The time dependence of the $h$-index is analyzed by considering the average behaviour of $h$ as a function of the academic age $A_A$ for about 1400 Italian physicists, with career lengths spanning from 3 to 46 years. The individual $h$-index is strongly correlated with the square root of the total citations $N_C$: $h approx0.53 N_C$. For academic ages ranging from 12 to 24 years, the distribution of the time scaled index $h/A_A$ is approximately time-independent and it is well described by the Gompertz function. The time scaled index $h/A_A$ has an average approximately equal to 3.8 and a standard deviation approximately equal to 1.6. Finally, the time scaled index $h/A_A$ appears to be strongly correlated with the contemporary $h$-index $h_c$.