Curvature tensor in discrete gravity

Abstract

We study numerically the curvature tensor in a three-dimensional discrete space. Starting from the continuous metric of a three-sphere, we transformed it into a discrete space using three integers $n_1, n_2$, and $n_3$. The numerical results are compared with the expected values in the continuous limit. We show that as the number of cells in the lattice increases, the continuous limit is recovered.

1 Introduction

The difficulties that arise in attempts to quantize gravity have motivated the development of discrete gravity theories. In discrete gravity, as the naming suggests, the continuous spacetime is discretized and is no longer treated as a smooth manifold, with no unique approach to how the discretization is carried out. For example, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice [6]. Regge calculus is another proposed discrete approximation to general relativity. The formalism involves dividing spacetime into simplices with polyhedrons (Euclidean simplices) as its basic building blocks. Subsequently, the curvature of spacetime is approximated within each simplex [4]. One other framework is the Euclidean dynamical triangulation (EDT), which approximates the spacetime as a triangulated lattice, where the distances between neighboring points on the lattice are the edges of the triangles [1]. One last example is Loop quantum gravity, which is a theory of quantum gravity that postulates that spacetime is represented as a network of finite loops [5]. These approaches are all attempts to develop a theory of gravity that can merge general relativity with quantum mechanics.

Lately, a new approach of discrete gravity was proposed [3]. Each elementary cell in the discrete space is completely characterized by displacement operators connecting a cell to its neighbors by the spin connection. The curvature of the discrete space was defined and it was shown that as the elementary volume vanishes, the standard results for the continuous curved differentiable manifolds are fully recovered. This proposed discretization of space was modeled after QCD where instead of the SU(3) color group, the rotation symmetry of the tangent group is kept intact. The main advantage of this approach over others is that the continuous limit is manifest.

Recently, and within the lattice-like approach, the scalar curvature of discrete gravity in two dimensions was studied [2], based on the model proposed in [3]. Our aim in this paper is to study numerically the curvature tensor in three dimensions in discrete space. In the three-dimensional case, each cell has six neighboring cells which share with it a common boundary. Three integers that can be positive and negative, $n_1, n_2, n_3$, are used to enumerate each of the elementary cells. In the continuous limit, these series of integers become coordinates on the manifold, this is presented in the first section of this paper. In the second section, we convert the continuous metric of a three-sphere into a lattice. We also investigate how the scalar curvature changes in the discrete space depending on the number of cells, and we demonstrate that it approaches the anticipated value in the continuous limit as the number of cells increases. In the third section, the expected values of the spin connections and the curvature tensor in the continuous case will be compared with the ones acquired in the discrete case.

2 Isotropic coordinates for a three-sphere

Starting from a three-sphere metric with a unit radius:

$$\begin{aligned} ds^{2} = d\chi ^{2}+\sin ^{2}\chi \left( d\theta ^2 + sin^2 \theta \ d\phi ^{2}\right) , \end{aligned}$$

we can rewrite it as

$$\begin{aligned} ds^{2} = \frac{dr^{2}}{1-r^2} + r^{2} \left( d\theta ^2 + sin^2 \theta \ d\phi ^{2}\right) , \end{aligned}$$

where $r = sin \chi$.

To plot the set of discrete points that will make the discrete three-sphere, we start from the regular coordinates given below:

$$\begin{aligned} x = r\sin \theta \cos \phi ,\quad y= r \sin \theta \sin \phi ,\quad z = r \cos \theta . \end{aligned}$$

We define $\left( x_1, x_2, x_3\right)$ in terms of $n_1$, $n_2$ and $n_3$ and in terms of $\bar{r} = 2 \tan \frac{\chi }{2}$

$$\begin{aligned} x_{1}= & {} \frac{2n_{1}}{N} = \bar{r} \sin \theta \cos \phi , \quad x_{2} = \frac{2n_{2}}{N} = \bar{r} \sin \theta \sin \phi , \quad \\ x_{3}= & {} \frac{2n_{3}}{N} = \bar{r} \cos \theta , \end{aligned}$$

where

$$\begin{aligned} n_{1}= & {} 0,\pm 1,\pm 2,\ldots ,\pm \left( N-1\right) ,\\ n_{2}= & {} 0,\pm 1,\pm 2,\ldots ,\pm \left( N-1\right) ,\qquad \\ n_{3}= & {} 0,\pm 1,\pm 2,\ldots ,\pm \left( N-1\right) . \end{aligned}$$

This finally gives

$$\begin{aligned} ds^2 = \frac{d x_1^2 + dx_2^2 + dx_3^2}{\left( 1+\frac{\bar{r}^2}{4}\right) ^2}, \end{aligned}$$

with the restriction $n_{1}^{2}+n_{2}^{2}+n_{3}^{2}\le N^{2}$. The coordinates (x, y, z) will be given by:

$$\begin{aligned} x= & {} \frac{2}{N} \frac{n_{1}}{1+ \frac{\left( n_{1}^{2}+n_{2}^{2}+n_3^2\right) }{N^{2}}},\quad y=\frac{2}{N} \frac{n_{2}}{1+ \frac{\left( n_{1}^{2}+n_{2}^{2}+n_3^2\right) }{N^{2}}},\quad \\ z= & {} \frac{2}{N} \frac{n_{3}}{1+ \frac{\left( n_{1}^{2}+n_{2}^{2}+n_3^2\right) }{N^{2}}}. \end{aligned}$$

The set of discrete points forming a three-sphere of radius one are displayed in Fig. 1.

