# The Common Solution Space of General Relativity

Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, Durban 4000, South Africa  Departamento de
Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280
Antofagasta, Chile

###### Abstract

In this review, we explore the solution space for the field equations of
Einstein’s General Relativity for various static, spherically symmetric
spacetimes. We consider the vacuum case, represented by the Schwarzschild
black hole; the de Sitter-Schwarzschild geometry, which includes a
cosmological constant; the Reissner-Nordström geometry, which accounts for the
presence of charge. Additionally we consider the homogenenous and anisotropic
locally rotational Bianchi II spacetime in the vacuum. Our analysis reveals
that the field equations for these scenarios share a common three-dimensional
group of point transformations, with the generators being the elements of the
$D\otimes_{s}T_{2}$ Lie algebra, known as the semidirect product of dilations
and translations in the plane. Due to this algebraic property the field
equations for the aforementioned gravitational models can be expressed in the
equivalent form of the null geodesic equations for conformally flat
geometries. Consequently, the solution space for the field equations is
common, and it is the solution space for the free particle in a flat space.
This appoach open new directions on the construction of analytic solutions in
gravitational physics and cosmology.

Exact solutions; General Relativity; Solution Space; Blach holes;
minisuperspace

##  I Introduction

The construction of exact and analytic solutions for differential equations is
essential in all areas of physics and applied mathematics. Closed-form
solutions provide critical insights into the dynamical behavior of the given
system and enhance our understanding of the initial value problem. The
importancy of the analytic solutions is well described by Arscott in the
introduction of his book asc .

Nowadays because of the high computing capacity allows for the numerical
treatment of nonlinear differential equations. While numerical methods offer
immediate information about the local behavior of the dynamical system near
the initial conditions, they do not always provide a global understanding of
the solution’s behavior and information regarding the initial value problem
ch1 .

A powerful approach for the analytic treatment of nonlinear differential
equations is the symmetry analysis established by Sophus Lie lie1 ; lie2 ;
lie3 . The key characteristic of symmetry analysis is the identification of
invariance properties of differential equations under finite transformations
related to continuous groups. The presence of a symmetry vector indicates the
existence of invariant functions, which can be used to simplify the given
differential equation by defining a new reduced equation. When feasible, this
approach allows for the construction of closed-form solutions, known as
similarity solutions, for more details we refer the reader to olver ; kumei ;
ibra ; Ovsi .

A pioneer application of the symmetry analysis established by Emmy Noether
Noether18 . Criteria have been established where the admitted symmetries are
directly related to the existence of conservation laws for a given dynamical
system. While conservation laws can be derived through various approaches
other than Noether’s theorems hj1 ; hj2 , the simplicity and systematic nature
of Noether’s algorithm make her work one of the most influential studies in
physical science noe1 ; noe2 ; noe3 ; noe4 ; noe5 ; noe8 .

In gravitational physics, symmetries play a crucial role at every stage of the
theory. The general form of the physical space is constrained by the existence
of symmetries sym1 ; sym2 ; sym3 ; sym4 ; sym5 . Moreover, the differential
equations governing the dynamical variables, as they are provided by General
Relativity, are nonlinear second-order differential equations. The symmetry
method has been widely applied to these equations to determine solutions see
for instance sym6 ; sym7 ; sym8 ; sym9 ; sym10 ; sym11  and references
therein.

In this study, we review the solution space for the field equations of several
well-known gravitational models in General Relativity. We explore the
dynamical behavior of Einstein’s field equations and identify common features
within the solution space for these spacetimes. This study extends the concept
of symmetries in gravitational models and opens new directions for
constructing analytic solutions. We demonstrate that the field equations for
the gravitational models under consideration can be solved using the
linearization approach ln1 ; ln2 . Specifically, we show that these field
equations can be reformulated as a set of linear equations, rendering the
dynamics trivial. Furthermore, we illustrate that the solutions to Einstein’s
field equations can be expressed in terms of linear functions. The structure
of the paper is as follows.

In Section II we present the basic properties and definitions for the
conformally related metrics and Lagrangians. In the following Sections we
investigate the solution space for the Einstein’s field equations in various
models. In Section III, we study the solution space for the field equations of
the static spherical symmetric spacetime in the vacuum leading to the
Schwarzschild black hole. Furthermore, in Section IV the cosmological constant
is introduced, where we show that the de Sitter-Schwarzschild solution it has
the same origin with that of the vacuum spacetime. In V we introduce charge
and we study the solution space for the Reissner-Nordström black hole. We
extend our analysis to the cosmological case and specifically to the
homogeneous and anisotropic locally rotational Bianchi II spacetime. The
solution space for this gravitational model is determined in Section VI.

