Explain Einstein's two postulates of special relativity (the principle of relativity and the constancy of the speed of light), and apply the time dilation formula $t = \gamma t_0$ where $\gamma = 1/\sqrt{1 - v^2/c^2}$ to predict the time experienced by moving observers

What this dot point is asking

QCAA wants you to explain Einstein's two postulates of special relativity, define proper time and the Lorentz factor, and apply the time dilation formula to calculate the time elapsed for observers in relative motion. Cross-link: see the relativity calculator.

The postulates of special relativity

Einstein's 1905 special relativity is built on two postulates:

Postulate 1: The principle of relativity. The laws of physics are the same in all inertial (non-accelerating) reference frames. No experiment can detect uniform motion.

Postulate 2: The constancy of the speed of light. The speed of light in vacuum is $c = 3 \times 10^8$ m s$^{-1}$ in every inertial frame, regardless of the motion of the source or the observer.

Postulate 2 is the surprising one. Classical (Galilean) addition of velocities would predict that a moving source's light should travel at $c + v$ in some frame. Einstein asserted (and the Michelson-Morley experiment confirmed) that this is not so: every observer measures $c$ for the speed of light.

Consequences

The two postulates have radical consequences that defy classical intuition:

• Time dilation. Moving clocks run slow.
• Length contraction. Moving objects are shortened along their direction of motion.
• Relativity of simultaneity. Events simultaneous in one frame are not simultaneous in another.
• Mass-energy equivalence. $E = mc^2$.
• No object with rest mass can reach $c$. The energy required would be infinite.

This dot point focuses on time dilation. The other consequences are covered in separate dot points.

The Lorentz factor

The Lorentz factor $\gamma$ quantifies the relativistic distortion:

Properties:

• IMATH_9 always.
• IMATH_10 at $v = 0$ (no distortion).
• IMATH_12 as $v \to c$ (extreme distortion).

Examples:

At everyday speeds ($v \ll c$), $\gamma$ is so close to 1 that relativistic effects are unmeasurable. They become important above about $v = 0.1 c$.

Time dilation

A clock at rest in some frame measures the proper time $t_0$ between two events at the same location in that frame. An observer in a different frame, moving relative to the first at speed $v$, measures a longer time:

The moving clock appears to run slowly compared to the stationary observer's clock.

Which time is "proper"?

The proper time is the time measured in the frame where the two events occur at the same place. For a moving clock, the two events (start tick and end tick) occur at the same place in the clock's own frame, so the clock-frame time is the proper time. From any other frame, the same two events occur at different places (the clock moved), and the elapsed time is $\gamma t_0$.

Time dilation is real

Time dilation is not an optical illusion. Direct experimental confirmations:

• Muon decay. Cosmic-ray muons reach Earth's surface despite their short rest-frame half-life because their lifetime as measured from Earth is dilated.
• Atomic clocks on aeroplanes. Hafele-Keating (1971) flew atomic clocks around the world. The travelling clocks showed measurable time differences from stationary clocks on the ground, in agreement with relativistic predictions.
• GPS satellites. GPS clocks are travelling at high speed (relative to the ground) and in weaker gravity. Without relativistic corrections (both special and general), GPS positions would drift by kilometres per day.
• Particle accelerators. Unstable particles travelling at relativistic speeds live measurably longer in the lab than in their own rest frame.

The twin paradox

A classic thought experiment. Twin A stays on Earth; Twin B travels at near-$c$ to a distant star and back. When they reunite, Twin B is younger than Twin A. Why?

The "paradox" is resolved by noting that the situation is not symmetric. Twin B undergoes acceleration (turnaround at the star), so Twin B is not in a single inertial frame. The asymmetry breaks the apparent symmetry of the situation.

In the simplest analysis, Twin A measures the round-trip time (no acceleration, all inertial). Twin B's elapsed time is shorter by a factor of $\gamma$. Twin B returns younger.

Worked example: round-trip space travel

A spaceship travels to a star 4 light-years away (as measured from Earth) at $v = 0.80 c$ and returns at the same speed.

Earth-frame round-trip time. Distance $= 8$ light-years; speed $= 0.80 c$; time $= 8 / 0.80 = 10$ years.

Astronaut-frame round-trip time. Apply time dilation. The astronaut's clock is the proper time; the Earth's clock is dilated. So $t_{\text{Earth}} = \gamma t_{\text{astronaut}}$, giving $t_{\text{astronaut}} = 10 / 1.667 \approx 6$ years.

The astronaut returns younger by 4 years.

Note: the astronaut would also need to account for the length contraction of the trip from their own frame, which gives the same answer self-consistently. Both effects are required for a complete relativistic picture.

Common errors

Confusing proper time with dilated time. Proper time is the shorter time, measured in the frame where the two events occur at the same place.

Wrong formula direction. $t = \gamma t_0$. The dilated time is larger by a factor $\gamma$.

Treating relativity as applicable only to extreme speeds. Even at $v / c = 0.1$, $\gamma \approx 1.005$, a 0.5 percent effect. The effects are real at all speeds; they are just very small at non-relativistic speeds.

Confusing $\gamma$ with $1/\gamma$. Length contraction has factor $1/\gamma$ (length shortens); time dilation has factor $\gamma$ (time lengthens).

Mixing inertial and non-inertial frames. Special relativity strictly applies only to inertial (non-accelerating) frames. Accelerating frames (like Twin B during turnaround) require care or general relativity.

In one sentence

Special relativity rests on two postulates (the laws of physics are the same in all inertial frames; the speed of light is constant for all observers), from which follows time dilation $t = \gamma t_0$ where $\gamma = 1/\sqrt{1 - v^2/c^2}$; moving clocks run slow by a factor $\gamma$ compared to stationary observers, an effect verified by muon decay, atomic clocks on aeroplanes, GPS satellites, and particle accelerators.