# New insights on null and timelike warped symmetric spacetime splittings

Alan Maciel⋆ Centro de Matemática, Computação e Cognição, Universidade Federal
do ABC,
Avenida dos Estados 5001, CEP 09210-580, Santo André, São Paulo, Brazil
Morgan Le Delliou∗,† -Institute of Theoretical Physics & Research Center of
Gravitation, Lanzhou University, Lanzhou 730000, China
-Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou 730000, China
-Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China  Instituto de Astrofísica e Ciências do Espaço, Universidade de Lisboa, Faculdade de Ciências, Ed. C8, Campo Grande, 1769-016 Lisboa, Portugal  Université de Paris, APC-Astroparticule et Cosmologie (UMR-CNRS 7164), F-75006 Paris, France.  José P. Mimoso‡ Instituto de Astrofísica e Ciências do Espaço, Universidade de Lisboa, Faculdade de Ciências, Ed. C8, Campo Grande, 1769-016 Lisboa, Portugal  Departamento de Física
Faculdade de Ciências da Universidade de Lisboa,
Campo Grande, Ed. C8 1749-016 Lisboa, Portugal
E-mail<EMAIL_ADDRESS>
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###### Abstract

We explore in detail the 2+2 and 1+1+2 formalism in spherically symmetric
spacetimes, spanning from deducing the dynamical equations to relating them to
the well-known generalised Painlevé-Gullstrand (GPG) coordinate system. The
evolution equations are the Raychaudhuri equations for null rays, including
those also known as cross-focusing equations whose derivation, to the best of
our knowledge, we present for the first time. We physically interpret the
scalars that arise in this scenario, namely the flow 2-expansion $\Theta_{n}$,
the flow acceleration $\mathcal{A}$, and the radial extrinsic curvature
$\mathcal{B}$. We derive a coordinate independent formula for the redshift
which shows that $\mathcal{B}$ is the sole source for the redshift in
spherically symmetric spacetimes. We also establish the correspondence between
the 1+1+2 scalars and the 1+3 splitting scalars, expansion and shear. We
further make a comparison with the Newman-Penrose formalism, in order to
clarify the context where each formalism is more useful, and finally, we
extend our results to planar and hyperbolic symmetric warped spacetimes as
well, in particular, the relationship between $\mathcal{B}$ and the redshift.

##  I Introduction

One of the fundamental pillars of the theory of General Relativity (GR) is the
Principle of Covariance Penrose (2005); d’Inverno (1992). It establishes that
the laws of Physics ought to be the same, independently of the choice of
coordinate system or observers. By the same token it enables one to adopt
choices of coordinates that best suit the comprehension of the questions under
investigation without contaminating the results.

Among the various systems of coordinates that can be found in the literature
Weinberg (1972), the consideration of two congruences of null vector fields
has been devised to address problems where we focus on the structure of light
cones. It was in due time perceived that null geodesics determine the
behaviour of the latter, and hence define the causal domains. This was
manifest in connection with the characterization of black holes, where, in
particular, null coordinates play an essential role in the compactification of
spacetime in the remarkable Penrose diagrams Kopczynski and Trautman (1992);
Hawking and Ellis (1973), and in recent results regarding both the
thermodynamics of black holes, and more general cosmological settings Hayward
(1994); Bak and Rey (2000); Cai and Cao (2007); Hayward (1998); Binetruy and
Helou (2015). The consideration of two congruences of null vectors forming a
tetrad together with two complex vectors also underlies the so-called Newman-
Penrose (NP) formalism Sachs (1964); Newman and Penrose (1962); Chandrasekhar
(1980), that is particularly adequate to establish the Petrov classification
of the Weyl tensor, and for the investigation of gravitational radiation.
However the latter formalism is technically involved and difficult to decode.
The dual-null formalism (alternatively dubbed double-null) has also been
adopted in studies of the initial value problem d’Inverno and Smallwood
(1980); Brady  _et al._ (1996), as well as in connection to the use of the so-
called geodesic light-cone (GLC) coordinates Fleury  _et al._ (2016); Nugier
(2016) applied to static black holes. The numerical applications of the
double-null-coordinates to a number of gravitational problems was also
recently the object of a comprehensive review Nakonieczna  _et al._ (2018).

In previous work we have utilised the dual null formalism Maciel  _et al._
(2015) to extend both the Birkhoff theorem Maciel  _et al._ (2018) and the
Tolman-Oppenheimer-Volkov (TOV) Maciel  _et al._ (2020a) equation to more
general geometric frameworks, such as planar or cylindrically symmetric space
times, as well as hyperbolically symmetric spacetimes.

In addition, when matter dominates the physics of the spacetime, it is useful
to unfold the 2+2 formalism into an 1+1+2 formalism that singles out the
matter flow. The 1+1+2 formalism has been used in order to study metric
perturbations in symmetric spacetimes Clarkson and Barrett (2003); Clarkson
(2007), variations of the Birkhoff Theorem Goswami and Ellis (2011, 2012);
Ellis and Goswami (2013), optical and light beam propagation in spacetimes
Głód (2020), as well as a way to define quasilocal universal horizons Maciel
(2016) in modified gravity theories. Our objective in this work is to connect
the characterizations of spacetime dynamics in both frameworks in a clear way,
to extend the spherically symmetric approach to other symmetries, as well as
to show we can obtain the usual metric, namely its generalised Painlevé-
Gullstrand (GPG) description Painlevé (1921); Gullstrand (1922); Gautreau
(1984); Lasky and Lun (2006, 2007a); Faraoni and Vachon (2020) and coordinate
based description, from the matter flow scalars, in a similar manner as done
in generalised Lemaître-Tolman-Bondi (gLTB) coordinates Sussman (2009). We
also aim to connect some of these scalars to physically meaningful
observables, such as the redshift Ellis (2009); Codur and Marinoni (2021) and
to point out the better fit of these formalisms compared with the NP approach
Chandrasekhar (1980) to our questions. Finally, we present in the appendix the
detailed derivations of the evolution and constraint equations in 2+2
formalism, for spherically symmetric spacetimes, first presented in Hayward
(1994, 1993), as its exposition seem to evade the literature.

In the present work, we revisit the 2-null formalism in a more comprehensive
way in order to fully display its structure, and aiming at to render its
advantages more apparent in the case of warped spaces with symmetric
foliations, namely spherical symmetry. We naturally obtain its relation with
the 1+1+2 splitting for the same classes of spacetimes, which shares many of
the virtues of the 2+2 formalism, while highlighting the existence of a
timelike preferred flow. Such clear relation should allow easy translation
between lightcones and matter flow problems, enabling causal conclusions for
fluid problems as well as a way to obtain an observer centered perspective in
optical problems. We also connect those formalisms with the GPG system of
coordinates in order to retrieve the usual description of those spacetimes,
and to provide us with a physical understanding of the flow related
quantities. Although we start focusing on spherically symmetric spacetimes, we
subsequently generalize our methods to cases of spaces with hyperbolic or
planar codimension-2 foliations Maciel  _et al._ (2018, 2020a); Herrera  _et
al._ (2010, 2021); Carot  _et al._ (1999).

The paper is organised as follows: Sec. II discusses the dynamics of gravity
from the 2+2 description to the GPG coordinates. In Sec. II.1, we describe
briefly the main assumptions of the 2+2 formalism, followed by the most
relevant scalar quantities that arise from the 2+2 setup in Sec. II.2. We then
proceed to unfold the dual null basis into a timelike and spacelike preferred
basis, obtaining the equivalent flow scalars and equations for the 1+1+2
formalism. We relate our quantities with the GPG coordinates, by retrieving
the GPG line element from the flow scalars in Sec. II.4.

Those results lead us to propose physically meaningful interpretations in Sec.
III. We give the physical interpretation of the Raychaudhuri-like evolution
and constraint equations in Sec. III.1. In Sec. III.2, we also relate the
scalars of the 1+1+2 with those of the most commonly used 1+3 formalism. In
Sec. III.3 we derive the relationship between the redshift and the flow
scalars.

Finally, generalisations and comparisons of our results are proposed in Sec.
IV, before concluding. We first obtain the analogous results for planar and
hyperbolic symmetric spacetimes, which are readily obtained in 1+1+2
formalism, in Sec. IV.1. Finally, we compare our formalism with the Newman-
Penrose formalism, in order to discuss the context where each of them is more
useful, in Sec. IV.2. In the appendix we show the detailed derivations of the
evolution and constraint equations in 2+2 formalism, for spherically symmetric
spacetimes.

##  II Dynamics from 2+2 to 1+1+2 and GPG

###  II.1 Basics of the dual null formalism

We consider spacetimes whose metric may be written in the following form:

$\displaystyle g_{ab}=n_{ab}+s_{ab}\,,$ (1)

where $s_{ab}=r^{2}\gamma_{ab}$, with $\gamma_{ab}$ the metric of 2-sphere and
$s^{ab}\partial_{a}r=0$, that is, $r$ constant along the tangent directions to
the spheres of symmetry. Those spacetimes admit a codimension-2 foliation
where the leaves correspond to spherical surfaces.

