Timelike geodesics in Schwarzschild Metric.

[Nilupa]

Please explain me how to derive the Timelike geodesics in Schwarzschild Metric.

Thank you.

 

[Bill_K]

To derive the geodesics for any metric, write out and solve the Euler-lagrange equations for the Lagrangian F = g_μν(dx^μ/ds)(dx^ν/ds).

For Schwarzschild this is (in the equatorial plane) F = (1-2m/r)^-1(dr/ds)² + r²(dθ/ds)² - (1-2m/r)(dt/ds)² . There are three first integrals: one for the t coordinate, one for the θ coordinate, and F itself. The result can be expressed as a differential equation for r(θ).

 

[Nilupa]

Thank you so much Bill K.

I derived the equations.