Substituting the above variables into the field equations (11.26), (11.27),
(11.69) we immediately obtain

$\displaystyle x^{\prime\prime}+\frac{f_{TT}}{f_{T}}x^{\prime}T^{\prime}=0$
(11.71) $\displaystyle
y^{\prime\prime}+\frac{f_{TT}}{f_{T}}y^{\prime}T^{\prime}=0$ (11.72)
$\displaystyle H=4f_{T}x^{\prime}y^{\prime}-\left(f-Tf_{T}\right)$ (11.73)

while the torsion scalar is given by

$T=4x^{\prime}y^{\prime}\;.$ (11.74)

Finally, the generalized Lagrangian (11.23) acquires the simple form

$L=4f_{T}x^{\prime}y^{\prime}+\left(f-Tf_{T}\right)\;.$ (11.75)

Since the analysis of the previous subsection revealed two classes of Noether
symmetries, namely for arbitrary $f(T)$, and $f(T)=T^{n}$, in the following
subsections we investigate them separately.

###### Arbitrary $f(T)$

In the case where $f(T)$ is arbitrary, a special solution of the system
(11.71)-(11.74) is

$\displaystyle x\left(\tau\right)$ $\displaystyle=$ $\displaystyle
c_{1}\tau+c_{2}$ (11.76) $\displaystyle y\left(\tau\right)$ $\displaystyle=$
$\displaystyle c_{3}\tau+c_{4}$ (11.77)

and the Hamiltonian constraint ($H=0$) reads

$4c_{1}c_{3}\frac{df}{dT}|_{T=4c_{1}c_{3}}-f+T\frac{df}{dT}|_{T=4c_{1}c_{3}}=0$
(11.78)

where $T=4c_{1}c_{3}$, and $c_{1-4}$ are integration constants. Utilizing
(11.70), (11.76) and (11.77), we get

$\displaystyle
b\left(\tau\right)=3^{\frac{1}{3}}\left(c_{3}\tau+c_{4}\right)^{\frac{1}{3}}$
$\displaystyle
a\left(\tau\right)=\frac{\sqrt{6}}{3^{\frac{2}{3}}}\left(c_{1}\tau+c_{4}\right)^{\frac{1}{2}}\left(c_{3}\tau+c_{4}\right)^{\frac{1}{6}}\;.$
(11.79)

For convenience, we can change variables from $b\left(\tau\right)$ to $r$
according to the transformation $b\left(\tau\right)=r$, where $r$ denotes the
radial variable. Inserting this into the above equations, we conclude that the
spacetime (11.28) in the coordinates $(t,r,\theta,\phi)$ can be written as

$ds^{2}=-A\left(r\right)dt^{2}+\frac{1}{c_{3}^{2}}\frac{1}{A\left(r\right)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (11.80)

with

$A\left(r\right)=\frac{2c_{1}}{3c_{3}}r^{2}-\frac{2c_{\mu}}{c_{3}r}=\lambda_{A}(1-\frac{r_{\star}}{r})R(r),$
(11.81)

and

$R(r)=\left(\frac{r}{r_{\star}}\right)^{2}+\frac{r}{r_{\star}}+1.$ (11.82)

In these expressions we have $c_{\mu}=c_{1}c_{4}-c_{2}c_{3}$,
$\lambda_{A}=\left(\frac{8c_{1}c_{\mu}^{2}}{3c_{3}^{3}}\right)^{1/3}$ and
$r_{\star}=(\frac{3c_{\mu}}{c_{1}})^{1/3}=(\frac{3c_{3}\lambda_{A}}{2c_{1}})^{1/2}$
is a characteristic radius with the restriction $c_{\mu}c_{1}>0$.

We observe that if we select the constant $c_{3}\equiv 1$ then we retain the
Schwarzschild-like metric. On the other hand, the function $R(r)$ can be
viewed as a distortion factor which quantifies the smooth deviation from the
pure Schwarzschild solution. Thus, the $f(T)$ gravity on small spherical
scales ($r\rightarrow r_{\star}^{+}$) tends to create a Schwarzschild
solution. In particular, the $f(T)$ spherical solution admits singularity only
at $r=r_{\star}$, which is also the case with the usual Schwarzschild
solution.

