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Consequences of the relation of the gravitational and the inertial mass

The gravitational mass

We begin with the gravitational mass of a body consisting of NP protons (P), NE eltons (E), Ne electrons (e) and Np positrons (p). Since the stable elementary particles carry the conserved gravitational charges gi = { – g∙me, + g∙me, + g∙mP, – g∙mE}, i= e,p,P,E, whereby the universal gravitation constant is G = g^2/4∙π and mP/me = 1836.1528, the gravitational mass of a body is

mg(NP,NE,Np,Ne) = |(Np – NE)∙mP + (Np – Ne)∙me|.

The gravitational mass is always greater or equal zero. Since the gravitational interaction is a product of two elementary charges, the different signs of gravitational charges can be put in the interaction part. We have chosen conventionally the gravitational charge of proton positive, gP = + g∙mP.

Now, we consider only electric neutral particle systems for which all the particles are bound. It is easy to see that an electron-positron pair, or a proton-elton pair, has the gravitational mass zero

mg(e,p) = me – me = 0; mg(P,E) = mP – mP = 0.

These two-particle systems correspond to the positronium and protonium and to the electron-neutrino and proton-neutrino. The proton-electron system, N0 = (P,e), has the same gravitational mass as the elton-positron system, N0 = (E,p),

mg(N0) = mg(P,e) = mg(N0) = mg(E,p) = mP – me,

and the same gravitational mass as the instable neutron, N = (P,e,p,e) with the decay N → P + e + electron-neutrino,

mg(N) = mg(P,e,p,e) = mg(P,e) = mP – me.

In absence of eltons, an electric neutral isotope, with NP = A protons, has the gravitational mass

mg(A) = A∙(mP – me).

The numbers of positrons don’t appear in the gravitational mass of an electric neutral isotope.

The inertial mass (at rest of center of mass)

The inertial mass of the stable neutron, N0 = (P,e), is to be calculate with the bound energy, E(N0, bound) = 2.04 MeV, to

mi(N0) = mP + me – E(N0, bound)/c^2 = mP + me – 2.04 MeV/c^2.

The inertial mass of the instable neutron, N = (P,e,p,e), is in nuclear physics measured,

mi(N) = mP + 3∙me – E(N, bound)/c^2 = 939.565 4133(58) MeV/c^2.

The bound energy, E(N, bound)/c^2, can be calculated with the proton mass, NP = 938.272 0813(58) MeV/c^2 and the electron mass Ne = 0.510998910(13) MeV/c^2 to

E(N, bound)/c^2 = 0.2396647 MeV/c^2.

For the two-particle systems, (e,p) and (P,E), the ground states are defined as states if their inertial masses are also zero: these are the electron-neutrino, νe, and the proton-neutrinos, νP, thus, if mi(νe) = 0 → 2∙me = E(νe,bound)/c^2 and if mi(νP) = 0 → 2∙mP = E(νP,bound)/c^2. From the neutrinos originate the Lagrange multiplier, h0, with the value (e is the elementary electric charge)

h0 = e^2/2∙c∙(2∙m’(neutrino)∙c^2/2∙E(neutrino,bound))^1/2 = e2/2∙c∙(1/8)^1/2= h/387.

Hereby the reduced masses are m’(νe) = me/2 and m’(νP) = mP/2, and h is the Planck constant. With h0 we can then calculate the bound energy of the stable neutron, N0. The Planck constant corresponds to the bound energy E(hydrogen,bound) = 13.8 eV.

The Lorentz-invariant formulation of the equations of motions

Lagrange multipliers, such as h and h0, appear in the Lorentz-invariant equations of particles motions as a consequence of the subsidiary conditions caused by the conserved numbers of the elementary particles, e, p, P and E. The Planck constant play the role of a Lagrange multiplier. The Lagrange multipliers occur in the equations of particles motions derived from an action integral formulated in finite ranges of the Minkowski space and taking the general uncertainty in to account that neither the positions, nor the velocities of particles are ecer exactly known. The action integral is constructed with a Lorentz-invariant Lagrange density, which is however not an expression for energy density, and from which the invariant equations of motions for the fields are also derived http://www.atomsz.com.

Further consequences of the conserved gravitational charges

We should mention, that the proton-electron system could also have inertial mass zero; that means

mi(P,e) = 0 = mP + me – E(bound)/c^2

with

E(bound) = (mP + me)∙c^2.

This state has the lowest bound energy of the proton-electron system. This state is the stable ground state of the (P,e)-system and this is not the ground state of the hydrogen atom.

For two-particle systems, the velocities of particles can be calculated according the formula

(v/c)^2/(1-(v/c)^2) = 2∙E(bound)/mij’c^2,

with the reduced masses mij’ = mi∙mj/(mi + mj). The radii of bound two-particle system are with the Lagrange multipliers, h and h0,

rij = h^2/(4∙π^2∙mij∙e^22),

r0ij = (h0)^2/(4∙π^2∙mij∙e^2).

In two-particle system the particles cannot approach each other under their mutual interactions below the relative distance of 10^-17cm. That means, the electron-positron pairs and the proton-elton pairs do never annihilate.