Fig. 1

Three-sphere of radius one formed from the set of the discrete points

3 Three dimensional lattice gravity

This case is simple because the rotation group $SO\left( 3\right)$ has the same Lie Algebra as $SU\left( 2\right)$. We take the connection:

$$\begin{aligned} \frac{1}{4}\omega _{\mu }^{ij}\left( n\right) \gamma _{ij}\equiv \frac{i}{2} \omega _{\mu }^{i} \left( n\right) \sigma _{i} \quad \omega _{\mu }^{i} \left( n\right) =\frac{1}{2} \epsilon ^{ijk}\omega _{\mu }^{\ jk}\left( n\right) . \end{aligned}$$

Thus the curvature $R_{\mu \nu }^{\quad ij}\left( n\right) =\epsilon ^{ijk}R_{\mu \nu }^{\quad k}\left( n\right)$ is given by (no summation on $\mu ,\nu$ in what follows)

$$\begin{aligned} R_{\mu \nu }^{\quad i}\left( n\right)= & {} \frac{2}{\mathcal {\ell }^{\mu }\mathcal {\ell }^{\nu }}\left( A_{\mu \nu }\left( n\right) B_{\mu \nu } ^{i}\left( n\right) \right. \nonumber \\{} & {} \left. -A_{\nu \mu }\left( n\right) B_{\nu \mu }^{i}\left( n\right) +\epsilon ^{ijk}B_{\mu \nu }^{j}\left( n\right) B_{\nu \mu }^{k}\left( n\right) \right) , \nonumber \\ \end{aligned}$$

(1)

where

$$\begin{aligned} A_{\mu \nu }\left( n\right)= & {} \left( \cos \frac{1}{2}\mathcal {\ell }^{\mu } \omega _{\mu }\left( n+\widehat{\nu }\right) \cos \frac{1}{2}\mathcal {\ell } ^{\nu }\omega _{\nu }\left( n\right) \right. \nonumber \\{} & {} \quad \left. -\widehat{\omega }_{\mu }^{j}\left( n{+}\widehat{\nu }\right) \widehat{\omega }_{\nu }^{j}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n{+}\widehat{\nu }\right) \sin \frac{1}{2}\mathcal {\ell }^{\nu }\omega _{\nu }\left( n\right) \right) , \nonumber \\ \end{aligned}$$

(2)

$$\begin{aligned} B_{\mu \nu }^{i}\left( n\right)= & {} \left( \widehat{\omega }_{\mu }^{i}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) \cos \frac{1}{2}\mathcal {\ell }^{\nu }\omega _{\nu }\left( n+\widehat{\mu }\right) \right. \nonumber \\{} & {} \quad \left. +\widehat{\omega }_{\nu }^{i}\left( n+\widehat{\mu }\right) \sin \frac{1}{2}\mathcal {\ell }^{\nu }\omega _{\nu }\left( n+\widehat{\mu }\right) \cos \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) \right. \nonumber \\{} & {} \quad \left. -\epsilon ^{ijk}\widehat{\omega }_{\mu }^{j}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) \widehat{\omega }_{\nu }^{k}\left( n+\widehat{\mu }\right) \right. \nonumber \\{} & {} \quad \left. \times \sin \frac{1}{2}\mathcal {\ell }^{\nu }\omega _{\nu }\left( n{+}\widehat{\mu }\right) \right) , \nonumber \\ \end{aligned}$$

(3)

and $\mathcal {\ell }_{\mu }$ is the length scale for each dimension. We are taking $\mathcal {\ell }_{1} = \mathcal {\ell }_{2} = \mathcal {\ell }_{3} = \mathcal {\ell }$. We have denoted

$$\begin{aligned} \quad \left( \omega _{\mu }\left( n\right) \right) ^{2}\equiv {\displaystyle \sum \limits _{i=1}^{3}} \omega _{\mu }^{i}\left( n\right) \omega _{\mu }^{i}\left( n\right) ,\quad \widehat{\omega }_{\mu }^{i}\left( n\right) \equiv \frac{\omega _{\mu } ^{i}\left( n\right) }{\omega _{\mu }\left( n\right) }. \nonumber \\ \end{aligned}$$

(4)

The torsion is given by

$$\begin{aligned} T_{\mu \nu }^{\quad i}\left( n\right)&=\frac{1}{\mathcal {\ell }^{\mu } }\left( \cos \mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) e_{\nu } ^{i}\left( n+\widehat{\mu }\right) \right. \nonumber \\&\quad \left. -\epsilon ^{ijk}\sin \mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) \widehat{\omega }_{\mu }^{j}\left( n\right) e_{\nu }^{k}\left( n+\widehat{\mu }\right) \right. \nonumber \\&\quad \left. +2\widehat{\omega }_{\mu }^{i}\left( n\right) \widehat{\omega } _{\mu }^{j}\left( n\right) \sin ^{2}\frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) e_{\nu }^{j}\left( n+\widehat{\mu }\right) -e_{\nu }^{i}\left( n\right) \right) \nonumber \\&\quad -\left( \mu \leftrightarrow \nu \right) \end{aligned}$$

(5)

where $e_{\nu }^{k}$ are the vierbeins (check [3]). The latter form a set of orthonormal vector fields where $\nu$ labels the general spacetime coordinate and k labels the local Lorentz spacetime. The vanishing of $T_{\mu \nu }^{\quad i}$ provides nine conditions to solve for the nine unknowns $\omega _{\mu }^{i}\left( n\right)$ (check appendix I). We then proceed to find the components of the curvature $R_{\mu \nu }^{i}$ using Eqs. 1–3 (check appendix II).

