Will a charged particle stationary in its reference frame appear to radiate to an accelerating observer? Suppose a charged particle, A is stationary in its frame of reference, and an observer, B passes parallel to A while accelerating in a straight line. Will the particle appear to B to emit radiation? 
 A: It depends on whether you consider general relativity or not.
The four-acceleration $\mathbf A$ of a body has two contributions:
$$ A^\alpha = \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} + \Gamma^\alpha_{\,\,\mu\nu}U^\mu U^\nu $$
The first term on the right hand side is the bit we instinctively think of as acceleration, i.e. the fact we can feel a g-force, while the second term is due to the curvature of spacetime. Generally speaking in electrodynamics we are working in flat spacetime and (assuming Cartesian coordinates) the Christoffel symbols $\Gamma^\alpha_{\,\,\mu\nu}$ are all zero and the four-acceleration takes the familiar form:
$$ A^\alpha = \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} $$
in this case it is always unambiguous which observer is accelerating and which is not, and a charge only radiates if its acceleration (i.e. the norm of the four-acceleration) is non-zero. So in your case of an accelerating observer and a non-accelerating charge no radiation is emitted by the charge and no radiation is detected by the accelerating observer.
The tricky bit is when we include GR, for example if the accelerating observer is falling freely towards the Earth while the charge is stationary on the Earth's surface. In this case the four-acceleration of the observer is zero while the four-acceleration of the charge is non-zero. I have to confess I am unsure what happens in this situation or even if the situation is fully understood.
