Gravitational Redshift

Gravitational Redshift

When light climbs out of a gravitational well, it loses energy. Since a photon's energy is $E = hf$, a lower energy means a lower frequency — the light is redshifted. This is gravitational redshift, first confirmed by Pound and Rebka in 1959 using the Mössbauer effect over just 22 metres in a Harvard building.

The Formula

Light emitted at radius $r$ from mass $M$ and received far away (at infinity) has a redshift parameter $z$:

$1 + z = \frac{\lambda_{\text{obs}}}{\lambda_{\text{emit}}} = \frac{1}{\sqrt{1 - r_s/r}} = \frac{1}{\sqrt{1 - 2GM/(c^2 r)}}$

So:

$z = \frac{1}{\sqrt{1 - 2GM/(c^2 r)}} - 1$

The observed wavelength and frequency are:

$\lambda_{\text{obs}} = \frac{\lambda_{\text{emit}}}{\sqrt{1 - r_s/r}}, \qquad f_{\text{obs}} = f_{\text{emit}} \cdot \sqrt{1 - r_s/r}$

Physical Examples

Your Task

Implement these functions with all constants defined inside each function:

• redshift_factor(M, r) — returns $z = 1/\sqrt{1 - 2GM/(c^2 r)} - 1$
• wavelength_observed(lambda_emit, M, r) — returns the redshifted wavelength
• frequency_observed(f_emit, M, r) — returns the blueshifted (lowered) frequency