# Notes on self-dual gravity

Pujian Mao and Weicheng Zhao

Notes on self-dual gravity

Pujian Mao and Weicheng Zhao

Center for Joint Quantum Studies and Department of Physics,

School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China

Abstract. In this paper, we study self-dual gravity in the Newman-Penrose
formalism. We specify the self-dual solution space from the Newman-Unti
solutions. We show that the asymptotic symmetries of the self-dual gravity are
still the (extended) BMS symmetries. We transform the self-dual Taub-NUT
solution into the Newman-Unti gauge in analytical form.

##  1 Introduction

Physics at the asymptotic boundary of spacetime has obtained renewed interest
in recent years since the seminal work of Barnich and Troessaert [1, 2, 3]
where the asymptotic symmetry algebra at null infinity in 4 dimensions is
shown to consist of the semidirect sum of supertranslations with
superrotations, the (local) conformal symmetry of the celestial sphere which
extends the original BMS symmetry [4, 5]. Considering asymptotic symmetries as
symmetries of S-Matrix in quantum gravity, the Ward identities of the
supertranslations and superrotations recover the leading and subleading soft
graviton theorems respectively [6, 7, 8, 9]. Alternatively, one can think that
any soft graviton theorem is indicating the existence of some symmetries of
quantum theory of gravity. If one considers only the tree level plus-helicity
soft particles, the entire tower of soft graviton theorems implies a chiral
Kac-Moody symmetry of the wedge algebra of $w_{1+\infty}$ [10, 11]. Those
symmetries even persist at all loop level [12] when restricted to the self-
dual sector, i.e., quantum self-dual gravity which is a theory of gravity with
half of the components of the Weyl tensor vanishing. Although the self-dual
structure in four dimensions excludes real solutions in Lorentzian signature,
it provides a good toy model to demonstrate some quantum aspects of gravity
from different perspectives such as integrablility, string theories and
scattering amplitudes (see, e.g., [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28] and references therein). However a detailed investigation
of asymptotic symmetries of the self-dual gravity in the standard asymptotic
analysis is still missing in the literature. Considering the importance of
asymptotic symmetries in the recent developments introduced above, the aim of
the present work is to explore this more in detail.

In [29], asymptotic symmetries of self-dual gravity were derived in a light-
cone gauge in the context of the classical double copy and only a subset of
the BMS symmetry is preserved by the self-dual condition. While this result is
somewhat surprising if one considers electromagnetism for a simpler analogue
of gravitational effects. The self-dual condition is on the field strength
which can not bring any constraint on a gauge transformation. In this paper,
we study the asymptotic symmetries of self-dual gravity in the Newman-Penrose
(NP) formalism. Adopting the widely used Newman-Unti (NU) gauge [30], we
recover the full (extended) BMS symmetry that preserves the self-dual
condition at null infinity.

Another line of research about asymptotic symmetries is their applications in
deriving the Bekenstein-Hawking entropy for black holes [31, 32, 33]. The
study of asymptotic symmetries normally relies on some particular coordinates
system with desired gauge conditions, e.g., the Bondi gauge [4] or the NU
gauge adopted in the present work. So it is very important to have the black
hole solutions in the coordinates system where the asymptotic symmetries are
studied. Another purpose of this paper is to give a generic treatment to
transform solutions into the NU gauge and we demonstrate the derivation in the
case of the self-dual Taub-NUT solution [34, 35].

This paper is organized as follows. In the next section, we specify the self-
dual conditions on the solutions of the NP equations. In section 3, we show
that the self-dual conditions do not bring any constraint on the asymptotic
symmetries. The asymptotic symmetries of the self-dual gravity are still the
full (extended) BMS symmetries. In section 4, we introduce a generic treatment
transforming solutions into the NU gauge and write the self-dual Taub-NUT
solutions precisely in the NU gauge. The last section is devoted to the
conclusion and discussion for future directions. There are three appendices
which provide useful information for the computations in the main text.

##  2 The self-dual sector in NP formalism

The Newman-Penrose formalism [36] is a special tetrad formalism with four null
basis vectors
$e_{1}=l=e^{2},\;e_{2}=n=e^{1},\;e_{3}=m=-e^{4},\;e_{4}=\bar{m}=-e^{3}$. In
Lorentzian signature, the basis vectors $m$ and $\bar{m}$ are complex and
conjugates of each other. The basis vectors are orthogonalized and normalized
as

$l\cdot m=l\cdot\bar{m}=n\cdot m=n\cdot\bar{m}=0,\quad l\cdot n=1,\quad
m\cdot\bar{m}=-1.$ (1)

The spacetime metric is constructed from the tetrad as

$g_{\mu\nu}=n_{\mu}l_{\nu}+l_{\mu}n_{\nu}-m_{\mu}{\bar{m}}_{\nu}-m_{\nu}\bar{m}_{\mu}.$
(2)

For the NP variables and equations, we would refer to [37] for the notations.

Newman and Unti [30] derived a generic solution of the NP system with certain
gauge and boundary conditions. First, by local Lorentz transformations, they
set

$\displaystyle\pi=\kappa=\epsilon=0,\,\,\;\;\rho=\bar{\rho},\;\;\,\,\tau=\bar{\alpha}+\beta.$
(3)

Such gauge choices mean that $l$ is tangent to a null geodesic with affine
parameter, the rest basis vectors are parallelly transported along $l$, and
$l$ is the gradient of a scalar field. In this case, one can choose the scalar
field and the affine parameter from $l$ as coordinates $x^{1}=u$ and
$x^{2}=r$. For the rest two angular coordinates, the complex stereographic
coordinates $x^{A}=(z,\bar{z})$ were applied. They are related to the usual
angular variables $(\theta,\phi)$ by $z=\cot\frac{\theta}{2}e^{i\phi}$. In
these coordinates, the tetrad and the co-tetrad satisfying conditions in (1)
must have the forms

$\begin{split}&n=\frac{\partial}{\partial u}+U\frac{\partial}{\partial
r}+X^{A}\frac{\partial}{\partial x^{A}},\\\ &l=\frac{\partial}{\partial r},\\\
&m=\omega\frac{\partial}{\partial r}+L^{A}\frac{\partial}{\partial x^{A}},\\\
&\bar{m}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\omega$\kern 0.0pt}}}\frac{\partial}{\partial
r}+\bar{L}^{A}\frac{\partial}{\partial x^{A}},\end{split}$ (4)

and

$\begin{split}&n=\left[-U-X^{A}(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\omega$\kern
0.0pt}}}L_{A}+\omega\bar{L}_{A})\right]\text{d}u+\text{d}r+(\omega\bar{L}_{A}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\omega$\kern
0.0pt}}}L_{A})\text{d}x^{A},\\\ &l=\text{d}u,\\\
&m=-X^{A}L_{A}\text{d}u+L_{A}\text{d}x^{A},\\\
&\bar{m}=-X^{A}\bar{L}_{A}\text{d}u+\bar{L}_{A}\text{d}x^{A},\end{split}$ (5)

where $L_{A}L^{A}=0,\;L_{A}\bar{L}^{A}=-1$. The asymptotic behaviors of the NP
variables imposed in [30] to derive the solution are111Originally, the
boundary conditions in [30] are
$\rho=-r^{-1}+O(r^{-2}),\tau,\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\tau$\kern 0.0pt}}}=O(r^{-1}),X^{A}=O(1)$. However
those conditions can be made stronger by pure gauge transformations. So we
think it is reasonable to impose them in the beginning. The rest boundary
conditions for the NP variables in [30] are not really needed for deriving the
solution. They could be the consequence of the NP equations.

