Relativistic Kinematics: Time Dilation, Length Contraction, and Velocity Addition

Manual: General · Subject: Physics

Develops the observable kinematic consequences of Lorentz symmetry and their operational interpretation.

Observable Consequences of Lorentz Symmetry

Time Dilation

A clock moving with speed $v$ relative to an inertial observer ticks more slowly by the factor $\gamma$ . If $\Delta \tau$ is the proper time between two events on the clock worldline, then $\Delta t = \gamma \Delta \tau.$

Length Contraction

A rod of proper length $L_0$ parallel to the direction of motion is measured to have length $L = \frac{L_0}{\gamma}$ in a frame where it moves at speed $v$ . The measurement must be made using simultaneous endpoint readings in the observer’s frame.

Velocity Addition

If an object moves with velocity $u'$ in a frame moving at velocity $v$ , the velocity in the lab frame is $u = \frac{u'+v}{1+u'v/c^2}.$ This ensures that no signal exceeds $c$ when physical velocities are subluminal.

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Operational Caution

Length contraction is not a visual distortion alone; it is a coordinate-dependent measurement requiring synchronized clocks in the observer’s frame.

Proper time is best described as:

The time measured by any distant observer The time between events in the rest frame of the object or along its worldline The coordinate time in Newtonian mechanics The time after applying length contraction

What is the relativistic velocity-addition formula for collinear velocities?

Key Kinematic Effects

Effect	Formula	Measurement Meaning
Time dilation	$\Delta t = \gamma \Delta \tau$	Moving clocks tick slower
Length contraction	$L = L_0/\gamma$	Moving rods are shorter along motion
Velocity addition	$u=(u'+v)/(1+u'v/c^2)$	Speeds do not simply add
Relativity of simultaneity	$\Delta t' = \gamma(\Delta t - v\Delta x/c^2)$	Simultaneous events depend on frame

Why can’t two subluminal velocities add to exceed $c$ in special relativity?

Because

\gamma

becomes complex below

c

Because the velocity-addition law saturates at

c

Because masses become infinite instantly Because acceleration is forbidden

How to Solve Kinematics Problems

1

Step 1: Identify the rest frame and the proper quantity.
2

Step 2: Compute $\gamma$ from the relative speed.
3

Step 3: Apply the correct invariant relation, such as $\Delta t = \gamma \Delta \tau$ .
4

Step 4: Check simultaneity conditions before comparing lengths or event times.

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