Velocity Addition using Metric Techniques

Hi all. I'm taking a course in GR and trying to get my intuition and mathematical techniques up to speed. I've been trying to derive the velocity addition formula in Minkowski space, but for some reason I can't do it. Here's what I have:



I'll use the Minkowski metric of signature (+1,-1,-1,-1). Then we have the invariant relation



[itex]



c^2d\tau^2=c^2dt^2-dz^2



[/itex]



Where [itex]\tau[/itex] is the proper time of an observer in the metric and [itex]t[/itex] is the coordinate time of some omniscient being looking at the Minkowski space from afar. For observer [itex]i[/itex] we have the relation



[itex]



c^2d\tau_i^2=c^2dt^2-dz_i^2



[/itex]



then dividing by [itex]c^2dt^2[/itex] we get



[itex]



(\frac{d\tau_i}{dt})^2=1-\frac{1}{c^2}(\frac{dz_i}{dt})^2=1-\frac{v_i^2}{c^2}=\frac{1}{\gamma_i^2}



[/itex]



where [itex]v_i=dz_i / dt[/itex] is the velocity of observer [itex]i[/itex] according to the omniscient being. If instead I had used the chain rule on the [itex]dz_i / dt[/itex] factor, I would get



[itex]



\frac{1}{\gamma_i^2}=1-\frac{1}{c^2}(\frac{dz_i}{d\tau_j}\frac{d\tau_j}{dt})^2=1-\frac{v_{rel}^2}{c^2}\frac{1}{\gamma_j^2}



[/itex]



where I defined the relative velocity as [itex]v_{rel}=dz_i / d\tau_j[/itex]. Rearranging for the relative velocity I find



[itex]



v_{rel}=v_j\gamma_i



[/itex]



which is certainly not the velocity addition formula. Where did I go wrong? And how should I change my thinking so it doesn't go wrong again?