Schwarzschild Gravitational Redshift

In general relativity, a clock deeper in a gravitational field ticks more slowly than a clock further away. In the exterior Schwarzschild spacetime describing a non-rotating black hole, the ticking rate of a static clock at coordinate radius r compared to a clock at infinity is given by the lapse function:
lapse = √(1 - 1/r)
where r is given in dimensionless units relative to the Schwarzschild radius.

Worked Example

Consider a photon emitted from a static source at r = 2 and received by an observer at infinity.

Emitter lapse = sqrt(1 - 1/2) = sqrt(0.5) ≈ 0.7071
Receiver lapse = 1 (at infinity)

Frequency ratio = Emitter lapse / Receiver lapse
                ≈ 0.7071 / 1.0 = 0.7071

Redshift z = 1 / Frequency ratio - 1
           ≈ 1 / 0.7071 - 1 ≈ 0.4142

The photon is redshifted (z > 0) because it climbs out of the gravitational well to reach the receiver at infinity.

Experiment

You can use the interactive module to explore gravitational redshift between different static observers.

Select an Emitter radius of 10.
Select a Receiver radius of 3.
Observe that the frequency ratio is greater than 1, resulting in a negative redshift (blueshift), because the photon falls deeper into the gravitational well.

Limitations

This model assumes the emitter and receiver are both strictly static (fixed coordinate radius r, θ, and φ) in the exterior Schwarzschild geometry. It does not account for the kinematic (special-relativistic) Doppler effect that would occur if either observer were moving, nor does it apply to cosmological expansion. Note also that no static observer can exist at the event horizon (r=1) because infinite thrust would be required to remain stationary.

References

The physics of static observers and photon exchange in the Schwarzschild geometry can be studied in standard general relativity literature.

General Relativity - https://arxiv.org/abs/gr-qc/9712019

Schwarzschild Gravitational Redshift