Quit[];

In this notebook we use a slightly different notation:
U -> AA
Y -> YY

Metric (Subscript[f, μν]) is given by qdd and metric (Subscript[g, μν]) by gdd.

Field Equations [no need to run this if the external file has been already generated]

In this notebook we use a slightly different notation:

U -> AA
Y -> YY

Metric (Subscript[f, μν]) is given by qdd and metric (Subscript[g, μν]) by gdd.

$Assumptions={r≥0,θ≥0,θ≤π,rh≥0,F[r]>0,B[r]>0,AA[r]>0,h[r]>0,p[r]>0,YY[r]>0,AA'[r]>0};
x={t,r,θ,φ};

gdd={{-((F[r])^2),0,0,0},{0,((B[r])^(-2)),0,0},{0,0,(r^2),0},{0,0,0,(r^2)((Sin[θ])^2)}};
qdd={{-((p[r])^2),0,0,0},{0,(((((AA'[r])/(YY[r]))))^2),0,0},{0,0,((AA[r])^2),0},{0,0,0,((AA[r])^2)((Sin[θ])^2)}};

Metric

MatrixForm[gdd]
MatrixForm[qdd]
guu=Inverse[gdd];
quu=Inverse[qdd];
DETg=Simplify[Normal[Series[Det[gdd],{eps,0,1}]]];
DETq=Simplify[Normal[Series[Det[qdd],{eps,0,1}]]];
dim=Length[gdd];
Do[g[i,j]=Simplify[gdd[[i]][[j]]];,{i,1,dim},{j,1,dim}];
Do[q[i,j]=Simplify[qdd[[i]][[j]]];,{i,1,dim},{j,1,dim}];

Basic definitions of the geometry

Do[Christq[i,j,k]=Simplify[Normal[Series[Sum[(1/2)quu[[k]][[l]](D[qdd[[l]][[i]],x[[j]]]+D[q[l,j],x[[i]]]-D[q[j,i],x[[l]]]),{l,1,dim}],{eps,0,1}]]];,{i,1,dim},{j,1,dim},{k,1,dim}];
Do[Christg[i,j,k]=Simplify[Normal[Series[Sum[(1/2)guu[[k]][[l]](D[gdd[[l]][[i]],x[[j]]]+D[g[l,j],x[[i]]]-D[g[j,i],x[[l]]]),{l,1,dim}],{eps,0,1}]]];,{i,1,dim},{j,1,dim},{k,1,dim}];

Do[Rudddq[a,i,j,k]=Simplify[Normal[Series[D[Christq[i,k,a],x[[j]]]-D[Christq[i,j,a],x[[k]]]+Sum[Christq[i,k,β]Christq[β,j,a]-Christq[i,j,β]Christq[β,k,a],{β,1,dim}],{eps,0,1}]]];,{a,1,dim},{i,1,dim},{j,1,dim},{k,1,dim}];

Do[Ricciq[i,j]=Simplify[Normal[Series[((Sum[Rudddq[l,i,l,j],{l,1,dim}]+Sum[Rudddq[l,j,l,i],{l,1,dim}])/2),{eps,0,1}]]];,{i,1,dim},{j,1,dim}];
Rq=Simplify[Normal[Series[Sum[quu[[i]][[j]]Ricciq[i,j],{i,1,dim},{j,1,dim}],{eps,0,1}]]];

Do[Rudddg[a,i,j,k]=Simplify[Normal[Series[D[Christg[i,k,a],x[[j]]]-D[Christg[i,j,a],x[[k]]]+Sum[Christg[i,k,β]Christg[β,j,a]-Christg[i,j,β]Christg[β,k,a],{β,1,dim}],{eps,0,1}]]];,{a,1,dim},{i,1,dim},{j,1,dim},{k,1,dim}];

Do[Riccig[i,j]=Simplify[Normal[Series[((Sum[Rudddg[l,i,l,j],{l,1,dim}]+Sum[Rudddg[l,j,l,i],{l,1,dim}])/2),{eps,0,1}]]];,{i,1,dim},{j,1,dim}];
Rg=Simplify[Normal[Series[Sum[guu[[i]][[j]]Riccig[i,j],{i,1,dim},{j,1,dim}],{eps,0,1}]]];

Coupling

XXX=DiagonalMatrix[Table[XXn[i,i],{i,1,dim}]];
MatrixForm[XXX]
eqs=Table[Sum[XXX[[i]][[l]]XXX[[l]][[j]],{l,1,dim}]==(guu.qdd)[[i]][[j]],{i,1,dim},{j,1,dim}]
syst=Solve[Flatten[eqs],Table[XXn[i,i],{i,1,dim}]]//Simplify;

XXXi=DiagonalMatrix[Table[XXin[i,i],{i,1,dim}]];
MatrixForm[XXXi]
eqs=Table[Sum[XXXi[[i]][[l]]XXXi[[l]][[j]],{l,1,dim}]==(quu.gdd)[[i]][[j]],{i,1,dim},{j,1,dim}]
systi=Solve[Flatten[eqs],Table[XXin[i,i],{i,1,dim}]]//Simplify;

Clear[ee,XX];
XX=XXX/.syst[[16]];
XXi=XXXi/.systi[[16]];
ee[X_]:={1,Tr[X],(1/2)(((Tr[X])^2)-Tr[X.X]),(1/6)(((Tr[X])^3)-3Tr[X]Tr[X.X]+2Tr[X.X.X]),Det[X]};