$\begin{split}&\rho=-r^{-1}+O(r^{-3}),\quad\sigma,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}=O(r^{-2}),\quad\tau,\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\tau$\kern 0.0pt}}}=O(r^{-2}),\\\
&L^{\bar{z}},\bar{L}^{z}=O(r^{-1}),\quad\bar{L}^{\bar{z}},L^{z}=O(r^{-2}),\quad
X^{A}=O(r^{-1}),\quad\Psi_{0}=O(r^{-5}).\end{split}$ (6)

Those conditions determine an asymptotically flat spacetime in Lorentzian
signature.

The NP formalism is originally designed to investigate systems with
gravitational radiation. Hence a pair of complex null bases is needed to keep
the spacetime in Lorentzian signature. Nevertheless it is more natural to
adopt the $(2,2)$ Kleinian signature in the NP formalism. In the Kleinian
signature, one can just work with four independent real basis vectors, i.e.,
$m$ and $\bar{m}$ are now real and independent. Correspondingly, $z$ and
$\bar{z}$ are independent real coordinates and all NP variables are
independent of the ones with an overhead bar. Analytic continuation from
Lorentzian to Kleinian signature has seen a surge in interest recently from
both quantum aspect (see, e.g., [38, 39]) and classical aspect [40, 41]. A
more relevant fact to our work is that it admits self-dual solutions in
Kleinian signature.

We will deal with vacuum Einstein gravity without cosmological constant in
this work. So the Riemann tensor and the Weyl tensor are the same. The
components of the Weyl tensor are either self-dual under Hodge’s star
operation

${}^{\ast}C_{abcd}=\frac{1}{2}\epsilon_{abef}{C^{ef}}_{cd},$ (7)

or anti-self-dual under Hodge’s star operation

${}^{\ast}C_{abcd}=-\frac{1}{2}\epsilon_{abef}{C^{ef}}_{cd}.$ (8)

It is easy to verify in NP variables that ${}^{\ast}\Psi_{i}=\Psi_{i}$ and
${}^{\ast}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{i}=-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{i}$. Hence the self-dual
sector in the NP formalism is just to impose that $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{i}=0$.
Clearly it is very convenient to study the self-dual gravity in the NP
formalism where the Weyl tensor is included in the equations of motion. One
can directly check the self-dual conditions in the solutions.

The self-dual solutions can be obtained from the analytic continuation of the
Newman-Unti solutions by simply setting $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{i}=0$. The self-dual solutions
in series expansion is given in Appendix A. In particular, we set
$L^{\bar{z}},\bar{L}^{z}=\frac{1}{r}+O(r^{-2})$, i.e., the null infinity as
the product of a null interval with a plane or a torus after compactification
in $z$ and $\bar{z}$ directions. It is a more convenient choice to demonstrate
the self-dual conditions in solution space. The generic case of
$L^{\bar{z}},\bar{L}^{z}=O(r^{-1})$ can be obtained by a Weyl transformation
at the infinity, see, e.g., the discussions in [42, 43, 44, 45]. The precise
constraints from the self-dual conditions are that $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}^{0}$ is
only a function of $\bar{z}$. Consequently, $\sigma^{0}$ is determined as

$\sigma^{0}=u\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern 0.0pt}}}^{0}(\bar{z})+\Theta(z,\bar{z}).$ (9)

It is worthwhile to point out that the non-vanishing of $\sigma^{0}$ and
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}^{0}$ under the self-dual conditions saves the full BMS symmetry as we
will present in the next section.

##  3 Asymptotic symmetries in self-dual sector

In 1960s, Bondi and collaborators formulated the Einstein equation as a
characteristic initial value problem for axisymmetric isolated systems to
understand the gravitational radiation in full Einstein theory [4].
Surprisingly, they found that the asymptotic symmetry group in this system has
an infinite dimensional subgroup, i.e., the supertranslations which are
translations in the time direction involving an arbitrary function of the
angular variable. Later on, Sachs extended this system to the generic
asymptotically flat case [5]. The asymptotic symmetries of Sachs’s system are
the semidirect sum of supertranslations with the Lorentz symmetries. This is
the original version of the BMS symmetry. In the NU gauge, the BMS symmetry
was recovered in [30]. More than ten years ago, Barnich and Toessaert reported
that if one removes some conditions from global aspects, the Lorentz part of
the BMS symmetry should be replaced by local conformal symmetries of the
celestial sphere which is referred to as superrotations. The full symmetry
algebra is the semidirect sum of supertranslations with two copies of Virasoro
algebra [1, 2, 3]. The connection of BMS symmetry in the Bondi gauge and in
the NU gauge was studied in [46]. Other extensions of the BMS symmetry can be
found, e.g., in [9, 47, 48, 49]. The emergence of the full infinite-
dimensional local conformal symmetries indicates that four dimensional quantum
gravity should be closely related to two dimensional conformal field theory.
Such idea was first realized by connecting 4d scattering amplitudes to 2d
correlators [50, 51]. The subset of 2d CFT correlators that is connected to 4d
amplitudes are normally referred to as celestial amplitudes [52] which is at
the core of the recently established flat holography [53, 54].

In the NP formalism, the gauge transformation is a combination of a
diffeomorphism and a local Lorentz transformation. Let ${\xi}^{\mu}$ and
${{\omega}^{a}}_{b}=-{\omega_{b}}^{a}$ denote parameters for the infinitesimal
transformations, they act on the NP variables as

$\begin{split}&\delta_{\xi,\omega}{e_{a}}^{\mu}={\xi}^{\nu}\partial_{\nu}{e_{a}}^{\mu}-\partial_{\nu}{\xi}^{\mu}{e_{a}}^{\nu}+{\omega_{a}}^{b}{e_{b}}^{\mu},\\\
&\delta_{\xi,\omega}\Gamma_{abc}={\xi}^{\nu}\partial_{\nu}\Gamma_{abc}-e_{c}^{\mu}\partial_{\mu}{\omega}_{ab}+{\omega_{a}}^{d}\Gamma_{dbc}+{\omega_{b}}^{d}\Gamma_{adc}+{\omega_{c}}^{d}\Gamma_{abd},\\\
&\delta_{\xi,\omega}C_{abcd}={\xi}^{\nu}\partial_{\nu}C_{abcd}+{{\omega}_{a}}^{f}C_{fbcd}+{{\omega}_{b}}^{f}C_{afcd}+{{\omega}_{c}}^{f}C_{abfd}+{{\omega}_{d}}^{f}C_{abcf}.\end{split}$
(10)

The parameters of residual gauge transformations that preserve the NU solution
space are entirely determined by asking the conditions (3) and (4) and (6) to
be preserved on-shell. This is worked out in detail in [55]. Adapted to our
conventions, the asymptotic symmetry parameters are characterized by three
arbitrary functions

$T(z,\bar{z}),\quad Y^{z}=Y(z),\quad Y^{\bar{z}}=\bar{Y}(\bar{z}).$ (11)

