Effective potentials for the Schwarzschild metric (BH Student Meetings, meeting #5)
Last edited by Jens Boos, Nov 6, 2019.

(* effective potential in terms of dimensionless variables *)
(* a maximum and minimum occur if ℓ > Sqrt[12] = 3.46... *)
Veff[ρ_,ℓ_]:=-1/ρ+ℓ^2/(2 ρ^2)(1-2/ρ);
Manipulate[Plot[Veff[ρ,ℓ],{ρ,0.1,20},Frame->True,PlotRange->{-3,3}],{ℓ,0,15}]

(* position of maximum and minimum *)

Solve[D[Veff[ρ,ℓ],ρ]==0,ρ]

ρmax[ℓ_]:=(1/2) ((ℓ^2)-ℓ (Sqrt[-12+(ℓ^2)]));
ρmin[ℓ_]:=(1/2) ((ℓ^2)+ℓ (Sqrt[-12+(ℓ^2)]));

(* Newtonian effective potential *)

VeffNewton[ρ_,ℓ_]:=-1/ρ+ℓ^2/(2ρ^2);
Manipulate[Plot[{Veff[ρ,ℓ],VeffNewton[ρ,ℓ]},{ρ,0.1,100},Frame->True,PlotRange->Automatic],{ℓ,0,15}]

(* there is always one maximum of the newtonian effective potential *)

Solve[D[VeffNewton[ρ,ℓ],ρ]==0,ρ]