\label{massive} Also the mass of the particles must be expressed by a property of the spacetime. We recall the general relativity where the space-time lattice (used by an observer at a given reference system) is curved by the mass of the objects. The mass thus becomes a physical characteristic (gravitational charge \(\Gamma\)) acting on the ST, through the gravitational field.
The equivalence principle \cite{einstein1908} leaded to the equivalence between the gravitational charge \(\Gamma\) and the gravitational mass \(m_g\) (in this case intended as resinstance to the gravitational force). We must recall that a lot of experiments \cite{agner_Gundlach_Adelberger_2008,ravari_Maccarone_Lucchesi_2009,Comandi_Saravanan_et_al__2012} shown the equivalence between the inertial mass, \(m_i\), and the gravitational mass, \(m_g\), so \(m_i = m_g = \Gamma\) and we can conclude that the mass (inertia) \(m\) of an object became a characteristics acting to the ST. So, taking into account of general relativity and of the connatural structure of the massive particles and gravitational field, we can derive that the mass of a particle must be expressed from a fundamental property of the field \(\Xi\). Now we conjecture that the mass is given by a particular elastic coupling (a massive coupling) between the oscillators of \(\Xi\).
We observe that if we define the mass of a particle with the massive coupling between IQuOs, there’s no dinstiction between inertial and gravitational mass \cite{einstein1908}.
In order to understand better this conjecture we use the relativistic invariant given by the proper time \(\tau\). In the Minkowsky spacetime we read the forth component of the speed, \(u_4 = ic\), with \(c\) speed of light, like an index for the movement in time.
So we could interpret the series of events in any system like the index of the passage of time in it. This characteristic is associated to the massive object, because when it takes existence, time is starting to move. It must then exist a characteristic of the object that is connected with this type of motion, rather it is its generator.
The uniform motion in time recalls the clock, so it must exist into every particle-object a periodical motion that produces the motion in time. So we call about an internal clock with \(\omega_0\) the proper frequency.
The new conception of motion in time, with proper time, is useful to understand the notion of rest energy \(E_0\): we refer to it like the energy of motion in time.
If the speed in time is \(c\), we can suppose that \(E_0 \propto mc^2\) (remembering the classical expression of kinetic energy); it follows than: \[\left\{\begin{matrix}
m & \propto & \omega_0^2\\
E_0 & \propto & m c^2
\end{matrix}\right.
\Rightarrow E_0 \propto \omega_0 \nonumber\] Following quantum mechanics: \[\label{eq:mass}
\left\{\begin{matrix}
E_0 & = & \hbar \omega_0\\
E_0 & = & m c^2
\end{matrix}\right.
\Rightarrow m = \frac{\hbar \omega_0}{c^2}\] Following this account, we conjecture that the mass is the expression of the proper frequency related to a particular elastic coupling, additional to the one existing between the oscillators of the field \(\Xi\).
If the frequency \(\omega_0\) generates the proper time \(\tau\) of the massive particle, for symmetry it must be exist a wavelength \(\lambda\) that generates the proper space of the particle.
Following De Broglie, we have: \[\label{eq:wave}
\left\{\begin{matrix}
p_0 & = & \frac{2\pi\hbar}{\lambda_0}\\
p_0 & = & m c
\end{matrix}\right.
\Rightarrow \lambda_0 = \frac{\hbar}{m c} \equiv \lambda_c\] We state that \(\lambda_0\), the Compton wavelength, defines the spatial rate of the proper spacetime lattice of the particle. So, only when the massive coupling is built between the oscillators of the field \(\Xi\), we can speak about proper time and proper space.
In this way we can assert that a massive object generates a spacetime lattice equivalent to a frame reference.
It is simple to deduce that lattices of particles of relative motion differ for a Lorentz transformation. Now, combining \eqref{eq:mass} and \eqref{eq:wave}, we have: \[E^2 = m^2 c^4 + p^2 c^2 \leftrightarrow \omega^2 = \omega_0^2 + k^2 c^2\] We observe that the second equation describe a set of pendulums coupled through springs. We can suppose that the massive coupling is a transverse coupling between two or more chains of IQuO of \(\Xi\). So the massive field could be pictured as in the figure \ref{fig_massive_lattice}.