quantity

spacetime interval

spacetime interval — from formula anatomy
9 related articles · 3 formulas

Description

The spacetime interval is a quantity in special relativity that measures the 'distance' between two events in four-dimensional spacetime. Unlike ordinary spatial distance, the interval is invariant under Lorentz transformations, meaning all observers agree on its value regardless of their relative motion. It is defined as Δs² = c²Δt² − Δx² − Δy² − Δz², where c is the speed of light.

History

The concept was introduced by Hermann Minkowski in 1908, who reformulated Einstein's special relativity in terms of a four-dimensional spacetime geometry. Minkowski's insight was crucial for the later development of general relativity.

How it works

The sign of Δs² determines the causal relationship between events: if Δs² > 0, the events are timelike-separated (one can causally influence the other); if Δs² < 0, they are spacelike-separated (no causal connection); if Δs² = 0, they lie on a light cone. The interval is the analog of the Pythagorean theorem in spacetime, but with a minus sign for spatial coordinates, reflecting the different geometry of spacetime.

In practice

The spacetime interval is fundamental in particle physics for analyzing decay processes and collisions, and in GPS satellite systems, where relativistic corrections are essential for accurate positioning.

Fun fact

The minus sign in the interval is why the geometry of spacetime is called 'pseudo-Riemannian'—it behaves like Euclidean geometry but with a flipped sign for time, leading to the famous 'light cone' structure.

Concept graph →

Formulas

$$ds^2 = -e^{2\rho} dt^2 + e^{2\lambda} dr^2 + r^2 d\Omega^2$$
Metric for spherically symmetric spacetime in general coordinates.
What is in the formula
ds^2variablespacetime interval — spacetime interval metre squared
dtvariablecoordinate time differential — time differential second
drvariableradial coordinate differential — radial coordinate differential metre
dΩ^2variablesolid angle element squared — solid angle element squared
ρvariablemetric function of r and t — metric function
λvariablemetric function of r and t — metric function
rvariableradial coordinate — radial coordinate metre
$$ds^2 = -e^{2\Phi(r)} dt^2 + \left(1 - \frac{b(r)}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$$
A static wormhole metric with shape function b(r) and redshift function Φ(r).
What is in the formula
ds^2variableSpacetime interval squared — spacetime interval metre squared
\Phi(r)variableRedshift function — redshift function
b(r)variableShape function — shape function metre
rvariableRadial coordinate — radial coordinate metre
tvariableTime coordinate — time coordinate second
d\Omega^2variableSolid angle element squared — solid angle element
doperatorExterior derivative — differential operator
$$ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\Omega^2$$
Describes spacetime around a spherically symmetric mass.
What is in the formula
ds^2variablespacetime interval — spacetime interval metre squared
dtvariablecoordinate time differential — time differential second
drvariableradial coordinate differential — radial coordinate differential metre
dΩ^2variablesolid angle element squared — solid angle element squared
rvariableradial coordinate — radial coordinate metre
tvariablecoordinate time — coordinate time second
GconstantNewton's constant of universal gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
cconstantspeed of light in vacuum — speed of light = 299792458 metre per second
Mconstantmass of the central object — mass kilogram
doperatorinfinitesimal change — differential

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