_{1}

This paper follows the earlier paper Part I: Veringa, H. J. (September 2016), New Basic Theory of Gravity, Journal of Modern Physics 7 (1818-1828) in which a new model to describe the gravitational interaction between particles and its consequences on the attractive force between two masses is proposed. The basis for the analysis is a merger of Quantum theory and Relativity. Nowhere in the analysis, there is any need to deviate from well proven and successful concepts of both theories and rules of calculation, and no exotic new particles will have to be introduced. By doing so, it is demonstrated that, next to its local interactions of a multi-particle system, the Schr ödinger equation leads to pairs of two and only two members. This solution is used as the invariant term in the quantized Dirac equation which gives gravitational interactions between members of the pairs. With this particular solution for the quantum-mechanical wave function, it is found that gravity is a second order effect operating over a long range. The emphasis of this paper is on the more precise justification of some of the basic assumptions made, on the historical context into which it should be placed, how it affects the ordering of our immediate environment and works on a cosmological scale. It is also found that the generator of gravity is contributing mass to particles that have gravitational interaction. This contribution is therefore related to cosmological parameters and will be further elucidated.

This paper elaborates on the analyses made in an earlier publication published in September 2016 [

The previous letter addresses the basic aspects, but some of the arguments can be expounded on and be put into a context of general insights into the micro-, as well as the macro cosmos. Then, later, some interesting clarifying analogies with daily life are offered. The analysis strongly connects with the famous debates between the founding fathers of Quantum theory and Relativity. Appreciation of these controversies shows that these historical, counter-intuitive, but at that time revolutionary, insights are all-important for the present development of the basic theory of gravity. When referring to formulas in the previous paper, Part I, the equations are indicated by (I. N).

In Equation (I.1) of Part I, a pair forming quantum mechanical wave equation is postulated, but some further justification can be given.

The total wave function describing a particle under its local influences,

Separating the local effect from the surroundings we can set:

The first Equation (2a) is the Schrödinger equation describing the behaviour of the particle in its local environment like in the nucleus or a solid where it has its individual interactions. The second Equation (2b) describes its movement or presence in the free space in which the particle or as part of a larger entity can move around. By taking

The splitting up as in Equations (2a) and (2b) disconnects the local interaction of separate particles, as is normally done in quantum mechanics, from the movement or presence of the particle individually or as part of a larger entity. In what follows we will only consider the second equation as this gives the generator for the gravitational interaction. As we are interested in the effects of masses outside the local interactions we will from now on take for the mass

A general solution of a wave equation describing independent particles in spherical symmetry is initiated by the operator:

But now we take the operator pair-wise with a pair identified by the set (ij) so that the operator becomes:

The factor 1/2 is to compensate for double counting and in the case of

The case of

consider independent pair potentials of the pair ij, and we will replace the

It has been shown in part I that _{i} and m_{j} find themselves, there is always a

There is freedom in the choice of the particles m_{i}, m_{j}, ・・・, m_{l}, ・・・. It can actuallly be anything like elementary particles, nuclei or even larger entities if, at least, we can describe such an entity by a single wave function in its own coordinate system and solve the equation to form a pair with another entity.

Consider the atom with one outer electron of mass m that is circulating at some distance from the nucleus of mass M and experiencing a net charge Ze. The Schrödinger equation which incorporates both electrostatic interaction, given by

from which it can be seen that the gravitational contribution is completely insignificant for any value of r. The general solution is:

where

In Part I, the quantized Dirac equation describes the interaction field in a pair ij which creates its own environment, or subspace, with coordinates

This problem can simply be overcome by adding to the Dirac equation, while preserving co-variance, the energy operator, rest masses and the kinetic term which reads:

where l runs over all indices for individual particles, and r_{l} is the coordinate system of the observer.

These extra terms to the Dirac equation extends the solution to:

The number of pairs in the first product is _{ij} and r_{ji}, gravitational interactions are occurring. Our observer only sees the separate interacting members of the pair with an energy due to this interaction as is shown schematically in

In the analysis in Part I going from Equations (I.12) to (I.14) terms are disappearing due to the solution proposed in Equation (1.11). But this has to be interpreted with caution. The pair probability density

The opposite situation in which the energy of the pair is positive, which in principle is allowed by the Dirac equation, is not possible when we assume that the energy of the vacuum, to be taken as the reference point, is zero. In this interpretation the interaction between mass and the surroundings is a means to transfer mass related energy (mc^{2}) to gravitational energy. This transfer changes the rest masses of the pair but does not create new mass. The consequences at a larger scale are worked out in paragraph 10 of this letter.

