109].
  * (63) P. Tod, _Spacetimes with all Penrose limits diagonalisable_ , _Class. Quant. Grav._ 37 (2020) 075021 [1909.07756].
  * (64) D.Z. Freedman and A. Van Proeyen, _Supergravity_ , Cambridge Univ. Press, Cambridge, UK (5, 2012), 10.1017/CBO9781139026833.
  * (65) R. Gueven, _Plane wave limits and T duality_ , _Phys. Lett. B_ 482 (2000) 255 [hep-th/0005061].
  * (66) A. Biggs and J. Maldacena, _Scaling similarities and quasinormal modes of D0 black hole solutions_ , _JHEP_ 11 (2023) 155 [2303.09974].
  * (67) C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, _Carroll symmetry of plane gravitational waves_ , _Class. Quant. Grav._ 34 (2017) 175003 [1702.08284].
  * (68) L. Ciambelli, L. Freidel and R.G. Leigh, _Null Raychaudhuri: canonical structure and the dressing time_ , _JHEP_ 01 (2024) 166 [2309.03932].
  * (69) G.T. Horowitz and A.R. Steif, _Space-Time Singularities in String Theory_ , _Phys. Rev. Lett._ 64 (1990) 260.
  * (70) G.T. Horowitz and A.R. Steif, _Strings in Strong Gravitational Fields_ , _Phys. Rev. D_ 42 (1990) 1950.
  * (71) E. Kiritsis and C. Kounnas, _String propagation in gravitational wave backgrounds_ , _Phys. Lett. B_ 320 (1994) 264 [hep-th/9310202].
  * (72) J.G. Russo and A.A. Tseytlin, _On solvable models of type 2B superstring in NS NS and RR plane wave backgrounds_ , _JHEP_ 04 (2002) 021 [hep-th/0202179].
  * (73) G. Papadopoulos, J.G. Russo and A.A. Tseytlin, _Solvable model of strings in a time dependent plane wave background_ , _Class. Quant. Grav._ 20 (2003) 969 [hep-th/0211289].
  * (74) D.E. Berenstein, J.M. Maldacena and H.S. Nastase, _Strings in flat space and pp waves from N=4 superYang-Mills_ , _JHEP_ 04 (2002) 013 [hep-th/0202021].
  * (75) L. Eberhardt and K. Ferreira, _The plane-wave spectrum from the worldsheet_ , _JHEP_ 10 (2018) 109 [1805.12155].
  * (76) T. Adamo, E. Casali, L. Mason and S. Nekovar, _Plane wave backgrounds and colour-kinematics duality_ , _JHEP_ 02 (2019) 198 [1810.05115].
  * (77) T. Adamo and A. Ilderton, _Classical and quantum double copy of back-reaction_ , _JHEP_ 09 (2020) 200 [2005.05807].
  * (78) I.P. Lobo and G.G. Carvalho, _The geometry of null-like disformal transformations_ , _Int. J. Geom. Meth. Mod. Phys._ 16 (2019) 1950180 [1707.01784].
  * (79) R. Gonzo and C. Shi, _Geodesics from classical double copy_ , _Phys. Rev. D_ 104 (2021) 105012 [2109.01072].
  * (80) A. Ball, A. Bencke, Y. Chen and A. Volovich, _Hidden symmetry in the double copy_ , _JHEP_ 10 (2023) 085 [2307.01338].
  * (81) S. Chawla and C. Keeler, _Black hole horizons from the double copy_ , _Class. Quant. Grav._ 40 (2023) 225004 [2306.02417].
  * (82) J.-L. He and J.-H. Huang, _Cosmological horizons from classical double copy_ , _Phys. Lett. B_ 851 (2024) 138579 [2312.00972].
  * (83) M. Walker and R. Penrose, _On quadratic first integrals of the geodesic equations for type [22] spacetimes_ , _Commun. Math. Phys._ 18 (1970) 265.
  * (84) C. Patricot, _Kaigorodov spaces and their Penrose limits_ , _Class. Quant. Grav._ 20 (2003) 2087 [hep-th/0302073].
  * (85) I. Bialynicki-Birula, _Particle beams guided by electromagnetic vortices: New solutions of the Lorentz, Schrodinger, Klein-Gordon and Dirac equations_ , _Phys. Rev. Lett._ 93 (2004) 020402 [physics/0403078].
