Puncture function for Kerr circular equatorial orbits

Assumptions and Setup

Assume circular equatorial orbits

$Assumptions=r1>3M&a<M&&a>-M&&M>0;

ut[r1_,a_,M_]:=(((r1(Sqrt[M r1])+a M))/(((Sqrt[M r1])(Sqrt[(r1^2)-3M r1+2 a(Sqrt[M r1])]))));
uphi[r1_,a_,M_]:=((Sqrt[M r1])/((r1(Sqrt[(r1^2)-3M r1+2 a(Sqrt[M r1])]))));

Ω[r1_,a_,M_]:=(M/(a M+(Sqrt[M (r1^3)])));

ur1=0;
uθ1=0;

Δϕ -> Cos[Δϕ],Sin[Δϕ]

CosMatrix[p_]:=Module[{x},Table[SeriesCoefficient[Cos[m x],{x,0,2n}],{m,0,p/2},{n,0,p/2}]];

SinMatrix[p_]:=Module[{x},Table[SeriesCoefficient[Sin[m x],{x,0,2n-1}],{m,1,(p+1)/2},{n,1,(p+1)/2}]];

CosSubRules[p_]:=Thread[Table[((x[[4]])^(2n)),{n,0,p/2}]-> Inverse[CosMatrix[p]].Table[Cos[m x[[4]]],{m,0,p/2}]]

SinSubRules[p_]:=Thread[Table[((x[[4]])^(2n+1)),{n,0,(p-1)/2}]-> Inverse[SinMatrix[p]].Table[Sin[m x[[4]]],{m,1,(p+1)/2}]]

SubRules[p_]:=Riffle[CosSubRules[p],SinSubRules[p]]

Coordinate expansions of σ and (Subscript[σ, α])

Define σ and (Subscript[σ, a]) as a coordinate expansion

ExpOrder=5;

Truncate[expr_,x_List,maxorder_]:=Module[{ϵ,tmp,truncrule},
tmp=expr/.Thread[x->ϵ x];
truncrule=Power[ϵ,order_]:>0/;order>maxorder;
tmp=Collect[tmp,ϵ]/.truncrule;
tmp/.ϵ->1]

x={Δt,Δr,Δθ,Δϕ};
ΔxRules={Δt->t-t1,Δr->r-r1,Δθ->θ-θ1,Δϕ->ϕ-ϕ1};
ΔxInvRules={t->Δt+t1,r->Δr+r1,θ-> Δθ+θ1,ϕ->Δϕ+ϕ1};