Y[n_,X_]:=Sum[(((-1))^jj)(X^(n-jj))ee[X][[jj+1]],{jj,0,n-1}]+(((-1))^n)IdentityMatrix[4]ee[X][[n+1]];

Field equations

Do[Vg[i,j]=Simplify[Normal[Series[Sum[(((-1))^n)β[n]gdd[[i]][[l]]Y[n,XX][[l]][[j]],{n,0,3},{l,1,dim}],{eps,0,1}]]];,{i,1,dim},{j,1,dim}];
Do[Vq[i,j]=Simplify[Normal[Series[Sum[(((-1))^n)β[4-n]qdd[[i]][[l]]Y[n,XXi][[l]][[j]],{n,0,3},{l,1,dim}],{eps,0,1}]]];,{i,1,dim},{j,1,dim}];

Do[Eqg[i,j]=Simplify[Normal[Series[Riccig[i,j]-(1/2)gdd[[i]][[j]]Rg+((m^4)/(mg^2))Vg[i,j],{eps,0,1}]]];,{i,1,dim},{j,1,dim}];
Do[Eqq[i,j]=Simplify[Normal[Series[Ricciq[i,j]-(1/2)qdd[[i]][[j]]Rq+((m^4)/(mf^2))Vq[i,j],{eps,0,1}]]];,{i,1,dim},{j,1,dim}];

Conservation equations

Do[Bianchi[a]=Simplify[Sum[D[Sum[Vg[a,k]guu[[k]][[b]],{k,1,dim}],x[[b]]],{b,1,dim}]+Sum[Christg[b,d,b]Vg[a,k]guu[[k]][[d]],{k,1,dim},{d,1,dim},{b,1,dim}]-Sum[Christg[d,a,b]Vg[d,k]guu[[k]][[b]],{k,1,dim},{d,1,dim},{b,1,dim}]];,{a,1,dim}]

Do[Bianchiq[a]=Simplify[Sum[D[Sum[Vq[a,k]quu[[k]][[b]],{k,1,dim}],x[[b]]],{b,1,dim}]+Sum[Christq[b,d,b]Vq[a,k]quu[[k]][[d]],{k,1,dim},{d,1,dim},{b,1,dim}]-Sum[Christq[d,a,b]Vq[d,k]quu[[k]][[b]],{k,1,dim},{d,1,dim},{b,1,dim}]];,{a,1,dim}]

Deriving the equations [no need to run this if the external file has been already generated]

We derive the eqs.[13]-[15] of Ref.[1]  using (Subscript[m, f])=(Subscript[m, g]).

mf=mg;
m=(μ^(1/2))((((β[1]+2β[2]+β[3])((1/(mf^2))+(1/(mg^2)))))^(-1/4));

β[n_]:=(((-1))^(n+1)) ((1/2) (3-n) (4-n)-(4-n) α[3]-α[4]);

(m^4)//Simplify

eqg11=Eqg[1,1]//Simplify
eqg12=Eqg[1,2]//Simplify
eqg13=Eqg[1,3]//Simplify
eqg14=Eqg[1,4]//Simplify
eqg22=Eqg[2,2]//Simplify
eqg23=Eqg[2,3]//Simplify
eqg24=Eqg[2,4]//Simplify
eqg33=Eqg[3,3]//Simplify
eqg34=Eqg[3,4]//Simplify
eqg44=Eqg[4,4]//Simplify
(***)
eqq11=Eqq[1,1]//Simplify
eqq12=Eqq[1,2]//Simplify
eqq13=Eqq[1,3]//Simplify
eqq14=Eqq[1,4]//Simplify
eqq22=Eqq[2,2]//Simplify
eqq23=Eqq[2,3]//Simplify
eqq24=Eqq[2,4]//Simplify
eqq33=Eqq[3,3]//Simplify
eqq34=Eqq[3,4]//Simplify
eqq44=Eqq[4,4]//Simplify
eq1=Bianchi[1]//Simplify
eq2=Bianchi[2]//Simplify
eq3=Bianchi[3]//Simplify
eq4=Bianchi[4]//Simplify
eeq1=Bianchiq[1]//Simplify
eeq2=Bianchiq[2]//Simplify
eeq3=Bianchiq[3]//Simplify
eeq4=Bianchiq[4]//Simplify

Check that Schwarzschild is a solution

AA[r_]:=r;
F[r_]:=(Sqrt[1-2M/r]);
B[r_]:=F[r];
p[r_]:=(Sqrt[1-2M/r]);
YY[r_]:=p[r];
rule=Union[];
Simplify[{eqg11,eqg12,eqg13,eqg14,eqg22,eqg23,eqg24,eqg33,eqg34,eeq2,eq2}//.rule]
Simplify[{eqq11,eqq12,eqq13,eqq14,eqq22,eqq23,eqq24,eqq33,eqq34}//.rule]
Clear[AA,F,B,p,YY]

Derivation of Eqs. [13]-[15]  (the juction of these solutions give the final result which is given in the next section)