They represent the supertranslations and superrotations respectively. The
associated residual gauge transformations are explicitly determined by the
above parameters as

$\begin{split}&\xi^{u}=f(u,z,\bar{z})=T(z,\bar{z})+\frac{1}{2}u(\partial_{z}Y+\partial_{\bar{z}}\bar{Y}),\\\
&\xi^{A}=Y^{A}(x^{A})-\partial_{B}f\int^{+\infty}_{r}dr[L^{A}\bar{L}^{B}+\bar{L}^{A}L^{B}],\\\
&\xi^{r}=-\partial_{u}fr+\partial_{z}\partial_{\bar{z}}f-\partial_{A}f\int^{+\infty}_{r}dr[\omega\bar{L}^{A}+\bar{\omega}L^{A}+X^{A}],\\\
\end{split}$ (12)

and

$\begin{split}\omega^{12}=&\partial_{u}f+X^{A}\partial_{A}f,\quad\omega^{23}=\bar{L}^{A}\partial_{A}f,\quad\omega^{24}=L^{A}\partial_{A}f,\\\
\omega^{13}=&-\partial_{u}\partial_{z}f+\partial_{A}f\int^{+\infty}_{r}dr[\lambda
L^{A}+\mu\bar{L}^{A}],\\\
\omega^{14}=&-\partial_{u}\partial_{\bar{z}}f+\partial_{A}f\int^{+\infty}_{r}dr[\bar{\lambda}\bar{L}^{A}+\bar{\mu}L^{A}],\\\
\omega^{34}=&\frac{1}{2}(\partial_{\bar{z}}\bar{Y}-\partial_{z}Y)-\partial_{A}f\int^{+\infty}_{r}dr[(\bar{\alpha}-\beta)\bar{L}^{A}+(\bar{\beta}-\alpha)L^{A}].\end{split}$
(13)

To specify the constraint from the self-dual conditions, one just needs to
work out the transformation law of the relevant NP variables. In particular,
for the $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{i}$ part, their transformations are

$\begin{split}&\delta_{s}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{0}=\xi^{\mu}\partial_{\mu}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{0}+2(\omega^{21}+\omega^{43})\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{0}+4\omega^{23}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{1},\\\
&\delta_{s}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{1}=\xi^{\mu}\partial_{\mu}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{1}+(\omega^{21}+\omega^{43})\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{1}+3\omega^{23}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{2}+\omega^{14}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{0},\\\
&\delta_{s}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{2}=\xi^{\mu}\partial_{\mu}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{2}+2\omega^{23}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{3}+2\omega^{14}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{1},\\\
&\delta_{s}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{3}=\xi^{\mu}\partial_{\mu}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{3}-(\omega^{21}+\omega^{43})\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{3}+\omega^{23}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{4}+3\omega^{14}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{2},\\\
&\delta_{s}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{4}=\xi^{\mu}\partial_{\mu}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{4}-2(\omega^{21}+\omega^{43})\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{4}+4\omega^{14}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{3}.\\\ \end{split}$ (14)

where $s=(T,Y,\bar{Y})$. Since there is no inhomogenous term, the conditions
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{i}=0$ can not bring any constraint on the residual gauge
transformations. One can show that the transformation laws of the rest fields
are

$\begin{split}&\delta_{s}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}=[Y\partial_{z}+\bar{Y}\partial_{\bar{z}}+f\partial_{u}+\partial_{u}f-\partial_{\bar{z}}\bar{Y}+\partial_{z}Y]\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}-\partial_{z}^{2}f,\\\
&\delta_{s}\lambda^{0}=[Y\partial_{z}+\bar{Y}\partial_{\bar{z}}+f\partial_{u}+2\partial_{u}f-\partial_{\bar{z}}\bar{Y}+\partial_{z}Y]\lambda^{0}-\frac{1}{2}\partial_{z}^{3}Y,\\\
&\delta_{s}\Psi^{0}_{0}=[Y\partial_{z}+\bar{Y}\partial_{\bar{z}}+f\partial_{u}+3\partial_{u}f+\partial_{\bar{z}}\bar{Y}-\partial_{z}Y]\Psi^{0}_{0}+4\Psi^{0}_{1}\partial_{\bar{z}}f,\\\
&\delta_{s}\Psi^{0}_{1}=[Y\partial_{z}+\bar{Y}\partial_{\bar{z}}+f\partial_{u}+3\partial_{u}f+\frac{1}{2}(\partial_{\bar{z}}\bar{Y}-\partial_{z}Y)]\Psi^{0}_{1}+3\Psi^{0}_{2}\partial_{\bar{z}}f,\\\
&\delta_{s}\Psi^{0}_{2}=[Y\partial_{z}+\bar{Y}\partial_{\bar{z}}+f\partial_{u}+3\partial_{u}f]\Psi^{0}_{2}+2\Psi^{0}_{3}\partial_{\bar{z}}f,\end{split}$
(15)

and

$\begin{split}&\delta_{s}\Theta=[Y\partial_{z}+\bar{Y}\partial_{\bar{z}}+2\partial_{\bar{z}}\bar{Y}]\Theta-\partial_{\bar{z}}^{2}T,\\\
&\delta_{s}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern
0.0pt}}}^{0}=[\bar{Y}\partial_{\bar{z}}+2\partial_{\bar{z}}\bar{Y}]\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}^{0}-\frac{1}{2}\partial_{\bar{z}}^{3}\bar{Y}.\end{split}$ (16)

It is clear from the transformation law that if $\Theta$ and
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}^{0}$ are vanishing, the asymptotic symmetries will be changed
significantly. One copy of the Virasoro symmetry will be reduced to $SL(2,R)$
and the supertranslations are reduced to two holomorphic functions which are
solutions of $\partial_{\bar{z}}^{2}T=0$. This will precisely recover the
symmetries obtained in [29].

The charges generating the BMS transformations in the self-dual case take the
form [55]

${\cal Q}_{s}=-\frac{1}{8\pi
G}\int_{\Gamma}f(\Psi^{0}_{2}+\sigma^{0}\lambda^{0}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern
0.0pt}}}^{0})+Y[\frac{1}{2}\partial_{z}(\sigma^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0})+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\sigma$\kern 0.0pt}}}^{0}\partial_{z}\sigma^{0}]\\\
+\bar{Y}[\frac{1}{2}\partial_{\bar{z}}(\sigma^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0})+\sigma^{0}\partial_{\bar{z}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern 0.0pt}}}^{0}+\Psi^{0}_{1}],$ (17)

where $\Gamma$ can be any two-surface at fix time at the null infinity. Two
immediate remarks follow from the expressions of the charges. First, the
charges associated with full superrotations are not vanishing. This means that
the self-dual condition can not reduce the asymptotic symmetries at the charge
level either. Second, it is very important to recover the full BMS symmetry
for the self-dual theory from the charge analysis. $\Psi_{1}^{0}$ as the
angular momentum aspect in the full theory arises only in the charges of the
anti-holomorphic part of the superrotations which are reduced to $SL(2,R)$ in
the subset BMS symmetry [29].

##  4 Self-dual Taub-NUT solutions in NU gauge

A special feature of the NU gauge is that it is based on the congruence of
null geodesics that is hypersurface orthogonal and the generator of the null
geodesics is the gradient of a scalar field. Once a solution admits such null
vector in explicit form, the solution can be written in the NU gauge in a
straightforward way, see also [56] for relevant discussion for the Kerr case.
So the crucial step to having solutions in the NU gauge is to find the null
geodesics generator $l$. This can be done systematically in the NP formalism.
One can split the task into two steps. Once the solution is written in the NP
formalism with four null basis vectors, the first step is to use three classes
of tetrad rotations to turn off certain spin coefficients, namely conditions
in (3). Note that there are the other half conditions for the variables with a
bar. Such conditions will guarantee that the null basis $l$ generates null
geodesics and is the gradient of a scalar field. The second step is to use
coordinates transformation to have the basis vectors in the NU gauge (5).
Since the spin coefficients are scalars, there are not changed under any
coordinates transformation.

According to the transformation property of the three classes of rotations
which are listed in Appendix C in Kleinian signature, the strategy for the
first step is as follows. First, one needs to perform a combined third and
second classes of rotations to set $\rho=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\rho$\kern 0.0pt}}}$ and $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\kappa$\kern 0.0pt}}}=\kappa=0$.
While only a second class of rotation should be involved to achieve so in some
special cases. Then one needs to use a third class of rotation to turn off
$\epsilon$ and  $\epsilon$ . Next, a first class of rotation will turn off
$\pi$ and  $\pi$ . At last, another third class of rotation may be involved to
set $\tau=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\alpha$\kern 0.0pt}}}+\beta$ and $\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\tau$\kern 0.0pt}}}=\alpha+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\beta$\kern 0.0pt}}}$. The last
operation will not turn on $\epsilon$ nor  $\epsilon$ . Once the spin
coefficient conditions are fulfilled, there must be a coordinates system that
will have the basis vectors in the form of (4) and (5).

We will test the above processes by transforming the self-dual Taub-NUT
solution into the NU gauge. The Taub-NUT solution in Lorentzian signature in
the complex stereographic coordinates adapted to the NP conventions is [57]

$\text{d}s^{2}=f(r)\left(\text{d}t+2iN\frac{z\text{d}\bar{z}-\bar{z}\text{d}z}{1+z\bar{z}}\right)^{2}-\frac{\text{d}r^{2}}{f(r)}-(r^{2}+N^{2})\frac{4\text{d}z\text{d}\bar{z}}{(1+z\bar{z})^{2}},$
(18)

where $f(r)=\frac{r^{2}-2Mr-N^{2}}{r^{2}+N^{2}}$. $N$ is the NUT parameter and
$M$ is the mass parameter. When setting $N=0$, one recovers the Schwarzschild
solution. One can obtain the solution in Kleinian signature simply by defining
a real function ${\cal N}=iN$ and considering $z$ and $\bar{z}$ as independent
real coordinates. The Taub-NUT solution is of Petrov type D. Hence the only
non-vanishing Weyl scalars are $\Psi_{2}$ and $\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{2}$ as shown
in Appendix B. In the Kleinian signature, one has

$\Psi_{2}=-\frac{M+{\cal N}}{(r+{\cal N})^{3}},\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{2}=-\frac{M-{\cal N}}{(r-{\cal N})^{3}}.$ (19)

The self-dual solution is simply to require ${\cal N}=M$. The line element of
the self-dual solution is

$\text{d}s^{2}=f(r)\left(\text{d}t+2M\frac{z\text{d}\bar{z}-\bar{z}\text{d}z}{1+z\bar{z}}\right)^{2}-\frac{\text{d}r^{2}}{f(r)}-(r^{2}-M^{2})\frac{4\text{d}z\text{d}\bar{z}}{(1+z\bar{z})^{2}}.$
(20)

Now one has $f(r)=\frac{r-M}{r+M}$. We define $\tilde{u}=t-r-2M\ln(r-M)$ and
choose the following four null bases

$\begin{split}&l=\text{d}\tilde{u}-2M\frac{\bar{z}\text{d}z}{1+z\bar{z}}+2M\frac{z\text{d}\bar{z}}{1+z\bar{z}},\\\
&n=f\left(\frac{1}{2}\text{d}\tilde{u}+\frac{\text{d}r}{f}-M\frac{\bar{z}\text{d}z}{1+z\bar{z}}+M\frac{z\text{d}\bar{z}}{1+z\bar{z}}\right),\\\
&m=\sqrt{2}(r-M)\frac{\text{d}z}{(1+z\bar{z})},\\\
&\bar{m}=\sqrt{2}(r+M)\frac{\text{d}\bar{z}}{(1+z\bar{z})}.\end{split}$ (21)

In the vector form, they are

$\begin{split}&l=\partial_{r},\\\
&n=\partial_{\tilde{u}}-\frac{f}{2}\partial_{r},\\\
&m=\frac{\sqrt{2}Mz}{r+M}\partial_{\tilde{u}}-\frac{1+z\bar{z}}{\sqrt{2}(r+M)}\partial_{\bar{z}},\\\
&\bar{m}=\frac{\sqrt{2}M\bar{z}}{M-r}\partial_{\tilde{u}}-\frac{1+z\bar{z}}{\sqrt{2}(r-M)}\partial_{z}.\end{split}$
(22)

The non-zero NP variables are

$\begin{split}&\Psi_{2}=-\frac{2M}{(r+M)^{3}},\\\
&\alpha=-\frac{\bar{z}}{2\sqrt{2}(r-M)},\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\alpha$\kern
0.0pt}}}=-\frac{z}{2\sqrt{2}(r+M)},\quad\beta=-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\alpha$\kern
0.0pt}}},\quad\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\beta$\kern 0.0pt}}}=-\alpha,\\\
&\gamma=\frac{M}{2(r+M)^{2}},\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\gamma$\kern
0.0pt}}}=\frac{M}{2(r+M)^{2}},\quad\epsilon=\frac{M}{r^{2}-M^{2}},\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}=-\frac{M}{r^{2}-M^{2}},\\\
&\mu=-\frac{1}{2}\frac{f}{r+M},\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}=-\frac{1}{2}\frac{f}{r-M},\quad\rho=-\frac{1}{r+M},\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\rho$\kern
0.0pt}}}=-\frac{1}{r-M}.\end{split}$ (23)

To have the spin coefficients in the NU gauge, we need the following three
rotations of the basis vectors. First, a second class of rotation

$b=\frac{2\sqrt{2}M(1+\bar{z}+z\bar{z}^{2})}{(r+M)\bar{z}},\quad\bar{b}=0,$
(24)

sets $\rho=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\rho$\kern 0.0pt}}}$ without turning on $\kappa$ nor  $\kappa$ . Then a
third class of rotation

$A=1,\quad\vartheta=-\ln(1+\bar{z}+z\bar{z}^{2}),$ (25)

turns off $\epsilon$ and  $\epsilon$ . The last one is a first class of
rotation

$a=-\frac{\sqrt{2}M}{\bar{z}(r+M)},\quad\bar{a}=0,$ (26)

which turns off  $\pi$ arisen by the previous second class of rotation. The
spin coefficients after the rotations become

$\displaystyle\Psi_{0}=-\frac{96M^{3}}{(r+M)^{5}\bar{z}^{2}},\quad\Psi_{1}=-\frac{12\sqrt{2}M^{2}}{(r+M)^{4}\bar{z}},\quad\Psi_{2}=-\frac{2M}{(r+M)^{3}},$
$\displaystyle\rho=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\rho$\kern 0.0pt}}}=\frac{1}{M-r},\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}=\frac{1}{2(M-r)},\quad\mu=\frac{M-r}{2(r+M)^{2}},$
$\displaystyle\tau=\frac{2\sqrt{2}M^{2}(3r-M)}{(r-M)(r+M)^{3}\bar{z}},\quad\gamma=\frac{Mr}{(r-M)(r+M)^{2}},\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\gamma$\kern
0.0pt}}}=-\frac{Mr}{(r-M)(r+M)^{2}},$ $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\nu$\kern
0.0pt}}}=\frac{\sqrt{2}M^{2}}{(r-M)(r+M)^{2}\bar{z}},\quad\alpha=\frac{\bar{z}}{2\sqrt{2}(M-r)},\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\beta$\kern 0.0pt}}}=-\alpha,$
$\displaystyle\beta=\frac{1}{2\sqrt{2}(r+M)^{3}\bar{z}(1+\bar{z}+z\bar{z}^{2})^{2})}\bigg{[}r^{2}\bar{z}(1+z+4z\bar{z}+3z^{2}\bar{z}^{2})-2Mr(2+3\bar{z}$
$\displaystyle\hskip
28.45274pt-z\bar{z}+2\bar{z}^{2}+4z\bar{z}^{3}-3z^{2}\bar{z}^{3}+2z^{2}\bar{z}^{4})+M^{2}(8+17\bar{z}+z\bar{z}+8\bar{z}^{2}+20z\bar{z}^{2}$
$\displaystyle\hskip
28.45274pt+16z\bar{z}^{3}+3z^{2}\bar{z}^{3}+8z^{2}\bar{z}^{4})\bigg{]},\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\alpha$\kern
0.0pt}}}=\tau-\beta,$ $\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}=\frac{M}{(M-r)(r+M)^{3}\bar{z}^{2}(1+\bar{z}+z\bar{z}^{2})^{2}}\bigg{[}4Mr(1+\bar{z}+z\bar{z}^{2})^{2}+r^{2}(1-2z\bar{z}^{2}$
$\displaystyle\hskip
28.45274pt-2z^{2}\bar{z}^{3})+M^{2}(2z\bar{z}^{2}+2z^{2}\bar{z}^{3}-1)\bigg{]},$
$\displaystyle\sigma=\frac{2M}{(r+M)^{4}\bar{z}^{2}(1+\bar{z}+z\bar{z}^{2})^{2}}\bigg{[}r^{2}(2z\bar{z}^{2}+2z^{2}\bar{z}^{3}-1)-2Mr(3+4\bar{z}+2\bar{z}^{2}$
$\displaystyle\hskip
28.45274pt+2z\bar{z}^{2}+4z\bar{z}^{3}-2z^{2}\bar{z}^{3}+2z^{2}\bar{z}^{4})+M^{2}(7+16\bar{z}+8\bar{z}^{2}+18z\bar{z}^{2}+16z\bar{z}^{3}$
$\displaystyle\hskip 28.45274pt+2z^{2}\bar{z}^{3}+8z^{2}\bar{z}^{4})\bigg{]}$

The resulting null basis

$l=\text{d}\tilde{u}-\frac{2M\bar{z}}{1+z\bar{z}}\text{d}z+\frac{2M\left[2+(2+z)\bar{z}+2z\bar{z}^{2}\right]}{\bar{z}(1+z\bar{z})}\text{d}\bar{z}$
(27)

should be the gradient of a scalar field. We can define $l=du$. Then $u$ can
be solved out by integrating the right hand side of (27). The solution is

$u=\tilde{u}+4M(\bar{z}+\ln\bar{z})-2M\ln(1+z\bar{z}).$ (28)

In the $(u,r,z,\bar{z})$ coordinates, the null bases are given by

$\begin{split}&l=\text{d}u,\\\
&n=\frac{f}{2}\text{d}u+\text{d}r-\frac{4Mr\left(z\bar{z}^{2}+\bar{z}+1\right)}{\bar{z}(r+M)(1+z\bar{z})}\text{d}\bar{z},\\\
&m=-\frac{2\sqrt{2}M^{2}}{\bar{z}(r+M)^{2}}\text{d}u+\frac{2\sqrt{2}M}{\bar{z}(r+M)}\text{d}r+\frac{\sqrt{2}(r-M)}{(1+z\bar{z})\left(z\bar{z}^{2}+\bar{z}+1\right)}\text{d}z\\\
&\hskip
113.81102pt+\frac{4\sqrt{2}M^{2}(M-r)\left(z\bar{z}^{2}+\bar{z}+1\right)}{\bar{z}^{2}(r+M)^{2}(1+z\bar{z})}\text{d}\bar{z},\\\
&\bar{m}=\frac{\sqrt{2}(r+M)\left(z\bar{z}^{2}+\bar{z}+1\right)}{(1+z\bar{z})}\text{d}\bar{z}.\end{split}$
(29)

The last operation is to define a new angular variable $\hat{z}$ to put the
tetrad $m$ in the NU gauge. Considering $z$ as function of
$(r,\hat{z},\bar{z})$, the only constraint on $z$ is

$\frac{\partial z}{\partial
r}=\frac{2M(1+z\bar{z})\left(z\bar{z}^{2}+\bar{z}+1\right)}{\bar{z}(M^{2}-r^{2})}.$
(30)

One possible solution is

$z=\frac{1-(1+\bar{z})e^{2\operatorname{arctanh}\left(\frac{r}{M}\right)+\frac{\bar{z}\hat{z}}{\sqrt{2}}}}{\bar{z}\left(\bar{z}e^{2\operatorname{arctanh}\left(\frac{r}{M}\right)+\frac{\bar{z}\hat{z}}{\sqrt{2}}}-1\right)}.$
(31)

In the new coordinates $(u,r,\hat{z},\bar{z})$, the basis $m$, now denoted by
$\hat{m}$, becomes

$\hat{m}=-\frac{2\sqrt{2}M^{2}}{\bar{z}(r+M)^{2}}\text{d}u+\frac{\sqrt{2}(r-M)}{(1+z\bar{z})\left(z\bar{z}^{2}+\bar{z}+1\right)}\frac{\partial
z}{\partial\hat{z}}\text{d}\hat{z}\\\
+\left[\frac{\sqrt{2}(r-M)}{(1+z\bar{z})\left(z\bar{z}^{2}+\bar{z}+1\right)}\frac{\partial
z}{\partial\bar{z}}+\frac{4\sqrt{2}M^{2}(M-r)\left(z\bar{z}^{2}+\bar{z}+1\right)}{\bar{z}^{2}(r+M)^{2}(1+z\bar{z})}\right]\text{d}\bar{z}.$
(32)

For the rest null bases and the spin coefficients, one just needs to insert
$z$ as function in the new coordinates into the expression.

##  5 Conclusion

In this paper, we find that the self-dual condition will not reduce the
asymptotic symmetries at null infinity in four dimensions which is still the
full BMS symmetry rather than a subset. The reduction of the asymptotic
symmetry in [29] is due to the light-cone conditions. The subset BMS symmetry
can be recovered from the full BMS symmetry by certain reductions in the self-
dual solution space (see also [58] for relevant discussions). The self-dual
gravity was known to have an infinite ladder of symmetries [59, 60, 61], which
can be seen as the origin of recently discovered celestial $w_{1+\infty}$
symmetries [62]. The subset of BMS symmetry in [29] represents two rungs in
the infinite ladder. Since the self-dual gravity has the full BMS symmetry at
null infinity, one would naturally expect if this indicates another copy of
$w_{1+\infty}$ symmetries. This is a very remarkable question that should be
stressed elsewhere.

Asymptotic symmetries are normally studied in certain gauge systems. It is
very important for the application of asymptotic symmetries to have exact
solutions in the corresponding gauge system. In this paper, we present a
generic treatment to transform exact solutions into the NU gauge which has
been tested by checking the self-dual Taub-NUT solution. It would be of
interest to test our proposal in more generic cases such as the Taub-NUT
solution where transformations in series expansion can be found in [63, 64] or
even the more generic one, the Kerr-Taub-NUT solution. If one is restricted in
the self-dual Kerr-Taub-NUT case, a surprising fact is that it can be obtained
from the self-dual Taub-NUT solution simply by coordinates transformation
[41]. It would be very meaningful to study such transformation in the
asymptotic framework and to investigate its relation to the BMS
transformations.

## Acknowledgments

The authors thank Glenn Barnich and Jun-Bao Wu for useful discussions. P.M.
would like to thank Glenn Barnich again for long term collaborations and
supports in relevant research topics. This work is supported in part by the
National Natural Science Foundation of China under Grant No. 11905156 and No.
11935009.

##  Appendix A Self-dual solutions in NP formalism

The asymptotic expansions of all NP variables are given by:

$\displaystyle\Psi_{0}=\frac{\Psi_{0}^{0}(u,z,\bar{z})}{r^{5}}+O(r^{-6}),\quad\Psi_{1}=\frac{\Psi_{1}^{0}(u,z,\bar{z})}{r^{4}}-\frac{\partial_{z}\Psi_{0}^{0}}{r^{5}}+O(r^{-6}),$
$\displaystyle\Psi_{2}=\frac{\Psi_{2}^{0}(u,z,\bar{z})}{r^{3}}-\frac{\partial_{z}\Psi_{1}^{0}}{r^{4}}+O(r^{-5}),\quad\Psi_{3}=\frac{\Psi_{3}^{0}(u,z,\bar{z})}{r^{2}}-\frac{\partial_{z}\Psi_{2}^{0}}{r^{3}}+O(r^{-4}),$
$\displaystyle\Psi_{4}=\frac{\Psi_{4}^{0}(u,z,\bar{z})}{r}-\frac{\partial_{z}\Psi_{3}^{0}}{r^{2}}+O(r^{-3}),$
$\displaystyle\rho=-\frac{1}{r}-\frac{\sigma^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}}{r^{3}}+O(r^{-5}),\quad\tau=-\frac{\Psi^{0}_{1}}{2r^{3}}+O(r^{-4}),\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\tau$\kern 0.0pt}}}=O(r^{-4}),$
$\displaystyle\sigma=\frac{\sigma^{0}(u,z,\bar{z})}{r^{2}}+\frac{\sigma^{0}\sigma^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}-\frac{1}{2}\Psi_{0}^{0}}{r^{4}}+O(r^{-5}),\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}=\frac{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\sigma$\kern
0.0pt}}}^{0}(u,z,\bar{z})}{r^{2}}+\frac{\sigma^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\sigma$\kern 0.0pt}}}^{0}}{r^{4}}+O(r^{-5}),$
$\displaystyle\alpha=O(r^{-4}),\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\alpha$\kern 0.0pt}}}=O(r^{-4}),$
$\displaystyle\beta=-\frac{\frac{1}{2}\Psi^{0}_{1}}{r^{3}}+O(r^{-4}),\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\beta$\kern 0.0pt}}}=O(r^{-4}),$
$\displaystyle\mu=-\frac{\sigma^{0}\lambda^{0}+\Psi^{0}_{2}}{r^{2}}+\frac{\partial_{z}\Psi_{1}^{0}}{2r^{3}}+O(r^{-4}),\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}=-\frac{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\sigma$\kern 0.0pt}}}^{0}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}^{0}}{r^{2}}+O(r^{-4}),$
$\displaystyle\lambda=\frac{\lambda^{0}(u,z,\bar{z})}{r}+\frac{\sigma^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}\lambda^{0}+\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}\Psi_{2}^{0}}{r^{3}}+O(r^{-4}),\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}=\frac{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern
0.0pt}}}^{0}(\bar{z})}{r}+\frac{\sigma^{0}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}\lambda^{0}}{r^{3}}+O(r^{-4}),$
$\displaystyle\gamma=-\frac{\Psi^{0}_{2}}{2r^{2}}+\frac{\partial_{z}\Psi_{1}^{0}}{3r^{3}}+O(r^{-4}),\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\gamma$\kern 0.0pt}}}=O(r^{-4}),$
$\displaystyle\nu=-\frac{\Psi^{0}_{3}}{r}+\frac{\partial_{z}\Psi^{0}_{2}}{2r^{2}}+O(r^{-3}),\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\nu$\kern 0.0pt}}}=O(r^{-3}),$
$\displaystyle X^{z}=\frac{\Psi_{1}^{0}}{6r^{3}}+O(r^{-4}),\quad
X^{\bar{z}}=O(r^{-4}),\quad\omega=\frac{\partial_{z}\sigma^{0}}{r}-\frac{\sigma^{0}\partial_{\bar{z}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}+\frac{1}{2}\Psi^{0}_{1}}{r^{2}}+O(r^{-3}),$
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\omega$\kern 0.0pt}}}=\frac{\partial_{\bar{z}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}}{r}-\frac{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}\partial_{z}\sigma^{0}}{r^{2}}+O(r^{-3}),\quad
U=-\frac{\Psi^{0}_{2}}{2r}+\frac{\partial_{z}\Psi_{1}^{0}}{6r^{2}}+O(r^{-3}),$
$\displaystyle
L^{z}=-\frac{\sigma^{0}}{r^{2}}-\frac{\sigma^{0}\sigma^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}-\frac{1}{6}\Psi_{0}^{0}}{r^{4}}+O(r^{-5}),\;\;\;\;L^{\bar{z}}=\frac{1}{r}+\frac{\sigma^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}}{r^{3}}+O(r^{-5}),$
$\displaystyle\bar{L}^{\bar{z}}=-\frac{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}}{r^{2}}-\frac{\sigma^{0}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern 0.0pt}}}^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}}{r^{4}}+O(r^{-5}),\quad\bar{L}^{z}=\frac{1}{r}+\frac{\sigma^{0}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}}{r^{3}}+O(r^{-5}),$ $\displaystyle L_{z}=-r+O(r^{-3}),\quad
L_{\bar{z}}=-\sigma^{0}+\frac{\Psi_{0}^{0}}{6r^{2}}+O(r^{-3}),$
$\displaystyle\bar{L}_{\bar{z}}=-r+O(r^{-3}),\;\;\;\;\;\;\bar{L}_{z}=-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}+O(r^{-3}),$

where

$\displaystyle\lambda^{0}=\partial_{u}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0},\quad\sigma^{0}=u\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}^{0}+\Theta(z,\bar{z}),$
$\displaystyle\Psi_{2}^{0}=\partial_{z}^{2}\Theta-\partial_{\bar{z}}^{2}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\sigma$\kern 0.0pt}}}^{0}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern 0.0pt}}}^{0}-(u\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}^{0}+\Theta)\partial_{u}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern 0.0pt}}}^{0},$
$\displaystyle\Psi^{0}_{3}=-\partial_{\bar{z}}\partial_{u}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}^{0},\quad\Psi^{0}_{4}=-\partial_{u}^{2}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern 0.0pt}}}^{0},$
$\displaystyle\partial_{u}\Psi^{0}_{0}=\partial_{\bar{z}}\Psi^{0}_{1}+3(u\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}^{0}+\Theta)\Psi^{0}_{2},\quad\partial_{u}\Psi^{0}_{1}=\partial_{\bar{z}}\Psi^{0}_{2}+2(u\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}^{0}+\Theta)\Psi^{0}_{3}.$

##  Appendix B Taub-NUT solutions in NP formalism

Defining

$\tilde{u}=t-r-(M+\sqrt{M^{2}+N^{2}})\ln(r-M-\sqrt{M^{2}+N^{2}})\\\
-(M-\sqrt{M^{2}+N^{2}})\ln(r-M+\sqrt{M^{2}+N^{2}}),$ (33)

we choose the null bases for the metric (18) as

$\begin{split}&l=\text{d}\tilde{u}-2iN\frac{\bar{z}\text{d}z}{1+z\bar{z}}+2iN\frac{z\text{d}\bar{z}}{1+z\bar{z}},\\\
&n=f\left(\frac{1}{2}\text{d}\tilde{u}+\frac{\text{d}r}{f}-iN\frac{\bar{z}\text{d}z}{1+z\bar{z}}+iN\frac{z\text{d}\bar{z}}{1+z\bar{z}}\right),\\\
&m=\sqrt{2}(r-iN)\frac{\text{d}z}{(1+z\bar{z})},\\\
&\bar{m}=\sqrt{2}(r+iN)\frac{\text{d}\bar{z}}{(1+z\bar{z})}.\end{split}$ (34)

In the vector form, they are

$\begin{split}&l=\partial_{r},\\\
&n=\partial_{\tilde{u}}-\frac{f}{2}\partial_{r},\\\
&m=\frac{i\sqrt{2}Nz}{r+iN}\partial_{\tilde{u}}-\frac{1+z\bar{z}}{\sqrt{2}(r+iN)}\partial_{\bar{z}},\\\
&\bar{m}=-\frac{i\sqrt{2}N\bar{z}}{r-iN}\partial_{\tilde{u}}-\frac{1+z\bar{z}}{\sqrt{2}(r-iN)}\partial_{z}.\end{split}$
(35)

The non-zero NP variables are

$\begin{split}&\alpha=-\frac{\bar{z}}{2\sqrt{2}(r-iN)},\quad\beta=-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\alpha$\kern
0.0pt}}},\quad\gamma=\frac{r^{2}M+2N^{2}r-MN^{2}}{2(r+N^{2})^{2}},\\\
&\epsilon=\frac{iN}{r^{2}+N^{2}},\quad\mu=-\frac{1}{2}\frac{f}{r+iN},\quad\rho=-\frac{1}{r+iN},\quad\Psi_{2}=-\frac{M+iN}{(r+iN)^{3}}.\end{split}$
(36)

The null basis vectors $l$ and $n$ are repeated principal null directions
without shear but with twist [57].

##  Appendix C Three classes of tetrad rotations in Kleinian signature

First class of rotation:

$l\to l,\quad m\to m+al,\quad\bar{m}\to\bar{m}+\bar{a}l,\quad n\to
n+a\bar{m}+\bar{a}m+a\bar{a}l.$ (37)

One should consider $a$ and $\bar{a}$ as independent real functions. The
effect of the transformation on the NP variables are as follows

$\begin{split}&\Psi_{0}\to\Psi_{0},\quad\Psi_{1}\to\Psi_{1}+\bar{a}\Psi_{0},\quad\Psi_{2}\to\Psi_{2}+2\bar{a}\Psi_{1}+\bar{a}^{2}\Psi_{0},\\\
&\Psi_{3}\to\Psi_{3}+3\bar{a}\Psi_{2}+3\bar{a}^{2}\Psi_{1}+\bar{a}^{3}\Psi_{0},\\\
&\Psi_{4}\to\Psi_{4}+4\bar{a}\Psi_{3}+6\bar{a}^{2}\Psi_{2}+4\bar{a}^{3}\Psi_{1}+\bar{a}^{4}\Psi_{0},\end{split}$
(38)

and

$\begin{split}&\kappa\to\kappa,\quad\sigma\to\sigma+a\kappa,\quad\rho\to\rho+\bar{a}\kappa,\quad\epsilon\to\epsilon+\bar{a}\kappa,\\\
&\tau\to\tau+a\rho+\bar{a}\sigma+a\bar{a}\kappa,\quad\pi\to\pi+2\bar{a}\epsilon+\bar{a}^{2}\kappa+l^{\mu}\partial_{\mu}\bar{a},\\\
&\alpha\to\alpha+\bar{a}\rho+\bar{a}\epsilon+\bar{a}^{2}\kappa,\quad\beta\to\beta+a\epsilon+\bar{a}\sigma+a\bar{a}\kappa,\\\
&\gamma\to\gamma+a\alpha+\bar{a}\beta+\bar{a}\tau+a\bar{a}(\rho+\epsilon)+\bar{a}^{2}\sigma+a\bar{a}^{2}\kappa,\\\
&\lambda\to\lambda+\bar{a}(2\alpha+\pi)+\bar{a}^{2}(\rho+2\epsilon)+\bar{a}^{3}\kappa+\bar{m}^{\mu}\partial_{\mu}\bar{a}+\bar{a}l^{\mu}\partial_{\mu}\bar{a},\\\
&\mu\to\mu+a\pi+2\bar{a}\beta+2a\bar{a}\epsilon+\bar{a}^{2}\sigma+a\bar{a}^{2}\kappa+m^{\mu}\partial_{\mu}\bar{a}+al^{\mu}\partial_{\mu}\bar{a},\\\
&\nu\to\nu+a\lambda+\bar{a}(\mu+2\gamma)+\bar{a}^{2}(\tau+2\beta)+\bar{a}^{3}\sigma+a\bar{a}(\pi+2\alpha)\\\
&\hskip
28.45274pt+a\bar{a}^{2}(\rho+2\epsilon)+a\bar{a}^{3}\kappa+(n^{\mu}\partial_{\mu}+\bar{a}m^{\mu}\partial_{\mu}+a\bar{m}^{\mu}\partial_{\mu}+a\bar{a}l^{\mu}\partial_{\mu})\bar{a}.\end{split}$
(39)

For the other half variables, one just needs to add an overhead bar and
swapping $a\rightleftharpoons\bar{a}$.