If, however the vacuum state is, as it is generally believed, a non-zero energy state there might be energy available which increases with the interaction area, the white area in

In Part I a solution for the Schrödinger equation of a pair of particles for an observer at distances

The Dirac equation in operator language now reads:

Setting the right hand side to zero, a mass-less particle, we see an equation for a travelling wave at the speed of light. To get rid of the singularity we set _{o} can be identified as the distance from the centre to where local influences have no impact.

We can solve this equation, but it is not necessary as it can immediately be seen that it dedicates mass to the field in the vicinity of the particle which is equal to_{0} = m_{i}:

The consequence is that either

It would be tempting to evaluate

Starting from

Estimates of the universe on the basis of the analysis in section 10 give a total mass of the universe of 4 × 10^{51} kg, There are 6 × 10^{26} protons in a kg; so we have 2 × 10^{78} protons. ^{4} J×m/kg^{4} and the proton mass is 1.7 × 10^{−27} kg. It leads to an estimate for the ^{−15} m, which is about the size of a proton (0.8 femtometers), [

Although the correspondence with measured data is surprisingly good, it is still a rough estimate and not without speculation. For instance, the sub-space due to the generator

Boys playing football next to a ditch are often unfortunate to find their ball in the water. Their usual course of action is to try throwing stones behind the ball from their point of view in the believe that the small water waves generated by the stones will, as they pass by the ball, push the ball a little bit towards them. A wise man passing by will tell them that it will not help as the only effect of the stones is moving the ball up and down.

However, it can be shown, after a lengthy calculation starting from the analysis of Mei [

It has been said before that the basic equation to form the pairs is valid at those locations where other influences like electromagnetic interactions are not important. In real life this means that gravitation interaction manifests itself where other forces are not the determining factor. Therefore in our real world we see that our direct vicinity has structures of forms that are changing over short distances like mountains, cities, sky scrapers, boats, forests etc. At larger distances, of the order of 100 kilometers the gravity becomes the dominant factor and bodies begin to take spherical shapes. Obviously the smaller the gravity is, as it is in smaller planets than earth, the structural variability will become larger. As has been said before, the electromagnetic interaction becoming insignificant in shaping the environment is not due to the form of the electrostatic interaction, which has basically the same shape as the gravitational interaction, but is due to the fact that positive and negative charges balance and compensate for their interaction starting already at short distances.

Consider a piece of matter m_{o}_{ }in a gravitational field generated by a larger mass M_{o}, and an observer far away out of this gravitational field, like on earth. This observer will interpret the real rest mass of m_{o }after he has taken it from a distance of R from the center of gravity to his free space. For the observer the rest mass is given by_{0}, generating the gravity field, as_{0} and m_{0} are encircling each other the observer will conclude that the force balance is given by:

It is important to note here that, for the observer outside the gravity field of the two masses, the experience of equality of the gravitational mass and inertial mass is not valid anymore.

Also the observer sees that the time lapse T in the M_{0}-m_{0} system is changed to

When the observer on the planet Mercury, knowing that his mass is much smaller than the mass of the sun, sees that he has made one complete revolution around the sun, so:

The model presented here started from the Schrödinger equation where no boundary conditions are imposed on a system of particles. The subsequent finding of the pair formation as a result of this has consequences. First it means that the entire space, in which the particle pairs are embedded, is necessary for the interaction. Secondly, it is only the pairs that create the forces between them such that in Newton’s law the product of the masses gives rise to a total gravitational force. Also, the particular form of the solution of the Schrödinger equation, in which the wave function amplitude itself is used as an operator in the Dirac equation, leads to the R^{2}-dependence. It is also important to remark that the gravitational interaction, leading to an interaction energy, is dependent on the product of the rest masses and independent of the speed in any direction in which one of the members of the pair or both are moving, even though the total mass of the pair changes relativistically.

