  The mechanism by which the radiation received from obliquely rotating neutron
stars is generated remains an open question half a century after the discovery
of pulsars. In contrast, considerable progress has recently been made in
determining the structure of the magnetosphere that surrounds these objects:
numerical computations based on the force-free, magnetohydrodynamic and
particle-in-cell formalisms have now established that the magnetosphere of an
oblique rotator entails a current sheet outside its light cylinder whose
rotating distribution pattern moves with linear speeds exceeding the speed of
light in vacuum. Here we insert the description of the current sheet provided
by the numerical simulations in the classical expression for the retarded
potential and thereby calculate the radiation field generated by this source in
the time domain. We find a radiation consisting of highly focused pulses whose
(i) spectrum can extend from radio waves to gamma rays, (ii) brightness
temperature can exceed 10^(40) K, (iii) linear polarization can be 100%, (iv)
two concurrent polarization position angles are approximately orthogonal often
and swing through 180 deg across the pulse profile in most cases, (v) circular
polarization reverses sense across some components of the pulse profile, (vi)
microstructure is determined by the thickness of the current sheet, and (vii)
whose flux density diminishes with the distance D from the star as D^(-3/2)
(rather than D^(-2)) in certain directions. The intrinsically transient
radiation process analysed here is thus capable of generating an emission whose
features are strikingly similar to those of the emissions received from pulsars
and magnetars and from the sources of fast radio bursts and gamma-ray bursts.