For all these gravitational models, the field equations can be linearized
through point transformations, which means that they share the solution space.
Specifically, the Einstein’s field equations can be written in the equivalent
form of the geodesic equations in a flat space. The origin for this common
feature is discussed in Section VII. Finally, in Section VIII we summarize our
results and we draw our conclusions.

##  II Conformally related metrics and Lagrangians

In this Section we briefly discuss the basic mathematical definitions
necessary for the rest of the study.

###  II.1 Conformally related metrics

Consider the two metric tensor $g_{ij},~{}\bar{g}_{ij}.$ We say that the
tensors $g_{ij},~{}\bar{g}_{ij}$ are conformally related if there exist a
function $\Omega\left(x^{k}\right)$, such that,
$\bar{g}_{ij}=\Omega^{2}g_{ij}~{}$sym4 .

As conformal symmetries are characterized the generators $X$ of the point
transformations which preserves the angles between two lines and the null
structure. In the case where not only angles and the null structure but also
the length is preserved, the CKV will be characterized as a killing symmetry
(KV) or isometry.

Let $X$ be a CKV for the metric tensor $g_{ij}$, the following mathematical
condition holds true sym4

$\mathcal{L}_{X}g_{ij}=2\psi\left(x^{k}\right)g_{ij}~{},~{}$ (1)

where $\psi\left(x^{k}\right)$ is known as the conformal factor defined as
$\psi\left(x^{k}\right)=\frac{1}{\dim g}X_{~{};k}^{k}$, and $\mathcal{L}_{X}$
is the Lie derivative with respect to the vector field $X$.

For the conformally related metric $\bar{g}_{ij}$ the symmetry condition for
the CKV reads sym4

$\mathcal{L}_{X}\bar{g}_{ij}=2\bar{\psi}\left(x^{k}\right)\bar{g}_{ij}~{},~{}\bar{\psi}\left(x^{k}\right)=\psi\left(x^{k}\right)+\left(\ln\Omega\right)_{,k}X^{k}\text{.}$
(2)

Consequently, conformally related spaces share the conformal symmetries
(CKVs). That is, the conformal structure remain invariant under conformal
transformations.

A $n-$dimensional space, for $n\geq 3$, which admits
$\frac{\left(n+1\right)\left(n+2\right)}{2}$ CKVs is a maximally symmetric
space and it is conformally flat. If a space is conformally flat, then there
exist a coordinate system such that
$\bar{g}_{ij}=\Omega\left(x^{k}\right)^{2}\eta_{ij}$, where $\eta_{ij}$ is the
diagonal flat space.

A main characteristic for the conformally flat spaces is that for $n=3$, the
Cotton-York tensor defined as sym4

$C_{ijk}=R_{ij;k}-R_{kj;i}+\frac{1}{4}\left(R_{;j}g_{ik}-R_{;k}g_{ij}\right),$
(3)

is always zero, while for $n>4$, the Weyl tensor is zero. The definition for
the Weyl tensor is as follows sym4

$C_{ijkl}=R_{ijkl}+\frac{2}{n-2}\left(\left(R_{i[l}g_{k]j}+R_{j[k}g_{l]i}\right)+\frac{1}{\left(n-1\right)}Rg_{i[k}g_{l]j}\right).$
(4)

Last but not least, all two-dimensional spacetimes are conformally flat and
admits infinity number of CKVs inf1 .

CKVs are important because they can be used to identify the geometric
characteristics of a spacetime as also are related with the existence of
conservation laws for the geodesic equations. Specifically, for every KV there
correspond a conservation law for the time-like geodesic equations, while
proper CKVs are related with conservation laws for the null geodesics ka1 ;
ka2 ; ka3 .

###  II.2 Conformally related Lagrangians

Let us assume the Action Integral $S$ given by the expression

$S=\int L\left(x^{k},\frac{dx^{k}}{ds}\right)ds~{},~{}$ (5)

where $L\left(x^{k},\frac{dx^{k}}{ds}\right)$ describes the geodesic
Lagrangian for the metric tensor $g_{ij}$, defined as

$L\left(x^{k},\frac{dx^{k}}{ds}\right)=\frac{1}{2}g_{ij}\frac{dx^{i}}{ds}\frac{dx^{j}}{ds}.$
(6)

Variation of the Action Integral (5) gives the equations of motion, i.e. the
Euler-Lagrange equations,

$\frac{d^{2}x^{i}}{ds^{2}}+\Gamma_{jk}^{i}\left(x^{k}\right)\frac{dx^{j}}{ds}\frac{dx^{k}}{ds}=0,$
(7)

where $\Gamma_{jk}^{i}\left(x^{k}\right)$ remarks for the the Levi-Civita
connection of the metric tensor $g_{ij}$.

