Einstein Relatively Easy

Advance of the perihelion of Mercury

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Category: General Relativity

Last Updated: 20 December 2022

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      Another test that Einstein suggested for testing his gravitational theory was the precession of perihelia. This reflects the fact that noncircular orbits in General Relativity are not perfect closed ellipses; to a good approximation they are ellipses that precess.

The strategy is like as in our previous article about light deflection to describe the evolution of the radial coordinate r as a function of the angular coordinate Φ; for a perfect ellipse, r(Φ) would be periodic with period 2π, reflecting the fact that perihelion occured at the same angluar position each orbit.

Using then perturbation theory, we can show how General Relativity introduces a slight alteration of the period, giving rise to precession.

We recall from our previous article Gravitational deflection of light the relativistic expression of the Binet's equation (in Newtonian physics, the last term in u² is absent) for a particule with mass (note the presence of the term GM/h² on the right-side of the equation)

If we now consider a circular orbit with constant radius r_c:  r_c should be solution of the previous equation so that, with u_c=1/r_c:

 If we now assume that the solution has the form

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