# Noether symmetries of the minimal surface Lagrangian for Gödel-type
spacetimes

U. Camci

###### Abstract

In this paper, we investigate the Noether symmetries of the minimal surface
Lagrangian for four classes of metrics in Gödel-type spacetimes. We calculate
the Noether symmetries for all classes, namely, classes I, II, III and IV.
Additionally, we determine the conserved fields corresponding to each classes,
allowing us to derive a comprehensive characterization of the minimal surface
equations for Gödel-type spacetimes.

##  1 Introduction

Gödel metric [1] which describes a homogeneous rotating universe containing
closed time-like curves (CTCs), is one of the most interesting and well-known
exact solution of Einstein’s field equations. This spacetime metric is the
best known example of causality violated universe model [2]. The rotational
symmetry of Gödel’s metric comes from the existence of CTCs corresponding to
circular orbits in specific coordinates. Furthermore these circular orbits
have discussed by Raychaudhuri and Thakurta [3]. In Refs. [4, 5], the authors
have pointed out that the causality features of the Gödel-type spacetimes are
related with two independent parameters: $m$ and $\omega$. They have shown
that there exists only one non-causal region if $0\leq m^{2}<4\omega^{2}$, and
there is no CTCs if $m^{2}\geq 4\omega^{2}$, in which the limiting case
$m^{2}=4\omega^{2}$ yields a completely causal and spacetime homogeneous Gödel
model. For $m^{2}<0$, there exists an infinite number of alternating causal
and noncausal regions. The Gödel metrics are mainly interesting for their high
degree of symmetry [6, 7, 8]. Since there exists a timelike Killing vector,
both the original Gödel metric and Gödel-type metrics are stationary. However,
a more detailed analysis of the Killing vectors shows, that those spacetimes
are not static and also not isotropic which is a feature that is due to the
existence of a rotational axis giving rise to a rotational symmetry in the
planes of constant $z$. All classes of Gödel-type spacetimes admit at least a
$G_{5}$ group of motions. It has been shown that the group of motions is
$G_{7}$, a maximal symmetry group of Gödel-type spacetimes [6], in a special
case $m^{2}=4\omega^{2}$.

To solve the Euler-Lagrange (EL) equations, symmetries of the Lagrangian
corresponding EL equations play an important role to find simple expressions
for conservation laws. The Noether symmetries are associated with differential
equations possessing a Lagrangian, and they are symmetries of the action
integral. So, these symmetries concern all problems involving an action
integral even if these problems do not involve the equations of motion. The
geodesic equations of motion are the EL equations for arc length minimizing
action. Therefore, they are important to study the evaluation of dynamical
system [9]. In the case of determination of the minimal surface area under
constant volume in a given Riemannian space, the action integral involves the
minimization of a surface and not of an arc length, as is the case with
geodesic equations of motion. Tsamparlis et al.[10] have applied the Noether
symmetry approach to a general Euclidian space, spaces of constant curvature
and to Schwarzschild spacetime, and also showed how the Noether symmetries of
minimal surface Lagrangian can be used to reduce the minimal surface equation
form a partial differential equation (PDE) to an ordinary differential
equation (ODE) in the Friedmann-Robertson-Walker (FRW) spacetime with dust
source. In Ref.[11] the authors found the Noether symmetries of minimal
surfaces Lagrangian with fixed volume for plane-fronted gravitational wave
(pp-wave) spacetimes. The Noether symmetry algebras admitted by wave equations
on pp-waves with parallel rays are determined in Ref. [12]. Noether symmetry
analysis of some special classes of FRW universe and nonlinear wave equations
in this geometry are performed by Camci _et al._ [13]. It has been examined
the the gravitational coupling of Klein-Gordon and Dirac fields to matter
vorticity and spacetime torsion, in the background of Gödel-type geometries
[14, 15], which admits the five Killing vectors.

The geodesic equations of motion for a space are important in the study of the
evaluation of dynamical systems because of that the geometry of the background
space determines the kinematics of the dynamical system in a unique manner.
The symmetries of the corresponding EL equation is called Lie symmetries.
Noether symmetries are also Lie symmetries, but the converse is not true in
general. The relation of both the Lie and Noether symmetries of the geodesic
equations for some spacetimes have been discussed by several authors [16, 17,
18]. The Noether symmetries are more important than Lie symmetries as it gives
double reduction of the differential equations and provides conserved
quantities. The Noether symmetries can be used to reduce the order of the ODEs
by considering the first integrals or the number of variable providing
conserved quantities in the case of PDEs. In addition, it may be possible that
one can linearize the nonlinear differential equations by means of the Noether
equations. In recent years it has been provided a classification according to
their geodesic Lagrangian considering Noether symmetry approach in the
background of FRW [16], Bertotti-Robinson like [19, 20, 21], plane symmetric
static [22] and pp-wave [18, 23] spacetimes. The geodesic equations of motion
for the general cylindrically symmetric stationary spacetimes together with
their Dirac’s constraint analysis and symplectic structure have been obtained,
and integrated in Ref. [24]. The geodesic equations of motion in Gödel-type
spacetimes have been analyzed by several authors. Firstly, Kunt [25] was
solved the geodesic equations for Gödel’s metric, where it was used the
Killing vectors and corresponding constants of motion. Later, Chandrasekhar
and Wright [26] presented an independent derivation of the solution for the
geodesic equations of Gödel’s metric. A detailed study on geodesic motion in
the original Gödel’s universe has been provided by Novello _et al._ [27]. The
geodesic equations for the special case $m^{2}=4\omega^{2}$ with seven
isometries have been integrated by Rebouças and Teixeira [28]. The geodesics
of the Som-Raychaudhuri spacetime [29] have examined by Paiva _et. al._ [30].
Grave _et al._ [31] derived the analytical solution of the geodesic equations
of Gödel’s universe for both particles and light, in which they have
generalized the work of Kajari _et. al._ [32] on the solution of lightlike
geodesic equations. Afterwards, Dautcourt [33] studied the lightcone of the
Gödel-type metrics by considering only the lightlike case. It is given a
complete discussion for timelike and null geodesics of Gödel-type spacetimes
by Calvão _et al._ [34] and Gleiser _et al._ [35]. Recently, Camci [36, 37]
calculated the Noether symmetries for four classes of Gödel-type metrics, and
explicitly integrated the geodesic equations of motion by using the first
integrals of corresponding classes. Furthermore, some spacetime symmetry
properties of the original Gödel metric and Gödel-type spacetimes have been
discussed in Refs.[38]-[41].

This study is designed as follows. In the following section, we give a short
review about Gödel-type spacetimes and their properties. In section 3, we
present briefly the Noether symmetry approach for the minimal surface
Lagrangian under constant volume and then apply this approach to the minimal
surface Lagrangian of Gödel-type spacetimes. In section 4, we give solution of
Noether symmetry equations and of the equations following from the
conservation relation in each detail. Finally, our conclusion with a brief
summary and discussions of finding is presented in Section 5.

##  2 Gödel-type Spacetimes

The line element for the Gödel-type spacetimes in cylindrical coordinates
$x^{a}=\\{t,r,\phi,z\\}$, $a=0,1,2,3$, can be written as [2, 3]

$ds^{2}=\left[dt+H(r)d\phi\right]^{2}-dr^{2}-D^{2}(r)d\phi^{2}-dz^{2}.$ (2.1)

It is found that the necessary and sufficient conditions for a Gödel-type
manifold to be spacetime homogeneous (STH, hereafter) are [4, 5, 6, 7]

$\displaystyle\frac{D^{\prime\prime}}{D}=const\equiv m^{2},$ (2.2)
$\displaystyle\frac{H^{\prime}}{D}=const\equiv-2\omega$ (2.3)

where prime denotes derivative with respect to the radial coordinate $r$.
Throughout this paper we have used the following property

$D^{2}\left(\frac{D^{\prime}}{D}\right)^{\prime}=-1,$ (2.4)

which is valid for STH Gödel-type spacetimes only. The four-dimensional
homogeneous Riemannian Gödel-type manifolds are locally characterized by two
independent parameters $m^{2}$ and $\omega$: the pair of ($m^{2},\omega$)
identically specify locally equivalent manifolds. All STH Riemannian manifolds
endowed with a Gödel-type spacetime (2.1) are listed in Table 1. We note that
the scalar curvature $R$ of the Gödel-type spacetimes becomes
$R=2(\omega^{2}-m^{2})$. Furthermore, if $m^{2}=\omega=0$, then the line
element (2.1) is clearly Minkowskian. Therefore, this particular case has not
been included in this study.

