Kerr Newman De Sitter (KNdS) Horizon and Ergosphere Plotter [geodesics.yukterez.net] / [knds.yukterez.net]

(* Syntax: Wolfram Mathematica *)

M=1; (* mass *)
a=9/10; (* spin *)
℧=2/5; (* charge *)

u=85 π/180; (* inclination *)
im=640; (* image size *)
PR=4; (* plot range *)
plp=Round[im/PR/5]; (* resolution *)

LaunchKernels[4]

rH1=1/(2 Sqrt[3]) (√(-2 a^2+6/Λ+(9+Λ (-42 a^2+a^4 Λ-36 ℧^2))/(Λ (486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3))+(486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3)/Λ)-√(-4 a^2+12/Λ+(-9+Λ (42 a^2-a^4 Λ+36 ℧^2))/(Λ (486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3))-(486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3)/Λ-(36 Sqrt[3] M)/(Λ √(-2 a^2+6/Λ+(9+Λ (-42 a^2+a^4 Λ-36 ℧^2))/(Λ (486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3))+(486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3)/Λ))));

RH1[A_]:={Sqrt[rH1^2+A^2] Sin[θ] Cos[φ], Sqrt[rH1^2+A^2] Sin[θ] Sin[φ], rH1 Cos[θ]};

rH2=1/(2 Sqrt[3]) (√(-2 a^2+6/Λ+(9+Λ (-42 a^2+a^4 Λ-36 ℧^2))/(Λ (486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3))+(486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3)/Λ)+√(-4 a^2+12/Λ+(-9+Λ (42 a^2-a^4 Λ+36 ℧^2))/(Λ (486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3))-(486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3)/Λ-(36 Sqrt[3] M)/(Λ √(-2 a^2+6/Λ+(9+Λ (-42 a^2+a^4 Λ-36 ℧^2))/(Λ (486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3))+(486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3)/Λ))));

RH2[A_]:={Sqrt[rH2^2+A^2] Sin[θ] Cos[φ], Sqrt[rH2^2+A^2] Sin[θ] Sin[φ], rH2 Cos[θ]};

rH3=1/(2 Sqrt[3]) (-√(-2 a^2+6/Λ+(9+Λ (-42 a^2+a^4 Λ-36 ℧^2))/(Λ (486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3))+(486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3)/Λ)+√(-4 a^2+12/Λ+(-9+Λ (42 a^2-a^4 Λ+36 ℧^2))/(Λ (486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3))-(486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3)/Λ+(36 Sqrt[3] M)/(Λ √(-2 a^2+6/Λ+(9+Λ (-42 a^2+a^4 Λ-36 ℧^2))/(Λ (486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3))+(486 M^2 Λ+(-3+a^2 Λ)^3+108 Λ (-3+a^2 Λ) (a^2+℧^2)+1/2 Sqrt[-4 ((-3+a^2 Λ)^2-36 Λ (a^2+℧^2))^3+(972 M^2 Λ+2 (-3+a^2 Λ)^3+216 Λ (-3+a^2 Λ) (a^2+℧^2))^2])^(1/3)/Λ))));

RH3[A_]:={Sqrt[rH3^2+A^2] Sin[θ] Cos[φ], Sqrt[rH3^2+A^2] Sin[θ] Sin[φ], rH3 Cos[θ]};

rE1=1/2 √(-((2 a^2)/3)+2/Λ+(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])/(3 Λ (-3888 M^2 Λ-8 (-3+a^2 Λ)^3+288 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+2 Sqrt[2] √((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2))^(1/3))-(1/(3 Λ))((-486 M^2 Λ-(-3+a^2 Λ)^3+9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2)+Sqrt[(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2))^2]/(2 Sqrt[2]))^(1/3)))-1/2 √(-((4 a^2)/3)+4/Λ+(1/(3 Λ))((-486 M^2 Λ-(-3+a^2 Λ)^3+36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+Sqrt[(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2]/(2 Sqrt[2]))^(1/3))-(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])/(3 Λ (-3888 M^2 Λ-8 (-3+a^2 Λ)^3+288 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+2 Sqrt[2] √((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2))^(1/3))-(12 M)/(Λ √(-((2 a^2)/3)+2/Λ+(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])/(3 Λ (-3888 M^2 Λ-8 (-3+a^2 Λ)^3+288 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+2 Sqrt[2] √((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2))^(1/3))-(1/(3 Λ))((-486 M^2 Λ-(-3+a^2 Λ)^3+9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2)+(1/(2 Sqrt[2]))(√((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2))^2)))^(1/3)))));

