Das Prinzip des Seins

(* Schwarzschild Bewegungsgleichung || yukterez.net *)

G = 1; M = 1; c = 1; rs = 2 G M/c^2;
r0 = 15/10 rs; v0 = 999/1000c c; φ = 0; θ0 = 0; z = 1; tau = 30;
vr0 = v0 Sin[φ]/j*k; vθ0 = v0/r0 Cos[φ]/j;
d1 = z/5; d2 = d1; wp = 24;

j = Sqrt[1 - v0^2/c^2];
k = Sqrt[1 - rs/r0];

Needs["DifferentialEquations`NDSolveProblems`"];
Needs["DifferentialEquations`NDSolveUtilities`"];

sol = NDSolve[{
    r''[t] == -((G M)/r[t]^2) + r[t] θ'[t]^2 - (3 G M)/c^2 θ'[t]^2,
    r'[0] == vr0,
    r[0] == r0,
    θ''[t] == -((2 r'[t] θ'[t])/r[t]),
    θ'[0] == vθ0,
    θ[0] == θ0,
    τ'[t] == Sqrt[c^2 r[t] + r[t] r'[t]^2 - c^2 rs + r[t]^3 θ'[t]^2 - r[t]^2 rs θ'[t]^2]/(c Sqrt[r[t] - rs] Sqrt[1 - rs/r[t]]),
    τ[0] == 0
    }, {r, θ, τ}, {t, 0, z},
   MaxSteps -> Infinity, Method -> {"StiffnessSwitching",
   Method -> {"ExplicitRungeKutta", Automatic}},
   WorkingPrecision -> wp,
   InterpolationOrder -> All];

t[ξ_] :=
  Quiet[χ /.
    FindRoot[
     Evaluate[τ[χ] /. sol][[1]] - ξ, {χ, 0},
     WorkingPrecision -> wp, Method -> Automatic]];
Τ := Quiet[t[ι]];

x[t_] := (Sin[Evaluate[θ[t] /. sol]] Evaluate[r[t] /. sol])[[1]]
y[t_] := (Cos[Evaluate[θ[t] /. sol]] Evaluate[r[t] /. sol])[[1]]

s[text_] := Style[text, FontSize -> font];  font = 11;

(* proper time *) Do[Print[
  Rasterize[Grid[{{Show[Graphics[{
      {Black, Circle[{0, 0}, rs]},
      {Lighter[Gray], Dashed, Circle[{0, 0}, r0]}},
        Frame -> True, ImageSize -> 400, PlotRange -> 2 r0],
       Graphics[{PointSize[0.01], Red, Point[{x[т], y[т]}]}],
       ParametricPlot[{x[η], y[η]}, {η, 0, т},
        PlotStyle->{LightGray}],
       ParametricPlot[{x[η], y[η]}, {η, 0, т},
        ColorFunction -> Function[{x, y, η},
          Hue[0, 1, 0.5, Max[Min[(-т + (η + d2))/d2, 1], 0]]],
        ColorFunctionScaling -> False]]},
     {Grid[{
        {" ", s["proper time"], " = ", s[N[т, 8]], s["GM/c³"]},
        {" ", s["coordinate time"], " = ", s[N[Evaluate[τ[т] /. sol][[1]], 8]], s["GM/c³"]},
        {" ", s["time dilation"], " = ", s[N[Evaluate[τ'[т] /. sol][[1]], 8]], s["dτ/dt"]},
        {" ", s["angle"], " = ", s[N[Evaluate[(θ[т] /. sol) 180/Pi][[1]], 8]], s["degree"]},
        {" ", s["radial distance"], " = ", s[N[Evaluate[r[т] /. sol][[1]], 8]], s["GM/c²"]},
        {" ", s["x-Axis"], " = ", s[N[x[т], 8]], s["GM/c²"]},
        {" ", s["y-Axis"], " = ", s[N[y[т], 8]], s["GM/c²"]}
        }, Alignment -> Left]}}, Alignment -> Left]]
  ], {т, T/300, T, T/300}]

(* coordinate time *) Do[Print[
  Rasterize[Grid[{{Show[Graphics[{
       {Black, Circle[{0, 0}, rs]},
      {Lighter[Gray], Dashed, Circle[{0, 0}, r0]}},
        Frame -> True, ImageSize -> 400, PlotRange -> 2 r0],
       Graphics[{PointSize[0.01], Red, Point[{x[Τ], y[Τ]}]}],
       ParametricPlot[{x[η], y[η]}, {η, 0, Τ},
       PlotStyle->{LightGray}],
       ParametricPlot[{x[η], y[η]}, {η, 0, Τ},
        ColorFunction ->
         Function[{x, y, η},
          Hue[0, 1, 0.5, Max[Min[(-Τ + (η + d2))/d2, 1], 0]]],
        ColorFunctionScaling -> False]]},
     {Grid[{
        {" ", s["proper time"], " = ", s[N[Τ, 8]], s["GM/c³"]},
        {" ", s["coordinate time"], " = ", s[N[ι, 8]], s["GM/c³"]},
        {" ", s["time dilation"], " = ", s[N[Evaluate[τ'[Τ] /. sol][[1]], 8]], s["dτ/dt"]},
        {" ", s["angle"], " = ", s[N[Evaluate[(θ[Τ] /. sol) 180/Pi][[1]], 8]], s["degree"]},
        {" ", s["radial distance"], " = ", s[N[Evaluate[r[Τ] /. sol][[1]], 8]], s["GM/c²"]},
        {" ", s["x-Axis"], " = ", s[N[x[Τ], 8]], s["GM/c²"]},
        {" ", s["y-Axis"], " = ", s[N[y[Τ], 8]], s["GM/c²"]}
        }, Alignment -> Left]}}, Alignment -> Left]]
  ], {ι, tau/300, tau, tau/300