In this dual null formalism we define two basis vectors $(k^{a},l^{a})$ that
span the subspace orthogonal to the spherical surfaces, formally

$\displaystyle k^{a}s_{ab}=l^{a}s_{ab}=0\,.$ (2)

Note that unlike the expansion of timelike flow, where if the flow is given by
a timelike vector field $u^{a}$, we can normalize it by imposing
$u^{a}u_{a}=-1$, in the null case we have no natural normalization, so that
the value of the expansion _is dependent on our choice of a null vector field
tangent to the flow_. Instead, in the latter case we can choose a kind of
normalization fixing the product $k^{a}l_{a}<0$, which we are so far keeping
arbitrary, albeit constant and negative, in order to make the basis future-
directed.

In terms of $k^{a}$ and $l^{a}$, the metric (1) can be written as

$g_{ab}=\frac{1}{k^{c}l_{c}}2k_{(a}l_{b)}+s_{ab}\,,$ (3)

where, by definition $x_{a}=g_{ab}x^{b}$ and $x_{(ab)}=(1/2)(x_{ab}+x_{ba})$.
Note that the induced metric is independent of the particular choice of
$k^{a}$ and $l^{a}$, provided they are null.

###  II.2 Flow scalars

The expansion associated with the spacelike two-surfaces along $k^{a}$ is
given by

$\Theta_{(k)}=\frac{1}{2}s^{ab}\mathcal{L}_{k}s_{ab}=s^{ab}\nabla_{a}k_{b}\,,$
(4)

where We denote $\mathcal{L}_{v}$ the Lie derivative along $v^{a}$.

We conveniently decompose the projected gradient of the null flow – say of
$k^{a}$ – in a scalar, a symmetric tensor, and an anti-symmetric tensor on the
two dimensional leaves:

$s\indices{{}^{a}_{c}}s\indices{{}^{b}_{d}}\nabla_{a}k_{b}=\frac{\Theta_{(k)}}{2}s_{cd}+\sigma_{cd}+\omega_{cd}\,,$
(5)

where the shear $\sigma_{ab}$ is trace-free and symmetric, and where the
vorticity $\omega_{ab}$ is also trace-free, but anti-symmetric. Remark that,
as it happens with the expansions, in the codimension-2 foliation case we have
two shear and vorticity tensors, one corresponding to evolution in each of the
null directions. In other words, there will be a corresponding decomposition
to (5) for $l_{a}$.

In the spherically symmetric case, the only non-vanishing component of the
gradient of null curves is the expansion as the other terms would violate
spherical symmetry.

We can redefine our null basis, by making

$\displaystyle k^{\prime a}=\lambda(u_{+},u_{-})k^{a}\,\quad\text{and}\quad
l^{\prime a}=\lambda^{-1}(u_{+},u_{-})l^{a}\,,$ (6)

where $u_{\pm}$ are coordinates along the null directions of $k^{a}$ and
$l^{a}$, while maintaining our normalization. Then the expansion transforms as

$\displaystyle\Theta_{(k^{\prime})}=\lambda\Theta_{(k)}\,,$ (7)

which will allow us to transform the equations involving flow scalars in
accordance to a transformation between pairs of dual null basis. The
transformation (6) is a local boost transformation with rapidity given by
$\ln\lambda$. This freedom is crucial in order to make the unfolding of the
2+2 formalism to the 1+1+2 formalism, since we need to have the liberty to
chose the correct null basis that will be related to the matter flow $n^{a}$
by the use of those local boosts.

The aim is to characterize the dynamics of the spacetime in terms of the
relationship of the flow scalars with the matter, instead of the metric, as
usual in GR. For this, we combine the well known Raychaudhuri equation for
null congruences, which provides de evolution of the null expansions along the
null flow, along with what we call the cross-focusing equations, that gives
the behaviour of one null expansion along the other null flow. They are given
by

$\displaystyle\mathcal{L}_{k}\Theta_{(k)}$
$\displaystyle=-\frac{\Theta_{(k)}^{2}}{2}+\nu_{k}\Theta_{(k)}-8\pi
T_{cd}k^{c}k^{d}\,,$ (8a) $\displaystyle\mathcal{L}_{l}\Theta_{(k)}$
$\displaystyle=-\Theta_{(k)}\Theta_{(l)}-\nu_{l}\Theta_{(k)}-\frac{k^{c}l_{c}}{r^{2}}+8\pi
T_{ab}k^{a}l^{b}\,,$ (8b)

where $T_{ab}$ is the energy-momentum tensor (EMT) of the sources, together
with two more equations obtained by exchanging the roles of $k$ and $l$ in
(8). A detailed derivation of those equations is given in Appendix (A and B).

We notice that $T_{cd}\,k^{c}k^{d}\equiv\mu$ is the complete projection of the
energy-momentum tensor of the sources along the null vector $k^{a}$. It can be
seen as a null dust energy density, interpreting $T_{ab}$ as if it were
representing null dust along the $l$ direction (Faraoni and Côté, 2019; Ellis
_et al._ , 2012, Sec. 7.4). Given the equations (9) below, relating the null
vectors $k^{a}$ and $l^{a}$ to both the 4-velocity of the matter flow and the
spatial vector defining the direction of local rotational symmetry, it is
possible to express the quantity $\mu$ under consideration as a combination of
the thermodynamical quantities devised by an observer comoving with the matter
fluid. The same applies to $T_{ab}\,k^{a}l^{b}\equiv Q$ that corresponds to
the flux of the vector $T_{ab}\,k^{a}$ accross the wave-surfaces orthogonal to
$l^{a}$, as well as equivalently to the flow of the vector $T_{ab}\,l^{a}$
accross the wave-surfaces orthogonal to $k^{a}$. The null dust interpretation
would yield 0 by definition, so it corresponds to a more complex picture.

Further notice that, as shown in the continuation of the paper, the sum and
differences of Eqs. (8) involve respectively the radial acceleration and the
redshift generator scalars.

Incidentally, we recognise that $\mu$ is associated with the so-called null
energy condition Hawking and Ellis (1973) which requires $\mu\geq 0$.

###  II.3 Equations in the 1+1+2 formalism

Although the dual null formalism, or 2+2, is very convenient in order to study
the geometric structure of the spacetime, specially in issues related with
strong gravity near dynamical black holes, when there is a matter flow
dominating the physics it is usually more relevant to single out the matter
flow $n^{a}$. This leads naturally to the well known $1+3$ formalism. However,
in order to keep the advantages of both $2+2$ and $1+3$ formalism, for
spacetimes that exhibit spherical symmetry, we apply the $1+1+2$ formalism. We
define, instead of a dual null basis for the subspace orthogonal to the
spherical leaves, a basis consisting of a timelike and a spacelike vector, in
the following way

$\displaystyle n^{a}=-\frac{1}{2k^{c}l_{c}}\left(k^{a}+l^{a}\right)\,,$
$\displaystyle e^{a}=-\frac{1}{2k^{c}l_{c}}\left(k^{a}-l^{a}\right)\,.$ (9)

such that

$\displaystyle n^{a}n_{a}=-1\,,\quad e^{a}e_{a}=1\,\quad\text{and}$
$\displaystyle g_{ab}=-n_{a}n_{b}+e_{a}e_{b}+s_{ab}\,.$ (10)

For the sake of simplicity, from now on we choose $k^{c}l_{c}=-1/2$ as the
normalization of the dual null basis. Therefore, we find for the Lie
Derivatives with respect to $n^{a}$ and $e^{a}$:

$\displaystyle\mathcal{L}_{n}=\mathcal{L}_{k}+\mathcal{L}_{l}\,,$
$\displaystyle\mathcal{L}_{e}=\mathcal{L}_{k}-\mathcal{L}_{l}\,.$ (11)

Although the expansion in relation to a two dimensional spatial orthogonal
subspace is natural for null congruences, here we extend this concept to any
vector orthogonal to $s_{ab}$. For any vector $v^{a}$ generated by the basis
$(n^{a},e^{a})$ this is achieved by defining

$\displaystyle\Theta_{(v)}=\frac{1}{2}s^{ab}\mathcal{L}_{v}s_{ab}=\frac{2}{r}v^{a}\nabla_{a}r\,.$
(12)

We call the latter as 2-expansions in order to distinguish them from the usual
expansions in 1+3 formalism.