For the conformal related metric
$\bar{g}_{ij}=\Omega^{2}\left(x^{k}\right)g_{ij}$, the Lagrangian function
which describes the geodesic equations is

$\bar{L}\left(x^{k},\frac{dx^{k}}{ds}\right)=\frac{1}{2}\Omega^{2}\left(x^{k}\right)g_{ij}\frac{dx^{i}}{ds}\frac{dx^{j}}{ds},$
(8)

where now the geodesic equations are expressed as

$\frac{d^{2}x^{i}}{ds^{2}}+\left(\Gamma_{jk}^{i}-\left(\ln\Omega\right)^{,i}g_{jk}\right)\frac{dx^{j}}{ds}\frac{dx^{k}}{ds}=0,$
(9)

in which
$\bar{\Gamma}_{jk}^{i}=\Gamma_{jk}^{i}-\left(\ln\Omega\right)^{,i}g_{jk}$ is
the Levi-Civita connection for the conformally related metric $\bar{g}_{ij}$.

It is straightforward to conclude that the geodesic equations are invariant
under a conformal transformation if and only if the Hamiltonian for the
geodesic equations is zero, that is,
$\frac{1}{2}g_{ij}\frac{dx^{i}}{ds}\frac{dx^{j}}{ds}=0~{}$ka4 . Hence, the
geodesic equations for Lagrangian (6) is invariant under conformal
transformations if and only if it is describes null geodesics.

Before we proceed with the main analysis of this study, we should remark that
for a conformally flat metric $\bar{g}_{ij}$, there exist always a coordinate
system where the null geodesic equations are expressed as terms of the linear
system

$\frac{d^{2}x^{i}}{ds^{2}}=0~{},~{}\eta_{ij}\frac{dx^{i}}{ds}\frac{dx^{j}}{ds}=0.$
(10)

That interesting invariant property for the null geodesics of conformally flat
spaces we employ later in this work to study the solution space in General
Relativity.

##  III The Schwarzschild spacetime

We begin our exploration by considering the static spherically symmetric
spacetime, described by the line element

$ds^{2}=-a^{2}\left(r\right)dt^{2}+n^{2}\left(r\right)dr^{2}+b^{2}\left(r\right)\left(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2}\right).$
(11)

Only two of the three scale factors $a\left(r\right)$, $b\left(r\right)$ and
$n\left(r\right)$ are essential, and they are determined by the solution of
the field equations. Therefore, choosing a functional form for one of the
scale factors is equivalent to selecting a coordinate system.

Within the framework of General Relativity and in vacuum, there exist a unique
analytic solution for the line element (11), derived in 1916 by Karl
Schwarzschild sch1 .

In the coordinate system where $b\left(r\right)=r$, Schwarzschild’s solution
reads

$ds^{2}=-\left(1-\frac{r_{s}}{r}\right)dt^{2}+\left(1-\frac{r_{s}}{r}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2}\right).$
(12)

This spacetime black hole solution, where $r_{s}$ marks as the Schwarzschild
radius.

Einstein’s field equations for the metric tensor (11) follows from the
variation of the point-like Lagrangian function

$L\left(n,a,a^{\prime},b,b^{\prime}\right)=\frac{1}{2n}\left(8ba^{\prime}b^{\prime}+4ab^{\prime
2}\right)+2na,$ (13)

where now prime denotes derivative with respect the radius parameter, that is,
$a^{\prime}=\frac{da}{dr}$.

The Euler-Lagrange equations of Lagrangian (13) leads to the gravitational
field equations

$\frac{1}{2n^{2}}\left(8ba^{\prime}b^{\prime}+4ab^{\prime 2}\right)-2a=0,$
(14)
$a^{\prime\prime}+\frac{1}{b}a^{\prime}b^{\prime}+\frac{n^{2}}{2}\frac{a}{b^{2}}-\frac{1}{2}\frac{a}{b^{2}}b^{\prime
2}-\frac{1}{n}a^{\prime}n^{\prime}=0,$ (15)
$b^{\prime\prime}+\frac{1}{2b}b^{\prime
2}-\frac{1}{a}b^{\prime}n^{\prime}-\frac{1}{2}\frac{n^{2}}{b}=0.$ (16)

The field equations (14), (15), (16) form a singular Hamiltonian system with
equation (14) to be the Hamiltonian constraint.

We introduce the momentum $p_{a}=\frac{\partial L}{\partial a^{\prime}}$,
$p_{b}=\frac{\partial L}{\partial b^{\prime}}$, and in the Hamiltonian
formalism the field equations (14), (15), (16) become

$n\left(\frac{p_{a}p_{b}}{4b}-\frac{a}{8b^{2}}p_{a}^{2}-2a\right)=0~{},$ (17)
$\frac{1}{n}a^{\prime}=\frac{ap_{a}-bp_{b}}{4b^{2}}~{},~{}\frac{1}{n}b^{\prime}=\frac{p_{a}}{4b}~{},$
(18)
$\frac{1}{n}p_{a}^{\prime}=2+\frac{p_{a}^{2}}{b^{2}}~{},~{}\frac{1}{n}p_{b}^{{}^{\prime}}=\frac{1}{4b^{2}}\left(p_{a}p_{b}-\frac{p_{a}^{2}}{2}\right)~{}.$
(19)

These equations are of the same form as those of a Hamiltonian system, which
describes the motion of two particles with varying mass under the action of a
conservative force. Here, the scale factors play the role of the particles,
while the spatial curvature term provides the interaction term.

