# Special relativity and invariant intervals

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

Course: [Introduction to quantum physics and relativity](https://studybrix.com/en/courses/quantum-and-relativity/index.md)
Level: university
Prerequisites: Algebra, spacetime diagrams and inertial frames.

## 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.

## Practice questions

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

Answer: 1.25

Explanation: 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.

Answer: 5 s

Explanation: Δt=γΔτ=5 s.

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

A. Any remote clock regardless of synchronization
B. The clock present at both events
C. Only a clock at infinity
D. No physical clock

Answer: The clock present at both events

Explanation: 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 +−−−.

Answer: 16

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

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

A. Equal to mass
B. Zero
C. Always negative
D. Always positive

Answer: Zero

Explanation: 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 / marking guidance: Lorentz transformations mix time with spatial separation. For separated events, simultaneity can differ between inertial frames.

Explanation: Credit spatial separation and frame transformation.

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

A. It is unrelated to energy
B. It is always greater than c
C. It is less than c
D. It must equal c

Answer: It is less than c

Explanation: 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 / 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.

Explanation: Credit operational event definitions.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.