We will calculate the energy balance of the universe assuming that, where matter is manifesting itself, the mass is distributed homogeneously. This assumption ignores any clustering of matter that will influence the energy, but it can be shown that this contribution is negligible against the energy that is dedicated to the relativistically defined masses (mc^{2}).

To start with, relativistic masses are not taken into consideration. If the density of the rest mass is given by ρ_{0} there are two contributions: potential energy, V, as the masses feel their gravitational pull to all other surrounding masses, and the kinetic energy, T, as the masses are moving relatively to each other [

where h is the so-called “Hubble constant”. It connects the expansion speed, v, of the matter with the distance, r, from the observer so that: v = hr.

We will take the sum as zero which means that, in the case of “flat, matter only” ultimately the universal expansion comes to rest, and we find:

This value of

It is interesting to evaluate this last equation, knowing that R/c is the radius of the universe in light seconds, that the average intergalactic density is about 1000 hydrogen atoms per cubic meter so that R/c = 0.3 × 10^{17} light seconds or about 10^{9} light years and that the generally accepted value of h is 2.3 × 10^{−18}. These values are in the right order of the values assumed on the basis of telescopic observations.

Now we want to play the same game as above, but in a relativistic context. For this we should take the rest mass M_{0} as the starting point in both calculations of the potential and the kinetic energy. First we will have to know what mass, M(R), is contained in a spherical volume with radius R:

In this equation the factor _{0}. This is significantly larger than the non-relativistic number.

Making use of the fact that the interaction energy is dependent on the separation distance of the members of the pair and independent of the speed at which the members are moving, the potential energy is as above in Equation (15):

The kinetic energy cannot be evaluated by “Mv^{2}/2” but more simply by:

Again, setting the sum equal to zero will give:

This clearly differs from the non-relativistic value since the relativistic kinetic energy is so much larger than in the non relativistic scenario and it appears that the relativistic universe seen by our observer is “larger” by a factor of 2.1 or more heavy.

There is more than one way of interpreting this result. First we can say that from our observer’s point of view the energy due to expansion speeds in the outer areas is so high that it will never balance the gravitational energy, whereas for the observer in the outer areas, however, it could. From another point of view our observer sees that time is progressing more and more slowly in the outer areas and even, if the expansion speed will slow down, he will never be able to see it.

There have been speculations [^{2}-term:

In that case, surprisingly, the universe should be significantly larger, or heavier, than on the basis of the analysis of the “non relativistic” universe. But this latter argument should be considered with scepticism. Rest mass m_{0} seen by an observer in free space changes from its value in a local gravity field by a factor of _{0} is interacting and R the distance, but mass is not newly created. The argument suggests that there is a mechanism by which gravitational energy can be changed into new rest mass.

The factor mentioned is a small correction that plays no significant role in the analysis here, but it has been shown in paragraph 9 that it is of significance in calculating the precession of the perihelion of the planet Mercury around the sun.

The model describing the gravitational interaction between particles has a some relation with the classical Bohr-Einstein dispute [

The controversy culminated in the well-known Einstein-Podolsky-Rosen Paradox, (EPR) of 1935 [

This is a superposition of two states of the ensemble. At some moment we do an experiment and find out that one of the particles is specified as “up”. From quantum theory we conclude that the other should be “down”. It might be that the system is influenced by the measurement so that the result “up” emerged, but the other particle is definitely not influenced and we know that it is characterized as “down”. It appears that the wave function has selected the option

We can now identify the solution for the pair potential in Equation (I.3) of Part I or, more generally, Equation (I.12), for a multi particle pair system, in a similar way as the “up/down” combination given above. As Max Born pointed out in a letter to Einstein: “There is a wholeness to a quantum events that persists over time and space and makes linkages possible”. These linkages, leading to the definition of the invariant in the Dirac equation, apparently, give rise to the gravitational interaction. So, nothing new proposed here, just a consequence of a classical debate between two scientists at the beginning of the previous century.

Apparently a single particle sees an environment and makes pairs with all of the particles around it. Suppose that at the other side of our galaxy two particles k, l annihilate. Suddenly the number of pairs reduces and this is seen by our particle. This change in the wave function:

generates a gravitational wave travelling through empty space at the speed of light and that adjusts to the new situation. But the information that the gravitational wave must start has already been exchanged between our particle with the observer and the annihilating pair. Again we end up in the same controversy as between Bohr and Einstein.

