---
title: "Spacetime Intervals and Four-Vectors"
description: "Observers disagree about distances and times but agree on the interval between two events, which is the invariant of spacetime. Grouping quantities into four-vectors makes that invariance automatic."
canonical: https://lightmysky.com/learn/science/spacetime-intervals-and-four-vectors-mt_CYdi7Qpvrf
source: https://lightmysky.com/learn/science/spacetime-intervals-and-four-vectors-mt_CYdi7Qpvrf.md
retrieved: 2026-09-12
---

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# Spacetime Intervals and Four-Vectors

Observers disagree about distances and times but agree on the interval between two events, which is the invariant of spacetime. Grouping quantities into four-vectors makes that invariance automatic.

Subject: Science · Area: Quantum & Modern Physics · Ages 19 to 20
Page: https://lightmysky.com/learn/science/spacetime-intervals-and-four-vectors-mt_CYdi7Qpvrf

## Ready when they can

- Computes the interval between two events and classifies it as timelike, spacelike or lightlike
- Shows that the interval is unchanged by a Lorentz transformation
- Explains why causality survives even though the time order of some events is frame dependent

## Lesson: The distance everyone agrees on

Observers argue about times and lengths, but they share one quantity: the interval. For two events take s squared equals c delta t squared minus the squared space gaps. Every frame computes the same s squared, since the Lorentz remix is built to preserve it.

**Example.** Take two flashes 5 s apart in time and 3 light s apart in space. Then s squared is 25 minus 9, which is 16, a positive number. Positive means timelike: one slow traveller could visit both events, and all frames agree on their order.

A negative s squared is spacelike: no signal could connect the events, so frames may disagree on their time order, and neither could have caused the other. Zero is lightlike: only light itself could connect them. Causes always precede effects, so causality survives.

**Tip.** Bundle time with space as the four-vector c t, x, y, z. With time scaled by c in the same bundle, the invariance comes along for free.

**Recap.** Compute the interval, read its sign, and you know who could have caused whom.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Inner Products, Length and Orthogonality](https://lightmysky.com/learn/mathematics/inner-products-length-and-orthogonality-mt_X0HPRGto4W)
- [Relativistic Momentum and Energy](https://lightmysky.com/learn/science/relativistic-momentum-and-energy-mt_zBq87F-j3B)

## Opens up

- [General Relativity: Curved Spacetime and Its Tests](https://lightmysky.com/learn/science/general-relativity-curved-spacetime-and-its-tests-mt_-rZZxZwU2L)
- [Field Quantisation: Creation Operators and Particle Number](https://lightmysky.com/learn/science/field-quantisation-creation-operators-and-particle-number-mt_BJyZWQA5cb)
- [The Klein-Gordon and Dirac Equations](https://lightmysky.com/learn/science/the-klein-gordon-and-dirac-equations-mt_njCW9OEevW)
- [Covariant Electrodynamics and the Field Tensor](https://lightmysky.com/learn/science/covariant-electrodynamics-and-the-field-tensor-mt_RoxM9z6DfK)