Thierry Coilhac, June 2022

A new approach to quantum gravity based on the philosophical assumption that ” Only what is observable exists” .

From our initial postulate that “only what is observable is real” we conclude that it is the interaction between elementary particles that creates the reality of the world. The belonging of macroscopic objects to the same reality requires that all the particles that compose them, and therefore ultimately all the particles of the universe, interact two by two via a new interaction in order to confirm their mutual existence and define the distance between them, which is necessary to define the relative distance of the macroscopic objects to which they belong.

A maximum coherence of the universe, considering a speed of propagation of information limited to C, is obtained if two particles interact every instant R/C where R is the distance which separates them. Their interaction gives rise to a change of state corresponding to one quantum of action ћ. Every pair of particles therefore performs a change of ћ every R/C instant.

The characteristics of a two particles system as well as the transition from a relative frame to an absolute frame conduct to Newton’s law from the above considerations.

The fact that the reality of the world is defined by the two-to-two interaction of all the particles of the universe applies to space-time and its geometry. An element of length is real only if it corresponds to the distance defined by the interaction of two particles. It follows that, as in general relativity, matter-energy defines the geometry of space-time.

 

1. An interaction between particles to sustain reality

The question of what is real or not has always animated both philosophical and scientific debates. We will see how an answer to this question leads to the conclusion that gravitation is a consequence of Special Relativity (SR) and Quantum Mechanics (QM).

We start from the very general principle that only what is observable is real . The term “observable” must be universal, both for the observer and for the observed. It must certainly apply to the researcher observing particles, but cannot require that a human being endowed with consciousness observes the object considered. The most general “observation” mechanism is interaction. We therefore adopt the new postulate that only what interacts is real.

The question that comes then is “which interacts, but with what?”. With something from the real world. This suggests that the notion of reality is symmetrical. Two things exist for each other if they interact, then they belong to the same reality.

The next question is “which interacts, but how?”. Any interaction will ultimately be through an elementary particle. In any case, a macroscopic object does not have a precisely defined existence, what exists is the set of particles that compose it. The notion of reality is defined at the level of elementary particles. The reality of the world is defined by the set of interacting particles. So what allows two objects to exist for each other, to belong to the same reality, is the fact that all the particles of one interact with all the particles of the other.

Nevertheless, the interactions already known would a priori be insufficient to maintain the reality of the objects that make up the world. We are therefore led to suppose that there is an interaction between particles which maintains the notion of reality. We call it the O interaction. The fact that two particles exist for each other is based on their O interaction.

2. Application to two masses

We apply this principle to two objects of masses M1 and M2, and to two particles P1 and P2 belonging respectively to M1 and M2. P1 and P2 are separated by a distance R and assumed to have no relative speed. P1 and P2 have masses m1 and m2. We will use the principle stated by De Broglie “to each piece of energy of proper mass m is linked a periodic phenomenon of frequency ν with h ν = mc²”. The De Broglie waves associated with P1 and P2 have the frequency, wave number and wavelength ν1, ν2, k1, k2 , l1 , l2 . For each particle we have k = 2 π ν / C, where C is the universal constant. The relation h ν = m C² therefore leads to k = m C / ћ, where ћ is the Dirac constant also called reduced Planck constant.

3. P1 and P2 interact all R/C timeslots

The interaction between P1 and P2 is necessarily accompanied by a change of state of the P1 – P2 system, and therefore of each particle, which requires an exchange of information . However, according to RR, information cannot go faster than C. The two particles cannot therefore interact once and then a second time immediately afterwards. From the interaction and during an R/C time no information coming from P2 can arrive in P1. They will only be able to interact again after an R/C time. They interact at most all R/C time intervals.

During the interaction, the state of P2 seen from P1 is that of P2 one time interval T=R/C earlier, T being the time it takes for the state of P2 to arrive at P1. P1 and P2 interact with a time lag equal to R/C .

The distance D which separates them in space-time at the moment of the interaction observed in P1 is therefore such that D² = C² T² – R² is zero. The O interaction is therefore local in the space-time in which the particles are separated by a zero distance.

Do P1 and P2 interact again as soon as the R/C time is up? We propose an affirmative answer which allows the link between P1 and P2 resulting from this interaction to be as frequent as possible. This link is even permanent but with a time lag, since there is then no instant in the two-particle system P1-P2 shared between P1 and P2 during which they do not interact. Indeed, apart from these moments of interaction, every R/C, P1 and P2 do not exist for each other since no information can link them.