The curvature tensor is given by

$$\begin{aligned} R_{\mu \nu }^{\quad ij}=\epsilon ^{ijk}R_{\mu \nu }^{k}, \end{aligned}$$

and the scalar curvature is

$$\begin{aligned} R= {\displaystyle \sum \limits _{\mu ,\nu }} e_{i}^{\mu }e_{j}^{\nu }R_{\mu \nu }^{\quad ij}= {\displaystyle \sum \limits _{\mu ,\nu }} \epsilon ^{ijk}e_{i}^{\mu }e_{j}^{\nu }R_{\mu \nu }^{k} . \end{aligned}$$

To show that the curvature tensor has the correct continuous limit, we note that as $\mathcal {\ell }\rightarrow 0$

$$\begin{aligned} A_{\mu \nu }\left( n\right)&\rightarrow 1+O\left( \mathcal {\ell } ^{2}\right) \\ B_{\mu \nu }^{i}\left( n\right)&\rightarrow \frac{\mathcal {\ell }}{2}\left( \omega _{\mu }^{i}\left( n\right) +\omega _{\nu }^{i}\left( n+\widehat{\mu }\right) \right) \\&\quad -\frac{\mathcal {\ell }^{2}}{4}\left( \epsilon ^{ijk} \omega _{\mu }^{j}\left( n\right) \omega _{\nu }^{k}\left( n\right) \right) +O\left( \mathcal {\ell }^{3}\right) . \end{aligned}$$

This implies that

$$\begin{aligned} R_{\mu \nu }^{\quad i}\left( n\right)&\rightarrow \frac{2}{\mathcal {\ell }^{2}}\frac{\mathcal {\ell }}{2}\left( \omega _{\mu }^{i}\left( n\right) +\omega _{\nu }^{i}\left( n+\widehat{\mu }\right) -\omega _{\nu }^{i}\left( n\right) -\omega _{\mu }^{i}\left( n+\widehat{\nu }\right) \right) \\&\quad +\left( \frac{2}{\mathcal {\ell }^{2}}\right) \left( -\frac{\mathcal {\ell }^{2}}{4}\right) 2\epsilon ^{ijk}\omega _{\mu }^{j}\left( n\right) \omega _{\nu } ^{k}\left( n\right) \\&\rightarrow \partial _{\mu }\omega _{\nu }^{i}-\partial _{\nu }\omega _{\mu } ^{i}-\epsilon ^{ijk}\omega _{\mu }^{j}\omega _{\nu }^{k}, \end{aligned}$$

where we have used $\omega _{\nu }^{i} \left( n + \widehat{\mu } \right) -\omega _{\nu }^{i} \left( n\right) = \mathcal {\ell }\partial _{\mu }\omega _{\nu }^{i} + O\left( \mathcal {\ell }^{2}\right) .$ Noting that

$$\begin{aligned} R_{\mu \nu }^{\quad ij}=\epsilon ^{ijk}R_{\mu \nu }^{k}=\partial _{\mu }\omega _{\nu }^{ij}-\partial _{\nu }\omega _{\mu }^{ij} + \omega _{\mu }^{ik}\omega _{\nu }^{kj}-\omega _{\nu }^{ik}\omega _{\mu }^{kj}, \end{aligned}$$

where $\omega _{\mu }^{ij} = \epsilon ^{ijk}\omega _{\mu }^{k}$ recovers the familiar expression for the curvature tensor.

In the case when $e_{\mu }^{i}$ is diagonal then

$$\begin{aligned} R=2\left( \frac{1}{e_{\overset{.}{1}}^{1} e_{\overset{.}{2}}^{2} }R_{\overset{.}{1}\overset{.}{2}}^{3}+\frac{1}{e_{\overset{.}{2}}^{2} e_{\overset{.}{3}}^{3}} R_{\overset{.}{2}\overset{.}{3}}^{1}+\frac{1}{e_{\overset{.}{3}}^{3} e_{\overset{.}{1}}^{1} } R_{\overset{.}{3}\overset{.}{1}}^{2}\right) , \end{aligned}$$

where the dotted index corresponds to the coordinate directions in the spacetime manifold while the undotted index represents the tangent space directions.

Considering the example of the three-sphere in isotropic coordinates we will have

$$\begin{aligned} e_{\overset{.}{1}}^{1}\left( n\right) =e_{\overset{.}{2}}^{2}\left( n\right) {=}e_{\overset{.}{3}}^{3}\left( n\right) {=}e\left( n\right) =\frac{1}{1+\frac{1}{N^{2}}\left( n_{1}^{2}+n_{2}^{2}+n_{3}^{2}\right) }. \end{aligned}$$

Since $\left( x^{1}\right) ^{2}+\left( x^{2}\right) ^{2}+\left( x^{3}\right) ^{2}=r^{2}=2\tan \frac{\theta }{2}$, one must use two coordinate charts. Therefore, one must have $r^{2}\le 4$ and thus $\left( n^{1}\right) ^{2}+\left( n^{2}\right) ^{2}+\left( n^{3}\right) ^{2}\le N^{2}.$ This restriction is imposed on the values of $n_1, n_2, n_3$.

4 Continuous case

It is helpful to list the values of the spin connections and curvatures in the continuous limit. In a numerical calculation to test the accuracy of the program, we can compare the numerical values of $\omega _{\mu }^{i}$, $R_{\mu \nu }^{\quad ij}$ to those of the continuous limit. In particular, one can test the homogeneity of the space by comparing the Ricci tensor $R_{\nu \sigma }$ to the metric $g_{\nu \sigma }$

$$\begin{aligned} R_{\nu \sigma }= & {} e_{i}^{\mu }R_{\mu \nu }^{\quad ij}e_{\sigma j}=2g_{\nu \sigma }\\ R_{\mu \nu }^{ij}= & {} e_{\mu }^{k}e_{\nu }^{l}R_{kl}^{\quad ij}=\left( e_{\mu }^{i}e_{\nu }^{j}-e_{\mu }^{j}e_{\nu }^{i}\right) =\epsilon ^{ijk}R_{\mu \nu }^{k}. \end{aligned}$$

Inverting we get

$$\begin{aligned} R_{\mu \nu }^{k}=\epsilon ^{kij}e_{\mu }^{i}e_{\nu }^{j}, \end{aligned}$$

and the only non-vanishing curvatures are:

$$\begin{aligned} R_{\overset{.}{1}\overset{.}{2}}^{\quad 12}=R_{\overset{.}{1}\overset{.}{2} }^{3}=\frac{1}{\left( 1+\frac{1}{4}r^{2}\right) ^{2}}=R_{\overset{.}{2} \overset{.}{3}}^{1}=R_{\overset{.}{3}\overset{.}{1}}^{2}. \end{aligned}$$