| $H(r)$ | $D(r)$ |  $m,\quad\omega$
---|---|---|---
Class I | $\quad\frac{2\omega}{m^{2}}\left[1-\cosh(mr)\right]\quad$ | $\quad\frac{1}{m}\sinh(mr)\quad$ | $\qquad m^{2}>0,\quad\omega\neq 0$
Class II | $\quad-\omega r^{2}\quad$ | $\quad r\quad$ | $\qquad m=0,\quad\omega\neq 0$
Class III | $\frac{2\omega}{\mu^{2}}\left[\cos(\mu r)-1\right]$ | $\quad\frac{1}{\mu}\sin(\mu r)\quad$ | $\qquad m^{2}\equiv-\mu^{2}<0,\,\,\mu^{2}>0,\quad\omega\neq 0$
Class IV | $0$ | $\quad\frac{1}{m}\sinh(mr)\quad$ | $\qquad m^{2}>0,\quad\omega=0$
| $0$ | $\quad\frac{1}{\mu}\sin(\mu r)\quad$ | $\qquad m^{2}\equiv-\mu^{2}<0,\quad\omega=0$
Table 1: A list of all STH Rimannian manifolds endowed with a Gödel-type
spacetime (2.1), where the general solution of Eqs. (2.2) and (2.3) is written
in each class. It is referred to the manifolds of Class IV as degenerated
Gödel-type manifolds, since the cross term in the line element, related to the
rotation $\omega$ in the Gödel model, vanishes which means that we can make
$H(r)=0$ by a trivial coordinate transformation.

The group of conformal motions generated by a conformal Killing vector (CKV)
field ${\bf K}$ is defined by $\pounds_{\bf K}g_{ab}=2\psi g_{ab}$, where
$\pounds_{\bf K}$ is the Lie derivative operator along the vector field ${\bf
K}$, and $\psi=\psi(x^{a})$ is a conformal factor. The vector field ${\bf K}$
is an isometry or a Killing vector (KV) field if $\psi=0$, and a homothetic
vector (HV) if $\psi_{,a}=0$. If $\psi_{;ab}\neq 0$, then the CKV field ${\bf
K}$ is said to be proper, otherwise it is a special conformal Killing vector
(SCKV) field when $\psi_{;ab}=0$. The set of all CKV (respectively SCKV, HKV
and KV) form a finite-dimensional Lie algebra denoted by $\mathcal{C}$
(respectively $\mathcal{S},\mathcal{H}$ and $\mathcal{G}$).

It is proved by Rebouças _et al._ [8] that the four-dimensional homogeneous
Riemannian Gödel-type manifolds admit a group of isometry $G_{r}$ with

(i) $r=5$ in classes I (where $m^{2}\neq 4w^{2}$), II and III;

(ii) $r=6$ in class IV;

(iii) $r=7$ in the special case of class I, where $m^{2}=4\omega^{2}$.
The KV fields of the classes I-IV of STH Gödel-type spacetimes (2.1) have been
determined as follows. In case (i), the five KVs ${\bf K}_{1},...,{\bf K}_{5}$
for class I are obtained as

$\displaystyle{\bf K}_{1}=\partial_{t},\quad{\bf K}_{2}=\partial_{z},\quad{\bf
K}_{3}=\frac{2\omega}{m}\partial_{t}-m\partial_{\phi},$ $\displaystyle{\bf
K}_{4}=-\frac{H}{D}\sin\phi\partial_{t}+\cos\phi\partial_{r}-\frac{D^{\prime}}{D}\sin\phi\partial_{\phi},$
(2.5) $\displaystyle{\bf
K}_{5}=-\frac{H}{D}\cos\phi\partial_{t}-\sin\phi\partial_{r}-\frac{D^{\prime}}{D}\cos\phi\partial_{\phi},$

For class III in case (i), where $m^{2}\equiv-\mu^{2}<0,\mu^{2}>0$ and
$\omega\neq 0$, it follows that the KVs ${\bf K}_{1},{\bf K}_{2},{\bf
K}_{4},{\bf K}_{5}$ are the same form as given the above, but only ${\bf
K}_{3}$ has the form $(2\omega/\mu)\partial_{t}+\mu\partial_{\phi}$. For class
II of case (i), where $H(r)=-\omega r^{2}$ and $D(r)=r$, the five KVs ${\bf
K}_{1},...,{\bf K}_{5}$ are given by

$\displaystyle{\bf K}_{1}=\partial_{t},\quad{\bf K}_{2}=\partial_{z},\quad{\bf
K}_{3}=\partial_{\phi},\,\,{\bf
K}_{4}=-\omega\,r\,\sin\phi\partial_{t}-\cos\phi\partial_{r}+\frac{1}{r}\sin\phi\partial_{\phi},$
$\displaystyle{\bf
K}_{5}=-\omega\,r\,\cos\phi\partial_{t}+\sin\phi\partial_{r}+\frac{1}{r}\cos\phi\partial_{\phi}\,.$
(2.6)

For class IV in case (ii), the six KVs ${\bf K}_{1},...,{\bf K}_{6}$ are

$\displaystyle{\bf K}_{1}=\partial_{t},\quad{\bf K}_{2}=\partial_{z},\quad{\bf
K}_{3}=z\,\partial_{t}+t\partial_{z},\quad{\bf
K}_{4}=\cos\phi\partial_{r}-\frac{D^{\prime}}{D}\sin\phi\partial_{\phi},$
$\displaystyle{\bf
K}_{5}=-\sin\phi\partial_{r}-\frac{D^{\prime}}{D}\cos\phi\partial_{\phi},\quad{\bf
K}_{6}=\partial_{\phi}\,.$ (2.7)

Finally, for the special class I in case (iii), where $m^{2}=4\,\omega^{2}$,
i.e. $m=+2\,\omega$, the seven KVs ${\bf K}_{1},...,{\bf K}_{7}$ are given by

$\displaystyle{\bf K}_{1}=\partial_{t},\quad{\bf K}_{2}=\partial_{z},\quad{\bf
K}_{3}=\partial_{t}-m\partial_{\phi},$ $\displaystyle{\bf
K}_{4}=-\frac{H}{D}\sin\phi\partial_{t}+\cos\phi\partial_{r}-\frac{D^{\prime}}{D}\sin\phi\partial_{\phi},$
$\displaystyle{\bf
K}_{5}=-\frac{H}{D}\cos\phi\partial_{t}-\sin\phi\partial_{r}-\frac{D^{\prime}}{D}\cos\phi\partial_{\phi},$
(2.8) $\displaystyle{\bf
K}_{6}=-\frac{H}{D}\cos(mt+\phi)\partial_{t}+\sin(mt+\phi)\partial_{r}+\frac{1}{D}\cos(mt+\phi)\partial_{\phi},$
$\displaystyle{\bf
K}_{7}=-\frac{H}{D}\sin(mt+\phi)\partial_{t}-\cos(mt+\phi)\partial_{r}+\frac{1}{D}\sin(mt+\phi)\partial_{\phi}\,.$

It is known that both the original Gödel metric and the STH Gödel-type
spacetimes does not admit HVs [38, 39]. The proper CKVs and complete conformal
algebra of a Gödel-type spacetime have been computed in Ref. [40]. The Ricci
collineations and the matter collineations of STH Gödel-type spacetimes have
been obtained in Refs. [39] and [41], respectively. In this work, we aim to
give a complete classification for STH Gödel-type spacetimes according to the
Noether symmetries of their minimal surface Lagrangian.

