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Physics · lesson 1 of 4

Special relativity and invariant intervals

Special relativity describes inertial-frame transformations with invariant vacuum light speed. It replaces Galilean transformations when relativistic effects matter.

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Prerequisites: Algebra, spacetime diagrams and inertial frames.

Postulates

Special relativity describes inertial-frame transformations with invariant vacuum light speed. It replaces Galilean transformations when relativistic effects matter.

Proper time

Proper time is measured by a clock along its own worldline. For uniform inertial relative speed, a coordinate interval satisfies Δt=γΔτ.

Interval

Using the +−−− convention, s²=c²Δt²−Δx²−Δy²−Δz². The sign distinguishes timelike, null and spacelike separations.

Simultaneity

Spatially separated simultaneous events in one inertial frame need not be simultaneous in another. Time dilation must not be used without identifying the clock and events.

Worked examples

Example 1

For v=0.6c, find the Lorentz factor.

γ=1/sqrt(1−0.6²)=1/0.8.

1.25

Example 2

A clock accumulates proper time 4 s. For γ=1.25, find the corresponding coordinate time.

Δt=γΔτ=5 s.

5 s

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 For v=0.6c, find the Lorentz factor.

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1.25

γ=1/sqrt(1−0.6²)=1/0.8.

2 A clock accumulates proper time 4 s. For γ=1.25, find the corresponding coordinate time.s

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5 s

Δt=γΔτ=5 s.

3 Which clock measures proper time between two events on its worldline?

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The clock present at both events

Proper time belongs to the path joining the events.

4 In units c=1, events have Δt=5 and Δx=3, with other separations zero. Find s² using +−−−.

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16

s²=25−9=16, a timelike separation.

5 For a lightlike interval in vacuum, what is s²?

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Zero

Null separations have zero invariant interval.

6 Explain why a frame-independent statement that distant events occur at the same time is generally invalid.

Sample answer and marking guidance

Lorentz transformations mix time with spatial separation. For separated events, simultaneity can differ between inertial frames.

Credit spatial separation and frame transformation.

7 In an inertial frame, a massive free particle's speed is constrained how?

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It is less than c

Finite-energy massive particles do not attain the vacuum light speed.

8 Why must one identify event pairs before applying a time-dilation formula?

Sample answer and marking guidance

The formula relates coordinate and proper intervals for a specific clock worldline. Mixing intervals from different clocks or event pairs produces invalid comparisons.

Credit operational event definitions.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.