Gravitational Redshift Calculator
Compute the observed frequency of light or electromagnetic radiation emitted near a massive object, accounting for gravitational redshift predicted by General Relativity. Essential for astrophysics, pulsar timing, and GPS system design.
About this calculator
Gravitational redshift describes how light loses energy — and therefore drops in frequency — as it climbs out of a gravitational well. According to General Relativity, the observed frequency f_obs received far from the source is f_obs = f_emit × √(1 − 2GM / (rc²)), where f_emit is the emitted frequency, G = 6.6743×10⁻¹¹ N·m²/kg² is the gravitational constant, M is the gravitational mass, r is the distance from the center of mass to the emission point, and c = 299,792,458 m/s. The term 2GM/rc² is recognizable as rₛ/r (Schwarzschild radius divided by distance), so light emitted right at the event horizon is redshifted to zero frequency. For weak gravitational fields (rₛ ≪ r), the formula approximates to a fractional shift Δf/f ≈ GM/(rc²). GPS satellites must apply gravitational blueshift corrections of ~45 µs/day because their clocks run faster in Earth's weaker gravitational field.
How to use
A signal is emitted at f_emit = 1,000,000,000 Hz (1 GHz) from the surface of a neutron star of mass M = 4×10³⁰ kg at radius r = 12,000 m (12 km). Compute 2GM/rc²: numerator = 2 × 6.6743×10⁻¹¹ × 4×10³⁰ = 5.339×10²⁰. Denominator = 12,000 × (2.998×10⁸)² = 12,000 × 8.988×10¹⁶ ≈ 1.0786×10²¹. Ratio = 5.339×10²⁰ / 1.0786×10²¹ ≈ 0.4951. Then √(1 − 0.4951) = √0.5049 ≈ 0.7106. So f_obs = 1,000,000,000 × 0.7106 ≈ 710,600,000 Hz (710.6 MHz). The signal is redshifted by nearly 30%.
Frequently asked questions
What is gravitational redshift and how is it different from Doppler redshift?
Gravitational redshift occurs because photons lose energy climbing out of a gravitational field — they do not slow down (light always travels at c) but their frequency decreases and wavelength increases, shifting them toward the red end of the spectrum. Doppler redshift, by contrast, arises from relative motion between source and observer and can be either a redshift (recession) or blueshift (approach). Gravitational redshift is a purely General Relativistic effect, confirmed by the Pound–Rebka experiment in 1959 and routinely measured in white dwarfs, neutron stars, and GPS satellites.
How does gravitational redshift affect GPS satellite clocks?
GPS satellites orbit at about 20,200 km altitude where Earth's gravitational field is weaker than at the surface. Clocks in a weaker gravitational field run faster (gravitational blueshift from the satellite's perspective). This causes satellite clocks to gain approximately 45.9 microseconds per day relative to ground clocks. Combined with the Special Relativistic time dilation of −7.2 µs/day due to their orbital speed, the net gain is about +38.4 µs/day. Without correcting for this, GPS position errors would accumulate at roughly 10 km per day, making the system useless for navigation.
What happens to light frequency when it falls into a gravitational field rather than escaping from it?
When light falls into a gravitational well — moving from a region of weaker gravity to stronger gravity — it gains energy and its frequency increases, a phenomenon called gravitational blueshift. This is the reverse of gravitational redshift. The same formula applies: f_obs = f_emit × √(1 − 2GM/(rc²)) with r now being the emission radius; if r_emission > r_observation, f_obs > f_emit. This effect has been confirmed experimentally and is precisely symmetric with the redshift case. It means a photon emitted far from a black hole and absorbed near it appears blueshifted to a local observer near the black hole.