# Scattering of classical gravitons with matter fields in the Null Surface
Formulation

Carlos N. Kozameh<EMAIL_ADDRESS>Emmanuel A. Tassone
<EMAIL_ADDRESS>FaMAF, Universidad Nacional de Córdoba, 5000,
Córdoba, Argentina

###### Abstract

Using a set of field equations in the Null Surface Formulation we obtain the
linearized coupling between the gravitational and matter fields. We first
derive a formula for the metric of the space time and then we use this formula
to study the scattering of incoming gravitational waves when matter is
present, obtaining explicit formulae relating the radiation modes at past and
future null infinity for a general asymptotically flat spacetime. An example
application is made at the end of this work when the matter field is a
massless real scalar field. The relevance of this result for a perturbation
procedure is discussed.

††preprint: APS/123-QED

##  I  Introduction

The general theory of relativity is a geometric theory, the main field is the
metric tensor from which the curvature and other geometrical structures are
constructed.
The null surface formulation (NSF) is an equivalent formulation of general
relativity written in terms of two scalar functions, one giving the conformal
structure of the spacetime and the other one providing the conformal factor of
the metric tensor. Thus, the main variables of the NSF are these two scalar
functions, and the curvature related quantities, such as the metric tensor,
are derived concepts.
Introducing a bundle of null directions for the points of the spacetime with
local coordinates $(x^{a},\zeta,\bar{\zeta})$, the two scalars functions are
the conformal factor of the theory, $\Omega(x^{a},\zeta,\bar{\zeta})$, and a
family of characteristic null surfaces of the metric,
$Z(x^{a},\zeta,\bar{\zeta})$ which satisfies a set of two differential
equations, the so-called ”metricity conditions”.
Mathematically speaking, the ten Einstein equations with components depending
of the spacetime points $x^{a}$ are exchanged by one equation for the scalars
$\Omega(x^{a},\zeta,\bar{\zeta})$ and $Z(x^{a},\zeta,\bar{\zeta})$ along with
the two metricity conditions[1]. The new system of differential equations
depends on six variables instead of the four spacetime points $x^{a}$. This
differential system equation continues being nonlinear and thus numerical and
perturbative approaches must be employed in order to find solutions.
Recently, there have been some further developments of geometrical structures
via the NSF, namely, a definition of center of mass and intrinsic angular
momentum[2], a study of global variables in black hole coalescence[3, 4], and
the field equations for the classical graviton were developed[5]. The
classical graviton is a Ricci flat spacetime that represents the self
interaction of incoming gravitational waves. In addition, a perturbation
procedure to obtain the solution was given and at each level of this
perturbation scheme the constructed solution was finite. This approach could
be of importance when dealing with quantization of pure gravity and hints of
this quantization were given before the perturbation scheme was implemented.
[6, 7]

It is also important to extend this approach to the interaction of classical
gravitons with other fields but a word of caution must be given. The formalism
is not adapted to study asymptotically flat spacetimes with singularities
since an ab initio assumption is broken, namely, that it is possible to find
regular null cone cuts at future or past null infinity. Thus, the scenario we
envisage is of a stress energy tensor constructed from suitable fields that
preserve the regularity of the null cone cuts, as for example the interaction
of gravitons with photons.

In this work we use the NSF to study the interaction between matter and the
gravitational field in the asymptotic limit and an explicit relationship
between the incoming and outgoing gravitational fields is derived. We thus
generalize the concept of classical gravitons previously given. The
gravitational radiation field at future and past null boundaries possess the
same mathematical structure as in the vacuum case[5] but now there is an
interaction term inside the spacetime. The results are equivalent to the
classical scattering of fields at the first non trivial level, usually called
tree level, where the first order interaction term between the interacting
fields is used to compute the incoming and outgoing fields.
The paper is organized as follows. In Sec.II we give an outline of the NSF and
particularly, we explain the linearized version of the formulation which is
simpler mathematically and allows us to study the interaction of the fields
analytically. Then, we derive the NSF equation including matter. In Sec.III,
we derive the relation between the metric and the NSF variables and we also
analyse the ingoing and outgoing characteristic null surfaces to linear order.
In case of no matter, we relate the modes in the decomposition of
gravitational radiation. We then perform a similar calculation incorporating
the matter field tensor and, in particular, we study the scattering of
gravitons with scalar waves, showing that we obtain an expected result at the
tree level approximation. Finally, we finish the work with a summary and
conclusions in Sec.IV.

##  II The linearized Null Surface Formulation

The Null Surface Formulation recasts general relativity as a theory of null
surfaces interacting with matter via the field equations. In NSF the
Lorentzian metric of the spacetime $g_{ab}$ is constructed from a conformally
invariant part $h_{ab}$ and a conformal factor $\Omega$. The conformal metric
is obtained from knowledge of special null surfaces obtained from a real
function $Z$ given on the sphere of null directions bundle. Given
$Z=Z(x^{a},\zeta,\bar{\zeta})$, with $x^{a}$ points of the spacetime and
$(\zeta,\bar{\zeta})$ stereographic coordinates on the sphere of null
directions, $Z=const.$ is a null surface on the spacetime. From knowledge of
$Z$ one then constructs a conformal metric $h_{ab}[Z]$. In addition, one gives
a second real function $\Omega=\Omega(x^{a},\zeta,\bar{\zeta})$, a conformal
factor, that plays a dual role: it is used to obtain a conformal metric that
only depends on the spacetime points and it yields an Einstein metric via the
field equations. From knowledge of these two functions one obtains

$g_{ab}(x^{a})=\Omega^{2}h_{ab}[Z].$

Since the equation $Z=const.$ defines a null hypersurface, $Z$ must satisfy

$g^{ab}\partial_{a}Z\partial_{b}Z=0.$ (1)

It is clear from (1) that the conformal structure does not depend on $\Omega$.
It also follows from this equation that the null vector
$Z^{a}=g^{ab}\partial_{b}Z,$ satisfies the homogeneous geodesic equation, thus
defining an affine length $s$.

Directly from (1), and taking $\eth$ and $\bar{\eth}$ derivatives [1], one
obtains the components of the conformal metric. The non trivial coefficients
of $h^{ab}$ are functions of a single scalar $\Lambda$ defined as
$\Lambda=\eth^{2}Z$. Once the conformal metric coefficients have been obtained
one finds a condition on $\Lambda$, namely

$\eth^{3}(g^{ab}\partial_{a}Z\partial_{b}Z)=0\implies g^{ab}(3\partial_{a}\eth
Z\partial_{b}\Lambda+\partial_{a}Z\partial_{b}\eth\Lambda)=0.$

This condition can be rewritten as

$\frac{\partial\eth\Lambda}{\partial s}+3\eth Z^{b}\partial_{b}\Lambda=0.$ (2)

Only for functions $\Lambda$ that satisfy condition (2) it is possible to
obtain a conformal metric. In what follows we assume condition (2) is
satisfied.
One can also show that directly from

$\eth^{2}\bar{\eth}^{2}(g^{ab}\partial_{a}Z\partial_{b}Z)=0,$

one obtains a relationship between $\Omega$ and $\Lambda$, namely,

$\frac{\partial\bar{\eth}^{2}\Lambda}{\partial
s}=\eth\bar{\eth}(\Omega^{2})+g^{ab}\partial_{a}\Lambda\partial_{b}\bar{\Lambda}.$
(3)

which can be formally integrated giving [5],

$\bar{\eth}^{2}\Lambda=\eth^{2}\bar{\sigma}(Z,\zeta,\bar{\zeta})+\bar{\eth}^{2}\sigma(Z,\zeta,\bar{\zeta})+\int_{Z}^{\infty}\dot{\sigma}\dot{\bar{\sigma}}du-\int_{s}^{\infty}(\eth\bar{\eth}(\Omega^{2})+g^{ab}\partial_{a}\Lambda\partial_{b}\bar{\Lambda})ds^{\prime},$
(4)

where $\sigma(u,\zeta,\bar{\zeta})$, the Bondi shear, is directly related to
the gravitational radiation reaching null infinity and $\dot{\sigma}$ is the
derivative with respect to the Bondi time $u$. It is worth mentioning that
conditions (2) and (3) have been used to obtain the free data at null
infinity[5], otherwise one should have more general free function when (3) is
integrated along a null geodesic from the point $x^{a}$ to null infinity. An
explicit calculation for the linearized approximation given in [8] serves to
illuminate this point.

In the NSF approach the function $\Lambda$ from eq. 4 plays an important role
since the conformal metric is completely given in terms of this function and
its vanishing yields a flat conformal metric. Thus, one can implement a
perturbation procedure directly from knowledge of $\Lambda$ and write down the
lowest non trivial formulation from a linearized approximation.

The conformal factor is fixed via the field equations. Directly from the
conformal relationship between the two metrics one finds[9]:

$\displaystyle 2\frac{\partial^{2}\Omega}{\partial
s^{2}}=Z^{a}Z^{b}(R_{ab}[h]-R_{ab}[g])\Omega.$ (5)

The scalar $R_{ab}[h]Z^{a}Z^{b}$ is a quadratic function of $\Lambda$ and it
thus vanishes at a linearized approximation, whereas $R_{ab}[g]$ can be
replaced by the trace free part of the energy-momentum tensor $T_{ab}$. The
three scalar equations (2), (3) and (5) are completely equivalent to the full
Einstein equations for a metric $g_{ab}$. The linearized version of eqs. 4 and
5 is obtained by first giving the zeroth-order solution that yields a flat
metric, namely,

$\displaystyle Z_{0}=x^{a}\ell_{a},\;\;\Omega_{0}=1.$ (6)

with $\ell^{a}$ a null vector defined as
$\ell^{a}=\frac{1}{\sqrt{2}}(1,r^{i})$ and $r^{i}$ the unit vector on the
sphere of null directions, and $x^{a}$ a point in the flat spacetime.
One then writes down a linearized departure from the zeroth order solution as

$\displaystyle Z=x^{a}\ell_{a}+Z_{1},\;\;\Omega=1+\Omega_{1},$ (7)

and its corresponding equations of motion

$\displaystyle\frac{\partial(\bar{\eth}^{2}\eth^{2}Z_{1})}{\partial
s}=2\eth\bar{\eth}(\Omega_{1}),$ (8) $\displaystyle
2\frac{\partial^{2}\Omega_{1}}{\partial s^{2}}=-T_{ab}Z^{a}Z^{b},$ (9)

which can be integrated following the null geodesic either to future or past
null infinity to obtain the advanced or retarded solutions. Note that
$\Omega_{1}$ only depends on the stress energy tensor and vanishes in absence
of matter fields. Assuming we are solving eqs. 8 and 9 for an asymptotically
flat spacetime, we can integrate eq. 8 and obtain

$\displaystyle\bar{\eth}^{2}\eth^{2}Z_{1}=\bar{\eth}^{2}\sigma(Z_{1},\zeta,\bar{\zeta})+\eth^{2}\bar{\sigma}(Z_{1},\zeta,\bar{\zeta})-2\int_{0}^{\infty}\eth\bar{\eth}(\Omega_{1})ds+\mathcal{O}(\Lambda^{2}).$
(10)

The function $Z_{1}$, on the other hand, depends both on the matter fields as
well as $\sigma$. Note also that the r.h.s. of eq. 10 contains non-linear
expressions of $Z_{1}$ since the Bondi shear is an arbitrary function at null
infinity. Thus one must linearized the above equation using a perturbation
procedure. Eq. (10) is a non-homogeneous 4th order elliptic equation on the
sphere and its solution can be found by convoluting the inhomogeneity with the
corresponding Green function [10]. Note that the above eq. 10 given in this
work differs from a previous derivation in [11]. The calculation showed in the
latter reference is more involved, as one can see from comparing the terms
containing the conformal factor as well as the extra integral along the Bondi
time that is not present in our derivation. The main reason for the difference
between these equations is that the one given on the previous work has been
derived combining the two metricity conditions. We have used our derivation
for simplicity. It is worth pointing out this difference since we heavily use
eq. (10) in the results presented in this work.

Higher order terms on a perturbation scheme can be implemented in eq. 4 by
making a perturbation series in the null foliation

$\displaystyle Z=Z_{0}+Z_{1}+Z_{2}+...$ (11)

where $Z_{0}$ correspond to a null foliation in Minkowski space as in eq. 6,
and $Z_{1}$ satisfies equations (9) and (10). This perturbation procedure is
carried on giving the $n-1$ and $n-2$ orders that were previously obtained to
solve for $Z_{n}$. At any order of the perturbation a conformal structure is
derived and the new solution is a field on this spacetime.

We will show later in Sec.III and III.6 this scheme to be useful when studying
gravitational radiation and its interaction with the energy-momentum tensor.
The linearized NSF allows to express $\Omega(x^{a},\zeta,\bar{\zeta})$ as an
explicit functional of the energy momentum tensor leaving the theory with just
one main scalar, $Z$, which can be obtained by iterating to higher orders.

##  III Scattering in the linearized NSF

In this Section, we derive the relation between the incoming and outgoing
radiation at the null boundary of a general asymptotic spacetime, solution to
eq. 10. With this end, we need to find a point in the spacetime where both
kind of radiations are well defined, relate this point to null infinity
through geodesic paths and, finally, apply the uniqueness of the metric to the
corresponding expansion order. Thus, we will use the subscripts $+$ and $-$ to
denote quantities associated to future and past null infinity respectively. In
this sense, $\sigma_{+}$ will stand for outgoing radiation at future null
infinity and $\sigma_{-}$ for ingoing radiation coming from past null
infinity.
The reader should remind that the linearized eq. 10 still remains nonlinear in
the variable $Z$, as the dependence of the shear $\sigma(Z,\zeta,\bar{\zeta})$
on $Z$ is not known. To tackle this problem, we use the perturbative approach
from eq. 11 at the end of this section.

###  III.1 Tensorial spin-$s$ harmonics

When working with gravitational radiation (or any problem involving spin-$s$
fields) it is common the use of spin-$s$ spherical harmonics $Y^{s}_{lm}$ to
expand the angular dependence of functions that are defined at null infinity.
In this work we’ll use instead an equivalent complete base of orthogonal
functions known as tensorial spin-s harmonics.

Given a Newman-Penrose (NP) null tetrad $(l^{a},n^{a},m^{a},\bar{m}^{a})$ for
a flat metric, we can construct a 3-dimensional orthonormal base
$(c^{i},m^{i},\bar{m}^{i})$ for any tangent space of a point $p$ in $S^{2}$.
This 3-dimensional base can be constructed as follows: First, we construct the
timelike and spacelike vectors $t^{a}=l^{a}+n^{a}$ and $c^{a}=l^{a}-n^{a}$.
Then, we take the projections of our 4-dimensional base normal to the vector
$t^{a}$. The three obtained independent projections will be called
$(c^{i},m^{i},\bar{m}^{i})$. These Euclidean vectors are the main ingredient
in the construction of the tensor spin-$s$ harmonics. Moreover, the three
dimensional Euclidean base can be used to set up a general $l$-dimensional
tensor base by doing the exterior product of the vectors
$c^{i},m^{i},\bar{m}^{i}$. This concept is what the tensor spin-$s$ harmonics
embrace. We mention in this subsection only a few properties of the
construction but the reader may find more in the following Ref. [12, 13].
The tensorial spin-$l$ harmonics, denoted $Y^{l}_{lI_{l}}$ with $I_{l}$ a set
of $l$ indices, are defined as the $l$-times product of the space-like vector
$m^{i}$. Then, the complex covariant derivative on the manifold defined in
$S^{2}$, known as $\eth$, along with its conjugate $\bar{\eth}$ behave as
”ladder operators” with respect to the spin index (the number in the
superscript). These ladder operators allow to obtain the whole set of
representations of the tensors $Y^{s}_{lI_{l}}$. A relevant property from
these tensors is

$\displaystyle\bar{Y}^{s}_{lI_{l}}=Y^{-s}_{lI_{l}}.$

###  III.2 Antipodal transformations on the sphere

To link incoming and outgoing radiation at null infinity, we need to introduce
the notion of antipodal points on the sphere. The antipodal points are those
diametrically opposite to each other in the sphere. Then, we define the
antipodal transformation to be the one that carries a point of the sphere to
its antipodal point.
In the usual spherical chart $(\theta,\phi)$, the antipodal transformation
reads

$\displaystyle(\theta,\phi)\rightarrow(\pi-\theta,\pi+\phi),$

or in stereographic coordinates

$\displaystyle(\zeta,\bar{\zeta})\rightarrow(-1/\bar{\zeta},-1/\zeta).$

We denote the antipodal transformation with the symbol $\widehat{}$ , i.e,
$\widehat{\zeta}=-1/\bar{\zeta}$. Given a tensorial spin-$s$ harmonic
$Y^{s}_{lI_{l}}(\zeta,\bar{\zeta})$, we have

$\displaystyle
Y^{s}_{lI_{l}}(\widehat{\zeta},\widehat{\bar{\zeta}})=(-1)^{l}Y^{-s}_{lI_{l}}(\zeta,\bar{\zeta})$
(12)

.

In particular, if we write $l_{-}^{a}=\frac{1}{\sqrt{2}}(-1,r^{i})$ with
$r^{i}$ the corresponding spatial vector, the antipodal transformation is

$\displaystyle\widehat{l}_{-}^{a}=\frac{1}{\sqrt{2}}(-1,\widehat{r}^{i})=\frac{1}{\sqrt{2}}(-1,-r^{i})=-\frac{1}{\sqrt{2}}(1,r^{i})=-l_{+}^{a}$
(13)

eq. 13 is the antipodal transformation applied to a null vector $l_{-}^{a}$
defined at past null infinity. Hence, this vector has its antipodal point at
minus the position of a vector defined at future null infinity.
The antipodal transformation on the derivative operator $\eth$, which will
appear later in the calculations, can be proven to be

$\widehat{\eth}=-\bar{\eth}\ .$

###  III.3 Relation between null cuts at $\mathcal{I}^{+}$ and
$\mathcal{I}^{-}$

As discussed in Sec.II, the condition of the null cone cuts to be null on a
general asymptotic spacetime

$g^{ab}\partial_{a}Z\partial_{b}Z=0$

can be written for each perturbation order from eq. 11. Thus, the n-order null
condition reads

$\displaystyle\sum_{n}^{\infty}\sum_{r+s=0}^{n}g_{n-r-s}^{ab}\partial_{a}Z_{r}\partial_{b}Z_{s}=0$
(14)

with,

$g_{0}^{ab}=\eta^{ab},$

and,

$g_{1}^{ab}=2\Omega_{1}\eta^{ab}+h_{1}^{ab},$

the first order correction to the flat metric.

The zero order null condition is just the vector $l^{a}$ to be null
$\eta^{ab}l_{a}l_{b}=0$. The first non-trivial null condition appears at order
$n=1$,

$\displaystyle h^{ab}l_{a}l_{b}+2\eta^{ab}l_{a}\partial_{b}Z=0,$ (15)

where we have dropped the subindex $1$ in $h_{1}^{ab}$ and $Z_{1}$ for
simplicity. Up to first order, we can use the Minkowski metric to raise or
lower indices, thus, $\eta^{ab}l_{a}=l^{b}$ and

$\displaystyle
h_{ab}(x^{a},\zeta)l^{a}l^{b}=-2l^{a}\partial_{a}Z(x^{a},\zeta)\ .$ (16)

eq. 16 relates the first order correction $h_{ab}$ with the first order
correction of the null foliation $Z$. Note that $Z$ plays the role of a
potential for $h_{ab}$. As we will see below the solution to the linearized
equation for $Z$ shows its explicit dependence on the gravitational radiation
$\sigma$ and the stress energy tensor $T_{ab}$, and it is completely
equivalent to a standard formulation using an advanced or retarded Green
function.
Eq.(14) can be continued to higher orders to obtain nonlinear terms. We will
only keep the first non trivial terms in this equation, since the purpose of
this work is to obtain the tree diagrams arising in a scattering formulation
of gravitational radiation interacting with matter fields.

###  III.4 Solution in absence of matter

In case of no matter fields, i.e $T^{ab}=0$, we have $\Omega_{1}=0$ in eq. 10.
Considering this simplification, we want to find the first order correction to
the null flat foliation $Z_{0}$. Hence, we need to solve eq. 10 to first order

$\displaystyle\bar{\eth}^{2}\eth^{2}Z(x^{a},\zeta,\bar{\zeta})=\bar{\eth}^{2}\sigma(Z_{0},\zeta,\bar{\zeta})+\eth^{2}\bar{\sigma}(Z_{0},\zeta,\bar{\zeta}).$
(17)

The solution can be found by convoluting the inhomogeneity of the equation
with the Green function on the sphere [10],

$\displaystyle
G_{00^{\prime}}(\zeta,\bar{\zeta},\zeta^{\prime},\bar{\zeta^{\prime}})=\frac{1}{4\pi}l^{+a}l^{\prime+}_{a}ln(l^{+a}l^{\prime+}_{a}).$
(18)

The solution reads

$\displaystyle
Z^{+}(x^{a},\zeta,\bar{\zeta})=\oint_{S^{2}}G_{00^{\prime}}\left(\bar{\eth}^{\prime
2}\sigma^{+}(x^{a}l^{\prime+}_{a},\zeta^{\prime},\bar{\zeta^{\prime}})+\eth^{\prime
2}\bar{\sigma}^{+}(x^{a}l^{\prime+}_{a},\zeta^{\prime},\bar{\zeta^{\prime}})\right)dS^{\prime},$
(19)

where we have used that $Z_{0}^{+}=u=x^{a}l^{+}_{a}$. The superscript in
$Z^{+}$ is used to denote that the future null directed foliation has been
used to construct the future null cone cut $x^{a}l^{+}_{a}$. Similarly,
$\sigma^{+}$ denotes the outgoing radiation at future null infinity.
Another solution to eq. 17 can be found in terms of ingoing radiation

$\displaystyle
Z^{-}(x^{a},\zeta,\bar{\zeta})=\oint_{S^{2}}G_{00^{\prime}}\left(\bar{\eth}^{\prime
2}\sigma^{-}(-x^{a}\ell^{\prime-}_{a},\zeta^{\prime},\bar{\zeta^{\prime}})+\eth^{\prime
2}\bar{\sigma}^{-}(-x^{a}\ell^{\prime-}_{a},\zeta^{\prime},\bar{\zeta^{\prime}})\right)dS^{\prime}.$
(20)

where we have used the null cone cut $Z^{-}_{0}=-x^{a}\ell^{-}_{a}$, given by
the intersection of the past null cone from $x^{a}$ with past null infinity.
Note also that at past null infinity the cut is described by $v=Z^{-}_{0}$,
with $v$ the usual advanced time coordinate.
Given a point in the spacetime $x^{a}$ and the metric $g^{ab}(x^{a})$ at that
point, we can write the metric in terms of the ingoing ($\sigma^{-}$) or
outgoing radiation ($\sigma^{+})$ using the scalar field $Z$. On the other
hand, from the uniqueness of the metric tensor $g^{ab}(x^{c})$ both
descriptions of the metric must coincide, i.e
$g^{ab}_{-}(x^{c})=g^{ab}_{+}(x^{c})$. Thus, one should have,

$\displaystyle
h^{+}_{1ab}(x^{c})l^{+a}l^{+b}=h^{-}_{1ab}(x^{c})\hat{l}^{-a}\hat{l}^{-b}.$
(21)

In absence of a stress energy tensor the first order deviation $h_{1}^{ab}$
only depends on $x^{a}$. Using eq. 16 we find a relation between first order
outgoing and ingoing null surfaces

$\displaystyle l^{\prime
a}\partial_{a}Z^{+}=\hat{l}^{+a}\partial_{a}\widehat{Z^{-}}$ (22)

which can also be written as

$\displaystyle l^{+a}\partial_{a}(Z^{+}+\widehat{Z^{-}})=0.$ (23)

Eq. (23) can be thought of as an equation for two different functions of the
same variables. Taking into account that $l^{-a}=-\hat{l}^{+a}$,
$G_{00^{\prime}}(\zeta,\zeta^{\prime})=G_{00^{\prime}}(\hat{\zeta},\hat{\zeta}^{\prime})$
and performing a change of variables $\zeta\to\hat{\zeta}$, one rewrites
$Z^{-}(x^{a},\hat{\zeta},\hat{\bar{\zeta}})$ in eq. 23 as

$\displaystyle
Z^{-}(x^{a},\hat{\zeta},\hat{\bar{\zeta}})=\oint_{S^{2}}\left(\eth^{\prime
2}G_{00^{\prime}}\sigma^{-}(x^{a}\ell^{\prime+}_{a},\widehat{\zeta^{\prime}},\widehat{\bar{\zeta^{\prime}}})+\bar{\eth}^{\prime
2}G_{00^{\prime}}\bar{\sigma}^{-}(x^{a}\ell^{+^{\prime}}_{a},\widehat{\zeta^{\prime}},\widehat{\bar{\zeta^{\prime}}})\right)dS^{\prime},$
(24)

where the function
$\sigma^{-}(x^{a}\ell^{+}_{a},\widehat{\zeta^{\prime}},\widehat{\bar{\zeta^{\prime}}})$
is now thought as a function given on future null infinity. A detailed
analysis taking into account the parity of this function when expanding in
spherical harmonics is given below. From this point of view, the action of
$l^{+a}\partial_{a}$ on $(Z^{+}+\widehat{Z^{-}})$ is to add a common factor
$l^{+a}l^{\prime+}_{a}$ to the whole equation. For example, its action on
$Z^{-}(x^{a},\hat{\zeta},\hat{\bar{\zeta}})$ yields

$\oint_{S^{2}}l^{+a}l^{\prime+}_{a}\left(\eth^{\prime
2}G_{00^{\prime}}\dot{\sigma}^{-}(u^{\prime},\widehat{\zeta^{\prime}},\widehat{\bar{\zeta^{\prime}}})+\bar{\eth}^{\prime
2}G_{00^{\prime}}\dot{\bar{\sigma}}^{-}(u^{\prime},\widehat{\zeta^{\prime}},\widehat{\bar{\zeta^{\prime}}})\right)dS^{\prime},$

with $u^{\prime}=x^{a}\ell^{\prime+}_{a}$ and now the radiation scalars are
derived with respect to this variable. Moreover, using the properties of the
spin weighted spherical harmonics, the antipodal transformation of each
eigenfunctions is related to its complex conjugate. Thus, we expect a
relationship between the Bondi shear at past null infinity and its complex
conjugate at future null infinity. An identical result can be obtained by
simply demanding that

$\displaystyle Z_{1}^{+}+\widehat{Z_{1}^{-}}=0.$ (25)

In this case an analogous formula is obtained for
$\sigma^{-}(u,\widehat{\zeta},\widehat{\bar{\zeta}})$ instead of
$\dot{\sigma}^{-}(u,\widehat{\zeta},\widehat{\bar{\zeta}})$. The potential
gauge supertraslation freedom has been fixed by demanding that in absence of
radiation the null cone cut is given by $u=x^{a}\ell^{+}_{a}$. Note that the
physical variables are gauge independent since the metric depends on
$\dot{\sigma}$.

###  III.5 Modes relation

To study the relation between the different modes of gravitational radiation,
we expand the dependence on the stereographic coordinates from all the
physical scalar quantities. That is to say, given a quantity
$\eta(u,\zeta,\bar{\zeta})$ with spin weight s, we have

$\eta(u,\zeta,\bar{\zeta})=\sum_{l}\eta(u)^{I_{l}}Y^{s}_{l,I_{l}}(\zeta,\bar{\zeta}),$

where $s=2$ if $\eta=\sigma^{0}$.
Also, the Green function (18) for the Laplacian on $S^{2}$ can be expanded as

$\displaystyle
G_{00^{\prime}}=\sum_{l=2}^{\infty}\frac{4\pi}{2l+1}Y_{l,I_{L}}^{0}(\zeta,\bar{\zeta})Y_{l,I_{L}}^{\prime
0}(\zeta^{\prime},\bar{\zeta}^{\prime}).$

Replacing the above formula in eq. 19 and using Stokes theorem we get the
$l,I_{L}$ contribution to $Z$, i.e.,

$Z^{+}_{l,I_{l}}=\oint\big{(}Y_{l,I_{l}}^{-2}\sigma^{+}(u,\zeta,\bar{\zeta})+Y_{l,I_{l}}^{2}\bar{\sigma}^{+}(u,\zeta,\bar{\zeta})\big{)}d^{2}S,$
(26)

where the factor $\frac{4\pi}{2l+1}$ has been absorbed in the definition of
$Z^{+}_{l,I_{l}}$. A similar calculation can be carried out to obtain the
right hand side in eq. 22 but we must also apply the antipodal transformations
from Sec. III.2, then

$\displaystyle\widehat{Z}^{-}_{l,I_{l}}=(-1)^{l}\oint\big{(}Y_{l,I_{l}}^{-2}\sigma^{-}(v,\zeta,\bar{\zeta})+Y_{l,I_{l}}^{2}\bar{\sigma}^{-}(v,\zeta,\bar{\zeta})\big{)}d^{2}S$
(27)

To obtain the relation between the different modes from outgoing radiation
$\sigma^{+}_{l,I_{L}}$ and incoming radiation $\sigma^{-}_{l,I_{l}}$, one does
the harmonic decomposition of both functions and makes a change of variables
as in eq. 13. The details of the derivation are shown in Appx. A. The
coefficients of the ingoing and outgoing radiation are given by the following
relations

$\displaystyle\sigma^{+}_{l,I_{l}}(w)+(-1)^{l}\bar{\sigma}^{-}_{l,I_{l}}(w)=0,$
(28)

using a positive frequency $w$ and integrating on $S^{2}$ or,

$\displaystyle\sigma^{+}(u,\zeta,\bar{\zeta})+\bar{\sigma}^{-}(u,\widehat{\zeta},\widehat{\bar{\zeta}})=0,$
(29)

as a function of the Bondi time $u$ and the stereographic coordinates on the
sphere $(\zeta,\bar{\zeta})$. The same result could have been obtained by
grouping terms under the same value of $\eth^{\prime 2}G_{00^{\prime}}$ and
c.c. in eq. 25. The alternative derivation given above gives the same result
and it can be extended to the case with a non trivial energy momentum tensor.