##  3 Noether Symmetries of the Minimal Surface Lagrangian

First, we will explain Noether symmetry approach for a first-order Lagrangian
$L(x^{i},q^{\alpha},q^{\alpha}_{i})$ as the following. Let $x^{i}$ and
$q^{\alpha}$ be $n$-independent and $m$-dependent variables, respectively. The
total derivative operator is given by

$D_{i}=\partial_{x^{i}}+q^{\alpha}_{i}\partial_{q^{\alpha}}+q^{\alpha}_{ij}\partial_{q^{\alpha}_{j}}+...,$
(3.1)

where
$q^{\alpha}_{i}=D_{i}(q^{\alpha}),q^{\alpha}_{ij}=D_{i}D_{j}(q^{\alpha}),...$
represent the derivatives of $q^{\alpha}$ with respect to $x^{i}$. Then the
Euler-Lagrange equations corresponding to the Lagrangian
$L(x^{i},q^{\alpha},q^{\alpha}_{i})$ are

$\frac{\partial L}{\partial q^{\alpha}}-\frac{\partial}{\partial
x^{i}}\left(\frac{\partial L}{\partial q^{\alpha}_{i}}\right)=0,$ (3.2)

where $i,j,...=1,...,n$ and $\alpha,\beta,...=1,...,m$. The infinitesimal
symmetry generator

${\bf
X}=\xi^{i}(x^{k},q^{\beta})\partial_{x^{i}}+\eta^{\alpha}(x^{k},q^{\beta})\partial_{q^{\alpha}}$
(3.3)

is called a Noether symmetry (NS) of the Lagrangian $L$ if there exits a
vector-valued gauge function ${\bf A}=A^{i}(x^{k},q^{\beta})\partial_{x^{i}}$
such that the following condition is satisfied,

$\displaystyle{\bf X}^{[1]}L+LD_{i}(\xi^{i})=D_{i}(A^{i}),$ (3.4)

where ${\bf X}^{[1]}$ is the first-order prolongation vector field defined by
${\bf X}^{[1]}={\bf X}+\eta^{\alpha}_{i}\partial_{q^{\alpha}_{i}}$ with
$\eta^{\alpha}_{i}=D_{i}\eta^{\alpha}-q^{\alpha}_{j}D_{i}(\xi^{j})$. The
components of the conserved vector field ${\bf T}=T^{i}\partial_{x^{i}}$ which
is also called Noether flow satisfying $D_{i}T^{i}=0$, is given by [42]

$\displaystyle
T^{i}=\xi^{i}L+\left(\eta^{\alpha}-\xi^{j}q^{\alpha}_{j}\right)\frac{\partial
L}{\partial q^{\alpha}_{i}}-A^{i}\,.$ (3.5)

Now, we will determine the Noether symmetries of the minimal surface
Lagrangian $L$ for the Gödel-type spacetime by applying Noether symmetry
approach summarized above. Thus, after a simple rearrangement, one can also
write the Gödel-type spacetime (2.1) as

$ds^{2}=h_{ij}dx^{i}dx^{j}-dz^{2}\,,$ (3.6)

where $i,j,...=1,2,3$, and the metric $h_{ij}$ is of the form

$h_{ij}=\left(\begin{array}[]{ccc}1&\,\ 0&H\\\ 0&-1&0\\\ H&\,\ 0&\,\
H^{2}-D^{2}\\\ \end{array}\right)$ (3.7)

which yields $h=\det(h_{ij})=D^{2}$. Therefore, the number of independent and
dependent variables are $n=3$ and $m=1$, respectively, that means
$x^{i}=\\{t,r,\phi\\}$ and $q=\\{z=f(t,r,\phi)\\}$. We note that the metric
$h_{ij}$ is $(1+3)$-decomposable because it is not a function of
$z=f(t,r,\phi)$, which gives rise to the metric (3.6) that admits the gradient
KV $\partial_{z}$. The Lagrangian of minimal surface for a 3-area enclosing a
constant volume is given by [10]

$\displaystyle
L=\sqrt{|h|+|h|h^{ij}f_{,i}f_{,j}}+\lambda\int{\sqrt{|h|}}df\,,$ (3.8)

which gives

$\displaystyle
L=\sqrt{D^{2}+(D^{2}-H^{2})f_{,t}^{2}+2Hf_{,t}f_{,\phi}-D^{2}f_{,r}^{2}-f_{,\phi}^{2}}+\lambda\,Df(t,r,\phi)\,,$
(3.9)

for the Gödel-type spacetime (3.6). The minimal surface equation for this
spacetime which results from the Lagrangian (3.9) is a second-order partial
differential equation in which independent variables are
$x^{i}=\\{t,r,\phi\\}$. Then, one may obtain the Euler-Lagrange equation of
minimal surface for the Lagrangian (3.9) by using (3.2) as

$\displaystyle(D^{2}-H^{2})\left(1-f^{2}_{,r}\right)f_{,tt}+2\left[\left(1-f^{2}_{,r}\right)H+f_{,t}f_{,\phi}\right]f_{,t\phi}$
$\displaystyle\,\,-\left(1+f^{2}_{,t}-f^{2}_{,r}\right)f_{,\phi\phi}-\left[D^{2}\left(1+f_{,t}^{2}\right)+2Hf_{,t}f_{,\phi}-H^{2}f^{2}_{,t}-f^{2}_{,\phi}\right]f_{,rr}$
$\displaystyle\,\,+2f_{,r}\left[(D^{2}-H^{2})f_{,t}+Hf_{,\phi}\right]f_{,tr}+2f_{,r}(Hf_{,t}-f_{,\phi})f_{,r\phi}-\frac{\lambda}{D}F^{3}$
$\displaystyle\,\,+f_{,r}\left[\left(DD^{\prime}(3-\frac{2H^{2}}{D^{2}})-HH^{\prime}\right)f^{2}_{,t}+H\left(\frac{H^{\prime}}{H}+\frac{4D^{\prime}}{D}\right)f_{,t}f_{,\phi}-DD^{\prime}\left(1+3f^{2}_{,r}\right)\right]=0,\quad\qquad$
(3.10)

where $F$ is defined by

$F\equiv\sqrt{D^{2}+(D^{2}-H^{2})f_{,t}^{2}+2Hf_{,t}f_{,\phi}-D^{2}f_{,r}^{2}-f_{,\phi}^{2}}\,.$
(3.11)

Using this definition, the Lagrangian (3.9) becomes
$L=F+\lambda\,Df(t,r,\phi)$. Introducing the momenta $P^{i}=\frac{\partial
L}{\partial f_{,i}}$, we have

$P^{t}=\frac{1}{F}\left[(D^{2}-H^{2})f_{,t}+Hf_{,\phi}\right]\,,\quad
P^{r}=-\frac{D^{2}}{F}f_{,r}\,\,,\quad
P^{\phi}=\frac{1}{F}\left(Hf_{,t}-f_{,\phi}\right)\,.$ (3.12)

One can define the Hamiltonian function corresponding to the minimal surface
Lagrangian such as $\mathcal{H}_{L}=f_{,i}P^{i}-L$, which becomes

$\mathcal{H}_{L}=-D\left(\frac{D}{F}+\lambda\,f\right)\,,$ (3.13)

by using the components of the momenta $P^{i}$ in (3.12).

Now, let us consider the Noether symmetry generator

${\bf
X}=\eta\,\partial_{f}+\xi^{1}\partial_{t}+\xi^{2}\partial_{r}+\xi^{3}\partial_{\phi}$
(3.14)

where $\eta,\xi^{1},\xi^{2}$ and $\xi^{3}$ are depend on
$x^{i}=\\{t,r,\phi\\}$ and $z=f(t,r,\phi)$. The first extension of the above
vector field is given by

${\bf X^{[1]}}={\bf
X}+\eta^{\prime}_{t}\,\partial_{f_{,t}}+\eta^{\prime}_{r}\,\partial_{f_{,r}}+\eta^{\prime}_{\phi}\,\partial_{f_{,\phi}}\,,$
(3.15)

in which $\eta^{\prime}_{t}=D_{t}\eta-f_{,j}D_{t}\xi^{j}$,
$\eta^{\prime}_{r}=D_{r}\eta-f_{,j}D_{r}\xi^{j}$ and
$\eta^{\prime}_{\phi}=D_{\phi}\eta-f_{,j}D_{\phi}\xi^{j}$, where $D_{i}$’s are
the operator of total differentiation with respect to $x^{i}$ such that
$D_{t}=\partial_{t}+f_{,t}\partial_{f}$,
$D_{r}=\partial_{r}+f_{,r}\partial_{f}$ and
$D_{\phi}=\partial_{\phi}+f_{,\phi}\partial_{f}$. Then, it follows from Eq.
(3.4) that for the minimal surface Lagrangian (3.9), the Noether symmetry
equations yield _14_ partial differential equations:

$\displaystyle\lambda D(\xi^{1}+f\xi^{1}_{,f})-A^{1}_{,f}=0,\quad\lambda
D(\xi^{2}+f\xi^{2}_{,f})-A^{2}_{,f}=0,\quad\lambda
D(\xi^{3}+f\xi^{3}_{,f})-A^{3}_{,f}=0,$
$\displaystyle(D^{2}-H^{2})\eta_{,t}+H\eta_{,\phi}+D^{2}\xi^{1}_{,f}=0,\quad\eta_{,r}-\xi^{2}_{,f}=0,\quad
H\eta_{,t}-\eta_{,\phi}+D^{2}\xi^{3}_{,f}=0,$
$\displaystyle\frac{D^{\prime}}{D}\xi^{2}+\eta_{,f}+\xi^{1}_{,t}+\xi^{3}_{,\phi}=0,\quad\eta_{,f}-\xi^{2}_{,r}=0\,,\quad\lambda
D\eta-A^{1}_{,t}-A^{2}_{,r}-A^{3}_{,\phi}=0\,,$ (3.16)
$\displaystyle\frac{D^{\prime}}{D}\xi^{2}-\xi^{2}_{,r}-H\xi^{3}_{,t}+\xi^{3}_{,\phi}=0,\,\,D^{2}\xi^{1}_{,r}-(D^{2}-H^{2})\xi^{2}_{,t}-H\xi^{2}_{,\phi}=0\,,\,\,\xi^{2}_{,\phi}-H\xi^{2}_{,t}+D^{2}\xi^{3}_{,r}=0,$
$\displaystyle\left(\frac{H^{\prime}}{H}-\frac{2D^{\prime}}{D}\right)\xi^{2}+2\xi^{2}_{,r}-\xi^{1}_{,t}+\frac{1}{H}\xi^{1}_{,\phi}-\xi^{3}_{,\phi}-\frac{(D^{2}-H^{2})}{H}\xi^{3}_{,t}=0,$
$\displaystyle
H^{2}\left(\frac{D^{\prime}}{D}-\frac{H^{\prime}}{H}\right)\xi^{2}+(D^{2}-H^{2})(\xi^{2}_{,r}-\xi^{1}_{,t})-H\xi^{1}_{,\phi}=0,$

where the subscripts with comma denotes partial derivatives. It is noted here
that the set of all NSs form a finite dimensional Lie algebra denoted by
$\mathcal{N}$. The significance of NS is clearly comes from the fact that if
${\bf X}$ is the Noether symmetry corresponding to the Lagrangian
$L(t,r,\phi,f,f_{,t},f_{,r},f_{,\phi})$ given by (3.9), then

$T^{i}=\xi^{i}L+\left(\eta-\xi^{j}f_{,j}\right)\frac{\partial L}{\partial
f_{,i}}-A^{i}\,,$ (3.17)

is a _vector-valued quantity_ associated with the vector field ${\bf X}$. Then
the Noether current ${\bf
T}=T^{1}\partial_{t}+T^{2}\partial_{r}+T^{3}\partial_{\phi}$, where the
components satisfies the relation

$D_{i}T^{i}=0\quad\Longleftrightarrow\quad
D_{t}T^{1}+D_{r}T^{2}+D_{\phi}T^{3}=0\,,$ (3.18)

is conserved along with the solutions of minimal surface equation (3.10).
Corresponding to each of ${\bf X}$, there exists a conserved vector ${\bf T}$
and then one can write the components of the Noether conserved vector from
(3.17) as

$\displaystyle T^{1}=\xi^{1}L+\left(\eta-
f_{,t}\xi^{1}-f_{,r}\xi^{2}-f_{,\phi}\xi^{3}\right)\frac{\partial L}{\partial
f_{,t}}-A^{1}\,,$ $\displaystyle T^{2}=\xi^{2}L+\left(\eta-
f_{,t}\xi^{1}-f_{,r}\xi^{2}-f_{,\phi}\xi^{3}\right)\frac{\partial L}{\partial
f_{,r}}-A^{2}\,,$ (3.19) $\displaystyle T^{3}=\xi^{3}L+\left(\eta-
f_{,t}\xi^{1}-f_{,r}\xi^{2}-f_{,\phi}\xi^{3}\right)\frac{\partial L}{\partial
f_{,\phi}}-A^{3}\,.$

Recently the Noether symmetries of minimal surface Lagrangian for some
spacetimes have been calculated, and classified according to their symmetry
generators [9, 10, 11]. In this study, we investigate the NSs of minimal
surface Lagrangian for the Gödel-type spacetimes. The general solution to the
above NS equations is introduced in the next section for each classes I-IV of
the Gödel-type spacetimes.

##  4 The Solution of Noether Symmetry Equations

After some algebra, we have calculated the general solution to Eqs. (3.16) in
order to get NSs of minimal surface Lagrangian for each of the classes I, II,
III and IV as the following.

###  4.1 Classes I, II and III

The components of NS vector field ${\bf X}$ and the vector-valued gauge
function ${\bf A}$ for classes I and III with the condition
$m^{2}\neq\omega^{2}$ are found as

$\displaystyle\eta=c_{1},\quad\xi^{1}=\frac{H}{D}(c_{2}\cos\phi-
c_{3}\sin\phi)+c_{4},\,\,$
$\displaystyle\xi^{2}=c_{2}\sin\phi+c_{3}\cos\phi,\quad\xi^{3}=\frac{D^{\prime}}{D}(c_{2}\cos\phi-
c_{3}\sin\phi)+c_{5},$ $\displaystyle A^{1}=\lambda\,f\left[H\,(c_{2}\cos\phi-
c_{3}\sin\phi)+c_{4}D\right]+\Phi^{1}(t,r,\phi),$ (4.1) $\displaystyle
A^{2}=\lambda\,f\,D\,(c_{2}\sin\phi+c_{3}\cos\phi)+\Phi^{2}(t,r,\phi),$
$\displaystyle A^{3}=c_{1}\lambda\phi
D+\lambda\,f\left[D^{\prime}(c_{2}\cos\phi-
c_{3}\sin\phi)+c_{5}D\right]-\int{(\Phi^{1}_{,t}+\Phi^{2}_{,r})d\phi}+\Phi^{3}(t,r),$

where $c_{1},...,c_{5}$ are constant parameters, $\Phi^{i}$’s are integration
functions. Hence, we have five NSs for classes I and III, which can be stated
by means of the KVs given in (2.5) for class I such as

${\bf X}_{1}={\bf K}_{1}\,,\quad{\bf X}_{2}={\bf K}_{2}\,,\quad{\bf
X}_{3}=\partial_{\phi}=\frac{2\omega}{m^{2}}{\bf K}_{1}-\frac{1}{m}{\bf
K}_{3}\,,\quad{\bf X}_{4}={\bf K}_{4}\,,\quad{\bf X}_{5}={\bf K}_{5}\,,$ (4.2)

together with the corresponding gauge vectors

$\displaystyle{\bf A}_{1}=\lambda\,fD\,{\bf X}_{1}\,,\,\,{\bf
A}_{2}=\lambda\phi D\,{\bf X}_{3}\,,\,\,{\bf A}_{3}=\lambda\,fD\,{\bf
X}_{3}\,,\,\,{\bf A}_{4}=\lambda\,fD\,{\bf X}_{4}\,,\,\,{\bf
A}_{5}=\lambda\,fD{\bf X}_{5}\,,\qquad$ (4.3)

where ${\bf K}_{1},...,{\bf K}_{5}$ are the KVs for classes I and III given in
(2.5). Also, it is seen from (4.1) that for any NS of these classes there
exists a generic gauge vector functions such as

${\bf
A}_{0}=\Phi^{1}(t,r,\phi)\partial_{t}+\Phi^{2}(t,r,\phi)\partial_{r}+\left[\Phi^{3}(t,r)-\int{(\Phi^{1}_{,t}+\Phi^{2}_{,r})d\phi}\right]\partial_{\phi}\,.$
(4.4)