Fig. 2

The mean value of the curvature $\bar{R}$ for different values of N

Fig. 3

$R_{\dot{1} \dot{2}, con}^{3}$, $R_{\dot{2} \dot{3},con}^{1}$ and $R_{\dot{3} \dot{1},con}^{2}$ in the continuous case (which are equal) are compared to $R_{\dot{1} \dot{2}}^{3}$, $R_{\dot{2} \dot{3}}^{1}$, and $R_{\dot{3} \dot{1}}^{2}$ in the discrete case as a function of the index which corresponds to a point in this discretized space

Fig. 4

The values of $R_{\dot{1} \dot{2}}^{1}$ and $R_{\dot{1} \dot{2}}^{2}$ in the discrete case are close to zero as expected

Fig. 5

$R_{\dot{1} \dot{1}, con}$ = $R_{\dot{2} \dot{2}, con}$ = $R_{\dot{3} \dot{3}, con}$ in the continuous case are compared to $R_{\dot{1} \dot{1}}$, $R_{\dot{2} \dot{2}}$, and $R_{\dot{3} \dot{3}}$ in the discrete case

Fig. 6

The values of $R_{\dot{1} \dot{2}}$ and $R_{\dot{1} \dot{3}}$ in the discrete case are close to zero as expected

Fig. 7

${\widehat{\omega }}^1_{\dot{1},con}$ in the continuous limit is compared to ${\widehat{\omega }}^1_{\dot{1}}$ in the discrete case

Fig. 8

${\widehat{\omega }}^2_{\dot{1},con}$ in the continuous limit is compared to ${\widehat{\omega }}^2_{\dot{1}}$ in the discrete case

Fig. 9

${\widehat{\omega }}^2_{\dot{3},con}$ in the continuous limit is compared to ${\widehat{\omega }}^2_{\dot{3}}$ in the discrete case

Fig. 10

${\omega }_{\dot{1}}$ in the continuous limit compared to ${\omega }_{\dot{1}}$ in the discrete case

Fig. 11

${\omega }_{\dot{2},con}$ in the continuous limit is compared to ${\omega }_{\dot{2}}$ in the discrete case

Fig. 12

${{\omega }}^2_{\dot{1},con}$ in the continuous limit is compared to ${{\omega }}^2_{\dot{1}}$ in the discrete case

Fig. 13

${{\omega }}^3_{\dot{2},con}$ in the continuous limit is compared to ${{\omega }}^3_{\dot{2}}$ in the discrete case

The total curvature scalar is then

$$\begin{aligned} R=2\left( R_{\overset{.}{1}\overset{.}{2}}^{\quad \overset{.}{1} \overset{.}{2}}+R_{\overset{.}{2}\overset{.}{3}}^{\quad \overset{.}{2} \overset{.}{3}}+R_{\overset{.}{1}\overset{.}{3}}^{\quad \overset{.}{1} \overset{.}{3}}\right) =6. \end{aligned}$$

We also note that the continuous limit of $\omega _{\mu }^{i}=\frac{1}{2}\epsilon ^{ijk}\omega _{\mu }^{jk}=\frac{1}{2}\epsilon ^{ijk}x^{j}e_{\mu }^{k}.$ In components we have

$$\begin{aligned} \omega _{\overset{.}{1}}^{1}&=0,\quad \omega _{\overset{.}{2}}^{1}=-\frac{1}{2}x^{3}e,\quad \omega _{\overset{.}{3}}^{1}=\frac{1}{2}x^{2}e\\ \omega _{\overset{.}{1}}^{2}&=\frac{1}{2}x^{3}e,\quad \omega _{\overset{.}{2} }^{2}=0,\quad \omega _{\overset{.}{3}}^{2}=-\frac{1}{2}x^{1}e\\ \omega _{\overset{.}{1}}^{3}&=-\frac{1}{2}x^{2}e,\quad \omega _{\overset{.}{2} }^{3}=\frac{1}{2}x^{1}e,\quad \omega _{\overset{.}{3}}^{3}=0. \end{aligned}$$

Fig. 14

The root-mean-squared error of $R_{1213} - R_{1312}$, $R_{1223} - R_{2312}$ and $R_{1323} - R_{2313}$ versus N is shown

The expression appearing in the numerical $\mathcal {\ell \omega }_{\mu }$ must correspond to

$$\begin{aligned} \left( \omega \overset{.}{_{1}}\right) ^{2}&=\frac{1}{4}\left( \left( x^{2}\right) ^{2}+\left( x^{3}\right) ^{2}\right) e^{2},\\ \left( \omega \overset{.}{_{2}}\right) ^{2}&=\frac{1}{4}\left( \left( x^{1}\right) ^{2}+\left( x^{3}\right) ^{2}\right) e^{2},\\ \left( \omega \overset{.}{_{3}}\right) ^{2}&=\frac{1}{4}\left( \left( x^{1}\right) ^{2}+\left( x^{2}\right) ^{2}\right) e^{2}, \end{aligned}$$

so that

$$\begin{aligned} \widehat{\omega }_{\overset{.}{1}}^{1}&=0, \quad \widehat{\omega }_{\overset{.}{1}}^{2} = \frac{x^{3}}{\sqrt{\left( x^{2}\right) ^{2} + \left( x^{3}\right) ^{2}}}, \quad \widehat{\omega }_{\overset{.}{1}}^{3} = -\frac{x^{2}}{\sqrt{\left( x^{2}\right) ^{2} + \left( x^{3}\right) ^{2}}} \\ \widehat{\omega }_{\overset{.}{2}}^{1}&=-\frac{x^{3}}{\sqrt{\left( x^{1}\right) ^{2} + \left( x^{3}\right) ^{2}}}, \quad \widehat{\omega }_{\overset{.}{2}}^{2}=0, \quad \widehat{\omega }_{\overset{.}{2}}^{3} =\frac{x^{1}}{\sqrt{\left( x^{1}\right) ^{2} +\left( x^{3}\right) ^{2}}}\\ \widehat{\omega }_{\overset{.}{3}}^{1}&=\frac{x^{2}}{\sqrt{\left( x^{1}\right) ^{2} +\left( x^{2}\right) ^{2}}}, \quad \widehat{\omega }_{\overset{.}{3}}^{2} = -\frac{x^{1}}{\sqrt{\left( x^{1}\right) ^{2}+\left( x^{2}\right) ^{2}}}, \quad \widehat{\omega }_{\overset{.}{3}}^{3}=0. \end{aligned}$$

