Gravitational Time Dilation Calculator

Q: Why do GPS satellites need relativistic corrections?

Gravity (weaker at satellite orbit) makes GPS clocks run 45.9 us/day FAST. Velocity (special relativity) makes them 7.2 us/day SLOW. Net: +38.4 us/day fast. Uncorrected, GPS would accumulate 10 km/day position error.

Q: How slow does time run at Earth surface?

About 60 microseconds per day slower than at infinity. After 100 years, a clock at sea level is behind one in space by about 2.2 seconds.

Q: What happens to time at a black hole event horizon?

From an outside observer, time appears to stop at the event horizon (infinite time dilation). The infalling observer experiences no such effect locally.

Q: How is gravitational time dilation measured?

Pound-Rebka experiment (1959): measured frequency shift of gamma rays between top and bottom of a tower. Confirmed to 1% accuracy.

Q: What is the equivalence principle?

Einstein equivalence: gravitational time dilation equals what an accelerating observer sees. A clock at the bottom of an accelerating rocket ticks slower than at the top.

Gravity slows time. Clocks at lower gravitational potential run slower than clocks far from massive objects. This effect is measurable on Earth and must be corrected in GPS satellites.

🌍 Relativity📐 t' = t√(1-rₛ/r)⚫ Astrophysics

Object Mass (kg)

Distance from center r (m)

Proper Time t (s)

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Dilated Time	t′	t′ = t√(1 − r_s/r)	s
Schwarzschild Radius	r_s	r_s = 2GM/c²	m
Gravitational Factor	g₀₀	g₀₀ = 1 − r_s/r	—
GPS Correction	—	~45.9 μs/day fast (gravity) −7.2 slow (velocity)	μs/day

Step-by-Step Examples

Example 1

GPS Satellite vs Earth Surface

GPS orbit r=26,560 km + 6371 km = 2.6956e7 m. Earth surface r=6.371e6 m.

Factor at surface: sqrt(1 - 8.87e-3/6.371e6) = 1 - 6.95e-10
GPS satellite clock runs ~45.9 us/day FASTER than Earth surface
(Velocity effect makes it 7.2 us/day SLOWER; net +38.7 us/day)

✓ GPS clocks are corrected by +38.4 us/day for combined relativistic effects

Example 2

Earth Surface vs Infinity

Earth mass=5.972e24 kg, r=6.371e6 m, t=1 day.

r_s = 2*6.674e-11*5.972e24/(9e16) = 8.87e-3 m = 8.87 mm
Factor = sqrt(1 - 8.87e-3/6.371e6) = 1 - 6.95e-10
dt = 6.95e-10 * 86400 = 6.0e-5 s/day

✓ 60 microseconds/day slower on Earth surface vs far away

Example 3

Near Neutron Star

Neutron star M=2 solar masses, r=10 km.

r_s = 2*6.674e-11*3.978e30/(9e16) = 5906 m
Factor = sqrt(1-5906/10000) = sqrt(0.4094) = 0.640
One second on surface = 0.640 s at infinity

✓ 40% time dilation on neutron star surface

Real-World Applications

📱

GPS Navigation

GPS requires time dilation correction or position errors accumulate at 10 km/day.

⚫

Black Hole Research

Near a black hole, time slows dramatically. At the event horizon, time stops from outside view.

⌛

Precision Timekeeping

Optical clocks now detect height differences of 1 cm via gravitational time dilation.

🚀

Space Missions

Gravitational time dilation matters for precise navigation near massive objects.

Common Mistakes to Avoid

⚠️

Distance r is from center, not surface

For Earth surface calculations, r = 6.371e6 m (Earth radius from center), not 0.

⚠️

r must be greater than Schwarzschild radius

If r < r_s, you are inside a black hole. The formula returns a complex number.

⚠️

This is gravitational effect only

GPS also has special relativistic time dilation (velocity). Total GPS correction requires both effects.

Frequently Asked Questions

Why do GPS satellites need relativistic corrections? ▾

Gravity (weaker at satellite orbit) makes GPS clocks run 45.9 us/day FAST. Velocity (special relativity) makes them 7.2 us/day SLOW. Net: +38.4 us/day fast. Uncorrected, GPS would accumulate 10 km/day position error.

How slow does time run at Earth surface? ▾

About 60 microseconds per day slower than at infinity. After 100 years, a clock at sea level is behind one in space by about 2.2 seconds.

What happens to time at a black hole event horizon? ▾

From an outside observer, time appears to stop at the event horizon (infinite time dilation). The infalling observer experiences no such effect locally.

How is gravitational time dilation measured? ▾

Pound-Rebka experiment (1959): measured frequency shift of gamma rays between top and bottom of a tower. Confirmed to 1% accuracy.

What is the equivalence principle? ▾

Einstein equivalence: gravitational time dilation equals what an accelerating observer sees. A clock at the bottom of an accelerating rocket ticks slower than at the top.

Related Physics Calculators

Formula Explorer connections

Interpretation: This relationship connects mass, distance, orbit or spacetime behavior through gravitation and astrophysical scaling. Assumption: Many calculations assume spherical bodies, point masses, circular orbits, weak fields or Newtonian gravity; relativistic regimes require the stated correction.

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