## How to analyze the solution of motion of particle with Schwarzchild metric? [closed]

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)});

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It's not clear what your question is. My first take is you're asking about basic mechanics in the Schwartzchild metric. Many basic GR textbooks cover this, like Misner, Thorne and Wheeler. But your question isn't clear and your pseudo-code is out of place on this forum. I've flagged your question to be closed as it seems both off-topic and inappropriately written for this forum. – Ryan Budney Jan 9 2011 at 3:21