The postulates of special relativity and time dilation

Key ideas

• Special relativity is Einstein's theory of how space and time behave for observers moving at constant velocity relative to each other (an inertial frame). It rests on two postulates — statements taken as true because every experiment testing them has confirmed them.
  • Postulate 1 (principle of relativity): the laws of physics are the same in every inertial frame. No experiment done inside a smoothly moving lab can tell you how fast, or even whether, it is moving.
  • Postulate 2 (constancy of $c$): the speed of light in a vacuum, $c = 3.00\times10^8$ m s⁻¹, is measured to be the same by every observer, regardless of the observer's motion or the motion of the light source.
• Postulate 2 is the strange one: if you chase a beam of light, classical physics says you should measure a slower speed for it. Experiment says you still measure $c$. Something classical has to give — and what gives is the assumption that time and length are absolute.
• Because every observer must agree on $c$, but $c = \dfrac{\text{distance}}{\text{time}}$, observers who disagree on relative velocity must also disagree on the distance and the time between the same two events.
• Time dilation is the result for time: a moving clock runs slow, as measured by an observer who sees it move.

Proper time and the time dilation equation

• Proper time, $t_0$, is the time interval between two events measured by an observer for whom both events happen at the same place — i.e. the observer moving with the clock (or the process being timed).
• Any other observer, who sees the clock moving past them at speed $v$, measures a longer interval, $t$:

$t = \gamma t_0$

• Each variable, with sub-bullets:
  • $t$ — the dilated time, measured by the observer who sees the clock/event moving (s)
  • $t_0$ — the proper time, measured in the frame where the two events occur at the same point in space (s)
  • $\gamma$ — the Lorentz factor (dimensionless, always $\ge 1$):

$\gamma = \frac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}$

• $v$ — the relative speed between the two observers (m s⁻¹)
• $c$ — the speed of light, $3.00\times10^8$ m s⁻¹
• Since $\gamma \ge 1$ always, $t \ge t_0$: the proper time is always the shortest time any observer can measure between the two events. Every other observer measures it as longer — this is what "moving clocks run slow" means.
• At everyday speeds ($v \ll c$), $\gamma \approx 1$ and time dilation is undetectably small — this is why it never shows up in daily life.

Tips

• Identify the proper time first, every time. Ask "in which frame do the two events happen at the same place?" — that observer has $t_0$. Plugging numbers into $t = \gamma t_0$ without deciding this is the single biggest source of wrong answers.
• $\gamma$ only depends on $v/c$, so it's often quickest to compute $v/c$ as a decimal first, then work through $\gamma$ — keep several decimal places, since $\gamma$ is sensitive to rounding when $v$ is close to $c$.