We can also understand the 2-expansions as projections of the mean curvature
vector, denoted $\Theta^{c}$, along the given direction. Since the second
fundamental form, also called shape tensor, is

$\displaystyle\mathcal{K}\indices{{}^{c}_{a}{}_{b}}=\left(s\indices{{}_{a}^{d}}s\indices{{}_{b}^{f}}\nabla_{d}N\indices{{}_{f}^{c}}\right)=-\left(k^{c}\mathcal{L}_{l}s_{ab}+l^{c}\mathcal{L}_{k}s_{ab}\right)\,,$
(13)

where $N\indices{{}_{a}^{b}}=-2\left(k_{a}l^{b}+l_{a}k^{b}\right)$ is the
projector onto the subspace orthogonal to $s_{ab}$, the mean curvature vector,
$\Theta^{c}$, is given by the trace of $\mathcal{K}\indices{{}^{c}_{a}{}_{b}}$
thus yielding

$\displaystyle\Theta^{c}=-2\Theta_{(k)}l^{c}-2\Theta_{(l)}k^{c}\,.$ (14)

So, we can now equivalently define the 2-expansions $\Theta_{(v)}$ as the
projections of $\Theta^{c}$ along $v^{a}$:

$\displaystyle\Theta_{(v)}\equiv v_{c}\Theta^{c}\,.$ (15)

In particular, the 2-expansions along $n^{a}$ and $e^{a}$ are given by

$\displaystyle\Theta_{(n)}=\Theta_{(k)}+\Theta_{(l)}\,,$
$\displaystyle\Theta_{(e)}=\Theta_{(k)}-\Theta_{(l)}\,.$ (16)

We aim at computing the Raychaudhuri-like equations along the directions
$n^{a}$ and $e^{a}$, as well as the mixed equations. By using Eqs. (8), we
obtain the relations

$\displaystyle\mathcal{L}_{n}\Theta_{(n)}=\left(\mathcal{L}_{k}+\mathcal{L}_{l}\right)\left(\Theta_{(k)}+\Theta_{(l)}\right)\,,$
(17a)
$\displaystyle\mathcal{L}_{e}\Theta_{(e)}=\left(\mathcal{L}_{k}-\mathcal{L}_{l}\right)\left(\Theta_{(k)}-\Theta_{(l)}\right)\,,$
(17b)
$\displaystyle\mathcal{L}_{n}\Theta_{(e)}=\left(\mathcal{L}_{k}+\mathcal{L}_{l}\right)\left(\Theta_{(k)}-\Theta_{(l)}\right)\,,$
(17c)
$\displaystyle\mathcal{L}_{e}\Theta_{(v)}=\left(\mathcal{L}_{k}-\mathcal{L}_{l}\right)\left(\Theta_{(k)}+\Theta_{(l)}\right)\,,$
(17d)

which lead to

$\displaystyle\mathcal{L}_{n}\Theta_{(n)}$
$\displaystyle=-\frac{3\Theta_{(n)}^{2}}{4}+\frac{\Theta_{(e)}^{2}}{4}-\frac{1}{r^{2}}+\mathcal{A}\Theta_{(e)}-8\pi
T_{ab}e^{a}e^{b}\,,$ (18a) $\displaystyle\mathcal{L}_{e}\Theta_{(e)}$
$\displaystyle=-\frac{3\Theta_{(e)}^{2}}{4}+\frac{\Theta_{(n)}^{2}}{4}+\mathcal{B}\Theta_{(n)}+\frac{1}{r^{2}}-8\pi
T_{ab}n^{a}n^{b}\,,$ (18b) $\displaystyle\mathcal{L}_{n}\Theta_{(e)}$
$\displaystyle=-\frac{\Theta_{(n)}\Theta_{(e)}}{2}+\mathcal{A}\Theta_{(n)}-8\pi
T_{ab}n^{a}e^{b}\,,$ (18c) $\displaystyle\mathcal{L}_{e}\Theta_{(n)}$
$\displaystyle=\mathcal{L}_{n}\Theta_{e}+\mathcal{B}\Theta_{(e)}-\mathcal{A}\Theta_{(n)}\,,$
(18d)

where we have defined

$\displaystyle\mathcal{A}=n^{a}e^{b}\nabla_{a}n_{b}=-2(k^{c}l_{c})\left(\nu_{k}-\nu_{l}\right)=\nu_{k}-\nu_{l}\,,$
$\displaystyle\mathcal{B}=e^{a}e^{b}\nabla_{a}n_{b}=-2(k^{c}l_{c})\left(\nu_{k}+\nu_{l}\right)=\nu_{k}+\nu_{l}\,.$
(19)

We recognize $\mathcal{A}$ as the flow scalar acceleration, and we also
realise that $\mathcal{B}$ can also be defined as
$\mathcal{B}=\mathcal{K}_{ab}e^{a}e^{b}$, where $\mathcal{K}_{ab}$ is the
extrinsic curvature of the hypersurface orthogonal to the flow111The extrinsic
curvature is equivalent to the shape tensor in codimension-one foliations, the
shape tensor in this case is given by
$\mathcal{K}\indices{{}^{c}_{a}{}_{b}}=n^{c}\mathcal{K}_{ab}$. and, as it will
be shown in Sec. III.3, it is also the scalar that acts as source for the
redshift.

###  II.4 From 1+1+2 to the GPG metric

The 1+1+2 formalism presents many advantages in order to analyse the dynamical
equations of the spacetime and predict its behavior. By using only coordinate
independent quantities that are adapted to the symmetries of the spacetime, it
not only highlights the matter flow, as in the 1+3 threading, but also puts in
evidence the spatial direction orthogonal to the symmetric surfaces. However,
for practical purposes it is important to relate this approach to well known
coordinates system for spherically symmetric spacetimes, such as the
generalized Painlevé-Gullstrand (GPG) coordinates. In order to achieve this
goal, we recall the definition of the Misner-Sharp mass-energy $M_{ms}$
(Misner and Sharp, 1964):

$\displaystyle g^{ab}\partial_{a}r\partial_{b}r=1-\frac{2M_{ms}}{r}\,,$ (20)

which, together with Eq. (10), gives us

$\displaystyle-(n^{a}\partial_{a}r)^{2}+(e^{b}\partial_{b}r)^{2}=1-\frac{2M_{ms}}{r}\,,$
(21)

which we may rearrange as

$\displaystyle
e^{b}\partial_{b}r=\pm\sqrt{1+(n^{a}\partial_{a}r)^{2}-\frac{2M_{ms}}{r}}\,.$
(22)

Interpreting the term $n^{a}\partial_{a}r$ as the fluid radial velocity, we
are then motivated to define

$\displaystyle E=(n^{a}\partial_{a}r)^{2}-\frac{2M_{ms}}{r}\,,$ (23)

which corresponds to twice the Newtonian mechanic energy per unit mass222In
some sources $E$ is defined as the Newtonian energy per unity mass, but here
we prefer to avoid a factor 2 appearing in the GPG metric..

Using the scalar function $E$, Eq. (22) becomes

$\displaystyle e^{b}\partial_{b}r=\pm\sqrt{1+E}\,.$ (24)

The GPG coordinates correspond to using a timelike coordinate $t$ chosen along
the flow, together with the areal radius as the spacelike coordinate
orthogonal to the spheres of symmetry. Therefore, translating from the
abstract index notation to a coordinate notation, we have

$\displaystyle n_{a}\rightarrow-\alpha(t,r)\mathrm{d}t\,,$ (25)

where we chose the minus sign in order to guarantee that $t$ is future
directed with positive $\alpha(t,r)$, which corresponds to the shift function
of the Arnowitt-Deser-Misner (ADM) decomposition Arnowitt  _et al._ (1959).
Since, by construction, $n_{a}e^{a}=0$, we realise that $e^{a}$ has the form
$(0,e^{r},0,0)$. From Eq. (24), we conclude that

$\displaystyle e^{r}=\sqrt{1+E}\Rightarrow
e^{a}\rightarrow\sqrt{1+E(t,r)}\partial_{r}\,,$ (26)

where we chose the plus sign in order to make $e^{a}$ directed outwards, that
is, towards increasing $r$.

In order to translate the full metric to coordinate notation we have to find
the form of $e_{a}$. Since $e^{a}$ is a unity vector, applying $e^{a}e_{a}=1$
gives us $e_{a}=(e_{t},\frac{1}{\sqrt{1+E}},0,0)$, with $e_{t}$ unconstrained.
Therefore, defining a function $\beta(t,r)=-e_{t}\sqrt{1+E}$, we write:

$\displaystyle
e_{a}\rightarrow\frac{1}{\sqrt{1+E(t,r)}}\left(-\beta(t,r)\mathrm{d}t+\mathrm{d}r\right)\,.$
(27)

Plugging Eqs. (25) and (27) into Eq. (10) and making the final translation

$\displaystyle s_{ab}\rightarrow
r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta^{2}\mathrm{d}\varphi^{2})=r^{2}\mathrm{d}\Omega^{2}\,,$
(28)

we obtain the well known GPG line element Lasky and Lun (2007b); Adler  _et
al._ (2005); Lasky and Lun (2006, 2007a); Gautreau (1984):

$\displaystyle\mathrm{d}s^{2}=-\alpha(t,r)^{2}\mathrm{d}t^{2}+\frac{(-\beta(t,r)\mathrm{d}t+\mathrm{d}r)^{2}}{1+E(t,r)}+r^{2}\mathrm{d}\Omega^{2}\,.$
(29)

In order to interpret the meaning of $\beta(t,r)$, we compute $n^{a}$ in GPG
coordinates

$\displaystyle
n^{a}\rightarrow\frac{1}{\alpha(t,r)}\left(\partial_{t}+\beta(t,r)\partial_{r}\right)\,,$
(30)

which is given by

$\displaystyle n^{a}\partial_{a}r=\frac{\beta(t,r)}{\alpha(t,r)}\,.$ (31)

Combining Eqs. (21) and (31), we obtain for $\beta$:

$\displaystyle\frac{\beta(t,r)}{\alpha(t,r)}=\pm\sqrt{\frac{2M_{ms}(t,r)}{r}+E(t,r)}\,,$
(32)

which we interpret as the radial velocity of the flow, where the
${}^{\prime}-^{\prime}$ (minus) sign corresponds to collapse and the
${}^{\prime}+^{\prime}$ (plus) sign to expansion333Note that our definition
for $\beta(t,r)$ adopts the opposite sign to that of Ref. Lasky and Lun
(2007b) .

