Pramana – J. Phys.          (2022) 96:121 © Indian Academy of Sciences
https://doi.org/10.1007/s12043-022-02361-8

Conservation laws and solution of the geodesic system of Gödel’s
metric via Lie and Noether symmetries

F ALKINDI1, A H KARA2 ,∗ and M ZIAD3

1General Directorate of Education of A’ddakhiliyah Province, Ministry of Education, Sultanate of Oman
2School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
3Department of Mathematics, College of Science, Sultan Qaboos University, Muscat, Sultanate of Oman
*Corresponding author. E-mail: Abdul.Kara@wits.ac.za

MS received 19 August 2021; revised 19 January 2022; accepted 7 February 2022

Abstract. We consider the geodesic system for Gödel’s metric as a toy model and solve it analytically using
its Lie point symmetries. It is shown that the differential invariants of these symmetries reduce the second-order
non-linear system to a single second-order ordinary differential equation (ODE). Invariance of the latter under a
one-dimensional Lie point symmetry group reduces it to an integrable first-order ODE. A complete solution of
the system is then achieved. The sub-algebras of Noether symmetries and isometries are then found with their
corresponding first integrals.

Keywords. Geodesic equations; Lie symmetries; Noether symmetries; isometries.

PACS Nos 04.25.-g; 02.30.Hq; 02.30.Jr; 02.40.-k

1. Introduction

Lie point symmetries provide operative tools to find
solutions of differential equations [1–9]. Various tech-
niques are applied using Lie point symmetries to inte-
grate ordinary differential equations (ODEs), which
under some considerations, can be applied for systems of
ODEs [1,7,8]. If a system admits a transitive solvable Lie
algebra of the Lie point symmetries of enough dimen-
sions, then the solution can be found by line integrals. If
there are several symmetries but they do not satisfy the
condition of solvability, method of successive reduction
of order using canonical forms of the symmetry gener-
ators may be purposive. Furthermore, invariants of Lie
point symmetries can, in some cases, provide invari-
ant solutions or first integrals obtained from differential
invariants. However, if the Lagrangian of the system is
known, then the variational symmetries (if exist) pro-
vide first integrals via Noether theorem. Similarly, if the
ambient space–time of the system admits enough isome-
tries, then according to Cartan theory [1] these readily
provide the reduction of order of the system.

We show the efficiency of using Lie point sym-
metries to find a complete analytic solution of the
geodesic system of Gödel’s metric. Gödel’s metric
belongs to the family of the Einstein and the de Sitter
space–times which are well known homogeneous Petrov

type-D static metric [10–14]. It admits five space–time
isometries and is stationary compared to the other two
members of its family. It gained popularity and attracted
much attention of many investigators [15–23] since its
emergence because of its unusual features. The stress–
energy tensor of this metric represents a rotating matter
without having a singularity. Another notable charac-
teristic of this space–time metric is that a portion of the
trajectories of the free particles in it appears as time-like
closed curve. Here we exhibit this feature of Gödel’s
metric explicitly.

We provide a complete solution of the geodesic sys-
tem of Gödel’s metric in a cylindrical coordinate system
(t, r, φ, z), where t < ∞, 0 ≤ r ≤ ∞, 0 ≤ φ ≤ 2π ,
−∞ < z < ∞, given by [15,18]

ds2 = a2([dt + √
2 sinh2 r dφ]2 − dr2 − dz2

− sinh2 r cosh2 r dφ2), (1)

with the aid of Lie point symmetries. Here a = 2/ω,
where ω is a measure of the constant rotation of the
matter flow. Assuming ω = 2, the geodesic equations
for this metric turn out to be

ẗ + 4 sinh r

cosh r
ṫṙ + 2

√
2 sinh3 r

cosh r
ṙ φ̇ = 0, (2a)

r̈ + 2
√

2 sinh r cosh r ṫ φ̇

0123456789().: V,-vol

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  121 Page 2 of 7 Pramana – J. Phys.          (2022) 96:121 

