Question

I use equations from math side and then substitute with Christoffel2 but do not know right hand side F*e

1. is right hand side = 0 is right? if not, what is F*e? and how to assume some cases?

2. use dsolve getting a complicated solution, How to analyze and understand the solution?

   ***** Schwarzchild metric ******** with(tensor): coord := [t, r, theta, Phi]:

   g_compts:=array(symmetric,sparse,1..4,1..4):

   g_compts[1,1]:=1 - 2*G*M/(r*c^2): g_compts[2,2]:=-(1 - 2*G*M/(r*c^2))^(-1): g_compts[3,3]:=-r^2: g_compts[4,4]:=-(r^2)*sin(theta)^2:

   g1 := create([-1,-1], eval(g_compts)): g1_inv := invert( g1, 'detg' ):

   D1g := d1metric( g1, coord ):

   Cf1_1 := Christoffel1(D1g): Cf2_1 := Christoffel2(g1_inv, Cf1_1): displayGR(Christoffel2,Cf2_1):

   ---

   template := expand((t1+t2+t3+t4)^2);

   • Diff(f1(t), t)^2
   • 2*Diff(f1(t), t)*Diff(f2(t), t)
   • 2*Diff(f1(t), t)*Diff(f3(t), t)

   • 2*Diff(f1(t), t)*Diff(f4(t), t)

   • Diff(f2(t), t$2)

   • 2*Diff(f2(t), t)*Diff(f3(t), t)

   • 2*Diff(f2(t), t)*Diff(f4(t), t)

   • Diff(f3(t), t$2)

   • 2*Diff(f3(t), t)*Diff(f4(t), t)

   • Diff(f4(t), t)^2;

   ---

   ex1 := { Diff(f1(t), t$2)

   • 2*(-G*M/(r*(-r*c^2+2*G*M)))*Diff(f(t), t1)*Diff(f2(t), t) = 0,

   Diff(f2(t), t$2)

   • (-(-r*c^2+2*G*M)*G*M/(r^3*c^4))*Diff(f1(t), t)^2
   • (G*M/(r*(-r*c^2+2*G*M)))*Diff(f2(t), t$2)
   • ((-r*c^2+2*G*M)/c^2)*Diff(f3(t), t$2)
   • ((-r*c^2+2*G*M)*sin(theta)^2/c^2)*Diff(f4(t), t)^2 = 0,

   Diff(f3(t), t$2)

   • (-sin(theta)*cos(theta))*Diff(f4(t), t)^2 = 0,

   Diff(f3(t), t$2)

   • 2*(1/r)*Diff(f2(t), t)*Diff(f4(t), t)
   • 2*(cos(theta)/sin(theta))*Diff(f3(t), t)*Diff(f4(t), t) = 0

   };

   dsolve(ex1, {f1(t),f2(t),f3(t),f4(t)});

   ---