Within this framework, for commoving observers, $u^{i}u_{i}=A^{2}(r)$, it is
easy to show that the Einstein’s tensor becomes

$G_{j}^{i}=\mathrm{diag}\left(2c_{1}c_{3}-\frac{1}{r^{2}},2c_{1}c_{3}-\frac{1}{r^{2}},2c_{1}c_{3},2c_{1}c_{3}\right).$

Therefore, from the 1+3 decomposition of $G_{ij}$we define the fluid physical
parameters

$\displaystyle\rho_{T}=\frac{1}{\left(u^{i}u_{i}\right)}G_{ij}u^{i}u^{j}=2c_{1}c_{3}-\frac{1}{r^{2}}$
(11.85) $\displaystyle
p_{T}=\frac{1}{3}h_{ij}G_{ij}=2c_{1}c_{3}-\frac{1}{3r^{2}}$ $\displaystyle
q^{i}=h^{ij}G_{jk}u^{k}=0$ $\displaystyle\pi_{\theta}^{\theta}$
$\displaystyle=$
$\displaystyle\pi_{\phi}^{\phi}=-\frac{\pi_{r}^{r}}{2}=\frac{1}{3r^{2}}$
(11.86)

where

$\pi_{ij}=(h_{i}^{r}h_{j}^{s}-\frac{1}{3}h_{ij}h^{rs})G_{rs}$ (11.87)

is the anisotropic stress tensor and $h_{ij}$ is the $u^{i}$ projection tensor
defined by

$h_{ij}=g^{ij}-\frac{1}{\left(u^{i}u_{i}\right)}u^{i}u^{j}.$ (11.88)

Furthermore the fluid is also anisotropic $(\pi_{ij}\neq 0$ ) but not heat
conducting $(q^{i}=0).$

In order to apply the above considerations for specific $f(T)$ forms, we
consider the following viable $f(T)$ models, motivated by cosmology555The
$f(T)$ models of Refs.[188, 189] are consistent with the cosmological data.

  * •

Exponential $f(T)$ gravity [189]:

$f(T)=T+f_{0}e^{-f_{1}T},$

where $f_{0}$ and $f_{1}$ are the two model parameters which are connected via
(11.78)

$f_{0}=\frac{4c_{1}c_{3}}{8f_{1}c_{1}c_{3}+1}\exp\left(4f_{1}c_{1}c_{3}\right).$

  * •

A sum of two different power law $f(T)$ gravity:

$f(T)=T^{m}+f_{0}T^{n}$

where from (11.78) we have

$f_{0}=\frac{1-2m}{2n-1}\left(4c_{1}c_{3}\right)^{m-n}\;.$

Note that in the case of $m=1$ we recover the $f(T)$ model by Bengochea &
Ferraro [188].

###### $f(T)=T^{n}$

In the $f(T)=T^{n}$ case, the field equations (11.26), (11.27), (11.69) and
the torsion scalar (11.74) give rise to the following dynamical system:

$\displaystyle T=4x^{\prime}y^{\prime},$ (11.89) $\displaystyle
4nT^{n-1}x^{\prime}y^{\prime}-\left(1-n\right)T^{n}=0,$ (11.90) $\displaystyle
x^{\prime\prime}+\left(n-1\right)x^{\prime}T^{-1}T^{\prime}=0,$ (11.91)
$\displaystyle
y^{\prime\prime}+\left(n-1\right)y^{\prime}T^{-1}T^{\prime}=0\;.$ (11.92)

It is easy to show that combining equation (11.89) with the Hamiltonian
(11.90), we can impose constraints on the value of $n$, namely $n=1/2$. Under
this condition, solving the system of equations (11.91) and (11.92) we arrive
at the solutions