− sinh r cosh r(1 − 2 sinh2 r)φ̇2 = 0, (2b)

φ̈ − 2
√

2

sinh r cosh r
ṫṙ + 2

sinh r cosh r
ṙ φ̇ = 0, (2c)

z̈ = 0, (2d)

where dot over the variables t , r , φ and z denote the
derivatives with respect to the arc-length parameter
s. In the next section, Lie point symmetry generators
of geodesic system (2) are found which form a ten-
dimensional Lie algebra and a brief analysis of the
resulting Lie algebra is provided in terms of its various
sub-algebras and their applicability to solve the system.
In §3, the system is reduced, with the aid of the admit-
ted Lie point symmetries, to an integrable autonomous
first-order ODE which is then analytically solved in
details. In the following section, Noether symmetries of
the geodesic system besides the isometries of the space
as well as their corresponding first integrals are found.

2. Lie point symmetries of the geodesic system

Finding Lie point symmetries (using Einstein summa-
tion convention)

X = ξ(s, xa)
∂

∂s
+ ηa(s, xa)

∂

∂xa
(3)

of a system of k second-order ordinary differential equa-
tions

Ei (s, x
a, ẋa, ẍa) = 0, i, a = 1, . . . , k, (4)

means finding the general solution ξ(s, xa) andηa(s, xa)
of the determining equations obtained from the symme-
try condition

X̂(Ei ) = 0, (5)

where X̂ is the extension of the symmetry operator X
written as

X̂ = ξ(s, xa)
∂

∂s
+ ηa(s, xa)

∂

∂xa
+ (ηa)′(s, xa, ẋa) ∂

∂ ẋa

+(ηa)′′(s, xa, ẋa) ∂

∂ ẍa

and

(ηa)(n) = d(ηa)(n−1)

ds
− (xa)(n) dξ

ds
.

The solution of symmetry condition (5) applied onto
system (2) gives a ten-dimensional Lie algebra (L10)

whose symmetry generators are

X1 = s
∂

∂s
, X2 = z

∂

∂s
, X3 = ∂

∂s
,

X4 = −√
2 tanh r sin φ

∂

∂t
+ cos φ

∂

∂r

− 2 cosh2 r − 1

sinh r cosh r
sin φ

∂

∂φ
,

X5 = √
2 tanh r cos φ

∂

∂t

+ sin φ
∂

∂r
+ 2 cosh2 r − 1

sinh r cosh r
cos φ

∂

∂φ
,

X6 = ∂

∂φ
, X7 = ∂

∂t
, X8 = s

∂

∂z
,

X9 = z
∂

∂z
, X10 = ∂

∂z
, (6)

with non-vanishing commutators given by

[X1, X2] = −X2, [X1, X3] = −X3,

[X1, X8] = X8, [X2, X8] = X9 − X1,

[X2, X9] = −X2, [X2, X10] = −X3,

[X3, X8] = X10, [X4, X5] = 2
√

2X7 + 4X6,

[X4, X6] = X5, [X5, X6] = X4,

[X8, X9] = X8, [X9, X10] = −X10.

The derived algebra for L10 is L10 itself showing that it
is neither solvable nor nilpotent. The sub-algebra

H7 = 〈X1, X2, X3, X6, X7, X9, X10〉 (7)

is the maximal solvable sub-algebra of L10. With a
change of basis, L10 can be written as

L10 =
〈
X1 + X9, X1 − X9, X2, X3, X4, X5, X6

+
√

2

2
X7, X7, X8, X10

〉
.

