# Discretization of space and time: how matter deforms space and time

Luca Roatta

<figure><img src="image_1.png" /></figure>

## To cite this version:

Luca Roatta. Discretization of space and time: how matter deforms space and time. 2017. hal-01481775

HAL Id: hal-01481775

https://hal.science/hal-01481775

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# Discretization of space and time: how matter deforms space and time

Luca Roatta

E-mail: lucaroa@gmail.com

## Abstract

Assuming that space and time can only have discrete values, it is shown how space and time are deformed in presence of matter. It's introduced a conceptual model to explain the procedure followed.

## 1. Introduction

Let's assume, as work hypothesis, the existence of both discrete space and discrete time, namely spatial and temporal intervals not further divisible; this assumption leads to some interesting consequences. Here we find how the presence of matter deforms space and time.

So, if we suppose that neither space nor time are continuous, but that instead both are discrete, and following the terminology used in a previous document[1], we call $l_0$ the fundamental length and $t_0$ the fundamental time.

## 2. Determining how matter deforms space and time

The existence of a minimum length below which it is impossible to descend, necessarily implies that in a discrete space (for conceptual simplicity we consider only one dimension), in a vacuum, for any value of the length $r$ there must be an integer number $n$ such that $r=nl_0$.

General Relativity affirms that matter deforms spacetime, that is a continuum in GR.

Here we try to understand how the *discrete* space behaves in presence of material bodies.

However, we can not start simply considering a material body A located in the origin of the discrete $x$ axis and a second material body B located at a distance $r$ from A, because if we want to understand how the two material bodies deform the discrete space, we can not think to move a body from somewhere to the desired position, because so doing the space would already be deformed right from the beginning.
We must in some way build our system so that the deformation of space appears only when the two bodies are in the desired position. Of course it is a conceptual construction, whose lawfulness we reserve to evaluate based on the results obtained.

The figure below shows the initial situation in presence of space not yet deformed:

<figure><img src="image_3.png" /><figcaption>Fig. 1</figcaption></figure>

Said n the number of space cells that are between the two bodies, it follows that:

$$r=nl_0 \qquad (1)$$

Let us now *expand* the body A until it reaches a linear dimension denoted by *d*. This results in a shortening of the cells.

To visually clarify this, we use an image that, although based on different physical concepts, can help to better understand the process.

Let us suppose that the two bodies are joined together by a spring; the expansion of the body A causes a shortening of the spring and thus a decrease of the distance between the coils of the spring, while the number of the coils does not change. Now, if the body A is rigid, the shortening of the spring will coincide exactly with the real size of the body A. If the body A is soft, such as a ball of dough that leavens, then a shortening of the spring will still occur, but it will no longer match the size of A, that being soft will extend up to incorporate part of the coils.

From this dual model (rigid body and soft body) it is evident that the size of A will be higher than, or at most equal to, *d*, the value that corresponds to the shortening of the spring, but not lower (it is difficult to imagine a super-rigid body, having dimension less than *d*, whose expansion causes a shortening equal or even superior to *d*). Basically, if the body A were soft, the shortening of the spring due to its expansion would correspond to the shortening caused by the expansion of a *rigid* body having dimension *d* smaller than the real dimension of the *soft* body A. Another observation to make is that *d* should only depend on the properties of the body A because no other element has been considered. Of course, also the linear dimension *d* must comply with the condition $d \ge l_0$. So there will be no expansion, and consequently no deformation in space, if *d* for any reason can not reach the value $l_0$.

In our case, therefore, no a priori assumption is made about the physical meaning of *d*, nor about his relationship with the real dimension of A. As a result of the expansion, there are no changes for the value of *r* (it is like fixing the body B at the distance *r*), nor for the value of *n*, because in this process no space cell is lost. The value of $l_0$ instead changes, because now depends on *r* (and *d*); let us denote this value by $l_0(r)$: the cells shrink to make room for the expansion of the body A.

The figure below shows the situation in which the body A is **not** enlarged up to incorporate the body B: this means $d<r$.
<figure><img src="image_4.png" /><figcaption>Fig. 2</figcaption></figure>

It is evident that

$$d+n l_{0}(r)=r \quad (2)$$

from which, given that for Eq. (1) is $n=r/l_0$, we obtain:

$$l_{0}(r)=l_{0} \frac{r-d}{r} \quad (3)$$

The figure below shows the situation in which the body A is enlarged up to incorporate the body B: this means $d>r$.

<figure><img src="image_5.png" /><figcaption>Fig. 3</figcaption></figure>

It is evident that

$$d-n l_{0}(r)=r \quad (4)$$

from which, given that for Eq. (1) is $n=r/l_0$, we obtain:

$$l_{0}(r)=l_{0} \frac{d-r}{r} \quad (5)$$

It has already been shown[1] that $l_0/t_0=c$. In presence of matter, $l_0(r)$ must be used instead of $l_0$. If we want to maintain $c$ constant, also $t_0$ must be replaced by $t_0(r)$, so we have $l_0(r)/t_0(r)=c$. It follows that:

$$t_{0}(r)=\frac{l_{0}(r)}{c} \quad (6)$$

### 3. Conclusion

The assumption that both space and time are discrete, using a simple conceptual model, has led to find how the presence of matter deforms space and time.

### References

[1] Luca Roatta. Discretization of space and time in wave mechanics: the validity limit. 2017. <hal-01476650>. Available at https://hal.archives-ouvertes.fr/hal-01476650