Four-vector

A four-vector is a vector whose four vector components transform under a Lorentz transformation as $U^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{U}^{\nu }$. Its individual vector components generally change between inertial frames; the Minkowski metric combines them into invariant scalar products. Listing four quantities is not enough to make a four-vector: their transformation law matters.

The most direct example is the displacement between two spacetime events, written $(c\Delta t,\Delta x,\Delta y,\Delta z)$. Multiplying the time difference by the speed of light gives all four vector components the same length units. A position four-vector describes an event relative to a chosen origin, while a displacement does not depend on a common translation of the origin. Four-momentum and four-velocity use the same transformation rule for different physical quantities.

A four-vector is a mathematical object used in the theory of relativity, which combines space and time into a single entity. In the context of physics, four-vectors help simplify the description of physical phenomena in a way that respects the principles of special relativity. A four-vector has four components, typically denoted as \( V^\mu \), where \( \mu = 0, 1, 2, 3 \).