Initial conditions, position: r0=Sqrt[3^2-a^2]; r1=r0+1; θ0=π/2; φ0=0; a=9/10; ℧=2/5; q=0; Velocity, relative to the local ZAMO: v0=2/5; α0=0; i0=ArcTan[5/6]; vr0=v0 Sin[α0]=0; vθ0=v0 Cos[α0] Sin[i0]=2/Sqrt=0.25607376; vφ0=v0 Cos[α0] Cos[i0]=12/5/Sqrt=0.30728851;
Above: animation, BL: particle corotating above the outer horizon, RD: particle corotating below the inner horizon. Below: right before (Boyer Lindquist, left) and right after (Raindrop Doran, right) the horizon, same initial conditions. r₊=1+√3/10=1.1732, r₋=1-√3/10=0.82679:
At proper time τ=7.7712857648 the particle reaches the outer horizon. In the frame of the external bookkeeper (BL), the particle freezes above the horizon and corotates with the BH. In the raindrop frame (RD) the particle gets fried before it exits the inner Cauchy horizon.
τ Plots: Radial coordinate r, longitude θ, latitude φ & velocity v in Boyer Lindquist (left) and Raindrop Doran (right) coordinates, plot by proper time τ in GM/c³. Radius in GM/c², angular coordinates in degree. Outer event horizon crossing at τ=7.7712857648 (BL and RD).
t Plots: r, θ, φ & v in Boyer Lindquist (left) and Raindrop Doran (right) coordinates, plot by coordinate time t in GM/c³. The velocity is relative to a ZAMO in both plots. Outer event horizon crossing at t=∞ (BL) and t=13.791594 (RD). Main site: kerr.newman.yukterez.net