$\displaystyle x(\tau)=\frac{\sigma(\tau)^{3}}{3}+c_{\sigma}$ (11.93)
$\displaystyle y(\tau)=\frac{\sigma(\tau)^{3}}{3}$ (11.94)

where $c_{\sigma}$ is the integration constant. Now using (11.70) we derive
$a$,$b$ as

$\displaystyle b(\tau)=\sigma(\tau)$ (11.95) $\displaystyle
a(\tau)=\sqrt{\frac{2\left[\sigma^{3}(\tau)+3c_{\sigma}\right]}{3\sigma(\tau)}}.$
(11.96)

Using the coordinate transformation $\sigma(\tau)=r$, which implies
$\tau=F(r)$ [with $F(\sigma(\tau))=\tau$], and using simultaneously (11.95),
the spherical metric (11.28) can be written as

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+r^{2}\left(d\theta+\sin^{2}\theta
d\phi^{2}\right)\;$ (11.97)

where

$A(r)=\frac{2}{3}r^{2}+\frac{2c}{r}$ (11.98)

and

$B(r)=\frac{F_{,r}^{2}}{A(r)r^{4}}.$ (11.99)

Furthermore, considering the commoving observers
$\left(u^{i}u_{i}\right)=-\frac{2\left(r^{3}+3c_{\sigma}\right)}{3r}$, we can
write the Einstein tensor components as

$\displaystyle G_{t}^{t}$ $\displaystyle=$
$\displaystyle-\frac{r}{3F_{r}^{3}}\left[4rF_{,rr}\left(r^{3}+3c_{\sigma}\right)-2F_{,r}\left(7r^{3}+12c_{\sigma}\right)+\frac{3}{r^{3}}F_{,r}^{3}\right]$
$\displaystyle G_{r}^{r}$ $\displaystyle=$
$\displaystyle\frac{2r^{4}}{F_{,r}^{2}}-\frac{1}{r^{2}}$ $\displaystyle
G_{\theta}^{\theta}$ $\displaystyle=$ $\displaystyle
G_{\phi}^{\phi}=-\frac{1}{3}\frac{r}{F_{,r}^{3}}\left[rF_{,rr}\left(4r^{3}+3c_{\sigma}\right)-F_{,r}\left(14r^{3}+6c_{\sigma}\right)\right]$

where $F_{,r}=dF/dr$ and $F_{,rr}=d^{2}F/dr^{2}$.

Similarly, based on the equalities (11.85)-(11.87), we compute the following
fluid parameters

$\rho_{T}=\frac{4r^{2}F_{,rr}}{F_{,r}^{3}}\left(\frac{1}{3}r^{3}+c_{\sigma}\right)-\frac{2r}{F_{,r}^{2}}\left(\frac{7}{3}r^{3}-4c_{\sigma}\right)+\frac{1}{r^{2}}$
(11.100)
$p_{T}=-\frac{2}{3}\frac{r^{2}F_{,rr}}{F_{,r}^{3}}\left(\frac{4}{3}r^{3}+c_{\sigma}\right)+\frac{2}{3}\frac{r}{F_{,r}^{2}}\left(\frac{17}{3}r^{3}+2c_{\sigma}\right)-\frac{1}{3r^{2}}$
(11.101)
$\pi_{,r}^{r}=\frac{2}{3}\frac{r^{3}F_{,rr}}{F_{,r}^{2}}\left(\frac{4}{3}r^{3}+c_{\sigma}\right)-\frac{2}{3}\frac{1}{F_{,r}^{2}}\left(\frac{8}{3}r^{3}+2c_{\sigma}\right)-\frac{2}{3r^{2}}$
$\pi_{\theta}^{\theta}=\pi_{\phi}^{\phi}=-\frac{1}{2}\pi_{r}^{r}\;$ $q^{i}=0$

####  11.8 Conclusion

In this chapter we studied the Noether symmetries of $f\left(T\right)$
gravity. We proved that for some diagonal frames the Lagrangian of the field
equations admits Noether symmetries for arbitrary $f\left(T\right)$ function.
However, in the case of power law $f\left(T\right)$, i.e.
$f\left(T\right)=T^{n}$ it is possible the Lagrangian to admits extra Noether
symmetries. We applied this results in order to classify the Noether
symmetries of the field equations in a spatially flat FRW spacetime and in a
static spherical symmetric spacetime. For each background spacetime we found
analytical solutions of the field equations.

