1 Introduction

The challenges encountered while attempting to quantize gravity have inspired the advancement of discrete gravity theories. Different approaches to lattice gravity were explored over the past years [1, 8, 11]. Lately, in [6], a new approach to discrete gravity was proposed where the manifold is taken to be discrete and consists of elementary cells. The dimension d is defined assuming each cell has 2d neighboring cells that share a common boundary with each individual cell, and a finite number of degrees of freedom is associated with each cell. This approach stands out from others primarily because it clearly reveals the continuous limit.

Recently and based on the above model, the scalar curvature of discrete gravity in two dimensions was investigated in [4], while the examination of the curvature tensor in three dimensions can be found in [5]. In the latter, a three-sphere was considered and its continuous metric was converted into a lattice. It was shown that the scalar curvature in the discrete space approaches the expected value in the continuous limit.

In this paper, we will study again the three-dimensional case, but now for a black hole coset space metric. To discretize, a length scale \(l_{\mu }\) and three integers \(n_1, n_2, n_3\) are used. The metric corresponding to a three-dimensional coset space, along with the spin connections and curvatures in the continuous case, are presented in the Sect. 2. In Sect. 3, we discretize the continuous metric of the black hole coset space. We investigate the domains in the proximity of the singularities and away from them, and numerically compare the discrete values of the curvature tensor with the expected continuous ones.

2 Black hole coset space metric

Consider the metric corresponding to a three-dimensional coset space SO(4)/SO(3) [3, 7]

$$\begin{aligned} ds^{2} = \frac{\tanh ^{2}z}{u}dx^{2} + \frac{\coth ^{2}z}{u} dy^{2} + dz^{2}, \end{aligned}$$
(1)

where

$$\begin{aligned} u=1-\left( x^{2}+y^{2}\right) ,\quad u\ge 0,\quad -\infty \le z\le \infty . \end{aligned}$$
(2)

A coset space of the form G/H is the space of elements where \(g\in G\) can be decomposed in the form:

$$\begin{aligned} g= & {} kh\in G,\quad k=\exp \left( i\theta ^{i4}J_{i4}\right) \in G/H,\\ h= & {} \exp \left( i\theta ^{ij}J_{ij}\right) ,\quad i,j=1,2,3. \end{aligned}$$

This metric was first obtained to represent a configuration of a metric in the presence of a dilation field, but we will not add a dilation here. This choice is of particular interest because the lattice is not trivial and not maximally symmetric. The z-behaviour is different than the x and y. Upon letting

$$\begin{aligned} \theta ^{1} = \frac{\tanh z}{\sqrt{u}}dx, \quad \theta ^{2} = \frac{\coth z}{\sqrt{u}}dy, \quad \theta ^{3} = dz \end{aligned}$$

be the three one-forms, the spin connections and curvature tensors are all computed. The scalar curvature is then given by

$$\begin{aligned} R=-4\left( 1+\frac{1}{\sinh ^{2}z\cosh ^{2}z} + \frac{1}{u} \left( y^{2}\tanh ^{2}z + x^{2} \coth ^{2}z\right) \right) . \nonumber \\ \end{aligned}$$
(3)

Since \(u\ge 0\), R will always have negative curvature. There is a singularity at all points on the circle defined by \(u=0\) and at \(z=0\). We can further confirm this by calculating the curvature invariant \(R^{ab}R_{ab}\)

$$\begin{aligned} R^{ab}R_{ab}= & {} \left( R_{11}\right) ^{2}+\left( R_{22}\right) ^{2}+\left( R_{33}\right) ^{2}+2\left( R_{12}\right) ^{2}\\{} & {} +2\left( R_{13}\right) ^{2}+2\left( R_{23}\right) ^{2}. \end{aligned}$$

It is worth noting that the coordinate system we are using cannot be extended. It is an Euclidean black hole with no horizon. The singularities are for \(z=0\) and on the boundary of the disc with \(x^2+y^2=1\), i.e. \(u=0\).