Pp=Solve[{eq2==0,eqq22==0},{p[r],p'[r]}][[1]][[1]]//Simplify
rule1=Union[{Pp,D[Pp,r]},{}];

e1=eqg11//.rule1//Simplify;
e2=eqg22//.rule1//Simplify;
FB=Solve[{e1==0,e2==0},{F'[r],B'[r]}][[1]]//Simplify
rule2=Union[rule1,{FB[[1]],FB[[2]],D[FB[[1]],r],D[FB[[2]],r]},{}];

e3=eqq11//Simplify;
e4=eqq22//Simplify;
AY=Solve[{e3==0,e4==0},{AA'[r],YY'[r]}][[1]]//Simplify

Run this to check that Eqs. [13]-[15] of Ref.[1] satisfy field equations

Eqs. [15]-[17] of Ref.[1]

rule={B'[r]->(1/(4 r B[r]))(2+6 (r^2) (μ^2)-2 ((B[r])^2)-4 (r^2) (μ^2) α[3]-(r^2) (μ^2) α[4]-(μ^2) ((AA[r])^2) (-1+2 α[3]+α[4])+2 r (μ^2) AA[r] (-3+3 α[3]+α[4])+((μ^2) (((AA[r])^2) (α[3]+α[4])-2 r AA[r] (-1+2 α[3]+α[4])+(r^2) (-3+3 α[3]+α[4])) (24 r ((AA[r])^3) ((B[r])^4) YY[r] (((AA[r])^3) (((α[3]+α[4]))^2)+r ((AA[r])^2) (α[3]-2 ((α[3])^2)+α[4]-3 α[3] α[4]-((α[4])^2))-(r^2) AA[r] (2+5 ((α[3])^2)-α[4]+((α[4])^2)+α[3] (-5+4 α[4]))+(r^3) (3+6 ((α[3])^2)-4 α[4]+((α[4])^2)+α[3] (-9+5 α[4])))-4 r ((AA[r])^2) ((B[r])^3) (-(μ^2) ((AA[r])^6) α[4] (((α[3]+α[4]))^2)+6 (r^4) (-1+((YY[r])^2)) (((-3+3 α[3]+α[4]))^2)+r (μ^2) ((AA[r])^5) (3 ((α[3])^3)+13 ((α[3])^2) α[4]+((α[4])^2) (-2+5 α[4])+α[3] α[4] (-2+15 α[4]))+(r^3) AA[r] (-3+3 α[3]+α[4]) (12 (-1+2 α[3]+α[4])+(r^2) (μ^2) (2+5 ((α[3])^2)-α[4]+((α[4])^2)+α[3] (-5+4 α[4])))+2 ((AA[r])^4) (-12 (r^2) (μ^2) ((α[3])^3)+((α[3])^2) (1+(r^2) (μ^2) (6-25 α[4]))+α[3] (2+(r^2) (μ^2) (7-20 α[4])) α[4]+3 ((YY[r])^2) (((α[3]+α[4]))^2)+α[4] (α[4]+(r^2) (μ^2) (2+5 α[4]-5 ((α[4])^2))))-(r^2) ((AA[r])^2) (8+48 (r^2) (μ^2) ((α[3])^3)-4 α[4]-3 (r^2) (μ^2) α[4]+4 ((α[4])^2)-14 (r^2) (μ^2) ((α[4])^2)+5 (r^2) (μ^2) ((α[4])^3)+12 ((YY[r])^2) (2+5 ((α[3])^2)-α[4]+((α[4])^2)+α[3] (-5+4 α[4]))+((α[3])^2) (20+(r^2) (μ^2) (-72+61 α[4]))+2 α[3] (-10+8 α[4]+(r^2) (μ^2) (12-25 α[4]+15 ((α[4])^2))))+2 r ((AA[r])^3) (27 (r^2) (μ^2) ((α[3])^3)+((α[3])^2) (-4+(r^2) (μ^2) (-27+41 α[4]))+α[4] (2-2 α[4]+(r^2) (μ^2) (-6-9 α[4]+5 ((α[4])^2)))+α[3] (2-6 α[4]+(r^2) (μ^2) (3-21 α[4]+25 ((α[4])^2)))))+2 AA[r] ((B[r])^2) YY[r] ((μ^2) ((AA[r])^8) (((α[3]+α[4]))^3)-6 r (μ^2) ((AA[r])^7) (((α[3]+α[4]))^2) (-1+2 α[3]+α[4])-3 (r^8) (μ^2) (((-3+3 α[3]+α[4]))^3)+4 (r^5) AA[r] (((-3+3 α[3]+α[4]))^2) (-3+3 ((YY[r])^2)+5 (r^2) (μ^2) (-1+2 α[3]+α[4]))+(r^2) (μ^2) ((AA[r])^6) (46 ((α[3])^3)+((α[3])^2) (-49+96 α[4])+α[4] (4-39 α[4]+10 ((α[4])^2))+α[3] (4-88 α[4]+60 ((α[4])^2)))-(r^4) ((AA[r])^2) (-3+3 α[3]+α[4]) (-12 (-1+2 α[3]+α[4])+12 ((YY[r])^2) (-1+2 α[3]+α[4])+(r^2) (μ^2) (81+214 ((α[3])^2)-101 α[4]+54 ((α[4])^2)+α[3] (-221+214 α[4])))+(r^2) ((AA[r])^4) (4 (2 ((α[3])^2)+(-1+α[4]) α[4]+α[3] (-1+3 α[4]))+12 ((YY[r])^2) (2 ((α[3])^2)+(-1+α[4]) α[4]+α[3] (-1+3 