We employ Eisenhart’s approach el  to write the field equations (17), (18),
(19) in the equivalent form of a Hamiltonian system which describes geodesic
equations. Indeed, we introduce the new scalar $\psi$, and the momentum
$p_{\psi}$, where the new Hamiltonian function is

$\mathcal{H}=n\left(\frac{p_{a}p_{b}}{4b}-\frac{a}{8b^{2}}p_{a}^{2}-2ap_{\psi}^{2}\right).$
(20)

Consequently, the equations of motion in terms of the momentum are written as
follows

$n\left(\frac{p_{a}p_{b}}{4b}-\frac{a}{8b^{2}}p_{a}^{2}-2ap_{\psi}^{2}\right)=0$
(21)
$\frac{1}{n}a^{\prime}=\frac{ap_{a}-bp_{b}}{4b^{2}}~{},~{}\frac{1}{n}b^{\prime}=\frac{p_{a}}{4b}~{},~{}\frac{1}{n}\psi^{\prime}=4ap_{\psi}~{},$
(22)
$\frac{1}{n}p_{a}^{\prime}=2p_{z}^{2}+\frac{p_{a}^{2}}{b^{2}}~{},~{}\frac{1}{n}p_{b}^{{}^{\prime}}=\frac{1}{4b^{2}}\left(p_{a}p_{b}-\frac{p_{a}^{2}}{2}\right)~{},~{}p_{\psi}^{\prime}=0~{}.$
(23)

From the latter expression, that is, equation (23), it follows that the
momentum $p_{\psi}$ is conserved. That is, $p_{\psi}$ represents a second
conservation law for the geodesic equations. Nevertheless, in order to recover
the original gravitational system (17), (18), (19), the following constraint
should be applied, $p_{\psi}=1$.

As we shall see in the following lines, the introduction of the scalar field
$\psi$ leads to the introduction of new conservation laws, and new dynamical
properties. These conservation laws are not lost when conservation law
$p_{\psi}=1$ is applied in the system, but they become nonlocal, that is,
hidden symmetries.

The Hamiltonian function (20) with the constraint (21) describes the null
geodesic equations for the three-dimensional space with line element

$ds^{2}=n\left(8b~{}da~{}db+4a~{}db^{2}-\frac{d\psi^{2}}{2a}\right).$ (24)

We will refer to the latter space as the extended minisuperspace.

The null geodesics are invariant under conformation, which is why parameter
$n$ plays no role in the dynamics.

Thus, for the line element (24) we calculate that all the components of the
Cotton-York tensor (3) are zero; that is $C_{ijk}=0$. This property states
that the three-dimensional space (24) is conformally flat, that is, there
exist a coordinate transformation
$\left\\{a,b,\psi\right\\}\rightarrow\left\\{x,y,z\right\\}$, where the line
element is of the form
$ds^{2}=n\Omega^{2}\left(x,y,z\right)\left(\alpha_{1}dx^{2}+\alpha_{2}dy^{2}+\alpha_{3}dz^{2}\right)$,
where $\alpha_{1},~{}\alpha_{2}$ and $\alpha_{3}$ are constants. Function
$\Omega\left(x,y,z\right)$ is known as the conformal factor. In the new
coordinates $\left\\{x,y,z\right\\}$, the equations of motion (21), (22), (23)
are linear. Recall that the unique linear geodesic equations are those of the
free particle in the flat space.

An equivalent way to verify this property is to calculate the number of
conservation laws for the null geodesics. It is known that Conformal Killing
Vectors (CKVs) generate conservation laws for null geodesics. Hence, the
conformal condition for the line element (24) leads to the derivation of ten
CKVs, which is the maximum number of CKVs for a three dimensional space; that
is, space (24) is conformally flat.

