Physical transverse velocity of a tilted string ($v_{\perp }=|k|/\sqrt{1+k^2}$)

For $X=(t,\sigma ,k(\sigma +t))$, the vertical coordinate velocity is $k$, but its normal projection has magnitude $|k|/\sqrt{1+k^2}<1$. Tangential motion merely changes string labels. The induced worldsheet metric has determinant $-1$ even when constant-label curves become spacelike for $|k|>1$.