α[4]))+(r^2) (μ^2) (108-204 ((α[3])^3)+((α[3])^2) (201-330 α[4])+39 α[4]+5 ((α[4])^2)-50 ((α[4])^3)+α[3] (-207+24 α[4]-220 ((α[4])^2))))-2 r ((AA[r])^5) (4 (((α[3]+α[4]))^2)+(r^2) (μ^2) (12+14 ((α[3])^3)+18 α[4]-38 ((α[4])^2)-4 ((α[4])^3)+((α[3])^2) (-39+28 α[4])+α[3] (-12-101 α[4]+4 ((α[4])^2))))+2 (r^3) ((AA[r])^3) (-6 ((YY[r])^2) (2+5 ((α[3])^2)-α[4]+((α[4])^2)+α[3] (-5+4 α[4]))+2 (-6-3 ((α[3])^2)-9 α[4]+((α[4])^2)+α[3] (3+4 α[4]))+(r^2) (μ^2) (-105+280 ((α[3])^3)+79 α[4]-99 ((α[4])^2)+37 ((α[4])^3)+((α[3])^2) (-442+417 α[4])+α[3] (301-379 α[4]+216 ((α[4])^2)))))+r B[r] ((r^10) (μ^4) (((-3+3 α[3]+α[4]))^4)-(μ^4) ((AA[r])^10) (((α[3]+α[4]))^2) (((α[3])^2)-2 α[3] α[4]-(-2+α[4]) α[4])-8 (r^7) (μ^2) AA[r] (((-3+3 α[3]+α[4]))^3) (-1+(r^2) (μ^2) (-1+2 α[3]+α[4]))-2 r (μ^4) ((AA[r])^9) (2 ((α[3])^4)+2 ((α[3])^2) α[4] (-7+16 α[4])+((α[3])^3) (-1+18 α[4])+2 ((α[4])^2) (1-4 α[4]+2 ((α[4])^2))+α[3] α[4] (2-21 α[4]+20 ((α[4])^2)))+(μ^2) ((AA[r])^8) (((α[3]+α[4]))^2) (4-8 α[3]-8 α[4]-4 ((YY[r])^2) (-1+2 α[3]+α[4])+(r^2) (μ^2) (20+101 ((α[3])^2)-68 α[4]+29 ((α[4])^2)+2 α[3] (-52+57 α[4])))+(r^4) ((AA[r])^2) (((-3+3 α[3]+α[4]))^2) (-8-8 (r^2) (μ^2) (1+6 α[3]+4 α[4])-8 ((YY[r])^2) (-1+(r^2) (μ^2) (-6+4 α[3]+α[4]))+(r^4) (μ^4) (35+115 ((α[3])^2)-68 α[4]+29 ((α[4])^2)+2 α[3] (-63+59 α[4])))-2 r (μ^2) ((AA[r])^7) (250 (r^2) (μ^2) ((α[3])^4)+α[4] (-4+(18+47 (r^2) (μ^2)) α[4]-16 (1+6 (r^2) (μ^2)) ((α[4])^2)+32 (r^2) (μ^2) ((α[4])^3))+((α[3])^3) (-28+(r^2) (μ^2) (-387+586 α[4]))-2 ((YY[r])^2) (29 ((α[3])^3)+((α[3])^2) (-29+57 α[4])+α[4] (8-13 α[4]+7 ((α[4])^2))+α[3] (8-42 α[4]+35 ((α[4])^2)))+((α[3])^2) (22-72 α[4]+(r^2) (μ^2) (165-676 α[4]+532 ((α[4])^2)))+α[3] (-4+40 α[4]-60 ((α[4])^2)+(r^2) (μ^2) (-18+164 α[4]-427 ((α[4])^2)+216 ((α[4])^3))))-2 r ((AA[r])^5) (8 (2 ((α[3])^2)+(-1+α[4]) α[4]+α[3] (-1+3 α[4]))+4 (r^2) (μ^2) (-9+11 ((α[3])^3)+30 α[4]-8 ((α[4])^2)-5 ((α[4])^3)-((α[3])^2) (64+3 α[4])+α[3] (51-56 α[4]-15 ((α[4])^2)))-4 (r^2) (μ^2) ((YY[r])^2) (-72+211 ((α[3])^3)+99 α[4]-62 ((α[4])^2)+15 ((α[4])^3)+((α[3])^2) (-422+253 α[4])+α[3] (291-316 α[4]+105 ((α[4])^2)))+(r^4) (μ^4) (90+910 ((α[3])^4)-246 α[4]+392 ((α[4])^2)-272 ((α[4])^3)+56 ((α[4])^4)+((α[3])^3) (-2227+1754 α[4])+2 ((α[3])^2) (974-1539 α[4]+662 ((α[4])^2))+α[3] (-717+1622 α[4]-1551 ((α[4])^2)+448 ((α[4])^3))))-2 (r^3) ((AA[r])^3) (-3+3 α[3]+α[4]) (-8 (-1+2 α[3]+α[4])-6 (r^2) (μ^2) (-15+2 ((α[3])^2)+5 α[4]+4 ((α[4])^2)+α[3] (23+10 α[4]))+(r^4) (μ^4) (-77+266 ((α[3])^3)+151 α[4]-128 ((α[4])^2)+32 ((α[4])^3)+((α[3])^2) (-493+404 α[4])+α[3] (316-501 α[4]+200 ((α[4])^2)))-2 ((YY[r])^2) (8 (-1+2 α[3]+α[4])+(r^2) (μ^2) (111+103 ((α[3])^2)-62 α[4]+11 ((α[4])^2)+α[3] (-206+66 α[4]))))+2 ((AA[r])^6) (4 (((α[3]+α[4]))^2)-24 (r^2) (μ^2) (((α[3]+α[4]))^2) (-1+2 α[3]+α[4])+(r^4) (μ^4) (12+611 ((α[3])^4)-53 α[4]+174 ((α[4])^2)-190 ((α[4])^3)+49 ((α[4])^4)+2 ((α[3])^3) (-623+622 α[4])+((α[3])^2) (825-1824 α[4]+1000 ((α[4])^2))+2 α[3] (-99+351 α[4]-492 ((α[4])^2)+182 ((α[4])^3)))-4 ((YY[r])^2) (-3 (((α[3]+α[4]))^2)+2 (r^2) (μ^2) (-6+42 ((α[3])^3)+15 α[4]-14 ((α[4])^2)+5 ((α[4])^3)+((α[3])^2) (-63+61 α[4])+α[3] (33-59 α[4]+30 ((α[4])^2)))))+2 (r^2) ((AA[r])^4) (-2 ((YY[r])^2) (16 (3 ((α[3])^2)+(-3+α[4]) α[4]+α[3] (-3+4 α[4]))+(r^2) (μ^2) (-327+518 ((α[3])^3)+353 α[4]-151 ((α[4])^2)+25 ((α[4])^3)+((α[3])^2) (-1295+549 α[4])+2 α[3] (556-435 α[4]+100 ((α[4])^2))))+(r^2) (μ^2) (2 (-93+70 ((α[3])^3)+154 α[4]-27 ((α[4])^2)-10 ((α[4])^3)+((α[3])^2) (-355+18 α[4])+α[3] (358-238 α[4]-30 ((α[4])^2)))+(r^2) (μ^2) (216+957 ((α[3])^4)-474 α[4]+557 ((α[4])^2)-288 ((α[4])^3)+49 ((α[4])^4)+2 ((α[3])^3) (-1293+914 α[4])+((α[3])^2) (2625-3624 α[4]+1322 ((α[4])^2))+2 α[3] (-606+1153 α[4]-883 ((α[4])^2)+210 ((α[4])^3))))))+AA[r] YY[r] ((μ^4) ((AA[r])^10) (((α[3]+α[4]))^3) (-1+2 α[3]+α[4])-8 r (μ^4) ((AA[r])^9) (((α[3]-2 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(μ^2) ((α[4])^2)+4 B[r] YY[r] (-1+2 α[3]+α[4])-2 ((YY[r])^2) (-1+2 α[3]+α[4]))-2 r (μ^2) ((AA[r])^3) (((α[3])^2)+4 α[3] α[4]+α[4] (-1+2 α[4])))))/(4 B[r] (r B[r]+((μ^2) AA[r] YY[r] (((((AA[r])^2) (α[3]+α[4])-2 r AA[r] (-1+2 α[3]+α[4])+(r^2) (-3+3 α[3]+α[4])))^2))/((μ^2) ((AA[r])^4) α[4] (α[3]+α[4])+((AA[r])^2) (-2 α[3]-5 (r^2) (μ^2) α[3]+10 (r^2) (μ^2) ((α[3])^2)-2 α[4]-8 (r^2) (μ^2) α[4]+18 (r^2) (μ^2) α[3] α[4]+6 (r^2) (μ^2) ((α[4])^2)+8 B[r] YY[r] (α[3]+α[4])-6 ((YY[r])^2) (α[3]+α[4]))+(r^2) (-3+3 α[3]+α[4]) (-2+2 ((YY[r])^2)+(r^2) (μ^2) (-1+2 α[3]+α[4]))-2 r AA[r] (2+(r^2) (μ^2)-4 α[3]-7 (r^2) (μ^2) α[3]+7 (r^2) (μ^2) ((α[3])^2)-2 α[4]-5 (r^2) (μ^2) α[4]+8 (r^2) (μ^2) α[3] α[4]+2 (r^2) (μ^2) ((α[4])^2)+4 B[r] YY[r] (-1+2 α[3]+α[4])-2 ((YY[r])^2) (-1+2 α[3]+α[4]))-2 r (μ^2) ((AA[r])^3) (((α[3])^2)+4 α[3] α[4]+α[4] (-1+2 α[4]))))),YY'[r]->(1/(4 AA[r] B[r]))((μ^2) (((AA[r])^2) (α[3]+α[4])-2 r AA[r] (-1+2 α[3]+α[4])+(r^2) (-3+3 α[3]+α[4]))-((-2+2 ((YY[r])^2)+(r^2) (μ^2) (-1+2 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α[3]+α[4])+(r^2) (μ^2) (188 ((α[3])^4)+4 ((α[3])^3) (-47+110 α[4])+α[3] α[4] (13-148 α[4]+184 ((α[4])^2))+((α[3])^2) (44-282 α[4]+412 ((α[4])^2))+α[4] (6-11 α[4]-32 ((α[4])^2)+32 ((α[4])^3))))+(r^2) (μ^2) ((AA[r])^6) (888 (r^2) (μ^2) ((α[3])^4)+2 ((α[3])^3) (58+(r^2) (μ^2) (-657+925 α[4]))-2 ((YY[r])^2) (222 ((α[3])^3)+((α[3])^2) (-207+440 