{σt1,σr1,σθ1,σϕ1}={((M^2*(a^2+r1*(-2*M+r1)))/(6*r1^6)+(M^2*(-3*a^2+(5*M-2*r1)*r1)*Δr)/(6*r1^7))*Δt^3+(2*a*M*Δϕ)/r1-(a*M*(-3*a^2+(5*M-6*r1)*r1)*Δr^2*Δϕ)/(6*r1^3*(a^2+r1*(-2*M+r1)))-(a*M*(3*a^4+a^2*r1*(-10*M+9*r1)+2*r1^2*(3*M^2-10*M*r1+6*r1^2))*Δr^3*Δϕ)/(12*r1^4*(a^2+r1*(-2*M+r1))^2)-(a*M*(3*a^2+r1*(2*M+3*r1))*Δθ^2*Δϕ)/(6*r1^3)+(a*M*(-((a^2*M)/r1^2)+r1)*(a^2+r1*(-2*M+r1))*Δϕ^3)/(6*r1^4)+Δt^2*(-(a*M^2*(a^2+r1*(-2*M+r1))*Δϕ)/(2*r1^6)-(a*M^2*(-3*a^2+(5*M-2*r1)*r1)*Δr*Δϕ)/(2*r1^7))+Δr*(-((a*M*Δϕ)/r1^2)+(a*M*(9*a^2+r1*(4*M+3*r1))*Δθ^2*Δϕ)/(12*r1^4)-(a*M*(10*a^2*M^2*r1+3*a^2*r1^3+r1^5-2*M*(3*a^4+2*a^2*r1^2+2*r1^4))*Δϕ^3)/(12*r1^7))+Δt*(1-(2*M)/r1+(M*(-3*a^2+(5*M-4*r1)*r1)*Δr^2)/(6*r1^3*(a^2+r1*(-2*M+r1)))+(M*(3*a^4+6*(M-r1)^2*r1^2+a^2*r1*(-10*M+7*r1))*Δr^3)/(12*r1^4*(a^2+r1*(-2*M+r1))^2)+(M*(3*a^2+(2*M-r1)*r1)*Δθ^2)/(6*r1^3)-(M*(3*a^2*M*r1*(-a^2+(2*M-r1)*r1)+r1^4*(a^2+r1*(-2*M+r1)))*Δϕ^2)/(6*r1^7)+Δr*(M/r1^2-(M*(9*a^2+(4*M-r1)*r1)*Δθ^2)/(12*r1^4)+(M*(30*a^2*M^2*r1+3*a^2*r1^3+r1^5-2*M*(9*a^4+6*a^2*r1^2+2*r1^4))*Δϕ^2)/(12*r1^7))),-(r1*(-3*a^2*r1-M^2*r1+2*M*(a^2+r1^2))*Δr^3)/(6*(a^2+r1*(-2*M+r1))^3)+(r1*(a^4+M^3*r1-4*a^2*r1^2-M^2*(3*a^2+2*r1^2)+M*(5*a^2*r1+2*r1^3))*Δr^4)/(8*(a^2+r1*(-2*M+r1))^4)+(M^2*(3*a^2+r1*(-5*M+2*r1))*Δt^4)/(24*r1^7)+((M-r1)*Δθ^4)/24+(a*M^2*(-3*a^2+(5*M-2*r1)*r1)*Δt^3*Δϕ)/(6*r1^7)+((-((a^2*M)/r1^2)+r1)*Δϕ^2)/2+((-5*a^4*M^3*r1-r1^8+M^2*(3*a^6+2*a^4*r1^2+4*a^2*r1^4)+M*(-3*a^4*r1^3-a^2*r1^5+r1^7))*Δϕ^4)/(24*r1^7)+Δθ^2*(r1/2+((9*a^4*M+4*a^2*M^2*r1+7*a^2*M*r1^2+2*M*r1^4-6*r1^5)*Δϕ^2)/(24*r1^4))+Δr*(-(r1^2/(a^2-2*M*r1+r1^2))+(r1*(-5*M+3*r1)*Δθ^2)/(6*(a^2+r1*(-2*M+r1)))+((-7*a^2*M^2*r1+3*r1^5+M*(3*a^4-2*a^2*r1^2-5*r1^4))*Δϕ^2)/(6*r1^3*(a^2+r1*(-2*M+r1))))+Δt^2*(-M/(2*r1^2)-(M*(-3*a^2+(7*M-2*r1)*r1)*Δr)/(6*r1^3*(a^2+r1*(-2*M+r1)))-(M*(9*a^4+a^2*r1*(-38*M+17*r1)+6*r1^2*(7*M^2-6*M*r1+r1^2))*Δr^2)/(24*r1^4*(a^2+r1*(-2*M+r1))^2)+(M*(9*a^2+(4*M-r1)*r1)*Δθ^2)/(24*r1^4)-(M*(30*a^2*M^2*r1+3*a^2*r1^3+r1^5-2*M*(9*a^4+6*a^2*r1^2+2*r1^4))*Δϕ^2)/(24*r1^7))+Δr^2*((r1*(-a^2+M*r1))/(2*(a^2+r1*(-2*M+r1))^2)+((6*a^2*r1-M*(5*a^2+r1^2))*Δθ^2)/(24*(a^2+r1*(-2*M+r1))^2)+((-42*a^2*M^3*r1^2+6*a^