In class III, the NSs ${\bf X}_{1},{\bf X}_{2},{\bf X}_{4}$ and ${\bf X}_{5}$
are the same form as for class I described in (4.2), but the NS ${\bf X}_{3}$
becomes ${\bf X}_{3}=\partial_{\phi}=-(2\omega/\mu^{2}){\bf K}_{1}+(1/\mu){\bf
K}_{3}$. The Lie algebra of the NSs (4.2) has the following non-vanishing
commutators:

$\displaystyle\left[{\bf X}_{3},{\bf X}_{4}\right]={\bf X}_{5},\quad\left[{\bf
X}_{3},{\bf X}_{5}\right]=-{\bf X}_{4},\quad\left[{\bf X}_{4},{\bf
X}_{5}\right]=2\omega{\bf X}_{1}-m^{2}{\bf X}_{3}.$

The conserved vector fields following from Eq.(3.17) for the Noether
symmetries ${\bf X}_{1},...,{\bf X}_{5}$ of classes I and III read,
respectively,

$\displaystyle{\bf T}_{1}=F\,{\bf X}_{1}-\frac{f_{,t}}{F}{\bf
T}_{0}\,,\quad{\bf T}_{2}=-\lambda\phi D\,{\bf X}_{3}+\frac{1}{F}{\bf
T}_{0}\,,\quad{\bf T}_{3}=F\,{\bf X}_{3}-\frac{f_{,\phi}}{F}{\bf T}_{0}\,,$
(4.5) $\displaystyle{\bf T}_{4}=F\,{\bf
X}_{4}+\frac{1}{F}\left[\frac{H}{D}f_{,t}\sin\phi-
f_{,r}\cos\phi+\frac{D^{\prime}}{D}f_{,\phi}\sin\phi\right]{\bf T}_{0}\,,$
(4.6) $\displaystyle{\bf T}_{5}=F\,{\bf
X}_{5}+\frac{1}{F}\left[\frac{H}{D}f_{,t}\cos\phi+f_{,r}\sin\phi+\frac{D^{\prime}}{D}f_{,\phi}\cos\phi\right]{\bf
T}_{0}\,,$ (4.7)

where the function $F$ has the form as given in (3.11) and we have defined
${\bf T}_{0}$ as follows:

${\bf
T}_{0}=[(D^{2}-H^{2})f_{,t}+Hf_{,\phi}]\partial_{t}-D^{2}f_{,r}\partial_{r}+(Hf_{,t}-f_{,\phi})\partial_{\phi}\,.$
(4.8)

Ultimately, the conservation relation $D_{i}T^{i}=0$ for the conserved vector
fields (4.5)-(4.7) yields

$\displaystyle f_{,r}\,f_{,t}\left(\frac{D^{2}}{F}\right)^{\prime}=0\,,\quad
f_{,r}f_{,\phi}\left(\frac{D^{2}}{F}\right)^{\prime}=0\,,\quad
f_{,r}\left(\frac{D^{2}}{F}\right)^{\prime}+\lambda\,D=0\,,$ (4.9)
$\displaystyle
f_{,r}f_{,t}\left[\left(\frac{HD}{F}\right)^{\prime}-\frac{H}{F}\right]+f_{,r}f_{,\phi}\left[\left(\frac{DD^{\prime}}{F}\right)^{\prime}+\frac{1}{F}\right]=0\,,$
(4.10) $\displaystyle
F^{\prime}-\frac{D^{\prime}}{D}F+f_{,t}^{2}\frac{H^{2}}{FD}+f_{,r}^{2}\left(\frac{D^{2}}{F}\right)^{\prime}-f_{,\phi}^{2}\frac{D^{\prime}}{FD}+f_{,t}f_{,\phi}\frac{H(D^{\prime}-1)}{FD}=0\,.$
(4.11)

Here, we have three different cases for classes I and III following from the
above equations:

$\displaystyle(a)\,\left(\frac{D^{2}}{F}\right)^{\prime}=0\,,\,\lambda=0;\quad(b)\,f_{,r}=0\,,\,\lambda=0;\quad(c)\,f_{,t}=f_{,\phi}=0\,,\,\lambda\neq
0\,.$ (4.12)

In case (a), the equations in (4.9) are identically satisfied, and the
condition $(D^{2}/F)^{\prime}=0$ has a solution as
$F=D^{2}/h(f_{{}_{,t}}\,,f_{{}_{,r}}\,,f_{{}_{,\phi}})$, where
$h(f_{{}_{,t}}\,,f_{{}_{,r}}\,,f_{{}_{,\phi}})$ is an integration function.
Then, Eq.(4.10) becomes $f_{,r}f_{,t}\left[(H/D)^{\prime}-H/D^{2}\right]=0$ if
$f_{,t},f_{,r}\neq 0$, which also identically satisfied for classes I and III.
For this case, the Eq. (4.11) yields

$D^{4}D^{\prime}+h^{2}\left[H^{2}f_{,t}^{2}-D^{\prime}f_{,\phi}^{2}+H(D^{\prime}-1)f_{,t}f_{,\phi}\right]=0\,.$
(4.13)

In case (b), Eqs. (4.9) and (4.10) are directly identities. Further, we can
write Eq.(4.11) as

$(F^{2})^{\prime}-\frac{2D^{\prime}}{D}F^{2}+\frac{2Hf_{,t}}{D}\left[Hf_{,t}+(D^{\prime}-1)f_{,\phi}\right]-\frac{2D^{\prime}}{D}f_{,\phi}^{2}=0\,,$
(4.14)

which gives $f_{,\phi}=-Hf_{,t}$ using
$F=\sqrt{D^{2}+(D^{2}-H^{2})f_{,t}^{2}+2Hf_{,t}f_{,\phi}-f_{,\phi}^{2}}$.
Since $f_{,r}=0$ (i.e., $f=f(t,\phi)$) in this case, the relation
$f_{,\phi}=-Hf_{,t}$ yields $H^{\prime}=0$, i.e. $\omega=0$. So the case (b)
reduces to the class IV. Therefore this case should be excluded as a
possibility for the classes I and III. In case (c), $f=f(r)$ and one of the
relations in (4.9) remains such that

$f_{,r}\left(\frac{D^{2}}{F}\right)^{\prime}+\lambda\,D=0\,.$ (4.15)

Additionally, Eq. (4.10) is an equality, and Eq.(4.11) becomes

$F^{\prime}-\frac{D^{\prime}}{D}F-\lambda Df_{,r}=0\,,$ (4.16)

regarding (4.15). The equation (4.16) is a linear first-order ordinary
differential equation in terms of $F$, and has a solution as $F=D(\lambda
f(r)+b_{1})$, where $b_{1}$ is a constant of integration. Using this solution
of $F$, the Eq.(4.15) yields

$f_{,r}^{2}-(\lambda f+b_{1})f_{,r}\frac{D^{\prime}}{\lambda D}-\left(\lambda
f+b_{1}\right)^{2}=0\,.$ (4.17)

Further, one can easily see from the definition of $F$ that for this case
$F=D\sqrt{1-f_{,r}^{2}}\,$, which gives directly that

$\sqrt{1-f_{,r}^{2}}-(\lambda f+b_{1})=0\,.$ (4.18)

Thus, introducing a new variable $U(r)=\lambda f+b_{1}$ and taking into
account the relation (4.18) in (4.17), it follows the following differential
equation

$\frac{D^{\prime}}{D}\left(U^{2}\right)^{\prime}+2\lambda^{2}(1-2U^{2})=0\,,$
(4.19)

which has a general solution as

$U(r)^{2}=\frac{1}{2}+b_{2}\left(D^{\prime}\right)^{-\frac{4\lambda^{2}}{m^{2}}}\,,$
(4.20)

where $b_{2}$ is an integration constant. Then, after substitution (4.20) into
$U(r)=\lambda f$ in which we take $b_{1}=0$ without loss of generality, one
can get for $f(r)$ that

$f(r)=\frac{1}{\lambda}\sqrt{\frac{1}{2}+b_{2}\left(D^{\prime}\right)^{-\frac{4\lambda^{2}}{m^{2}}}}\,.$
(4.21)