The symmetries involving the arc length s would nec-
essarily be gauge-dependent and those that correspond
to zero gauge form a sub-algebra as, viz., [20]

L7 =
〈
X3, X4, X5, X6 +

√
2

2
X7, X7,X8, X10

〉
. (8)

Equation (2d) is exclusive from the rest of the ODEs
of the system and can be integrated readily to give z as
a linear function of s. Thus, we consider eqs (2a)–(2c)
which form a coupled system admitting

H6 = 〈X1, X3, X4, X5, X6, X7〉. (9)

In the next section, we show the employment of H6 on
solving geodesic systems (2a)–(2c).



Pramana – J. Phys.          (2022) 96:121 Page 3 of 7   121 

3. Solving the geodesic system using H6

The generators X6 and X7 show that the variables φ

and t do not appear explicitly in the system. Thus, these
generators will be no more useful for applying further
reduction of order. So is the case with the generators X4
and X5 as their coefficients are dependent on the variable
φ. Thus, we have only two generators X1 and X3 to be
taken in the reduction process. The commutator

[X1, X3] = −X3,

sets the order X3, then X1 is to be taken into considera-
tion. Using X3, which is in the normal form with respect
to arc-length parameter s, the system can be reduced to
first order with respect to all dependent variables by
change of variables

τ = r, v1 = t, v2 = s, v3 = φ. (10)

Then, transforming the system into the space of the
canonical variables (τ, u1, u2, u3) with respect to gen-
erator X1 using the transformation:

τ = τ, u1 = ln v̇2, u2 = v̇1, u3 = v̇3, (11)

where v̇i = (dvi/dτ) reduces the system into the form

u̇1 = 2
√

2 sinh τ cosh τ u2u3

− sinh τ cosh τ(1 − 2 sinh2 τ)u2
3, (12a)

u̇2 = −4 sinh τ u2 + 2
√

2 sinh3 τ u3

cosh τ

+ 2
√

2 sinh τ cosh τ u2
2u3

− sinh τ cosh τ(1 − 2 sinh2 τ)u2u
2
3, (12b)

u̇3 = 2
√

2 u2 − 2 u3

cosh τ
+ 2

√
2 sinh τ cosh τ u2u

2
3

− sinh τ cosh τ(1 − 2 sinh2 τ)u3
3, (12c)

where u̇i = dui/dτ . The last two equations of sys-
tem (12) form a system of two first-order ODEs in the
two variables u2 and u3. In principle, solving this lat-
ter system for u2 and u3, substituting into eq. (12a)
and transforming back into the original variables give
the complete solution of geodesic system (2). Practi-
cally, this is a non-autonomous system of two nonlinear
first-order ODEs which are in a non-integrable form.
Moreover, it does not admit a Lie point symmetry. Thus,

we turn to use the differential invariants which appear
to be practically operative [7,8].

The differential invariants of the transitive Lie sub-
group H6, which form the solution 	i (x) of the system

X̂ N	i (x) = 0, N = 2, . . . , 6,

where x = (s, t, r, φ, ṫ, ṙ , φ̇) and X̂ N is the prolonged
form of XN up to the first derivatives of the dependent
variables, are

	1 = ṫ + √
2 sinh2 r φ̇, (13a)

	2 = ṙ2 + sinh2 r cosh2 r φ̇2. (13b)

Invariants (13) satisfy geodesic system (2). Thus, they
are first integrals, which are analogous to the first inte-
grals found in ref. [15] according to his form of Gödel’s
metric. To reduce the order of the system, one requires
a full set of first integrals which needs to be three in this
case. However, one can use them to reduce the number
of equations in the system. Substituting invariant solu-
tions (13) into the geodesic system reduces the system of
three equations to a single ordinary differential equation
given as

r̈ + 2
√

2	1

√
	2 − ṙ2 − 2(	2 − ṙ2)

tanh 2r
= 0, (14)

where 	i are constants. The above equation admits the
only symmetry generator ∂/∂s by which it is reduced,
via the transformation v = ṙ , x = r , to the first-order
ODE

vv̇ + 2
√

2	1

√
	2 − v2

2
(
	2 − v2

) = 2

tanh 2x
, (15)

where v̇ = (∂v/∂x). The solution of eq. (15) is given
implicitly as
√

2	1 cosh 2x −
√

	2 − v2 sinh 2x + 	3 = 0, (16)

where 	3 is a constant. Substituting back into the space
of the original variables s, r, ṙ gives

ṙ2 = 	2 −
(√

2	1 cosh 2r + 	3

)2

sinh2 2r
. (17)

Equation (17) offers the third functionally independent
first integral of system (2) which, with eqs (13), provide
a complete solution of system (2). This is given in detail
as follows.