The value of $\ell _{\mu }$ has to be properly chosen for it to yield the correct values of the curvature tensors. For that reason, we iterated over multiple values of $\ell _{\mu }$. $\ell _{\mu } = 2/N$ turned out to be the optimal choice leading to the discretized form of ${\frac{\mathcal {\ell }}{2} \omega }_{i}^{j}\rightarrow \frac{\epsilon ^{ijk}n^{k}}{1+\frac{n^{l}n^{l}}{N^{2}}}$, upon replacing $x^{i}$ by $2\frac{n^{i}}{N}$.

Figure 2 illustrates the convergence of the mean curvature as a function of N to its expected value of 6. Figures 3 and 4 show the curvature in the continuous and discrete limit. In the continuous limit, $R_{\dot{1} \dot{2}}^{3}$ has the same value as $R_{\dot{2} \dot{3}}^{1}$ and $R_{\dot{3} \dot{1}}^{2}$. Their plot is compared with each of the other plots in the discrete case in Fig. 3, while Fig. 4 presents $R_{\dot{1} \dot{2}}^{1}$ and $R_{\dot{1} \dot{2}}^{2}$ that vanish in the continuous case. Also, the Ricci curvature is plotted in Figs. 5 and 6. In Fig. 5, the values of $R_{\dot{1}\dot{1}}$, $R_{\dot{2}\dot{2}}$, and $R_{\dot{3}\dot{3}}$ are equal in the continuous case and are compared to their values in the discrete case, while Fig. 6 shows the plots of $R_{\dot{1}\dot{2}}$ and $R_{\dot{1}\dot{3}}$ in the discrete case. The latter is expected to vanish in the continuous case. To compare the numerical values of the spin connection, Figs. 7, 8, 9, 10, 11, 12 and 13 are represented. These figures are a sample of the spin connections plotted in the discrete case versus the plots of the expected values in the continuous case, for $N=12$. The plots in the discrete case were obtained by solving Eqs. 7–15 combined with Eq. 4 using nleqslv, which is an R package for solving nonlinear equations.

In the above figures, we compared the continuum to the discrete values. It is worth noting that the deviation from zero in Figs. 4 and 6 is not due to numerical errors but is rather linked to discretization. The limit we are reporting the results of is far from the continuum limit. The numerical solver nleqslv is based on Newton’s method whose precision is set to $10^{-12}.$ In order to avoid getting stuck in local solutions of the system we iterated over 500 different initial guesses, all of which returned the solutions we reported.

The similarities between the continuum and the discrete are shown in all the figures. The discrepancies between them have to do with the choice of discretization, in this case $N = 12$. We opted to report the errors in the scalar curvature R for which the continuum value is known to be 6. The scalar curvature is the evaluator of the goodness of the numerics.

Having established the rapid convergence of the scalar curvature to the continuous limit, we examine the properties of the discrete values of the tensors and whether they exhibit the same symmetries as the continuous case. We thus define:

$$\begin{aligned} R_{kl}^{\quad ij} = \sum _{\mu = 1}^3 \sum _{\nu = 1}^3 e_k^{\mu }e_l^{\nu } R_{\mu \nu }^{\quad ij}, \end{aligned}$$

and

$$\begin{aligned} R_{klij} = R_{kl}^{\quad mn} \delta _{mi} \delta _{nj} \quad R_{ki} = R_{klij} \delta ^{lj}. \end{aligned}$$

In the continuous case, we have:

$$\begin{aligned} R_{klij} = R_{ijkl}, \quad R_{kl} = R_{lk}. \end{aligned}$$

To test whether the symmetries of the Riemann tensor hold, we examine, in the discrete case, the differences:

$$\begin{aligned} R_{1213} - R_{1312}, \quad R_{1223} - R_{2312}, \quad R_{1323} - R_{2313}, \end{aligned}$$

(6)

for the small values of N. Figure 14 presents the root-mean-squared error of each of the differences given in Eq. 6 versus N. This shows that symmetries in the discrete limit do not hold, but the difference is getting smaller as N increases. The least-square fit to the RMS error as a function of N, using the fit function $0.5 \times N^{-A}$, returns $A = 1.97$ with $r^2 = 0.98$.

5 Conclusion

In this paper, we used the definition of the curvature of the discrete space in the recently proposed model of discrete gravity. Three integers (positive or negative) were used to enumerate each of the elementary cells in this lattice-like approach. Considering the example of a three-sphere, the curvature was computed numerically. It was shown that as the number of cells increases, the continuous limit is recovered. In this formalism, the metric is given as input. To account for all geometries we must allow for metric fluctuations. Our future work includes studying the link between metric changes and quantum fluctuations and also handling the more difficult problem of studying four-dimensional gravity on the lattice. Our ultimate goal is to apply this method to quantum gravity.

