Hartle - Thorne solution

(on equatorial plane + functions for the transformation to the z - axis)

In[39]:=

Am1 = (15 r (r - 2 Mm) (1 - 3 u^2))/(16 Mm^2) Log[r/(r - 2 Mm)] ;

Am2 = (15 (r^2 - 2 Mm^2) (3 u^2 - 1))/(16 Mm^2) Log[r/(r - 2 Mm)] ;

L1 = 80 Mm^6 + 8 Mm^4 r^2 + 10 Mm^3 r^3 + 20 Mm^2 r^4 - 45 Mm r^5 + 15 r^6 ;

p1 = (8 Mm r^4 (r - 2 Mm))^(-1) ;

W1 = (r - Mm) (16 Mm^5 + 8 Mm^4 r - 10 Mm^2 r^3 - 30 Mm r^4 + 15 r^5) + u^2 (48 Mm^6 - 8 Mm^5 r - 24 Mm^4 r^2 - 30 Mm^3 r^3 - 60 Mm^2 r^4 + 135 Mm r^5 - 45 r^6) ;

Fm1[Mm_, r_, u_] = -p1 W1 + Am1 ;

Fm2[Mm_, r_, u_] = 5 r^3 p1 (3 u^2 - 1) (r - Mm) (2 Mm^2 + 6 Mm r - 3 r^2) - Am1 ;

Gm1[Mm_, r_, u_] = p1 ((L1 - 72 Mm^5 r) - 3 u^2 (L1 - 56 Mm^5 r)) - Am1 ;

Hm1[Mm_, r_, u_] = Am2 + (8 Mm r^4)^(-1) (1 - 3 u^2) (16 Mm^5 + 8 Mm^4 r - 10 Mm^2 r^3 + 15 Mm r^4 + 15 r^5) ;

Hm2[Mm_, r_, u_] = -Am2 + (8 Mm r)^(-1) 5 (1 - 3 u^2) (2 Mm^2 - 3 Mm r - 3 r^2) ;

gttHT[Mm_, jj_, q_, r_, u_] = -(1 - 2 Mm/r) (1 + jj^2 Fm1[Mm, r, u] - q Fm2[Mm, r, u]) ;

grrHT[Mm_, jj_, q_, r_, u_] = (1 - 2 Mm/r)^(-1) (1 + jj^2 Gm1[Mm, r, u] + q Fm2[Mm, r, u]) ;

gθθHT[Mm_, jj_, q_, r_, u_] = r^2 (1 + jj^2 Hm1[Mm, r, u] - q Hm2[Mm, r, u]) ;

gφφHT[Mm_, jj_, q_, r_, u_] = r^2 (1 + jj^2 Hm1[Mm, r, u] - q Hm2[Mm, r, u]) (1 - u^2) ;

gtφHT[Mm_, jj_, r_, u_] = -2 Mm^2/r jj (1 - u^2) ;

RcircHTNew[Mm_, jj_, q_, r_] = gφφHT[Mm, jj, q, r, 0]^(1/2) ;

DEtildaHTNew[Mm_, jj_, q_, r_] = -ΩHTNew[Mm, jj, q, r] D[EtildaHTNew[Mm, jj, q, r], r]/D[ΩHTNew[Mm, jj, q, r], r] ;

ρHT[Mm_, jj_, q_, r_, u_] = (gtφHT[Mm, jj, r, u]^2 - gttHT[Mm, jj, q, r, u] gφφHT[Mm, jj, q, r, u])^(1/2) ;

gzzHT[Mm_, jj_, q_, r_, u_] = (D[ρHT[Mm, jj, q, r, u], r]^2/grrHT[Mm, jj, q, r, u] + ((1 - u^2) D[ρHT[Mm, jj, q, r, u], u]^2)/gθθHT[Mm, jj, q, r, u])^(-1) ;

In[60]:=

In[87]:=

AΩ[Mm_, r_] = (15 (r^3 - 2 Mm^3))/(32 Mm^3) Log[r/(r - 2 Mm)] ;

FΩ1[Mm_, r_] = (48 Mm^7 - 80 Mm^6 r + 4 Mm^5 r^2 - 18 Mm^4 r^3 + 40 Mm^3 r^4 + 10 Mm^2 r^5 + 15 Mm r^6 - 15 r^7) (16 Mm^2 (r - 2 Mm) r^4)^(-1) + AΩ[Mm, r] ;

FΩ2[Mm_, r_] = 5 (6 Mm^4 - 8 Mm^3 r - 2 Mm^2 r^2 - 3 Mm r^3 + 3 r^4) (16 Mm^2 (r - 2 Mm) r)^(-1) - AΩ[Mm, r] ;

ΩHT[Mm_, jj_, q_, r_] = Mm^(1/2)/r^(3/2) (1 - jj Mm^(3/2)/r^(3/2) + jj^2 FΩ1[Mm, r] + q FΩ2[Mm, r]) ;