We introduce the new variable $A$, with the transformation rule
$a=\sqrt{\frac{A}{b}}$, then the line element (24) reads

$ds^{2}=\frac{1}{n}\left(\frac{b}{A}\right)^{\frac{1}{2}}\left(8~{}dAdb-d\psi^{2}\right).$
(25)

By introducing the diagonal coordinates
$A=\frac{x+y}{2\sqrt{2}}~{},~{}b=\frac{x-y}{2\sqrt{2}}$, it follows

$ds^{2}=\frac{1}{n}\left(\frac{x-y}{x+y}\right)^{\frac{1}{2}}\left(dx^{2}-dy^{2}-d\psi^{2}\right),$
(26)

Coordinates $\left\\{x,y,\psi\right\\}$ are the canonical coordinates for the
Hamiltonian system (20). In the coordinate system$\left\\{x,y,\psi\right\\}$,
the field equations (21), (22), (23) are written in the following linearized
form

$\frac{1}{\tilde{n}}x^{\prime}=p_{x}~{},~{}\frac{1}{\tilde{n}}y^{\prime}=p_{y}~{},~{}\frac{1}{\tilde{n}}\psi^{\prime}=p_{\psi},$
(27) $p_{x}^{\prime}=0~{},~{}p_{y}^{\prime}=0~{},~{}p_{\psi}^{\prime}=0,$ (28)

with constraints

$x^{\prime 2}-y^{\prime 2}-\psi^{\prime 2}=0,~{}p_{\psi}=1,$ (29)

and $\tilde{n}=n\left(\frac{x-y}{x+y}\right)^{-\frac{1}{2}}$.

Without loss of generality we can select $\tilde{n}=1$ and the latter
dynamical system takes the form of the free particle in a three dimensional
flat space, that is,

$x^{\prime\prime}=0~{},~{}y^{\prime\prime}=0~{},~{}\psi^{\prime\prime}=0,$
(30)

with constraints (29).

We have demonstrated that the solution space for the Einstein field equations
for this problem corresponds to that of the three-dimensional free particle in
a flat space. It is important to note that we are referring to the dynamics of
the scale factors driven by the gravitational theory, and not on the test
particles of the physical space. While the transformation applied to linearize
the field equations is not unique, the uniqueness lies in the solution of the
field equations itself.

##  IV De Sitter-Schwarzschild spacetime

The introduction of the cosmological constant $\Lambda$ within the framework
of the static spherical symmetric spacetime (11) leads to the de Sitter-
Schwarzschild metric with line element sch2

$ds^{2}=-\left(1-\frac{r_{s}}{r}-\frac{\Lambda}{3}r^{2}\right)dt^{2}+\left(1-\frac{r_{s}}{r}-\frac{\Lambda}{3}r^{2}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2}\right).$
(31)

We observe that the Schwarzschild spacetime (12) is recovered in the limit
where the cosmological constant vanishes.

The point-like Lagrangian which describes the evolution of the scale factors,
leading to the analytic solution (31), is as follows

$L^{\Lambda}\left(n,a,a^{\prime},b,b^{\prime}\right)=\frac{1}{2n}\left(8ba^{\prime}b^{\prime}+4ab^{\prime
2}\right)+2na\left(1+\Lambda b^{2}\right).$ (32)

We employ the same procedure as before. The equivalent geodesic Hamiltonian,
which describes the field equations for the de Sitter-Schwarzschild geometry,
is

$\mathcal{H}^{\Lambda}=n\left(\frac{p_{a}p_{b}}{4b}-\frac{a}{8b^{2}}p_{a}^{2}-2a\left(1+\Lambda
b^{2}\right)p_{\psi}^{2}\right),$ (33)

with constraints $\mathcal{H}^{\Lambda}=0$ and $p_{\psi}=1$.

Furthermore, the line element for the corresponding extended minisuperspace is

$ds^{\Lambda~{}2}=\frac{1}{n}\left(8b~{}da~{}db+4a~{}db^{2}-\frac{d\psi^{2}}{2a\left(1+\Lambda
b^{2}\right)}\right).$ (34)

For the three-dimensional space (34) the Cotton-York tensor (3) has zero
components, that is, space (34) has the maximum conformal algebra and it is
conformally flat.

We consider the same change of variables as before $a=\sqrt{\frac{A}{b}}$,
such that the line element (34) is expressed as follows

$ds^{\Lambda~{}2}=\frac{1}{\left(1+\Lambda
b^{2}\right)n}\left(\frac{b}{A}\right)^{\frac{1}{2}}\left(8\left(1+\Lambda
b^{2}\right)dAdb-d\psi^{2}\right).$ (35)

Under the second change of variables $dB=\int\left(1+\Lambda b^{2}\right)db,$
it follows
$ds^{\Lambda~{}2}=\frac{1}{\hat{n}}\left(8dAdB-d\psi^{2}\right)~{}$where
$\hat{n}=\left(1+\Lambda b^{2}\right)n\left(\frac{b}{A}\right)^{\frac{1}{2}}$.

