x0=1
dx[x_,ct_]=Sqrt[x^2-ct^2]-x0
ctau[x_,ct_]=x0 ArcTanh[ct/x]
x[dx_,ctau_]=(x0+dx) Cosh[ctau/x0]
ct[dx_,ctau_]=(x0+dx) Sinh[ctau/x0]

xs1=x0
v1=3 10^-6
theta1=Pi/4
ParametricPlot[{v1 Cos[theta1] t,dx[xs1+v1 Sin[theta1] t,t]},{t,0,5 10^-6},AspectRatio->Automatic]
ParametricPlot[{ctau[xs1+v1 Sin[theta1] t,t],dx[xs1+v1 Sin[theta1] t,t]},{t,0,5 10^-6}]

v1=.5
theta1=Pi/4
ParametricPlot[{v1 Cos[theta1] t,dx[xs1+v1 Sin[theta1] t,t]},{t,0,1.45},AspectRatio->Automatic]
ParametricPlot[{ctau[xs1+v1 Sin[theta1] t,t],dx[xs1+v1 Sin[theta1] t,t]},{t,0,1.6}]

see light bend:
theta1=.3
v1=1
ParametricPlot[{v1 Cos[theta1] t,dx[xs1+v1 Sin[theta1] t,t]},{t,0,.6},AspectRatio->Automatic]
ParametricPlot[{ctau[xs1+v1 Sin[theta1] t,t],dx[xs1+v1 Sin[theta1] t,t]},{t,0,1.419 }]

vdx[xs_,v_,theta_,t_]=D[dx[xs+v Sin[theta] t,t],t]/D[ctau[xs+v Sin[theta] t,t],t]
vdy[xs_,v_,theta_,t_]=v Cos[theta]/D[ctau[xs+v Sin[theta] t,t],t]

speed of light:
Simplify[vdy[xs,1,theta,t]^2+vdx[xs,1,theta,t]^2]
show: =(xs+Sin[theta] t)^2-t^2=(x0+dx)^2

xs1=1
v1=.5
theta1=Pi/4
solution= NDSolve[{
x''[t]== -(1+x[t]/x0)/x0 (1-2 x'[t]^2/(1+x[t]/x0)^2),
y''[t]== 2 x'[t] y'[t]/(x0 +x[t]),
  x[0]== 0,
  y[0]== 0,
 x'[0]== N[vdx[xs1,v1,theta1,0]],
 y'[0]== N[vdy[xs1,v1,theta1,0]]},{x,y},{t,0,1.3}]

ParametricPlot[Evaluate[{y[t],x[t]} /. solution],{t,0,1.3},AspectRatio->Automatic]
ParametricPlot[Evaluate[{t,x[t]} /. solution],{t,0,1.3}]