###  Chapter 12 Discussion

####  12.1 Discussion

In this thesis we study the Lie point symmetries and the Noether point
symmetries of second order differential equations usinng a geometric approach
and we apply the results to systems which are relevant to relativistic
physics.

In particular, we have studied the point symmetries of the equations of motion
of dynamical systems in a Riemannian space with Lagrangian

$L\left(x^{i},\dot{x}^{k}\right)=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}-V\left(x^{k}\right)$
(12.1)

where $g_{ij}=g_{ij}\left(x^{k}\right)$ is the metric of the space and we
proved that the Lie point symmetries of the Euler-Lagrange equations, i.e.
$E^{i}\left(L\right)=0,$ of Lagrangian (12.1) are generated from the elements
of the special Projective algebra of the Riemannian manifold with metric
$g_{ij}$ whereas the Noether point symmetries are generated from the
homothetic algebra of the space with metric $g_{ij}$. Therefore we have
transfer the problem of determination of the Lie/Noether symmetries of
differential equations to the determination of the collineations of the
underlying manifold; hence, we are able to use the plethora of existing
results of differential geometry.

We have applied this geometric approach in many directions. In particular, we
have classified the Lie and the Noether symmetries of the geodesic Lagrangian
for some important spacetimes, such as the FRW spacetime, the Gödel space, the
Taub space and the 1+3 decomposable spacetimes. Moreover we proved that for
Einstein spaces the point symmetries of the geodesic equations are generated
from the elements of the Killing algebra of the metric.

Furthermore we have determined all the two and the three dimensional Newtonian
systems which admit Lie and Noether point symmetries. We note that, due to the
geometric derivation and the tabular presentation, the results can be extended
easily to higher dimensional flat spaces. We applied these results in the
study of the symmetries of the Hènon - Heiles potential and of the Kepler-
Ermakov potential in a two dimensional space. Moreover, we determined the
potentials which admit Noether symmetries in a two dimensional sphere $S^{2}$.

We proved that a dynamical system admits as Lie symmetries the
$sl\left(2,R\right)$ Lie algebra if and only if the underlying manifold admits
a gradient Homothetic vector. The Newtonian system which is invariant under
the Lie group $sl\left(2,R\right)$ is the well known Kepler-Ermakov system.
Therefore, the requirement for a dynamical system with Lagrangian of the form
of (12.1) to admit as Lie and Noether symmetries the generators of the
$sl\left(2,R\right)$ Lie algebra leads us to the generalization of the
Newotian Kepler-Ermakov system in a Riemannian manifold; that is, we found
that the general autonomous Kepler-Ermakov system follows from the Lagrangian

$L\left(u,\dot{u},y^{A},\dot{y}^{A}\right)=\frac{1}{2}\left(\dot{u}^{2}+\dot{u}^{2}h_{AB}\dot{y}^{A}\dot{y}^{B}\right)+\frac{\mu^{2}}{2}u^{2}+\frac{1}{u^{2}}V\left(y^{C}\right)$
(12.2)

In additionally, we studied the Liouville integrability of the three
dimensional Newtonian Kepler-Ermakov via Noether point symmetries and we
investigated the application of Lagrangian (12.2) in dynamical systems
emerging from alternative theories of gravity. In particular we showed that
the field equations in a Bianchi I spacetime for an exponential scalar field
and for $f\left(R\right)$ gravity when
$f\left(R\right)=\left(R-2\Lambda\right)^{\frac{7}{8}}$ follow from
Lagrangians of the form of (12.2) and for these models we proved that the
field equations are Liouville integrable.