2*r1^5+M^2*(38*a^4*r1+20*a^2*r1^3)-M*(9*a^6+13*a^4*r1^2-a^2*r1^4+r1^6))*Δϕ^2)/(24*r1^4*(a^2+r1*(-2*M+r1))^2))+Δt*((a*M*Δϕ)/r1^2+(a*M*(-3*a^2+7*M*r1)*Δr*Δϕ)/(3*r1^3*(a^2+r1*(-2*M+r1)))+(a*M*(9*a^4+14*M*(3*M-2*r1)*r1^2+a^2*r1*(-38*M+15*r1))*Δr^2*Δϕ)/(12*r1^4*(a^2+r1*(-2*M+r1))^2)-(a*M*(9*a^2+r1*(4*M+3*r1))*Δθ^2*Δϕ)/(12*r1^4)+(a*M*(10*a^2*M^2*r1+3*a^2*r1^3+r1^5-2*M*(3*a^4+2*a^2*r1^2+2*r1^4))*Δϕ^3)/(12*r1^7)),(M*r1*Δr^2*Δθ)/(6*(a^2+r1*(-2*M+r1)))+(M*(a^2-r1^2)*Δr^3*Δθ)/(12*(a^2+r1*(-2*M+r1))^2)+(r1*(-2*M+r1)*Δθ^3)/6+Δt^2*((M*(-3*a^2+(2*M-r1)*r1)*Δθ)/(6*r1^3)+(M*(9*a^2+r1*(-4*M+r1))*Δr*Δθ)/(12*r1^4))+Δt*(-(a*M*(-3*a^2+(2*M-3*r1)*r1)*Δθ*Δϕ)/(3*r1^3)+(a*M*(-9*a^2+(4*M-3*r1)*r1)*Δr*Δθ*Δϕ)/(6*r1^4))+Δθ*(-r1^2+(M*(-3*a^4+a^2*(2*M-5*r1)*r1-2*r1^4)*Δϕ^2)/(6*r1^3))+Δr*(((-M+r1)*Δθ^3)/6+Δθ*(-r1+(M*(9*a^4-2*r1^4+a^2*r1*(-4*M+5*r1))*Δϕ^2)/(12*r1^4))),(-(a*M^2*(a^2+r1*(-2*M+r1)))/(6*r1^6)-(a*M^2*(-3*a^2+(5*M-2*r1)*r1)*Δr)/(6*r1^7))*Δt^3-((2*a^2*M+a^2*r1+r1^3)*Δϕ)/r1+((5*a^2*M^2*r1-3*a^2*r1^3+M*(-3*a^4-8*a^2*r1^2+r1^4))*Δr^2*Δϕ)/(6*r1^3*(a^2+r1*(-2*M+r1)))+((6*a^2*M^3*r1^2+6*a^2*r1^5-2*M^2*(5*a^4*r1+14*a^2*r1^3)+M*(3*a^6+11*a^4*r1^2+13*a^2*r1^4-r1^6))*Δr^3*Δϕ)/(12*r1^4*(a^2+r1*(-2*M+r1))^2)+((2*a^2*M^2*r1+3*r1^3*(a^2+r1^2)+M*(3*a^4+7*a^2*r1^2-2*r1^4))*Δθ^2*Δϕ)/(6*r1^3)+((-(a^2*M)+r1^3)^2*(a^2+r1*(-2*M+r1))*Δϕ^3)/(6*r1^6)+Δt^2*(-(M*(3*a^2*M*r1*(-a^2+(2*M-r1)*r1)+r1^4*(a^2+r1*(-2*M+r1)))*Δϕ)/(6*r1^7)+(M*(30*a^2*M^2*r1+3*a^2*r1^3+r1^5-2*M*(9*a^4+6*a^2*r1^2+2*r1^4))*Δr*Δϕ)/(12*r1^7))+Δr*(((a^2*M)/r1^2-r1)*Δϕ-((9*a^4*M+4*a^2*M^2*r1+7*a^2*M*r1^2+2*M*r1^4-6*r1^5)*Δθ^2*Δϕ)/(12*r1^4)+((5*a^4*M^3*r1+r1^8-M^2*(3*a^6+2*a^4*r1^2+4*a^2*r1^4)+M*(3*a^4*r1^3+a^2*r1^5-r1^7))*Δϕ^3)/(6*r1^7))+Δt*((2*a*M)/r1-(a*M*(-3*a^2+(5*M-6*r1)*r1)*Δr^2)/(6*r1^3*(a^2+r1*(-2*M+r1)))-(a*M*(3*a^4+a^2*r1*(-10*M+9*r1)+2*r1^2*(3*M^2-10*M*r1+6*r1^2))*Δr^3)/(12*r1^4*(a^2+r1*(-2*M+r1))^2)-(a*M*(3*a^2+r1*(2*M+3*r1))*Δθ^2)/(6*r1^3)+(a*M*(-((a^2*M)/r1^2)+r1)*(a^2+r1*(-2*M+r1))*Δϕ^2)/(2*r1^4)+Δr*(-((a*M)/r1^2)+(a*M*(9*a^2+r1*(4*M+3*r1))*Δθ^2)/(12*r1^4)-(a*M*(10*a^2*M^2*r1+3*a^2*r1^3+r1^5-2*M*(3*a^4+2*a^2*r1^2+2*r1^4))*Δϕ^2)/(4*r1^7)))};

