# Discretization of space and time: consequences of modified gravitational law

Luca Roatta

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## To cite this version:

Luca Roatta. Discretization of space and time: consequences of modified gravitational law. 2017. [hal-01494151](https://hal.science/hal-01494151)

HAL Id: hal-01494151

https://hal.science/hal-01494151

Preprint submitted on 22 Mar 2017

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# Discretization of space and time: consequences of modified gravitational law

Luca Roatta

E-mail: lucaroa@gmail.com

## Abstract

Assuming that space and time can only have discrete values, it is shown that the modified law of gravitational attraction implies that the third principle of dynamics is not fully respected and that only bodies with sufficient mass can exert gravitational attraction.

## 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 that in this discrete context the modified gravitational law obtained in a previous document[1] causes a (little) violation of the third principle of dynamics. We also find that not all bodies can exert gravitational attraction: the condition that allows a body to deform the discrete space, and consequently exert a gravitational force, is that its mass must not be lower than a minimum mass.

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[2], we call $l_0$ the fundamental length and $t_0$ the fundamental time.

## 2. Difference between discrete gravitational force and Newtonian

In a previous document[1] we obtained that in a discrete context the modified expression of the gravitational force exerted by a body A having mass $M$ on a body B having mass $m$ at the distance $r>d$ from A is:

$$f(r) = -mc^2 \frac{r}{(r-l_0)} \frac{d}{(r-d)^2} \quad (1)$$

where

$$d = \frac{GM}{c^2} \quad (2)$$

is the radius of the black hole of mass $M$.

If $r \gg l_0$ Eq. (1) becomes:
$$f(r) = -\frac{GMm}{(r-d)^2} \qquad (3)$$

The Newtonian force acting on B is:

$$f_N(r) = -\frac{GMm}{r^2} \qquad (4)$$

The difference between Eq. (3) and Eq. (4) $f(r)-f_N(r)$, when $r \gg d$, is:

$$f(r) - f_N(r) = -\frac{2GMmd}{r^3} \qquad (5)$$

that is the exact expression of a tidal force.

It seems that in this discrete context the new expression of the gravitational force contains a term which takes account of tidal effects. It is an aspect that should be further investigated.

### 3. A (little) violation of the third principle of dynamics

The term $d$ in Eq. (2) is also present in the expression[3] of the discrete space deformed by matter:

$$l_0(r) = l_0 \frac{r-d}{r} \qquad (6)$$

Depending $d$ on the mass of only one body (in this case the body A), it's evident that the value of $l_0(r)$ is different if we consider how the body B modifies the discrete space, because in this case we have:

$$d = \frac{Gm}{c^2} \qquad (7)$$

The difference in Eq. (6) when $d$ is expressed by Eq. (2) or by Eq. (7) becomes irrelevant when $r \gg d$, but it evidences a quite surprising fact: also Eq. (1), that expresses the force that the body A exerts on the body B, depends on $d$ so, if we consider the force exerted by B on A, we obtain a different result, due to the presence of a different value of $d$ at the denominator of Eq. (1).

This means that the gravitational force exerted by A on B is different from the gravitational force exerted by B on A. This difference (that when calculated for $r \gg d$ has the exact form of two tidal forces, one acting on A and one acting on B) tends to disappear when $r \ll d$, but it implies the violation of the third principle of mechanics, at least in this gravitational context.

Many other documents[4][5][6][7] speak about the violation of the action-reaction principle in particular circumstances, so it doesn't seem to be a completely new concept.

In this discrete context, however, the expected violation can be useful to design an experiment that could confirm or disprove this theory: with a sufficiently sensitive torsion balance, it would be possible to test this difference.

### 4. The minimum mass for gravitational attraction

In a previous document[2] we have already determined the existence of a maximum frequency and its value. For black holes there is a similar limit: as for any length, also for $d=GM/c^2$, the radius of the black hole of mass $M$, must be $d \ge l_0$. This means $M \ge l_0 c^2 / G$; so there is a minimum mass a body must have to be a black hole:
$$m_{min} = \frac{l_0 c^2}{G} \qquad (8)$$

A mass lower than $m_{min}$ can not generate a black hole, because its radius should be less than $l_0$. Also in this case, as can be noted, in the continuous context ($l_0 \to 0$) there is no lower limit for the mass of a black hole.

The fact that must be $d \ge l_0$ implies that in Eq. (1) it is impossible to have arbitrarily small masses, because the limit $M \ge m_{min}$ must always be satisfied.

Of course it does not mean that masses lower than $m_{min}$ can not exist: it means however that lower masses can not originate gravitational attraction and, or rather *because*, they can not warp space, as shown in a previous document[3].

This has the consequence that General Relativity is not valid for masses lower than $m_{min}$, which so constitutes the lower limit of applicability of General Relativity, similarly to the mass indicated in [2], which constitutes the upper limit of applicability of wave mechanics and therefore also of quantum physics.

It is known or supposed that General Relativity breaks down at the Planck scale[8], or inside of a black hole[9], or at high energies[10], or at singularities[11]; so also the limit of applicability of General Relativity found in this discrete context does not seem to be a completely new concept.

## 5. Conclusion

Assuming that space and time can only have discrete values, it has been shown that the modified law of gravitational attraction implies that the third principle of dynamics is not fully respected and that only bodies with sufficient mass can exert gravitational attraction. Both arguments do not represent completely new concepts.

## References

[1] Luca Roatta. Discretization of space and time: a slight modification to the Newtonian gravitation which implies the existence of black holes. 2017. <hal-01493148>. Available at https://hal.archives-ouvertes.fr/hal-01493148

[2] 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

[3] Luca Roatta. Discretization of space and time: how matter deforms space and time. 2017. Available at https://hal.archives-ouvertes.fr/hal-01481775

[4] Harvey R. Brown, Dennis Lehmkuhl. Einstein, the reality of space and the action-reaction principle. 2013. Available at https://arxiv.org/pdf/1306.4902.pdf

[5] Mario J. Pinheiro. On Newton's Third Law and its Symmetry-Breaking Effects. 2011. Available at https://arxiv.org/pdf/1104.5011.pdf

[6] Carlos Lopez. Relativistic locality and the action reaction principle predict de Broglie fields. 2016. Available at https://arxiv.org/pdf/1605.00844.pdf

[7] Pascal R. Buenzli, Rodrigo Soto. Violation of action-reaction and self-forces induced by nonequilibrium fluctuations. Available at https://arxiv.org/pdf/0804.3106.pdf

[8] Ruth Durrer, Roy Maartens. Dark energy and modified gravity. 2008. Available at https://arxiv.org/pdf/0811.4132.pdf

[9] Lee Smolin. The fate of black hole singularities and the parameters of the standard models of particle physics and cosmology. 1994. Available at https://arxiv.org/pdf/gr-qc/9404011v1.pdf
[10] Jun Tao, Peng Wang, Haitang Yang. Free-fall Frame Black Hole in Gravity's Rainbow. 2016. Available at https://arxiv.org/pdf/1602.08686v2.pdf

[11] Ovidiu Cristinel Stoica. Spacetimes with Singularities. An. Stiint. Univ. Ovidius Constanta, Ser. Mat. 2012. Available at https://hal.archives-ouvertes.fr/hal-00617027