  * (86) R.H. Boyer and R.W. Lindquist, _Maximal analytic extension of the kerr metric_ , _Journal of mathematical physics_ 8 (1967) 265.
  * (87) V.P. Frolov, P. Krtous and D. Kubiznak, _Black holes, hidden symmetries, and complete integrability_ , _Living Rev. Rel._ 20 (2017) 6 [1705.05482].
  * (88) E.T. Newman and A.I. Janis, _Note on the Kerr spinning particle metric_ , _J. Math. Phys._ 6 (1965) 915.
  * (89) N. Arkani-Hamed, Y.-t. Huang and D. O’Connell, _Kerr black holes as elementary particles_ , _JHEP_ 01 (2020) 046 [1906.10100].
  * (90) A. Guevara, B. Maybee, A. Ochirov, D. O’connell and J. Vines, _A worldsheet for Kerr_ , _JHEP_ 03 (2021) 201 [2012.11570].
  * (91) B. Carter, _Global structure of the Kerr family of gravitational fields_ , _Phys. Rev._ 174 (1968) 1559.
  * (92) J.M. Bardeen, _Timelike and null geodesics in the Kerr metric_ , _Proceedings, Ecole d’Eté de Physique Théorique: Les Astres Occlus : Les Houches, France, August, 1972, 215-240_ (1973) 215.
  * (93) S. Chandrasekhar, _The mathematical theory of black holes_ (1985).
  * (94) S.E. Gralla and A. Lupsasca, _Null geodesics of the Kerr exterior_ , _Phys. Rev. D_ 101 (2020) 044032 [1910.12881].
  * (95) G. Compère, Y. Liu and J. Long, _Classification of radial Kerr geodesic motion_ , _Phys. Rev. D_ 105 (2022) 024075 [2106.03141].
  * (96) A. Cieślik, E. Hackmann and P. Mach, _Kerr geodesics in terms of Weierstrass elliptic functions_ , _Phys. Rev. D_ 108 (2023) 024056 [2305.07771].
  * (97) T. Igata, H. Ishihara and Y. Yasunishi, _Observability of spherical photon orbits in near-extremal Kerr black holes_ , _Phys. Rev. D_ 100 (2019) 044058 [1904.00271].
  * (98) K. Fransen, _Quasinormal modes from Penrose limits_ , _Class. Quant. Grav._ 40 (2023) 205004 [2301.06999].
  * (99) T.J. Hollowood, G.M. Shore and R.J. Stanley, _The Refractive Index of Curved Spacetime II: QED, Penrose Limits and Black Holes_ , _JHEP_ 08 (2009) 089 [0905.0771].
  * (100) G. Papadopoulos, _Separability, plane wave limits and rotating black holes_ , _Class. Quant. Grav._ 38 (2021) 195018 [2007.04702].
  * (101) D. Kubiznak, V.P. Frolov, P. Krtous and P. Connell, _Parallel-propagated frame along null geodesics in higher-dimensional black hole spacetimes_ , _Phys. Rev. D_ 79 (2009) 024018 [0811.0012].
  * (102) M. Cariglia, T. Houri, P. Krtous and D. Kubiznak, _On Integrability of the Geodesic Deviation Equation_ , _Eur. Phys. J. C_ 78 (2018) 661 [1805.07677].
  * (103) R. Penrose, _In $\\{$2 2$\\}$ vacuums, the null-datum on null rays is exactly r- 5_, _General Relativity and Gravitation_ 27 (1995) 1323.
  * (104) R. Penrose, _Golden oldie: null hypersurface initial data for classical fields of arbitrary spin and for general relativity_ , _General Relativity and Gravitation_ 12 (1980) 225.
  * (105) S. Aksteiner, _Geometry and analysis on black hole spacetimes_ , Ph.D. thesis, Leibniz U., Hannover, 2014.  10.15488/8214.
  * (106) R. Geroch, A. Held and R. Penrose, _A space-time calculus based on pairs of null directions_ , _Journal of Mathematical Physics_ 14 (1973) 874.
  * (107) K.E. Kunze, _Behavior of curvature and matter in the Penrose limit_ , _Phys. Rev. D_ 71 (2005) 063518 [gr-qc/0411115].
