Gravitational Redshift Calculator
Compute gravitational redshift using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Gravitational Redshift Calculator
Calculator
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Formula: z = 1/sqrt(1 - 2GM/Rc^2) - 1
Worked example — z = 2.12e-6 | Observed: 656.3014 nm | Shift: 0.0014 nm (velocity equiv: 636 m/s)
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 Gravitational potential: GM/Rc^2 = (6.674e-11 * 1.989e30) / (6.957e8 * (3e8)^2) = 2.12e-6 Schwarzschild radius: Rs = 2 * 6.674e-11 * 1.989e30 / (3e8)^2 = 2954 m Compactness: 2954 / 6.957e8 = 4.25e-6 Exact z = 1/sqrt(1 - 4.25e-6) - 1 = 2.12e-6 Observed wavelength: 656.3 * (1 + 2.12e-6) = 656.3014 nm Wavelength 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 R = 0.012 * 6.957e8 = 8.349e6 m Schwarzschild radius: 2 * 6.674e-11 * 1.193e30 / (3e8)^2 = 1773 m Compactness: 1773 / 8.349e6 = 2.12e-4 z = 1/sqrt(1 - 2.12e-4) - 1 = 1.06e-4 For H-alpha: observed = 656.3 * 1.000106 = 656.370 nm Velocity 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.
What is the Schwarzschild radius and why is it important?
The Schwarzschild radius is the radius of the event horizon of a non-rotating black hole, given by Rs = 2GM/c^2, where G is the gravitational constant, M is the mass, and c is the speed of light. If an object is compressed to within its Schwarzschild radius, it becomes a black hole from which nothing, not even light, can escape. For the Sun, the Schwarzschild radius is about 3 kilometers, while for the Earth it is only about 9 millimeters. The ratio of the Schwarzschild radius to the actual radius of an object, called the compactness parameter, determines how strong relativistic effects are at its surface. When this ratio approaches 1, the gravitational redshift becomes infinite.
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.
Can gravitational blueshift occur?
Yes, gravitational blueshift occurs when light falls into a gravitational well, gaining energy and shifting to shorter wavelengths (higher frequencies). This happens when an observer is deeper in a gravitational field than the source of light. For example, light from distant stars or cosmic microwave background radiation is slightly blueshifted as it falls toward Earth. In the Pound-Rebka experiment, gamma rays were observed to be blueshifted when falling downward and redshifted when traveling upward. Gravitational blueshift is the exact reverse of gravitational redshift and follows the same formula but with opposite sign. In accretion disks around black holes, infalling material experiences gravitational blueshift that partially converts gravitational potential energy into thermal radiation.
How does gravitational redshift relate to time dilation?
Gravitational redshift and gravitational time dilation are two aspects of the same physical phenomenon described by general relativity. The redshift factor (1 + z) equals the time dilation factor, meaning that if a clock runs slower by a factor of sqrt(1 - 2GM/Rc^2) at a given gravitational potential, then light emitted from that location is redshifted by the same factor. A distant observer sees atomic transitions occurring at lower frequencies from atoms in strong gravity, exactly as if those atoms clocks were running slow. This connection was beautifully demonstrated by comparing atomic clocks at different elevations, where the clock at lower altitude (stronger gravity) ticks measurably slower and emits light at correspondingly lower frequency, in perfect agreement with the predicted redshift.
What astronomical objects produce the strongest gravitational redshifts?
The strongest gravitational redshifts are produced by the most compact objects in the universe. Neutron stars, with masses of 1.4 to 2 solar masses compressed into spheres only 10 to 12 kilometers in radius, produce surface redshifts of z approximately 0.2 to 0.4, which significantly affects the observed spectra of X-ray bursts. Black holes produce theoretically infinite redshift at their event horizons. White dwarfs produce more modest but easily measurable redshifts of z approximately 0.0001 to 0.001. The Sun produces a tiny but measurable redshift of z approximately 2.1 times 10^-6, first detected in solar spectral lines. Even the Earth produces a gravitational redshift of about z = 7 times 10^-10, measurable only with the most precise modern atomic clocks.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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