4. The O interaction between P1 and P2 causes them to approach a quantum of action

The interaction between P1 and P2 gives rise to a change of state whose action A cannot be less than the Dirac constant ћ in accordance with the principle stated by Niels Bohr. We consider that this action follows the principle of least action, that is to say that it is the smallest possible. Its value is therefore A = ћ.

Recall that action A is the product of displacement times momentum . In our case the momentum is worth ћ k. We do not use the momentum defined as the product of the mass times the velocity because we consider that the particles make a jump. Prior to the interaction the particles have not begun to move in preparation for the interaction. It is the interaction that instantly causes a change of state, so the notion of speed is not relevant. Hence A = ћ k d, where d is a displacement representing the change of state resulting from the interaction. Action A must have this value both in a reference frame associated with P1 and a reference frame associated with P2 because the action is an invariant by transformation.

We evaluate this displacement seen from P1. The particle P1 can perceive this displacement only by noting a change of position of P2 compared to it.

We first observe the change from P2

For the displacement of P2 to have an action of value ћ it must perform a displacement d2 such that A = ћ k2 d2 is equal to ћ. Hence d2 = 1/k2, or d2 = l2/2π , or d2= ћ / m2 C . In the reference frame associated with P2, the new position of the particle P2 is therefore on the sphere centered on its previous position and of radius d2 = ћ / m2 C.

We now observe the change from P1

We observe that the two-particles system P1 – P2 has a cylindrical symmetry but does not have enough particles to define an angular variation. To illustrate this, note that the two configurations in the picture below are identical.

First consequence, a change of position of P2 is always seen from P1 as either a rapprochement or a moving away. We favor the hypothesis of a rapprochement because if we consider that the interaction must maximize the coherence of the P1-P2 system, this objective will be achieved with a reduction in the distance which separates the particles, which will increase the R/C frequency which they interfere.

Second consequence, among all the new possible positions of P2 seen from P2 we must eliminate those which are not on the P1 – P2 axis because seen from P1 they will result in a displacement less than d2 and therefore in an action less than ћ. In the picture below, the new positions P2′, P2” are not possible because the variation in distance seen from P1 equaling d2 cos(θ) leads to an action ћ k2 d2 cos( θ) which is less than ћ.

However, the probability of a position being exactly on the axis is zero because it corresponds to a single point. We take into account the fact that a distance is only defined up to the Planck length λ . All the possible positions of P2 distant from the axis P1-P2 by a distance less than λ are considered possible because they also give rise to an action equal to ћ seen from P1. The other positions are considered impossible, therefore of zero probability. In summary, on the diagram above:

• All the positions on the sphere of radius d2 centered on P2, which like P”’2 and P2”” are distant from the axis P1-P2 by a value less than λ are considered to be on the axis P1-P2, and therefore correspond to a displacement of d2 measured in R1. These positions occupy a surface π λ² which is the surface of the disc of radius λ.
• All the other positions on the sphere of radius d2 are equivalent to a displacement d2 cos( θ) smaller than d2 seen from P1, they therefore lead to an action less than the action quantum. They therefore have zero probability. Let us notice that this point imposes a conservation of the cylindrical symmetry around the initial axis P1-P2.

The particle P2 therefore performs a displacement d2 with a probability π λ²/ 2 π d2² (positions on the disc of radius λ on the hemisphere of radius d2), and performs a displacement d2 with probability zero (other positions). The average displacement is therefore worth d= d2 πλ²/ 2 πd2², i.e. d= λ²/2d2, which we can write in the form d=λ² k2/ 2, or d = π λ² / l2 or d = λ² m2 c / 2 ћ.

The distance d is the displacement of P2 towards P1 observed by P1, that is to say the displacement of P1 observed by P1 in a reference frame associated with P2. At each interaction, i.e. every R/C instant, the particle P1 moves by the average distance d =π  λ² / l2 towards the P2 particle.

Note: We can verify that the average speed between two interactions is much lower than C since it is (π  λ²/ l2)/(R/C ), i.e. C π  λ² / l2 R. This verification is necessary due to the fact that we took ћ k as the value of the momentum.

5. This displacement is equivalent to a gravitational acceleration

Let us now show that this displacement of a constant distance d = π λ²/ l2 every time interval t = R/C is equivalent to an acceleration and calculate the value of this acceleration. For this, let’s see what happens over several successive time intervals.

*) At time 0, the particle P1 has zero speed in the reference frame R1(0) attached to it. After an interval t= R/C it has move a distance d and has acquired a speed v.

