Does an extreme electric/magnetic field dilate time rates for a charged/magnetized clock? I want to know if general relativity (or any defined metric in general relativity) predicts any change for the clock rates in the electromagnetic fields (potentials) for some charged or magnetic clocks?
 A: The Reissner–Nordström (RN) metric described the spacetime outside a charged and non-rotating spherical source. It has a static EM field, and an EM field has a stress energy tensor, so the metric is not a vacuum metric and hence it differs from the Schwarzschild spacetime without violating Birkhoff’s theorem.
In the RN spacetime the metric is different from the standard Schwarzschild spacetime. In particular, to your question it has different gravitational time dilation. $$\gamma=\frac{dt}{d\tau}=\left( 1- \frac{R_S}{r}+\frac{R_Q^2}{r^2}\right)^{-1/2}$$ where $R_S$ is a parameter that parameterizes the curvature from the mass and $R_Q$ parameterizes the curvature from the charge.
As you see, this differs from the standard Schwarzschild gravitational time dilation. The resulting time dilation applies to all clocks of any construction.
A clock that is also charged would have a force acting on it. Depending on the design of the clock, that force may distort the structure of the clock or otherwise result in a reading that deviates from the above equation. If so, that would not be described as additional time dilation, but as a deviation of such clocks from ideal behavior. In other words, such a clock would no longer be recording proper time.