σ=((3*a^4+2*a^2*(M-6*r1)*r1-M*(M-8*r1)*r1^2)*Δr^4)/(24*(a^2+r1*(-2*M+r1))^3)+((-9*a^4*r1-M^3*r1^2+12*a^2*r1^3+M^2*(3*a^2*r1+2*r1^3)+M*(4*a^4-5*a^2*r1^2-6*r1^4))*Δr^5)/(24*(a^2+r1*(-2*M+r1))^4)+(-(M^2*(a^2+r1*(-2*M+r1)))/(24*r1^6)+(M^2*(3*a^2+r1*(-5*M+2*r1))*Δr)/(24*r1^7))*Δt^4+((3*a^2+(2*M-r1)*r1)*Δθ^4)/24+((2*a^2*M+a^2*r1+r1^3)*Δϕ^2)/(2*r1)-((-(a^2*M)+r1^3)^2*(a^2+r1*(-2*M+r1))*Δϕ^4)/(24*r1^6)+Δt^3*((a*M^2*(a^2+r1*(-2*M+r1))*Δϕ)/(6*r1^6)+(a*M^2*(-3*a^2+(5*M-2*r1)*r1)*Δr*Δϕ)/(6*r1^7))+Δθ^2*(r1^2/2-((2*a^2*M^2*r1+3*r1^3*(a^2+r1^2)+M*(3*a^4+7*a^2*r1^2-2*r1^4))*Δϕ^2)/(12*r1^3))+Δr^2*(r1^2/(2*(a^2+r1*(-2*M+r1)))+((3*a^2-M*r1)*Δθ^2)/(12*(a^2+r1*(-2*M+r1)))+((-5*a^2*M^2*r1+3*a^2*r1^3+M*(3*a^4+8*a^2*r1^2-r1^4))*Δϕ^2)/(12*r1^3*(a^2+r1*(-2*M+r1))))+Δr^3*((r1*(a^2-M*r1))/(2*(a^2+r1*(-2*M+r1))^2)+((-6*a^2*r1+M*(5*a^2+r1^2))*Δθ^2)/(24*(a^2+r1*(-2*M+r1))^2)+((-6*a^2*M^3*r1^2-6*a^2*r1^5+2*M^2*(5*a^4*r1+14*a^2*r1^3)+M*(-3*a^6-11*a^4*r1^2-13*a^2*r1^4+r1^6))*Δϕ^2)/(24*r1^4*(a^2+r1*(-2*M+r1))^2))+Δr*(((M-r1)*Δθ^4)/24+((-((a^2*M)/r1^2)+r1)*Δϕ^2)/2+((-5*a^4*M^3*r1-r1^8+M^2*(3*a^6+2*a^4*r1^2+4*a^2*r1^4)+M*(-3*a^4*r1^3-a^2*r1^5+r1^7))*Δϕ^4)/(24*r1^7)+Δθ^2*(r1/2+((9*a^4*M+4*a^2*M^2*r1+7*a^2*M*r1^2+2*M*r1^4-6*r1^5)*Δϕ^2)/(24*r1^4)))+Δt^2*(-1/2+M/r1-(M*(-3*a^2+(5*M-4*r1)*r1)*Δr^2)/(12*r1^3*(a^2+r1*(-2*M+r1)))-(M*(3*a^4+6*(M-r1)^2*r1^2+a^2*r1*(-10*M+7*r1))*Δr^3)/(24*r1^4*(a^2+r1*(-2*M+r1))^2)-(M*(3*a^2+(2*M-r1)*r1)*Δθ^2)/(12*r1^3)+(M*(3*a^2*M*r1*(-a^2+(2*M-r1)*r1)+r1^4*(a^2+r1*(-2*M+r1)))*Δϕ^2)/(12*r1^7)+Δr*(-M/(2*r1^2)+(M*(9*a^2+(4*M-r1)*r1)*Δθ^2)/(24*r1^4)-(M*(30*a^2*M^2*r1+3*a^2*r1^3+r1^5-2*M*(9*a^4+6*a^2*r1^2+2*r1^4))*Δϕ^2)/(24*r1^7)))+Δt*((-2*a*M*Δϕ)/r1+(a*M*(-3*a^2+(5*M-6*r1)*r1)*Δr^2*Δϕ)/(6*r1^3*(a^2+r1*(-2*M+r1)))+(a*M*(3*a^4+a^2*r1*(-10*M+9*r1)+2*r1^2*(3*M^2-10*M*r1+6*r1^2))*Δr^3*Δϕ)/(12*r1^4*(a^2+r1*(-2*M+r1))^2)+(a*M*(3*a^2+r1*(2*M+3*r1))*Δθ^2*Δϕ)/(6*r1^3)+(a*M*(a^2*M-r1^3)*(a^2+r1*(-2*M+r1))*Δϕ^3)/(6*r1^6)+Δr*((a*M*Δϕ)/r1^2-(a*M*(9*a^2+r1*(4*M+3*r1))*Δθ^2*Δϕ)/(12*r1^4)+(a*M*(10*a^2*M^2*r1+3*a^2*r1^3+r1^5-2*M*(3*a^4+2*a^2*r1^2+2*r1^4))*Δϕ^3)/(12*r1^7)));

