We investigated recording dynamics of a holographic grating in the photorefractive LiNbO$_3$ crystal under the low-intensity condition of recording beams. New expressions for the space-charge field and the recording time constant were obtained by solving the Kukhtarev equations under the global space -charge field, which is induced in the previous process of recording and erasing. Their validity can be confirmed by considering the limit that the period of the grating goes to infinity both theoretically and experimentally. It was found that the new expression for the recording time constant allows us to determine acceptor concentration to be $1.2${ imes}$10^{21}m^{-3}$</TEX> for pure LiNbO$_3$ crystal and 2.5${ imes}$$10^{21}m^{-3}$ for the 0.1 mol% iron doped LiNbO$_3$ crystal from the measured ratio of the recording time constant under the extremely large grating condition, in which the diffusion effect can be neglected, to that under the small grating condition.