Finally in the diagonal variables $A=\frac{X+Y}{2\sqrt{2}}$ and
$B=\frac{X-Y}{2\sqrt{2}}$, the extended minisuperspace is written in the
canonical form of a conformally flat space, that is,

$ds^{\Lambda~{}2}=\frac{1}{2\hat{n}}\left(dX^{2}-dY^{2}-d\psi^{2}\right).$
(36)

Consequently the gravitational field equations are written in the equivalent
form of the free particle in a three-dimensional flat space, i.e.,

$X^{\prime\prime}=0~{},~{}Y^{\prime\prime}=0~{},~{}\psi^{\prime\prime}=0,$
(37)

with constraint equation

$X^{\prime 2}-Y^{\prime 2}-\psi^{\prime 2}=0~{},~{}p_{\psi}=1.$ (38)

We remark that the field equations for the Schwarzschild black hole, whether
in a Minkowski or a de Sitter background, share a common solution space, which
is that of the null geodesic equations in a conformally flat extended
minisuperspace. Consequently, there exists a one-to-one transformation that
relates the two solutions. It’s important to note that this transformation
does not relate the physical space but rather the space of solutions for the
scale factors of spacetime (11).

At this point we want to mention that this is not the unique approach to
extract the de Sitter-Schwarzschild from the Schwarzschild geometry. Another
geometric construction can be found in jb1 .

We now proceed with our investigation into the solution space when an
electromagnetic fluid is introduced into the physical space.

##  V The Reissner-Nordström black hole

The analytic solution of Einstein’s General Relativity for a static spherical
symmetric spacetime (11) with charge is the Reissner-Nordström black hole sch3
; sch4

$ds^{2}=-\left(1-\frac{r_{s}}{r}+\frac{r_{Q}^{2}}{r^{2}}\right)dt^{2}+\left(1-\frac{r_{s}}{r}+\frac{r_{Q}^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta~{}d\phi^{2}\right),$
(39)

in which $r_{Q}$ is the characteristic length scale related to the charge.

The field equations are described by the point-like Lagrangian dim1

$L^{RN}\left(n,a,a^{\prime},b,b^{\prime},\zeta,\zeta^{\prime}\right)=\frac{1}{2n}\left(8ba^{\prime}b^{\prime}+4ab^{\prime
2}+4\frac{b^{2}}{a}\zeta^{\prime 2}\right)+2na,$ (40)

where $\zeta\left(r\right)$ is the potential of the electromagnetic tensor.

For this dynamical system, the corresponding geodesic equivalent Hamiltonian
is

$\mathcal{H}^{RN}=n\left(\frac{p_{a}p_{b}}{4b}-\frac{a}{8b^{2}}p_{a}^{2}+\frac{a}{8b^{2}}p_{\zeta}^{2}-2ap_{\psi}^{2}\right)$
(41)

with constraints $\mathcal{H}^{\Lambda}=0$ and $p_{\psi}=1$. The extended
minisuperspace is defined as

$ds^{RN~{}2}=\frac{1}{n}\left(8b~{}da~{}db+4a~{}db^{2}+4\frac{b^{2}}{a}d\zeta^{2}-\frac{d\psi^{2}}{2a}\right).$
(42)

For the latter element the Weyl tensor is calculated to be always zero.
Consequently, the extended minisuperspace (42) is conformally flat.

We introduce the change of variables
$a=\sqrt{\frac{A}{b}+\frac{z^{2}}{b^{2}}}$, $~{}\zeta=\frac{z}{b^{2}}$. Hence,
the extended minisuperspace (42) is expressed

$ds^{RN~{}2}=\frac{1}{n}\frac{b}{\sqrt{bA+z^{2}}}\left(4dA~{}db+4dz-d\psi^{2}\right),$
(43)

where easily it can be written in the diagonal form

$ds^{RN~{}2}=\frac{1}{\check{n}}\left(dU^{2}-dV^{2}-dZ^{2}-d\psi^{2}\right),$
(44)

and the field equations take the linear form

$U^{\prime\prime}=0~{},\
V^{\prime\prime}=0~{},~{}Z^{\prime\prime}=0~{},~{}\psi^{\prime\prime}=0~{},$
(45)

with constraints

$U^{\prime 2}-V^{\prime 2}-Z^{\prime 2}-\psi^{\prime 2}=0~{},~{}p_{\psi}=1.$
(46)

We observe that the solution space for the field equations of the Reissner-
Nordström black hole consists once again of the equations of motion for a free
particle in a flat geometry. This property is similar to the solution space
for the field equations of the Schwarzschild and de Sitter-Schwarzschild
spacetimes. However, the dimension of the solution space is higher due to the
additional degrees of freedom related to the charge.