Ar[Mm_, r_] = (16 Mm^3 (r - 6 Mm))^(-1) (15 r (r - 2 Mm) (2 Mm^2 + 13 Mm r - 4 r^2)) Log[r/(r - 2 Mm)] ;

Fr1[Mm_, r_] = (6 Mm^(3/2) (r + 2 Mm))/(r^(3/2) (r - 6 Mm)) ;

Fr2[Mm_, r_] := (8 Mm^2 r^4 (r - 2 Mm) (r - 6 Mm))^(-1) (384 Mm^8 - 720 Mm^7 r - 112 Mm^6 r^2 - 76 Mm^5 r^3 - 138 Mm^4 r^4 - 130 Mm^3 r^5 + 635 Mm^2 r^6 - 375 Mm r^7 + 60 r^8) + Ar[Mm, r]

Fr3[Mm_, r_] := (8 Mm^2 r (r - 2 Mm) (r - 6 Mm))^(-1) (5 (48 Mm^5 + 30 Mm^4 r + 26 Mm^3 r^2 - 127 Mm^2 r^3 + 75 Mm r^4 - 12 r^5)) - Ar[Mm, r]

vHTr[Mm_, jj_, q_, r_] := (Mm (r - 6 Mm) r^(-4) (1 + jj Fr1[Mm, r] - jj^2 Fr2[Mm, r] - q Fr3[Mm, r]))^(1/2)

rISCOHT[Mm_, jj_, q_] := 6 Mm (1 - jj (2/3)^(3/2) + jj^2 (251647/2592 - 240 Log[3/2]) + q (-9325/96 + 240 Log[3/2]))

Bz[Mm_, r_] := (15 (2r - Mm) (r - 2 Mm)^2)/(16 Mm^3) Log[r/(r - 2 Mm)]

G1[Mm_, r_] := (6 Mm^(3/2))/r^(3/2)

G2[Mm_, r_] := (8 Mm^2 r^4 (r - 2 Mm))^(-1) (48 Mm^7 - 244 Mm^6 r + 28 Mm^5 r^2 + 6 Mm^4 r^3 - 170 Mm^3 r^4 + 295 Mm^2 r^5 - 165 Mm r^6 + 30 r^7) - Bz[Mm, r]

G3[Mm_, r_] := (8 Mm^2 r (r - 2 Mm))^(-1) (5 (6 Mm^4 + 34 Mm^3 r - 59 Mm^2 r^2 + 33 Mm r^3 - 6 r^4)) + Bz[Mm, r]

vHTz[Mm_, jj_, q_, r_] := (Mm r^(-3) (1 - jj G1[Mm, r] + jj^2 G2[Mm, r] + q G3[Mm, r]))^(1/2)

A2[Mm_, r_] := -15 (r^2 - 2 Mm^2)/(16 Mm^2) Log[r/(r - 2 Mm)]

H2[Mm_, r_] := (8 Mm r)^(-1) 5 (2 Mm^2 - 3 Mm r - 3 r^2) - A2[Mm, r]

H1[Mm_, r_] := (8 Mm r^4)^(-1) (16 Mm^5 + 8 Mm^4r - 10 Mm^2 r^3 + 15 Mm r^4 + 15 r^5) + A2[Mm, r]

RcircHT[Mm_, jj_, q_, r_] = (r^2 (1 + jj^2 H1[Mm, r] + q H2[Mm, r]))^(1/2) ;

Be[Mm_, r_] := (15 r (8 Mm^2 - 7 Mm r + 2 r^2))/(32 Mm^2 (r - 3 Mm)) Log[r/(r - 2 Mm)]

Fe1[Mm_, r_] := Mm^(5/2)/(r^(1/2) (r - 2 Mm) (r - 3 Mm))

Fe2[Mm_, r_] := (16 Mm r^4 (r - 2 Mm) (r - 3 Mm)^2)^(-1) (144 Mm^8 - 144 Mm^7 r - 28 Mm^6 r^2 - 58 Mm^5 r^3 - 176 Mm^4 r^4 + 685 Mm^3 r^5 - 610 Mm^2 r^6 + 225 Mm r^7 - 30 r^8) + Be[Mm, r]

Fe3[Mm_, r_] := (5 (r - Mm) (6 Mm^3 - 20 Mm^2 r - 21 Mm r^2 + 6 r^3))/(16 Mm r (r - 2 Mm) (r - 3 Mm)) - Be[Mm, r]

EtildaHT[Mm_, jj_, q_, r_] := (r - 2 Mm)/(r (r - 3 Mm))^(1/2) (1 - jj Fe1[Mm, r] + jj^2 Fe2[Mm, r] + q Fe3[Mm, r])

DEtildaHT[Mm_, jj_, q_, r_] = -ΩHT[Mm, jj, q, r] D[EtildaHT[Mm, jj, q, r], r]/D[ΩHT[Mm, jj, q, r], r] ;

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