The electromagnetic field of a charge supported in a uniform gravitational field is examined from the viewpoint of an observer falling freely in the gravitational field. It is argued that such a charge, which from the principle of equivalence is moving with a uniform acceleration with respect to the (inertial) observer, could not be undergoing radiation losses at a rate implied by Larmor's formula. It is explicitly shown that the total energy in electromagnetic fields, including both velocity and acceleration fields, of a uniformly accelerated charge, at any given instant of the inertial observer's time, is just equal to the self-energy of a non-accelerated charge moving with a velocity equal to the instantaneous ldquopresentrdquo velocity of the accelerated charge. At any given instant of time, and as seen with respect to the ldquopresentrdquo position of the uniformly accelerated charge, although during the acceleration phase there is a radially outward component of the Poynting vector, there is throughout a radially inward Poynting flux component during the deceleration phase, and a null Poynting vector at the instant of the turn around. From Poynting's theorem, defined for any region of space strictly in terms of fixed instants of time, it is shown that a uniformly accelerated charge does not emit electromagnetic radiation, in contrast to what is generally believed.