Gravitational time dilation

From the equivalence between gravity and acceleration alone, combined with the fact that light travels in a straight line in the instantaneously inertial frame, a surprising result follows at once: clocks run slower where gravity is stronger.

A photon emitted from the bottom of the tower rises towards the top “losing energy”, much as a stone would: its frequency decreases.

Here is a heuristic version of the argument. A photon emitted upward from below, rising through a height $H$, “loses energy” exactly as a rising stone does, by conservation of energy. But for a photon $E = h f$: losing energy means the frequency $f$ decreases. And frequency is nothing but “how many cycles per second”. If the receiver above gets fewer cycles per second than the source below emits, then the clock above must be running faster than the one below. Restated from the source’s point of view: the clock below runs slower than the one above.

Gravitational redshift

A photon of frequency $f_\text{em}$ emitted where the Newtonian gravitational potential is $\phi_\text{em}$, arriving where the potential is $\phi_\text{ric}$, has frequency $\ev{\frac{f_\text{ric}}{f_\text{em}} \approx 1 + \frac{\phi_\text{ric} - \phi_\text{em}}{c^2}}$ valid for $\abs{\phi}\ll c^2$. If the receiver is higher up ($\phi_\text{ric} > \phi_\text{em}$) then $f_\text{ric} < f_\text{em}$: the light shifts towards the red, gravitational redshift.

Near the Earth’s surface the Newtonian potential is $\phi = -GM_T/R_T$, and for a height difference $\Delta h$, $\Delta\phi = g\,\Delta h$. The formula thus reduces to the practical estimate $\frac{\Delta f}{f} \approx \frac{g\,\Delta h}{c^2}.$

The Pound–Rebka experiment

In 1959 Pound and Rebka measured gravitational redshift in the laboratory (Pound and Rebka 1960). A Mössbauer source ($\gamma$ rays from $^{57}$Fe, with $E_\gamma = 14.4$ keV) was placed at the base of a tower $H = 22.5$ m tall, with a detector at the top. What is the predicted shift?

$\frac{\Delta f}{f} \approx \frac{gH}{c^2} = \frac{9.81\cdot 22.5}{(3\cdot 10^8)^2} \approx 2.45\cdot 10^{-15}.$

A tiny effect, revealed only thanks to the extreme spectroscopic precision of the Mössbauer effect. It is direct confirmation: gravity affects clocks, in full agreement with general relativity.

GPS satellites

GPS satellites orbit at about $h \approx 20\,200$ km with speed $v \approx 3.87$ km/s. Two opposing effects act on them.

• Special relativity (the clock in flight runs slow relative to Earth because of $v$): $\Delta t_\text{RR}/t \approx -v^2/(2c^2) \approx -8.3\cdot 10^{-11}$, i.e. about $-7\;\mu$s per day.
• General relativity (the clock higher up, where the potential is less negative, runs faster): with $\Delta\phi = -GM/R_T + GM/(R_T+h)$ one finds $\Delta t_\text{RG}/t \approx +5.3\cdot 10^{-10}$, i.e. about $+45\;\mu$s per day.

Net effect: about $+38\;\mu$s per day. If left uncorrected, in a single day the GPS-triangulated position would drift by $c\cdot 38\,\mu\text{s} \approx 11.4$ km. This is why the atomic clocks on GPS satellites are corrected on the ground, before launch.

Links

Topics: Relatività ristretta Concepts: Effetto Doppler Skills: Analisi dimensionale

Related exercises: Problema — Un semaforo rosso visto verde · Problema — Composizione per via Doppler · Problema — Redshift di una galassia