s, (OverBar[r]), Ruσuσ, etc.

σ

Collect[σ/.Δt->0,{Δr,Δθ,Δϕ},Simplify]

Collect[Truncate[σ,x,3],x,Together]

s

We need to help Mathematica along with the square root coming from ut, uϕ

s=√Truncate[2 σ + Expand[(((ut1 σt1+ur1 σr1 + uθ1 σθ1 + uϕ1 σϕ1))^2)],x,ExpOrder];

When truncating, we go one order higher because the square root reduces the order by 1

sO4=Truncate[s,x,5];

sO3=Truncate[s,x,4];

sO2=Truncate[s,x,3];

sO1=Truncate[s,x,2];

(OverBar[r])

rbar2=Truncate[Expand[(((ut1 σt1+ur1 σr1 + uθ1 σθ1 + uϕ1 σϕ1))^2)],x,ExpOrder];

rbar1=Truncate[Expand[(ut1 σt1+ur1 σr1 + uθ1 σθ1 + uϕ1 σϕ1)],x,ExpOrder];

rbar2O4=Truncate[rbar2,x,4];

rbar2O3=Truncate[rbar2,x,3];

rbar2O2=Truncate[rbar2,x,2];

rbar1O5=Truncate[rbar1,x,5];

rbar1O4=Truncate[rbar1,x,4];

rbar1O3=Truncate[rbar1,x,3];

rbar1O2=Truncate[rbar1,x,2];

rbar1O1=Truncate[rbar1,x,1];

Ruσuσ

This was calculated in RiemannTerms.nb

Ruσuσ=Truncate[-(1/(r1^7))M (-(a^2)+(2 M-r1) r1) (6 (a^3) ut1 uϕ1+2 a r1 (-4 M+3 r1) ut1 uϕ1-3 (a^4) ((uϕ1)^2)+(a^2) (-3 (ut1^2)+4 (M-r1) r1 ((uϕ1)^2))-r1 (-4 M (ut1^2)+2 r1 (ut1^2)+(r1^3) ((uϕ1)^2))) ((σr1)^2)+((M (((-2 a M ut1+(r1^3) uϕ1+(a^2) (2 M+r1) uϕ1))^2) ((σt1)^2))/((r1^5) ((a^2)+r1 (-2 M+r1))))-(1/(r1^7))M (6 (a^3) ut1 uϕ1+2 a r1 (-2 M+3 r1) ut1 uϕ1-3 (a^4) ((uϕ1)^2)+(a^2) (-3 (ut1^2)+(2 M-5 r1) r1 ((uϕ1)^2))-r1 (-2 M (ut1^2)+r1 (ut1^2)+2 (r1^3) ((uϕ1)^2))) ((σθ1)^2)+(1/((r1^5) ((a^2)+r1 (-2 M+r1))))2 M ((r1^3) (-2 M+r1) ut1 uϕ1+(a^2) (-8 (M^2)+(r1^2)) ut1 uϕ1+2 (a^3) M (2 M+r1) ((uϕ1)^2)+2 a M (2 M (ut1^2)-r1 (ut1^2)+(r1^3) ((uϕ1)^2))) σt1 σϕ1+((M (((-2 M ut1+r1 ut1+2 a M uϕ1))^2) ((σϕ1)^2))/((r1^5) ((a^2)+r1 (-2 M+r1)))),x,ExpOrder];