To relate it to the lattice setting, the metric used here gives the curvature tensor in flat coordinates (anholonomic system). This is the system where:

$$\begin{aligned} \theta ^{1}=e_{\overset{.}{1}}^{1}dx,\quad \theta ^{2}=e_{\overset{.}{2}} ^{2}dy,\quad \theta ^{3}=e_{\overset{.}{3}}^{3}dz \end{aligned}$$

so that:

$$\begin{aligned} e_{\overset{.}{1}}^{1}=\frac{\tanh z}{\sqrt{u}},\quad e_{\overset{.}{2}} ^{2}=\frac{\coth z}{\sqrt{u}},\quad e_{\overset{.}{3}}^{3}=1. \end{aligned}$$
(4)

Thus, we will have:

$$\begin{aligned} R_{\mu \nu }^{\quad ij}=e_{\mu }^{k}e_{\nu }^{l}R_{kl}^{\quad ij}. \end{aligned}$$

In terms of components, we will have:

$$\begin{aligned}{} & {} R_{\overset{.}{1}\overset{.}{2}}^{\quad ij} = e_{\overset{.}{1}}^{1} e_{\overset{.}{2}}^{2}R_{12}^{\quad ij},\quad R_{\overset{.}{1}\overset{.}{3}}^{\quad ij} = e_{\overset{.}{1}}^{1}e_{\overset{.}{3}}^{3}R_{13}^{\quad ij},\quad \\{} & {} R_{\overset{.}{2}\overset{.}{3}}^{\quad ij} = e_{\overset{.}{2}}^{2}e_{\overset{.}{3}}^{3}R_{23}^{\quad ij}. \end{aligned}$$

For the non-vanishing components, we have

$$\begin{aligned} R_{\overset{.}{1}\overset{.}{3}}^{\quad 13}&= \frac{2\sinh z}{\sqrt{u} \cosh ^{3}z},\quad R_{\overset{.}{1}\overset{.}{2}}^{\quad 13} = -\frac{2y}{u^{\frac{3}{2}}\cosh ^{2}z}, \\ R_{\overset{.}{2}\overset{.}{3}}^{\quad 23}&= -\frac{2\cosh z}{\sqrt{u}\sinh ^{3}z},\quad R_{\overset{.}{1}\overset{.}{2}}^{\quad 23} = -\frac{2x}{u^{\frac{3}{2}}\sinh ^{2}z},\\ R_{\overset{.}{1}\overset{.}{3}}^{\quad 12}&= -\frac{2y\sinh z}{u\cosh ^{3}z},\quad R_{\overset{.}{2}\overset{.}{3}}^{\quad 12} = -\frac{2x\cosh z}{u\sinh ^{3}z},\quad \\ R_{\overset{.}{1}\overset{.}{2}}^{\quad 12}&= -\frac{2}{u}\left( 1+\frac{1}{u}\left( y^{2}\tan ^{2}z+x^{2}\cot ^{2}z\right) \right) . \end{aligned}$$

We note that we can compare with the discrete case by observing that:

$$\begin{aligned} R_{\overset{.}{1}\overset{.}{3}}^{\quad 13}&= -R_{\overset{.}{1} \overset{.}{3}}^{\quad 2}, \quad R_{\overset{.}{1}\overset{.}{2}}^{\quad 13}=-R_{\overset{.}{1}\overset{.}{2}}^{\quad 2},\\ R_{\overset{.}{2}\overset{.}{3}}^{\quad 23}&= R_{\overset{.}{2} \overset{.}{3}}^{\quad 1},\quad R_{\overset{.}{1}\overset{.}{2}}^{\quad 23} = R_{\overset{.}{1}\overset{.}{2}}^{\quad 1},\\ R_{\overset{.}{1}\overset{.}{3}}^{\quad 12}&= R_{\overset{.}{1}\overset{.}{3}}^{\quad 3},\quad R_{\overset{.}{2}\overset{.}{3}}^{\quad 12} = R_{\overset{.}{2}\overset{.}{3}}^{\quad 3},\quad R_{\overset{.}{1}\overset{.}{2}}^{\quad 12} = R_{\overset{.}{1}\overset{.}{2}}^{\quad 3}. \end{aligned}$$

3 Discretizition and numerics

In order to discretize, we follow the same methodology we used in [5]. Explicitly, we define:

$$\begin{aligned} \mathcal {\ell }_1 = \mathcal {\ell }_2 = \mathcal {\ell }_3 = \frac{1}{N},\quad x=\frac{n_{1}}{N},\quad y=\frac{n_{2}}{N},\quad z=\frac{n_{3}}{N}, \end{aligned}$$
(5)

where

$$\begin{aligned} n_{1}= 1, 2,\ldots N,\quad n_{2}= 1, 2,\ldots N,\quad n_{3} = 1, 2,\ldots N,\nonumber \\ \end{aligned}$$
(6)

which together with the constraint (from Eq. 2) give:

$$\begin{aligned} u=1-\frac{n_{1}^{2}+n_{2}^{2}}{N^{2}} > 0. \end{aligned}$$
(7)

Discretization is required in all three dimensions, x, y, and z; therefore, in terms of numerical complexity, the code is \(\mathcal {O}(N^3)\). Hence, expanding the lattice would scale as \(N^3\).