  * (108) R. Monteiro, I. Nicholson and D. O’Connell, _Spinor-helicity and the algebraic classification of higher-dimensional spacetimes_ , _Class. Quant. Grav._ 36 (2019) 065006 [1809.03906].
  * (109) T.J. Hollowood and G.M. Shore, _The Refractive index of curved spacetime: The Fate of causality in QED_ , _Nucl. Phys. B_ 795 (2008) 138 [0707.2303].
  * (110) A.I. Harte, _Strong lensing, plane gravitational waves and transient flashes_ , _Class. Quant. Grav._ 30 (2013) 075011 [1210.1449].
  * (111) A.I. Harte, _Optics in a nonlinear gravitational plane wave_ , _Class. Quant. Grav._ 32 (2015) 175017 [1502.03658].
  * (112) A.I. Harte, _Tails of plane wave spacetimes: Wave-wave scattering in general relativity_ , _Physical Review D_ 88 (2013) 084059.
  * (113) P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, _The Memory Effect for Plane Gravitational Waves_ , _Phys. Lett. B_ 772 (2017) 743 [1704.05997].
  * (114) P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, _Soft gravitons and the memory effect for plane gravitational waves_ , _Phys. Rev. D_ 96 (2017) 064013 [1705.01378].
  * (115) E.E. Flanagan, A.M. Grant, A.I. Harte and D.A. Nichols, _Persistent gravitational wave observables: Nonlinear plane wave spacetimes_ , _Phys. Rev. D_ 101 (2020) 104033 [1912.13449].
  * (116) A.I. Harte and T.D. Drivas, _Caustics and wave propagation in curved spacetimes_ , _Phys. Rev. D_ 85 (2012) 124039 [1202.0540].
  * (117) E.S. Perlman, S.A. Rappaport, W.A. Christiansen, Y.J. Ng, J. DeVore and D. Pooley, _New Constraints on Quantum Gravity from X-ray and Gamma-Ray Observations_ , _Astrophys. J._ 805 (2015) 10 [1411.7262].
  * (118) V. Cardoso, E. Franzin and P. Pani, _Is the gravitational-wave ringdown a probe of the event horizon?_ , _Phys. Rev. Lett._ 116 (2016) 171101 [1602.07309].
  * (119) T. Baker, E. Bellini, P.G. Ferreira, M. Lagos, J. Noller and I. Sawicki, _Strong constraints on cosmological gravity from GW170817 and GRB 170817A_ , _Phys. Rev. Lett._ 119 (2017) 251301 [1710.06394].
  * (120) P.V.P. Cunha and C.A.R. Herdeiro, _Shadows and strong gravitational lensing: a brief review_ , _Gen. Rel. Grav._ 50 (2018) 42 [1801.00860].
  * (121) M.D. Johnson et al., _Universal interferometric signatures of a black hole’s photon ring_ , _Sci. Adv._ 6 (2020) eaaz1310 [1907.04329].
  * (122) S. Hadar, D. Kapec, A. Lupsasca and A. Strominger, _Holography of the photon ring_ , _Class. Quant. Grav._ 39 (2022) 215001 [2205.05064].
  * (123) H. Paugnat, A. Lupsasca, F. Vincent and M. Wielgus, _Photon ring test of the Kerr hypothesis: Variation in the ring shape_ , _Astron. Astrophys._ 668 (2022) A11 [2206.02781].
  * (124) A. Cárdenas-Avendaño and A. Lupsasca, _Prediction for the interferometric shape of the first black hole photon ring_ , _Phys. Rev. D_ 108 (2023) 064043 [2305.12956].
  * (125) V.S.H. Lee, K.M. Zurek and Y. Chen, _Astronomical image blurring from transversely correlated quantum gravity fluctuations_ , _Phys. Rev. D_ 109 (2024) 084005 [2312.06757].
  * (126) H. Jia, E. Quataert, A. Lupsasca and G.N. Wong, _Photon Ring Interferometric Signatures Beyond The Universal Regime_ , 2405.08804.
