Gravitational Time Dilation

Equivalent forms

d\tau = dt\sqrt{1 - \frac{r_s}{r}}

\Delta\tau / \Delta t = (1 - r_s/r)^{1/2}

A single square root, parameterized by the Schwarzschild radius, encodes how gravity warps the flow of time near any spherical mass.

Symbol	Name	SI unit	Dimension	Range
$dtau_dt$	Proper Time Rateoutput Ratio of clock rate at radius r to distant clock	dimensionless	1	0 – 1
$M$	Central Mass Mass of the gravitating body	kg	M	100000000000000000000 – 1e+40
$r$	Radial Distance Distance from center of mass	m	L	1 – 1000000000000
$G$	Gravitational Constant Newton's gravitational constant	m^3·kg^-1·s^-2	M^-1L^3T^-2	6.674e-11 – 6.674e-11
$c$	Speed of Light Speed of light in vacuum	m/s	LT^-1	299792458 – 299792458

Unit systems

Symbol	SI	CGS	Imperial
$dtau_dt$	1	1	1
$M$	kg	g	lb
$r$	m	cm	ft
$G$	m^3·kg^-1·s^-2	cm^3·g^-1·s^-2	ft^3·lb^-1·s^-2
$c$	m/s	cm/s	ft/s

Where it holds

Valid outside a non-rotating spherical mass, r >

r_s = 2GM/c^2

. Static observers only; rotating bodies require Kerr metric.

Dimensional analysis

dtau/

dt \to [1]

\sqrt(1 - [M^-1L^3T^-2][M] / ([L][LT^-1]^2)) \to \sqrt{1 - [1]}

Both sides dimensionless.

\checkmark

Discovery

Albert Einstein · 1915

Predicted in Einstein's 1915 general relativity. The Schwarzschild solution (1916) gave the exact form for spherical masses; Pound-Rebka (1959) verified it experimentally.

Try this

Why do GPS satellite clocks tick faster than ground clocks?

Clocks higher in a gravitational potential tick faster. For GPS at 20200 km altitude, the gravitational effect is +45 microseconds/day; velocity effect is −7 microseconds/day. Net +38 microseconds/day correction.

Research status: stable

Real-world applications

GPS positioning (38 microseconds/day correction)
Pulsar timing arrays for gravitational waves
Black hole imaging (Event Horizon Telescope)
Optical lattice clock geodesy (sensitive to 1 cm altitude changes)

Common misconceptions

Gravitational time dilation is NOT a Doppler effect — it persists for static observers
It depends on potential difference, not on local g
Clocks don't 'know' they are dilated — locally each clock runs at 1 second per second

Experimental verification

Pound-Rebka (1959) measured 22.5 m height redshift at Harvard. Hafele-Keating (1971) flew atomic clocks. Gravity Probe A (1976) verified to 70 ppm. GPS depends on it continuously.

Derivation

The Schwarzschild metric in static coordinates is

ds^2 = -(1 - r_s/r) c^2\,\mathrm{dt}^2 + (1 - r_s/r)^{-1} dr^2 + r^2

dOmega^2.

For a stationary observer

(dr = 0

, dOmega

= 0)

, proper time satisfies c^2 dtau^

2 = -ds^2 = (1 - r_s/r) c^2\,\mathrm{dt}^2

, giving dtau/

dt = \sqrt(1 - 2GM/(rc^2))

.

Limiting cases

r \to \infty

⟶ dtau/

dt \to 1

Far from any mass, clocks tick at the coordinate rate.

r \to r_s = 2GM/c^2

⟶ dtau/

dt \to 0

At the Schwarzschild radius, time appears frozen to a distant observer (event horizon).

weak field⟶ dtau/

dt \approx 1 - GM/(rc^2)

First-order expansion gives the Newtonian potential / c^2 correction.

What if…

What if you fell toward the event horizon?

A distant observer sees your clock slow to a halt at r_s; you yourself see nothing unusual locally as you cross.

What if Earth's mass were doubled?

Surface time dilation roughly doubles $to \sim 1.4e-9$ ; GPS correction would grow proportionally.

What if r < r_s?

Inside the horizon, the t and r coordinates swap roles — 'time' becomes radial; no static observer exists.

1

GPS satellite vs Earth surface

Given ·

M:: 5.972e+24
r:: 26571000
G:: 6.674e-11

Find · dtau_dt

Steps

Surface: $GM/(R_E c^2) \approx 6.96e-10$
Orbit: $GM/(r c^2) \approx 1.67e-10$
$\Delta$ (dtau/ $dt) \approx 5.29e-10 \to 45.7$ microseconds/day

Result · At r_orbit

= 26571\,\mathrm{km}

, dtau/

dt \approx 1 - GM/(rc^2) \approx 1 + 1.67e-10

relative to surface — clocks gain

\sim 45

microseconds/day.

2

Clock near a stellar-mass black hole

Given ·

M:: 2e+30
r:: 10000
G:: 6.674e-11

Find · dtau_dt

Steps

$r_s = 2 * 6.674e-11 * 2e30 / (3e_{8})^2 \approx 2967\,\mathrm{m}$
$r_s/r = 0.297$
dtau/ $dt = \sqrt{0.703} \approx 0.838$

Result ·

r_s = 2GM/c^2 \approx 2970\,\mathrm{m}

. At

r = 10\,\mathrm{km}

, dtau/

dt = \sqrt{1 - 0.297} \approx 0.838

— clock runs at 84% of distant rate.

Schwarzschild Radius→Gravitational Redshift→Time Dilation→Spacetime Interval→