To derive the three-dimensional discrete curvature, we start with the definition [6]

$$\begin{aligned} \frac{i}{2}R_{\mu \nu }^{i}\sigma ^{i}&=\frac{1}{2\mathcal {\ell }^{\mu }\mathcal {\ell }^{\nu }}\left( \Omega _{\mu }\left( n\right) \Omega _{\nu }\left( n+\widehat{\mu }\right) \Omega _{\mu }^{-1} \left( n+\widehat{\nu }\right) \right. \\&\left. \quad \Omega _{\nu }^{-1}\left( n\right) -\mu \leftrightarrow \nu \right) \\&=\frac{1}{2\mathcal {\ell }^{\mu } \mathcal {\ell }^{\nu }}\left\{ \left( \cos \frac{1}{2} \mathcal {\ell }^{\mu } \omega _{\mu }\left( n\right) + i\widehat{\omega }_{\mu }^{i} \left( n\right) \right. \right. \\&\left. \quad \sin \frac{1}{2} \mathcal {\ell }^{\mu } \omega _{\mu }\left( n\right) \sigma ^{i}\right) \\&\quad \cdot \left( \cos \frac{1}{2}\mathcal {\ell }^{\nu }\omega _{\nu }\left( n+\widehat{\mu }\right) +i\widehat{\omega }_{\nu }^{i}\right. \\&\left. \quad \left( n\right) \sin \frac{1}{2} \mathcal {\ell }^{\nu } \omega _{\nu }\left( n+\widehat{\mu }\right) \sigma ^{i}\right) \\&\quad \cdot \left( \cos \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n+\widehat{\nu }\right) -i\widehat{\omega }_{\mu }^{i}\left( n+\widehat{\nu }\right) \right. \\&\left. \quad \sin \frac{1}{2} \mathcal {\ell }^{\mu } \omega _{\mu }\left( n + \widehat{\nu }\right) \sigma ^{i}\right) \\&\quad \cdot \left( \cos \frac{1}{2}\mathcal {\ell }^{\nu }\omega _{\nu }\left( n\right) -i\widehat{\omega }_{\nu }^{i} \left( n\right) \sin \frac{1}{2} \mathcal {\ell }^{\nu }\right. \\&\left. \left. \quad \omega _{\nu }\left( n\right) \sigma ^{i} \right) -\mu \leftrightarrow \nu \right\} . \end{aligned}$$

Consider the product

$$\begin{aligned}&\left( \cos \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) +i\widehat{\omega }_{\mu }^{i}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) \sigma ^{i}\right) \\&\left( \cos \frac{1}{2}\mathcal {\ell }^{\nu }\omega _{\nu }\left( n+\widehat{\mu }\right) +i\widehat{\omega }_{\nu }^{i}\left( n+\widehat{\mu }\right) \sin \frac{1}{2}\mathcal {\ell }^{\nu }\omega _{\nu }\left( n+\widehat{\mu }\right) \sigma ^{i}\right) , \end{aligned}$$

this can be rewritten as:

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

where

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

and:

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

Similarly, the next pair gives

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

Thus, the total product is

$$\begin{aligned} \frac{i}{2}R_{\mu \nu }^{i}\sigma ^{i}&=\left( \frac{1}{2\mathcal {\ell }^{\mu }\mathcal {\ell }^{\nu }}\left( A_{\nu \mu }+iB_{\mu \nu }^{i}\sigma ^{i}\right) \left( A_{\mu \nu }-iB_{\nu \mu }^{i}\sigma ^{i}\right) \right. \\&\quad \left. -\mu \leftrightarrow \nu \right) \\&=\frac{i}{\mathcal {\ell }^{\mu }\mathcal {\ell }^{\nu }}\left( A_{\mu \nu } B_{\mu \nu }^{i}-A_{\nu \mu }B_{\nu \mu }^{i}+i\epsilon ^{ijk}B_{\mu \nu }^{j}B_{\nu \mu }^{k}\right) \sigma ^{i}, \end{aligned}$$

giving the below result

$$\begin{aligned} R_{\mu \nu }^{i}\left( n\right)= & {} \frac{2}{\mathcal {\ell }^{\mu }\mathcal {\ell }^{\nu }}\left( A_{\mu \nu }\left( n\right) B_{\mu \nu }^{i}\left( n\right) -A_{\nu \mu }\left( n\right) B_{\nu \mu }^{i}\left( n\right) \right. \nonumber \\{} & {} \left. +\epsilon ^{ijk}B_{\mu \nu }^{j}\left( n\right) B_{\nu \mu }^{k}\left( n\right) \right) . \end{aligned}$$
(8)

