# Discretization of space and time: determining the speed limit

Luca Roatta

► To cite this version:

Luca Roatta. Discretization of space and time: determining the speed limit. 2017. hal-01480553

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# Discretization of space and time: determining the speed limit

Luca Roatta

E-mail: lucaroa@gmail.com

## Abstract

Assuming that space and time can only have discrete values, it is shown that necessarily there must be a superior limit for the speed and that this limit is coincident with the speed of light c.

## 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 the upper limit for the speed that a particle or a body can reach.

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 the maximum speed

The existence of a minimum time below which it is impossible to descend, necessarily implies the existence of a maximum speed: in fact, if *ad absurdum* it were not so, and the speed could take arbitrarily large values, it would always be possible to find a speed $v$ at which the time taken to travel a certain space would be smaller than $t_0$. But this is not possible because it is in contrast with the initial hypothesis that $t_0$ is the smallest time interval.

So there is a maximum speed; let's call it $v_{max}$.

Directly from the definition of constant speed $v = s/t$, we have $t = s/v$.

The minimum value for $t$ in a discrete context is of course $t_0$ that can be obtained minimizing the numerator $s$ and maximizing the denominator $v$.

The minimum value for $s$ is by hypothesis $l_0$ and the maximum value for $v$ is $v_{max}$ as shown above. So we have:

$$t_0 = \frac{l_0}{v_{max}} \qquad (1)$$
or

We have already obtained[1] that

so we must conclude that

$$v_{max} = \frac{l_0}{t_0} \qquad (2)$$

$$\frac{l_0}{t_0} = c \qquad (3)$$

$$v_{max} = c \qquad (4)$$

### 3. Conclusion

The assumption that both space and time are discrete has led to find the existence of a maximum speed that can not be exceeded. This upper limit results to be coincident with the light speed c.

### 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