Near a mass, clocks run slow and light loses energy as it climbs out. This is a second, separate kind of time dilation from the one in special relativity, which comes from motion. It is set by the Schwarzschild radius and by how close you are to it.
💡 Why it works
A photon of energy $E = hf$ carries mass-equivalent $E/c^2$. Climbing out of a gravitational well from radius $R$ to far away costs it about $(E/c^2)(GM/R)$, so its frequency drops by a fraction $$\frac{\Delta f}{f} \approx \frac{GM}{Rc^2}.$$ Equivalently, a clock deep in the well ticks slower than one far away. In general relativity the exact rate for a clock hovering at radius $r$ outside a nonspinning mass is $$\frac{d\tau}{dt} = \sqrt{1 - \frac{r_s}{r}},$$ which reduces to $1 - GM/(rc^2)$ when $r \gg r_s$.
🔑 Key idea
Light leaving radius $r$ arrives far away stretched by $1 + z = 1/\sqrt{1 - r_s/r}$ (so $z \approx GM/(rc^2)$ in a weak field). At the horizon the factor becomes infinite. This effect depends on where a clock is; special relativity's time dilation depends on how fast it moves.
Clocks at different distances from a black hole (in units of $r_s$). Clocks closer in tick more slowly as seen from far away.
Worked examples
✏️ Worked example: Sirius B's redshift
Sirius B has $1.02\,M_\odot$ and $R = 5{,}840$ km. Find $GM/(Rc^2)$ and the equivalent Doppler speed $cz$.
Answer: About 77 km/s; measurements of Sirius B's spectral lines give about 80 km/s. (For the Sun the shift is only 0.64 km/s.)
✏️ Worked example: GPS clocks
GPS satellites orbit at $r = 2.66 \times 10^{7}$ m. How much faster does gravity make their clocks run than ground clocks, per day? ($GM_\oplus = 3.99 \times 10^{14}$ m³/s².)
Answer: About 45.7 μs per day fast. Their orbital speed slows them by about 7.2 μs per day (special relativity), so the net is about 38.5 μs per day, which the system corrects for.
✏️ Worked example: Near a black hole
A clock hovers at $r = 2r_s$. How long does a distant observer see it take to tick off 1 hour?
Rate $= \sqrt{1 - 1/2} = 0.707$.
$\Delta t = 1\text{ h}/0.707$.
Answer: About 1.41 hours; its light arrives with wavelengths stretched by the same factor 1.41.
⚠️ Common mistake
Mixing up gravitational time dilation with the time dilation of motion. Special relativity's $\gamma$ depends on speed and is symmetric (each observer sees the other's clock slow). Gravitational time dilation depends on position in a gravitational field and is not symmetric: the clock lower in the well runs slow as judged by both. GPS needs both effects, and they have opposite signs. Check the sign: a higher clock always runs fast compared with a lower one.
In 1960 Robert Pound and Glen Rebka reported measuring the shift for gamma rays climbing a 22.5 m tower at Harvard: $\Delta f/f = gh/c^2 = 2.5 \times 10^{-15}$, as predicted.
Practice
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