  This work establishes the time-dependent relativistic Hartree-Fock (TD-RHF)
model with spherical symmetry for the first time. The time-dependent
integro-differential Dirac equations are solved by expanding Dirac spinors on
the spherical Dirac Woods-Saxon (DWS) basis. The numerical verification
demonstrates the high conservation qualities for both the total binding energy
and the particle number, as well as the time-reversal invariance of the system,
which ensures the precision and reliability of the newly developed TD-RHF
model. Subsequently, the isoscalar giant monopole resonance (ISGMR) mode of
$^{208}$Pb is investigated using the RHF Lagrangian PKO1. The constrained
energy of the ISGMR calculated by PKO1 is found to be in close agreement with
the experimental data, and the strength function is similar to the results
given by the relativistic Hartree-Fock plus random phase approximation. Based
on the advantage of the TD-RHF model in avoiding complicated calculations of
the residual interactions, the ISGMR mode of $^{208}$Pb is calculated by twelve
relativistic effective Lagrangians. The results indicate that the value of the
incompressibility of nuclear matter $K_\infty$ constrained by relativistic
effective Lagrangians is in the range of $237\sim246$ MeV, which is lower than
the previous investigations based on the relativistic models.