  * (127) R. Gueven, _Plane Waves in Effective Field Theories of Superstrings_ , _Phys. Lett. B_ 191 (1987) 275.
  * (128) D. Amati and C. Klimcik, _Nonperturbative Computation of the Weyl Anomaly for a Class of Nontrivial Backgrounds_ , _Phys. Lett. B_ 219 (1989) 443.
  * (129) M. Blau, M. Borunda, M. O’Loughlin and G. Papadopoulos, _Penrose limits and space-time singularities_ , _Class. Quant. Grav._ 21 (2004) L43 [hep-th/0312029].
  * (130) M. Blau, M. Borunda, M. O’Loughlin and G. Papadopoulos, _The Universality of Penrose limits near space-time singularities_ , _JHEP_ 07 (2004) 068 [hep-th/0403252].
  * (131) S.B. Giddings, D.J. Gross and A. Maharana, _Gravitational effects in ultrahigh-energy string scattering_ , _Phys. Rev. D_ 77 (2008) 046001 [0705.1816].
  * (132) B. Craps, F. De Roo and O. Evnin, _Can free strings propagate across plane wave singularities?_ , _JHEP_ 03 (2009) 105 [0812.2900].
  * (133) T.J. Hollowood and G.M. Shore, _The Effect of Gravitational Tidal Forces on Renormalized Quantum Fields_ , _JHEP_ 02 (2012) 120 [1111.3174].
  * (134) E.J. Martinec and N.P. Warner, _The Harder They Fall, the Bigger They Become: Tidal Trapping of Strings by Microstate Geometries_ , _JHEP_ 04 (2021) 259 [2009.07847].
  * (135) I. Bena, A. Houppe and N.P. Warner, _Delaying the Inevitable: Tidal Disruption in Microstate Geometries_ , _JHEP_ 02 (2021) 103 [2006.13939].
  * (136) M. Dodelson and H. Ooguri, _Singularities of thermal correlators at strong coupling_ , _Phys. Rev. D_ 103 (2021) 066018 [2010.09734].
  * (137) K. Nishii and D. Yoshida, _String excitation by initial singularity of inflation_ , _JHEP_ 10 (2021) 025 [2105.12339].
  * (138) A. Balivada, P.R. Padhi and A. Virmani, _Tidal forces in Kerr-AdS and Grey galaxies_ , _Class. Quant. Grav._ 41 (2024) 125008 [2309.14672].
  * (139) G.T. Horowitz, M. Kolanowski, G.N. Remmen and J.E. Santos, _Sudden breakdown of effective field theory near cool Kerr-Newman black holes_ , _JHEP_ 05 (2024) 122 [2403.00051].
  * (140) B. Guo, S.D. Hampton and N.P. Warner, _Inscribing geodesic circles on the face of the superstratum_ , _JHEP_ 05 (2024) 224 [2401.17366].
  * (141) J.M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * (142) E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253 [hep-th/9802150].
  * (143) O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, _Large N field theories, string theory and gravity_ , _Phys. Rept._ 323 (2000) 183 [hep-th/9905111].
  * (144) J.C. Plefka, _Lectures on the plane wave string / gauge theory duality_ , _Fortsch. Phys._ 52 (2004) 264 [hep-th/0307101].
  * (145) D. Sadri and M.M. Sheikh-Jabbari, _The Plane wave / superYang-Mills duality_ , _Rev. Mod. Phys._ 76 (2004) 853 [hep-th/0310119].
  * (146) A. Pound and B. Wardell, _Black hole perturbation theory and gravitational self-force_ , 2101.04592.
  * (147) W. Kinnersley, _Type D Vacuum Metrics_ , _J. Math. Phys._ 10 (1969) 1195.
  * (148) J.F. Plebanski and M. Demianski, _Rotating, charged, and uniformly accelerating mass in general relativity_ , _Annals Phys._ 98 (1976) 98.
  * (149) J.B. Griffiths and J. Podolsky, _A New look at the Plebanski-Demianski family of solutions_ , _Int. J. Mod. Phys. D_ 15 (2006) 335 [gr-qc/0511091].