Also, we find from Eq.(4.18) that $\lambda^{2}=-m^{2}/4$ and $b_{2}=\pm 1/2$
for class I, and $\lambda^{2}=\mu^{2}/4$ and $b_{2}=\pm 1/2$ for class III.
Therefore, the final form of $f(r)$ in case (c) becomes

$f(r)=\left\\{\begin{array}[]{ll}\frac{\sqrt{2}}{m}\sqrt{-1\pm\cosh(mr)}&\quad\hbox{for
class I;}\\\ \\\ \frac{\sqrt{2}}{\mu}\sqrt{1\pm\cos(\mu r)}&\quad\hbox{for
class III.}\end{array}\right.$ (4.22)

In the special class I case, where $m^{2}=4\omega^{2}$, i.e. $\omega=+m/2$, we
found that the components $\eta,\xi^{1},\xi^{2},\xi^{3}$ of the Noether
symmetry generator ${\bf X}$, and the gauge vector field components
$A^{1},A^{2}$ and $A^{3}$ are

$\displaystyle\eta=c_{1},\quad\xi^{1}=-\frac{H}{D}\left[c_{2}\sin\phi-
c_{3}\cos\phi+c_{4}\sin(mt+\phi)+c_{5}\cos(mt+\phi)\right]+c_{6}\,$
$\displaystyle\xi^{2}=c_{2}\cos\phi+c_{3}\sin\phi-
c_{4}\cos(mt+\phi)+c_{5}\sin(mt+\phi)\,,$
$\displaystyle\xi^{3}=-\frac{D^{\prime}}{D}(c_{2}\sin\phi-
c_{3}\cos\phi)+\frac{1}{D}\left[c_{4}\sin(mt+\phi)+c_{5}\cos(mt+\phi)\right]+c_{7}\,,$
(4.23) $\displaystyle A^{1}=-\lambda
fH\left[-c_{2}\sin\phi+c_{3}\cos\phi+c_{4}\sin(mt+\phi)+c_{5}\cos(mt+\phi)\right]+c_{6}\lambda
fD+\Phi^{1}(t,r,\phi),$ $\displaystyle
A^{2}=\lambda\,fD\,\left[c_{2}\cos\phi+c_{3}\sin\phi-
c_{4}\cos(mt+\phi)+c_{5}\sin(mt+\phi)\right]+\Phi^{2}(t,r,\phi),$
$\displaystyle A^{3}=c_{1}\lambda\phi
D+\lambda\,f\left[D^{\prime}(-c_{2}\sin\phi+c_{3}\cos\phi)+c_{4}\sin(mt+\phi)+c_{5}\cos(mt+\phi)+c_{7}D\right]$
$\displaystyle\qquad\quad-\int{(\Phi^{1}_{,t}+\Phi^{2}_{,r})d\phi}+\Phi^{3}(t,r)\,.$

Then, one finds that there are seven NSs of minimal surface Lagrangian which
are given by

$\displaystyle{\bf X}_{1}={\bf K}_{1}\,,\quad{\bf X}_{2}={\bf
K}_{2}\,,\quad{\bf X}_{3}=\partial_{\phi}=\frac{1}{m}\left({\bf K}_{1}-{\bf
K}_{3}\right)\,,$ (4.24) $\displaystyle{\bf X}_{4}={\bf K}_{4}\,,\quad{\bf
X}_{5}={\bf K}_{5}\,,\quad{\bf X}_{6}={\bf K}_{6}\,,\quad{\bf X}_{7}={\bf
K}_{7}\,.$ (4.25)

The gauge vectors of ${\bf X}_{1},{\bf X}_{2}$ and ${\bf X}_{3}$ are the same
given in (4.3), and the gauge vectors for the NSs ${\bf X}_{4},{\bf
X}_{5},{\bf X}_{6}$ and ${\bf X}_{7}$ yield

$\displaystyle{\bf A}_{4}=\lambda\,fD\,{\bf K}_{4}\,,\quad{\bf
A}_{5}=\lambda\,fD\,{\bf K}_{5}\,,\quad{\bf A}_{6}=\lambda\,fD\,{\bf
K}_{6}\,,\quad{\bf A}_{7}=\lambda\,fD\,{\bf K}_{7}\,.$ (4.26)

The corresponding Lie algebra has the following non-vanishing commutators:

$\displaystyle\left[{\bf X}_{1},{\bf X}_{6}\right]=-m{\bf
X}_{7},\quad\left[{\bf X}_{1},{\bf X}_{7}\right]=m{\bf X}_{6},\quad\left[{\bf
X}_{3},{\bf X}_{4}\right]={\bf X}_{5},\quad\left[{\bf X}_{3},{\bf
X}_{5}\right]=-{\bf X}_{4},$ $\displaystyle\left[{\bf X}_{3},{\bf
X}_{6}\right]=-{\bf X}_{7},\quad\,\,\left[{\bf X}_{3},{\bf X}_{7}\right]={\bf
X}_{6},\quad\left[{\bf X}_{4},{\bf X}_{5}\right]=m{\bf X}_{1}-m^{2}{\bf
X}_{3},\quad\left[{\bf X}_{6},{\bf X}_{7}\right]=m{\bf X}_{1}.$

The conserved vector fields of this special class I for ${\bf X}_{1},{\bf
X}_{2},{\bf X}_{3},{\bf X}_{4}$ and ${\bf X}_{5}$ by taking $\omega=m/2$ are
the same form obtained in (4.5), (4.6) and (4.7). The remaining conserved
quantities associated with ${\bf X_{6}}$ and ${\bf X_{7}}$ given in (4.25) are

$\displaystyle{\bf T}_{6}=F\,{\bf
X}_{6}+\frac{1}{F}\left[\frac{H}{D}f_{,t}\cos(mt+\phi)-f_{,r}\sin(mt+\phi)-\frac{1}{D}f_{,\phi}\cos(mt+\phi)\right]{\bf
T}_{0}\,,$ (4.27) $\displaystyle{\bf T}_{7}=F\,{\bf
X}_{7}+\frac{1}{F}\left[\frac{H}{D}f_{,t}\sin(mt+\phi)+f_{,r}\cos(mt+\phi)-\frac{1}{D}f_{,\phi}\sin(mt+\phi)\right]{\bf
T}_{0}\,,$ (4.28)

where $F$ and ${\bf T}_{0}$ are the same as Eqs. (3.11) and (4.8),
respectively. Further, the conservation law given in (3.18) for the Noether
currents ${\bf T}_{1},...,{\bf T}_{5}$ of the special class I case gives rise
to the same equations with (4.9), (4.10) and (4.11). Additionally, for the
Noether currents ${\bf T}_{6}$ and ${\bf T}_{7}$, the conservation law reads

$\displaystyle
F^{\prime}+\frac{(H-1)}{D}F-f_{,t}^{2}\frac{H}{FD}\left[H+m(D^{2}-H^{2})\right]+f_{,r}^{2}\left(\frac{D^{2}}{F}\right)^{\prime}$
$\displaystyle\qquad\qquad\qquad\qquad\quad\,\,\,+f_{,\phi}^{2}\frac{(mH-1)}{FD}+\frac{f_{,t}f_{,\phi}}{FD}\left[2H+m(D^{2}-2H^{2})\right]=0\,,$
(4.29) $\displaystyle
f_{,r}f_{,t}\left[\left(\frac{HD}{F}\right)^{\prime}+\frac{H}{F}+\frac{m(D^{2}-H^{2})}{F}\right]-f_{,r}f_{,\phi}\left[\left(\frac{D}{F}\right)^{\prime}+\frac{1}{F}-\frac{mH}{F}\right]=0\,.$
(4.30)

For the special class I, three distinct cases arise from the conservation
relations delineated in class I. While the outcomes derived in class I remain
applicable to the special class I, Eq. (4.29) further yields $m=1$ in case
(c).