α[4])+α[4] (42-107 α[4]+54 ((α[4])^2))+α[3] (42-314 α[4]+272 ((α[4])^2)))+((α[3])^2) (-110+256 α[4]+(r^2) (μ^2) (615-1814 α[4]+1566 ((α[4])^2)))+2 α[4] (10-37 α[4]+20 ((α[4])^2)+(r^2) (μ^2) (2+28 α[4]-92 ((α[4])^2)+49 ((α[4])^3)))+α[3] (4 (5-46 α[4]+45 ((α[4])^2))+(r^2) (μ^2) (-86+413 α[4]-924 ((α[4])^2)+630 ((α[4])^3))))+(r^4) ((AA[r])^2) (-3+3 α[3]+α[4]) (16 (-1+2 α[3]+α[4])-24 ((YY[r])^4) (-1+2 α[3]+α[4])+2 (r^2) (μ^2) (17-78 ((α[3])^2)+α[3] (35-102 α[4])+45 α[4]-28 ((α[4])^2))+(r^4) (μ^4) (-33+232 ((α[3])^3)+97 α[4]-97 ((α[4])^2)+29 ((α[4])^3)+((α[3])^2) (-370+351 α[4])+α[3] (193-377 α[4]+176 ((α[4])^2)))+2 ((YY[r])^2) (4 (-1+2 α[3]+α[4])+(r^2) (μ^2) (-35+6 ((α[3])^2)-9 α[4]+10 ((α[4])^2)+α[3] (37+30 α[4]))))-2 r ((AA[r])^5) (-4 (((α[3]+α[4]))^2)+2 (r^2) (μ^2) (-6+130 ((α[3])^3)+55 α[4]-69 ((α[4])^2)+17 ((α[4])^3)+3 ((α[3])^2) (-65+71 α[4])+α[3] (75-242 α[4]+108 ((α[4])^2)))+(r^4) (μ^4) (652 ((α[3])^4)+16 ((α[3])^3) (-77+81 α[4])+12 ((α[3])^2) (70-139 α[4]+87 ((α[4])^2))+α[3] (-231+642 α[4]-848 ((α[4])^2)+392 ((α[4])^3))+2 (9-33 α[4]+69 ((α[4])^2)-80 ((α[4])^3)+28 ((α[4])^4)))+((YY[r])^2) (12 (((α[3]+α[4]))^2)-2 (r^2) (μ^2) (-18+300 ((α[3])^3)+103 α[4]-117 ((α[4])^2)+37 ((α[4])^3)+15 ((α[3])^2) (-28+31 α[4])+α[3] (165-467 α[4]+228 ((α[4])^2)))))+(r^2) ((AA[r])^4) (-16 (2 ((α[3])^2)+(-1+α[4]) α[4]+α[3] (-1+3 α[4]))+24 ((YY[r])^4) (2 ((α[3])^2)+(-1+α[4]) α[4]+α[3] (-1+3 α[4]))+2 (r^2) (μ^2) (-90+396 ((α[3])^3)+279 α[4]-149 ((α[4])^2)+10 ((α[4])^3)+((α[3])^2) (-885+446 α[4])+α[3] (549-774 α[4]+140 ((α[4])^2)))+(r^4) (μ^4) (126+1348 ((α[3])^4)-330 α[4]+490 ((α[4])^2)-380 ((α[4])^3)+98 ((α[4])^4)+((α[3])^3) (-2920+2682 α[4])+((α[3])^2) (2413-4110 α[4]+2098 ((α[4])^2))+α[3] (-909+2033 α[4]-2110 ((α[4])^2)+742 ((α[4])^3)))-2 ((YY[r])^2) (-12 (2 ((α[3])^2)+(-1+α[4]) α[4]+α[3] (-1+3 α[4]))+(r^2) (μ^2) (-162+796 ((α[3])^3)+399 α[4]-245 ((α[4])^2)+50 ((α[4])^3)+((α[3])^2) (-1529+970 α[4])+α[3] (903-1276 α[4]+380 ((α[4])^2)))))-2 (r^3) ((AA[r])^3) (12 ((YY[r])^4) (2+5 ((α[3])^2)-α[4]+((α[4])^2)+α[3] (-5+4 α[4]))-2 ((YY[r])^2) (2 (-6-3 ((α[3])^2)-9 α[4]+((α[4])^2)+α[3] (3+4 α[4]))+(r^2) (μ^2) (-129+222 ((α[3])^3)+153 α[4]-46 ((α[4])^2)+4 ((α[4])^3)+((α[3])^2) (-579+196 α[4])+α[3] (486-349 α[4]+52 ((α[4])^2))))+(r^2) (μ^2) (2 (-87+56 ((α[3])^3)+92 α[4]+10 ((α[4])^2)-13 ((α[4])^3)-29 ((α[3])^2) (10+α[4])+α[3] (311-106 α[4]-54 ((α[4])^2)))+(r^2) (μ^2) (75+548 ((α[3])^4)-196 α[4]+265 ((α[4])^2)-160 ((α[4])^3)+32 ((α[4])^4)+12 ((α[3])^3) (-107+90 α[4])+2 ((α[3])^2) (564-941 α[4]+402 ((α[4])^2))+α[3] (-457+1057 α[4]-948 ((α[4])^2)+264 ((α[4])^3))))))))};
ruleT=Union[rule,D[rule,r],D[rule,{r,2}]];

