Gravitational Time Dilation
Gravitational time dilation is the phenomenon in which time is measured to pass more slowly for a clock located deeper within a gravitational field (closer to a massive object), compared to a clock further away, a direct consequence of the equivalence principle and the curvature of spacetime near mass.
Curious
Just as time dilation from special relativity says a moving clock ticks slower, general relativity predicts that a clock sitting deeper within a gravitational field — closer to a massive object — also ticks slower than an identical clock further away. This isn't a small correction dreamed up to patch a discrepancy; it follows directly and necessarily from the equivalence principle, which says gravity and acceleration are locally indistinguishable, combined with the fact that special relativity already predicts time dilation for accelerating observers.
The effect is genuinely tiny at everyday scales — clocks at different floors of a building differ by only fractions of a nanosecond per day — but it has been measured directly, and it becomes practically essential to account for at the precision required by GPS satellites, which experience meaningfully weaker gravity than clocks on Earth's surface.
Exploring
| Symbol | Name | SI Unit |
|---|---|---|
| t_∞ | Time interval measured far from the mass (weak field) | second (s) |
| t_r | Time interval measured at radius r from the mass's centre | second (s) |
| r_s | Schwarzschild radius of the mass | metre (m) |
Worked example: Using the Sun's Schwarzschild radius (r_s ≈ 2,950 m from the earlier worked example) and Earth's orbital radius (r ≈ 1.496×10¹¹ m), the fractional time dilation at Earth's distance from the Sun is √(1 − 2950/1.496×10¹¹) ≈ √(1 − 1.97×10⁻⁸) ≈ 1 − 9.86×10⁻⁹ — an extraordinarily small effect at this distance, illustrating why gravitational time dilation from the Sun is far less significant for Earth-based clocks than Earth's own, much closer gravitational field.
Deep Dive
The Pound-Rebka experiment (1959–1960), conducted at Harvard using a 22.5-metre-tall tower, provided the first direct laboratory confirmation of gravitational time dilation (equivalently, gravitational redshift) using the Mössbauer effect to detect the extraordinarily tiny frequency shift of gamma rays travelling up or down the tower — a shift of only about 2.5 parts in 10¹⁵, yet measured to agree with general relativity's prediction to within about 10% precision, later refined further.
Relation to gravitational redshift: Gravitational time dilation and gravitational redshift are two descriptions of the same underlying effect: light emitted from deeper within a gravitational field, where clocks run slower, appears redshifted (lower frequency, longer wavelength) when observed from further away, exactly matching the ratio of the two locations' proper time rates. This effect has since been measured with extreme precision in numerous experiments, including comparisons between atomic clocks at different altitudes and, most dramatically, in the extreme gravitational fields near neutron stars and black holes, where gravitational redshift becomes large enough to be directly significant for astronomical observations.
The full general-relativistic formula for gravitational time dilation generalises the flat-spacetime formula given above and becomes considerably more complex for rotating masses (requiring the Kerr metric rather than Schwarzschild) or for strong, rapidly varying fields — the simple formula above applies specifically to the static, spherically symmetric Schwarzschild case.
Concept Relationship Map
Requires: Schwarzschild metric, equivalence principle
Enables: gravitational redshift [PENDING FULL ENTRY], GPS clock corrections
Commonly confused with: special-relativistic time dilation from velocity (a physically distinct effect that happens to combine with this one in GPS satellites)
Sources
- Pound, R.V., Rebka, G.A. (1960). Physical Review Letters, 4(7), 337–341.
- Ashby, N. (2003). "Relativity in the Global Positioning System." Living Reviews in Relativity, 6, 1.