##  III Physical interpretations

###  III.1 Interpreting the flow equations

The Raychaudhuri equations (18) encode the evolution and constraint equations
for spacetime and matter, given the symmetry assumptions. In order to render
explicit these relations we will show that Eq. (18a) is equivalent to the
generalised TOV (gTOV) functional Mimoso  _et al._ (2013) that provides an
effective acceleration of matter shells in this spacetime. We assume in this
section that matter is adequately modeled by a perfect fluid, with energy-
momentum tensor given by

$\displaystyle T_{ab}=\rho n_{a}n_{b}+P(e_{a}e_{b}+s_{ab})\,,$ (33)

where $\rho$ and $P$, respectively, are the energy-density and the isotropic
pressure measured by the observer comoving with velocity $n^{a}$. Thus we have

$\displaystyle T_{ab}n^{a}n^{b}=\rho\,,\quad T_{ab}e^{a}e^{b}=P\,,\quad
T_{ab}n^{a}e^{b}=0\,.$ (34)

and, by applying the conservation of the energy-momentum $\nabla_{a}T^{ab}=0$
along the $n^{a}$ and $e^{a}$ directions we derive the equations

$\displaystyle n^{a}\nabla_{a}\rho$
$\displaystyle+(\rho+P)(\mathcal{B}+\Theta_{n})=0\,,$ (35a)
$\displaystyle\mathcal{A}$ $\displaystyle=-\frac{e^{a}\nabla_{a}P}{\rho+P}\,,$
(35b)

which relate the $1+1+2$ scalars to the latter hydrodynamic variables. These
Eqs. (35a) and (35b) complement Eqs. (18) to form the set of field equations.

Before proceeding it is appropriate to write up the anticipated relation
between the scalar quantities $\mu$ and $Q$, associated with the dual null
directions $k^{a}$ and $l^{a}$, and the thermodynamic quantities (34). Indeed,
notice that $\mu$ and $Q$ in Eqs. (8) reduce to
$\mu=T_{cd}k^{c}k^{d}=\frac{\rho+P}{4}$, which appears related to the fluid
enthalpy $h=E/V+P$, and $Q=T_{ab}k^{a}l^{b}=\frac{\rho-P}{4}$, that could be
interpreted as related to a fluid enthalpy for negative pressure
$"h"=E/V+(-P)$, so that, as expected the null energy condition $\mu\geq 0$
translates into $\rho+P\geq 0$.444Incidentally, if $T_{ab}$ coincides with the
EMT of a cosmological constant the null energy condition is marginally
satisfied, since $\rho+P=0$.

We remark that the Misner-Sharp mass can be related to the 2-expansions as

$\displaystyle-\Theta_{(n)}^{2}+\Theta_{(e)}^{2}=\frac{4}{r^{2}}\left(1-\frac{2M_{ms}}{r}\right)\,,$
(36)

which, applied to Eq. (18a) along with (34) leads to

$\displaystyle\mathcal{L}_{n}\Theta_{(n)}=-\frac{\Theta_{n}^{2}}{2}+\frac{1}{r^{2}}\left(1-\frac{2M_{ms}}{r}\right)+\frac{1}{r^{2}}+\mathcal{A}\Theta_{(e)}-8\pi
P\,,$ (37)

which can be further simplified by using Eq. (35b) and

$\displaystyle e^{a}\partial_{a}=\sqrt{1+E}\partial_{r}\,,$ (38)

leading to

$\displaystyle\mathcal{L}_{n}\Theta_{(n)}+\frac{\Theta_{(n)}^{2}}{2}=-\frac{2}{r}\left(\frac{1+E}{\rho+P}\partial_{r}P+\frac{M_{ms}}{r^{2}}+4\pi
Pr\right)=-\frac{2}{r}\text{gTOV}\,,$ (39)

which was already shown in Maciel  _et al._ (2015), although here the result
is obtained in a more straightforward procedure.

We can also see that the Raychaudhuri equations along $e^{a}$, Eqs (18b) and
(18d) are constraints equations. By computing

$\displaystyle\mathcal{L}_{e}(-\Theta_{(n)}^{2}+\Theta_{(e)}^{2})=-2\Theta_{(n)}\mathcal{L}_{e}\Theta_{(n)}+2\Theta_{(e)}\mathcal{L}_{e}\Theta_{(e)}\,,$
(40)

we find, after using Eqs. (18b), (18d), (34) and (36) and some manipulation:

$\displaystyle\mathcal{L}_{e}M_{ms}=2\Theta_{(e)}\rho r^{3}\,,$ (41)

which, using the translation to GPG coordinates, Eq. (26), leads to

$\displaystyle\partial_{r}M_{ms}=4\pi\rho r^{2}\,,$ (42)

as expected from the mass-energy definition.

###  III.2 Flow scalars from 2D to 3D

In order to compute the extrinsic curvature of the spatial sections orthogonal
to the flow $n^{a}$, we use the definition

$\displaystyle
K_{ab}=\frac{1}{2}\mathcal{L}_{n}h_{ab}=\frac{1}{2}\mathcal{L}_{n}(e_{a}e_{b}+s_{ab})\,.$
(43)

The easiest way to translate this equation in terms of the dual null flow
scalars is to project it by making use of the symmetries:

$\displaystyle s^{ab}K_{ab}=\Theta_{n}\,,\quad n^{a}K_{ab}=0\,,\quad
e^{a}K_{ab}=\frac{1}{2}\left(\mathcal{L}_{n}e_{b}+e^{a}e_{b}\mathcal{L}_{n}e_{a}\right)=\mathcal{B}e_{b}\,,$
(44)

which results in

$\displaystyle K_{ab}=\mathcal{B}e_{a}e_{b}+\frac{\Theta_{n}}{2}s_{ab}\,.$
(45)

From the extrinsic curvature, we compute the 3D expansion $\Theta$:

$\displaystyle\Theta=g^{ab}\nabla_{a}n_{b}=K\indices{{}_{a}^{a}}=\mathcal{B}+\Theta_{n}\,.$
(46)

Since the shear tensor $\sigma_{ab}$ is defined as the trace free part of
$K_{ab}=h\indices{{}_{a}^{c}}h\indices{{}_{b}^{d}}\nabla_{c}n_{d}$, we obtain

$\displaystyle\sigma_{ab}=\mathcal{B}e_{a}e_{b}+\frac{\Theta_{n}}{2}s_{ab}-\frac{1}{3}\left(\mathcal{B}+\Theta_{n}\right)\left(e_{a}e_{b}+s_{ab}\right)=$
$\displaystyle=\frac{\Theta_{n}-2\mathcal{B}}{6}\left(-2e_{a}e_{b}+s_{ab}\right)\,,$
(47)

where we recognize the symmetric trace free projector (STFPr)
$P_{ab}=-2e_{a}e_{b}+s_{ab}$, (check Ref.Maciel  _et al._ (2015)). In the
latter equation (47) we define the shear scalar as

$\displaystyle\sigma=\frac{\Theta_{n}-2\mathcal{B}}{6}\,,$ (48)

which, combined with Eq. (46), leads to the relation between 2D and 3D
expansions and shear

$\displaystyle\Theta_{n}$ $\displaystyle=\frac{2\Theta}{3}+2\sigma\ ,$ (49a)
$\displaystyle\mathcal{B}$ $\displaystyle=\frac{\Theta}{3}-2\sigma\,,\ $ (49b)

that shows that the pair of scalars $\Theta_{n}$ and
$\mathcal{B}=K_{ab}e^{a}e^{b}$ is equivalent to the pair $\Theta$ and $\sigma$
in order to describe the flow in spherical symmetry.

###  III.3 Redshift

To compute the redshift suffered by a photon travelling radially, we have to
write the wave vector $K^{a}$ in our formalism. Since the frequency $\omega$
measured by the observer whose four velocity is $n_{a}$ is defined as

$\displaystyle\omega=-K^{a}n_{a}\,,$ (50)

and the wave vector is null, it can be written as

$\displaystyle K^{a}=\omega\left(n^{a}\pm e^{a}\right)\,,$ (51)

provided that the wave vector satisfies the geodesic equation,
$K^{a}\nabla_{a}K_{b}=0$. The geodesic equation applied to (51) gives us

$\displaystyle\omega(n^{a}\pm e^{a})\nabla_{a}\left[\omega(n_{b}\pm
e_{b})\right]=0\,,$ (52)

which, after using the definitions (19), implies the following equality

$\displaystyle(n^{a}\pm
e^{a})\frac{\nabla_{a}\omega}{\omega}=\mp\mathcal{A}-\mathcal{B}\,,$ (53)

that we use to write

$\displaystyle\nabla_{a}\ln\omega=\mathcal{B}n_{a}-\mathcal{A}e_{a}\,.$ (54)

By defining the redshift as $1+z=\frac{\omega}{\omega_{0}}$, where
$\omega_{0}$ is the emitted frequency, we obtain the relation

$\displaystyle\mathcal{L}_{n}\ln{(1+z)}=-\mathcal{B}\,.$ (55)

Eq. (55) gives us the interpretation of the $\mathcal{B}$ scalar as the source
for redshift for general spherically symmetric spacetimes. Note that by Eq.
(49b), $\mathcal{B}$ corresponds to a linear combination of the 3-dimensional
expansion $\Theta$ and 3-dimensional shear scalar $\sigma$. It geometrically
encapsulates previous expressions Ellis (2009); Plebanski and Krasinski
(2006), and for an isotropic and homogeneous model it just corresponds to the
Hubble constant $H$, hence returning the well know relation
$1+z=\frac{a_{0}}{a}$.