Check (can take some time)

e1=eqg11//.ruleT//Simplify
e2=eqg22//.ruleT//Simplify
e3=eqq11//.ruleT//Simplify
e4=eqq22//.ruleT//Simplify
e5=eq2//.ruleT//Simplify 
e6=eeq2//.ruleT//Simplify

Save equations for future times

EQY=YY'[r]-(YY'[r]//.rule)//Simplify;
EQB=B'[r]-(B'[r]//.rule)//Simplify;
EQA=AA'[r]-(AA'[r]//.rule)//Simplify;
EQF=F'[r]-(F'[r]//.rule)//Simplify;
EQp=p[r]-(p[r]//.rule)//Simplify;

SetDirectory[NotebookDirectory[]];
Put[{rule,EQY,EQB,EQA,EQF,EQp},"bimetric_eqs.mx"]

Numerical Solution

We have 3 parameters (μ, α3 and α4). To be able to compute the series at the horizon easily, fixing α3 and α4 now is very efficient. Otherwise it can take a considerable amount of time.

α[3]=2;α[4]=-2;

Equations are written in the external file “bimetric_eqs.mx”

$Assumptions={r≥0,rh≥0,ah[1]>0,bh[1]>0,c>0,Cη>0,Sη>0,μ>0,{bh[1],ah[1]}∈Reals};
{r≥0,rh≥0,ah[1]>0,bh[1]>0,c>0,Cη>0,Sη>0,μ>0,(bh[1]|ah[1])∈Reals};

SetDirectory[NotebookDirectory[]];
EQY=Get["./bimetric_eqs.mx"][[2]]//Simplify;
EQB=Get["./bimetric_eqs.mx"][[3]]//Simplify;
EQA=Get["./bimetric_eqs.mx"][[4]]//Simplify;
EQF=Get["./bimetric_eqs.mx"][[5]]//Simplify;
EQp=Get["./bimetric_eqs.mx"][[6]]//Simplify;
rule=Get["./bimetric_eqs.mx"][[1]]//Simplify;

Boundary Conditions

(*These definitions help mathematica compute the series*)
B[r_]:=(Sqrt[bb[r]]);
YY[r_]:=(Sqrt[yy[r]]);
F[r_]:=(Sqrt[ff[r]]);
p[r_]:=(Sqrt[pp[r]]);
AA[r_]:=u rh+aa[r];

EQBB=Simplify[EQB(r-rh) ];
EQAA=Simplify[EQA(r-rh)];
EQYY=Simplify[EQY(r-rh)];
EQFF=Simplify[EQF(r-rh)];
EQpp=Simplify[EQp];
Clear[bb,yy,aa,B,YY,AA,F,u,μ,ff,p]

Series

ORDH=1;
rh=2;
(**)
ss=Series[({EQYY,EQAA,EQBB}//.Union[{bb-> Function[{r},Sum[ah[i](((r-rh))^i),{i,1,ORDH}]],
yy-> Function[{r},Sum[bh[i](((r-rh))^i),{i,1,ORDH}]],
aa-> Function[{r},Sum[ch[i](((r-rh))^i),{i,1,ORDH}]]}]),{r,rh,ORDH}];
eqs=Union[Table[SeriesCoefficient[ss[[1]],i]==0,{i,0,ORDH,(1/2)}],Table[SeriesCoefficient[ss[[2]],i]==0,{i,0,ORDH}],Table[SeriesCoefficient[ss[[3]],i]==0,{i,0,ORDH,(1/2)}]];
sss=Simplify[eqs//.{ah[1]->(AA^2),bh[1]->(BB^2),ch[1]->CC},{AA>0,BB>0}];
systH1=Simplify[Solve[sss,{AA,BB,CC}]];
Length[systH1]

For each choice of the parameters, there exist two branches (in most of the cases only one will give an asymptotically flat solution). Note that for some choices of coefficients the series becomes imaginary (no solutions exist).

N[(AA^2)/.systH1//.{μ->0.2,u->1.1}]
N[CC/.systH1//.{μ->0.1,u->1}]
N[(BB^2)/.systH1//.{μ->0.1,u->1}]

Integrator

u0 -> parameter u;
μ0 -> mass μ;
EPS1 -> deviation from the horizon  (rh=);
EPS2 -> infinity;
solH-> choose branch (solH=1 for assymptotically flat solutions in most cases)

INT[u0_,μ0_,EPS_,EPS2_,solH_]:=(
ruleP={AccuracyGoal->80,PrecisionGoal->13,MaxSteps->(10^6)};

param={u-> u0,μ->μ0,dh[1]->1};
rh=2;
rhn=rh(1+EPS);
INF=EPS2;

(**)
ORDH1=2;
ss=Series[({EQYY,EQAA,EQBB}//.Union[{bb-> Function[{r},Sum[ah[i](((r-rh))^i),{i,1,ORDH1}]],
yy-> Function[{r},Sum[bh[i](((r-rh))^i),{i,1,ORDH1}]],
aa-> Function[{r},Sum[ch[i](((r-rh))^i),{i,1,ORDH1}]]
},param,systH1[[solH]],{ah[1]->(AA^2),bh[1]->(BB^2),ch[1]->CC}]),{r,rh,ORDH1}];
eqs=Union[Table[SeriesCoefficient[ss[[1]],i]==0,{i,(3/2),ORDH1,(1/2)}],Table[SeriesCoefficient[ss[[2]],i]==0,{i,2,ORDH1}],Table[SeriesCoefficient[ss[[3]],i]==0,{i,(3/2),ORDH1,(1/2)}]];
yh=Union[Table[ah[i],{i,2,ORDH1}],Table[bh[i],{i,2,ORDH1}],Table[ch[i],{i,2,ORDH1}]];
systH=Solve[eqs,yh][[1]];

