Gravitational Time Dilation

Gravitational Time Dilation

One of the most striking predictions of General Relativity is that gravity slows time. A clock deep in a gravitational well ticks more slowly than a clock far away. This is not a measurement artifact — it is a real physical effect confirmed to extraordinary precision by atomic clocks, GPS satellites, and the Pound–Rebka experiment.

The Formula

For a clock at radius $r$ from a mass $M$ (outside the event horizon, $r > r_s$), compared to a clock at infinity, the time dilation factor is:

$\frac{d\tau}{dt_\infty} = \sqrt{1 - \frac{r_s}{r}} = \sqrt{1 - \frac{2GM}{c^2 r}}$

Equivalently, one local second ($d\tau = 1$) corresponds to a longer interval at infinity:

$dt_\infty = \frac{d\tau}{\sqrt{1 - r_s/r}} = \frac{d\tau}{\sqrt{1 - 2GM/(c^2 r)}}$

The factor $1/\sqrt{1 - r_s/r} \geq 1$ tells you how much the remote observer's time is stretched relative to the local clock.

GPS Satellites

GPS satellites orbit at $r \approx 26{,}560$ km. Because they are higher in Earth's gravitational field, their clocks run faster than clocks on the surface by about $45\,\mu$s per day due to gravitational dilation (partially offset by the special-relativistic slowdown from orbital speed). Without GR corrections, GPS would drift by kilometres per day.

Your Task

Implement these functions with all constants defined inside each function:

• time_dilation_factor(M, r) — returns $1/\sqrt{1 - 2GM/(c^2 r)}$, the factor $> 1$ by which a remote observer sees the local clock run slow
• time_at_infinity(dt_local, M, r) — returns the elapsed time at infinity for a local interval $dt_{\text{local}}$
• gravitational_time_shift(dt_local, M, r) — returns the extra time at infinity: $dt_\infty - dt_{\text{local}}$