###  III.6 Scattering in presence of matter

When a matter field is present in the spacetime, there is an additional term
affecting the form of the foliation $Z$ in eq. 10. The presence of the matter
term yields an additional complication since the calculations are technically
more involved. Nevertheless, it is possible to derive an explicit formula for
each mode.

The future directed solution of the NSF equation can be written as

$\displaystyle
Z^{+}_{l,I_{l}}=\oint\big{[}Y_{l,I_{l}}^{-2}\sigma^{+}(u,\zeta,\bar{\zeta})+c.c.-2Y_{l,I_{l}}^{0}\int_{0}^{\infty}ds\bar{\eth}\eth\Omega_{1}(y^{c}(s),\zeta,\bar{\zeta})\big{]}d^{2}S,$
(30)

with $y^{c}(s)=x^{c}+sl^{+c}$. A geometrical formula can be written by
defining $N^{+}_{x}$ as the future null cone coming out from the point
$x^{c}$, and $C^{+}_{x}$ as the intersection of $N^{+}_{x}$ with future null
infinity. With these definitions, we have

$\displaystyle
Z^{+}_{l,I_{l}}=\oint_{C^{+}_{x}}d^{2}S\big{(}Y_{l,I_{l}}^{-2}\sigma^{+}(u,\zeta,\bar{\zeta})+c.c.\big{)}-2\int_{N^{+}_{x}}Y_{l,I_{l}}^{0}\bar{\eth}\eth\Omega_{1}(x^{c},s,\zeta,\bar{\zeta})dsd^{2}S.$
(31)

The corresponding formulae for $Z^{-}_{l,I_{l}}$ are

$\displaystyle
Z^{-}_{l,I_{l}}=\oint\big{[}Y_{l,I_{l}}^{-2}\sigma^{-}(v,\zeta,\bar{\zeta})+c.c.-2Y_{l,I_{l}}^{0}\int_{0}^{\infty}ds\bar{\eth}\eth\Omega_{1}(y^{c},\zeta,\bar{\zeta})\big{]}d^{2}S,$
(32)

and

$\displaystyle
Z^{-}_{l,I_{l}}=\oint_{C^{-}_{x}}d^{2}S\big{(}Y_{l,I_{l}}^{-2}\sigma^{+}(v,\zeta,\bar{\zeta})+c.c.\big{)}-2\int_{N^{-}_{x}}Y_{l,I_{l}}^{0}\bar{\eth}\eth\Omega_{1}(x^{c},s,\zeta,\bar{\zeta})dsd^{2}S.$
(33)

###  III.7 The conformal factor in the linearized NSF

The relation between the Ricci tensor field $R_{ab}$ for a general asymptotic
spacetime and its conformal version $\tilde{R}_{ab}$ is given by [9]

$\displaystyle
R_{ab}=\tilde{R}_{ab}+2\Omega^{-1}\tilde{\nabla}_{a}\tilde{\nabla}_{b}\Omega+\tilde{g}_{ab}\tilde{g}^{cd}\left(\Omega^{-1}\tilde{\nabla}_{c}\tilde{\nabla}_{d}\Omega-3\Omega^{-2}\tilde{\nabla}_{c}\Omega\tilde{\nabla}_{d}\Omega\right).$
(34)

As we defined in Sec.II, $\Omega$ is the conformal factor of the metric and
one of the main scalars from NSF. For the linearized version of NSF, we can
expand $\Omega=1+\Omega_{1}$ and
$g_{01}=1+2\Omega_{1}(x^{a},\zeta,\bar{\zeta})$. Recalling
$\tilde{R}_{ab}=\mathcal{O}(\Lambda^{2})$ and contracting with the null tensor
$l^{a}l^{b}$, we get

$R_{ab}l^{a}l^{b}=2l^{a}l^{b}\partial_{a}\partial_{b}\Omega_{1}\ ,$ (35)

where we have replaced the covariant by the partial derivative, valid at this
level of approximation. We then use the Fourier version of eq. 35

$\displaystyle\Omega_{1}(k^{a},\zeta,\bar{\zeta})$
$\displaystyle=\frac{l^{a}l^{b}R_{ab}(k^{c})}{2(k^{c}l_{c})^{2}}$ (36)
$\displaystyle=\frac{T_{ab}(k)l^{a}l^{b}}{2(k^{c}l_{c})^{2}},$ (37)

where $k^{a}$ is the 4-vector from the transformation and the last equality
follows from the fact that $l_{a}$ is the first vector in the NP null base.
Replacing eq. 37 in eq. 10, we obtain

$\displaystyle\bar{\eth}\eth\Omega_{1}(x^{c},\zeta,\bar{\zeta})$
$\displaystyle=\int
d^{4}k\theta(k^{o})[T_{ab}(k^{c})e^{-ix^{c}k_{c}}+\bar{T}_{ab}(k^{c})e^{ix^{c}k_{c}}]\bar{\eth}\eth\left(\frac{l^{a}l^{b}}{(k^{c}l_{c})^{2}}\right),$

where $\theta(k^{o})$ is the step function.

Hence, each advanced mode solution to the NSF equation in presence of matter
reads

$Z^{+}_{l,I_{l}}=\oint\bigg{(}Y_{l,I_{l}}^{-2}\sigma^{+}(x^{c}\ell^{+}_{c},\zeta,\bar{\zeta})+iY_{l,I_{l}}^{0}\int
d^{3}k\bar{\eth}\eth\left(\frac{T_{ab}(k^{c})\ell^{+a}\ell^{+b}}{(k^{c}\ell^{+}_{c})^{2}}\right)\frac{e^{-ix^{c}k_{c}}}{k^{c}\ell^{+}_{c}}+c.c.\bigg{)}d^{2}S.$
(38)

The positive frequency component of this equation is given by

$Z^{+}(w)_{l,I_{l}}=\sigma(w)_{l,I_{l}}+i\int_{0}^{\infty}dk^{o}\oint
d^{2}\hat{k}T_{ab}(k^{o},w\hat{k}^{i})\oint
Y_{l,I_{l}}^{0}\bar{\eth}\eth\left(\frac{l^{a}l^{b}}{(k^{c}l_{c})^{2}}\right)\frac{1}{k^{c}l_{c}},$
(39)

with $k^{a}=(k^{o},w\hat{k}^{i})$.

Likewise, the retarded solution for $Z$ yields

$Z^{-}_{l,I_{l}}=(-1)^{l}\oint\bigg{(}Y_{l,I_{l}}^{-2}\sigma^{-}(-x^{c}\ell^{-}_{c},\zeta,\bar{\zeta})-iY_{l,I_{l}}^{0}\int
d^{3}k\bar{\eth}\eth\left(\frac{\bar{T}_{ab}(k^{c})\ell^{+a}\ell^{-b}}{(k^{c}\ell^{-}_{c})^{2}}\right)\frac{e^{-ix^{c}k_{c}}}{k^{c}\ell^{-}_{c}}+c.c.\bigg{)}d^{2}S,$
(40)

and

$Z^{-}(w)_{l,J_{l}}=(-1)^{l}\bigg{(}\sigma^{-}(w)_{l,J_{l}}-i\int_{0}^{\infty}dk^{o}\oint
d^{2}\hat{k}\bar{T}_{ab}(k^{o},w\hat{k}^{i})\oint
Y_{l,J_{l}}^{0}\bar{\eth}\eth\left(\frac{l^{-a}l^{-b}}{(k^{c}l^{-}_{c})^{2}}\right)\frac{1}{k^{c}l^{-}_{c}}\bigg{)},$
(41)

Finally (see Appendix), from
$Z^{+}(x^{a},\zeta,\bar{\zeta})+Z^{-}(x^{a},\widehat{\zeta},\widehat{\bar{\zeta}})=0$,
we get

$\sigma^{+}_{l,I_{l}}(w)+(-1)^{l}\bar{\sigma}^{-}_{l,I_{l}}(w)=i\oint
d^{2}SY_{l,I_{l}}^{0}\int_{0}^{\infty}dk^{o}\oint
d^{2}\hat{k}T_{ab}\left(\frac{\bar{\eth}\eth\big{(}\frac{l^{a}l^{b}}{(k^{c}l_{c})^{2}}\big{)}^{+}}{k^{c}l^{+}_{c}}+(-1)^{l}\frac{\bar{\eth}\eth\big{(}\frac{l^{a}l^{b}}{(k^{c}l_{c})^{2}}\big{)}^{-}}{k^{c}l^{-}_{c}}\big{)}\right).$
(42)

The last term on the integral is a non vanishing form factor that yields the
non trivial part of $\sigma^{+}_{l,J_{l}}(w)$. It does not depend on the
stress energy tensor but it could give a vanishing contribution for some
special cases, as for example, $T_{ab}=f(k)k_{a}k_{b}+g(k)\eta_{ab}$.