Equation (17) can be written as

ṙ2 = 4	2 sinh2 r cosh2 r − (
√

2	1(sinh2 r + cosh2 r) + 	3)
2

4 sinh2 r cosh2 r
(18)

or

(2ṙ sinh r cosh r)2 = 4	2 sinh2 r cosh2 r

−[√2	1(sinh2 r + cosh2 r) + 	3]2.



  121 Page 4 of 7 Pramana – J. Phys.          (2022) 96:121 

The left-hand side of this equation is nothing but( d
ds sinh2 r

)2
. Thus, we can write it as(

d

ds
y

)2

= 4	2y(1 + y)

−[√2	1(y + y + 1) + 	3]2,

where y = sinh2 r . Hence

ẏ2 = 4	2(y
2 + y) − [√2	1(2y + 1) + 	3]2. (19)

Expanding and regrouping the terms, this can be written
as

ẏ2 = −4(2	2
1 − 	2)

×
[
y2 − 	2 − 2	2

1 − √
2	1	3

2	2
1 − 	2

y

+(
√

2	1 + 	3)
2

4(2	2
1 − 	2)

]
. (20)

From eqs (13), it is clear that 	2 ≥ 0, and

2	2
1 − 	2 = 2(ṫ + √

2φ̇2 sinh2 r)2 − ṙ2

−φ̇2 sinh2 r cosh2 r. (21)

Dividing eq. (1) by ds2

1 = (ṫ + √
2φ̇2 sinh2 r)2 − ṙ2

−φ̇2 sinh2 r cosh2 r − ż2 (22)

and subtracting from eq. (21) gives

2	2
1 − 	2 = (ṫ + √

2φ̇2 sinh2 r)2 + 1 + ż2 > 1. (23)

This proves that 2	2
1 − 	2 is a positive quantity. More-

over, from eq. (20)

	2 − 2	2
1 − √

2	1	3

= ẏ2 + 4(2	2
1 − 	2)y2 + (

√
2	1 + 	3)

2

4y
> 0. (24)

Thus, one can write eq. (20) as

ẏ2 = −4A2 [
y2 − B2y + C2] , (25)

where

A2 = 2	2
1 − 	2, B2 = 	2 − 2	2

1 − √
2	1	3

2	2
1 − 	2

,

C2 = (
√

2	1 + 	3)
2

4(2	2
1 − 	2)

. (26)

An alternative form of eq. (25) is

ẏ2 = 4A2

[
B4 − 4C2

4
−

(
y − B2

2

)2
]

, (27)

from which we can deduce that

B4 − 4C2 > 0. (28)

Hence

ẏ = 2A

√
B4 − 4C2

4
−

(
y − B2

2

)2

, (29)

whose integral

y = B2

2
+

√
B4 − 4C2

2
cos (2As + s◦) = sinh2 r.

(30)

In terms of 	s,

y(s) = 	2 − 2	2
1 − √

2	1	3

2
(
2	2

1 − 	2
)

+
√

	2
(
	2 − 2	2

1 + 	2
3

)
2

(
2	2

1 − 	2
)

× cos

(
2
√

2	2
1 − 	2 s + s0

)
. (31)

To find ṫ and φ̇ we use eqs (17) and (13b) from which
we can find that

φ̇2 =
[√

2	1(sinh2 r + cosh2 r) + 	3

]2

4 sinh4 r cosh4 r
. (32)

Taking the square root of both sides and regrouping the
terms we can write it as

φ̇ =
√

2	1 − 	3

2 cosh2 r
+

√
2	1 + 	3

2 sinh2 r
. (33)

Substituting eq. (30) gives

φ(s) =
√

2	1 − 	3

2
I +

√
2	1 − 	3

2
J, (34)

where

I = ds

1 + B2

2 +
√
B4−4C2

2 cos(2As + s0)

and

J = ds
B2

2 +
√
B4−4C2

2 cos(2As + s0)
.

