Gravitational Time Dilation Calculator
Compute how much slower a clock runs in a gravitational field or at high velocity, with annual drift readout.
🕰️ What is the Gravitational Time Dilation Calculator?
Gravitational time dilation is one of the central predictions of Einstein's general theory of relativity: clocks in stronger gravitational fields run slower than clocks in weaker ones. This calculator computes the time dilation factor for two distinct physical scenarios, the Schwarzschild gravitational case and the special relativistic velocity case, giving you both the dimensionless factor and the practical annual clock drift in human-readable units.
In the gravitational (Schwarzschild) mode you enter a central mass (a star, neutron star, or black hole) and an observation distance. The calculator computes the dilation factor alpha = sqrt(1 - r_s/r), where r_s is the Schwarzschild radius of the mass. This tells you how much slower a clock at that distance runs compared to a clock infinitely far away. Practical applications include understanding GPS satellite corrections, estimating time passage near neutron stars, and studying the environment around black holes.
In the velocity (Lorentz) mode you enter a velocity as a percentage of the speed of light. The calculator returns the Lorentz factor gamma and the dilation factor alpha = 1/gamma, telling you how much slower a moving clock runs compared to a stationary one. This is special relativistic time dilation (no gravity), relevant to particle accelerators, hypothetical interstellar travel, and thought experiments about twins.
A common confusion is between the dilation factor and the clock rate percentage. A factor of 0.766 means the clock runs at 76.6% of normal speed, losing 23.4% of time compared to a reference at infinity. The annual drift converts this to concrete terms: a factor of 0.766 means the clock loses about 85.5 days every year. This calculator makes that connection explicit so the effect is immediately tangible.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Neutron star surface (M = 1.4 M☉, r = 10 km)
Clock on a typical neutron star surface
Example 2 — Sun's surface (M = 1 M☉, r = 695700 km)
Clock on the surface of the Sun
Example 3 — Velocity dilation at 99% of c
A spacecraft travelling at 99% of the speed of light
❓ Frequently Asked Questions
🔗 Related Calculators
What is gravitational time dilation?
Gravitational time dilation is the phenomenon by which a clock in a stronger gravitational field (closer to a massive object) runs slower than a clock farther away. Predicted by Einstein's general theory of relativity, it has been confirmed by experiments such as the Pound-Rebka experiment (1959) and GPS satellite corrections applied daily.
What is the formula for gravitational time dilation?
The Schwarzschild metric gives the time dilation factor alpha = sqrt(1 - r_s/r), where r_s = 2GM/c^2 is the Schwarzschild radius of the mass M and r is the distance from the center of mass. A clock at distance r ticks at rate alpha relative to a clock at infinity.
How do I calculate the Lorentz factor for velocity time dilation?
The special relativistic dilation factor is alpha = sqrt(1 - v^2/c^2) = 1/gamma, where gamma is the Lorentz factor and v is the velocity as a fraction of c. A clock moving at velocity v ticks at rate alpha relative to a stationary observer.
What does the annual drift result mean?
Annual drift is the cumulative time difference between two clocks after one year. For example, a clock on a neutron star surface loses about 85.5 days per year compared to a distant clock. A GPS satellite clock gains about 45 microseconds per day due to gravity (partially offset by velocity dilation).
Why is the Schwarzschild radius shown in the results?
The Schwarzschild radius r_s = 2GM/c^2 is an intermediate result in the time dilation formula. Knowing it lets you check whether your chosen distance is safely outside the event horizon. If you enter a distance less than or equal to r_s, the calculator returns an error because time dilation is undefined inside the horizon.
How much does gravity slow time on Earth's surface?
On Earth's surface (M = 5.972e24 kg, r = 6.371e6 m), the dilation factor is approximately 1 - 6.95e-10. Clocks on Earth's surface lose about 21.9 microseconds per year compared to a hypothetical clock at infinite distance. This is why GPS systems must apply relativistic corrections.
Can time dilation be combined for gravity and velocity?
Yes, but this calculator handles them separately for clarity. The combined dilation (as in a satellite orbit) requires integrating both: alpha_total = sqrt(1 - r_s/r) * sqrt(1 - v^2/c^2). The gravitational component makes clocks run faster at higher altitude; the velocity component makes them run slower. For GPS satellites the gravitational effect dominates.
What happens at the Schwarzschild radius?
At r = r_s the dilation factor alpha = 0, meaning a clock there appears frozen to an outside observer. Proper time still passes for an infalling observer, but from the outside the clock never crosses the event horizon. The formula breaks down at and below r_s.
What is the time dilation at 99.9% of the speed of light?
At 99.9% c the Lorentz factor gamma = 1/sqrt(1 - 0.999^2) approximately 22.37. The dilation factor alpha = 1/22.37 approximately 0.04472. A clock at that speed ticks once for every 22.37 ticks of a stationary clock, losing about 0.955 years per calendar year (about 348 days per year).
What is the dilation on a neutron star surface?
For a typical neutron star (1.4 solar masses, radius 10 km), r_s = 4.135 km and r = 10 km, giving alpha = sqrt(1 - 4.135/10) = 0.7658. A clock on the surface runs at 76.6% of the rate of a distant clock, losing about 85.5 days per year.