# Fuzzy Based Wellness Analyzer

Rohit Ner, Dibya Prakash Das, Rishabh Kumar, Shubhika Garg  Department of
Mathematics, Indian Institute of Technology Kharagpur, India
{rohitner, dibyadas, rishabhkumar<EMAIL_ADDRESS>
###### Abstract

Social health and emotional wellness is a matter of concern in today’s urban
world. Being the part of a metropolis has an effect on mental health through
the influence of increased stressors and factors such as overcrowded and
polluted environment, high levels of violence, and reduced social support. It
is important to realize that only healthy citizens can constitute together a
smart city. In this paper, we present a fuzzy-based approach for analyzing the
well being of a person. We track the general day to day activities of a person
and analyze its performance. To do so, we divide the factors affecting the
wellness of a person into three components which are the physical, productive
and social. Using these parameters, we output a coefficient for the overall
well being of a person.

###### Index Terms:

Fuzzy Logic, Public health, Smart cities

##  I Introduction

Over half of the world’s population lives in urban areas. Urbanization
presents opportunities and risks, as well as enormous challenges for
maintaining and improving human health and well being. Degrading wellness is
proving to be an important issue in the modern world. Cases of suicide and
depression have been rising over the years. One of the main barriers in
handling these situations is the improper reporting of the problem by the
victim and their lack of communication with the concerned close ones. In
recent times several such issues have been addressed by researchers using
fuzzy logic. Guntuku et al. [1] identify mentally ill users by patterns in
their language and online activity and propose automated analysis by large-
scale passive monitoring. Kelley et al. [2] discuss the challenges and
opportunities in leveraging self-tracking for mental wellness. A fuzzy
optimization model based on chanced constrained optimization to give health
related recommendations is proposed by Mezei et al. [3]. Liu et al. [4]
examine the correlation between depression and speech using acoustic features
of patients. Lutz et al. [5] propose a framework that uses fuzzy-set theory to
measure human well-being in consistence with Sen’s Capability Approach.

From the review, it has been seen that not many references address the problem
of online wellness tracking using fuzzy logic. We introduce a novel approach
to analyze the well being of a person by his tracking the daily activities. We
investigate the common symptoms of a depressed person which include a loss of
interest in daily activities, loss of appetite, change in the sleep cycle,
change in health and lack of social interactions. Our approach compensates for
the inability to define ideal health, social and work conditions due to their
subjective nature by using a fuzzy inference mechanism.

###  I-A About Fuzzy Logic

Fuzzy logic is used widely in all domains since the past few decades. Its
cross-platform utility since its inception is one of the primary reasons for
such instantaneous growth. Fuzzy logic also sometimes called as grey logic
discards the principles of Propositional Logic which revolves around two truth
values and instead proposes a new theory of ‘Partial Truth’, also referred as
the degree of membership as per the suggestions of Zadeh [6].

Let $\mathbf{S}$ be a non empty set, called the universe set. Now, consider
any crisp set $A\subset\mathbf{S}$. A characteristic function
$\mathbold{\chi}_{A}$ is defined as

$\mathbold{\chi}_{A}(x)=\begin{cases}1,&\text{if }x\in A\\\
0,&\text{otherwise}\end{cases}$

$\mathbold{\chi}_{A}$ maps every element of $\mathbf{S}$ to a value of either
$1$ or $0$. Let $B$ be a fuzzy set such that $B\subset\mathbf{S}$. For any
$x\in S$ the function $\mathbold{\mu}_{B}(x)$ defined as
$\mathbold{\mu}_{B}\mathrel{\mathop{\mathchar
58\relax}}\mathbf{S}\rightarrow[0,1]$ yields the degree of belonging of $x$ to
the set $B$ ranging between $0$ and $1$. $\mathbold{\mu}_{B}(x)$ is known as
the membership function. The fuzzy set[7] theory provides a new dimension
which is the limitation of classical theory and helps us to express
uncertainty in a bold manner.

###  I-B Problem Formulation

The problem can be divided into three major parts:

  * •

Context Recognition: Collecting sensor measurements and labels describing the
user’s context through various methods including self-reporting.

  * •

Base Level Inference: Fuzzify the crisp input labels and calculate the values
membership functions of broadly categorized components.

  * •

Top Level Inference: The components are defuzzified and fed into the top
controller to get a output the level of percentage well being of the person.

Figure 1: Fuzzy logic system architecture for determining a person’s well
being.

##  II Working Principle

The general work flow of this paper is shown in Figure 1.

###  II-A Context Recognition

Context recognition is used to identify the user’s real-time contextual
information from data of sensors, using relevant signal processing approaches.
There are several approaches mentioned in the literature that incorporate
stimuli from smartphone sensors which include audio-based context recognition
[8], involving dynamics-based using smart-watches [9]. Most of the approaches
need data that is collected under strictly constrained conditions with the
investigator instructing subjects to execute fabricated chores. There is a
need for an approach capable of performing given the practical in-the-wild
conditions. Vaizman et al. [10] introduce an approach which recognizes the
context on the basis of natural behavioral content. The system consists of
single-sensor classifiers and sensor-fusion. We use this approach for our
needs.

On investigation, the common symptoms of depression turn out to be a loss of
interest in daily activities, appetite loss, altered sleep cycle and lack of
social interaction. We divided the factors affecting the overall wellness of
the person into three components namely physical, productive and social. A
person in an urban area carries one’s smartphone almost everywhere. We use the
ExtraSensory App [11] for the purpose of collecting the data using the in-the-
wild approach. The application provides a self-reporting interface through
which users can report labels signifying their current activity.

TABLE I: Factors affecting physical component Factor | Label
---|---
Sleep |  LYING_DOWN, SLEEPING
Diet |  RESTAURANT, EATING
Exercise |  RUNNING, WALKING, BICYCLING, GYM,
| EXERCISE
TABLE II: Factors affecting productive component Factor | Label
---|---
Work |  IN_CLASS, IN_A_MEETING, AT_SCHOOL,
| LOC_main_workplace
Leisure |  WATCHING_TV, SINGING, SHOPPING
TABLE III: Factors affecting social component Factor | Label
---|---
Interaction |  WITH_FRIENDS, WITH_CO-WORKERS,
|  TALKING, AT_A_PARTY
Online Activity |  SURFING_THE_INTERNET,
| PHONE_IN_HAND

####  II-A1 Physical

Physical health ensures that the body is functioning properly. One must gain
fiber, use one’s parts and maintain an amenable environment for stable growth.
The physical health of a person is analyzed by subdividing the component into
three categories sleep, diet and exercise. Table I shows the labels in the
application used to determine the quantity of these categories.

####  II-A2 Productive

The productivity of a person depends on several factors like the capacity of
the person to reach a certain level of efficiency by putting in energy in the
form of manual or cognitive activity. According to the findings of Mokana et
al. [12], work-life balance was found to be significantly dependent on work
culture and technological improvements. An appropriate balance between the
work time and the leisure time is suggested for good health. We hence choose
the categories of work and leisure as the primarily affecting factors of
productivity. Table II shows the labels used for determining the amount of
workload and leisure time spent by the person.

####  II-A3 Social

Social wellness is the positive effect relationships have on one’s mental and
physical health. Quantitatively, it is a measure of health based on the number
of close and personal friends one has as well as how often one spends time
with them in person. Online social networking has become a part of health
promoting interventions. Peer pressure, social support and data exchange in
these virtual communities are seen to affect health behaviors. We divide the
categories affecting this component as interaction and online activity. The
corresponding labels are shown in Table III.

###  II-B Base Level Inference

Fuzzy inference is the process of mapping from a given input to an output
using fuzzy logic. This system helps to model linguistic variables in terms of
a well-established quantitative input and providing a crisp output. Multiple
fuzzy control models exist in theory. The general architecture is shown in
Figure 2. The two most commonly used classes of fuzzy controllers are Mamdani
controller and Sugeno controller.