Concerning the second order partial differential equations we considered
equations of the generic form

$A^{ij}\left(x^{k}\right)u_{ij}-B^{i}(x^{k},u)u_{i}-f(x^{k},u)=0$ (12.3)

and we proved a theorem which relates the Lie pont symmetries of equation
(12.3) with the elements of the Conformal Killing vectors of the second order
tensor $A_{ij}\left(x^{k}\right)$ (considered to be a metric). We have applied
this result in order to study the Lie point symmetries of the Heat equation
and the Poisson equation. It has been shown that the Lie symmetries of the
Heat equation follow from the Killing and the homothetic algebras of
$A_{ij}\left(x^{k}\right),$ whereas the Lie symmetries of the Poisson equation
follow from the Killing, the homothetic and the conformal algebras of
$A_{ij}\left(x^{k}\right).$ In each case we have determoned the form of the
Lie symmetry vectors.

Furthermore, we have determined the Lie symmetries of the Schrödinger equation

$\Delta u-u_{,t}=V\left(x^{k}\right)u$ (12.4)

and the Klein Gordon equation

$\Delta u=V\left(x^{k}\right)u$ (12.5)

in a general Riemannian space. It has been shown that these symmetries are
related to the Noether symmetries of the classical Lagrangian for which the
metric $g_{ij}$ is the kinematic metric. More precisely, for the Schrödinger
equation (12.4) it has been shown that if a KV or a HV of the metric $g_{ij}$
produces a Lie symmetry for the Schrödinger equation, then it produces a
Noether symmetry for the Classical Lagrangian in the space with metric
$g_{ij}$ and potential $V(x^{k})$. For the Klein Gordon equation (12.5) the
situation is different; the Lie symmetries of the Klein Gordon are generated
by the elements of the conformal group of the metric $g_{ij}.$ The KVs and the
HV of this group produce a Noether symmetry of the classical Lagrangian with a
constant gauge function. However the proper CKVs produce a Noether symmetry
for the conformal Lagrangian if there exists a conformal factor
$N\left(x^{k}\right)$ such that the CKV becomes a KV/HV of $g_{ij}$.

We have applied these results to three cases of practical interest: the motion
in a central potential, the classification of all potentials in two and three
dimensional Euclidian spaces for which the Schrödinger equation and the Klein
Gordon equation admit a Lie symmetry and finally we have considered the Lie
symmetries of the Klein Gordon equation in the static, spherically symmetric
empty spacetime. In the last case, we have demonstrated the role of the Lie
symmetries and that of the conformal Lagrangians in the determination of the
closed form solution of Einstein equations. Furthermore, we investigated the
Lie point symmetries of the null Hamilton Jacobi equation and we proved that
if a CKV generates a point symmetry for the Klein Gordon equation, then it
also generates a point symmetry for the null Hamilton Jacobi equation.

We also studied the problem of Type II hidden symmetries of second order
partial differential equations in $n~{}$ dimensional Riemannian spaces from a
geometric of view. We have considered the reduction of the Laplace and of the
homogeneous heat equation and the consequent possibility of existence of Type
II hidden symmetries in some general classes of spaces which admit some kind
of symmetry; hence, they admit nontrivial Lie symmetries.

The Type II hidden symmetries of Laplace equation are directly related to the
transition of the CKVs from the space where the original equation is defined
to the space where the reduced equation resides. In this sense, we related the
Lie symmetries of PDEs with the basic collineations of the metric i.e. the
CKVs.

Concerning the Type II hidden symmetries of the homogeneous heat equation we
considered the problem in the spaces which admit a gradient KV or a gradient
HV and finally spacetime which admits a HV which acts simply and transitively.
For the reduction of the homogeneous heat equation and the existence of Type
II hidden symmetries, we found the following general geometric results: (a) If
we reduce the homogeneous heat equation via the symmetries which are generated
by a gradient KV $\left(S^{,i}\right)$ the reduced equation is a heat equation
in the nondecomposable space. In this case we have the Type II hidden symmetry
$\partial_{t}-\frac{1}{2t}w\partial_{w}$ provided if we reduce the heat
equation with the symmetry $tS^{,i}-\frac{1}{2}Su\partial_{u}$. (b) If we
reduce the homogeneous heat equation via the symmetries which are generated by
a gradient HV the reduced equation is Laplace equation for an appropriate
metric. In this case the Type II hidden symmetries are generated from the
proper CKVs and (c) in Petrov type III spacetime, the reduction of the
homogeneous heat equation via the symmetry generated from the nongradient HV
gives PDE that inherit the Lie symmetries, hence no Type II hidden symmetries
are admitted.

