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#### Welcome to Einstein Relatively Easy!

This web site is aimed at the general reader who is keen to discover Einstein's theories of special and general relativity, and who may also like to tackle the essential underlying mathematics.

Einstein's Relativity is too beautiful and too engaging to be restricted to the professionals!

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Albert Einstein

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## Advance of the perihelion of Mercury

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- Category: General Relativity
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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 u^{2} is absent) for a particule with mass (note the presence of the term GM/h^{2} 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

## Gravitational deflection of light

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In 1911 the The Equivalence Principle led Einstein to believe that light would be deflected in the presence of a gravitational field.

It wasn't until 1915, however, after he had successfully incorporated curved spacetime into a gravitational theory of relativity, that he was able to make an accurate prediction as to the magnitude of such a deflection.

Returning to the mathematics, we have seen in our previous article Geodesics in Schwarzschild spacetime that the Schwarzschild geodesic equations can be used to derive **null geodesic equations** that describe the path of a light ray in spacetime.

Ultimately, for this purpose, we would like to express r as a function of Φ

Let's define **u = 1/r**

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