Historical question: Equations of motion from lagrangian

[solveforX]

Hey, in general relativity, essentially I am asking how any metric (I.e. schwarzschild metric) was found. are the metrics derived or are they extrapolated from the correct lagrange equations of motion? If there is a derivation available, please provide a link.

thanks

 

[tom.stoer]

Hilbert showed how to derive the field equations for GR from a Lagrangian density. Soon after Einstein & Hilbert the first solutions have been found - but forgetting about the physicalcontent this is nothing else but solving non-linear partial differential equations; of course it can be done but there is no complete theory as for linear partial differential equations

 

[Mentz114]

Hey, in general relativity, essentially I am asking how any metric (I.e. schwarzschild metric) was found. are the metrics derived or are they extrapolated from the correct lagrange equations of motion? If there is a derivation available, please provide a link.

thanks

The Einstein tensor comes from an action which is extremised according to the least action principle.
http://en.wikipedia.org/wiki/Einstein_field_equations

The Schwarzschild metric can then be derived using nothing more than the spherical symmetry and the requirement that the Einstein tensor be zero.
http://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

Given a solution of the EFE the equations of motion can then be found using Euler-Lagrange method for the Lagrangian of a body moving in curved space.
Any textbook on GR has this derivation.