##  VI Bianchi II vacuum spacetime

Let us proceed our discussion with the consideration of cosmological
spacetimes. We consider the locally rotational Bianchi II geometry with the
line element

$ds^{2}=-N^{2}\left(t\right)dt^{2}+a\left(t\right)^{2}\left(dr-\theta
d\phi\right)^{2}+b\left(t\right)^{2}\left(d\theta^{2}+d\phi^{2}\right).$ (47)

For this gravitational model the point-like Lagrangian which reproduces the
field equations is defined as

$L^{II}\left(N,a,\dot{a},b,\dot{b}\right)=\frac{1}{N}\left(2b\dot{a}\dot{b}+a\dot{b}^{2}\right)+N\frac{a^{3}}{b^{2}},$
(48)

where a dot denotes derivative with respect to the time parameter, i.e.
$\dot{a}=\frac{da}{dt}$. The vacuum solution derived before in va1 .

The Hamiltonian function for the geodesic description of the field equations
is

$\mathcal{H}^{II}=N\left(\frac{p_{a}p_{b}}{2b}-\frac{a}{4b^{2}}p_{a}^{2}-\frac{a^{3}}{b^{2}}p_{\psi}^{2}\right),$
(49)

with the constraints $\mathcal{H}^{II}=0$ , $p_{\psi}=1$.

Therefore the extended minisuperspace has the following line element

$ds^{II~{}2}=\frac{1}{N}\left(4b~{}da~{}db+2a~{}db^{2}-\frac{b^{2}}{a^{3}}d\psi^{2}\right),$
(50)

where easily it follows that the line element (50) is conformally flat.

In the terms of the new dynamical variables
$a\rightarrow\left(AB\right)^{\frac{1}{4}}$ and $b\rightarrow
B^{-\frac{1}{2}}$, the extended minisuperspace becomes

$ds^{II~{}2}=\frac{1}{A^{\frac{3}{4}}B^{\frac{7}{4}}N}\left(\frac{1}{2}dA~{}dB+d\psi^{2}\right).$
(51)

Therefore, the field equations can be written in the equivalent form of the
linearized system

$\ddot{x}=0~{},~{}\ddot{y}=0~{},~{}\ddot{\psi}=0~{},$ (52)

in which we have introduced the second change of variables
$A=\sqrt{2}\left(x+y\right)~{},~{}B=\sqrt{2}\left(x-y\right)$. Finally, the
following constraints hold true $\dot{x}^{2}-\dot{y}^{2}-\dot{\psi}^{2}=0$,
$p_{\psi}=1$.

##  VII The Lie algebra $D\otimes_{s}T_{2}$

In this Section, we investigate the common geometric property of the field
equations discussed in the previous sections. This property leads to the
unification and linearization of Einstein’s field equations, resulting in the
trivial derivation of the analytic solutions.

The field equations generated by the point-like Lagrangian (13) of the
Schwarzschild spacetime are invariant under the point transformations with
generators, the vector fields.

$X^{1}=\frac{1}{ab}\partial_{a}~{},~{}X^{2}=-a\partial_{a}+b\partial_{b}~{},$
$X^{3}=\left(-\frac{a}{2b}\partial_{a}+\partial_{b}\right).$

For these three vector fields we calculate the the commutators

$\left[X^{1},X^{2}\right]=-X^{1}~{},~{}\left[X^{1},X^{3}\right]=0~{},~{}\left[X^{2},X^{3}\right]=-X^{2}\,.$

Therefore, the vector fields $\left\\{X^{1},X^{2},X^{3}\right\\}$ form the Lie
algebra $A_{3,3}$ in the Patera et al. classification scheme patera . It is a
solvable Lie algebra commonly known as the semidirect product of dilations and
translations in the plane, i.e., $D\otimes_{s}T_{2}\equiv
A_{1}\otimes_{s}2A_{1}$.

In the appearance of the cosmological constant in the physical background
space, the field equations for the de Sitter-Schwarzschild are invariant under
the point transformations with generators

$X_{\Lambda}^{1}=\frac{1}{ab}\partial_{a}~{},~{}X_{\Lambda}^{2}=\frac{1}{1+\Lambda
b^{2}}\left(-a\left(1+\frac{2}{3}\Lambda
b^{2}\right)\partial_{a}+b\left(1+\frac{\Lambda}{3}b^{2}\right)\partial_{b}\right)~{},$
$X_{\Lambda}^{3}=\frac{1}{\left(1+\Lambda
b^{2}\right)}\left(-\frac{a}{2b}\partial_{a}+\partial_{b}\right).$

The vector fields
$\left\\{X_{\Lambda}^{1},X_{\Lambda}^{2},X_{\Lambda}^{3}\right\\}$ have the
same commutator rules with that for $\Lambda=0$, which means that they form
the $D\otimes_{s}T_{2}$ Lie algebra, expressed in a different representation.