It is also useful to write down an equation relating
$\sigma^{+}(u,\zeta,\bar{\zeta})$ with $\sigma^{-}(u,\zeta,\bar{\zeta})$ and
the stress energy tensor $T_{ab}(x)$. To do this, we invert the frequency
decomposition of equation (82) and multiply by the corresponding
$Y_{l,J}^{2}(\zeta,\bar{\zeta})$ obtaining

$\sigma^{+}(u,\zeta,\bar{\zeta})+\bar{\sigma}^{-}(u,\widehat{\zeta},\widehat{\bar{\zeta}})=i\int
d^{4}ke^{iwu}T_{ab}(k)\oint
d^{2}S^{\prime}\left(\frac{\eth^{2}G_{00^{\prime}}\bar{\eth^{\prime}}\eth^{\prime}\big{(}\frac{l^{\prime
a}l^{\prime
b}}{(k^{c}l^{\prime}_{c})^{2}}\big{)}^{+}}{k^{c}l^{\prime+}_{c}}+\frac{\eth^{2}G_{\widehat{0}0^{\prime}}\bar{\eth^{\prime}}\eth^{\prime}\big{(}\frac{l^{\prime
a}l^{\prime
b}}{(k^{c}l^{\prime}_{c})^{2}}\big{)}^{-}}{k^{c}l^{\prime-}_{c}}\right),$ (43)

where
$G_{\widehat{0}0^{\prime}}=G_{00^{\prime}}(\widehat{\zeta},\widehat{\bar{\zeta}},\zeta^{\prime},\bar{\zeta}^{\prime})$,
$T_{ab}$ is written in spherical coordinates as
$T_{ab}(k)=T_{ab}(k^{o},w,\hat{k})$, and $w=\sqrt{\vec{k}.\vec{k}}$. Note that
a generic $T_{ab}(x)$ does not satisfy the homogeneous wave equation, and
thus, $k^{a}$ is not a null vector. As we can see in eq. (43), the coordinates
$(\zeta,\bar{\zeta})$ and $(\hat{\zeta},\bar{\hat{\zeta}})$ are associated
with the advanced and retarded solution respectively.

###  III.8 The scattering with a massless scalar field

In this section we will apply the formula eq. 43 to analyze the particular
case when the matter field present is due to a scalar field. This implies the
Lagrangian density from the field is

$\displaystyle\mathcal{L}=-\frac{1}{2}g^{ab}\partial_{a}\phi\partial_{b}\phi$

and the energy-momentum tensor

$\displaystyle T^{ab}$
$\displaystyle=g^{ab}(-\frac{1}{2}\partial_{c}\phi\partial^{c}\phi)+\partial^{a}\phi\partial^{b}\phi$
(44)

Contracting eq. 44 with the null vectors $l^{a}$ , we obtain that the only
term surviving the contraction is

$\displaystyle T_{ab}(x)l^{a}l^{b}=(l_{a}\partial^{a}\phi(x))^{2}.$

On the other hand, the scalar fields, which are solutions to the wave
equation, can be written in terms of the Fourier transform

$\phi(y^{a})=\int
d^{4}k\left(e^{-ik^{b}y_{b}}a(k)+e^{ik^{b}y_{b}}\bar{a}(k)\right)\delta(k^{a}k_{a})$
(45)

since the real scalar $\phi(y^{a})$ satisfies the wave equation. The energy-
momentum tensor component that is needed for this calculation can be computed
in momentum space as

$\displaystyle T_{ab}(k^{c})l^{a}l^{b}$ $\displaystyle=\int
d^{4}xe^{ik^{b}x_{b}}(l^{a}\partial_{a}\phi(x^{c}))^{2}$

then,

$T_{ab}(k^{a})l^{a}l^{b}=\int
d^{4}k^{\prime}a(k^{\prime})a(k-k^{\prime})(l_{a}k^{\prime a})l_{b}(k^{\prime
b}-k^{b})\delta(k^{\prime
a}k^{\prime}_{a})\delta((k-k^{\prime})^{b}(k-k^{\prime})_{b}).$

Note that both $k^{\prime a}$ and $(k-k^{\prime})^{b}$ are null vectors
whereas $k^{b}$ is not since $T_{ab}(x)l^{a}l^{b}$ does not satisfy the
homogeneous wave equation, as one can see from a direct calculation.

The amplitude $a(k)$ in eq. 45 is directly related to the free data of the
scalar field given at the boundary of the space time. One can thus directly
compute the scattering between gravitational and scalar waves in terms of the
different modes of the corresponding radiation fields at null infinity. To do
that we first give the relationship between the Fourier transform
$a(k^{\prime})$ and the free data of the scalar field.

Assuming one has incoming scalar waves, one can write the solution of the wave
equation as

$\phi(y^{a})=\oint d^{2}S\dot{A}(v,\zeta,\bar{\zeta}),$ (46)

with $v=-y^{a}l^{-}_{a}$ and $A(v,\zeta,\bar{\zeta})$ the free data given at
past null infinity. Furthermore, using a Fourier transform to obtain a
frequency decomposition one has

$A(v,\zeta,\bar{\zeta})=\int_{0}^{\infty}dw\left(A(w,\zeta,\bar{\zeta})e^{-iwv}+\bar{A}(w,\zeta,\bar{\zeta})e^{iwv}\right).$
(47)

Thus,

$\phi(y^{a})=i\int_{0}^{\infty}w\;dw\oint
d^{2}S\left(A(w,\zeta,\bar{\zeta})e^{-iwv}-\bar{A}(w,\zeta,\bar{\zeta})e^{iwv}\right),$
(48)

which can be regarded as spherical coordinates in momentum space. One then has

$\phi(y^{a})=i\int\frac{d^{3}k}{2w}2i\left(\bar{A}(w,\zeta,\bar{\zeta})e^{-ik^{a}x_{a}}-A(w,\zeta,\bar{\zeta})e^{ik^{a}x_{a}}\right),$
(49)

where we have used $v=-l^{-a}x_{a}$ to obtain the above formula. Equation (45)
can be written as

$\phi(y^{a})=\int\frac{d^{3}k}{2w}\left(a(k)e^{-ik^{a}x_{a}}+\bar{a}(k)e^{ik^{a}x_{a}}\right),$
(50)

by a straight calculation using $\delta(k^{2})=\delta((k^{o}-w)(k^{o}-w))$. It
then follows that,

$a(w,\zeta,\bar{\zeta})=2i\bar{A}(w,\zeta,\bar{\zeta}),$

or

$a(w)_{lJ}=2i\bar{A}(w)_{lJ}.$

One can also show that using the positive frequency decomposition for $\phi$,
together with $\delta(k^{2})=\frac{1}{2w_{k}}\delta((k^{o}-w))$ one obtains,

$T_{ab}(k^{a})l^{a}l^{b}=\int\frac{d^{3}k_{1}}{4w_{1}w_{2}}a(\vec{k}_{1})a(\vec{k}-\vec{k}_{1})(l_{a}k_{1}^{a})l_{b}(k_{1}^{b}-k^{b})\delta(k^{o}-(w_{1}+w_{2})),$
(51)

with $w_{2}=\sqrt{(\vec{k}-\vec{k}_{1}).(\vec{k}-\vec{k}_{1})}$.

We can use the formula derived in the previous section to describe the
scattering of incoming gravitational and scalar waves. For that, we give free
radiation data for both fields and use formula eq. 43 to obtain the outgoing
gravitational waves at future null infinity represented by the radiation field
$\sigma^{+}(u,\zeta,\bar{\zeta})$. In particular, we would like to compute the
gravitational tail of the outgoing wave when both the incoming waves have
compact support at past null infinity.

Selecting appropriate times $v_{i}$ and $v_{f}$ so that there are no incoming
waves for $v<v_{i}$ or $v_{f}<v$, then we would like to obtain the
gravitational radiation at future null infinity at sufficient large value of
$u$ so that the ”free” outgoing gravitational and scalar waves have died out.
A Bondi time $u_{f}$ can be obtained such that a null plane that starts at
$(v_{f},\hat{\zeta}\hat{\bar{\zeta}})$, ends up at
$(u_{f},\zeta,\bar{\zeta})$. This plane is given by
$y^{a}\ell^{+}_{a}=u_{f}=const.$ but it can also be written as
$u_{f}=-y^{a}\hat{\ell}^{-}_{a}=v_{f}$. Thus, for times $u>u_{f}$, there are
no ”free” outgoing waves, and directly from eq. 43 one gets,

$\displaystyle\sigma^{+}(u,\zeta,\bar{\zeta})=i\int
d^{3}k_{1}d^{3}k_{2}e^{-iwu}a^{-}(\vec{k}_{1})a^{-}(\vec{k}_{2})\oint
d^{2}S^{\prime}\frac{\eth^{2}G_{00^{\prime}}}{(k_{1}+k_{2})^{c}l^{\prime}_{c}}\bar{\eth^{\prime}}\eth^{\prime}\big{(}\frac{k_{1a}k_{2b}l^{\prime
a}l^{\prime b}}{((k_{1}+k_{2})^{c}l^{\prime}_{c})^{2}}\big{)},$ (52)

with $w=\sqrt{(\vec{k}_{1}+\vec{k}_{2}).(\vec{k}_{1}+\vec{k}_{2})}$ and we
have used $l^{-a}=-\widehat{l}^{a}$,
$G_{00^{\prime}}=\widehat{G}_{00^{\prime}}$ together with the coordinate
transformation $\zeta\rightarrow\widehat{\zeta}$ to get the final form of eq.
(52). Note that $\sigma^{-}(u,\zeta,\bar{\zeta})$ is absent in the above
equation since it has compact support and vanishes for
$u=x^{a}l^{+}_{a}>u_{f}$. Likewise, $\phi(x)$ vanishes on the future null cone
of $x^{a}$ and gives no contribution to the the stress energy tensor on the
integral on that cone. The corresponding $(l,I_{l})$ mode for the positive
frequency decomposition is given by

$\displaystyle\sigma^{+}_{l,I_{l}}(w)=i\oint d^{2}\hat{k}\int
d^{3}k_{1}d^{3}k_{2}\mathcal{S}_{l,I_{l}}(k,k_{1},k_{1})a^{-}(\vec{k}_{1})a^{-}(\vec{k}_{2}),$
(53)

with,

$\displaystyle\mathcal{S}_{l,I_{l}}(k,k_{1},k_{2})=\delta(w-(w_{1}+w_{2}))\frac{Y_{l,I_{l}}^{0}(\hat{k})}{(k_{1}+k_{2})^{c}l_{c}}\bar{\eth}\eth\big{(}\frac{k_{1}^{a}k_{2}^{b}l_{a}l_{b}}{((k_{1}+k_{2})^{c}l_{c})^{2}}\big{)}$
(54)

and $k^{a}=wl^{a}(\hat{k})$, $k_{1}^{a}=w_{1}l^{a}(\hat{k}_{1})$,
$k_{2}^{a}=w_{2}l^{a}(\hat{k}_{2})$. Using a particle physics interpretation
of the plane waves , the delta function can be interpreted as the conservation
of energy of incoming and outgoing particles. Energy conservation, as opposed
to full 4- momenta conservation, is a characteristic feature of the
interaction with an external field, as one can also see in Section:8.7 of
[14].

It is worth making some remarks regarding these results. Eq. (52) exhibits the
tail of the gravitational wave due to the interaction with the scalar field
since it is evaluated for times $u>u_{f}$. The equation for the modes relation
(53) is also interesting since it shows that one can obtain different harmonic
components for the outgoing gravitational wave that may not be present in the
incoming wave. Eq. (53) is also interesting in a quantum field theory approach
since it gives the probability scattering amplitude that an incoming graviton
with a given value of quantum numbers end up with a different set of outgoing
quantum numbers. In this case the scalar field is considered as an external
classical field.

####  III.8.1 Minimal coupling between the gravitational and a matter field

Here we review the standard Lagrangian formulation for minimal coupling
between matter and gravity since the two methods yield analogous results at a
linearized level. However, as we will see below, there is a big difference
between our formulation and the Green function approach based on the flat
conformal structure. One can be extended in perturbation procedure, the other
one cannot.

The Lagrangian for minimum coupling is given by

$\displaystyle\mathcal{L}=\mathcal{L}_{G}+\kappa_{M}\mathcal{L}_{M}$ (55)

with

$\displaystyle\mathcal{L}_{G}=\sqrt{-g}R,$

$\kappa_{M}$ a suitable coupling constant, and $\mathcal{L}_{M}$ the
lagrangian of the associated matter field. Writing down the Euler-Lagrange
equations and then linearizing the obtained equations yields the first order
set of equations from which a perturbation procedure can be implemented.

As a particular example we consider here a real massless scalar field $\phi$,

$\displaystyle\mathcal{L}_{M}=\frac{1}{2}\sqrt{-g}g^{ab}\partial_{a}\phi\partial_{b}\phi,$
(56)

It is straightforward to show that the trace free linearized perturbation of a
flat metric obeys the wave equation with or without sources, i.e.,

$\displaystyle\Box
h_{ab}=T_{ab}=\partial_{a}\phi\partial_{b}\phi-\frac{1}{2}g_{ab}\partial_{c}\phi\partial^{c}\phi,$
(57)

with the massless field satisfying.

$\displaystyle\Box\phi=0.$ (58)

Using the Green functions for the D’Alambertian operator in flat space one
obtains

$\displaystyle
g^{+}_{ab}=\oint_{C^{+}_{x}}d^{2}S\big{(}\dot{\sigma}^{+}(u,\zeta,\bar{\zeta})\bar{m}_{a}\bar{m}_{b}+c.c.\big{)}+\int_{N^{-}_{x}}d^{4}x^{\prime}G^{+}(x,x^{\prime})T_{ab}(x^{\prime}),$
(59) $\displaystyle
g^{-}_{ab}=\oint_{C^{-}_{x}}d^{2}S\big{(}\dot{\sigma}^{-}(v,\zeta,\bar{\zeta})\bar{m}_{a}\bar{m}_{b}+c.c.\big{)}+\int_{N^{-}_{x}}d^{4}x^{\prime}G^{-}(x,x^{\prime})T_{ab}(x^{\prime}),$
(60)

with $G^{+}(x,x^{\prime})$ ($G^{-}(x,x^{\prime})$), the advanced (retarded)
Green function of the wave operator. We thus obtain a similar derivation to
the one presented in the previous sections.

A natural question is then why should one follow the somewhat involved
approach to obtain a standard result? The main reason is that there is no
suitable perturbation calculation to obtain the higher order terms using the
Green function of the flat D’Alambertian operator. If one tries to do so, one
discovers that the null geodesics of the flat spacetime become either timelike
or spacelike curves of the non trivial metric. Thus, they either end up at
spacelike or at timelike infinity, i.e. never at null infinity and thus one
cannot find a perturbation procedure that converges to the solution one is
looking for. Our formulation, on the other hand, is amenable to a perturbation
approach that is well defined at every step of the perturbation scheme. All
the variables involved that were defined and used in the previous sections can
be redefined at every step of the perturbation series since they have
geometrical meanings, i.e, null cone cuts, affine parameters, Bondi
coordinates, etc. For example, in the field equation for $Z$

$\displaystyle\bar{\eth}^{2}\eth^{2}Z=\eth^{2}\bar{\sigma}(Z,\zeta,\bar{\zeta})+\bar{\eth}^{2}\sigma(Z,\zeta,\bar{\zeta})+\int_{Z}^{\infty}\dot{\sigma}\dot{\bar{\sigma}}du-\int_{s}^{\infty}(\eth\bar{\eth}(\Omega^{2})+g^{ab}\partial_{a}\Lambda\partial_{b}\bar{\Lambda})ds^{\prime},$

all the variables involved can be obtained in a perturbation series leaving
the geometrical meaning of the equation unchanged.

In addition, our formulation has a clear distinction between the true
gravitational field i.e. the conformal structure, and the matter field. As one
can see from the field equations, the function $\eth^{2}Z$ defines the
conformal structure of the spacetime and only vanishes for a flat spacetime
whereas the conformal factor $\Omega$ depends on the stress energy tensor.

##  IV Summary and conclusions

In this work we use the NSF approach to study the coupling of matter and
gravity for the incoming/outgoing gravitational radiation given on the null
boundaries of an asymptotically flat spacetime.

We first derive a new field equation for the main variable since the original
formulae contained some errors[11]. This equation is a non-linear 4th order
elliptic equation on the sphere with the matter term taking part in the
inhomogeneity. Using an ab-initio assumption that the matter field preserves
the regularity of the null cone cuts we solve this equation by a perturbation
scheme in eq. 11.

We then consider the dispersion relation between incoming and outgoing
gravitational waves when matter is present at the lowest non trivial level.
This is done in Sec.III by first obtaining a relation between the incoming
$Z^{-}$ and outgoing $Z^{+}$ null foliations up to first order in the
iterative process. This relation directly follows from the uniqueness of the
first order metric.
Formal solutions to the elliptic equation on $S^{2}$ are given for null and
non-null matter fields. The former case yields, by means of eq. 25, a relation
for every mode in the decomposition of the gravitational shear $\sigma$.
Although this relation gives the trivial scattering in every mode of the
gravitational radiation in the absence of matter, it is important to obtain
since it gives a non trivial relation between the incoming and outgoing
radiation fields.
When matter is present, a similar relation can be derived for every mode.
Despite being more difficult than in the non-matter case, this expression
still gives a practical formula to work with. Indeed, the calculation of the
last term in eq. 10 can be directly implemented in numerical integration and
get the desired mode of scattering. When the energy-momentum tensor $T_{ab}$
depends on the free data given on null infinity the calculations are greatly
simplified. As an example, we consider a real, massless scalar field and write
the mode relations for that particular case.
The above results could be thought of as the tree level approximation that one
gets in field theory but using a different approach. A correlation with a more
standard linear approximation using advanced or retarded Green functions of
the flat wave equation is also given to emphasize the close relationship
between the two approaches. In most scattering situations the tree level
approximation yields the dominant terms of a perturbation series.

However, there are several reasons why our formulation has several advantages
over the standard approach. First, the perturbation approach based on null
surfaces correctly incorporates the null free data for gravity given on past
an future null infinities together with the conformal structure of the
spacetime. The n-order term is a field given on a spacetime with null surfaces
computed to the n-1 order. The Green function approach, on the other hand,
always keeps the flat metric to define the null cone structure at each level
of the approximation. Thus, these null lines of Minkowski space fail to reach
either future or past null infinity and therefore never really reach the null
free data and the perturbation procedure never converges to a solution of the
problem. A second point is that the same free data in our formulation is used
for any order of the perturbation procedure, i.e., the phase space is
constructed once and for all in our approach. This is extremely important at a
classical or quantum level since one can introduce either a canonical form
with Poisson brackets or quantum commutation relations for fields given on the
null boundaries that will not be modified as one proceeds with a perturbation
calculation[15]. There is however an assumption that was used in this work
that will have to be modified when higher order perturbations are considered.
In our calculations we were assuming that the conformal completion of
Minkowski spacetime is the Einstein universe, where spacelike infinity $i_{o}$
is a single point and the conformal metric is regular everywhere. This
assumption must be modified in the next orders of perturbation calculations
since they should include a non vanishing ADM mass. This in turns, forces to
reexamine the notion of both the definition of $i_{o}$ and the regularity
structure near spacelike infinity. A new mathematical treatment of $i_{o}$ has
to be given to go along with higher order perturbation terms in order to map
the kinematic structures of past and future null infinities. Although some
progress has been made in this area[16, 17] there is still much work to be
done to get a full understanding of the conformal completion of spacelike
infinity for non trivial spacetimes[18, 19].

###### Acknowledgements.

This research has been supported by grants from CONICET and the Agencia
Nacional de Ciencia y Tecnología.