(**)
ssf=Series[({EQFF,EQpp}//.Union[{bb-> Function[{r},Sum[ah[i](((r-rh))^i),{i,1,ORDH1}]],
yy-> Function[{r},Sum[bh[i](((r-rh))^i),{i,1,ORDH1}]],
aa-> Function[{r},Sum[ch[i](((r-rh))^i),{i,1,ORDH1}]],
ff-> Function[{r},Sum[dh[i](((r-rh))^i),{i,1,ORDH1}]],pp-> Function[{r},Sum[dh[1]eh[i](((r-rh))^i),{i,1,ORDH1}]]},param,systH1[[solH]],systH,{ah[1]->(AA^2),bh[1]->(BB^2),ch[1]->CC}]),{r,rh,ORDH1}];
eqsf=Union[Table[SeriesCoefficient[ssf[[1]],i]==0,{i,(3/2),ORDH1,(1/2)}],Table[SeriesCoefficient[ssf[[2]],i]==0,{i,(1/2),ORDH1,(1/2)}]];
yhf=Union[Table[dh[i],{i,2,ORDH1}],Table[eh[i],{i,1,ORDH1}]];
systHf=Solve[(eqsf),yhf][[1]];
(**)


RULEH=Union[param,systH1[[solH]],systH,systHf,{ah[1]->(AA^2),bh[1]->(BB^2),ch[1]->CC}];
(***INTEGRATION FROM THE HORIZON****)
Bg=(Sqrt[Sum[ah[i](((r-rh))^i),{i,1,ORDH1}]])//.RULEH;YYg=(Sqrt[Sum[bh[i](((r-rh))^i),{i,1,ORDH1}]])//.RULEH;AAg=u r+Sum[ch[i](((r-rh))^i),{i,1,ORDH1}]//.RULEH;Fg=(Sqrt[Sum[dh[i](((r-rh))^i),{i,1,ORDH1}]])//.RULEH;pg=(Sqrt[Sum[dh[1]eh[i](((r-rh))^i),{i,1,ORDH1}]])//.RULEH;

(**)
BCs={B[rhn]==Bg/.r->rhn,YY[rhn]==YYg/.r->rhn,AA[rhn]==AAg/.r->rhn,F[rhn]==Fg/.r->rhn}//.param;
EQs={EQA==0,EQB==0,EQY==0,EQF==0}//.param;


sol1=NDSolve[Union[EQs,BCs],{B[r],YY[r],AA[r],F[r]},{r,rhn,INF},ruleP];
Bn=B[r]/.sol1[[1]];
YYn=YY[r]/.sol1[[1]];
AAn=AA[r]/.sol1[[1]];
Fn=F[r]/.sol1[[1]];
dAAn=D[AAn,r];
dBn=D[Bn,r];
ddBn=D[dBn,r];
pn=p[r]//.rule//.Union[sol1[[1]],param,{F'[r]-> D[Fn,r]}];


(*match to expansion at infinity*)
int=Solve[{(-(((E^(-r μ)) (Aa (E^(r μ))+Bb (1+r μ (1+r μ))))/(2 (r^2) (Sqrt[1+((Aa+Bb (E^(-r μ)) (1+r μ))/r)])))//.μ->μ0)== dBn/.r-> INF,((Sqrt[1+((Bb (E^(-r μ)) (1+r μ)+Aa)/r)])//.μ-> μ0)== Bn/.r-> INF},{Aa,Bb}][[1]];
AAa=Aa//.int;
BBb=Bb//.int;

dAinf=1-(((E^(-r μ)) (1+r μ) (2+(r^2) (μ^2)) BBb)/((r^3) (μ^2)))//.{μ->μ0};

(*compute the temperatures*)
(*Tf=(1/(4π ))(Sqrt[((bh[1]eh[1])/((ch[1])^2))])//.RULEH;
Tg=(1/(4π))(Sqrt[ah[1]])//.RULEH;*)

{dAAn/.r-> INF,(dAAn/.r-> INF)-(dAinf/.r-> INF)});

MemoryConstrained[INT[1,.1,(10^(-3)),10,1],(10^8)]

guess=1.186;(*μ=0.4*)
f4[y1_?NumericQ]:=INT[y1,0.4,(10^(-6)),20,1][[2]];
f5[y1_?NumericQ]:=INT[y1,0.4,(10^(-6)),20,1][[1]];
SOL1=FindRoot[f4[y1]==0,{y1,guess},WorkingPrecision->10] (*Using expansion at infinity*)
SOL2=FindRoot[f5[y1]==1,{y1,guess},WorkingPrecision->10] 
(*Without expansion at infinity*)

yn1=y1/.SOL1;yn2=y1/.SOL2;
check={f4[yn1],f5[yn2]}

plot=Plot[{pn,Fn,YYn,Bn,dAAn},{r,rhn,INF},Frame-> True,PlotRange->All,FrameLabel->{"2r/(Subscript[r, H])","U '"}]