Using GPG coordinates, Eq. (55) acquires the form

$\displaystyle\frac{1}{\alpha}\left(\partial_{t}+\beta\partial_{r}\right)\ln(1+z)=$
$\displaystyle-\frac{1}{\alpha}\frac{\beta\partial_{r}E+\partial_{t}E}{2(1+E)}+\frac{\partial_{r}\beta}{\alpha}$
$\displaystyle=$
$\displaystyle-\frac{\mathcal{L}_{n}E}{2\left(1+E\right)}-\frac{\beta^{\prime}}{\alpha}\,,$
(56)

in which we recognise the left hand side of the momentum constraint from the
EFE.

That momentum constraint in GPG coordinates reads

$\displaystyle-\frac{\mathcal{L}_{n}E}{2\left(1+E\right)}-\frac{\beta^{\prime}}{\alpha}$
$\displaystyle=\left(\mathcal{L}_{n}r\right)^{\prime},$ (57)

and we recall the GPG expressions for the 3-expansion and 3-shear scalars

$\displaystyle\Theta=$
$\displaystyle\left(\mathcal{L}_{n}r\right)^{\prime}+2\frac{\mathcal{L}_{n}r}{r},$
(58) $\displaystyle\sigma=$
$\displaystyle-\frac{1}{3}\left[\left(\mathcal{L}_{n}r\right)^{\prime}-\frac{\mathcal{L}_{n}r}{r}\right].$
(59)

Therefore, the redshift source can be recast to relate it evidently to the
3-expansion and 3-shear as

$\displaystyle\frac{1}{\alpha}\left(\partial_{t}+\beta\partial_{r}\right)\ln(1+z)$
$\displaystyle=$ $\displaystyle\left(\mathcal{L}_{n}r\right)^{\prime}\;,$ (60)
$\displaystyle=$
$\displaystyle\frac{1}{3}\left[\left(\mathcal{L}_{n}r\right)^{\prime}+2\frac{\mathcal{L}_{n}r}{r}\right]+\frac{2}{3}\left[\left(\mathcal{L}_{n}r\right)^{\prime}-\frac{\mathcal{L}_{n}r}{r}\right]\;,$
$\displaystyle=$ $\displaystyle\frac{\Theta}{3}-2\sigma\;.$

In the restricted case of the Lemaître-Tolman-Bondi (LTB) dust model, we get

$\partial_{t}\ln(1+z)=-\partial_{t}\ln R^{\prime}$

so that we are led to

$1+z=\frac{R^{\prime}_{0}}{R^{\prime}}\;$

(See for instance Codur and Marinoni (2021).)

Notice that this fits again our expression for $\mathcal{B}$, as

$\frac{\dot{R}^{\prime}}{R^{\prime}}=\frac{1}{3}\left(2\frac{\dot{R}}{R}+\frac{\dot{R}^{\prime}}{R^{\prime}}\right)-2\,\frac{1}{3}\,\left(\frac{\dot{R}}{R}-\frac{\dot{R}^{\prime}}{R^{\prime}}\right)=\Theta/3-2\sigma.$

##  IV Extensions and comparisons

###  IV.1 Hyperbolic and planar symmetries

The 1+1+2 treatment is also convenient for spacetimes with codimension-two
hyperbolic and rigid planar foliations. We can express spacetimes of the three
types of symmetries, spherical, planar and hyperbolic in one unified notation
using

$\displaystyle
s_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}=r^{2}\left(\mathrm{d}\theta^{2}+S^{2}_{\epsilon}(\theta)\mathrm{d}\phi^{2}\right)\equiv
r^{2}\mathrm{d}\Sigma^{2}\,,$ (61)

where

$\displaystyle S_{\epsilon}(\theta)=\left\\{\begin{tabular}[]{c
l}$\sin\theta\,,$&for $\epsilon=1\,,$\\\ $\theta\,,$&for $\epsilon=0\,,$\\\
$\sinh\theta\,,$&for $\epsilon=-1\,.$\end{tabular}\right.$ (65)

It is worth noting that the quantities $r$, $\theta$ and $\phi$ do not keep
their intuitive geometrical meanings in the cases of planar and hyperbolic
symmetry. Another important remark is that the planar symmetry considered here
is not the general case, but only that associated with the "rigid" planar
symmetric spacetimes, which we can define as the cases where the 2-shear,
defined as,
$\sigma^{(v)}_{ab}=s\indices{{}_{a}^{c}}s\indices{{}_{b}^{d}}\nabla_{c}v_{d}-\frac{s_{ab}}{2}\Theta_{(v)}$
vanishes for any vector $v^{a}$ generated by $(n^{a}\,,e^{a})$.

We can generalize Eqs. (18) for the three symmetries as

$\displaystyle\mathcal{L}_{n}\Theta_{(n)}$
$\displaystyle=-\frac{3\Theta_{(n)}^{2}}{4}+\frac{\Theta_{(e)}^{2}}{4}+\frac{\epsilon}{r^{2}}+\mathcal{A}\Theta_{(e)}-8\pi
T_{ab}e^{a}e^{b}\,,$ (66a) $\displaystyle\mathcal{L}_{e}\Theta_{(e)}$
$\displaystyle=-\frac{3\Theta_{(e)}^{2}}{4}+\frac{\Theta_{(n)}^{2}}{4}+\mathcal{B}\Theta_{(n)}-\frac{\epsilon}{r^{2}}-8\pi
T_{ab}n^{a}n^{b}\,,$ (66b) $\displaystyle\mathcal{L}_{n}\Theta_{(e)}$
$\displaystyle=-\frac{\Theta_{(n)}\Theta_{(e)}}{2}+\mathcal{A}\Theta_{(n)}-8\pi
T_{ab}n^{a}e^{b}\,,$ (66c) $\displaystyle\mathcal{L}_{e}\Theta_{(n)}$
$\displaystyle=-\frac{\Theta_{(n)}\Theta_{(e)}}{2}+\mathcal{B}\Theta_{(e)}-8\pi
T_{ab}n^{a}e^{b}\,,$ (66d)

by following the same procedure we used for treating the spherically symmetric
case.

We can also generalize Eq. (20) by using the generalization of Hawking-Hayward
mass-energy (and by extension, of the Misner-Sharp mass-energy) proposed in
Maciel  _et al._ (2020b) in order to obtain

$\displaystyle g^{ab}\partial_{a}r\partial_{b}r=\epsilon-\frac{2\mu}{r}\,,$
(67)

where $\mu$ corresponds to the mass parameter in the planar and hyperbolic
cases, similar to a superficial mass density, as explained in Maciel  _et al._
(2020b) and it corresponds the Misner-Sharp mass-energy in the spherical case.
Using Eq. (67) and following the procedure of Secs. II.4 and III.1, we obtain

$\displaystyle\mathrm{d}s^{2}=-\alpha^{2}\mathrm{d}t^{2}+\frac{(-\beta\mathrm{d}t+\mathrm{d}r)^{2}}{\epsilon+E}+r^{2}\mathrm{d}\Sigma^{2}\,,$
(68)
$\displaystyle\text{gTOV}=\frac{\epsilon+E}{\rho+P}\partial_{r}P+\frac{M_{ms}}{r^{2}}+4\pi
Pr\,.$ (69) $\displaystyle\partial_{r}\mu=4\pi r^{2}\rho\,.$ (70)

When the solution is static, $\Theta_{(n)}=0$, which corresponds to
$E=\frac{2\mu}{r}$ in GPG language, Eq. (69) reduces to the generalized TOV
equation analysed in Ref. Maciel  _et al._ (2020b).

The results of Secs. III.2 and III.3 depend only on the warped symmetric
foliation and are valid for planar and hyperbolic symmetries without changes,
including Eqs. (49) and (55).

###  IV.2 Advantages of 1+1+2/2+2 over Newman-Penrose formalism

The Raychaudhuri equations correspond to the trace of the evolution of the $k$
or $l$ congruences covariant derivatives, while these covariant derivatives
are themselves represented as a series of transport equations in the NP
formalism Sachs (1964); Newman and Penrose (1962). The complexity of obtaining
the Raychaudhuri equations in NP formalism appears therefore higher than using
the optical scalars in the 2+2 approach, demonstrating its better fit to our
domain of interest.

NP formalism uses the entirely null complex tetrad to represent the metric as

$\displaystyle z_{A}^{a}=$
$\displaystyle\left(l_{NP}^{a},n_{NP}^{a},m^{a},\overline{m}^{a}\right),$
(71a) $\displaystyle\eta_{AB}=\eta^{AB}=$
$\displaystyle\left(\begin{array}[]{cccc}0&-1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{array}\right),$ (71f) $\displaystyle\Rightarrow g^{ab}=$
$\displaystyle\eta^{AB}z_{A}^{a}z_{B}^{b},$ (71g)

the NP spin coefficients, taking $l_{NP}=\sqrt{2}k$ and $n_{NP}=\sqrt{2}l$ to
correct normalisation, that account for projected parallel transports of the
tetrad members, involved in transport equations, that regulate their parallel
transports and scalars for projections of the Weyl and Ricci tensors.

