Gravitational Redshift Calculator
Compute gravitational redshift using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Formula
z = 1/sqrt(1 - 2GM/Rc^2) - 1
Where z = redshift parameter, G = gravitational constant (6.674e-11), M = mass of the object, R = radius where light is emitted, c = speed of light. The observed wavelength equals the emitted wavelength times (1+z). When 2GM/Rc^2 approaches 1 (the Schwarzschild radius), z approaches infinity.
Worked Examples
Example 1: Gravitational Redshift from the Sun
Problem: Calculate the gravitational redshift of the hydrogen-alpha line (656.3 nm) emitted from the surface of the Sun (M = 1 solar mass, R = 1 solar radius).
Solution: M = 1.989e30 kg, R = 6.957e8 m\nGravitational potential: GM/Rc^2 = (6.674e-11 * 1.989e30) / (6.957e8 * (3e8)^2) = 2.12e-6\nSchwarzschild radius: Rs = 2 * 6.674e-11 * 1.989e30 / (3e8)^2 = 2954 m\nCompactness: 2954 / 6.957e8 = 4.25e-6\nExact z = 1/sqrt(1 - 4.25e-6) - 1 = 2.12e-6\nObserved wavelength: 656.3 * (1 + 2.12e-6) = 656.3014 nm\nWavelength shift: 0.0014 nm
Result: z = 2.12e-6 | Observed: 656.3014 nm | Shift: 0.0014 nm (velocity equiv: 636 m/s)
Example 2: White Dwarf Gravitational Redshift
Problem: Calculate the gravitational redshift for a white dwarf with M = 0.6 solar masses and R = 0.012 solar radii (about Earth-sized).
Solution: M = 0.6 * 1.989e30 = 1.193e30 kg\nR = 0.012 * 6.957e8 = 8.349e6 m\nSchwarzschild radius: 2 * 6.674e-11 * 1.193e30 / (3e8)^2 = 1773 m\nCompactness: 1773 / 8.349e6 = 2.12e-4\nz = 1/sqrt(1 - 2.12e-4) - 1 = 1.06e-4\nFor H-alpha: observed = 656.3 * 1.000106 = 656.370 nm\nVelocity equivalent: 31.8 km/s
Result: z = 1.06e-4 | Wavelength shift: 0.070 nm | Velocity equivalent: 31.8 km/s
Frequently Asked Questions
What is gravitational redshift and what causes it?
Gravitational redshift is the phenomenon where light or electromagnetic radiation emitted from a region of strong gravity has its wavelength stretched (shifted toward the red end of the spectrum) as it climbs out of the gravitational field. This effect is a direct prediction of Einstein general theory of relativity and arises because time runs slower in stronger gravitational fields. A photon emitted at a certain frequency near a massive object will be observed at a lower frequency (longer wavelength) by a distant observer in weaker gravity. The stronger the gravitational field at the point of emission, the greater the redshift. This effect has been confirmed experimentally using atomic clocks at different altitudes and by observing spectral lines from white dwarf stars.
How is gravitational redshift different from Doppler redshift?
Gravitational redshift and Doppler redshift both cause wavelength shifts, but they arise from fundamentally different physical mechanisms. Doppler redshift occurs when a light source moves away from the observer, stretching the wavelength due to the relative motion. Gravitational redshift occurs even when the source and observer are stationary relative to each other, arising purely from the difference in gravitational potential between the emission and observation points. In practice, astronomers must carefully separate these effects when analyzing spectra of stars and galaxies. Cosmological redshift is yet another distinct effect caused by the expansion of space itself, which stretches photon wavelengths during their journey through the expanding universe.
How was gravitational redshift first experimentally confirmed?
The first precise laboratory confirmation of gravitational redshift was the Pound-Rebka experiment in 1959 at Harvard University. Robert Pound and Glen Rebka measured the frequency shift of gamma rays traveling 22.5 meters vertically in the Jefferson Tower, using the Mossbauer effect to achieve the extreme frequency precision required. The measured redshift agreed with the general relativity prediction to within 10%. A refined version by Pound and Snider in 1964 achieved 1% accuracy. Since then, gravitational redshift has been confirmed with much higher precision using hydrogen maser clocks on rockets (Gravity Probe A, 1976) and more recently using optical atomic clocks at different elevations, achieving agreement with theory at the parts-per-million level.
What is the gravitational redshift on the surface of a white dwarf?
White dwarf stars, with masses comparable to the Sun compressed into a volume the size of Earth, produce significant gravitational redshifts that are directly observable in their spectra. A typical white dwarf with a mass of 0.6 solar masses and a radius of about 0.01 solar radii has a gravitational redshift of z approximately equal to 3 times 10^-4, corresponding to a velocity equivalent of about 90 km/s. This redshift was first measured by Walter Adams in 1925 for Sirius B, providing one of the earliest confirmations of general relativity. Modern spectroscopic surveys of white dwarfs routinely measure gravitational redshifts to determine their masses independently of binary orbit observations.
How does gravitational redshift affect GPS satellites?
GPS satellites orbit at about 20,200 km altitude where gravity is weaker than on Earth surface, causing their onboard atomic clocks to tick faster by about 45 microseconds per day due to reduced gravitational time dilation. This is partially offset by special relativistic time dilation (clocks on moving satellites tick slower by about 7 microseconds per day), giving a net gain of about 38 microseconds per day. Without correcting for these relativistic effects, GPS position errors would accumulate at roughly 10 kilometers per day, making the system useless for navigation. The GPS system applies a frequency offset to satellite clocks before launch, setting them to tick slightly slow so they match ground clocks after accounting for both gravitational and velocity time dilation.
What happens to gravitational redshift near a black hole?
As light is emitted closer and closer to the event horizon of a black hole (the Schwarzschild radius), the gravitational redshift increases without bound, approaching infinity at the horizon itself. At the photon sphere (1.5 times the Schwarzschild radius), the redshift factor z equals approximately 0.41, meaning wavelengths are stretched by 41%. At the innermost stable circular orbit for a non-rotating black hole (3 times the Schwarzschild radius), z is about 0.22. Light emitted exactly at the event horizon would be infinitely redshifted and never reach a distant observer, which is equivalent to saying that time appears to stop at the horizon from an external perspective. This infinite redshift is what makes black hole event horizons effectively invisible to outside observers.