From this equation came an exact solution the
Schwarzschild metric from which the perihelion advance was calculated and agreed with observation.
The Schwarzschild metric is an exact solution for the Einstein field equations.
Since this is a vacuum solution the Einstein field equations reduce to;
Rₐₑ = 0.
For a planetary orbit θ = π /2 the metric simplifies to;
Plugging this simplified metric into vacuum field equation reduces to the general relativity equation for an orbit;
Where u = 1/r and b = r²dϕ is a constant.
Without the final term in the equation this is the equation for a Newtonian orbit around the Sun of mass M;
This has the solution;
Where e is the eccentricity or flatness of the orbit, ϕ is the angle subtended by the orbit and varies from 0 to 360 degrees, ϕ₀ is the angle where perihelion or the closest approach of the planet to the Sun occurs.
There is no direct solution for the general relativity equation for an orbit but an approximation can be obtained by substituting the solution of the Newtonian equation into the u² term of the equation.
This gives;
An approximate solution for this equation is;
where;
Δϕ increases with ϕ and is the constantly increasing correction to ϕ₀ or the perihelion advance with each orbit.
The calculated theoretical advance Δϕ per century for Mercury is around 43” and agrees with the observational result.
The calculation ‘falls into place’ using the Schwarzschild metric unlike the earlier methods used by Einstein before the field equations were fully developed.
