Solar Redshift

The light emitted from the Sun shows a redshift of the spectral lines even though our distance to the Sun is fixed during the measurement process. Why so?

Answer

Even though there may be no relative radial motion between the Sun and the observer on Earth, there is still a gravitational redshift dictated by the general theory of relativity (GTR). Recall that the infinitesimal distance ds in a flat Euclidean space with coordinates (r, θ, φ) is defined by ds² = c² dt² – dr² – r² dθ² – r² sin² θ dφ². In the gravitational field of mass M, this infinitesimal distance in the GTR becomes the Schwarzschild line element ds2 = (1 – r_g/r) c² dt² – (1 – r_g/r)–1 dr² – r² dθ² – r² sin² θ dφ², where r_g = 2GM/c² and G is the gravitational constant.

We see that near a massive body such as the Sun, the time coordinate includes a factor , where r is the position of the light measured from the center of the Sun. By evaluating this factor at the Sun’s surface and at Earth’s distance, one finds that the clocks at the two distances are ticking at different rates, faster for bigger r. One approach is to assume that the photon does not change its inherent physical properties for example, it maintains its characteristic frequency established during the emission process at the surface of the Sun. Then the observer on Earth, who has the faster-ticking reference clock with respect to the stars, will measure a lower photon frequency and see the light as redshifted.

A second approach determines that the change in r for the photon can be shown to correspond to a change in gravitational potential. The photon essentially begins in a gravitational potential energy valley and climbs upward to reach Earth. Its total energy must remain constant, so the increase in gravitational potential energy is matched by the decrease in photon energy that is, a redshifted photon because its energy E = hν.

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