Mass Calculus

Part 1

Spacetime Model


Mass Calculus

This webpage proposes a calculation in 4D of the mass effect m = f(x,y,z,t).

This webpage is part of the website
To understand the demonstration, it is necessary to read the whole theory.

Expression of "m"

In the Newton webpage we have seen that (fig. 1):

  • The displacement of spacetime VR is equal to that of the closed volume, V, that produces this displacement,
  • The curvature of spacetime, DVR, is calculated from the displacement of spacetime VR, and the coefficient of elasticity e v,
  • The radial curvature of spacetime (concerning a volume M with a spherical symmetry), DR, at the surface of M, is calculated dividing the volume by the surface: DR = DVR / S,
  • Finally, since DVR = ev VR, and V = VR, the linear curvature of spacetime becomes:
equ1 - mass calculus
fig1 - mass calculus

On the other hand, we have calculated the Newton Law starting with the following expression:

equ2 - mass calculus

...where K is an unknown constant having the dimensional quantity of [L/M].

Porting (2) in (1) gives:

equ3 - mass calculus


equ4 - mass calculus

Combining equation (20) of the Newton Law webpage and (4) gives the expression of the "mass effect". A parameter has been added, ρ, which is the density of external spacetime in the point of measurement, with a flat spacetime as reference. This parameter increases or decreases the mass effect.

equ5 - mass calculus


  • M = Mass effect (kg)
  • V = Volume of the closed volume (m3)
  • S = Surface of the closed volume (m2)
  • ev = Coefficient of elasticity of spacetime
  • c = Speed of the light (m/s)
  • G0 = Universal constant of gravitation
  • ρ = Density of external spacetime in the point of measurement, compared to a flat spacetime (coefficient without dimensional quantity)

The coefficient of elasticity of spacetime ev can be calculated from spherical particles. However, this calculation is not easy because the only supposed spherical particles known are leptons. To have accurate data, we must know exactly what electrons, muons, and taus are, and their accurate diameter. Nuclei with a null quadripolar moment (magic nuclei) could also be used but we do not know exactly the arrangement of nucleons inside the nucleus, nor the arrangement of quarks inside each nucleon. The space between elements may vary from one nucleon to another, and in this case, may give wrong results.

Since the mass effect is calculated from the closed volume, it is necessary to know the latter with accuracy before computing data.

As a direct consequence of the proposed theory, it could be possible that a relationship exists between the sphericity of particles and the accuracy of measurements. For example, leptons could be spherical since their mass is known with an excellent accuracy. This is not the case of quarks.

Case of a Sphere

In the particular case of a sphere, we have V = 4/3 pR3 and S = 4pR2, and expression (5) becomes:

equ6 - mass calculus


Nuclei aren't spherical. Therefore, it is not possible to apply equation (6) to calculate their mass effect since we do not know exactly their shape. Empirical formulae based on A, Z and N, as the Bethe-Weizsacker formula, give accurate results but it is important to note that the law in A1/3 does not mean that the mass is proportional to the volume. It means that the arrangement of nucleons inside the nucleus follows a A1/3 law. Results are equivalent but the signification is totally different.

To calculate the mass effect we must
know ev and the ratio volume/surface
of the closed volume that produces
the spacetime curvature.