Is the Mass-to-Charge Ratio Equivalent to the Speed of Light?

During my time at Western Washington University, two laboratory experiments left a particularly strong impression on me: one that measured the speed of light directly, and another that measured the electron charge-to-mass ratio (\(e/m_e\)). In particular, J. J. Thomson's cathode-ray measurement (1897) can be described using the electric field \(E\), the magnetic field \(B\), and the curvature radius \(r\) of the electron-beam trajectory:

\[ \frac{e}{m_e}=\frac{E}{B^2 r} \]

Today, precision techniques such as the Penning trap determine \(e/m_e\) with about ten decimal digits of precision. A commonly quoted value is

\[ \frac{e}{m_e}=1.75882001076(89)\times 10^{11}\ \mathrm{C/kg}. \]

Below, we assume typical laboratory-scale values to check the order of magnitude. For example, if \(E=2\times10^4\ \mathrm{N/C}\) and \(B=1\ \mathrm{mT}\), then

\[ r=\frac{E}{B^2(e/m_e)} =\frac{2\times10^4}{(1\ \mathrm{mT})^2\cdot 1.75882001\times10^{11}} =11.371260205\ \mathrm{cm}. \]

Thus the curvature radius \(r\) is a realistic value of about 11.4 cm.

Reinterpretation in the Space-Time (ST) Unit System

We revisit the gravitational law in Newtonian mechanics and Coulomb's law, and reconsider the roles of the gravitational constant \(G\), vacuum permittivity \(\varepsilon_0\), and vacuum permeability \(\mu_0\). We propose that the kilogram and the coulomb are conventional (human-defined) units and can be represented, in essence, as derived units built from the basic units of length (m) and time (s). We refer to this redefinition framework as the Space-Time (ST) unit system (see Reference).

In the ST unit system, the kilogram and coulomb are converted by transformation coefficients as follows:

\[ \delta\cdot \mathrm{kg}=2.780252259\times10^{23}\ \mathrm{m^3/s^2} \] \[ \varepsilon\cdot \mathrm{C}=5.272809743\times10^{12}\ \mathrm{m^2/s}. \]

Since both the electric field \(E\) and the magnetic field \(B\) contain \(\mathrm{kg}\) in the numerator and \(\mathrm{C}\) in the denominator in their dimensions, the ST conversion is obtained by applying the ratio \(\delta/\varepsilon\). For the same \(E\) and \(B\) used above, we define

\[ E_0=E\cdot\frac{\delta}{\varepsilon},\qquad B_0=B\cdot\frac{\delta}{\varepsilon}. \]

Numerically, this gives

\[ E_0= 2\times10^4\cdot \frac{2.780252259\times10^{23}}{5.272809743\times10^{12}} =1.054561949\times10^{24}\ \mathrm{m^2/s^3} \] \[ B_0= 1\ \mathrm{mT}\cdot \frac{2.780252259\times10^{23}}{5.272809743\times10^{12}} =5.272809743\times10^{16}\ \mathrm{m/s^2}. \]

Here, \(E_0\) and \(B_0\) are the ST-unit representations of the electric and magnetic fields. Next, using Thomson's relation, we evaluate the inverse charge-to-mass ratio in the ST unit system:

\[ \frac{m_0}{e_0}=\frac{B_0^2 r}{E_0}. \]

Substituting the values above, we obtain

\[ \frac{m_0}{e_0} = \frac{(5.272809743\times10^{16})^2\cdot 11.371260205\ \mathrm{cm}} {1.054561949\times10^{24}} = 2.9979246\times10^{8}\ \mathrm{m/s}. \]

This value matches the speed of light \(c\) up to the 7th decimal place. In the ST unit system, we therefore have

\[ c=\frac{m_0}{e_0}, \]

and the numerical agreement above can be understood as a direct consequence of this relation.

Penning Traps and Frequency-Based Determination

In Penning-trap measurements, \(e/m_e\) is determined through magnetic-field calibration and frequency measurement. In the ST unit system, since \(c\) is an invariant constant, the angular frequency \(\omega_f\) is determined solely by the ST magnetic field \(B_0\):

\[ \omega_f=\frac{B_0}{c}. \]

From the ST-unit viewpoint, both Thomson's cathode-ray experiment and Penning-trap precision measurements can be interpreted as measuring quantities directly tied to the speed of light, but from different angles.

For nearly five centuries since the time of Copernicus, mass has been accepted as a fundamental unit. However, when physical formulas are reexamined at a fundamental level, it becomes clear that the kilogram and the coulomb are not intrinsic features of nature, but units defined by human convention. Furthermore, the proportional constants associated with them-such as the gravitational constant, the vacuum permittivity, and the vacuum permeability-can be understood as non-essential constants introduced to preserve the internal consistency of a unit system based on the kilogram and the coulomb. Once the true nature of the kilogram and the coulomb is redefined, physical formulas begin to reveal a different structure, and new aspects of physical law emerge before us.