Figure 2: General architecture of a fuzzy controller

Mamdani Controller The Mamdani controller [13] is a universal fuzzy controller
i.e., it can take the form of any non-linearized controller depending on the
membership functions. Mamdani controller is generally used to convert
linguistic variables to a fuzzy $IF-THEN$ rule-base using $t-norm$ as $min(x)$
and $t-conorm$ as $max(x)$ as per the antecedent to compute the take logical
decisions on the antecedents. Then the output is converted to a defuzzified
value for producing the final score using an optimal defuzzification method
like center-of-gravity defuzzifier. Multiple research shows that Mamdani
controller is well suited for human inputs than the Sugeno counterpart.
Although the later is more computationally efficient, Mamdani controller is
much more intuitive and simplifies rule base formulation.

For calculating the scores of the three components, the labels of the
categories on which each component depends are fed into the base-level
inference system. The membership functions obtained as outputs are then
defuzzified using the center-of-gravity criterion. The defuzzified output
provides the percentage of the respective component exhibited by the person.

###  II-C Top Level Inference

The crisp values of the three components are fed into the subsequent
controller to obtain the membership function of the overall wellness of the
person which is then converted to a percentage using the center-of-gravity
defuzzification.

##  III Experiment

As stated earlier, we test our algorithm on data that is in its organic form
and obtained in the natural conditions of urban living. We use data of 60
users that was obtained by Vaizman et al. [11]. Every user was allotted a
universally unique identifier (UUID). The dataset has thousands of examples,
typically taken in approximate intervals of 1 minute for every user. The
measurements were obtained from the sensors on user’s smart-phone and a smart-
watch. Self-reported context labels are present in most of the examples. All
the relevant context labels could be reported by the user for the immediate
future, and it was also possible to choose the amount of time for which the
same labels will stay relevant.

Figure 3: Captured trend of the online subcomponent using STL decomposition TABLE IV: Rule base for top level inference with memberships as H=High, M=Medium, L=Low Health | Productive | Social | Overall
---|---|---|---
L | L | L | L
M | L | L | L
H | L | L | L
M | M | L | M
H | M | L | M
… | … | … | …
L | M | H | M
H | M | H | H
L | H | H | M
M | H | H | H
H | H | H | H

The membership functions of some of the linguistic terms used in the study are
shown in Figure 4. The rule base entries were decided after an extensive
lookout for the dependence of the three components physical, productive and
social on the overall wellness. An excerpt from the rule base for the top
layer inference is shown in Table IV.

STL Decomposition One of the issues faced was the irregularities in the
reporting done by the participant. To determine the number of times a person
does an activity, we use Seasonal and Trend decomposition using Loess. As seen
in Figure 3, this method helps in ruling out the noise generated in the data
during activity tracking.

TABLE V: Scores and list of moods of a sample from the dataset. Total |  Health |  Productive |  Social
---|---|---|---
UUID | 8023FE1A-D3B0-4E2C-A57A-9321B7FC755F
Labels |  HAPPY, ATTENTIVE, DREAMY
82.067 |  81.877 |  48.550 |  48.137
UUID | CA820D43-E5E2-42EF-9798-BE56F776370B
Labels |  HAPPY, HUNGRY, ATTENTIVE
58.075 |  80.721 |  48.550 |  25.294
UUID | 806289BC-AD52-4CC1-806C-0CDB14D65EB6
Labels |  STRESSED, TIRED, SLEEPY
86.826 |  81.877 |  48.550 |  81.877
UUID | B09E373F-8A54-44C8-895B-0039390B859F
Labels |  ACTIVE, CALM, TIRED
50.634 |  81.877 |  52.262 |  20.416

(a) $\mu_{sleep}$ (b) $\mu_{eat}$ (c) $\mu_{exercise}$ (d) $\mu_{work}$ (e)
$\mu_{leisure}$ (f) $\mu_{interaction}$ (g) $\mu_{online}$

Figure 4: Membership functions for linguistic terms of all categories of base-
layer inference input with respect to time.