Second class of rotation:

$n\to n,\quad m\to m+bn,\quad l\to l+b\bar{m}+\bar{b}m+b\bar{b}n.$ (40)

Here $b$ and $\bar{b}$ are independent real functions. The effect of the
transformation on the NP variables are as follows

$\begin{split}&\Psi_{4}\to\Psi_{4},\quad\Psi_{3}\to\Psi_{3}+b\Psi_{4},\quad\Psi_{2}\to\Psi_{2}+2b\Psi_{3}+b^{2}\Psi_{4},\\\
&\Psi_{1}\to\Psi_{1}+3b\Psi_{2}+3b^{2}\Psi_{3}+b^{3}\Psi_{4},\\\
&\Psi_{0}\to\Psi_{0}+4b\Psi_{1}+6b^{2}\Psi_{2}+4b^{3}\Psi_{3}+b^{4}\Psi_{4},\end{split}$
(41)

and

$\begin{split}&\nu\to\nu,\quad\lambda\to\lambda+\bar{b}\nu,\quad\mu\to\mu+b\nu,\quad\gamma\to\gamma+b\nu,\\\
&\pi\to\pi+b\lambda+\bar{b}\mu+b\bar{b}\nu,\quad\tau\to\tau+2b\gamma+b^{2}\nu-n^{\mu}\partial_{\mu}b,\\\
&\alpha\to\alpha+b\lambda+\bar{b}\gamma+b\bar{b}\nu,\quad\beta\to\beta+b\gamma+b\mu+b^{2}\nu,\\\
&\epsilon\to\epsilon+\bar{b}\beta+b\alpha+b\pi+b\bar{b}(\mu+\gamma)+b^{2}\lambda+\bar{b}b^{2}\nu,\\\
&\sigma\to\sigma+b(2\beta+\tau)+b^{2}(\mu+2\gamma)+b^{3}\nu-m^{\mu}\partial_{\mu}b-bn^{\mu}\partial_{\mu}b,\\\
&\rho\to\rho+\bar{b}\tau+2b\alpha+2b\bar{b}\gamma+b^{2}\lambda+\bar{b}b^{2}\nu-\bar{m}^{\mu}\partial_{\mu}b-\bar{b}n^{\mu}\partial_{\mu}b,\\\
&\kappa\to\kappa+\bar{b}\sigma+b(\rho+2\epsilon)+b^{2}(\pi+2\alpha)+b^{3}\lambda+b\bar{b}(\tau+2\beta)\\\
&\hskip
28.45274pt+\bar{b}b^{2}(\mu+2\gamma)+\bar{b}b^{3}\nu-(l^{\mu}\partial_{\mu}+\bar{b}m^{\mu}\partial_{\mu}+b\bar{m}^{\mu}\partial_{\mu}+b\bar{b}n^{\mu}\partial_{\mu})b.\end{split}$
(42)

For the other half variables, one just needs to add an overhead bar and
swapping $b\rightleftharpoons\bar{b}$.

Third class of rotation:

$l\to\frac{l}{A},\quad n\to An,\quad m\to e^{\vartheta}m,\quad\bar{m}\to
e^{-\vartheta}\bar{m},$ (43)

where $A$ and $\vartheta$ are real functions. The effect of the transformation
on the NP variables are as follows

$\begin{split}&\Psi_{0}\to\frac{e^{2\vartheta}}{A^{2}}\Psi_{0},\quad\Psi_{1}\to\frac{e^{\vartheta}}{A}\Psi_{1},\quad\Psi_{2}\to\Psi_{2},\\\
&\Psi_{3}\to\frac{A}{e^{\vartheta}}\Psi_{3},\quad\Psi_{4}\to\frac{A^{2}}{e^{2\vartheta}}\Psi_{4},\\\
&\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{0}\to\frac{1}{A^{2}e^{2\vartheta}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{0},\quad\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{1}\to\frac{1}{Ae^{\vartheta}}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{1},\quad\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{2}\to\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{2},\\\ &\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{3}\to
Ae^{\vartheta}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{3},\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\Psi$\kern 0.0pt}}}_{4}\to
A^{2}e^{2\vartheta}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\Psi$\kern 0.0pt}}}_{4},\end{split}$ (44)

and

$\begin{split}&\kappa\to\frac{e^{\vartheta}}{A^{2}}\kappa,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\kappa$\kern
0.0pt}}}\to\frac{1}{e^{\vartheta}A^{2}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\kappa$\kern
0.0pt}}},\quad\sigma\to\frac{e^{2\vartheta}}{A}\sigma,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}}\to\frac{1}{Ae^{2\vartheta}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern
0.0pt}}},\quad\rho\to\frac{1}{A}\rho,\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\rho$\kern 0.0pt}}}\to\frac{1}{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\rho$\kern 0.0pt}}},\\\
&\nu=\frac{A^{2}}{e^{\vartheta}}\nu,\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\nu$\kern
0.0pt}}}=A^{2}e^{\vartheta}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\nu$\kern
0.0pt}}},\quad\lambda=\frac{A}{e^{2\vartheta}}\lambda,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}=Ae^{2\vartheta}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}},\quad\mu=A\mu,\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}}=A\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern 0.0pt}}},\\\
&\tau=e^{\vartheta}\tau,\quad\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\tau$\kern
0.0pt}}}=\frac{1}{e^{\vartheta}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\tau$\kern
0.0pt}}},\quad\pi=\frac{1}{e^{\vartheta}}\pi,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}=e^{\vartheta}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}},\\\ &\gamma\to
A\gamma-\frac{1}{2}n^{\mu}\partial_{\mu}A+\frac{1}{2}An^{\mu}\partial_{\mu}\vartheta,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\gamma$\kern 0.0pt}}}\to
A\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\gamma$\kern
0.0pt}}}-\frac{1}{2}n^{\mu}\partial_{\mu}A-\frac{1}{2}An^{\mu}\partial_{\mu}\vartheta,\\\
&\epsilon\to\frac{1}{A}\epsilon+\frac{1}{2}l^{\mu}\partial_{\mu}\frac{1}{A}+\frac{1}{2A}l^{\mu}\partial_{\mu}\vartheta,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}\to\frac{1}{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\epsilon$\kern
0.0pt}}}+\frac{1}{2}l^{\mu}\partial_{\mu}\frac{1}{A}-\frac{1}{2A}l^{\mu}\partial_{\mu}\vartheta,\\\
&\alpha\to\frac{1}{e^{\vartheta}}\alpha-\frac{\bar{m}^{\mu}}{2Ae^{\vartheta}}\partial_{\mu}A+\frac{\bar{m}^{\mu}}{2e^{\vartheta}}\partial_{\mu}\vartheta,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\alpha$\kern 0.0pt}}}\to
e^{\vartheta}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\alpha$\kern
0.0pt}}}-\frac{e^{\vartheta}m^{\mu}}{2A}\partial_{\mu}A-\frac{1}{2}e^{\vartheta}m^{\mu}\partial_{\mu}\vartheta,\\\
&\beta\to
e^{\vartheta}\beta-\frac{e^{\vartheta}m^{\mu}}{2A}\partial_{\mu}A+\frac{1}{2}e^{\vartheta}m^{\mu}\partial_{\mu}\vartheta,\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\beta$\kern
0.0pt}}}\to\frac{1}{e^{\vartheta}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\beta$\kern
0.0pt}}}-\frac{\bar{m}^{\mu}}{2Ae^{\vartheta}}\partial_{\mu}A-\frac{\bar{m}^{\mu}}{2e^{\vartheta}}\partial_{\mu}\vartheta.\end{split}$
(45)

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