Appendices

Appendix I

Writing the nine equations in detail, we have:

$$\begin{aligned} \mathcal {\ell }T_{\overset{.}{1}\overset{.}{2}}^{1}&=0=\widehat{\omega }_{\overset{.}{1}}^{3}\left( n\right) \sin \mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{1}\right) \nonumber \\&\quad +2\widehat{\omega }_{\overset{.}{1}}^{1}\left( n\right) \widehat{\omega }_{\overset{.}{1}}^{2}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{1}\right) \nonumber \\&\quad -\cos \mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{2}\right) \nonumber \\&\quad -2\widehat{\omega }_{\overset{.}{2}}^{1}\left( n\right) \widehat{\omega }_{\overset{.}{2}} ^{1}\left( n\right) \sin ^{2}\frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}\ }\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{2}\right) +e_{\overset{.}{1}}^{1}\left( n\right) \end{aligned}$$

(7)

$$\begin{aligned} \mathcal {\ell }T_{\overset{.}{1}\overset{.}{2}}^{2}&=0=\widehat{\omega }_{\overset{.}{2}}^{3}\left( n\right) \sin \mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{2}\right) \nonumber \\&\quad +2\widehat{\omega }_{\overset{.}{1}}^{2}\left( n\right) \widehat{\omega }_{\overset{.}{1}}^{2}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{1}\right) \nonumber \\&\quad +\cos \mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{1}\right) -2\widehat{\omega }_{\overset{.}{2}}^{2}\left( n\right) \widehat{\omega }_{\overset{.}{2}} ^{1}\left( n\right) \nonumber \\&\quad \times \sin ^{2}\frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2} }\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{2}\right) -e_{\overset{.}{2}}^{2}\left( n\right) \end{aligned}$$

(8)

$$\begin{aligned} \mathcal {\ell }T_{\overset{.}{1}\overset{.}{2}}^{3}&=0=-\widehat{\omega }_{\overset{.}{2}}^{2}\left( n\right) \sin \mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{2}\right) \nonumber \\&\quad +2\widehat{\omega }_{\overset{.}{1}}^{2}\left( n\right) \widehat{\omega }_{\overset{.}{1}}^{3}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{1}\right) \nonumber \\&\quad -\widehat{\omega }_{\overset{.}{1}}^{1}\sin \mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{1}\right) \nonumber \\&\quad -2\widehat{\omega }_{\overset{.}{2}}^{1}\left( n\right) \widehat{\omega }_{\overset{.}{2}}^{3}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{2}\right) \end{aligned}$$

(9)

$$\begin{aligned} \mathcal {\ell }T_{\overset{.}{2}\overset{.}{3}}^{1}&=0=-\widehat{\omega }_{\overset{.}{2}}^{2}\left( n\right) \sin \mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{2}\right) \nonumber \\&\quad +2\widehat{\omega }_{\overset{.}{2}}^{1}\left( n\right) \widehat{\omega }_{\overset{.}{2}}^{3}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{2}\right) \nonumber \\&\quad -\widehat{\omega }_{\overset{.}{3}}^{3}\sin \mathcal {\ell }\omega _{\overset{.}{3}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{3}\right) \nonumber \\&\quad -2\widehat{\omega }_{\overset{.}{3}}^{1}\left( n\right) \widehat{\omega }_{\overset{.}{3}}^{2}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{3}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{3}\right) \end{aligned}$$

(10)

$$\begin{aligned} \mathcal {\ell }T_{\overset{.}{2}\overset{.}{3}}^{2}&=0=\widehat{\omega }_{\overset{.}{2}}^{1}\left( n\right) \sin \mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{2}\right) \nonumber \\&\quad +2\widehat{\omega }_{\overset{.}{2}}^{2}\left( n\right) \widehat{\omega }_{\overset{.}{2}}^{3}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{2}\right) \nonumber \\&\quad -\cos \mathcal {\ell }\omega _{\overset{.}{3}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{3}\right) \nonumber \\&\quad -2\widehat{\omega }_{\overset{.}{3}}^{2}\left( n\right) \widehat{\omega }_{\overset{.}{3}} ^{2}\left( n\right) \sin ^{2}\frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{3}\ }\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{3}\right) +e_{\overset{.}{2}}^{2}\left( n\right) \end{aligned}$$

(11)

$$\begin{aligned} \mathcal {\ell }T_{\overset{.}{2}\overset{.}{3}}^{3}&=0=\widehat{\omega }_{\overset{.}{3}}^{1}\left( n\right) \sin \mathcal {\ell }\omega _{\overset{.}{3}}\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{3}\right) \nonumber \\&\quad +2\widehat{\omega }_{\overset{.}{2}}^{3}\left( n\right) \widehat{\omega }_{\overset{.}{2}}^{3}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{2}\right) \nonumber \\&\quad +\cos \mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{2}\right) \nonumber \\&\quad -2\widehat{\omega }_{\overset{.}{3}}^{3}\left( n\right) \widehat{\omega }_{\overset{.}{3}} ^{2}\left( n\right) \sin ^{2}\frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{3} }\left( n\right) e_{\overset{.}{2}}^{2}\left( n+\overset{.}{3}\right) -e_{\overset{.}{3}}^{3}\left( n\right) \end{aligned}$$

(12)

$$\begin{aligned} \mathcal {\ell }T_{\overset{.}{1}\overset{.}{3}}^{1}&=0=-\widehat{\omega }_{\overset{.}{1}}^{2}\left( n\right) \sin \mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{1}\right) \nonumber \\&\quad +2\widehat{\omega }_{\overset{.}{1}}^{1}\left( n\right) \widehat{\omega }_{\overset{.}{1}}^{3}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{1}\right) \nonumber \\&\quad -\cos \mathcal {\ell }\omega _{\overset{.}{3}}\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{3}\right) -2\widehat{\omega }_{\overset{.}{3}}^{1}\left( n\right) \widehat{\omega }_{\overset{.}{3}} ^{1}\left( n\right) \nonumber \\&\quad \times \sin ^{2}\frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{3} }\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{3}\right) +e_{\overset{.}{1}}^{1}\left( n\right) \end{aligned}$$

