Relativity lecture

A few minutes from the last class:

Relativity Computations


Variation 1 
r = {(5+2 Sin[v])*Cos[u], (5+2 Sin[v])*Sin[u], Cos[v]};
ru=D[r,u];rv=D[r,v];n=Cross[ru,rv];n=n/Sqrt[n.n];
nu=D[n,u];nv=D[n,v];drT={ru,rv};dr=Transpose[drT];
g=drT.dr;gi=Inverse[g];dnT={nu,nv};dn=Transpose[dnT];
h=-dnT.dr;  e=dnT.dn; K=Det[h]/Det[g];
f=FullSimplify[K*Sqrt[Det[g]]];
CheckGaussBonnet:=Integrate[f,{u,0,2Pi},{v,0,Pi}];

XX={u,v};X[a_]:=XX[[a]];d=2; G[a_,b_]:=g[[a,b]]; 
c[i_,j_,k_]:=(D[G[j,k],X[i]]+D[G[k,i],X[j]]-D[G[i,j],X[k]])/2;
Ch[i_,j_,k_]:=Sum[gi[[k,l]]*c[i,j,l],{l,d}]; 
Riemann[i_,k_,j_,s_]:=D[Ch[i,j,s],X[k]]-D[Ch[i,k,s],X[j]]+
   Sum[Ch[i,j,r]*Ch[r,k,s]-Ch[i,k,r]*Ch[r,j,s],{r,d}];
R[i_,k_]:=Sum[Riemann[i,j,k,j],{j,d}];
S = Sum[gi[[i,j]] R[i,j],{i,d},{j,d}];
Einstein[i_,j_]:=R[i,j]-S G[i,j]/2; 
Table[Simplify[Riemann[i,j,k,l]],{i,d},{j,d},{k,d},{l,d}]
Table[Simplify[Einstein[i,j]],{i,d},{j,d}]

Variation 2 
g={{-(1-2M/r), 0         ,    0   ,           0    },
   {       0 , (1-2M/r)^(-1), 0   ,           0    },
   {       0 , 0         ,    r^2 ,           0    },
   {       0 , 0         ,    0   , r^2*Sin[phi]^2 }};
gi=Simplify[Inverse[g]]; 
XX={t,r,phi,theta};X[a_]:=XX[[a]];d=4; G[a_,b_]:=g[[a,b]]; 
c[i_,j_,k_]:=(D[G[j,k],X[i]]+D[G[k,i],X[j]]-D[G[i,j],X[k]])/2;
Ch[i_,j_,k_]:=Sum[gi[[k,l]]*c[i,j,l],{l,d}]; 
Riemann[i_,k_,j_,s_]:=D[Ch[i,j,s],X[k]]-D[Ch[i,k,s],X[j]]+
       Sum[Ch[i,j,m]*Ch[m,k,s]-Ch[i,k,m]*Ch[m,j,s],{m,d}];
R[i_,k_]:=Sum[Riemann[i,j,k,j],{j,d}];
S = Sum[gi[[i,j]] R[i,j],{i,d},{j,d}];
Einstein[i_,j_]:=R[i,j]-S G[i,j]/2; 
Table[Simplify[Riemann[i,j,k,l]],{i,d},{j,d},{k,d},{l,d}]
Table[Simplify[Einstein[i,j]],{i,d},{j,d}]

Variation 3 
g={{-(1-2M/r + e^2/r^2), 0         ,    0   ,           0    },
   {       0 , (1-2M/r+e^2/r^2)^(-1), 0   ,           0    },
   {       0 , 0         ,    r^2 ,           0    },
   {       0 , 0         ,    0   , r^2*Sin[phi]^2 }};
gi=Simplify[Inverse[g]]; 
XX={t,r,phi,theta};X[a_]:=XX[[a]];d=4; G[a_,b_]:=g[[a,b]]; 
c[i_,j_,k_]:=(D[G[j,k],X[i]]+D[G[k,i],X[j]]-D[G[i,j],X[k]])/2;
Ch[i_,j_,k_]:=Sum[gi[[k,l]]*c[i,j,l],{l,d}]; 
Riemann[i_,k_,j_,s_]:=D[Ch[i,j,s],X[k]]-D[Ch[i,k,s],X[j]]+
       Sum[Ch[i,j,m]*Ch[m,k,s]-Ch[i,k,m]*Ch[m,j,s],{m,d}];
R[i_,k_]:=Sum[Riemann[i,j,k,j],{j,d}];
S = Sum[gi[[i,j]] R[i,j],{i,d},{j,d}];
Einstein[i_,j_]:=R[i,j]-S G[i,j]/2; 
Table[Simplify[Riemann[i,j,k,l]],{i,d},{j,d},{k,d},{l,d}]
Table[Simplify[Einstein[i,j]],{i,d},{j,d}]

