Discretization of space and time: determining the speed limit

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Preprint submitted on 1 Mar 2017

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Discretization of space and time: determining the speed

limit

Luca Roatta

To cite this version:

(2)

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 t0 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 υ at which the time taken to travel a certain space would be smaller than t0. But this

is not possible because it is in contrast with the initial hypothesis that t0 is the smallest

time interval.

So there is a maximum speed; let’s call it υmax.

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

The minimum value for t in a discrete context is of course t0 that can be obtained

minimizing the numerator s and maximizing the denominator υ.

The minimum value for s is by hypothesis l0 and the maximum value for υ is υmax as

shown above. So we have:

t₀=_υl0

max (1)

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or

υ_max=l0 t0

(2) We have already obtained[1]_that

l₀ t0

=c (3)

so we must conclude that

υ_max=c (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