For the class II, where $H(r)=-\omega\,r^{2}$ and $D(r)=r$, the components of
the symmetry generator ${\bf X}$ and the gauge vector ${\bf A}$ are obtained
as

$\displaystyle\eta=c_{1},\quad\xi^{1}=\omega\,r(c_{2}\sin\phi-
c_{3}\cos\phi)+c_{4},\,\,$
$\displaystyle\xi^{2}=c_{2}\cos\phi+c_{3}\sin\phi,\quad\xi^{3}=\frac{1}{r}(-c_{2}\sin\phi+c_{3}\cos\phi)+c_{5}\,,$
$\displaystyle A^{1}=\lambda\,\omega\,r^{2}f(c_{2}\sin\phi-
c_{3}\cos\phi)+c_{4}\,\lambda\,rf+\Phi^{1}(t,r,\phi),$ (4.31) $\displaystyle
A^{2}=\lambda\,rf\,(c_{2}\cos\phi+c_{3}\sin\phi)+\Phi^{2}(t,r,\phi),$
$\displaystyle
A^{3}=c_{1}\lambda\,r\,\phi+\lambda\,f(-c_{2}\sin\phi+c_{3}\cos\phi+c_{5}\,r)-\int{(\Phi^{1}_{,t}+\Phi^{2}_{,r})d\phi}+\Phi^{3}(t,r)\,,$

which yields that the five KVs ${\bf K}_{1},{\bf K}_{2},{\bf K}_{3},{\bf
K}_{4}$ and ${\bf K}_{5}$ given in (2.6) are equivalent to the NSs ${\bf
X}_{1},{\bf X}_{2},{\bf X}_{3},{\bf X}_{4}$ and ${\bf X}_{5}$, respectively.
Afterwards, we can write that the corresponding gauge vector fields are of the
form

$\displaystyle{\bf A}_{1}=\lambda\,r\,f\,{\bf X}_{1}\,,\qquad{\bf
A}_{2}=\lambda\,r\,\phi\,{\bf X}_{3}\,,\quad\,\,{\bf
A}_{3}=\lambda\,r\,f\,{\bf X}_{3}\,,$ (4.32) $\displaystyle{\bf
A}_{4}=\lambda\,r\,f\,{\bf X}_{4}\,,\qquad{\bf A}_{5}=\lambda\,r\,f\,{\bf
X}_{5}\,.$ (4.33)

The Lie algebra of NSs for the class II will have the following non-vanishing
commutators:

$\displaystyle\left[{\bf X}_{3},{\bf X}_{4}\right]={\bf X}_{5},\quad\left[{\bf
X}_{3},{\bf X}_{5}\right]={\bf X}_{4},\quad\left[{\bf X}_{4},{\bf
X}_{5}\right]=2\omega{\bf X}_{1}.$

Hence, the conserved vector fields associated with Noether symmetries are

$\displaystyle{\bf T}_{1}=F\,{\bf X}_{1}+\frac{r^{2}}{F}f_{,t}{\bf
T}_{0}\,,\quad{\bf T}_{2}=-\lambda\,r\,\phi{\bf X}_{3}-\frac{r^{2}}{F}{\bf
T}_{0}\,,\quad{\bf T}_{3}=F\,{\bf X}_{3}+\frac{r^{2}}{F}f_{,\phi}{\bf
T}_{0}\,,$ (4.34) $\displaystyle{\bf T}_{4}=F\,{\bf
X}_{4}-\frac{r^{2}}{F}\left[\omega\,rf_{,t}\sin\phi+f_{,r}\cos\phi-\frac{1}{r}f_{,\phi}\sin\phi\right]{\bf
T}_{0}\,,$ (4.35) $\displaystyle{\bf T}_{5}=F\,{\bf
X}_{5}-\frac{r^{2}}{F}\left[\omega\,rf_{,t}\cos\phi-
f_{,r}\sin\phi-\frac{1}{r}f_{,\phi}\cos\phi\right]{\bf T}_{0}\,,$ (4.36)

where $F$ and ${\bf T}_{0}$ are of the form

$\displaystyle F=r\sqrt{1+(1-\omega^{2}r^{2})f^{2}_{,t}-2\omega
f_{,t}f_{,\phi}-f_{,r}^{2}-\frac{1}{r^{2}}f^{2}_{,\phi}}\,,$ (4.37)
$\displaystyle{\bf T}_{0}=\left[(\omega^{2}r^{2}-1)f_{,t}+\omega
f_{,\phi}\right]\partial_{t}+f_{,r}\partial_{r}+\left(\omega
f_{,t}+\frac{1}{r^{2}}f_{,\phi}\right)\partial_{\phi}\,.$ (4.38)

Then, applying the conservation law ${\bf D}.{\bf T}=0$ to the conserved
vector fields in (4.34), (4.35) and (4.36), we found the following constraint
equations:

$\displaystyle f_{,t}\,f_{,r}\left(\frac{r^{2}}{F}\right)^{\prime}=0\,,\quad
f_{,r}f_{,\phi}\left(\frac{r^{2}}{F}\right)^{\prime}=0\,,\quad
f_{,r}\left(\frac{r^{2}}{F}\right)^{\prime}+\lambda\,r=0\,,$ (4.39)
$\displaystyle
F^{\prime}-\frac{1}{r}F+\frac{\omega^{2}r^{3}}{F}f_{,t}^{2}+f_{,r}^{2}\left(\frac{r^{2}}{F}\right)^{\prime}-\frac{f_{,\phi}^{2}}{r\,F}=0\,,$
(4.40) $\displaystyle\omega
f_{,r}f_{,t}\left[\left(\frac{r^{3}}{F}\right)^{\prime}-\frac{r^{2}}{F}\right]-f_{,r}f_{,\phi}\left[\left(\frac{r}{F}\right)^{\prime}+\frac{1}{F}\right]=0\,.$
(4.41)

For class II, three distinct scenarios arise from the equations provided in
(4.12) as follows:

$\displaystyle(a)\,\left(\frac{r^{2}}{F}\right)^{\prime}=0\,,\,\lambda=0;\quad(b)\,f_{,r}=0\,,\,\lambda=0;\quad(c)\,f_{,t}=f_{,\phi}=0\,,\,\lambda\neq
0\,.$ (4.42)

In case (a), we derive a solution
$F=r^{2}/h(f_{{}_{,t}}\,,f_{{}_{,r}}\,,f_{{}_{,\phi}})$, where
$h(f_{{}_{,t}}\,,f_{{}_{,r}}\,,f_{{}_{,\phi}})$ is an integration function.
Consequently, Eq. (4.40) reads

$\displaystyle f_{,\phi}=0\,,\qquad\omega^{2}h^{2}f_{,t}^{2}+1=0\,,$ (4.43)

while Eq.(4.41) becomes an identity. By substituting the relation $F=r^{2}/h$
into (4.37) and employing (4.43), after some algebraic manipulation, we
ascertain that $f_{,r}^{2}-f_{,t}^{2}-1=0$ and
$F=\pm\,\omega\,r^{2}\sqrt{1-f_{,r}^{2}}$. In case (b), all constraint
equations (4.39)-(4.41) become identities. In case (c), no solution satisfying
the constraint equations is found.

###  4.2 Class IV

In this class, where $m^{2}\neq 0,\omega=0$, the metric functions are taken as
$H(r)=0$ and $D(r)=\frac{1}{m}\sinh(mr)$ for $m^{2}>0$, or
$D(r)=\frac{1}{\mu}\sin(\mu r)$ for $\mu^{2}=-m^{2}>0$. The general solution
to the Noether symmetry equations (3.16) for this class gives

$\displaystyle\eta=c_{1}-c_{2}t,\quad\xi^{1}=c_{2}f+c_{3}\,,\,\,\xi^{2}=c_{4}\sin\phi+c_{5}\cos\phi\,,$
$\displaystyle\xi^{3}=\frac{D^{\prime}}{D}(c_{4}\cos\phi-
c_{5}\sin\phi)+c_{6}\,,\quad
A^{1}=\lambda\,fD(c_{2}f+c_{3})+\Phi^{1}(t,r,\phi),$ (4.44) $\displaystyle
A^{2}=\lambda\,fD\,(c_{4}\sin\phi+c_{5}\cos\phi)+\Phi^{2}(t,r,\phi),$
$\displaystyle A^{3}=\lambda\,\phi
D(c_{1}-c_{2}t)+\lambda\,fD^{\prime}(c_{4}\cos\phi-
c_{5}\sin\phi)+c_{6}\lambda\,fD-\int{(\Phi^{1}_{,t}+\Phi^{2}_{,r})d\phi}+\Phi^{3}(t,r)\,.$

where $c_{i}$’s are constant parameters. So, one can write that there are six
NSs given by