RuσuσO4=Collect[Truncate[Ruσuσ,x,4],{Δt,Δr,Δθ,Δϕ},Simplify];

RuσuσO3=Collect[Truncate[Ruσuσ,x,3],{Δt,Δr,Δθ,Δϕ},Simplify];

RuσuσO2=Collect[Truncate[Ruσuσ,x,2],{Δt,Δr,Δθ,Δϕ},Simplify];

Ruσuσσ

Ruσuσσ=Truncate[-(1/(r1^10))3 M ((((a^2)+r1 (-2 M+r1)))^2) (10 (a^3) ut1 uϕ1+8 a r1 (-M+r1) ut1 uϕ1-5 (a^4) ((uϕ1)^2)+(a^2) (-5 (ut1^2)+2 (2 M-3 r1) r1 ((uϕ1)^2))-r1 (-4 M (ut1^2)+2 r1 (ut1^2)+(r1^3) ((uϕ1)^2))) ((σr1)^3)-(1/(r1^8))3 M (-8 a M (r1^3) ut1 uϕ1-8 (a^3) M (3 M+r1) ut1 uϕ1+(r1^6) ((uϕ1)^2)+(a^4) (12 (M^2)+8 M r1+(r1^2)) ((uϕ1)^2)+2 (a^2) (6 (M^2) (ut1^2)+4 M (r1^3) ((uϕ1)^2)+(r1^4) ((uϕ1)^2))) σr1 ((σt1)^2)-(1/(r1^10))3 M ((a^2)+r1 (-2 M+r1)) (-30 (a^3) ut1 uϕ1+2 a (6 M-11 r1) r1 ut1 uϕ1+15 (a^4) ((uϕ1)^2)+r1 (-6 M (ut1^2)+3 r1 (ut1^2)+4 (r1^3) ((uϕ1)^2))+(a^2) (15 (ut1^2)+r1 (-6 M+19 r1) ((uϕ1)^2))) σr1 ((σθ1)^2)-(1/(r1^8))6 M (2 (r1^3) (-2 M+r1) ut1 uϕ1+2 (a^2) (-12 (M^2)+2 M r1+(r1^2)) ut1 uϕ1+(a^3) (12 (M^2)+2 M r1-(r1^2)) ((uϕ1)^2)+a (12 (M^2) (ut1^2)-6 M r1 (ut1^2)+4 M (r1^3) ((uϕ1)^2)-(r1^4) ((uϕ1)^2))) σr1 σt1 σϕ1-((3 M ((r1^2) ut1 (3 ut1-2 a uϕ1)+12 (M^2) (((ut1-a uϕ1))^2)-4 M r1 (3 (ut1^2)-4 a ut1 uϕ1+(a^2) ((uϕ1)^2))) σr1 ((σϕ1)^2))/(r1^8)),x,ExpOrder];

RuσuσσO5=Collect[Truncate[Ruσuσσ,x,5],{Δt,Δr,Δθ,Δϕ},Simplify];

RuσuσσO4=Collect[Truncate[Ruσuσσ,x,4],{Δt,Δr,Δθ,Δϕ},Simplify];

RuσuσσO3=Collect[Truncate[Ruσuσσ,x,3],{Δt,Δr,Δθ,Δϕ},Simplify];

Ruσuσu

Ruσuσu=Truncate[-((6 M (-(a^2)+(2 M-r1) r1) uϕ1 (-ut1+a uϕ1) (-2 a M ut1+(r1^3) uϕ1+(a^2) (2 M+r1) uϕ1) σr1 σt1)/(r1^7))-((6 M (-(a^2)+(2 M-r1) r1) uϕ1 (-ut1+a uϕ1) (-2 M ut1+r1 ut1+2 a M uϕ1) σr1 σϕ1)/(r1^7)),x,ExpOrder];

RuσuσuO3=Simplify[Truncate[Ruσuσu,x,3]];

RuσuσuO2=Simplify[Truncate[Ruσuσu,x,2]];

(Subscript[Φ, S])

Format[at]:=(OverTilde[a]);
Format[r0t]:=(Subscript[(OverTilde[r]), 0]);
Format[Δ0t]:=(Subscript[(OverTilde[Δ]), 0]);
Format[uϕt]:=((OverTilde[u])^ϕ);