This gives

φ(s) = tan−1

⎛
⎝

√
2+B2−√

B4−4C2

2+B2+√
B4−4C2

tan
(
As+s0

2

)⎞
⎠

+ tan−1

⎛
⎝

√
B2−√

B4−4C2

B2+√
B4−4C2

tan
(
As+s0

2

)⎞
⎠ +φ0.

(35)



Pramana – J. Phys.          (2022) 96:121 Page 5 of 7   121 

Now substituting eq. (33) into eq. (13a) we get

t (s) = −
√

2
(
	3 − √

2	1

)
√
B4 − 4C2

I − 	1s, (36)

yielding

t (s) = √
2 tan−1

⎛
⎝

√
2+B2−√

B4−4C2

2+B2+√
B4−4C2

× tan
(
As+s0

2

)⎞
⎠ − 	1s + t0. (37)

In terms of the constants 	s, the time-like trajectories
in the Gödel Universe in (t, r, φ)-hypersurface are given
as
(t, r, φ)

=
⎛
⎝√

2 tan−1

⎛
⎝

√
2+B2−√

B4−4C2

2+B2+√
B4 − 4C2

×tan
(
As+ s0

2

)⎞
⎠ − 	1s + t0,

sinh−1

⎛
⎝

√
B2

2
+

√
B4 − 4C2

2
cos (2As + s0)

⎞
⎠ ,

tan−1

⎛
⎝

√
2 + B2 − √

B4 − 4C2

2 + B2 + √
B4 − 4C2

tan
(
As + s0

2

)⎞
⎠

+tan−1

⎛
⎝

√
B2−√

B4−4C2

B2+√
B4−4C2

tan
(
As+ s0

2

)⎞
⎠+φ0,

(38)
where A, B and C are as defined in eq. (26).

4. Noether/gauge symmetries, conservation laws
and isometries

It is well known that for a variational system, the Lie pint
symmetry that leaves the action integral invariant leads
to a conservation law via Noether’s theorem. In this case,
they are called variational or Noether symmetries [1,3,
4,12]. Since the geodesic equations are derived from a
metric, we have a ‘natural’ Lagrangian, with a = 1,
arising from (1), viz.,

L = ṫ2 − ṙ2 − ż2 + (2 sinh4 r − sinh2 r cosh2 r)φ̇2

+(2
√

2 sinh2 r)ṫ φ̇. (39)

L is substituted into a Killing-type equation given by

XL + LDsξ = Ds f, (40)

where Ds is the total derivative and f (s, t, r, z, φ) is a
gauge term. The conserved quantity or first integral is
then given by

T = Lξ + (ηα − ẋαξ)
∂L

∂ ẋα
− f, (41)

where, by definition,

DsT = 0 (42)

along the solutions of geodesic equations (2).
Alternative to determining X by eq. (40), one may

utilise the ‘multiplier approach’ [4] which determines
the multipliers that render a system of equations to be
closed/conserved. Here, each multiplier is of the form
(ηα − ẋαξ) and is a ‘symmetry’ since the system is
variational. The multiplier approach leads to table 1.
The method does not explicitly generate the gauge
functions.

The ‘energy’, from table 1 isT = −( ṫ+ √
2φ̇ sinh2 r).

The T s in table 1 are constants along the solutions of
the geodesic equations.