RE1[A_]:={Sqrt[rE1^2+A^2] Sin[θ] Cos[φ], Sqrt[rE1^2+A^2] Sin[θ] Sin[φ], rE1 Cos[θ]};

rE2=1/2 √(-((2 a^2)/3)+2/Λ+(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])/(3 Λ (-3888 M^2 Λ-8 (-3+a^2 Λ)^3+288 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+2 Sqrt[2] √((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2))^(1/3))-(1/(3 Λ))((-486 M^2 Λ-(-3+a^2 Λ)^3+9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2)+Sqrt[(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2))^2]/(2 Sqrt[2]))^(1/3)))+1/2 √(-((4 a^2)/3)+4/Λ+(1/(3 Λ))((-486 M^2 Λ-(-3+a^2 Λ)^3+36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+Sqrt[(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2]/(2 Sqrt[2]))^(1/3))-(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])/(3 Λ (-3888 M^2 Λ-8 (-3+a^2 Λ)^3+288 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+2 Sqrt[2] √((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2))^(1/3))-(12 M)/(Λ √(-((2 a^2)/3)+2/Λ+(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])/(3 Λ (-3888 M^2 Λ-8 (-3+a^2 Λ)^3+288 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+2 Sqrt[2] √((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2))^(1/3))-(1/(3 Λ))((-486 M^2 Λ-(-3+a^2 Λ)^3+9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2)+(1/(2 Sqrt[2]))(√((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2))^2)))^(1/3)))));

RE2[A_]:={Sqrt[rE2^2+A^2] Sin[θ] Cos[φ], Sqrt[rE2^2+A^2] Sin[θ] Sin[φ], rE2 Cos[θ]};

rE3=-(1/2) √(-((2 a^2)/3)+2/Λ+(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])/(3 Λ (-3888 M^2 Λ-8 (-3+a^2 Λ)^3+288 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+2 Sqrt[2] √((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2))^(1/3))-(1/(3 Λ))((-486 M^2 Λ-(-3+a^2 Λ)^3+9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2)+Sqrt[(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2))^2]/(2 Sqrt[2]))^(1/3)))+1/2 √(-((4 a^2)/3)+4/Λ+(1/(3 Λ))((-486 M^2 Λ-(-3+a^2 Λ)^3+36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+Sqrt[(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2]/(2 Sqrt[2]))^(1/3))-(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])/(3 Λ (-3888 M^2 Λ-8 (-3+a^2 Λ)^3+288 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+2 Sqrt[2] √((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2))^(1/3))+(12 M)/(Λ √(-((2 a^2)/3)+2/Λ+(-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])/(3 Λ (-3888 M^2 Λ-8 (-3+a^2 Λ)^3+288 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2)+2 Sqrt[2] √((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-36 Λ (-3+a^2 Λ) (-3 (a^2+℧^2)+a^2 (3+a^2 Λ Cos[θ]^2) Sin[θ]^2))^2))^(1/3))-(1/(3 Λ))((-486 M^2 Λ-(-3+a^2 Λ)^3+9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2)+(1/(2 Sqrt[2]))(√((-18+48 a^2 Λ-5 a^4 Λ^2+72 Λ ℧^2+36 a^2 Λ Cos[2 θ]+3 a^4 Λ^2 Cos[4 θ])^3+8 (486 M^2 Λ+(-3+a^2 Λ)^3-9 Λ (-3+a^2 Λ) (-12 (a^2+℧^2)+12 a^2 Sin[θ]^2+a^4 Λ Sin[2 θ]^2))^2)))^(1/3)))));

RE3[A_]:={Sqrt[rE3^2+A^2] Sin[θ] Cos[φ], Sqrt[rE3^2+A^2] Sin[θ] Sin[φ], rE3 Cos[θ]};

