Gravitational Redshift
Fractional stretching of light climbing out of gravity.
Formula
Variables
Example
Light leaving the Sun is redshifted by about 2×10⁻⁶
How it works
Gravitational redshift is the tiny increase in a photon’s wavelength as it climbs out of a gravitational well. In general relativity, gravity slows time more strongly deeper in the field, so light emitted lower down is measured higher up as having a lower frequency; for weak fields this fractional change is well approximated by z ≈ GM/(rc²). It is largest for compact, massive objects and becomes especially important when the source radius r is small compared with GM/c².
When to use it
Use this formula when estimating the gravitational shift of light from stars, white dwarfs, neutron stars, or near black holes in the weak-field limit. It is also useful in precision astrophysics whenever you need the order of magnitude of a gravitational contribution to a spectral line shift.
Worked example, step by step
Estimate the gravitational redshift of light emitted from the surface of a white dwarf with mass M = 1.0×10^30 kg and radius r = 5.0×10^6 m. Step 1: Write the formula: z ≈ GM/(rc²) Step 2: Substitute values: z ≈ (6.67×10^-11 × 1.0×10^30) / [(5.0×10^6)(3.00×10^8)^2] Step 3: Calculate the numerator: 6.67×10^19 Step 4: Calculate the denominator: (5.0×10^6)(9.00×10^16) = 4.50×10^23 Step 5: Divide: z ≈ 6.67×10^19 / 4.50×10^23 ≈ 1.48×10^-4 Final answer: the light is redshifted by about 1.5×10^-4, meaning its wavelength increases by roughly 0.015%.
Common mistakes to avoid
- Using the distance to the observer instead of the radius from the mass center to the emission point
- Forgetting that the formula is a weak-field approximation and breaks down for very compact objects
- Mixing up redshift z with an actual change in wavelength without converting via Δλ/λ ≈ z
Did You Know?
The Pound–Rebka experiment in 1959 confirmed it by measuring light climbing just 22.5 m up a Harvard tower.
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