## References

  * Frittelli, Kozameh, and Newman [1995a] S. Frittelli, C. Kozameh, and E. T. Newman, “Gr via characteristic surfaces,” Journal of Mathematical Physics 36, 4984–5004 (1995a).
  * Kozameh and Quiroga [2016] C. N. Kozameh and G. D. Quiroga, “Center of mass and spin for isolated sources of gravitational radiation,” Physical Review D 93, 064050 (2016).
  * Tassone  _et al._ [2021] E. A. Tassone, P. A. Mandrilli, C. N. Kozameh, G. D. Quiroga, and J. I. Nieva, “Numerical evolution of the center of mass and angular momentum for binary black holes,” Physical Review D 104, 084038 (2021).
  * Tassone and Kozameh [2022] E. A. Tassone and C. N. Kozameh, “Final velocity and radiated energy in numerical simulations of binary black holes,” Physical Review D 105, 044044 (2022).
  * Bordcoch, Kozameh, and Rojas [2016] M. Bordcoch, C. N. Kozameh, and T. A. Rojas, “Asymptotic structure of the null surface formulation and the classical graviton,” Physical Review D 94, 104051 (2016).
  * Frittelli  _et al._ [1997a] S. Frittelli, C. N. Kozameh, E. T. Newman, C. Rovelli, and R. S. Tate, “Fuzzy spacetime from a null-surface version of general relativity,” Classical and Quantum Gravity 14, A143 (1997a).
  * Frittelli  _et al._ [1997b] S. Frittelli, C. N. Kozameh, E. T. Newman, C. Rovelli, and R. S. Tate, “Quantization of the null-surface formulation of general relativity,” Physical Review D 56, 889 (1997b).
  * Frittelli, Newman, and Kozameh [1995] S. Frittelli, E. T. Newman, and C. N. Kozameh, “On the dynamics of characteristic surfaces,” Journal of Mathematical Physics 36, 6397–6416 (1995).
  * Wald [2010] R. M. Wald,  _General relativity_ (University of Chicago press, 2010).
  * Ivancovich, Kozameh, and Newman [1989] J. Ivancovich, C. Kozameh, and E. Newman, “Green’s functions of the edh operators,” Journal of mathematical physics 30, 45–52 (1989).
  * Frittelli, Kozameh, and Newman [1995b] S. Frittelli, C. N. Kozameh, and E. T. Newman, “Linearized einstein theory via null surfaces,” Journal of Mathematical Physics 36, 5005–5022 (1995b).
  * Newman and Silva-Ortigoza [2005] E. T. Newman and G. Silva-Ortigoza, “Tensorial spin-s harmonics,” Classical and Quantum Gravity 23, 497 (2005).
  * Mandrilli, Nieva, and Quiroga [2020] P. Mandrilli, J. Nieva, and G. Quiroga, “Correspondence between tensorial spin-s and spin-weighted spherical harmonics,” General Relativity and Gravitation 52, 1–14 (2020).
  * Mandl and Shaw [2010] F. Mandl and G. Shaw,  _Quantum field theory_ (John Wiley & Sons, 2010).
  * Ashtekar [1981] A. Ashtekar, “Asymptotic quantization of the gravitational field,” Physical Review Letters 46, 573 (1981).
  * Dominguez, Kozameh, and Ludvigsen [1997] A. E. Dominguez, C. N. Kozameh, and M. Ludvigsen, “The phase space of radiative spacetimes,” Classical and Quantum Gravity 14, 3377 (1997).
  * Strominger [2018] A. Strominger,  _Lectures on the infrared structure of gravity and gauge theory_ (Princeton University Press, 2018).
  * Friedrich [1998] H. Friedrich, “Gravitational fields near space-like and null infinity,” Journal of Geometry and Physics 24, 83–163 (1998).
  * Friedrich and Kannar [2000] H. Friedrich and J. Kannar, “Calculating asymptotic quantities near space-like and null infinity from cauchy data,” Annalen der Physik 9, 321–330 (2000).

##  Appendix A Derivation of $\sigma$ modes relation

We first derive the relationship for the ”free” part of Z, that satisfies the
wave equation, since it corresponds to undisturbed gravitational waves that
propagate from minus to plus null infinity. $Z^{+}_{l,I_{l}}$ is given by

$\displaystyle
Z^{+}_{l,I_{l}}=\oint\big{(}Y_{l,I_{l}}^{-2}(\zeta,\bar{\zeta})\sigma^{+}(u,\zeta,\bar{\zeta})+Y_{l,I_{l}}^{2}(\zeta,\bar{\zeta})\bar{\sigma}^{+}(u,\zeta,\bar{\zeta})\big{)}d^{2}S,$
(61)

and we assume the free data $\sigma^{+}(u,\zeta,\bar{\zeta})$ admits a
positive frequency decomposition,

$\sigma^{+}(u,\zeta,\bar{\zeta})=\int_{0}^{\infty}w^{2}\;dw\sigma^{+}(w,\zeta,\bar{\zeta})e^{-iwu}.$

We then rewrite the free data part of $Z$ as

$Z^{+}_{l,I_{l}}=\int_{0}^{\infty}w^{2}\;dw\oint
d^{2}S[Y^{-2}_{l,I_{l}}(\zeta,\bar{\zeta})\sigma^{+}(w,\zeta,\bar{\zeta})e^{-iwx^{a}l_{a}}+Y^{2}_{l,I_{l}}(\zeta,\bar{\zeta})\bar{\sigma}^{+}(w,\zeta,\bar{\zeta})e^{iwx^{a}l_{a}}],$
(62)

which can be rewritten as

$Z^{+}_{l,I_{l}}=\int
d^{3}k[Y^{-2}_{l,I_{l}}(\hat{k})\sigma^{+}(k)e^{-ix^{a}k_{a}}+Y^{2}_{l,I_{l}}(\hat{k})\bar{\sigma}^{+}(k)e^{ix^{a}k_{a}}],$
(63)

using a spherical decomposition of the 3-dim momentum space. In the above
equation the coordinates $(\zeta,\bar{\zeta})$ on the sphere have been
rewritten as $(\hat{k})$, and $k^{a}=wl^{a}(\hat{k})$. Note that
$k^{a}k_{a}=0$, and we have a field that satisfies the wave equation. Using
the eigenfunctions $e^{ix^{a}k_{a}}$ we obtain the Fourier transform of the
above equation as

$\int
d^{3}xe^{ix^{a}k_{a}}\overleftrightarrow{\partial_{t}}Z^{+}_{l,I_{l}}=Y^{-2}_{l,I_{l}}(\hat{k})\sigma^{+}(w,\hat{k})\
.$ (64)

The operator $\overleftrightarrow{\partial_{t}}$ is commonly used in Quantum
field theory and given two scalars $a$ and $b$, it is defined as
$a\overleftrightarrow{\partial_{t}}b=a(\partial_{t}b)-(\partial_{t}a)b$.
Finally, integrating on the sphere eq. 64 yields

$\oint d^{2}\hat{k}\int
d^{3}xe^{ix^{a}k_{a}}\overleftrightarrow{\partial_{t}}Z^{+}_{l,I_{l}}=\sigma^{+}(w)_{l,I_{l}}.$
(65)

$\widehat{Z^{-}}_{l,I_{l}}$, on the other hand, is given by

$\displaystyle\widehat{Z^{-}}_{l,I_{l}}=(-1)^{l}\oint\big{(}Y_{l,I_{l}}^{-2}(\zeta,\bar{\zeta})\sigma^{-}(v,\zeta,\bar{\zeta})+Y_{l,I_{l}}^{2}(\zeta,\bar{\zeta})\bar{\sigma}^{-}(v,\zeta,\bar{\zeta})\big{)}d^{2}S,$
(66)

thus, to compare both expressions one must first rewrite eq.(66) as a function
of $u=x^{a}l^{+}_{a}$ instead of $v=-x^{a}l^{-}_{a}$. From
$l^{-}_{a}=-\widehat{l^{+}}_{a}$ we obtain $v=\widehat{u}$. Performing a
change of variables $\zeta\rightarrow\widehat{\zeta}$ in Eq.(66) we obtain

$\displaystyle\widehat{Z^{-}}_{l,I_{l}}=\oint\big{(}Y_{l,I_{l}}^{-2}(\zeta,\bar{\zeta})\bar{\sigma}^{-}(u,\widehat{\zeta},\widehat{\bar{\zeta}})+Y_{l,I_{l}}^{2}(\zeta,\bar{\zeta})\sigma^{-}(u,\widehat{\zeta},\widehat{\bar{\zeta}})\big{)}d^{2}S,$
(67)

where we have used
$Y_{l,I_{l}}^{2}(\widehat{\zeta},\widehat{\bar{\zeta}})=(-1)^{l}Y_{l,I_{l}}^{-2}(\zeta,\bar{\zeta})$.
Note that $\bar{\sigma}^{-}(u,\widehat{\zeta},\widehat{\bar{\zeta}})$ is a s.w
2 function and plays the same role as $\sigma^{+}(u,\zeta,\bar{\zeta})$ in
$Z^{+}_{l,I_{l}}$. Thus,

$\widehat{Z^{-}}_{l,I_{l}}=\int_{0}^{\infty}w^{2}\;dw\oint
d^{2}S[Y^{-2}_{l,I_{l}}(\zeta,\bar{\zeta})\bar{\sigma}^{-}(w,\widehat{\zeta},\widehat{\bar{\zeta}})e^{-iwx^{a}l_{a}}+Y^{2}_{l,I_{l}}(\zeta,\bar{\zeta})\sigma^{-}(w,\widehat{\zeta},\widehat{\bar{\zeta}})e^{iwx^{a}l_{a}}],$
(68)

Finally,

$\oint d^{2}\hat{k}\int
d^{3}xe^{ix^{a}k_{a}}\overleftrightarrow{\partial_{t}}\widehat{Z^{-}}_{l,I_{l}}=(-1)^{l}\bar{\sigma}^{-}(w)_{l,I_{l}},$
(69)

and from eq. 25 we get

$\sigma^{+}(w)_{l,I_{l}}+(-1)^{l}\bar{\sigma}^{-}(w)_{l,I_{l}}=0.$ (70)

We now address the mode decomposition of the full term either at future or
past null infinity including the integration on the future or past null cones
from the point $x^{a}$ starting with the advanced solution, i.e.,

