On the Non-flatness Nature of Noncommutative
Minkowski Spacetime

Manali<EMAIL_ADDRESS>and B<EMAIL_ADDRESS>
Department of Physics,
School of Physical, Chemical and Applied Sciences,
Pondicherry University, Puducherry-605014, India

###### Abstract

In the framework of twisted-diffeomorphism approach to noncommutative gravity
with canonical/Moyal-Weyl type noncommutative (NC) coordinate structure, we
show that the NC Minkowski spacetime parametrized either with spherical polar
coordinates or with parabolic coordinates has nontrivial NC corrections to
Riemann curvature tensor, Ricci tensor and curvature scalar. Apparently, there
are no such corrections if we choose rectilinear coordinates or even
cylindrical coordinates, for which the metric in the commutative-counterpart
is dependent on at most one coordinate only. We present both first order and
second order calculations. The emergent curvature corrections might seem to
raise the question of whether the statement of curvature is coordinate-
dependent, but note, for example, that the transformation from spherical polar
system to Cartesian system is not a diffeomorphism as such since its non-
injective nature makes it a local diffeomorphism. In other words, if the flat-
spacetime metric tensor in the commutative case depends on more than one
curvilinear coordinates, the introduction of noncommutativity among these
coordinates can possibly make the spacetime curved. It is worth remarking that
such a curvature emerges in the context of NC Minkowski spacetime in the
absence of any gauge or matter fields. It is purely an NC geometric effect.

##  1 Introduction

Spacetime noncommutativity is expected to play a significant role in the
physics of Planck length scale [1]. In addition to the possibility of a
minimal length associated with such noncommutative (NC) spacetimes, the
quantization of length also shows up in certain cases [2]. On the one hand,
the idea of spacetime noncommutativity emerges from the attempts to unify the
gravity and the quantum [3], the emergent gravity phenomena in noncommutative
gauge field theories, on the other hand, take the quantum field theories a
step closer to bring the gravitational effects into the quantum field theories
[4]-[13], and in the same vein, the gravitational theories in NC spacetimes
are expected to bring the quantum effects into the gravitational theories.

There are a number of different approaches to NC gravity. With the idea of the
Seiberg-Witten map [14] being extended to enveloping algebra [15], gauge
theories of gravity in NC spacetime have been studied by mapping the
commutative gauge theories of gravitation to their NC counterparts. In one
approach, the noncommutative SO(4,1) de sitter group is gauged and contracted
to the Poincare group ISO(3,1) using the Seiberg-Witten map [16]. Using the
concept of twisted Poincare algebra [17, 18] and extending the idea to general
coordinate transformations, gravity theory in NC spacetime is also constructed
[19]. When commutation relations between the spacetime coordinates have a Lie
Algebra structure, there is a class of volume preserving general coordinate
transformation which preserves the symmetry of the noncommutative algebra [16,
20, 21, 22]. The theory of gravity corresponding to volume preserving
diffeomorphism is unimodular theory of gravity. In this case NC General theory
of relativity is constructed by gauging the Lorenz gauge group SO(3,1) in the
enveloping algebra framework. Two dimensional NC gravity theories have also
been proposed based on NC gauge theories [23].

Since the general theory of relativity is invariant under diffeomorphism, a
theory of noncommutative gravity is formulated using the mathematical
framework of twist-deformed diffeomorphism [19]. In this theory of
gravitation, which is invariant under deformed diffeomorphism, Christoffel
symbol, Riemann curvature tensor, Ricci tensor and curvature scalar etc., are
mathematically constructed in NC spacetime and the effect of non-commutation
of spacetime coordinates manifests as the NC corrections to these geometrical
elements.

In our work, we consider the twist-deformed diffeomorphism framework of NC
gravity in Moyal-Weyl-like canonical noncommutative spacetime. We briefly
review the idea of twist-deformed diffeomorphism in Section 2. The key point
in Section 3 is that when the metric tensor components are functions of only a
subset of coordinates that commute among themselves, there are no NC
corrections to curvature tensors and curvature scalar even if the spacetime
has other noncommuting coordinate pairs. In this case the NC gravity theory is
the same as the classical theory in the pure gravity/geometry sector. Only the
interaction with other gauge or matter can make the noncommutativity
observable. Noncommutative effects take place in the pure gravity/geometry
sector only if the metric tensor depends on at least two coordinates that do
not commute.

In Section 4, we deal with the NC curvature corrections to NC Minkowski
spacetime parametrized by spherical polar coordinates. We have done both first
order and second order noncommutative corrections to inverse metric tensor,
Christoffel symbols, Riemann curvature tensor, Ricci tensor and curvature
scalar in the twist-deformed diffeomorphism framework of NC gravity. We repeat
these calculations in Section 5 for the NC Minkowski spacetime parametrized by
parabolic coordinates. All of these calculations are based on the formulas
derived in [19].

We have extensively used Maple 2021 to get our results. All of the results
presented in Sections 4 and 5 are the ones displayed as outputs in Maple
(except for the indicial values that we denote like $(r,\theta,\phi,t)$ for
the ease of understanding). Some of these results have been cross-checked by
manual calculations to verify the Maple code.

The main result of our calculations is that the NC corrections to Riemann
curvature tensor, Ricci tensor and the curvature scalar do not vanish for the
NC Minkowski spacetime parametrized either by spherical polar or by parabolic
coordinates. This turns out to be the general feature of a space in which the
metric tensor is a function of at least two noncommuting coordinates. The
concluding remarks are given in Section 6.

##  2 Deformed Diffeomorphism on NC Spacetime

NC spacetime is defined by the commutation relations among spacetime
coordinates:

$\displaystyle\left[\hat{x}^{\mu},~{}\hat{x}^{\nu}\right]=i\Theta^{\mu\nu},$
(1)

where $(\Theta_{\mu\nu})$ is a constant and real antisymmetric matrix. The
above operator algebra can be realized on a linear space of complex functions
$f(x)$ of commuting variables if the algebra of functions is given a
noncommutative multiplication structure called Moyal-Weyl star-product
(*-product) [24]:

$\displaystyle
f(x)*g(x)=\displaystyle\left.e^{\frac{i}{2}\Theta^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\frac{\partial}{\partial
x^{\prime\nu}}}f(x)g(x^{\prime})\right|_{x^{\prime}=x}.$ (2)

The concept of deformed diffeomorphism relies on the idea that the general
coordinate transformations in general relativity are, in general,
diffeomorphism transformations. If $\delta_{\xi}$ denotes the infinitesimal
change in a scalar field $\phi(x)$ under the infinitesimal general coordinate
transformation $x^{\mu}\to x^{\prime\mu}=x^{\mu}+\xi^{\mu}(x)$, then the
invariance relation $\phi^{\prime}(x^{\prime})=\phi(x)$ implies the field
transformation
$\phi(x)\to\phi^{\prime}(x)=\phi(x)+\delta_{\xi}\phi=\phi(x)-\xi^{\mu}(x)\partial_{\mu}\phi$.
Also, for the pointwise multiplication of two transformed fields, the
following equality holds:

$\displaystyle\phi^{\prime}(x)\cdot\chi^{\prime}(x)=\phi(x)\cdot\chi(x)+\delta_{\xi}\left(\phi(x)\cdot\chi(x)\right)$
(3)

because of the Leibniz rule

$\displaystyle(\delta_{\xi}\phi)\cdot\chi+\phi\cdot(\delta_{\xi}\chi)=\delta_{\xi}(\phi\cdot\chi).$
(4a) If $\mu$ denotes the map that maps a tensor product to the pointwise
multiplication of commutative algebra, i.e.,
$\mu(\phi\otimes\chi)=\phi\cdot\chi$, then using the coproduct
$\triangle(\delta_{\xi})=\delta_{\xi}\otimes 1+1\otimes\delta_{\xi}$, Eq.(4a)
from right to left can be written as
$\displaystyle\delta_{\xi}(\phi\cdot\chi)=\delta_{\xi}\mu\\{\phi\otimes\chi\\}=\mu\\{(\delta_{\xi}\otimes
1+1\otimes\delta_{\xi})(\phi\otimes\chi)\\}=\mu\\{\triangle(\delta_{\xi})(\phi\otimes\chi)\\}.$
(4b)

The key point here is the commutation relation
$\delta_{\xi}\mu=\mu\triangle(\delta_{\xi})$. For three successive
infinitesimal transformations involving $-\xi,\eta$ and $\xi$, the commutator
$[\delta_{\xi},\delta{\eta}]=\delta_{\xi\times\eta}$ implies the closure
property of the algebra of vector fields $\xi^{\mu}\partial_{\mu}$ acting on a
differential manifold. Essentially, the commutator denotes the algebra of
diffeomorphism, and its coproduct-counterpart is worked out to be
$[\triangle(\delta_{\xi}),\triangle(\delta_{\eta})]=\triangle(\delta_{\xi\times\eta})$.
The Leibniz rule (4) holds regardless of whether $\delta_{\xi}$, represented
by $-\xi^{\mu}(x)\partial_{\mu}$, is independent of $x$ or not. But if
$\delta_{\xi}$ depends on $x$ and the pointwise product of the fields is
replaced by *-product, then Eq.(4) does not hold, i.e.,
$(\delta_{\xi}\phi)*\chi+\phi*(\delta_{\xi}\chi)\neq\delta_{\xi}(\phi*\chi)$.
This is exactly where the concept of twist-deformed diffeomorphism comes into
the picture.

If $\mu_{*}$ denotes the mapping of tensor product to the *-product of
noncommutative coordinate algebra, i.e.,

$\displaystyle\mu_{*}\\{\phi\otimes\chi\\}=\mu\mathcal{F}\\{\phi\otimes\chi\\}=\phi*\chi,$
(5)

where $\mathcal{F}=e^{\frac{i}{2}\Theta^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial x^{\nu}}}$ is called the Drinfeld
twist, then using the twisted coproduct defined by

$\displaystyle\triangle_{t}(\delta_{\xi})=e^{-\frac{i}{2}\Theta^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial x^{\nu}}}\left(\delta_{\xi}\otimes
1+1\otimes\delta_{\xi}\right)e^{\frac{i}{2}\Theta^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial
x^{\nu}}}=\mathcal{F}^{-1}\left(\delta_{\xi}\otimes
1+1\otimes\delta_{\xi}\right)\mathcal{F},$ (6)

the analogue of (4b) in noncommutative algebra can be written as

$\displaystyle\delta_{\xi}u_{*}\\{\phi\otimes\chi\\}=\delta_{\xi}\mu\\{\mathcal{F}(\phi\otimes\chi)\\}=\mu\triangle(\delta_{\xi})\\{\mathcal{F}(\phi\otimes\chi)\\}=\mu_{*}\\{\triangle_{t}(\delta_{\xi})(\phi\otimes\chi)\\},$
(7)

where an $\mathcal{F}\mathcal{F}^{-1}(=1)$ has been inserted into the last
equality. In order to take the $\delta_{\xi}$ out of the tensor product on the
rightmost side of Eq.(7), to the left of $\mu_{*}$, as in the leftmost side of
Eq.(7), the coproduct needs to be twisted:
$\mu_{*}\triangle_{t}(\delta_{\xi})=\delta_{\xi}\mu_{*}$. The above rule
Eq.(7) is easily generalized to the product of arbitrary tensor fields leading
to a covariant construction of theories.

Our idea here is not to review the entire formulation of twist-deformed
diffeomorphism algebra, but to stress how the twist-deformation of the tensor
product is at the core of this formalism and how it is linked to the
*-product.

In this twisted-diffeomorphism approach to NC gravity, the basic element is
the classical vierbein $e^{~{}a}_{\mu}(x)$, also called tetrad in 3+1
dimensions, and this is taken to be the full NC version of vierbein to all
orders in the NC parameter $\Theta$ [19]. But the metric in NC spacetime is
defined to have the following symmetric form with $\Theta$-dependency:

$G_{\mu\nu}=\frac{1}{2}\left(e^{~{}a}_{\mu}*e^{~{}b}_{\nu}+e^{~{}a}_{\nu}*e^{~{}b}_{\mu}\right)\,\eta_{ab},$
(8)

with its zeroth order being the metric in the commutative case, i.e.,
$G^{(0)}_{\mu\nu}=g_{\mu\nu}=e^{~{}a}_{\mu}\cdot e^{~{}a}_{\nu}\,\eta_{ab}$,
where $\eta_{ab}$ is the constant symmetric metric of flat Minkowski
spacetime.

With the conditions that the metric is a covariant tensor of rank two, that
its inverse $G^{\mu\nu*}$ is also a tensor of rank two such that
$G_{\mu\nu}*G^{\nu\rho*}=\delta_{\mu}^{\rho}$, and that the covariant
derivative of $G_{\mu\nu}$ is zero, the other elements like Christoffel
symbols, covariant derivative, Riemann curvature tensor, Ricci tensor and
curvature scalar are constructed in terms of the metric tensor and the inverse
metric tensor [19].

##  3 Remarks on the Tetrads and the Metrics

The metric tensors of the four-dimensional spacetime considered in this paper
are diagonal and the tetrads associated with these metrics are also diagonal.

###  3.1 At most One Coordinate Dependency of Tetrads

When the tetrads are constants, it is clear from (8) that the metric tensor
$G_{\mu\nu}=g_{\mu\nu}$ to all orders in $\Theta$. Other geometrical elements
are also not different from the commutative counterparts. The Cartesian system
of coordinates falls under this category.

