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Instead of solving the geodesic equations for the Schwarzschild metric, in many books (nearly in all books that I consulted), conserved quantities are looked at instead.

So take for eg. Carroll, he looks at the killing equation and extracts the equation

[tex] K_\mu \frac{dx^\mu}{d \lambda}= constant, [/tex]

and he then writes:"In addition we have another constant of the motion for geodesics", and he writes the normalization condition:

[tex] \epsilon = -g_{\mu \nu} \frac{dx^\mu}{d \lambda} \frac{dx^\mu}{d \lambda}. [/tex]

Now I don't understand why this set of equations is equivalent to the geodesic equations. And I do not understand why we are allowed to use these equations to extract information about the geodesics.

Maybe the questions are the same, but I hope you get my point.

Any help would be greatly appreciated!!

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# Killing vectors and Geodesic equations for the Schwarschild metric.

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