  The relativistic local density approximation (LDA) for the Coulomb exchange
functional in nuclear systems is presented. This approximation is composed of
the well-known Slater approximation in the non-relativistic scheme and the
corrections due to the relativistic effects. Its validity in finite nuclei is
examined by comparing with the exact treatment of the Coulomb exchange term in
the relativistic Hartree-Fock-Bogoliubov theory. The relativistic effects are
found to be important and the exact Coulomb exchange energies can be reproduced
by the relativistic LDA within 5% demonstrated by the semi-magic Ca, Ni, Zr,
Sn, and Pb isotopes from proton drip line to neutron drip line.