(13)

$$\begin{aligned} \mathcal {\ell }T_{\overset{.}{1}\overset{.}{3}}^{2}&=0=\widehat{\omega }_{\overset{.}{1}}^{1}\left( n\right) \sin \mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{1}\right) \nonumber \\&\quad +2\widehat{\omega }_{\overset{.}{1}}^{2}\left( n\right) \widehat{\omega }_{\overset{.}{1}}^{3}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{1}\right) \nonumber \\&\quad +\widehat{\omega }_{\overset{.}{3}}^{3}\sin \mathcal {\ell }\omega _{\overset{.}{3}}\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{3}\right) \nonumber \\&\quad -2\widehat{\omega }_{\overset{.}{3}}^{1}\left( n\right) \widehat{\omega }_{\overset{.}{3}}^{2}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{3}}\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{3}\right) \end{aligned}$$

(14)

$$\begin{aligned} \mathcal {\ell }T_{\overset{.}{1}\overset{.}{3}}^{3}&=0=-\widehat{\omega }_{\overset{.}{3}}^{2}\left( n\right) \sin \mathcal {\ell }\omega _{\overset{.}{3}}\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{3}\right) \nonumber \\&\quad +2\widehat{\omega }_{\overset{.}{1}}^{3}\left( n\right) \widehat{\omega }_{\overset{.}{1}}^{3}\left( n\right) \sin ^{2} \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{1}\right) \nonumber \\&+\cos \mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) e_{\overset{.}{3}}^{3}\left( n+\overset{.}{1}\right) -2\widehat{\omega }_{\overset{.}{3}}^{3}\left( n\right) \widehat{\omega }_{\overset{.}{3}} ^{1}\left( n\right) \nonumber \\&\quad \times \sin ^{2}\frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{3} }\left( n\right) e_{\overset{.}{1}}^{1}\left( n+\overset{.}{3}\right) -e_{\overset{.}{3}}^{3}\left( n\right) \end{aligned}$$

(15)

In this notation we have $e_{\overset{.}{1}}^{1}\left( n+\overset{.}{2}\ \right) \equiv e_{\overset{.}{1}}^{1}\left( n_{1},n_{2}+1,n_{3}\right)$ thus shifting only the second coordinate.

Appendix II

$$\begin{aligned} R_{\overset{.}{1}\overset{.}{2}}^{\quad 1}\left( n\right)&=\frac{2}{\mathcal {\ell }^{2}}\left( A_{\overset{.}{1}\overset{.}{2}} B_{\overset{.}{1}\overset{.}{2}}^{1}-A_{\overset{.}{2}\overset{.}{1} }B_{\overset{.}{2}\overset{.}{1}}^{1}+B_{\overset{.}{1}\overset{.}{2}} ^{2}\left( n\right) B_{\overset{.}{2}\overset{.}{1}}^{3}\left( n\right) \right. \nonumber \\&\quad \left. -B_{\overset{.}{1}\overset{.}{2}}^{3}\left( n\right) B_{\overset{.}{2} \overset{.}{1}}^{2}\left( n\right) \right) \end{aligned}$$

(16)

$$\begin{aligned} R_{\overset{.}{1}\overset{.}{2}}^{\quad 2}\left( n\right)&=\frac{2}{\mathcal {\ell }^{2}}\left( A_{\overset{.}{1}\overset{.}{2}} B_{\overset{.}{1}\overset{.}{2}}^{2}-A_{\overset{.}{2}\overset{.}{1} }B_{\overset{.}{2}\overset{.}{1}}^{2}+B_{\overset{.}{1}\overset{.}{2}} ^{3}\left( n\right) B_{\overset{.}{2}\overset{.}{1}}^{1}\left( n\right) \right. \nonumber \\&\quad \left. -B_{\overset{.}{1}\overset{.}{2}}^{1}\left( n\right) B_{\overset{.}{2} \overset{.}{1}}^{3}\left( n\right) \right) \end{aligned}$$

(17)

$$\begin{aligned} R_{\overset{.}{1}\overset{.}{2}}^{\quad 3}\left( n\right)&=\frac{2}{\mathcal {\ell }^{2}}\left( A_{\overset{.}{1}\overset{.}{2}} B_{\overset{.}{1}\overset{.}{2}}^{3}-A_{\overset{.}{2}\overset{.}{1} }B_{\overset{.}{2}\overset{.}{1}}^{3}+B_{\overset{.}{1}\overset{.}{2}} ^{1}\left( n\right) B_{\overset{.}{2}\overset{.}{1}}^{2}\left( n\right) \right. \nonumber \\&\quad \left. -B_{\overset{.}{1}\overset{.}{2}}^{2}\left( n\right) B_{\overset{.}{2} \overset{.}{1}}^{1}\left( n\right) \right) \end{aligned}$$

(18)

$$\begin{aligned} R_{\overset{.}{2}\overset{.}{3}}^{\quad 1}\left( n\right)&=\frac{2}{\mathcal {\ell }^{2}}\left( A_{\overset{.}{2}\overset{.}{3}} B_{\overset{.}{2}\overset{.}{3}}^{1}-A_{\overset{.}{3}\overset{.}{2} }B_{\overset{.}{3}\overset{.}{2}}^{1}+B_{\overset{.}{2}\overset{.}{3}} ^{2}\left( n\right) B_{\overset{.}{3}\overset{.}{2}}^{3}\left( n\right) \right. \nonumber \\&\quad \left. -B_{\overset{.}{2}\overset{.}{3}}^{3}\left( n\right) B_{\overset{.}{3} \overset{.}{2}}^{2}\left( n\right) \right) \end{aligned}$$

(19)

$$\begin{aligned} R_{\overset{.}{2}\overset{.}{3}}^{\quad 2}\left( n\right)&=\frac{2}{\mathcal {\ell }^{2}}\left( A_{\overset{.}{2}\overset{.}{3}} B_{\overset{.}{2}\overset{.}{3}}^{2}-A_{\overset{.}{3}\overset{.}{2} }B_{\overset{.}{3}\overset{.}{2}}^{2}+B_{\overset{.}{2}\overset{.}{3}} ^{3}\left( n\right) B_{\overset{.}{3}\overset{.}{2}}^{1}\left( n\right) \right. \nonumber \\&\quad \left. -B_{\overset{.}{2}\overset{.}{3}}^{1}\left( n\right) B_{\overset{.}{3} \overset{.}{2}}^{3}\left( n\right) \right) \end{aligned}$$

