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Physics · University

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.

Study notes

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

For v=0.6c, find the Lorentz factor.

γ=1/sqrt(1−0.6²)=1/0.8.

1.25

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

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

    1. Any remote clock regardless of synchronization
    2. The clock present at both events
    3. Only a clock at infinity
    4. No physical clock

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

  3. For a lightlike interval in vacuum, what is s²?

    1. Equal to mass
    2. Zero
    3. Always negative
    4. Always positive

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

Answers and marking guidance

Answers and marking notes
  1. The clock present at both events

    Proper time belongs to the path joining the events.

  2. 16

    s²=25−9=16, a timelike separation.

  3. Zero

    Null separations have zero invariant interval.

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

    Credit spatial separation and frame transformation.

Further reading and review references

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