How to analyze the solution of motion of particle with Schwarzchild metric? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T16:53:46Z http://mathoverflow.net/feeds/question/51513 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51513/how-to-analyze-the-solution-of-motion-of-particle-with-schwarzchild-metric How to analyze the solution of motion of particle with Schwarzchild metric? MetricAnalyst 2011-01-09T02:24:17Z 2011-01-09T02:24:17Z <p>I use equations from math side and then substitute with Christoffel2 but do not know right hand side F*e</p> <ol> <li>is right hand side = 0 is right? if not, what is F*e? and how to assume some cases?</li> <li><p>use dsolve getting a complicated solution, How to analyze and understand the solution?</p> <p><em><strong></em>**<em>*</em>**<em></strong> Schwarzchild metric <strong></em>**<em>*</em>**<em>*</em>**</strong> with(tensor): coord := [t, r, theta, Phi]:</p> <p>g_compts:=array(symmetric,sparse,1..4,1..4):</p> <p>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:</p> <p>g1 := create([-1,-1], eval(g_compts)): g1_inv := invert( g1, 'detg' ):</p> <p>D1g := d1metric( g1, coord ):</p> <p>Cf1_1 := Christoffel1(D1g): Cf2_1 := Christoffel2(g1_inv, Cf1_1): displayGR(Christoffel2,Cf2_1):</p> <hr> <p>template := expand((t1+t2+t3+t4)^2);</p> <ul> <li>Diff(f1(t), t)^2</li> <li>2*Diff(f1(t), t)*Diff(f2(t), t) </li> <li>2*Diff(f1(t), t)*Diff(f3(t), t) </li> <li><p>2*Diff(f1(t), t)*Diff(f4(t), t) </p></li> <li><p>Diff(f2(t), t\$2)</p></li> <li>2*Diff(f2(t), t)*Diff(f3(t), t) </li> <li><p>2*Diff(f2(t), t)*Diff(f4(t), t) </p></li> <li><p>Diff(f3(t), t\$2)</p></li> <li><p>2*Diff(f3(t), t)*Diff(f4(t), t) </p></li> <li><p>Diff(f4(t), t)^2;</p></li> </ul> <hr> <p>ex1 := { Diff(f1(t), t\$2)</p> <ul> <li>2*(-G*M/(r*(-r*c^2+2*G*M)))*Diff(f(t), t1)*Diff(f2(t), t) = 0,</li> </ul> <p>Diff(f2(t), t\$2)</p> <ul> <li>(-(-r*c^2+2*G*M)*G*M/(r^3*c^4))*Diff(f1(t), t)^2</li> <li>(G*M/(r*(-r*c^2+2*G*M)))*Diff(f2(t), t\$2)</li> <li>((-r*c^2+2*G*M)/c^2)*Diff(f3(t), t\$2)</li> <li>((-r*c^2+2*G*M)*sin(theta)^2/c^2)*Diff(f4(t), t)^2 = 0,</li> </ul> <p>Diff(f3(t), t\$2)</p> <ul> <li>(-sin(theta)*cos(theta))*Diff(f4(t), t)^2 = 0,</li> </ul> <p>Diff(f3(t), t\$2)</p> <ul> <li>2*(1/r)*Diff(f2(t), t)*Diff(f4(t), t) </li> <li>2*(cos(theta)/sin(theta))*Diff(f3(t), t)*Diff(f4(t), t) = 0</li> </ul> <p>};</p> <p>dsolve(ex1, {f1(t),f2(t),f3(t),f4(t)});</p> <hr></li> </ol>