(* black hole horizons and ergospheres *)

horizons1[A_, mesh_]:=Show[
ParametricPlot3D[Chop[RE1[A]], {φ, 0, 2 π}, {θ, 0, π}, Mesh->mesh, PlotPoints->plp, PlotStyle->Directive[Red, Opacity[0.15]],
ImageSize->im, Boxed->False, Axes->False, PlotRange->PR, ViewPoint->{0, 50 Sin[u], 50 Cos[u]}],
ParametricPlot3D[Chop[RE2[A]], {φ, 0, 2 π}, {θ, 0, π}, Mesh->None, PlotPoints->plp, PlotStyle->Directive[Blue, Opacity[0.06]]],
ParametricPlot3D[Chop[RH1[A]], {φ, 0, 2 π}, {θ, 0, π}, Mesh->None, PlotPoints->plp, PlotStyle->Directive[Cyan, Opacity[0.12]]],
ParametricPlot3D[Chop[RH3[A]], {φ, 0, 2 π}, {θ, 0, π}, Mesh->None, PlotPoints->plp, PlotStyle->Directive[Red, Opacity[0.22]]],
If[A==0, {}, ParametricPlot3D[{Sin[prm] A, Cos[prm] A, 0}, {prm, 0, 2π}, PlotStyle->{Thickness[0.0015], Orange}]]];

(* cosmic horizon and ergosphere *)

horizons2[A_, mesh_]:=Show[
ParametricPlot3D[Chop[RE3[A]], {φ, 0, 2 π}, {θ, 0, π}, Mesh->None, PlotPoints->plp, PlotStyle->Directive[Blue, Opacity[0.04]],
ImageSize->im, Boxed->False, Axes->False, PlotRange->PR, ViewPoint->{0, 50 Sin[u], 50 Cos[u]}],
ParametricPlot3D[Chop[RH2[A]], {φ, 0, 2 π}, {θ, 0, π}, Mesh->None, PlotPoints->plp, PlotStyle->Directive[Gray, Opacity[0.05]]]];

s[text_]:=Style[text,FontFamily->"Consolas", Bold]; w=Style[Y^Y,FontFamily->"Consolas", White];
rq[n_]:=Chop[n];

ParallelDo[Print[Rasterize[Grid[{{Show[
horizons1[a, 0],
horizons2[a, 0]]},
{},
{s["Hubble Constant"]},
{},
{s["H"->N[Sqrt[Λ/3]]], w},
{s["Λ"->N[Λ]], w},
{},
{s["BH Parameters"]},
{},
{s["M"->N[M]], w},
{s["a"->N[a]], w},
{s["℧"->N[℧]], w},
{s["Θ"->u/Pi*180"°"], w},
{},
{s["Boyer Lindquist r"]},
{},
{s["rH"->Chop[N[Block[{θ=π/2, φ=0}, rq[RH3[0][[1]]]]]]], w},
{s["rH"->Chop[N[Block[{θ=π/2, φ=0}, rq[RH1[0][[1]]]]]]], w},
{s["rH"->Chop[N[Block[{θ=π/2, φ=0}, rq[RH2[0][[1]]]]]]], w},
{},
{s["rE"->Chop[N[Block[{θ=π/2, φ=0}, rq[RE1[0][[1]]]]]]], w},
{s["rE"->Chop[N[Block[{θ=π/2, φ=0}, rq[RE2[0][[1]]]]]]], w},
{s["rE"->Chop[N[Block[{θ=π/2, φ=0}, rq[RE3[0][[1]]]]]]], w},
{},
{s["Cartesian R"]},
{},
{s["RH"->Chop[N[Block[{θ=π/2, φ=0}, rq[RH3[a][[1]]]]]]], w},
{s["RH"->Chop[N[Block[{θ=π/2, φ=0}, rq[RH1[a][[1]]]]]]], w},
{s["RH"->Chop[N[Block[{θ=π/2, φ=0}, rq[RH2[a][[1]]]]]]], w},
{},
{s["RE"->Chop[N[Block[{θ=π/2, φ=0}, rq[RE1[a][[1]]]]]]], w},
{s["RE"->Chop[N[Block[{θ=π/2, φ=0}, rq[RE2[a][[1]]]]]]], w},
{s["RE"->Chop[N[Block[{θ=π/2, φ=0}, rq[RE3[a][[1]]]]]]], w},
{}},
Alignment->Left]]],
{Λ, 24/1000, 12/100, 24/1000}]

Quit[]

The radii in the numeric display are in the equatorial plane θ=90°=π/2. Units: G=c=kε=1
Metric tensor in Boyer Lindquist, Null & Kerr Schild coordinates; FullHD video: Youtube
Simon Tyran, Vienna          Index: [[ Metric Collection ]]     Backup: [[ Archived Version ]]     Main site: [[ www.yukterez.net ]]