The connection \(\omega _{\nu }\left( n\right) \) is determined from the zero torsion condition, which is given by

$$\begin{aligned} T_{\mu \nu }\left( n\right)= & {} \frac{1}{\mathcal {\ell }^{\mu }}\left( \Omega _{\mu }\left( n\right) e_{\nu }\left( n+\widehat{\mu }\right) \Omega _{\mu }^{-1}\left( n\right) -e_{\nu }\left( n\right) \right) \nonumber \\{} & {} -\mu \leftrightarrow \nu , \end{aligned}$$
(9)

where this equation is written in a contracted form where the vielbeins \(e_{\alpha }^{a}\) are incorporated into the Clifford algebra. Upon computing this, we get

$$\begin{aligned} 0&=\frac{1}{\mathcal {\ell }^{\mu }} \left( \left( \cos \frac{1}{2}\mathcal {\ell }^{\mu } \omega _{\mu }\left( n\right) +i\widehat{\omega }_{\mu }^{i}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) \sigma ^{i}\right) \sigma ^{k}\right. \\&\quad \left( \cos \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) -i\widehat{\omega }_{\mu }^{j}\left( n\right) \sin \frac{1}{2}\mathcal {\ell }^{\mu }\omega _{\mu }\left( n\right) \sigma ^{j}\right) e_{\nu }^{k}\left( n+\widehat{\mu }\right) \\&\quad \left. -e_{\nu }^{i}\left( n\right) \sigma ^{i} \right) -\left. \mu \leftrightarrow \nu \right) . \end{aligned}$$

By expanding and grouping terms, we get the following result:

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

The vanishing of \(T_{\mu \nu }^{\quad i}\) provides 9 conditions to solve for the 9 unknowns \(\omega _{\mu }^{i}\left( n\right) .\) The values of the spin connections are obtained numerically, and hence, the three-dimensional discrete curvatures are obtained using Eq. (8).

Equation 3 gives the expression of the scalar curvature in the continuous case. We are going to compare it with what we get from the discrete case. In the latter, the expression of the scalar curvature is given by:

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

which holds when \(e_{\mu }^{i}\) is diagonal, which is the case here as it is evident from Eq. 4. It is clear from (3) that there is a singularity at \(z=0\) and at all points on the circle defined by \(u=0\). First, we will examine the domain where z is large, followed by an analysis of the domain where z is small.

3.1 The limit of large z

$$\begin{aligned} R\rightarrow -\frac{4}{u} \left( 1 + \frac{x^2 + y^2}{u} \right) . \end{aligned}$$
(10)

The singularity at \(u=0\) is evident. Therefore, in order to determine the permissible range over which the curvature is well-behaved, we define \(1 - x^2 - y^2 > \epsilon \), the range of coordinates in the vicinity of the singularity. We note that \(\epsilon = 0\) is the locus of \(u=0\) or equivalently \(x^2 + y^2 = 1\), and thus finding the right value of \(\epsilon \) is equivalent to finding the proper cut-off to avoid the singularity. Therefore, we iterate over different values of \(\epsilon \) between 0 and 0.5 in increments of 0.05, and calculate the mean-square error between the continuous value of the scalar curvature and its discrete numerical counterpart defined to be \(\frac{1}{n} \sum \nolimits _i^n (R_{i,con} - R_{i,numerical})^2\). We repeat the process for different values of N. The results are plotted in Fig. 1. The different colors correspond to different values of N. We conclude from the plot that the error is converging to zero for \(\epsilon = 0.15\).

Fig. 1
figure 1

The rms error between the continuous and discrete values of the scalar curvature in the limit of large z are plotted as a function of \(\epsilon \) for different N, where \(\epsilon \) denotes the distance away from \(u=0\) where the discretization is expected to hold. We note that the approach to \(\epsilon = 0\) where the locus of the singularity leads to a blow-up in the RMS

Full size image

For \(\epsilon \) smaller than 0.15, we are close to the singularity (\(u = 0\)) and the discretization method fails to agree with the continuous limit as the RMS error between \(R_{con}\) and \(R_{discrete}\) blows up. In the limit where they are in agreement, we show a sample in Fig. 2 for \(N=35\) and \(\epsilon = 0.15\), where we plot the values of \(R_{con}\) and those of \(R_{discrete}\) on the same graph with the index i denoting the element number in the curvature vector associated with given coordinates \(x_i,y_i\) and \(z_i\).

Fig. 2
figure 2

R continuous and discrete for \(N=35\) and \(\epsilon = 0.15\)

Full size image

3.2 The limit of small z

For a small value of z, we are in the region near the singularity \(z = 0\). Therefore, and following the same reasoning we did to find \(\epsilon \) near \(u=0\), we limit z between \(\epsilon _z /N\) and 1 and follow the RMS between the continuous and discrete values of the scalar curvature. In Fig. 3, we show the RMS for \(N = 35\) and \(\epsilon = 0.15\), which allows us to define the lower cutoff near \(z = 0\) below which the scalar curvature will blow up.

Fig. 3
figure 3

The RMS error is followed as a function of \(\epsilon _z\) for \(N = 35\) and \(\epsilon = 0.15\)

Full size image

It is evident from the plot that as \(\epsilon _z\) approaches 5, the error tends to zero. Further, in order to show the well-behavedness of the discrete scalar curvature, we follow it together with its continuous counterpart in this domain. Figure 4 shows the finiteness of \(R_{discrete}\) as opposed to the large, near-singular behavior of \(R_{con}\). The inset shows their agreement away from the jumps.

Fig. 4
figure 4

R continuous and discrete are followed for \(N=35\), \(\epsilon = 0.15\), \(z<1\), and \(\epsilon _z = 5\). The inset shows their local agreement and makes it obvious that \(R_{con}\) exhibits large near-singular jumps, which are avoided in the discrete case

Full size image

For very small z, we have \(\sinh z\rightarrow z\) and \(\cosh z\rightarrow 1\),

$$\begin{aligned} R&\rightarrow -4\left( 1+\frac{1}{z^{2}}+\frac{1}{u}\left( y^{2} z^{2}+\frac{x^{2}}{z^{2}}\right) \right) \end{aligned}$$
(11)
$$\begin{aligned}&\simeq - 4\left( \frac{1-y^{2} }{uz^{2}} \right) \end{aligned}$$
(12)

with obvious asymmetry. Figure 5 is obtained by plotting \(R_{discrete}\) and \(R_{limit} = - 4\left( \frac{1-y^{2} }{uz^{2}} \right) \) for small values of z, showing their agreement.

Fig. 5
figure 5

\(R_{discrete}\) is compared with its limiting behavior \(4( \frac{1-y^{2} }{uz^{2}})\) for small z

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Further, when \(x=0\) we get \(R\rightarrow -4N^{2}\), displaying clearly the singularity of the black hole. In Fig. 6, the discrete value of the scalar curvature is plotted for \(z \simeq \frac{1}{N}\) and \(x = 0\) along with its expected limiting behavior, which is \(-4 N^2\).

Fig. 6
figure 6

\(R_{discrete}\) is compared with its limiting value \(-4N^2\) for \(z \approx \frac{1}{N}\) and \(x=0\)

Full size image

4 Conclusion

In this paper, we applied the methodology used in [5] to define the curvature of the discrete space in the proposed model of discrete gravity. We considered the black hole coset space metric. The singularities were studied, and the curvatures in the discrete and continuous limits were compared close and far away from the singularity. We found that as we move away from the singularity, the rms error between the two curvatures tends to zero. We also showed that near the singularity, the discrete method is more reliable than the continuous.

Our contribution to the literature on black holes and numerical relativity is that it introduces a discrete approach complementing the prevailing continuum counterpart. This sets the ground for the calculations of curvatures for arbitrary surfaces. Further, the introduction of our method in the context of naked singularities establishes a mathematical and numerical framework to be generalized to 4D. Unlike the general concern of numerical relativity in answering questions about dynamics around black holes and their properties [2, 9, 10, 12], our work is an experimentation on the limitation of discrete gravity and its accompanying numerics and whether it is generalizable to 4D. The next step is to address how to calculate the metric and curvature of an arbitrary closed surface.