*) At time t, in the reference frame R1(t) which is attached to P1 and which therefore moves at the speed v compare to R1(0), P1 has zero speed. So its evolution is the same as before. After a new instant t, which brings us to time 2 t, it has again covered a distance d and has acquired a speed v. So in the initial reference frame R1(0),

• It has acquired the speed 2 v, corresponding to its speed in R1(t) to which we add the speed of R1(t) in reference R1(0).
• It has performed a new displacement of d + v t, that is to say its displacement in R1(t) to which we add the displacement of R1(t) in R1(0) .

*) At time 2 t, in the reference frame R1(2 t) which is attached to P1 and which therefore moves at the speed v compare to R1(t), P1 has zero speed. So its evolution is the same as before. After a new instant t, which brings us to time 3 t, it has again covered a distance d and has acquired a speed v. So in the initial reference frame R1(0),

• It has acquired the speed 3 v, corresponding to its speed in R1(2 t) to which we add the speed of R1(2 t) in reference R1(0).
• It has performed a new displacement of d + 2 v t, that is to say its displacement in R1(2 t) to which we add the displacement of R1(2 t) in R1(0) .

*) And so on.

So at time T = n t, in the initial reference frame R1(0) :

• The particle has acquired the speed nv which we note V, V = n v
• It has performed a displacement D equal to the sum of displacements during all time intervals t,

D = d + ( d + vt ) + (d + 2 vt) + …. + (d + (n-1) vt

D = nd + vt ( 1 + 2 + …. + (n-1))

D = nd + vt n (n-1)/2

The velocity v is the final velocity of the particle after traveling a distance d in a time t. As an approximation, we can estimate that this distance d is equal to the average speed vm multiplied by time, d = vm t, vm being the average between the zero initial speed and the final speed v and thus being worth vm =v/2. Hence d = v t / 2, hence v = 2 d / t

By using the relation v = 2 d /t and n = T/t in the expression describing D we obtain:

D = n d + (2 d / t) t n (n-1) / 2

D = n d + d n (n-1)

D = d n²

D = d T²/t²     which we put in the form
D = 2 ( ½ d/t²) T²

 

In the same way we have V = n v, V = n 2 d / t, V = 2 d T / t², V = 2 d/t² T

So, after a time T, the particle has traveled the distance D = ½ ( 2 d/t²) T² , and it has acquired the velocity V = 2 d/t² T. That means that the particle undergoes an acceleration of value g1 = 2 d / t²

Note : we have demonstrated that a displacement of a constant distance at constant time intervals in the local frame is equivalent to a constant acceleration in an absolute frame. The fundamental principle of dynamics is demonstrated in the same way.

In our case we saw above that d = λ ² m2 C / 2 ћ  and t = R/C.
The acceleration of the particle P1 is therefore g1 = λ² m2 C / ћ x C²/R², or g1 = λ² m2 C ³ / ћ R²

Taking into account the relationship λ² = G ћ / C ³ where G is the gravitational constant, we have
g1 = m2 G / R².

The movement of the particle P1 observed in its own reference frame is therefore identical to the movement resulting from the gravitational acceleration generated by the particle P2.

6. Newtonian attraction between the two masses M1 and M2

Let us now see that applied to all the particles which compose them, this phenomenon leads to the Newtonian attraction of the massive objects M1 and M2.

We took the case of a P1 particle of M1, and a P2 particle of M2. Let us now consider all the particles of M1 which we call P1i, i going from 1 to N1 where N1 is the number of particles of M1, and all the particles of M2 which we call P2j, j going from 1 to N2 where N2 is the number of particles of M2.

Each of the particles P2j will exert on each of the particles P1i an attraction g1ij = m2j G / R²

For each of the particles P1i all the accelerations g1 ij add up, since P1i will interact with each of the particles P2j, each time causing a displacement. The global acceleration undergone by P1i is thus:

g1i = (m21 + m22 .. + m2j … + m2N2) G / R², where m2j is the mass of the particle P2j

As m21 + m22 .. + m2j … + m2N2 is the sum of the masses of all the particles in M2, it is the total mass of M2. From where g1i = M2 G/ R², where g1i is the attraction of M2 on the particle P1.

All the particles of P1i of M1 undergo the same acceleration, therefore M2 G / R² is also the acceleration of the mass M1 resulting from the gravitational attraction exerted by M2.

7. Openings to General Relativity

Our postulate that “only what is observable exists” also applies to Space-Time. A point in space-time only exists if it is observable, which requires it to be able to interact and for that to correspond to the position of an elementary particle.

