ConverterHub

Gravitational Redshift Calculator

Find the gravitational redshift near a mass.

How it is calculated

The weak-field redshift is z ≈ G·M / (r·c²), where M is the mass in kilograms, r is the distance from the centre of the mass in metres (the radius of the body when light leaves its surface), G = 6.674×10−11 m³ kg−1 s−2 and c = 299,792,458 m/s. The result is a dimensionless number, printed in scientific notation with four decimals, such as 2.1230e-6.

Both mass and radius must be above zero. The formula assumes a weak field, so it is accurate when z is much smaller than 1, as for planets, the Sun and most stars. It is not suitable close to a black hole's horizon, where the full Schwarzschild expression is required. The value is the fractional shift in wavelength, Δλ/λ.

Worked example

For the Sun, M = 1.989e30 kg and r = 6.957e8 m. G·M = 6.674e-11 × 1.989e30 ≈ 1.3275e20. r·c² = 6.957e8 × (299,792,458)² ≈ 6.253e25. Dividing gives z ≈ 2.1230e-6. Light from the Sun's surface is therefore stretched by about two parts per million, corresponding to a Doppler speed of roughly 0.64 km/s.

Try this example →

Common mistakes

  • Entering the mass in solar masses or Earth masses and the radius in kilometres: the tool expects kilograms and metres, so the answer is off by many orders of magnitude.
  • Using the body's diameter instead of its radius: z is inversely proportional to the distance, so a diameter halves the answer.
  • Reading the result as a percentage: 2.1230e-6 is a fraction, so as a percentage it would be 0.00021%.
  • Applying the result to a compact object when r is only a few times its Schwarzschild radius: the weak-field value then understates the true redshift.

About this calculator

This calculator uses the weak-field approximation of gravitational redshift, z ≈ G·M/(r·c²), derived from the Schwarzschild metric of general relativity for the case where the gravitational field is not extreme (far from a black hole's event horizon). It describes how light climbing out of a gravitational well loses energy and shifts to longer (redder) wavelengths, because a photon's frequency, as measured by a distant observer, depends on how deep in the gravity well it was emitted.

Astrophysicists use this formula to predict the redshift of light from white dwarfs and neutron stars, where the effect is measurable, and it's the same physics behind the tiny but critical clock corrections applied to GPS satellites, whose onboard clocks run fast relative to ground clocks because they sit higher in Earth's gravitational field, an effect only partly offset by the slowing caused by their orbital speed. Physics students studying general relativity use it as an accessible entry point before tackling the full Schwarzschild formula.

Was this helpful?

You can pick up to 3 reactions.

Comments (0)

  • Be the first to comment.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators