3.12.3.3 - Time Dilation

One clock and two events

The proper time $t_0$ between two events is the interval measured in the inertial frame where those events occur at the same position. One clock at that position can register both. For a spacecraft clock, two ticks occur at one position in the spacecraft frame, even though they occur at different Earth-frame positions.

Earth-frame observers can measure the interval $t$ using synchronised clocks at rest at the two event positions. They obtain $t>t_0$ when the spacecraft moves at non-zero speed relative to them. This is time dilation: a longer frame-measured interval between successive ticks means the moving clock runs more slowly relative to that frame's clocks.

“An observer is at rest” does not identify proper time: every observer is at rest in their own coordinates. First name the two events, then ask whether they share a position in that frame. Two distant lamps flashing simultaneously in a laboratory are not a pair of ticks on one clock, so the elementary time-dilation equation cannot be applied to their zero laboratory interval.

These are measurements corrected for light's travel time from the events to the observer. Simply watching a receding clock adds signal-travel and Doppler effects, which are different from the comparison made here.

Deriving time dilation with a light clock

A light clock sends a pulse from its lower mirror to an upper mirror and back. The mirrors are separated by $d$, perpendicular to the clock's horizontal motion. A tick is the entire round trip, from emission at the lower mirror to return to that same mirror.

[DIAGRAM: asset_name: D.3.3 - Time Dilation - Diagram 1; asset_slug: D.3.3 - Time Dilation - Diagram 1; recommended_method: generated_png; caption: A light clock: vertical rest-frame path and diagonal laboratory path; description: Light clock rest-frame vertical round trip and laboratory-frame diagonal round trip through emission A, reflection B and return C. Half-trip sides are d, vt/2 and ct/2.]

The right panel shows mirror positions at three different events, not three fixed mirrors. The two panels are schematic; their common transverse separation is $d$.

In the clock frame, $ct_0=2d$. In the laboratory frame, the lower mirror moves horizontally by $vt$ during the tick. On each half-trip the pulse travels a diagonal distance $ct/2$, while the mirror moves $vt/2$. The speed of light is $c$ in both frames. Pythagoras gives

Substituting $d=ct_0/2$ gives $c^2t^2=c^2t_0^2+v^2t^2$. Rearrangement therefore yields

The Lorentz factor $\gamma$ is dimensionless. It equals 1 at $v=0$, exceeds 1 for $0<v<c$, and grows without limit as $v$ approaches $c$. At $0.10c$, $\gamma=1.005$: the tick interval increases by about $0.5\%$, a small but measurable effect with suitable equipment. At $0.80c$, $\gamma=1.667$: the moving clock advances by $0.600\,\mathrm{s}$ per laboratory second. A percentage increase in the interval is not the same percentage decrease in the clock rate.

Calculating intervals and journey times

Frame labels determine which way to use the equation. Multiplying a proper interval by $\gamma$ gives the laboratory interval; dividing a laboratory interval by $\gamma$ gives the elapsed time on the moving clock.

From ticks to relative speed

An onboard clock advances $12.0\,\mathrm{s}$ between two events, while synchronised laboratory clocks assign $20.0\,\mathrm{s}$ between those same events. Both happen at the onboard clock, so

Only a ratio of times occurs, so matching units cancel. Do not put $t/t_0$ inside that square root.

A delivery deadline travels with the cargo

Two stations at rest in one inertial frame are $6.00\times10^{12}\,\mathrm{m}$ apart. A spacecraft coasts between them at $0.80c$. Ignore acceleration stages. The station-frame time is

The cargo clock records $t_0=t\sqrt{1-0.80^2}=1.50\times10^4\,\mathrm{s}=4.17\,\mathrm{h}$. Cargo lasting $4.50\,\mathrm{h}$ on its own clock survives because $4.17<4.50$. Comparing its deadline directly with the $6.94\,\mathrm{h}$ station time would mix frames. State the numerical comparison as well as “yes”.