Theory of relativity

Time dilation and length contraction

A moving clock accumulates less time between meetings, and a moving object contracts along the direction of motion.

Proper time Δτ is the time between two events measured in the frame where they occur at the same position, such as the clock’s rest frame. An observer seeing the clock move measures Δt=γΔτ.

\Delta t=γ\Delta\tau,\qquad L=\frac{L_0}{γ},\qquad γ=\frac{1}{\sqrt{1−v^2/c^2}}

Definition: Definition

Proper time Δτ is measured by a clock accompanying the events; proper length $L_0$ is measured in the object’s rest frame. Δt and L are measured in a frame where the clock or object moves.

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Physical meaning

Length contraction occurs only along the direction of motion; transverse dimensions are unchanged. The object’s endpoints must be measured simultaneously in the same frame.

Example: Worked example

A clock moves at 0.8c and measures proper time Δτ=10 s. A rod with proper length L₀=5 m moves at the same speed. Find Δt and L.

Solution

γ=1/√(1−0.8²)=5/3. Thus Δt≈16.7 s and L=5/(5/3)=3.0 m.

Example: A moving clock and rod

A clock with a rest-frame period of $1.0\,s$ moves at $v=0.6c$ relative to a laboratory. What period does the laboratory measure? If a rod of proper length $10\,m$ moves at the same speed, what length is measured?

Solution

$γ=1/\sqrt{1-0.6^2}=1.25$ . The laboratory period is $γ\Delta\tau=1.25\,s$ ; the length parallel to motion is $L=L_0/γ=8.0\,m$ . Time intervals dilate while longitudinal lengths contract.

To measure the length of a moving object, record both endpoints simultaneously in the measuring frame; measurements at different times do not define its length there. Proper length $L_0$ is measured in the object’s rest frame, whereas $L=L_0/γ$ is measured in a frame where it moves. A clock measures proper time along its worldline between meetings; laboratory coordinate time uses synchronized clocks distributed through the frame. Keeping these measurement procedures distinct prevents apparent paradoxes.

Both effects are tested: atomic clocks on aircraft and satellites differ from ground clocks, while atmospheric muons survive longer in Earth’s frame because of time dilation.

Quick check

A rod has proper length L₀=6 m and moves with γ=2. What length is measured along its motion?

Which quantity is invariant under Lorentz transformations?

References

Edwin F. Taylor, John Archibald Wheeler (1992). Spacetime Physics
Robert Resnick (1968). Introduction to Special Relativity