For all electric neutral isotopes with the mass number, A, and with the nuclear charge, Z, the inertial masses, mi(A,Z), are measured by mass spectrometers, thus in the electromagnetic field. From the formula

mi(A,Z) = A∙(mP + me) +Np∙2∙me – E(A,Z; bound)/c^2,

we could calculate the bound energy of the isotopes, E(A,Z; bound), if we would know the numbers of positrons, Np, in the nuclei. In order to solve this problem, we have a variation principle for the bound energies for many body problems, with A = NP protons, Np positrons and NP + Np electrons, and applying h0 and h. This is a well defined model calculation in order to get the Np numbers from the calculated bound energy, E(A,Z; bound) at known mi(A,Z). The sizes of the nuclei are somewhat greater than the sizes of the electron-neutrino and of the stable neutron, which are r(νe) = 0.703∙10^-13cm and d =2∙r(N0) = 0.702∙10^-13cm.

Since we know experimentally the inertial masse of the isotopes, mi(A,Z), and also the gravitational masses, mg(A,Z) = A∙(mP – me), we can calculate the relation of these masses

mg(A,Z)/mi(A,Z) = 1 + Delta(A,Z).

The mass defect, Delta(A,Z), are varied for different types of isotopes

– 0.109% (hydrogen) < Delta(A,Z) < 0.784% (56Fe isotope).

The static force between two stable elementary particles is always the sum of two interactions

F(rij) = + qi∙qj∙rij/4∙π∙rij^3 – gi∙gj∙rij/4∙π∙rij^3,

whereby the conserved elementary electric charges are qi = {± e}. The modified Newton’s equation of motion for a body composed of different (electric neutral) isotopes is, if the other electric uncharged body, BODY, has the same sign of gravitational charge

mi(body)∙a(body) = – G∙mg(BODY)∙mg(body)∙r/r^3.

The acceleration of the body, a(body), is composition dependent

a(body) = – a0∙ mg(body)/ mi(body) = – a0∙(1 + Delta(body).

Therefore, the Universality of Free Fall (UFF) is violated and it can be measured by fall experiments with different composed test bodies. An experimental verification of the UFF violation is performed by Gy. I. Szász at the drop tower of the University Bremen on the 21.06.2004, and is reported in his book, Physics of Elementary Processes; Basic Approach in Physics and Astronomy, ISBN: 963 219 791 7 (2005) and in his lecture on YouTube https://www.youtube.com/watch?v=WsyJjxC7SRc . However, the editors/reviewers of physical journals, PRD, EJPC, ZNA and Foundation of Physics, rejected the articles that attempted to publish Szász’ results. Therefore, his theory did not came in circulation of physical science.

A comparison with the special relativity theory

All inertial masses discussed up to now are calculated if the center of masses (COM) is at rest. These masses are the so called rest masses, m0(body), which should appear in Einstein’s equation of the special relativity theory

E^2 = (m0(body))^2∙c^2 + p^2∙c^2.

However in particular, the rest inertial masses of composed systems are not invariant masses of bodies. Only the elementary particles e, p, P and E have invariant masses. The equation above is not verified in our theory. Since the bound energies of composed systems are caused finally by interactions between the elementary particles which propagate with c, at velocities of the COM nearby c, the composed systems would decay in the composing elementary particles.

In our theory, we can also not verify Einstein’s energy-mass-equivalence relation

E = mi(v)∙c^2.

The elementary masses mP and me are the invariant masses and they are not equivalent to energies. However, we get the equation for the inertial masses at rest of COM

mi(NP,NE,Np,Ne) = (NP +NE) ∙mP +(Np+Ne)∙ me – E(NP,NE,Np,Ne; bound)/c^2,

which are not invariants. On the other side, the gravitational masses are conserved

mg(NP,NE,Np,Ne) = |(Np – NE)∙mP + (Np – Ne)∙me|.

Conclusions

Here, it is demonstrated that the conclusions of the special relativity for masses and energy are erroneous.

Also the conclusions derived from the general relativity for gravity are erroneous then the gravity is obviously caused by conserved gravitational charges, gi = { – g∙me, + g∙me, + g∙mP, – g∙mE}, of the stable elementary particles i= e,p,P,E. A proof for the existence of elementary gravitational charges is given for instance by the confirmed violation of the UFF. But, these did not fit to the convinced meaning of editors/reviewers of physical journals about their imagination to the validity of the weak equivalence principle.

The physical science needs neither the deformation of space-time in order to explain the gravitation, nor the quantization of energy for the construction of a quantum field theory. After that all, the UFF is violated and the Planck constant h plays the role of a Lagrange multiplier.

The quantum field theory based on two conserved elementary charges of four kinds of stable elementary particles does not need Higgs-particle in order to explain the gravitational and the inertial masses of all particles.

This theory state also that the stable elementary particles are not composed of quarks. In the diploma thesis, SU(3) Symmetry in der starken Wechselwirkung; Ein Vergleich mit den Experimenten, at the University of Mainz, Gy. I. Szász has 1967 shown, that the prognoses of the SU(3) symmetry model of particles are scientifically not acceptable.

In his doctor thesis, Zur quantenmechanischen Beschreibung von Resonanzphänomenen, Szász proposed a new variation principle for the determination of instable states. The work was published in Z. Physik, A275, 403 (1975) and A278, 165 (1976), Fortschr. Physik, 24, 327 (1976), Phys Lett. A55, 327 (1976) and A62, 313 (1977). But, a further article with the significant recognition that the Planck constant play a role of Lagrange multiplier, submitted 1977 in Phys.Lett., has been rejected by the editors.

Gyula I. Szász