Finally, we applied the point symmetries and especially the Noether point
symmetries in modified theories of gravity in order to probe the nature of
dark energy. We used the Noether symmetries as a geometric criterion or
”selection rule” in order to select the scalar field potential in scalar-
tensor theories, and the functions $f\left(R\right)$ and $f\left(T\right)$ in
the corresponding alternative theories of gravity.

In the context of scalar-tensor cosmology we have found that to every non-
minimally coupled scalar field, we can associate a unique minimally coupled
scalar field in a conformally related space with an appropriate potential.
This result can be used in order to study the dynamical properties of the
various cosmological models, since the field equations of a non-minimally
coupled scalar field are the same, at the conformal level, with the field
equations of the minimally coupled scalar field. Furthermore, we have
identified the Noether point symmetries and the analytic solutions of the
equations of motion in the context of a minimally coupled and a non minimally
coupled scalar field in a FRW spacetime and we have classified the Noether
point symmetries of the field equations in Bianchi class A models with a
minimally coupled scalar field. We find that there is a rather large class of
hyperbolic and exponential potentials which admit extra (beyond the
$\partial_{t}$) Noether symmetries which lead to integral of motions. For
these potentials we used the corresponding Noether integrals in order to solve
analytically the field equations and find the functional form of the scalar
factor.

Concerning the $f\left(R\right)$ models we applied the Noether point
symmetries with the aim to utilize the existence of non-trivial Noether
symmetries as a selection criterion that can distinguish the $f(R)$ models on
a more fundamental level. We proved that in a spatially flat FRW background
the $f\left(R\right)$ theories which admit Noether point symmetries are
the$~{}R^{n},~{}R^{\frac{7}{8}},~{}\left(R-2\Lambda\right)^{\frac{7}{8}},~{}R^{\frac{3}{2}}~{}$and$~{}\left(R-2\Lambda\right)^{\frac{3}{2}}$.
The last two functional forms of the $f\left(R\right)$ function admit Noether
point symmetries also in the case of a non spatially flat FRW background. It
is interesting to note that the $\frac{7}{8}$ models are equivalent with the
Newtonian Kepler-Ermakov system whereas the $\frac{3}{2}$ models are
equivalent with the anisotropic hyperbolic oscillator. For these functional
forms we use the Noether integrals in order to find exact solutions of the
modified field equations.

Furthermore, in $f\left(T\right)$ gravity we have proved a Lemma that for
diagonal frames the only functional form of $f\left(T\right)$ which admits
extra Noether symmetries is the $f\left(T\right)=T^{n}$. We applied this
functional form in a spatially flat FRW background and we determined the
analytic solution of the field equations for each case. Finally, we studied
the field equations for the $f\left(T\right)=T^{n}$ model in a static
spherically symmetric spacetime and we determined a family of analytic
solutions.

The geometric approach is a new method for studying the symmetries of
differential equations and has shown that gives directly results by using only
the results (usually existing) of differential geometry without the need to
use of a computer library in order to determine the Lie or the Noether point
symmetries. This approach implies a better understanding of the nature of
symmetries and of the conservation laws and can be used in order to find
analogues of classical Newtonian systems in relativistic physics. It is of
interest that this method would be extendented in other classes of
differential equations and in other transformations which are not necessary
point transformations. Concerning the applications in Cosmology, it is of
interest the classification of modified theories of gravity with geometric
selection rules. In this thesis we studied some of the basic modified theories
of gravity; however there are other more such theories which could be studied
further either by means of point symmetries or by some new geometric criteria.

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