ϕrule=-3M r1+(r1^2)+2a(Sqrt[M r1])->term/.Solve[uphi[r1,a,M]==uϕ1/.{-3M r1+(r1^2)+2a(Sqrt[M r1])->term},term][[1]]

alpha4=FullSimplify[Simplify[(rbar2O2-(sO1^2))RuσuσO2/6/.{Δt->0}]/.{(a^2)+r1(-2M+r1)->Δ,uϕ1->uphi[r1,a,M],ut1->ut[r1,a,M]}]//.{r1->r,ϕrule}

alpha5=FullSimplify[Simplify[(rbar2O3-(sO2^2))RuσuσO3/6/.{Δt->0}]/.{(a^2)+r1(-2M+r1)->Δ,uϕ1->uphi[r1,a,M],ut1->ut[r1,a,M]}]//.{r1->r,ϕrule}

beta5a=FullSimplify[FullSimplify[Simplify[(rbar2O2-3(sO1^2))rbar1O1 RuσuσuO2/24/.Δt->0]/.{(a^2)+r1(-2M+r1)->Δ,(a^2)-2M r1+(r1^2)->Δ}]/.{(a^2)+r1(-2M+r1)->Δ,(a^2)+-2M r1+(r1^2)->Δ}]//.{r1->r,ϕrule,ut1->ut,uϕ1->uϕ}

beta5b=FullSimplify[FullSimplify[Simplify[Simplify[-(rbar2O2-(sO1^2))RuσuσσO3/24/.Δt->0]/.{(a^2)+r1(-2M+r1)->Δ,(a^2)+-2M r1+(r1^2)->Δ}]/.{(a^2)+r1(-2M+r1)->Δ,(a^2)+-2M r1+(r1^2)->Δ}]/.{(a^2)+r1(-2M+r1)->Δ}]//.{r1->r,ϕrule,ut1->ut,uϕ1->uϕ}

epsilon=FullSimplify[Simplify[CoefficientRules[(sO4^2)/.Δt->0,{Δr,Δθ,Δϕ}]/.{(a^2)+r1(-2M+r1)->Δ,(a^2)+-2M r1+(r1^2)->Δ}]/.{(a^2)+r1(-2M+r1)->Δ,(a^2)+-2M r1+(r1^2)->Δ},Assumptions->r1>3M&&M>0];

epsilon=FullSimplify[Simplify[(epsilon/.{ut1->ut[r1,a,M],uϕ1->uphi[r1,a,M]})]/.ϕrule/.{(a^2)+r1(-2M+r1)->Δ,(a^2)+-2M r1+(r1^2)->Δ},Assumptions->r>3M&&M>0]//.{r1->r,ϕrule,ut1->ut,uϕ1->uϕ}

epsilon/.(pows_List->val_):>((Subscript[𝒮, Sequence@@pows])==val)

Adimensionalize:

epsilon/.(pows_List->val_):>((Subscript[𝒮, Sequence@@pows])==FullSimplify[(val/(M^(2-pows[[1]]))/.{a->at M,r->r0t M,Δ->Δ0t (M^2),Δθ->δθ,Δϕ->δϕ,uϕ->uϕt/M})/.M->1,Assumptions->r0>0&&Δ0>0&&M>0])

TeXForm/@%

Map[Collect[#,{δr,δθ,δϕ},Simplify[#,Assumptions->r0>0&&Δ0>0&&M>0]&]&,Simplify[(alpha5/.ϕrule)/.{a->at M,r->r0t M,Δ->Δ0t (M^2),Δr->M δr,Δθ->δθ,Δϕ->δϕ,uϕ1->uϕ/M}]/(M^2)](M^2)

TeXForm[%]

Map[Collect[#,{δr,δθ,δϕ},Simplify[#,Assumptions->r0>0&&Δ0>0&&M>0]&]&,Simplify[(beta5a/.ϕrule)/.{a->at M,r->r0t M,Δ->Δ0t (M^2),Δr->M δr,Δθ->δθ,Δϕ->δϕ,uϕ1->uϕ/M}]/(M^2)](M^2)+Map[Collect[#,{δr,δθ,δϕ},Simplify[#,Assumptions->r0>0&&Δ0>0&&M>0]&]&,Simplify[(beta5b/.ϕrule)/.{a->at M,r->r0t M,Δ->Δ0t (M^2),Δr->M δr,Δθ->δθ,Δϕ->δϕ,uϕ1->uϕ/M}]/(M^2)](M^2)

TeXForm[%]