We use the following definitions for spin coefficients,

$\displaystyle\varrho\equiv$
$\displaystyle-\sqrt{2}m^{a}\overline{m}^{b}\nabla_{b}k_{a},$
$\displaystyle\varkappa\equiv$ $\displaystyle-2m^{a}k^{b}\nabla_{b}k_{a},$
$\displaystyle\varepsilon\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\left(\overline{m}^{a}k^{b}\nabla_{b}m_{a}-2l^{a}k^{b}\nabla_{b}k_{a}\right),$
(72a) $\displaystyle\mu\equiv$
$\displaystyle\sqrt{2}\overline{m}^{a}m^{b}\nabla_{b}l_{a},$
$\displaystyle\nu\equiv$ $\displaystyle
2\overline{m}^{a}l^{b}\nabla_{b}l_{a},$ $\displaystyle\gamma\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\left(\overline{m}^{a}l^{b}\nabla_{b}m_{a}-2l^{a}l^{b}\nabla_{b}k_{a}\right),$
(72b) $\displaystyle\pi\equiv$ $\displaystyle
2\overline{m}^{a}k^{b}\nabla_{b}l_{a},$ $\displaystyle\tau\equiv$
$\displaystyle-\sqrt{2}m^{a}n^{b}\nabla_{b}k_{a},$ (72c)

for transport equations,

$\displaystyle D\equiv$ $\displaystyle\sqrt{2}k^{a}\nabla_{a},$
$\displaystyle\Delta\equiv$ $\displaystyle\sqrt{2}l^{a}\nabla_{a},$ (73a)
$\displaystyle D\sqrt{2}k=$
$\displaystyle\left(\varepsilon+\overline{\varepsilon}\right)\sqrt{2}k-\overline{\varkappa}m-\varkappa\overline{m},$
$\displaystyle D\sqrt{2}l=$
$\displaystyle-\left(\varepsilon+\overline{\varepsilon}\right)\sqrt{2}l+\pi
m+\overline{\pi m},$ (73b) $\displaystyle\Delta\sqrt{2}k=$
$\displaystyle\left(\gamma+\overline{\gamma}\right)\sqrt{2}k-\overline{\tau}m-\tau\overline{m},$
$\displaystyle\Delta\sqrt{2}l=$
$\displaystyle-\left(\gamma+\overline{\gamma}\right)\sqrt{2}l+\nu
m+\overline{\nu m},$ (73c)

and for Ricci scalars

$\displaystyle\Phi_{00}\equiv$ $\displaystyle
R_{ab}k^{a}k^{b}\vphantom{\frac{R_{ab}1}{2R_{ab}}},$
$\displaystyle\Phi_{22}\equiv$ $\displaystyle
R_{ab}l^{a}l^{b}\vphantom{\frac{R_{ab}1}{2R_{ab}}},$
$\displaystyle\Phi_{11}\equiv$
$\displaystyle\frac{1}{4}R_{ab}\left(2k^{a}l^{b}+m_{a}\overline{m}_{b}\right)\vphantom{\frac{R_{ab}1}{2R_{ab}}},$
(74a) and $\displaystyle\Phi_{11}+3\varLambda=$ $\displaystyle
R_{ab}k^{a}l^{b},\textrm{ as}$ $\displaystyle\varLambda\equiv$
$\displaystyle\frac{R}{4!}\vphantom{\frac{R_{ab}1}{2R_{ab}}}=-2\left(2k_{a}l_{b}-m_{a}\overline{m}_{b}\right)\frac{R^{ab}}{4!}.$
(74b)

We then can link the NP spin coefficients to the 2+2 inaffinities as

$\displaystyle k.\Delta k=$ $\displaystyle 0,$ $\displaystyle l.Dk=$
$\displaystyle\left(\varepsilon+\overline{\varepsilon}\right)k.l\equiv\sqrt{2}\nu_{k}k.l,$
(75a) $\displaystyle l.Dl=$ $\displaystyle 0,$ $\displaystyle k.\Delta l=$
$\displaystyle-\left(\gamma+\overline{\gamma}\right)k.l\equiv\sqrt{2}\nu_{l}k.l.$
(75b)

We can define the expansions in NP formalism as

$\displaystyle\Theta_{\left(k\right)}=$
$\displaystyle\nabla_{a}k^{a}=g^{ab}\nabla_{a}k_{b}$ $\displaystyle=$
$\displaystyle\eta^{AB}z_{A}^{a}z_{B}^{b}\nabla_{a}k_{b}$ $\displaystyle=$
$\displaystyle\left[-2k^{a}l^{b}-2l^{a}k^{b}+\left(m^{a}\overline{m}^{b}+\overline{m}^{a}m^{b}\right)\right]\nabla_{a}k_{b}$
$\displaystyle=$
$\displaystyle-2\left(k^{a}l^{b}+l^{a}k^{b}\right)\nabla_{a}k_{b}+\left(m^{a}\overline{m}^{b}+\overline{m}^{a}m^{b}\right)\nabla_{a}k_{b}$
$\displaystyle=$ $\displaystyle-\sqrt{2}\left(l^{b}Dk_{b}+k^{b}\Delta
k_{b}\right)-\frac{\left(\varrho+\overline{\varrho}\right)}{\sqrt{2}}=\frac{\left(\varepsilon+\overline{\varepsilon}\right)-\left(\varrho+\overline{\varrho}\right)}{\sqrt{2}},$
(76) $\displaystyle\Theta_{\left(l\right)}=$
$\displaystyle-\sqrt{2}\left(l^{b}Dl_{b}+k^{b}\Delta
l_{b}\right)+\left(m^{a}\overline{m}^{b}+\overline{m}^{a}m^{b}\right)\nabla_{a}l_{b}=\frac{\mu+\overline{\mu}-\left(\gamma+\overline{\gamma}\right)}{\sqrt{2}}.$
(77)

The evolution traces left-hand-sides can then be written as

$\displaystyle k^{a}\nabla_{a}\Theta_{\left(k\right)}=$
$\displaystyle\frac{D\Theta_{\left(k\right)}}{\sqrt{2}},$ $\displaystyle
k^{a}\nabla_{a}\Theta_{\left(l\right)}=$
$\displaystyle\frac{D\Theta_{\left(l\right)}}{\sqrt{2}},$ (78a) $\displaystyle
l^{a}\nabla_{a}\Theta_{\left(k\right)}=$
$\displaystyle\frac{\Delta\Theta_{\left(k\right)}}{\sqrt{2}},$ $\displaystyle
l^{a}\nabla_{a}\Theta_{\left(l\right)}=$
$\displaystyle\frac{\Delta\Theta_{\left(l\right)}}{\sqrt{2}}.$ (78b)

Using the 2+2 approach in Eqs. (8), and the sources in the language of NP
formalism, the Raychaudhuri equations in spherical symmetry can be expressed
as

$\displaystyle
D\left(\frac{\left(\varepsilon+\overline{\varepsilon}\right)-\left(\varrho+\overline{\varrho}\right)}{2}\right)=$
$\displaystyle\frac{\left(\left(\varepsilon+\overline{\varepsilon}\right)-\left(\varrho+\overline{\varrho}\right)\right)^{2}}{4}-\Phi_{00},$
(79a)
$\displaystyle\Delta\left(\frac{\mu+\overline{\mu}-\left(\gamma+\overline{\gamma}\right)}{2}\right)=$
$\displaystyle\frac{\left(\mu+\overline{\mu}-\left(\gamma+\overline{\gamma}\right)\right)^{2}}{4}-\Phi_{22},$
(79b)
$\displaystyle\Delta\left(\frac{\left(\varepsilon+\overline{\varepsilon}\right)-\left(\varrho+\overline{\varrho}\right)}{2}\right)=$
$\displaystyle-\frac{\left(\left(\varepsilon+\overline{\varepsilon}\right)-\left(\varrho+\overline{\varrho}\right)\right)\left(\mu+\overline{\mu}-2\left(\gamma+\overline{\gamma}\right)\right)}{2}-\frac{k^{a}l_{a}}{r^{2}}+\Phi_{11}-3\varLambda,$
(79c) $\displaystyle
D\left(\frac{\mu+\overline{\mu}-\left(\gamma+\overline{\gamma}\right)}{2}\right)=$
$\displaystyle-\frac{\left(2\left(\varepsilon+\overline{\varepsilon}\right)-\left(\varrho+\overline{\varrho}\right)\right)\left(\mu+\overline{\mu}-\left(\gamma+\overline{\gamma}\right)\right)}{2}-\frac{k^{a}l_{a}}{r^{2}}+\Phi_{11}-3\varLambda.$
(79d)

The NP formalism does not provide an easy way to compute the right-hand-side
of the evolution traces, showing the 2+2 approach to be better fit to tackle
the problems under consideration, namely those concerning the interplay
between matter and lightcones, as well as observer centered radiation
problems.

##  V Conclusion

This work started by reviewing the fundamentals of the dual null formalism,
its projector on 2-space, the optical geometric decomposition of its kinematic
and its dynamics, reduced in spherical symmetry to the Raychaudhuri equations.