Similarly, a distance R in space only exists if it is observable, and therefore if it corresponds to a distance separating two particles. Let’s call these particles P1 and P2. Apart from the instants during which P1 and P2 interact, P1 and P2 no longer exist for each other because no information can link P1 to P2 as discussed in paragraph 3. Therefore, the distance R exists only during the interactions between P1 and P2, it is defined by these interactions. This reasoning applies to points and distances in space, but also in space-time.

We thus join the point of view of GR according to which space-time is defined by matter-energy, which is composed of particles. We can represent us the space-time as flickering points, each flicker corresponding to one particle interacting with another. Only the flickering points exist in the space-time. Each flicker is accompanied by a modification of the distance between the two particles which interact so that the change represents a quantum of action, and therefore of the distance defined by these two particles. Hence the deformation of the space-time by particles, that is to say by matter energy.

We note that in this approach the density of the universe is zero as when it was a singularity and that all the particles remain separated by a zero distance in the space-time.

8. Analysis

Our approach provides an elementary mechanism for gravitation. This allows us to formulate some comments relating to universal constants and physical quantities.

• Action is the fundamental quantity involved in gravitation. Gravity results from the fact that the interaction O between two particles causes a change whose action is one quantum of action ћ. The principles of minimum action, least action and conservation of action by change of reference frame play a key role.
• The constant C first appears as the speed at which the reality of a particle arrives in another particle.
• R/C represents the time lag between two particles separated by a distance R. One particle sees the other particle as it was an R/C time earlier.
• Its squaring, at the origin of the term R² in Newton’s law, results from the transition from a local reference frame to an absolute one. So the term 1/R² is doubly of relativistic origin.
• The constant C then appears in the momentum of the particles, which intervenes in the action relating to the change of state following an interaction O. The term C³ that we find in G = λ² C ³ / ћ is therefore derived, for a factor C² from the time lag of two interacting particles, for a factor C from the action associated with the change of state.
• Regarding the proportionality between the mass and the gravitational acceleration , the wavelength l2 of the De Broglie wave associated with the particle P2, inversely proportional to its mass appears at two levels. In paragraph 4:
  • When we consider P2 alone, the result d2 = l2/ 2π reflects the fact that the higher l2, the more uncertainty there will be about its new position, so the greater the probability that its new position will be far from the initial position. . The wavelength l2 appears as the quantification of the average displacement of P2. The mass m2, inversely proportional to l2, appears as the anchoring of the particle in space, its resistance to displacement, which joins the notion of inertia.
  • When we consider P2 in the P1-P2 system, the result d = π λ² / l2 reflects the fact that the higher l2, the greater the probability that the new position of P2 seen from P1 corresponds to an action less than the quantum of action, and therefore that the displacement does not take place since the action cannot be less than ћ.
  • We note that the value of the action seen from the particle P1 changes the coefficient l2 from the numerator to the denominator, because this time it is a surface and not a length that is involved in the calculation. Moreover it imposes a displacement on the axis P1-P2, therefore a conservation of the symmetry
• The Planck length λ appears as the precision with which a length is defined, it represents the grain of space. It can be considered as a quantum of distance. The greater its value, the greater the attraction, since the greater the number of possible final positions on the surface disk π  λ² will be big. The term λ² found in G reflects this surface.
• Newton’s law is expressed by the equation g1 = λ ² m2 C ³ / ћ R². It does not go through the notion of force which is only a calculation tool without physical reality. What is measurable is a displacement and not a force. If the displacement is made impossible because of other interactions, the effect of the elementary mechanism at the origin of gravity will relate to a change of state of another nature.

Regarding General Relativity, a more rigorous study of the proposed elementary mechanism could lead to Einstein’s equation establishing a relationship between the metric tensor and the energy-momentum tensor. This would require in particular taking into account in a relativistic approach the relative speed between two interacting particles and the fact that the distance separating the particles is not R but evolves with time.

This elementary mechanism opens ways to explain the divergences between the GR and the observations. Dark matter could for example be explained by the fact that under certain conditions the O interaction gives rise to a displacement representing several quantums of action and not just one, or by the fact that our elementary mechanism does not apply only to particles but to any “piece of matter”. The dark energy could be explained by the fact under certain conditions the O interaction gives rise to a distancing and no longer a rapprochement for which we have opted.

This elementary mechanism makes it possible to imagine gravitational barriers based on a blocking of the O interaction, which opens prospects for future of transport.