# Charge invariance

Charge invariance refers to the fixed electrostatic potential of a particle, regardless of speed. For example, an electron has a specific rest charge. Accelerate that electron, and the charge remains the same (as opposed to the relativistic mass and energy increasing). The key word here is relativistic. Some particle characteristics are relativistically invariant (charge, spin, and magnetic moment). Others are relativistic (mass, energy, and de Broglie wavelength). The key is the Lorentz factor,

${\displaystyle \gamma ={\frac {1}{\sqrt {1-({\frac {v}{c}})^{2}}}}}$

where ${\displaystyle v}$ is speed, and ${\displaystyle c}$ is the speed of light. Typically, we write:

${\displaystyle E=\gamma E_{0}\,\!}$
${\displaystyle m=\gamma m_{0}\,\!}$
${\displaystyle e=e_{0}\,\!}$
${\displaystyle {\frac {\hbar }{2}}={\frac {\hbar _{0}}{2}}\,\!}$

where the first equation describes relativistic energy, the second – relativistic mass, the third – charge invariance, and fourth – spin invariance. Notice that ${\displaystyle \gamma }$ does not appear in the third or fourth equations. ${\displaystyle \gamma }$ is a function of speed (relative to some rest frame where ${\displaystyle E_{0}}$ and ${\displaystyle m_{0}}$ refer to rest energy and mass). So, to say "rest charge" or "rest spin" is redundant. Notice also that as ${\displaystyle v}$ (speed) increases, ${\displaystyle \gamma }$ increases, and so ${\displaystyle E}$ (and ${\displaystyle m}$) increases too. The origin of charge invariance (indeed, all relativistic invariants) is under speculation presently. There may be some hints proposed by string/M-theory. It is possible the concept of charge invariance may provide a key to unlocking the mystery of unification in physics (the theoretical unification of gravity, electromagnetism, the strong, and weak nuclear forces).

The property of charge invariance follows from the vanishing divergence of the charge-current four-vector ${\displaystyle j^{\mu }=(c\rho ,{\vec {j}})}$, with ${\displaystyle \partial _{\mu }j^{\mu }=0}$.