$\displaystyle{\bf X}_{1}={\bf K}_{1}\,,\quad{\bf X}_{2}={\bf
K}_{2}\,,\quad{\bf X}_{3}=f\partial_{t}-t\partial_{f}={\bf K}_{3}-2t{\bf
K}_{2}\,,$ (4.45) $\displaystyle{\bf X}_{4}={\bf K}_{4}\,,\quad{\bf
X}_{5}={\bf K}_{5}\,,\quad{\bf X}_{6}={\bf K}_{6}\,,$ (4.46)

with the corresponding gauge vectors

$\displaystyle{\bf A}_{1}=\lambda\,fD\,{\bf X}_{1}\,,\quad\qquad{\bf
A}_{2}=\lambda\,\phi D\,{\bf X}_{6}\,,\quad\qquad{\bf A}_{3}=\lambda
D\,(f^{2},0,-t\phi)\,,$ (4.47) $\displaystyle{\bf A}_{4}=\lambda\,fD\,{\bf
X}_{4}\,,\quad\qquad{\bf A}_{5}=\lambda\,fD\,{\bf X}_{5}\,,\quad\qquad{\bf
A}_{6}=\lambda\,fD\,{\bf X}_{6}\,,$ (4.48)

where the KVs ${\bf K}_{1},...,{\bf K}_{6}$ for this class are given in (2.7).
Here, the vector fields ${\bf X}_{1},{\bf X}_{2},{\bf X}_{4},{\bf X}_{5}$ and
${\bf X}_{6}$ given above have the same form with the corresponding KVs. The
non-vanishing commutators of the NSs are

$\displaystyle\left[{\bf X}_{1},{\bf X}_{3}\right]=-{\bf
X}_{2}\,,\qquad\left[{\bf X}_{2},{\bf X}_{3}\right]={\bf X}_{1}\,,$
$\displaystyle\left[{\bf X}_{4},{\bf X}_{5}\right]=-m^{2}{\bf
K}_{6},\quad\left[{\bf X}_{4},{\bf X}_{6}\right]=-{\bf
X}_{5}\,,\quad\left[{\bf X}_{5},{\bf X}_{6}\right]={\bf X}_{4}\,.$

Furthermore, the conserved vector fields for each NSs of this class read

$\displaystyle{\bf T}_{1}=F\,{\bf X}_{1}-\frac{1}{F}f_{,t}{\bf
T}_{0}\,,\quad{\bf T}_{2}=-\lambda\phi D\,{\bf X}_{6}+\frac{1}{F}{\bf
T}_{0}\,,$ (4.49) $\displaystyle{\bf T}_{3}=fF{\bf X}_{1}+\lambda\ t\,\phi
D\,{\bf X}_{6}-\frac{1}{F}(t+ff_{,t}){\bf T}_{0}\,,$ (4.50) $\displaystyle{\bf
T}_{4}=F\,{\bf
X}_{4}-\frac{1}{F}\left(f_{,r}\cos\phi-\frac{D^{\prime}}{D}f_{,\phi}\sin\phi\right){\bf
T}_{0}\,,$ (4.51) $\displaystyle{\bf T}_{5}=F\,{\bf
X}_{5}-\frac{1}{F}\left(f_{,r}\sin\phi+\frac{D^{\prime}}{D}f_{,\phi}\cos\phi\right){\bf
T}_{0}\,,\quad{\bf T}_{6}=F\,{\bf X}_{6}-\frac{1}{F}f_{,\phi}{\bf T}_{0}\,,$
(4.52)

where ${\bf
T}_{0}=D^{2}f_{,t}\partial_{t}-D^{2}f_{,r}\partial_{r}-f_{,\phi}\partial_{\phi}$
and the function $F$ given by (3.11) becomes

$\displaystyle
F=\sqrt{D^{2}\left(1+f_{,t}^{2}-f_{,r}^{2}\right)-f_{,\phi}^{2}}\,.$ (4.53)

One can eventually apply the conservation law (3.18) for the Noether currents
obtained in (4.49)-(4.52), and find the the following relations:

$\displaystyle f_{,r}f_{,t}\left(\frac{D^{2}}{F}\right)^{\prime}=0\,,\qquad
f_{,r}\left(\frac{D^{2}}{F}\right)^{\prime}+\lambda D=0\,,$ (4.54)
$\displaystyle
F^{\prime}-\frac{D^{\prime}}{D}F+f_{,r}^{2}\left(\frac{D^{2}}{F}\right)^{\prime}-f_{,\phi}^{2}\frac{D^{\prime}}{FD}=0\,,\quad
f_{,r}f_{,\phi}\left[\left(\frac{DD^{\prime}}{F}\right)^{\prime}+\frac{1}{F}\right]=0\,,$
(4.55) $\displaystyle
f_{,r}(ff_{,t}+t)\left(\frac{D^{2}}{F}\right)^{\prime}+f_{,t}\frac{2D^{2}}{F}+\lambda\,tD=0\,.$
(4.56)

From the above equations, it is evident that this class also exhibits the same
three possibilities outlined in (4.12). In case (a), we offer solutions for
the constraint equations (4.54)-(4.56) as follows:

$\displaystyle F=\frac{D^{2}}{h}\,,\quad f_{,t}=0\,,\quad
f_{,r}^{2}=\pm\sqrt{1-\frac{2D^{2}}{h^{2}}}\,,\quad
f_{,\phi}=\pm\frac{D^{2}}{h}\,,$ (4.57)

where $h=h(f_{{}_{,t}}\,,f_{{}_{,r}}\,,f_{{}_{,\phi}})$. In case (b),
Eq.(4.56) results $f_{,t}=0$, indicating $f$ as a function of $\phi$ only.
Consequently, the function $F$ defined in (4.53) reduces to
$F=\sqrt{D^{2}-f_{,\phi}^{2}}$. When considering this $F$ in (4.55), both
equations are identically satisfied. For case (c), Eqs. (4.54)-(4.56) yield
the same equations as (4.15) and (4.16). Hence, the solutions for these
equations remain consistent, as provided in (4.22), such that

$f(r)=\left\\{\begin{array}[]{ll}\frac{\sqrt{2}}{m}\sqrt{-1\pm\cosh(mr)}&\quad\hbox{for
$m^{2}>0$;}\\\ \\\ \frac{\sqrt{2}}{\mu}\sqrt{1\pm\cos(\mu r)}&\quad\hbox{for
$\mu^{2}=-m^{2}>0$.}\end{array}\right.$ (4.58)

##  5 Conclusions

In this study, we employed the classical Noether symmetry approach to derive
symmetries of minimal surface Lagrangian and the conserved vector fields
corresponding to the Noether symmetry generators for Gödel-type spacetimes.
For the minimal surface Lagrangian $L$ in Gödel-type spacetimes, we obtained
the Noether symmetries for classes I, II, and III, resulting in five NS
generators. Consequently, Gödel-type spacetimes corresponding to these classes
exhibit the algebra $\mathcal{N}_{5}$, equivalent to the Killing algebra
$\mathcal{G}_{5}$. Notably, in special class I (where $m^{2}=4\omega^{2}$) and
class IV, we identified seven and six NS generators, respectively. The special
class I admits the algebra $\mathcal{N}_{7}\equiv\mathcal{G}_{7}$, while class
IV spacetime admits the algebra $\mathcal{N}_{6}\equiv\mathcal{G}_{6}$. We
observed that the Lagrangian for minimizing the 3-area enclosing a constant
three-volume in Gödel-type spacetimes shares a Lie algebra of Noether
symmetries identical to the Lie algebra of KVs of those spacetimes. The
corresponding gauge vectors are proportional to the KVs, which are equivalent
to the Noether symmetries, although they do not mirror the exact form to the
symmetries proposed in Ref. [9]. Moreover, we determined the Noether currents
resulting from the existence of Noether symmetries for minimal surface
Lagrangians across each class I, II, III, and IV. Using the obtained Noether
currents for all classes of Gödel-type spacetimes, we applied the conservation
law (3.18) to derive the analytical solutions of minimal surface equation
(3.10), showcasing the practicality and efficacy of Noether symmetries in this
context.

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