  As the title implies the article describes the possibility of taking into
account the relativistic correction to the field current density of the field
emission of electrons from the metal. The article provides the reader with some
analytic generalization of the Fowler-Nordheim equation with the relativistic
correction. The relativistic correction to the Fowler-Nordheim equation makes
it possible to take into account the influence of the relativism on the field
emission current. It is especially noted that the consideration of this
correction is necessary in the case of sufficiently strong electric fields and
relatively large interelectrode distances. It should be stressed that this
correction is valid for fixed interelectrode distances that decrease with
increasing electric field strength. It means that for the electric field
strength of 0.1 to 1 GV/m the interelectrode distance should not exceed values
of 1 to 0.1 cm. First in the article it is spoken in detail about finding of
the electron wave function. Next the field emission current calculations are
given. As a result the transmission coefficient of the potential step from the
Klein-Gordon equation within the framework of the Fowler-Nordheim approximation
is found. It is shown that in the case of the interelectrode distance less than
1 cm, an analytical expression for the field electron emission current density
is obtained. The conclusion that usually relativistic correction does not
exceed a tenth of a percent is made. But in the case of the field electron
emission from pulsars (where the work function and electric field strength are
much higher) the contribution of the relativistic correction about 10 % has
been established.