Moreover, according to Cartan’s theory, there exists a
first integral, Xa ẋa for each Lie point symmetry gener-
ator X = ξa∂a satisfying the Killing equations [1].

Xa;b + Xb;a = 0, (43)

where solution X of eq. (43) is an isometry of the space–
time metric and is called a Killing vector. It was shown
that for a natural Lagrangian, the Killing vectors of the
metric form a sub-algebra of the Noether symmetries
[23]. If the metric admits enough isometries, they are
used to instantly reduce the order of the geodesic system
by one, leading to a first-order system. Gödel’s metric
possesses five isometries which are the symmetry gener-
ators Xi where i = 4, . . . , 7, 10 in eq. (6), providing five
corresponding first integrals. This reduces the geodesic
system to

ṫ = c2

[
1 − 2 sinh2 r

cosh2 r

]
+

√
2c1

cosh2 r
, (44a)

φ̇ =
√

2c2

cosh2 r
− c1

sinh2 r cosh2 r
, (44b)

ṙ2 = c2
2 − c3 −

(√
2c2 sinh r

cosh r
− c1

sinh r cosh r

)2

,

(44c)

ṙ = − (a cos φ + b sin φ) , (44d)

where c1, c2, c3, a and b are constants. It is notable that



  121 Page 6 of 7 Pramana – J. Phys.          (2022) 96:121 

Table 1. First integrals corresponding to Noether/gauge symmetries of system (2).

X T

X3 ṫ2 − ṙ2 − ż2 + φ̇2 sinh2 r(sinh2 −1) + 2
√

2ṫ φ̇ sinh2 r
X4 ṙ cos φ + 2

√
2ṫ sin φ sinh r cosh r + φ̇ sin φ sinh r cosh r(2 sinh2 r − 1)

X5 ṙ sin φ − 2
√

2ṫ cos φ sinh r cosh r − φ̇ cos φ sinh r cosh r(2 sinh2 r − 1)

X6 − sinh2 r((sinh2 r − 1)φ̇ + √
2ṫ)

X7 −( ṫ + √
2φ̇ sinh2 r)

X8 sż
X10 sż − z

eq. (44c) is identical to eq. (17) with

	1 = c2, 	2 = c2
2 − c3, 	3 = −(2c1 + √

2c2).

Thus, solving this equation with eqs (44a) and (44b)
gives the complete solution for the geodesic system.

5. Conclusion

The geodesic system for Gödel’s metric is shown
to admit a 10-dimensional Lie algebra which con-
tains a solvable sub-algebra of dimension seven. We
focussed our attention on solving this system in the
hypersurface z = constant by making use of the admit-
ted Lie point symmetries. Applying Lie’s integration
theorem requires the admitted transitive solvable Lie
algebra to be of at least six dimensions, which is
not the case here and thus the solution cannot be
found through line integrals. Method of successive
reduction reduces the problem to a system of two
nonlinear first-order ODEs. Differential invariants pro-
vide two first integrals which reduce the problem to
an autonomous second-order ODE. The invariance of
this latter ODE under translation along the indepen-
dent variable reduces it to an integrable first-rder ODE,
which form an additional functionally-independent first
integral for the geodesic system. Accordingly, these
first integrals provide a complete analytic solution for
the time-like geodesics. Besides, variational symme-
tries via Noether theorem are found which constitute
a sub-algebra of Lie point symmetries. These provide
conserved quantities (first integrals) of the geodesic sys-
tem. Furthermore, isometries of the space–time, which
form a sub-algebra of Noether symmetries, are singled
out of the Lie point symmetries and used to find the
first integrals, thus reducing the order of the geodesic
system by one.

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	Conservation laws and solution of the geodesic system of Gödel's metric via Lie and Noether symmetries
	Abstract.
	1. Introduction
	2. Lie point symmetries of the geodesic system
	3. Solving the geodesic system using H6
	4. Noether/gauge symmetries, conservation laws and isometries
	5. Conclusion
	References