It is like a person somewhere far away sends me a message that something will be put into water so that I can go to the beach in The Netherlands and observe that there is some small rise of the sea level. If I will see anything, I have no idea where and when and in which way it originated, only that something has happened. Most likely the signal will be too small compared with the random disturbances. This analogy, however, is different from gravitational waves in that I have at least the sand of the beach as a reference level, whereas with gravitational waves such a thing does not exist.

The present letter follows the previous one to illustrate some of the particular aspects of the theory of gravity developed, and which is based on well known and proven quantum mechanical- and relativity related aspects. Also some of the assumptions made would have needed justification and further working out. The first justification is the basic quantum mechanical wave equation describing the behaviour of particles without their direct short range interactions, which are usually taken care of in normal quantum mechanical considerations. An important first conclusion was that particles with a mass can be described as a single non interaction pair containing only two members. The individual members can make pairs with all other mass in its surrounding. This, already peculiar pair effect, is used in the Dirac equation which, in a quantum-mechanical representation, describes a field around these members. The second conclusion made is that energy is subtracted from the pair and gives rise to an attractive force between the two members of a pair. By setting this force equal to the well-known parameters of Newton’s third law, numerical values can be given to the main parameters found with the Dirac equation. It is then found that an observer watching the pair will see that the pair has two members, but he cannot see how they interact or exchange information, only that it leads to a force between them. This force is independent of the movement of the members.

The field that occurs due to the Dirac equation is not only present outside the particle but must also has its influence in areas where the particle mass manifests itself. Not much is known about what this field looks like and its local interactions, but the most simple approach would be to assume that the amplitude of the generator of this field is constant. The dependence on space coordinates of this field inside the outer boundary of the particle leads to the attribution of mass. This mass is then found to be a consequence of all the interactions which the single particle has with the surrounding mass in which the distance, apparently, plays no role. If we start from the values of the parameters derived from the gravitational interaction, and the known mass of a proton, its outer boundaries can be calculated which agree surprisingly well with the data found experimentally. However, it must be stressed that this last reasoning is speculative.

Going from small to large the theory that is developed has its consequences. One is that mass is to be attributed to a rest mass in a gravity field which in normal cases is merely negligible, but it has consequences on, for instance, the movement of the perihelion of Mercury during its encircling of the sun. It is also shown how mass and gravity are intimately connected and that the description of the cosmos at large distances is governed by the specific gravitational interaction between bodies constituting the universe.

One point, difficult to accept from logical point of view, is that members of a pair seem to have instantaneous contact no matter how far they are apart and therewith generate the interaction field that gives rise to gravity. Gravity waves move at the speed of light or slightly less, depending on the mass density it is moving through, but its generator works apparently without delay. The situation is the same as the classical debate in the previous century between Einstein and Bohr and have remained to be an issue which is hard to believe but more than once shown to be true.

Another point to remark here is the occurrence of a generator creating a subspace. It follows unambiguously from the Schrödinger equation but nothing can be said about its internal structure where particles are entangled and apparently exchange information. This might be close by the idea of Einstein about “hidden variables”.

Further causality is of importance to keep in mind. The model starts from the fact that there are masses, and it is seen that they can form pairs and generate gravity. It yields numerical data about the masses following gravitational parameters. The strength of the model is the consistency of the data with what we observe in reality.

A new theory about gravity is developed which explains the particular aspects Newton’s third law. It is found that nowhere in the development of the theory there was any need to deviate from well established existing concepts in quantum-mechanics and relativity and rules of calculation. Some of the debates that originated from these insights could be given a place in the concepts developed and helped in the interpretation of the results. According to the theory developed gravity is a relativistic quantum effect that manifests itself at both small scale, the scale of our daily life and also on a cosmological scale. Taking the gravity constant as starting point, both microscopic and cosmological parameters can be derived that correspond well with the observations.

Veringa, H.J. (2016) Part II of New Basic Theory of Gravity. Journal of Modern Physics, 7, 2266-2280. http://dx.doi.org/10.4236/jmp.2016.716195