The complete derivation including the cross focusing equation is presented for
the fir st time, as far as we know, in the appendices.

We proposed a dictionary to translate between this splitting and the 1+1+2
approach in a structured way, in order to highlight the physical
interpretation of the flow scalars that arise from this formalism, as well as
connect it to the coordinated dependent GPG formalism and the 1+3 splitting of
the spacetime.

Such dictionary clearly relates Raychadhuri-like equations to the dynamics of
the flow or null fields and present transparently the role of flow
acceleration and the shape tensor associated with the foliation on the
dynamics. Such clear spelling can prove useful for the fields interested in
the interplay between lightcones and matter flow, as well as observer centered
point of view to radiation problems, such as Gravitational wave effects on
environment.

Physically, we recalled that the flow equation relates to a generalised TOV
hydrostatic equilibrium, while the cross focusing and radial Raychaudhuri
equations produce the Misner-Sharpe mass conservation. We transparently
spelled out the interpretation of 1+1+2 scalars as combination of flow
3-expansion and -shear, as well as an interesting connection to redshift that
unveils that the $\mathcal{B}$ scalar, which appears naturally on the 1+1+2
evolution equations, is also the source of the redshift in warped symmetric
spacetimes. To the best of our knowledge, the gauge invariant scalar
$\mathcal{B}$ is introduced for the first time. This is a strong argument in
favor of the realisation that the 1+1+2 formalism deals with the physically
meaningful scalars in the context of matter dominated warped symmetric
spacetimes.

Finally we addressed the generalisation of the above framework from the
symmetry of spherical foliations to planar and hyperbolic foliations of the
spacetime. We presented a thorough comparison showing the advantage of using
the present framework rather than the more convoluted NP formalism in such
problems. This work should render obvious the interest of the presented
framework for future explorations of warped, foliated spacetimes .

## Acknowledgements

JPM acknowledges funding from the Fundação para a Ciência e a Tecnologia (FCT,
Portugal) under research projects UIDB/04434/2020, UIDP/04434/2020, EXPL/FIS-
AST/1368/2021 and PTDC/FIS-AST/0054/2021. MLeD acknowledges the financial
support by the Lanzhou University starting fund and the Fundamental Research
Funds for the Central Universities (Grant No. lzujbky-2019-25).

##  Appendix A Raychaudhuri equation for null, foliatedly symmetric,
congruences

Our aim in this section is to compute $\mathcal{L}_{k}\Theta_{(k)}$. It is
important to define the quantities $\nu_{k}$ and $\nu_{l}$ as

$\displaystyle\nabla_{a}k_{b}=\frac{1}{k^{c}l_{c}}\left(\nu_{k}{l_{a}k_{b}}-\nu_{l}{k_{a}k_{b}}\right)+\frac{\Theta_{(k)}}{2}s_{ab}\,,$
(80)

where the last term comes from the definition of expansion, Eq. (4). Since we
keep $k^{a}l_{a}$ constant, we have $l^{a}\nabla k_{a}=-k^{a}\nabla l_{a}$,
therefore

$\displaystyle\nabla_{a}l_{b}=\frac{1}{k^{c}l_{c}}\left(\nu_{l}{k_{a}l_{b}}-\nu_{k}l_{a}l_{b}\right)+\frac{\Theta_{(l)}}{2}s_{ab}\,.$
(81)

Those quantities are interpreted as the inaffinities of the vectors $k^{a}$
and $l^{a}$, since,

$\displaystyle k^{a}\nabla_{a}k_{b}=\nu_{k}k_{b}\,,\quad\text{and}\quad
l^{a}\nabla_{a}l_{b}=\nu_{l}l_{b}\,,$ (82)

which means that when $\nu_{k}=\nu_{l}=0$ the dual null basis $(k^{a},l^{a})$
corresponds to a geodesic null field. Consider the $k$-directed covariant
derivative of $\nabla_{a}k_{b}$:

$\displaystyle
k^{c}\nabla_{c}\nabla_{a}k_{b}=R\indices{{}_{c}{}_{a}{}_{b}^{d}}k^{c}k_{d}+k^{c}\nabla_{a}\nabla_{c}k_{b}=$
$\displaystyle=-R_{cadb}k^{c}k^{d}+k^{c}\nabla_{a}\left[\frac{1}{k^{c}l_{c}}(\nu_{k}l_{c}k_{b}-\nu_{l}k_{c}k_{b})+\frac{\Theta_{(k)}}{2}s_{cb}\right]=$
$\displaystyle=-R_{cadb}k^{c}k^{d}+\nu_{k}\nabla_{a}k_{b}+\frac{\Theta_{(k)}}{2}k^{c}\nabla_{a}s_{cb}+(\dots)_{a}k_{b}\,,$
(83)

where $R\indices{{}_{a}{}_{b}{}_{c}^{d}}$ is the Riemann curvature tensor and
we gathered under $(\dots)_{a}k_{b}$ all terms that are $k$ directed in the
second index. We have also used the symmetry of the Riemann tensor and the
fact that $k^{c}s_{cb}=0$.

Thus, contracting (83) with $s^{ab}$, we obtain

$\displaystyle
s^{ab}k^{c}\nabla_{c}\nabla_{a}k_{b}=-s^{ab}R\indices{{}_{c}{}_{a}{}_{d}{}_{b}}k^{c}k^{d}+\nu_{k}s^{ab}\nabla_{a}k_{b}+\frac{\Theta_{(k)}}{2}s^{ab}k^{c}\nabla_{a}s_{bc}\Rightarrow$
$\displaystyle
s^{ab}k^{c}\nabla_{c}\nabla_{a}k_{b}=-R\indices{{}_{c}{}_{m}{}_{d}^{m}}k^{c}k^{d}+\nu_{k}\Theta_{(k)}-\frac{\Theta_{(k)}}{2}s^{ab}s\indices{{}^{c}_{b}}\nabla_{a}k_{c}\Rightarrow$
$\displaystyle
s^{ab}k^{c}\nabla_{c}\nabla_{a}k_{b}=-R_{cd}k^{c}k^{d}+\nu_{k}\Theta_{(k)}-\frac{\Theta_{(k)}^{2}}{2}\,,$
(84)

where $R_{ab}$ is the Ricci tensor and we used
$s^{ab}R_{cadb}k^{c}k^{d}=g^{ab}R_{cabd}k^{c}k^{d}$, since
$R\indices{{}_{c}{}_{a}{}_{b}{}_{d}}k^{(a}l^{b)}k^{c}k^{d}=0$ due to the
symmetries of the Riemann tensor.

On the other hand, we can express the left hand side of Eq. (84) as

$\displaystyle
s^{ab}k^{c}\nabla_{c}\nabla_{a}k_{b}=k^{c}\nabla_{c}\left(s^{ab}\nabla_{a}k_{b}\right)-\nabla_{a}k_{b}k^{c}\nabla_{c}s^{ab}\Rightarrow$
$\displaystyle
s^{ab}k^{c}\nabla_{c}\nabla_{a}k_{b}=\mathcal{L}_{k}\Theta_{(k)}-\nabla_{a}k_{b}k^{c}\nabla_{c}s^{ab}=\mathcal{L}_{k}\Theta_{(k)}\,,$
(85)

since, by Eq. (80), we have

$\displaystyle\nabla_{a}k_{b}k^{c}\nabla_{c}s^{ab}=\left[\frac{1}{k^{c}l_{c}}\left(\nu_{k}{l_{a}k_{b}}-\nu_{l}{k_{a}k_{b}}\right)+\frac{\Theta_{(k)}}{2}s_{ab}\right]k^{c}\nabla_{c}s^{ab}\Rightarrow$
$\displaystyle\nabla_{a}k_{b}k^{c}\nabla_{c}s^{ab}=-\frac{1}{k^{c}l_{c}}\left(\nu_{k}{l_{a}s^{ab}}-\nu_{l}{k_{a}s^{ab}}\right)k^{c}\nabla_{c}k_{b}+\frac{\Theta_{(k)}}{2}s_{ab}k^{c}\nabla_{c}s^{ab}=0\,,$
(86)

where we used the fact that $k_{b}s^{ab}$ and $s^{ab}s_{ab}$ are both
constants in order to manipulate the derivatives in the Eq. (86). Generalizing
for both null directions and using the Einstein’s equations:

$\displaystyle R_{ab}-\frac{1}{2}Rg_{ab}=8\pi T_{ab}\,.$ (87)

we finally obtain

$\displaystyle\mathcal{L}_{k}\Theta_{(k)}=-\frac{\Theta_{(k)}^{2}}{2}+\nu_{k}\Theta_{(k)}-8\pi
T_{cd}k^{c}k^{d}.$ (88)

##  Appendix B The cross focusing equation

In order to deduce the other evolution equation, relating the change of one
expansion as we move along the _other_ null direction, namely
$\mathcal{L}_{k}\Theta_{(l)}$ and $\mathcal{L}_{l}\Theta_{(k)}$. They are
closely related since

$\displaystyle\mathcal{L}_{k}\Theta_{(l)}=\mathcal{L}_{k}\left(\frac{2}{r}\mathcal{L}_{l}r\right)=\mathcal{L}_{k}\mathcal{L}_{l}\ln
r^{2}=\left(\mathcal{L}_{l}\mathcal{L}_{k}+\mathcal{L}_{[k,l]}\right)\ln
r^{2}=$
$\displaystyle=\mathcal{L}_{l}\Theta_{(k)}+\Theta_{([k,l])}=\mathcal{L}_{l}\Theta_{(k)}-\nu_{k}\Theta_{(l)}+\nu_{l}\Theta_{(k)}\,,$
(89)

where $[k,l]^{c}=\mathcal{L}_{k}l^{c}=\nu_{l}k^{c}-\nu_{k}l^{c}$ is the
commutator between the vectors $k^{a}$ and $l^{a}$.