As far as the dynamical system described by the point-like Lagrangian (40) is
concerned; that is, the field equations for the Reissner-Nordström spacetime,
they are invariant under the point transformations with generators the
following vector fields

$X_{RN}^{1}=\frac{1}{ab}\partial_{a}~{},~{}X_{RN}^{2}=-a\partial_{a}+b\partial_{b}-z\partial_{\zeta}~{},~{}$
$X_{RN}^{3}=-\left(\frac{a}{2b}+\frac{z^{2}}{ab}\right)\partial_{a}+\partial_{b}-\frac{\zeta}{b}\partial_{\zeta}~{},$
$X_{RN}^{4}=-a\zeta\partial_{a}+b\zeta\partial_{b}+\left(\frac{a^{2}}{4}-\frac{z^{2}}{2}\right)\partial_{\zeta},$
$X_{RN}^{5}=\frac{2\zeta}{ab}\partial_{a}+\frac{1}{b}\partial_{\zeta}~{},~{}X_{RN}^{6}=\partial_{\zeta}.$

The vector fields $\left\\{X_{RN}^{1},X_{RN}^{2},X_{RN\ }^{3}\right\\}$, form
the $D\otimes_{s}T_{2}$ subalgebra.

Finally, the cosmological field equations for the Bianchi II geometry
described by the point-like Lagrangian function (48) are also invariant under
the family of point transformations with generators the vector fields

$X_{II}^{1}=\frac{1}{a^{3}b^{2}}\partial_{a}~{},~{}X_{II}^{2}=b\partial_{b}~{},$
$X_{II}^{3}=-\frac{ab^{2}}{2}\partial_{a}+b^{3}\partial_{b}.$

The latter vector fields form again the $D\otimes_{s}T_{2}$ algebra.

We conclude that the common feature of these four-different models that we
proved that they are linearisable is the existence of the Lie symmetry vectors
which form the three-dimensional $A_{3,3}$ or equivalent, the
$D\otimes_{s}T_{2}$ Lie algebra. However, the natural question which arise is
what is the origin of the $D\otimes_{s}T_{2}$ Lie algebra, and how it is
related with the linearization process.

Consider for instance the maximum symmetric linear system (30). The dynamical
system admits ten symmetry vectors, due to the constraint equations. However,
the application of the constraint $p_{\psi}=1$, in order to determine the
original system, indicates that only three of the ten symmetries remain
points, while the rest six vector fields become nonlocal. The three symmetries
which survive are those which form the $D\otimes_{s}T_{2}$ Lie algebra.

In the case of the higher-dimensional linear system (45), the admitted Lie
symmetries are fifteen, where only the six symmetries remain point symmetries
when the constraint $p_{\psi}=1$ is applied.

Therefore, when a gravitational system is invariant under point
transformations with generators the elements of the $D\otimes_{s}T_{2}$ Lie
algebra, we have a strong indication that this given dynamical system can be
linearized, and the closed-form solution of the field equations can be written
in analytic form.

##  VIII Conclusions

In this piece of work, we delved into the solution space of Einstein’s General
Relativity for several well-known spacetimes. Specifically, we focused on
investigating the solution space for the gravitational field equations
governing the following spacetimes, Schwarzschild vacuum solution, with or
without the cosmological constant term, the Reissner-Nordström black hole with
a charge and the vacuum solution for the locally rotational Bianchi II
cosmology.

For the aforementioned geometries, the Einstein’s field equations are
invariant under the action of point transformations which form the same Lie
algebra, the $D\otimes_{s}T_{2}$ Lie algebra, also known as the
$A_{3,3}~{}$Lie algebra. This specific Lie algebra originates from a higher-
dimensional equivalent dynamical system that describes geodesic equations in
the solution space for the field equations.

For each gravitational model in our study, the equivalent higher-dimensional
dynamical system is found to correspond to the null geodesic equations of a
conformally flat geometry. Hence, we were able to determine coordinate
transformations where the field equations for each gravitational model can be
expressed in terms of the equations of motion for the Newtonian free particle
in three- (or four-) dimensional space. The families of these coordinate
transformations are those that relate the different representations of the
admitted symmetries for the $D\otimes_{s}T_{2}$ Lie algebra.

The static spherically symmetric spacetime considered previously is directly
related to the Kantowski-Sachs geometry and the locally rotational Bianchi III
geometry. Thus, the results of this analysis are valid not only for the
Einstein’s field equations governing the static spherically symmetric
spacetime but also for these two other spacetimes.

We conclude that the common solution space for these gravitational models is
the solution to the linear equations of the Newtonian free particle. This new
geometric approach opens new directions for deriving analytic solutions in
gravitational physics. Furthermore, this method can be applied to modified
theories of gravity and dark energy cosmological models. In future work, we
plan to further investigate these considerations.

Data Availability Statements: Data sharing is not applicable to this article
as no datasets were generated or analyzed during the current study.

###### Acknowledgements.

AP thanks the support of VRIDT through Resolución VRIDT No. 096/2022 and
Resolución VRIDT No. 098/2022.

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