The ExtraSensory application has an option to select mood labels when self-
reporting. For the purpose of evaluation, we aggregate the three most reported
labels provided by the user within the time frame of which the observations
are considered. Table V indicates the similarity between the reported moods
and the scores generated by our approach.

##  IV Concluding Remarks

A new fuzzy inference based approach is proposed in the context of wellness
analysis. The proposed algorithm is simple and easy to implement in practice.
The accuracy of the algorithm can be improved by enhancing the passive
reporting of the user using Internet-of-Things and sensor integration at
public spaces. Hybrid neuro-fuzzy models can be used for better prior
estimation. There is a need for a society where one can leverage the
connectivity on a social level owing to the respective platforms. The fuzzy-
based system discussed here has potential applications in the development of
self-awareness and proper circulation of data for an increased mutual
understanding.

## Acknowledgment

We would like to thank Professor Sudhirkumar Barai of the Department of Civil
Engineering, IIT Kharagpur for his continued and unconditional guidance. An
exemplary teacher and a magnificent person, we consider ourselves lucky to
have been taught by him and to have worked under his supervision. Without his
course, “Soft Computing Tools in Engineering”, and his co-operation the
preparation of this paper would not have been possible.

## References

  * [1] S. C. Guntuku, D. B. Yaden, M. L. Kern, L. H. Ungar, and J. C. Eichstaedt, “Detecting depression and mental illness on social media: an integrative review,” Current Opinion in Behavioral Sciences, vol. 18, pp. 43 – 49, 2017.  Big data in the behavioural sciences.
  * [2] C. Kelley, B. Lee, and L. Wilcox, “Self-tracking for mental wellness: Understanding expert perspectives and student experiences,” in Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems, CHI ’17, (New York, NY, USA), pp. 629–641, ACM, 2017.
  * [3] J. Mezei and S. Nikou, “Fuzzy optimization to improve mobile health and wellness recommendation systems,” Knowledge-Based Systems, vol. 142, pp. 108 – 116, 2018.
  * [4] Z. Liu, B. Hu, L. Yan, T. Wang, F. Liu, X. Li, and H. Kang, “Detection of depression in speech,” in 2015 International Conference on Affective Computing and Intelligent Interaction (ACII), pp. 743–747, Sept 2015.
  * [5] M. Baliamoune‐Lutz, “On the measurement of human well-being: Fuzzy set theory and sen’s capability approach,” World Institute for Development Economic Research (UNU-WIDER), Working Papers, 01 2004.
  * [6] L. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338 – 353, 1965.
  * [7] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications.  Upper Saddle River, NJ, USA: Prentice-Hall, Inc., 1995.
  * [8] H. Choi, S. Lee, J. Sung, and S. Park, “Sound-based real-time context recognition on smartphone,” in 2013 IEEE International Conference on Consumer Electronics (ICCE), pp. 669–670, Jan 2013.
  * [9] S. Faye, R. Frank, and T. Engel, “Adaptive activity and context recognition using multimodal sensors in smart devices,” in Mobile Computing, Applications, and Services (S. Sigg, P. Nurmi, and F. Salim, eds.), (Cham), pp. 33–50, Springer International Publishing, 2015.
  * [10] Y. Vaizman, K. Ellis, and G. Lanckriet, “Recognizing detailed human context in the wild from smartphones and smartwatches,” IEEE Pervasive Computing, vol. 16, pp. 62–74, October 2017.
  * [11] Y. Vaizman, K. Ellis, G. Lanckriet, and N. Weibel, “Extrasensory app: Data collection in-the-wild with rich user interface to self-report behavior,” in Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems, CHI ’18, (New York, NY, USA), pp. 554:1–554:12, ACM, 2018.
  * [12] M. Muthu, F. Pangil, and M. Isa, “Individual, organizational and environmental factors affecting work-life balance,” Asian Social Science, 01 2015.
  * [13] E. Mamdani and S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” International Journal of Man-Machine Studies, vol. 7, no. 1, pp. 1 – 13, 1975.