It will be convenient at this point to evoke the existence of coordinate null
basis along our $k^{a}$ and $l^{a}$ directions, which means a pair of
commuting vectors parallel to this direction. We denote those vectors as
$\delta^{a}_{\alpha}$, where we the Greek indices denote coordinate labels
instead of tensor indices. We also define the covariant vectors
$\delta_{a}^{\alpha}$ by the relation
$\delta^{a}_{\alpha}\delta_{a}^{\beta}=\delta_{\alpha}^{\beta}$, where in the
right hand side we interpret $\delta$ as the Krönecker symbol. In particular,
for the null directions $\delta^{a}_{+}$ and $\delta^{a}_{-}$, respectively,
defined such that

$\displaystyle[\delta^{a}_{+},\delta^{b}_{-}]=0\,,\quad
k^{a}=K\delta^{a}_{+}\,\quad l^{a}=L\delta^{a}_{-}\,,$ (90)

where $K$ and $L$ are scalar functions that can vary only along $k^{a}$ and
$l^{a}$ directions, that is $s^{ab}\nabla_{b}K=s^{ab}\nabla_{b}L=0$, due to
spherical symmetry. The functions $K$ and $L$ which relate the original dual
null basis with a coordinate basis must satisfy:

$\displaystyle[\delta^{a}_{+},\delta^{b}_{-}]=[K^{-1}k^{a},L^{-1}l^{b}]=0\,,\quad\text{which
implies}$ $\displaystyle K\mathcal{L}_{k}\ln L=-\nu_{k}\,,\quad\text{and}\quad
L\mathcal{L}_{l}\ln K=-\nu_{l}\,.$ (91)

We define the metric components
$g_{\alpha\beta}=g_{ab}\delta^{a}_{\alpha}\delta^{b}_{\beta}$, in particular,
$g_{+-}=g_{ab}\delta^{a}_{+}\delta^{b}_{-}=-e^{f}$, where $f$ is also a
function that only varies along directions orthogonal to the spheres of
symmetry, so the metric tensor may be written as:

$\displaystyle
g_{ab}=-2e^{f}\delta^{+}_{(a}\delta^{-}_{b)}+s_{\alpha\beta}\delta^{\alpha}_{a}\delta^{\beta}_{b}\,.$
(92)

Therefore, we obtain for the covariant vectors:

$\displaystyle k_{a}=g_{ab}k^{b}=-Ke^{f}\delta_{a}^{-}\,,\quad
l_{a}=g_{ab}l^{b}=-Le^{f}\delta_{a}^{+}\,.$ (93)

In particular, we obtain the relation

$\displaystyle k_{c}l^{c}=KLe^{f}\,.$ (94)

We define the expansions $\Theta_{\pm}$ as

$\displaystyle\Theta_{\pm}=\frac{1}{2}s^{ab}\mathcal{L}_{\pm}s_{ab}=2\frac{\partial_{\pm}r}{r}\,,$
(95)

where we denote $\partial_{\pm}=\delta^{c}_{\pm}\partial_{c}$.

In terms of the basis vectors, the Christoffel symbols are written as

$\displaystyle\Gamma_{\alpha\beta}^{\gamma}=\delta^{\gamma}_{b}\delta^{a}_{\alpha}\nabla_{a}\delta^{b}_{\beta}$
(96)

We obtain the following components for the Christoffel symbols (92) in terms
of the quantities we defined:

$\displaystyle\Gamma_{\pm\pm}^{\pm}=\partial_{\pm}f\,;\quad
s^{\alpha}_{\mu}s^{\nu}_{\gamma}\Gamma_{\nu\pm}^{\mu}=\frac{\Theta_{\pm}}{2}s^{\alpha}_{\gamma}\,;\quad
s^{\mu}_{\alpha}s^{\nu}_{\beta}\Gamma_{\mu\nu}^{\pm}=e^{-f}\frac{\Theta_{\pm}}{2}s_{\alpha\beta}\,;$
(97)

where we used $s^{\mu}_{\nu}$ as a projector to the spatial directions. The
other components are null or just correspond to the components of the two-
dimensional sphere whose metric is $s_{ab}$.

Computing the space-space components of the Ricci tensor we obtain

$\displaystyle
R_{\beta\gamma}=\left[e^{-f}\partial_{(-}\Theta_{+)}+e^{-f}\Theta_{+}\Theta_{-}+\frac{1}{r^{2}}\right]s_{\beta\gamma}\,.$
(98)

The Ricci scalar is

$\displaystyle
R=g^{ab}R_{ab}=2g^{+-}R_{+-}+s^{ab}R_{ab}=-2e^{-f}R_{+-}+2\left[e^{-f}\partial_{(-}\Theta_{+)}+e^{-f}\Theta_{+}\Theta_{-}+\frac{1}{r^{2}}\right]\,.$
(99)

The $(+-)$ component of the Einstein tensor is given by

$\displaystyle G_{+-}=R_{+-}-\frac{1}{2}g_{+-}R=$
$\displaystyle=R_{+-}+e^{f}\left[-e^{-f}R_{+-}+\left[e^{-f}\partial_{(-}\Theta_{+)}+e^{-f}\Theta_{+}\Theta_{-}+\frac{1}{r^{2}}\right]\right]=$
$\displaystyle=\partial_{(-}\Theta_{+)}+\Theta_{+}\Theta_{-}+\frac{e^{f}}{r^{2}}$
(100)

using the fact that $\partial_{-}\Theta_{+}=\partial_{+}\Theta_{-}$ and
Einstein’s equations, we finally obtain

$\displaystyle\mathcal{L}_{\pm}\Theta_{\mp}=-\Theta_{+}\Theta_{-}-\frac{e^{f}}{r^{2}}+8\pi
T_{+-}\,,$ (101)

Generalizing the expression (101) for our null basis $k^{a}=K\delta^{a}_{+}$,
and $l^{a}=L\delta^{a}_{-}$, we find

$\displaystyle\mathcal{L}_{l}\Theta_{(k)}=L\left[\mathcal{L}_{-}(K\Theta_{+})\right]=L\left(K\mathcal{L}_{-}\Theta_{+}+\Theta_{+}\mathcal{L}_{-}K\right)=$
$\displaystyle=\Theta_{+}L\mathcal{L}_{-}K+KL\left(-\Theta_{+}\Theta_{-}-\frac{e^{f}}{r^{2}}+8\pi
T_{+-}\right)=$
$\displaystyle=\frac{\mathcal{L}_{l}K}{K}\Theta_{(k)}+\left(-\Theta_{(k)}\Theta_{(l)}-\frac{k^{c}l_{c}}{r^{2}}+8\pi
T_{ab}k^{a}l^{b}\right)\,.$ (102)

The last step is identifying the term $\frac{\mathcal{L}_{l}K}{K}$. We can
write $K=k^{b}\delta^{+}_{b}$, therefore

$\displaystyle\frac{\mathcal{L}_{l}K}{K}=\frac{1}{K}l^{a}\nabla_{a}(K)=\frac{1}{K}l^{a}\nabla_{a}(k^{b}\delta_{b}^{+})=$
$\displaystyle=\frac{1}{K}\left(\delta_{b}^{+}l^{a}\nabla_{a}k^{b}+l^{a}k^{b}\nabla_{a}\delta^{+}_{b}\right)=$
$\displaystyle=\frac{1}{e^{f}KL}l_{b}l^{a}\nabla_{a}k^{b}=\frac{1}{k_{c}l^{c}}l^{a}l^{b}\nabla_{a}k_{b}=-\nu_{l}\,,$
(103)

where we used
$l^{a}k^{b}\nabla_{a}\delta^{+}_{b}=LK\delta^{a}_{-}\delta^{b}_{+}\nabla_{a}\delta^{+}_{b}=LK\Gamma_{+-}^{+}=0$,
since this Christoffel symbols vanishes.

Therefore, we find finally

$\displaystyle\mathcal{L}_{l}\Theta_{(k)}=-\Theta_{(k)}\Theta_{(l)}-\nu_{l}\Theta_{(k)}-\frac{k^{c}l_{c}}{r^{2}}+8\pi
T_{ab}k^{a}l^{b}\,,$ (104)

or, in the symmetrical form

$\displaystyle\mathcal{L}_{l}\Theta_{(k)}+\mathcal{L}_{k}\Theta_{(l)}=-2\Theta_{(k)}\Theta_{(l)}-\nu_{l}\Theta_{(k)}-\nu_{k}\Theta_{(l)}-\frac{2k^{c}l_{c}}{r^{2}}+16\pi
T_{ab}k^{a}l^{b}\,,$ (105)

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