What Happened to Potential Energy in E=m[tex]c^2[tex]

[bhthiang]

I have been wondering why is potential energy not included in the calculation of total energy E=m[tex]c^2[tex]?
I know that m[tex]c^2[tex] includes the rest energy and the kinetic energy. Suppose the particle or object is charged and is moving in a strong electric field, its potential energy will be sdgnificant wouldn't it?
If we have to include the PE, how should we do it?
Thanks

 

[quantumdude]

The energy-momentum relation that you see in a standard general physics textbook is:

$$E^2=(pc)^2+(mc^2)^2$$

It is typically advertised as the energy-momentum relation for a *free* particle (so $V=0$). If memory serves, Goldstein's "Classical Mechanics" treats relativistic mechanics with a potential, via the Lagrangian formalism.

 

[krab]

Properly,
$$(E-V)^2=(pc)^2+(mc^2)^2$$
where V is the potential energy and m, the rest mass, does NOT contain the kinetic energy.

 

[krab]

Properly,
$$(E-V)^2=(pc)^2+m^2c^4$$
where m is the rest mass and V the potential energy