--- title: "Spacetime Intervals and Four-Vectors" description: "Inertial observers can measure different time and space separations while agreeing on the spacetime interval. A four-vector transforms by the Lorentz rule, which preserves its Minkowski squared norm." 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Spacetime Intervals and Four-Vectors Inertial observers can measure different time and space separations while agreeing on the spacetime interval. A four-vector transforms by the Lorentz rule, which preserves its Minkowski squared norm. 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 interval between two events In flat spacetime, s squared = (c delta t) squared - (delta x) squared - (delta y) squared - (delta z) squared. Inertial frames agree on this quantity. A light-second is the distance light travels in one second. With seconds and light-seconds, c delta t has the numerical value of delta t and s squared is in light-seconds squared. **Example.** For gaps measured in seconds and light-seconds, subtract squared spatial distance from the squared time gap. A 4-second gap and 2-light-second distance give 16 - 4 = 12 light-seconds squared. Positive means timelike and permits a traveller below light speed. Negative means spacelike: no causal signal can connect the events. A permitted connection does not prove causation. For distinct events, positive s squared means timelike, negative means spacelike, and zero means lightlike. Timelike pairs allow a signal below light speed; lightlike pairs allow one at light speed. Spacelike pairs would require a faster-than-light signal and can reverse time order between inertial frames. Future-oriented inertial frames agree on timelike and lightlike order. *(drawing: Using seconds and light-seconds, square each numerical gap and subtract. The positive result permits a slower-than-light connection.)* In an inertial frame, a force-free object moves at constant velocity. The position four-vector X = (ct, x, y, z) has length-valued components that follow the Lorentz law; four entries alone do not make a four-vector. Differences of position four-vectors follow that law too. Their Minkowski squared norm is the spacetime interval. Unlike the positive-definite inner product for ordinary lengths, this signed norm can be negative or zero for a nonzero separation. For an x-direction boost, ct' = gamma(ct-beta x) and x' = gamma(x-beta ct), where beta = v/c and gamma = 1/sqrt(1-beta squared). Use these equations for event separations; y and z stay unchanged. Calculate the interval before and after the boost. Expanding (ct') squared - (x') squared cancels the mixed terms and gives gamma squared times (1-beta squared) times ((ct) squared - x squared). This prefactor is one, so the interval is preserved. A possible cause must reach its effect at speed c or below. For a future-directed timelike or lightlike separation, ct > 0 and ct >= the spatial distance. Since |beta| < 1, ct-beta x stays positive, so ct' stays positive. Only spacelike pairs can reverse order. A timelike or lightlike interval allows causal influence; it does not prove that influence occurred. **Recap.** Use one sign convention and consistent units. Lorentz boosts preserve the interval; its sign limits possible causal connections. *Lesson and assessment text authored for this dataset.* ## 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) ## 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)