On the Importance of Solar Eclipses

As has been well documented over recent days, there has been great excitement over the recent activity of full and partial solar eclipses, with students, astronomy enthusiasts, etc… all over showing great enthusiasm. However, a vast majority seem to be completely unaware as to why these solar eclipses are so important. Their most important purpose is that one can directly confirm as Eddington did in 1919 the validity of Einstein’s theory of General Relativity. When the sun is eclipsed, one can directly observe starlight from behind the sun being bent as should happen according to Einstein’s theory, namely that light bends according to the curvature of spacetime. I document here a brief calculation that demonstrates this:

We will assume that the spacetime under consideration is spherically symmetric and static, and so by Birkhoff’s theorem, outside the spherically symmetric body, the solution to Einstein’s equations is the well-known Schwarzschild metric:

eq1 To determine an equation for the path that light should follow in this spacetime, one writes the associated Lagrangian of geodesics as:
latexit-drag

Applying the Euler-Lagrange equations and exploiting the fact that the spacetime is spherically symmetric and static, one obtains the orbit equation for light as:

eq3

This ordinary differential equation is a nightmare to solve!! For example, Mathematica gives:MSP20931eb0034d37c3393a00003d4df7157eg76h00

That is, a complicated function of elliptic integrals. The easier approach is to consider perturbations as per the text of Gron and Hervik, which we do here:

Essentially, without getting into too much detail, taking this approach (See Ch. 10 of Gron and Hervik), we obtain the solution to the ODE above as:

latexit-drag

where b is defined to be an impact parameter. At u = 0, the photon flies out towards radial infinity, and so the deflection angle, can be calculated from the relationship:

latexit-drag

Let us perform a Taylor expansion about Pi/2, to obtain the angle by which an astrophysical body causes light to deflect by as:

latexit-drag

For the Sun, the deflection angle turns out to be approximately

latexit-drag

That is, the curvature of spacetime induced by the Sun causes starlight from behind the sun to be “bent”/deflected by 1.75”. This is precisely what Eddington’s team observed in their 1919 expedition. This test was what finally confirmed Einstein’s theory of General Relativity, and is the true reason why solar eclipses are so important!

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Published by

ikjyotsinghkohli24

Sikh, Theoretical and Mathematical Physicist, main research in the structure and dynamics of Einstein's field equations.

3 thoughts on “On the Importance of Solar Eclipses”

  1. Appreciating the commitment you put into your blog and detailed information you provide.

    It’s nice to come across a blog every once in a while that isn’t
    the same out of date rehashed information.
    Excellent read! I’ve saved your site and I’m including your RSS feeds to my Google account.

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