tret[x_,y_,t_]:=tprime/.FindRoot[tprime+Sqrt[(x-1/2Log[1+E^tprime])^2+y^2]-t,{tprime,-10}]

r[x_,y_,t_]:={x-1/2Log[1+E^tret[x,y,t]],y}

nhat[x_,y_,t_]:=r[x,y,t]/Norm[r[x,y,t]]

beta[x_,y_,t_]:={1/2(E^tret[x,y,t]/(1+E^tret[x,y,t])),0}

betadot[x_,y_,t_]:=(beta[x,y,t+0.01]-beta[x,y,t])/0.01

e[x_,y_,t_]:=(1-Norm[beta[x,y,t]]^2)*(nhat[x,y,t]-beta[x,y,t])/(Norm[r[x,y,t]]^2*(1-nhat[x,y,t].beta[x,y,t])^3)+((nhat[x,y,t]-beta[x,y,t])*(nhat[x,y,t].betadot[x,y,t])-(betadot[x,y,t])*(nhat[x,y,t].(nhat[x,y,t]-beta[x,y,t])))/( Norm[r[x,y,t]]*(1-nhat[x,y,t].beta[x,y,t])^3)

VectorPlot[e[x,y,41],{x,-50,50},{y,-50,50},VectorPoints->{25,25},VectorScale->{Tiny,Automatic,None}]