If the tetrads depend on only one coordinate, then also
$e^{~{}a}_{\mu}*e^{~{}b}_{\nu}=e^{~{}a}_{\mu}\cdot e^{~{}b}_{\nu}$ and all the
geometrical elements are the same as in the commutative theory. Because any
nontrivial $\Theta$-corrections require the basic element $e^{~{}a}_{\mu}$ to
be dependent on at least two distinct coordinates such that the corresponding
$\Theta^{\alpha\beta}\neq 0$. An example for this type is the cylindrical
system of coordinates.

In the above two cases, only the interactions with other fields can reveal the
underlying NC structure of spacetime.

###  3.2 More than One Coordinate Dependency of Tetrads

If the vierbein depends only on $x^{1}$ and $x^{2}$ and if these two
coordinates commute, then in the pure gravity sector there are no NC
corrections even if the spacetime has other noncommuting coordinate pairs.
Because all the basic geometric elements are constructed out of the *-product
of tetrads as in Eq.(8) and

$\displaystyle
e^{~{}a}_{\mu}(x^{1},x^{2})*e^{~{}b}_{\nu}(x^{1},x^{2})=e^{~{}a}_{\mu}e^{~{}b}_{\nu}~{}~{}\mathrm{if}~{}~{}\Theta^{12}=0.$
(9)

So the metric tensor will have no $\Theta$-dependency. The inverse metric,
which has an expansion in $\Theta$ that depends only on the derivatives of
$e^{~{}a}_{\mu}$ in its expressions for $\Theta$-corrections [19], will also
have no $\Theta$-corrections. The reason is both derivatives with respect to
$x^{\alpha}$ and $x^{\beta}$ appear as multiplicative factors in each of the
terms involving $\Theta^{\alpha\beta}$, and it is either that for non-trivial
derivative $\Theta^{\alpha\beta}$ is zero or that for nontrivial
$\Theta^{\alpha\beta}$ the derivative is zero, making each term to vanish. So
the metric and the inverse metric will be the same as their commutative-
counterparts. By the same token, the Christoffel symbols and Riemann tensor
will also be the same as their commutative-counterparts. Only the interaction
with other gauge or matter fields, which in general depend on all coordinates,
can reveal such underlying noncommuting structure of coordinates if
$\Theta^{\alpha\beta}\neq 0$ for other pairs of distinct values of $\alpha$
and $\beta$.

_But if the vierbein $e^{~{}a}_{\mu}$ is a function of, for example, $x^{1}$
and $x^{2}$, and if these two coordinates are promoted as noncommuting
operators, then nontrivial NC curvature corrections do show up._

##  4 NC Minkowski Spacetime in Spherical Polar Coordinates

The convention we adopt is
$x^{\mu}=(x^{1},x^{2},x^{3},x^{4})=(r,\theta,\phi,t),$ in which the spacetime
indices take the values $1,2,3$ and $4$. The signature of $\eta_{ab}$ is taken
to be $(+,+,+,-)$. In the commutative case, the tetrad, the metric tensor and
the inverse of the metric tensor in this system of coordinates coordinates are
given by

$\displaystyle e^{~{}a}_{\mu}=\\!\left[\begin{array}[]{cccc}1&0&0&0\\\
0&r&0&0\\\ 0&0&r\sin\theta&0\\\
0&0&0&-c\end{array}\right]\\!\\!;~{}g_{\mu\nu}=\\!\left[\begin{array}[]{cccc}1&0&0&0\\\
0&r^{2}&0&0\\\ 0&0&r^{2}\sin^{2}\theta&0\\\
0&0&0&-c^{2}\end{array}\right]\\!\\!;~{}g^{\mu\nu}\\!=\\!\left[\begin{array}[]{cccc}1&0&0&0\\\
0&\frac{1}{r^{2}}&0&0\\\ 0&0&\frac{1}{r^{2}\sin^{2}\theta}&0\\\
0&0&0&\frac{-1}{c^{2}}\end{array}\right]\\!\\!.$ (22)

There are two reasons why this coordinate system is chosen for the curvature
analysis: 1) The tetrad depends on more than one coordinate. 2) The Jacobian
determinant of the transformation
$(r,\theta,\phi,t)\longrightarrow(r\sin\theta\cos\phi,\,r\sin\theta\sin\phi,\,r\cos\theta,\,t)$,
which is equal to $r^{2}\sin\theta$, vanishes in a nontrivial subset of the
domain of the transformation.

In the commutative case, the non-zero Christoffel symbols of the second kind
are given by (see for example [25]):

$\displaystyle\Gamma_{\theta\theta}^{(0)}{}^{r}$ $\displaystyle=-r,$
$\displaystyle\qquad\Gamma_{\phi\phi}^{(0)}{}^{r}$
$\displaystyle=-r\sin^{2}\theta,$
$\displaystyle\qquad\Gamma_{r\theta}^{(0)}{}^{\theta}$
$\displaystyle=\frac{1}{r},$ (23a)
$\displaystyle\Gamma_{\phi\phi}^{(0)}{}^{\theta}$
$\displaystyle=-\sin\theta\cos\theta,$
$\displaystyle\Gamma_{r\phi}^{(0)}{}^{\phi}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\Gamma_{\theta\phi}^{(0)}{}^{\phi}$ $\displaystyle=\cot\theta,$
(23b)

with the proviso that
$\Gamma_{\mu\nu}^{(0)}{}^{\rho}=\Gamma_{\nu\mu}^{(0)}{}^{\rho}$. The Riemann
tensor $R_{\mu\nu\rho}^{(0)}{}^{\sigma}$, the Ricci tensor $R_{\mu\nu}^{(0)}$
and the curvature scalar $R^{(0)}$ are all zero.

Since the tetrad depends only on $r$ and $\theta$, when the coordinates are
promoted as noncommuting operators, the only noncommutation relation that will
lead to NC curvature effects is

$\displaystyle[\hat{r},~{}\hat{\theta}]=i\Theta^{r\theta}.$ (24)

No other nontrivial $\Theta^{\mu\nu}$ will result in any change in the nature
of curvature of the spacetime. If $\Theta^{r\theta}=0$, then the NC Minkowski
spacetime is flat regardless of the presence of any other nontrivial
$\Theta^{\mu\nu}$ in the theory.

###  4.1 First Order NC Corrections

####  4.1.1 Inverse Metric Tensor and Christoffel Symbols

The metric tensor in the NC case, Eq.(8), is symmetric and real by its
construction. Therefore, there is no first order correction to it.

The first order NC correction to the inverse of metric tensor is given by [19]

$\displaystyle
G^{(1)\mu\nu*}=-\frac{i}{2}\Theta^{\alpha\beta}\partial_{\alpha}\left(g^{\gamma\mu}\right)\partial_{\beta}\left(g_{\delta\gamma}\right)g^{\delta\nu}.$
(25)

Substituting for the zeroth order metric and inverse metric components from
Eq.(22) into Eq.(25) gives $G^{(1)\mu\nu*}$=0, i.e., there are no first order
corrections to the inverse metric.

