The causality of waves propagating in a double-negative (DNG) metamaterial (epsilon (r)<0 and mu(r)<0) has been investigated both analytically and numerically. By considering the one-dimensional electromagnetic problem of a pulsed current sheet radiating into a DNG medium, it is shown that causality is maintained in the presence of a negative index of refraction only if the DNG medium is dispersive. A Drude model DNG medium is used in this study. Spectrograms of the wave phenomena in the dispersive DNG medium show that the higher frequency components, which create the leading edge of the electromagnetic signals and see a double positive (DPS) medium (epsilon (r)>0 and mu(r)>0), arrive causally before the negative index effects germinate completely. Comparisons with approximate analytical results demonstrate the presence of the negative index of refraction properties in the continuous wave portion of the signals. This dynamic pulse reshaping between the positive and negative index of refraction wave components causes an apparent delay in the realization of the negative index of refraction properties. Pulse broadening of the signal tails is associated with both dispersion and a larger negative index of refraction seen by the associated wave components.