$\displaystyle
Z^{+}_{l,I_{l}}=\oint\big{(}Y_{l,I_{l}}^{-2}\sigma^{+}(u,\zeta,\bar{\zeta})+c.c.\big{)}+Y_{l,I_{l}}^{0}\int_{0}^{\infty}ds\bar{\eth}\eth
g_{01}(y^{c},\zeta,\bar{\zeta})\big{)}d^{2}S,$ (71)

where the last term is integrated on the future null cone from $x^{a}$ and
$c.c$ stands for the complex conjugate term. It is useful to have a spherical
harmonic decomposition before the integral on the null cone is performed. for
this we expand $T_{ab}(y^{a})$ in a Fourier decomposition,

$T_{ab}(y^{c})=\int d^{4}kT_{ab}(k^{c})e^{-ik^{a}y_{a}}=\int
d^{4}k\bar{T}_{ab}(k^{c})e^{ik^{a}y_{a}},$ (72)

where we have used the real condition on the last equation. We now write

$\bar{\eth}\eth
g_{01}(y^{+c},\zeta,\bar{\zeta})=T_{ab}(y^{+c})\bar{\eth}\eth\left(\frac{l^{+a}l^{+b}}{(k^{c}l^{+}_{c})^{2}}\right)$

where $y^{+c}=x^{c}+sl^{+c}$. Inserting (72) into (71) yields,

$\int_{0}^{\infty}ds\oint d^{2}SY_{l,I_{l}}^{0}\bar{\eth}\eth
g_{01}(y^{+c},\zeta,\bar{\zeta})=i\int
d^{4}ke^{-ik^{a}x_{a}}T_{ab}(k^{c})\oint
Y_{l,I_{l}}^{0}\bar{\eth}\eth\left(\frac{l^{+a}l^{+b}}{(k^{c}l^{+}_{c})^{2}}\right)\frac{d^{2}S}{k^{c}l_{c}},$

To compute the value of $\sigma^{+}_{l,I_{l}}(w)$ at future null infinity, and
calling $Z_{free}$ to part of $Z$ without the matter field term, it is
convenient to have a 4-dim Fourier decomposition of
$Z_{free}(x^{a})_{l,I_{l}}$ as,

$Z_{free}(x^{a})_{lJ_{l}}=\int
d^{4}k\delta(k^{a}k_{a})\theta(k^{o})[Y^{-2}_{l,I_{l}}(\hat{k})\sigma(w_{k},\hat{k})e^{-ix^{a}k_{a}}+Y^{2}_{l,I_{l}}(\hat{k})\bar{\sigma}(w_{k},\hat{k})e^{ix^{a}k_{a}}].$
(73)

Taking the Fourier transform of the above equation gives,

$\int
d^{4}xe^{ikx}Z_{free}(x^{a})_{lJ_{l}}=\frac{\delta(k^{o}-w_{k})}{2w_{k}}Y^{-2}_{l,I_{l}}(\hat{k})\sigma(w_{k},\hat{k}).$
(74)

Using spherical coordinates on momentum space, integrating on the frequency
and on the sphere in momentum space gives,

$\int_{o}^{\infty}dk^{o}\oint
d^{2}\hat{k}\frac{\delta(k^{o}-w_{k})}{2w_{k}}Y^{-2}_{l,I_{l}}(\hat{k})\sigma(w_{k},\hat{k})=\frac{1}{2w_{k}}\sigma(w_{k})_{l,J_{l}}.$
(75)

Finally, starting with (71), and following the same steps as before yields

$Z^{+}_{l,I_{l}}(w)=\sigma^{+}_{l,I_{l}}(w)+i\int_{0}^{\infty}dk^{o}\oint
d^{2}\hat{k}T^{+}_{ab}(k^{o},w\hat{k}^{i})\oint
Y_{l,I_{l}}^{0}\bar{\eth}\eth\left(\frac{l^{a}l^{b}}{(k^{c}l_{c})^{2}}\right)\frac{1}{k^{c}l_{c}},$
(76)

with $k^{a}=(k^{o},w\hat{k}^{i})$.

To obtain the contribution of the retarded solution we first write

$\displaystyle
Z^{-}(x^{a},\zeta,\bar{\zeta})=\oint_{S^{2}}\left(\bar{\eth}^{\prime
2}G^{-}_{00}(\zeta,\bar{\zeta},\zeta^{\prime},\bar{\zeta}^{\prime})\sigma^{-}(v,\zeta^{\prime},\bar{\zeta}^{\prime})+c.c.\right)dS^{\prime},$
(77)

with $v=-x^{a}l^{-}_{a}=-x^{a}\hat{l}^{+}_{a}$. In order to compare with
$Z^{-}(x^{a},\zeta,\bar{\zeta})$ we perform a change of variables
$\zeta^{\prime},\bar{\zeta}^{\prime}\rightarrow\widehat{\zeta}^{\prime},\widehat{\bar{\zeta}}^{\prime}$
giving

$\displaystyle
Z^{-}_{free}(x^{a},\widehat{\zeta},\widehat{\bar{\zeta}})=\oint_{S^{2}}\left(\bar{\eth}^{\prime
2}G^{-}_{00}(\widehat{\zeta},\widehat{\bar{\zeta}},\widehat{\zeta}^{\prime},\widehat{\bar{\zeta}}^{\prime})\sigma^{-}(-u,\widehat{\zeta}^{\prime},\widehat{\bar{\zeta}}^{\prime})+c.c.\right)dS^{\prime}.$
(78)

Thus, the positive frequency decomposition of
$Z^{-}(x^{a},\widehat{\zeta},\widehat{\bar{\zeta}})$ is given by

$\displaystyle
Z^{-}_{free}(x^{a},\widehat{\zeta},\widehat{\bar{\zeta}})=\int_{0}^{\infty}dw\oint_{S^{2}}dS^{\prime}\left(\bar{\eth}^{\prime
2}G^{-}_{00}(\widehat{\zeta},\widehat{\bar{\zeta}},\widehat{\zeta}^{\prime},\widehat{\bar{\zeta}}^{\prime})\sigma^{-}(w,\widehat{\zeta}^{\prime},\widehat{\bar{\zeta}}^{\prime})e^{iwx^{a}l^{\prime}_{a}}+c.c.\right),$
(79)

and,

$Z^{-}_{free}(x^{a})_{l,I_{l}}=(-1)^{l}\int_{0}^{\infty}dw\oint
d^{2}S^{\prime}[Y^{\prime
2}_{l,I_{l}}(\zeta^{\prime},\bar{\zeta}^{\prime})\bar{\sigma}^{-}(w,\zeta^{\prime},\bar{\zeta}^{\prime})e^{-iwx^{a}l^{\prime}_{a}}+Y^{\prime-2}_{lJ_{l}}(\zeta^{\prime},\bar{\zeta}^{\prime})\sigma^{-}(w,\zeta^{\prime},\bar{\zeta}^{\prime})e^{iwx^{a}l^{\prime}_{a}}],$
(80)

where we have used $\widehat{Y^{2}}_{l,I_{l}}=(-1)^{l}Y^{-2}_{l,I_{l}}$ and
$\widehat{Y^{2}_{l,I_{l}}Y^{-2}_{l^{\prime},I_{l}^{\prime}}}=Y^{2}_{l,I_{l}}Y^{-2}_{l^{\prime},I_{l}^{\prime}}$.
Thus, directly from

$Z^{+}_{free}(x^{a},\zeta,\bar{\zeta})+Z^{-}_{free}(x^{a},\widehat{\zeta},\widehat{\bar{\zeta}})=0,$

we get

$\sigma^{+}_{l,I_{l}}(w)+(-1)^{l}\bar{\sigma}^{-}_{l,I_{l}}(w)=0.$

To obtain the past null cone contribution to (76) we have to replace
$y^{+c}=x^{c}+sl^{+c}$ by $y^{-c}=x^{c}+sl^{-c}$,

$\int_{0}^{\infty}ds\oint d^{2}SY_{l,I}^{0}\bar{\eth}\eth
g_{01}(y^{c-},\zeta,\bar{\zeta})=i\int
d^{4}ke^{-ik^{a}x_{a}}T_{ab}(k^{c})\oint
Y_{l,J_{l}}^{0}\bar{\eth}\eth\left(\frac{l^{-a}l^{-b}}{(k^{c}l^{-}_{c})^{2}}\right)\frac{d^{2}S}{k^{c}l^{-}_{c}}.$

Thus,

$(-1)^{l}Z^{-}_{l,I_{l}}(w)=\bar{\sigma}^{-}_{l,I_{l}}(w)-i\int_{0}^{\infty}dk^{o}\oint
d^{2}\hat{k}T_{ab}(k^{c})\oint
Y_{l,I_{l}}^{0}\bar{\eth}\eth\left(\frac{l^{a}l^{b}}{(k^{c}l_{c})^{2}}\right)^{-}\frac{1}{k^{c}l^{-}_{c}},$
(81)

where the relative minus sign arises from the odd number of $l_{c}$ terms in
the stress energy tensor term.

Finally, from
$Z^{+}(x^{a},\zeta,\bar{\zeta})+Z^{-}(x^{a},\widehat{\zeta},\widehat{\bar{\zeta}})=0$,
we get

$\sigma^{+}(w)_{l,I_{l}}+(-1)^{l}\bar{\sigma}^{-}(w)_{l,I_{l}}+i\oint
d^{2}SY_{l,I_{l}}^{0}\int_{0}^{\infty}dk^{o}\oint
d^{2}\hat{k}T_{ab}\left(\frac{\bar{\eth}\eth\big{(}\frac{l^{a}l^{b}}{(k^{c}l_{c})^{2}}\big{)}^{+}}{k^{c}l^{+}_{c}}+(-1)^{l}\frac{\bar{\eth}\eth\big{(}\frac{l^{a}l^{b}}{(k^{c}l_{c})^{2}}\big{)}^{-}}{k^{c}l^{-}_{c}}\big{)}\right)=0.$
(82)