The Christoffel symbols of the second kind have the following expression for
their first order correction [19]:

$\displaystyle\Gamma_{\mu\nu}^{(1)\rho}=\frac{i}{2}\Theta^{\alpha\beta}\left(\partial_{\alpha}\Gamma_{\mu\nu}^{(0)\sigma}\right)g_{\sigma\tau}(\partial_{\beta}g^{\rho\tau}),$
(26)

and these are symmetric under the exchange of $\mu$ and $\nu$. For the
spacetime under consideration, the independent non-zero symbols are calculated
to be

$\displaystyle\Gamma_{\phi\phi}^{(1)\theta}=i\Theta^{r\theta}\frac{(1-2\cos^{2}\theta)}{r},\qquad\Gamma_{r\phi}^{(1)\phi}=i\Theta^{r\theta}\frac{\cos\theta}{r^{2}\sin\theta},$
$\displaystyle\qquad\Gamma_{\theta\phi}^{(1)\phi}=-i\Theta^{r\theta}\frac{1}{r\sin^{2}\theta},$
(27)

i.e., only the zeroth order symbols corresponding to Eq.(23b) have first order
$\Theta$-corrections.

####  4.1.2 Curvature Tensors and Curvature Scalar

Riemann tensor has the following first order expression [19]

$\displaystyle\begin{aligned}
R_{\mu\nu\rho}^{(1)}{}^{\sigma}&=-\frac{i}{2}\Theta^{\kappa\lambda}\bigg{(}(\partial_{\kappa}R_{\mu\nu\rho}^{(0)}{}^{\tau})(\partial_{\lambda}g_{\tau\gamma})g^{\gamma\sigma}\bigg{)}+\frac{i}{2}\Theta^{\kappa\lambda}\bigg{(}(\partial_{\kappa}\Gamma_{\nu\rho}^{(0)}{}^{\beta})\Big{(}\Gamma_{\mu\beta}^{(0)}{}^{\tau}(\partial_{\lambda}g_{\tau\gamma})g^{\gamma\sigma}\\\
&~{}~{}~{}-\Gamma_{\mu\tau}^{(0)}{}^{\sigma}(\partial_{\lambda}g_{\beta\gamma})g^{\gamma\tau}+\partial_{\mu}\big{(}(\partial_{\lambda}g_{\beta\gamma})g^{\gamma\sigma}\big{)}+(\partial_{\lambda}\Gamma_{\mu\beta}^{(0)}{}^{\sigma})\Big{)}-\left(\mu\leftrightarrow\nu\right)\bigg{)}.\end{aligned}$
(28)

It is antisymmetric under the exchange of $\mu$ and $\nu$. There are as many
as 14 nonzero components and 7 of them are independent for the spacetime under
consideration. They are the following:

$\displaystyle\begin{aligned}
R_{r\theta\phi}^{(1)}{}^{\phi}&=\frac{-3\,i\,\Theta^{r\theta}}{r^{2}\sin^{2}\theta}~{},&R_{r\phi\theta}^{(1)}{}^{\phi}&=\frac{i\,\Theta^{r\theta}(\cos^{2}\theta-2)}{r^{2}\sin^{2}\theta}~{},\\\
R_{r\phi\phi}^{(1)}{}^{\theta}&=\frac{i\,\Theta^{r\theta}(2-3\cos^{2}\theta)}{r^{2}}~{},&R_{\theta\phi
r}^{(1)}{}^{\phi}&=\frac{i\,\Theta^{r\theta}(1+\cos^{2}\theta)}{r^{2}(1-\cos^{2}\theta)}~{},\\\
R_{\theta\phi\theta}^{(1)}{}^{\phi}&=\frac{i\Theta^{r\theta}\cot\theta}{r}~{},&R_{\theta\phi\phi}^{(1)}{}^{r}&=-i\Theta^{r\theta}\sin^{2}\theta~{},\\\
R_{\theta\phi\phi}^{(1)}{}^{\theta}&=\frac{i\Theta^{r\theta}\sin\theta\cos\theta}{r}~{}.\end{aligned}$
(29)

The first order correction to the Ricci tensor is simply given by the sum over
two indices in the first order Riemann tensor,
$R_{\mu\nu}^{(1)}=R_{\mu\sigma\nu}^{(1)}{}^{\sigma}$, and it has 4 nonzero
components:

$\displaystyle\begin{aligned}
R_{r\theta}^{(1)}&=\frac{i\,\Theta^{r\theta}(\cos^{2}\theta-2)}{r^{2}\,\sin^{2}\theta}~{},&R_{\theta
r}^{(1)}&=\frac{i\,\Theta^{r\theta}(1+\cos^{2}\theta)}{r^{2}\,(1-\cos^{2}\theta)}~{},\\\
R_{\theta\theta}^{(1)}&=i\,\Theta^{r\theta}\,\frac{\cot\theta}{r}~{},&R_{\phi\phi}^{(1)}&=-i\,\Theta^{r\theta}\,\frac{\sin\theta\cos\theta}{r}.\end{aligned}$
(30)

The first order correction to the curvature scalar is, however, zero. This can
be easily seen since in our case
$R^{(1)}=G^{\mu\nu*(1)}R_{\nu\mu}^{(0)}+g^{\mu\nu}R_{\nu\mu}^{(1)}+\tfrac{i}{2}\Theta^{\alpha\beta}(\partial_{\alpha}g^{\mu\nu})(\partial_{\beta}R_{\nu\mu}^{(0)})=g^{\theta\theta}R_{\theta\theta}^{(1)}+g^{\phi\phi}R_{\phi\phi}^{(1)}=0$.

###  4.2 Second Order NC Corrections

####  4.2.1 Metric, Inverse Metric and Christoffel Symbols

Because of the nature of the definition Eq.(8) of the metric tensor in the NC
spacetime, it has the second order correction [19]:

$\displaystyle
G_{\mu\nu}^{(2)}=-\frac{1}{8}\theta^{\alpha_{1}\beta_{1}}\theta^{\alpha_{2}\beta_{2}}(\partial_{\alpha_{1}}\partial_{\alpha_{2}}e_{\mu}^{\
a})(\partial_{\beta_{1}}\partial_{\beta_{2}}e_{\nu}^{\ b})\eta_{ab}\,.$ (31)

For the case under consideration, the only nonzero component is

$\displaystyle
G_{\phi\phi}^{(2)}=\frac{1}{4}(\Theta^{r\theta})^{2}\cos^{2}\theta.$ (32)

The second order correction to the inverse metric tensor in the general case
is given by [19]:

$\displaystyle\begin{aligned}
G^{(2)\mu\nu\star}&=\frac{1}{8}\theta^{\alpha_{1}\beta_{1}}\theta^{\alpha_{2}\beta_{2}}\bigg{(}(\partial_{\alpha_{1}}\partial_{\alpha_{2}}g^{\mu\gamma})(\partial_{\beta_{1}}\partial_{\beta_{2}}g_{\gamma\eta})+g^{\mu\gamma}(\partial_{\alpha_{1}}\partial_{\alpha_{2}}e_{\gamma}{}^{a})(\partial_{\beta_{1}}\partial_{\beta_{2}}e_{\eta}{}^{b})\eta_{ab}\\\
&~{}~{}~{}-2\partial_{\alpha_{1}}\left((\partial_{\alpha_{2}}g^{\mu\gamma})(\partial_{\beta_{2}}g_{\gamma\delta})g^{\delta\epsilon}\right)(\partial_{\beta_{1}}g_{\epsilon\eta})\bigg{)}g^{\eta\nu},\end{aligned}$
(33)

and for the NC Minkowski spacetime in spherical polar coordinates, the only
component that has nontrivial correction is

$\displaystyle
G^{(2)\phi\phi\star}=-\left(\Theta^{r\theta}\right)^{2}\frac{(4+\cos^{2}\theta)}{4\,r^{4}\,\sin^{4}\theta}.$
(34)

The Christoffel symbols of the second kind have a rather lengthy expression
for its second order correction [19]. We give here only the independent
nonzero components:

$\displaystyle\Gamma_{\phi\phi}^{(2)\theta}=\frac{-13\,(\Theta^{r\theta})^{2}\sin
2\theta}{8\,r^{2}},~{}~{}\Gamma_{r\phi}^{(2)\phi}=\frac{(\Theta^{r\theta})^{2}(2-5\cos^{2}\theta)}{4\,r^{3}\,\sin^{2}\theta},~{}~{}\Gamma_{\theta\phi}^{(2)\phi}=\frac{-3\,(\Theta^{r\theta})^{2}\cos\theta}{4\,r^{2}\,\sin^{3}\theta}.$
(35)

So it turns out that only those Christoffel symbols that have first order
corrections acquire nontrivial second order corrections as well.

####  4.2.2 Curvature Tensors and Curvature Scalar

The following general expression for the second order correction to Riemann
tensor [19] involves only the Christoffel symbols and their derivatives,

$\displaystyle\begin{aligned}
R_{\mu\nu\rho}^{(2)}{}^{\sigma}&=\partial_{\nu}\Gamma_{\mu\rho}^{(2)}{}^{\sigma}+\Gamma_{\nu\rho}^{(2)}{}^{\gamma}\Gamma_{\mu\gamma}^{(0)}{}^{\sigma}+\Gamma_{\nu\rho}^{(0)}{}^{\gamma}\Gamma_{\mu\gamma}^{(2)}{}^{\sigma}\\\
&~{}~{}~{}~{}+\frac{i}{2}\theta^{\alpha\beta}\Big{(}(\partial_{\alpha}\Gamma_{\nu\rho}^{(1)}{}^{\gamma})(\partial_{\beta}\Gamma_{\mu\gamma}^{(0)}{}^{\sigma})+(\partial_{\alpha}\Gamma_{\nu\rho}^{(0)}{}^{\gamma})(\partial_{\beta}\Gamma_{\mu\gamma}^{(1)}{}^{\sigma})\Bigr{)}\\\
&~{}~{}~{}~{}-\frac{1}{8}\theta^{\alpha_{1}\beta_{1}}\theta^{\alpha_{2}\beta_{2}}(\partial_{\alpha_{1}}\partial_{\alpha_{2}}\Gamma_{\nu\rho}^{(0)}{}^{\gamma})(\partial_{\beta_{1}}\partial_{\beta_{2}}\Gamma_{\mu\gamma}^{(0)}{}^{\sigma})\quad-(\mu\leftrightarrow\nu).\end{aligned}$
(36)

These are antisymmetric under the exchange of $\mu$ and $\nu$. On the whole,
there are 18 nonzero components in the present case and 9 of them are
independent. They are as follows:

$\displaystyle R_{r\theta\phi}^{(2)}{}^{\phi}$
$\displaystyle=\frac{-(\Theta^{r\theta})^{2}\,\cos\theta}{r^{3}\,\sin^{3}\theta},$
$\displaystyle R_{r\phi r}^{(2)}{}^{\phi}$
$\displaystyle=\frac{9\,(\Theta^{r\theta})^{2}\,(2-5\cos^{2}\theta)}{4\,r^{4}\,\sin^{2}\theta},$
$\displaystyle R_{r\phi\theta}^{(2)}{}^{\phi}$
$\displaystyle=\frac{-(\Theta^{r\theta})^{2}\,(5\cos^{2}\theta+16)\cos\theta}{4\,r^{3}\,\sin^{3}\theta},$
$\displaystyle R_{r\phi\phi}^{(2)}{}^{r}$
$\displaystyle=\frac{(\Theta^{r\theta})^{2}\,(7\,\cos^{2}\theta+2)}{4\,r^{2}},$
$\displaystyle R_{r\phi\phi}^{(2)}{}^{\theta}$
$\displaystyle=\frac{5\,(\Theta^{r\theta})^{2}\,(21\,\cos^{2}\theta-20)\cos\theta}{4\,r^{3}\,\sin\theta},$
$\displaystyle R_{\theta\phi r}^{(2)}{}^{\phi}$
$\displaystyle=\frac{-(\Theta^{r\theta})^{2}\,(5\,\cos^{2}\theta+12)\cos\theta}{4\,r^{3}\,\sin^{3}\theta},$
$\displaystyle R_{\theta\phi\theta}^{(2)}{}^{\phi}$
$\displaystyle=\frac{(\Theta^{r\theta})^{2}\,(5\,\cos^{4}\theta-21\cos^{2}\theta+1)}{4\,r^{2}\,\sin^{4}\theta},$
$\displaystyle R_{\theta\phi\phi}^{(2)}{}^{r}$
$\displaystyle=\frac{-(\Theta^{r\theta})^{2}\,(5\,\cos^{2}\theta+6)\cos\theta}{4\,r\,\sin\theta},$
$\displaystyle R_{\theta\phi\phi}^{(2)}{}^{\theta}$
$\displaystyle=\frac{(\Theta^{r\theta})^{2}(55\cos^{2}\theta-43\cos^{4}\theta-15)}{4\,r^{2}\,\sin^{2}\theta}.$
(37)

The sum over the repeated index in $R_{\mu\sigma\nu}^{(2)}{}^{\sigma}$ leads
to nontrivial second order corrections to five Ricci tensor components:

$\displaystyle\begin{aligned}
R_{rr}^{(2)}&=\frac{9(\Theta^{r\theta})^{2}\,(2-5\cos^{2}\theta)}{4\,r^{4}\,\sin^{2}\theta},\qquad\qquad
R_{r\theta}^{(2)}=\frac{-(\Theta^{r\theta})^{2}(5\cos^{2}\theta+16)\cos\theta}{4\,r^{3}\,\sin^{3}\theta},\\\
R_{\theta
r}^{(2)}&=\frac{-(\Theta^{r\theta})^{2}\,(12+5\cos^{2}\theta)}{4\,r^{3}\,\sin^{3}\theta},\quad\qquad~{}R_{\theta\theta}^{(2)}=\frac{(\Theta^{r\theta})^{2}\,(5\cos^{4}\theta-21\cos^{2}\theta+1)}{4\,r^{2}\,\sin^{4}\theta},\\\
\\\
R_{\phi\phi}^{(2)}&=\frac{(\Theta^{r\theta})^{2}}{4\,r^{2}}\left[\frac{(43\cos^{4}\theta-55\cos^{2}\theta+15)}{\sin^{2}\theta}-(7\cos^{2}\theta+2)\right].\end{aligned}$
(38)

The curvature scalar in the second order is expressed as [19]

$\displaystyle\begin{aligned}
R^{(2)}&=G^{(2)\mu\nu\star}R_{\nu\mu}^{(0)}+g^{\mu\nu}R_{\nu\mu}^{(2)}+G^{(1)\mu\nu\star}R_{\nu\mu}^{(1)}\\\
&~{}~{}~{}+\frac{i}{2}\theta^{\alpha\beta}(\partial_{\alpha}g^{\mu\nu})(\partial_{\beta}R_{\mu\nu}^{(1)})-\frac{1}{8}\theta^{\alpha_{1}\beta_{1}}\theta^{\alpha_{2}\beta_{2}}(\partial_{\alpha_{1}}\partial_{\alpha_{2}}g^{\mu\nu})(\partial_{\beta_{1}}\partial_{\beta_{2}}R_{\mu\nu}^{(0)}).\end{aligned}$
(39)

Unlike the first order, this second order picks up nontrivial curvature scalar
correction and the result is

$\displaystyle R^{(2)}=\frac{4\,(\Theta^{r\theta})^{2}(7\cos 4\theta-11\cos
2\theta-2)}{r^{4}\,(\cos 4\theta-4\,\cos 2\theta+3)}.$ (40)

##  5 NC Minkowski Spacetime in Parabolic Coordinates

For the definition and discussion of this coordinate system in the 3-D
commutative case, see for example [26]. If we denote the Minkowski spacetime
coordinates as $x^{\mu}=(x^{1},x^{2},x^{3},x^{4})=(u,v,\phi,t),$ then the
tetrad can be compactly specified by

$\displaystyle
e^{~{}a}_{\mu}=\mathrm{diag}\left(\sqrt{u^{2}+v^{2}},~{}\sqrt{u^{2}+v^{2}},~{}uv,~{}-c\right),$
(41)

and the corresponding metric tensor and the inverse metric tensor are given by

$\displaystyle\begin{aligned}
\\!\\!g_{\mu\nu}=\mathrm{diag}\left(u^{2}+v^{2}\\!,\,u^{2}+v^{2}\\!,\,u^{2}v^{2}\\!,\,-c^{2}\right)\\!,~{}g^{\mu\nu}\\!=\mathrm{diag}\left(\frac{1}{u^{2}+v^{2}},\frac{1}{u^{2}+v^{2}},\frac{1}{u^{2}v^{2}},\frac{-1}{c^{2}}\right).\end{aligned}$
(42)

We choose this coordinate system because the corresponding tetrad is dependent
on two coordinates. Also, for the transformation
$(u,v,\phi,t)\longrightarrow(uv\cos\phi,uv\sin\phi,\tfrac{1}{2}(u^{2}-v^{2}),t)$
from parabolic system to Cartesian system, the Jacobian determinant, which is
equal to $(u^{2}+v^{2})uv$, vanishes in a nontrivial subset of the domain of
the transformation.

In addition to the symmetric nature of the metric, we also have
$g_{uu}=g_{vv}$. This results in 6 distinct nonzero Christoffel symbols in the
zeroth order case, and they are worked out to be

$\displaystyle\begin{aligned}
\Gamma_{uu}^{(0)}{}^{u}=-\Gamma_{vv}^{(0)}{}^{u}=\Gamma_{uv}^{(0)}{}^{v}&=\frac{u}{u^{2}+v^{2}},\qquad\Gamma_{uv}^{(0)}{}^{u}=-\Gamma_{uu}^{(0)}{}^{v}=&\Gamma_{vv}^{(0)}{}^{v}&=\frac{v}{u^{2}+v^{2}},~{}~{}\\\
\Gamma_{\phi\phi}^{(0)}{}^{u}&=\frac{-uv^{2}}{u^{2}+v^{2}},&\Gamma_{\phi\phi}^{(0)}{}^{v}&=\frac{-u^{2}v}{u^{2}+v^{2}},\\\
\Gamma_{u\phi}^{(0)}{}^{\phi}&=\frac{1}{u},&\Gamma_{v\phi}^{(0)}{}^{\phi}&=\frac{1}{v}.\end{aligned}$
(43)

But in this commutative case, the Riemann tensor $R_{\mu\nu\rho}^{(0)}{}^{u}$,
the Ricci tensor $R_{\mu\nu}^{(0)}$ and the curvature scalar $R^{(0)}$ are all
zero.

Let’s promote the two coordinates $u$ and $v$ as noncommuting operators with
the commutation relation

$\displaystyle[\hat{u},~{}\hat{v}]=i\Theta^{uv}.$ (44)

Any other nontrivial commutation relation in the theory is not of our concern
as they won’t be changing the curvature properties of the geometry.

###  5.1 First Order NC Corrections

####  5.1.1 Inverse Metric Tensor and Christoffel Symbols

Substitution of the zeroth order metric components Eq.(42) into Eq.(25)
results in $G^{(1)\mu\nu*}=0$, i.e., there are no first order corrections to
the inverse metric tensor.