(20)

$$\begin{aligned} R_{\overset{.}{2}\overset{.}{3}}^{\quad 3}\left( n\right)&=\frac{2}{\mathcal {\ell }^{2}}\left( A_{\overset{.}{2}\overset{.}{3}} B_{\overset{.}{2}\overset{.}{3}}^{3}-A_{\overset{.}{3}\overset{.}{2} }B_{\overset{.}{3}\overset{.}{2}}^{3}+B_{\overset{.}{2}\overset{.}{3}} ^{1}\left( n\right) B_{\overset{.}{3}\overset{.}{2}}^{2}\left( n\right) \right. \nonumber \\&\quad \left. -B_{\overset{.}{2}\overset{.}{3}}^{2}\left( n\right) B_{\overset{.}{3} \overset{.}{2}}^{1}\left( n\right) \right) \end{aligned}$$

(21)

$$\begin{aligned} R_{\overset{.}{1}\overset{.}{3}}^{\quad 1}\left( n\right)&=\frac{2}{\mathcal {\ell }^{2}}\left( A_{\overset{.}{1}\overset{.}{3}} B_{\overset{.}{1}\overset{.}{3}}^{1}-A_{\overset{.}{3}\overset{.}{1} }B_{\overset{.}{3}\overset{.}{1}}^{1}+B_{\overset{.}{1}\overset{.}{3}} ^{2}\left( n\right) B_{\overset{.}{3}\overset{.}{1}}^{3}\left( n\right) \right. \nonumber \\&\quad \left. -B_{\overset{.}{1}\overset{.}{3}}^{3}\left( n\right) B_{\overset{.}{3} \overset{.}{1}}^{2}\left( n\right) \right) \end{aligned}$$

(22)

$$\begin{aligned} R_{\overset{.}{1}\overset{.}{3}}^{\quad 2}\left( n\right)&=\frac{2}{\mathcal {\ell }^{2}}\left( A_{\overset{.}{1}\overset{.}{3}} B_{\overset{.}{1}\overset{.}{3}}^{2}-A_{\overset{.}{3}\overset{.}{1} }B_{\overset{.}{3}\overset{.}{1}}^{2}+B_{\overset{.}{1}\overset{.}{3}} ^{3}\left( n\right) B_{\overset{.}{3}\overset{.}{1}}^{1}\left( n\right) \right. \nonumber \\&\quad \left. -B_{\overset{.}{1}\overset{.}{3}}^{1}\left( n\right) B_{\overset{.}{3} \overset{.}{1}}^{3}\left( n\right) \right) \end{aligned}$$

(23)

$$\begin{aligned} R_{\overset{.}{1}\overset{.}{3}}^{\quad 3}\left( n\right)&=\frac{2}{\mathcal {\ell }^{2}}\left( A_{\overset{.}{1}\overset{.}{3}} B_{\overset{.}{1}\overset{.}{3}}^{3}-A_{\overset{.}{3}\overset{.}{1} }B_{\overset{.}{3}\overset{.}{1}}^{3}+B_{\overset{.}{1}\overset{.}{3}} ^{1}\left( n\right) B_{\overset{.}{3}\overset{.}{1}}^{2}\left( n\right) \right. \nonumber \\&\left. \quad -B_{\overset{.}{1}\overset{.}{3}}^{2}\left( n\right) B_{\overset{.}{3} \overset{.}{1}}^{1}\left( n\right) \right) \end{aligned}$$

(24)

We will also give an example of how the components of $A_{\mu \nu }\left( n\right)$ and $B_{\mu \nu }^{i}\left( n\right)$ look like

$$\begin{aligned} A_{\overset{.}{1}\overset{.}{2}}&=\cos \frac{1}{2}\mathcal {\ell } \omega _{\overset{.}{1}}\left( n+\overset{.}{2}\right) \cos \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) \nonumber \\&\quad - {\displaystyle \sum \limits _{j=1}^{3}} \widehat{\omega }_{\overset{.}{1}}^{j}\left( n+\overset{.}{2}\right) \widehat{\omega }_{\overset{.}{2}}^{j}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n+\overset{.}{2}\right) \nonumber \\&\quad \times \sin \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n\right) \end{aligned}$$

(25)

$$\begin{aligned} A_{\overset{.}{2}\overset{.}{1}}&=\cos \frac{1}{2}\mathcal {\ell } \omega _{\overset{.}{2}}\left( n+\overset{.}{1}\right) \cos \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) \nonumber \\&\quad - {\displaystyle \sum \limits _{j=1}^{3}} \widehat{\omega }_{\overset{.}{2}}^{j}\left( n+\overset{.}{2}\right) \widehat{\omega }_{\overset{.}{1}}^{j}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n+\overset{.}{2}\right) \nonumber \\&\quad \times \sin \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) \end{aligned}$$

(26)

$$\begin{aligned} B_{\overset{.}{1}\overset{.}{2}}^{i}\left( n\right)&=\left( \widehat{\omega }_{\overset{.}{1}}^{i}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) \cos \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n+\overset{.}{1}\right) \right. \nonumber \\&\quad \left. +\widehat{\omega }_{\overset{.}{2}}^{i}\left( n+\overset{.}{1}\right) \sin \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n+\overset{.}{1}\ \right) \cos \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) \right. \nonumber \\&\quad \left. -\epsilon ^{ijk}\widehat{\omega }_{\overset{.}{1}}^{j}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{1}}\left( n\right) \widehat{\omega }_{\overset{.}{2}}^{k}\left( n+\overset{.}{1} \right) \right. \nonumber \\&\quad \times \left. \sin \frac{1}{2}\mathcal {\ell }\omega _{\overset{.}{2}}\left( n+\overset{.}{1}\right) \right) \end{aligned}$$

(27)