Variation 4 
r={Cos[u] Cos[w],Sin[u] Cos[w], Cos[v] Sin[w],Sin[v] Sin[w]};
ru=D[r,u]; rv=D[r,v]; rw=D[r,w];   A=Transpose[{ru,rv,rw}]; 
g=Transpose[A].A;  gi=Inverse[g];  n=r; 
volume:=Integrate[Sqrt[Det[g]],{u,0,2Pi},{v,0,Pi},{w,-Pi/2,Pi/2}]
XX={u,v,w};X[a_]:=XX[[a]];d=3; G[a_,b_]:=g[[a,b]]; 
c[i_,j_,k_]:=(D[G[j,k],X[i]]+D[G[k,i],X[j]]-D[G[i,j],X[k]])/2;
Ch[i_,j_,k_]:=Sum[gi[[k,l]]*c[i,j,l],{l,d}]; 
Riemann[i_,k_,j_,s_]:=D[Ch[i,j,s],X[k]]-D[Ch[i,k,s],X[j]]+
   Sum[Ch[i,j,r]*Ch[r,k,s]-Ch[i,k,r]*Ch[r,j,s],{r,d}];
R[i_,k_]:=Sum[Riemann[i,j,k,j],{j,d}];
S = Sum[gi[[i,j]] R[i,j],{i,d},{j,d}];
Einstein[i_,j_]:=R[i,j]-S G[i,j]/2; 
Table[Simplify[Riemann[i,j,k,l]],{i,d},{j,d},{k,d},{l,d}]
Table[Simplify[Einstein[i,j]],{i,d},{j,d}]

Variation 5 
r = { Sin[v] Cos[u], Sin[v] Sin[u],Cos[v] + Log[Tan[v/2]]}; 
ru=D[r,u]; rv=D[r,v]; {{ru.ru,ru.rv},{rv.ru,rv.rv}}
ru=D[r,u];rv=D[r,v];n=Cross[ru,rv];n=n/Sqrt[n.n];
nu=D[n,u];nv=D[n,v];drT={ru,rv};dr=Transpose[drT];
g=drT.dr;gi=Inverse[g];dnT={nu,nv};dn=Transpose[dnT];
h=-dnT.dr;  e=dnT.dn; K=Det[h]/Det[g];
f=FullSimplify[K*Sqrt[Det[g]]];
CheckGaussBonnet:=Integrate[f,{u,0,2Pi},{v,0,Pi}];

XX={u,v};X[a_]:=XX[[a]];d=2; G[a_,b_]:=g[[a,b]]; 
c[i_,j_,k_]:=(D[G[j,k],X[i]]+D[G[k,i],X[j]]-D[G[i,j],X[k]])/2;
Ch[i_,j_,k_]:=Sum[gi[[k,l]]*c[i,j,l],{l,d}]; 
Riemann[i_,k_,j_,s_]:=D[Ch[i,j,s],X[k]]-D[Ch[i,k,s],X[j]]+
   Sum[Ch[i,j,r]*Ch[r,k,s]-Ch[i,k,r]*Ch[r,j,s],{r,d}];
R[i_,k_]:=Sum[Riemann[i,j,k,j],{j,d}];
S = Sum[gi[[i,j]] R[i,j],{i,d},{j,d}];
Einstein[i_,j_]:=R[i,j]-S G[i,j]/2; 
Table[Simplify[Riemann[i,j,k,l]],{i,d},{j,d},{k,d},{l,d}]
Table[Simplify[Einstein[i,j]],{i,d},{j,d}]
P.S. The Schwarzschild computer algebra computation appeared already (for one of the earliest Mathematica implementations) on page 97 of my text "Introduction to Geometry and geometric analysis" from the fall term 1995 taught at Caltech. But this code has been rewritten from scratch for this course (the old code had been written 29 years ago! ).