The first order nonzero Christoffel symbols Eq.(26) are worked out to be

$\displaystyle\begin{aligned}
\Gamma_{uu}^{(1)}{}^{u}&=\frac{-i\Theta^{uv}\,v}{(u^{2}+v^{2})^{2}}=\Gamma_{uv}^{(1)}{}^{v}=-\,\Gamma_{vv}^{(1)}{}^{u},\quad&\Gamma_{uu}^{(1)}{}^{v}&=\frac{-i\Theta^{uv}\,u}{(u^{2}+v^{2})^{2}}=-\,\Gamma_{uv}^{(1)}{}^{u}=-\,\Gamma_{vv}^{(1)}{}^{v},\\\
\Gamma_{u\phi}^{(1)}{}^{\phi}&=\frac{i\Theta^{uv}}{u^{2}v},&\Gamma_{v\phi}^{(1)}{}^{\phi}&=\frac{-i\Theta^{uv}}{uv^{2}},\\\
\Gamma_{\phi\phi}^{(1)}{}^{u}&=\frac{i\Theta^{uv}\,v\,(v^{2}-2u^{2})}{(u^{2}+v^{2})^{2}},&\Gamma_{\phi\phi}^{(1)}{}^{u}&=\frac{i\Theta^{uv}\,v\,(v^{2}-2u^{2})}{(u^{2}+v^{2})^{2}}.\end{aligned}$
(45)

####  5.1.2 Curvature Tensors and Curvature Scalar

There are 7 distinct nonzero components of Riemann tensor in the first order:

$\displaystyle\begin{aligned}
R_{uvu}^{(1)}{}^{u}&=R_{uvv}^{(1)}{}^{v}=\frac{2i\Theta^{uv}}{(u^{2}+v^{2})^{2}},\quad&R_{uv\phi}^{(1)}{}^{\phi}&=\frac{-3i\Theta^{uv}}{u^{2}v^{2}},\\\
R_{u\phi
v}^{(1)}{}^{\phi}&=\frac{-i\Theta^{uv}(u^{2}+2v^{2})}{(u^{2}+v^{2})u^{2}v^{2}},&R_{u\phi\phi}^{(1)}{}^{u}&=R_{v\phi\phi}^{(1)}{}^{v}=\frac{-4i\Theta^{uv}(u^{2}-v^{2})uv}{(u^{2}+v^{2})^{3}},\\\
R_{u\phi\phi}^{(1)}{}^{v}&=\frac{-i\Theta^{uv}(u^{2}-7v^{2})u^{2}}{(u^{2}+v^{2})^{3}},&R_{v\phi
u}^{(1)}{}^{\phi}&=\frac{i\Theta^{uv}(2u^{2}+v^{2})}{(u^{2}+v^{2})u^{2}v^{2}},\\\
R_{v\phi\phi}^{(1)}{}^{u}&=\frac{-7i\Theta^{uv}\,v^{2}\,(u^{2}-v^{2}/7)}{(u^{2}+v^{2})^{3}}.\end{aligned}$
(46)

There are only 3 nonzero first order components for Ricci tensor:

$\displaystyle\begin{aligned}
R_{uv}^{(1)}&=\frac{-i\Theta^{uv}(u^{4}+u^{2}v^{2}+2v^{4})}{(u^{2}+v^{2})^{2}u^{2}v^{2}},\qquad
R_{vu}^{(1)}=\frac{i\Theta^{uv}(2u^{4}+u^{2}v^{2}+v^{4})}{(u^{2}+v^{2})^{2}u^{2}v^{2}},\\\
R_{\phi\phi}^{(1)}&=\frac{8i\Theta^{uv}(u^{2}-v^{2})uv}{(u^{2}+v^{2})^{3}}.\end{aligned}$
(47)

Unlike the case of spherical polar coordinates, the parabolic system leads to
nontrivial first order correction to the curvature scalar:

$\displaystyle R^{(1)}=\frac{8i\Theta^{uv}(u^{2}-v^{2})}{(u^{2}+v^{2})^{3}}.$
(48)

###  5.2 Second Order NC Corrections

####  5.2.1 Metric, Inverse Metric and Christoffel Symbols

The only component of the metric tensor that picks up the second order NC
correction is

$\displaystyle G_{\phi\phi}^{(2)}=\frac{(\Theta^{uv})^{2}}{4}.$ (49)

But there are three nonzero second order components of the inverse metric
tensor and all of them are diagonal:

$\displaystyle\begin{aligned}
G^{(2)uu*}=\frac{(\Theta^{uv})^{2}}{(u^{2}+v^{2})^{3}},\qquad
G^{(2)vv*}=\frac{(\Theta^{uv})^{2}}{(u^{2}+v^{2})^{3}},\qquad
G^{(2)\phi\phi*}=\frac{-5(\Theta^{uv})^{2}}{u^{4}v^{4}}.\end{aligned}$ (50)

The Christoffel symbols pick up the following 6 distinct nonzero second order
corrections:

$\displaystyle\begin{aligned}
\Gamma_{uu}^{(2)}{}^{u}&=\frac{(\Theta^{uv})^{2}\,u}{(u^{2}+v^{2})^{3}}=\Gamma_{uv}^{(2)}{}^{v}=-\,\Gamma_{vv}^{(2)}{}^{u},\qquad&-\Gamma_{uu}^{(2)}{}^{v}&=\frac{(\Theta^{uv})^{2}\,v}{(u^{2}+v^{2})^{3}}=\Gamma_{uv}^{(2)}{}^{u}=\Gamma_{vv}^{(2)}{}^{v},\\\
\Gamma_{u\phi}^{(2)}{}^{\phi}&=\frac{-3(\Theta^{uv})^{2}}{4u^{3}v^{2}},&\Gamma_{v\phi}^{(2)}{}^{\phi}&=\frac{-3(\Theta^{uv})^{2}}{4u^{2}v^{3}},\\\
\Gamma_{\phi\phi}^{(2)}{}^{u}&=\frac{(\Theta^{uv})^{2}\,u\,(3u^{2}-11v^{2})}{2(u^{2}+v^{2})^{3}},&\Gamma_{\phi\phi}^{(2)}{}^{v}&=\frac{(\Theta^{uv})^{2}\,v\,(11u^{2}-3v^{2})}{2(u^{2}+v^{2})^{3}}.\\\
\end{aligned}$ (51)

####  5.2.2 Curvature Tensors and Curvature Scalar

In NC Minkowski spacetime in parabolic coordinates, there are 11 Riemann
tensor components that receive second corrections, apart from the
antisymmetric ones under the exchange of first two indices:

$\displaystyle R_{uvu}^{(2)}{}^{v}$
$\displaystyle=-\,R_{uvv}^{(2)}{}^{u}=\frac{12(\Theta^{uv})^{2}}{(u^{2}+v^{2})^{3}},$
$\displaystyle R_{u\phi v}^{(2)}{}^{\phi}$
$\displaystyle=\frac{-\,(\Theta^{uv})^{2}(29u^{4}+42u^{2}v^{2}+21v^{4})}{4(u^{2}+v^{2})^{2}u^{3}v^{3}},$
$\displaystyle R_{uv\phi}^{(2)}{}^{\phi}$
$\displaystyle=\frac{2\,(\Theta^{uv})^{2}(-u^{2}+v^{2})}{(u^{2}+v^{2})u^{3}v^{3}},$
$\displaystyle R_{u\phi\phi}^{(2)}{}^{v}$
$\displaystyle=\frac{(\Theta^{uv})^{2}(101u^{6}-393u^{4}v^{2}+39u^{2}v^{4}-3v^{6})}{4(u^{2}+v^{2})^{4}u^{2}},$
$\displaystyle R_{u\phi u}^{(2)}{}^{\phi}$
$\displaystyle=\frac{-\,3(\Theta^{uv})^{2}(13u^{2}+5v^{2})}{4(u^{2}+v^{2})u^{4}v^{2}},$
$\displaystyle R_{u\phi\phi}^{(2)}{}^{v}$
$\displaystyle=\frac{(\Theta^{uv})^{2}(9u^{6}-341u^{4}v^{2}+175u^{2}v^{4}-11v^{6})}{4(u^{2}+v^{2})^{4}uv},$
$\displaystyle R_{v\phi v}^{(2)}{}^{\phi}$
$\displaystyle=\frac{-\,3(\Theta^{uv})^{2}(5u^{2}+13v^{2})}{4(u^{2}+v^{2})u^{2}v^{4}},\quad$
$\displaystyle R_{v\phi u}^{(2)}{}^{\phi}$
$\displaystyle=\frac{-\,(\Theta^{uv})^{2}(21u^{4}+42u^{2}v^{2}+29v^{4})}{4(u^{2}+v^{2})^{2}u^{3}v^{3}},$
$\displaystyle R_{v\phi\phi}^{(2)}{}^{u}$
$\displaystyle=\frac{-\,(\Theta^{uv})^{2}(11u^{6}+175u^{4}v^{2}+341u^{2}v^{4}-9v^{6})}{4(u^{2}+v^{2})^{4}uv}~{},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$
$\displaystyle R_{v\phi\phi}^{(2)}{}^{u}$
$\displaystyle=\frac{-\,(\Theta^{uv})^{2}(3u^{6}-39u^{4}v^{2}+393u^{2}v^{4}-101v^{6})}{4(u^{2}+v^{2})^{4}v^{2}}~{}.$
(52)

The sum over the repeated index in $R_{\mu\sigma\nu}^{(2)}{}^{\sigma}$ results
in the second order Ricci tensor, and there are 5 nontrivial components:

$\displaystyle\begin{aligned}
R_{uu}^{(2)}&=\frac{-\,3(\Theta^{uv})^{2}(13u^{6}+15u^{4}v^{2}+23u^{2}v^{4}+5v^{6})}{4(u^{2}+v^{2})^{3}u^{4}v^{2}},\\\
R_{uv}^{(2)}&=\frac{-\,(\Theta^{uv})^{2}(29u^{4}+42u^{2}v^{2}+21v^{4})}{4(u^{2}+v^{2})^{2}u^{3}v^{3}},\\\
R_{vu}^{(2)}&=\frac{-\,(\Theta^{uv})^{2}(21u^{4}+42u^{2}v^{2}+29v^{4})}{4(u^{2}+v^{2})^{2}u^{3}v^{3}},\\\
R_{vv}^{(2)}&=\frac{-\,3(\Theta^{uv})^{2}(5u^{6}+23u^{4}v^{2}+15u^{2}v^{4}+13v^{6})}{4(u^{2}+v^{2})^{3}u^{2}v^{4}},\\\
R_{\phi\phi}^{(2)}&=\frac{(\Theta^{uv})^{2}(3u^{8}-140u^{6}v^{2}+786u^{4}v^{4}-140u^{2}v^{6}+3v^{8})}{4(u^{2}+v^{2})^{4}u^{2}v^{2}}.\end{aligned}$
(53)

The correction to the scalar curvature is worked out to be

$\displaystyle
R^{(2)}=\frac{(\Theta^{uv})^{2}(3u^{8}-2u^{6}v^{2}-54u^{4}v^{4}-2u^{2}v^{6}+3v^{8})}{(u^{2}+v^{2})^{4}u^{4}v^{4}}.$
(54)

##  6 Concluding Remarks

An integral representation of *-product has been given in [27] (see also [28])
which holds for any odd or even dimension $n>1$:

$(f*g)(x)=\frac{1}{(2\pi)^{n}}\int d^{n}y~{}d^{n}z~{}f(x-\tfrac{1}{2}\Theta
y)~{}g(x+z)~{}e^{-iyz}.$ (55)

This representation clearly implies that the value of $f*g$ at a particular
point $x$ depends on the values of $f$ and $g$ at all points in the domain.
Therefore, any coordinate transformation cannot restrict the domain in the new
coordinate system, else the nature of $f*g$ will change. In the commutative
case, in which the value of $f\cdot g$ at a point $x$ depends only on the
values of $f$ and $g$ at $x$, the restricted domain in the new system doesn’t
lead to any interesting consequences. But with the inclusion of the points
$r=0$ and $\theta=0$ and $\theta=\pi$, the transformation from spherical polar
to Cartesian coordinates is not a bijective local diffeomorphism. Only a
bijective local diffeomorphism is a diffeomorphism [29]. Since the Jacobian
determinant of the transformation is equal to $r^{2}\sin\theta$ which vanishes
at the values of $r=0$, $\theta=0$ and $\theta=\pi$, the transformation does
not have unique inverse resulting in the non-injective nature of the mapping
$(r,\theta,\phi,t)\mapsto(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta,t)$.
Hence, the NC Minkowski spacetime parametrized with spherical polar
coordinates and the NC Minkowski spacetime parametrized with Cartesian
coordinates are not $\Theta$-diffeomorphic to each other, and hence the
introduction of noncommutativity among $r$ and $\theta$ leads to an entirely
different NC spacetime with curvature effects. These curvature effects vanish
completely in the limit $\Theta^{r\theta}\to 0$. Similar arguments hold for
the case of parabolic coordinates for which the Jacobian determinant of the
surjective-only transformation
$(u,v,\phi,t)\longrightarrow(uv\cos\phi,uv\sin\phi,\tfrac{1}{2}(u^{2}-v^{2}),t)$
is equal to $(u^{2}+v^{2})uv$. Also, in the non-relativistic limit, similar NC
corrections would show up for the physical 3-D space parametrized with such
curvilinear coordinates.

In general, if the metric tensor in the commutative case depends on more then
one curvilinear coordinates and the Jacobian determinant of the transformation
to another coordinate system is vanishing in a nontrivial subset of the domain
of the transformation, thereby making the transformation non-injective, then
the introduction of coordinate-noncommutativity in the two cases can lead to
different curvature properties. A detailed study on this aspect of different
spaces, including the NC versions of curved spaces of the commutative case,
will be published elsewhere [30].

## Acknowledgments

MR would like to thank the DST for the INSPIRE fellowship. The authors
gratefully acknowledge the funding No.PU/PS2/PHYS/Minor-Equip/21-22/241 by
Pondicherry University under the Minor Equipment Grant No.PU/PD2/Minor-
Equip/2022/536. Thanks are also due to Dr.RSK for his efforts to develop the
computational lab in the department.

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