4vector

A 4-vector is a mathematical object used in physics, particularly in special relativity, to represent events or quantities that have four components. These components are typically three spatial coordinates and one time coordinate. The concept of the 4-vector is crucial for understanding how physical laws remain consistent in different inertial frames of reference.

The components of a 4-vector are usually denoted by $A^\mu$, where the Greek index $\mu$ takes values 0, 1, 2, and 3. The index 0 often represents the time component, while indices 1, 2, and 3 represent the spatial components (e.g., x, y, and z). The order of components can vary depending on the convention used.

A key feature of 4-vectors is how their components transform under Lorentz transformations, which are the transformations between different inertial frames in special relativity. Unlike ordinary vectors in Euclidean space, the time and space components of a 4-vector mix under these transformations. This mixing ensures that physical quantities like the speed of light are invariant.

Examples of 4-vectors in physics include the spacetime position 4-vector ($x^\mu = (ct, x, y, z)$), the four-momentum ($p^\mu = (E/c, p_x, p_y, p_z)$), and the four-current ($J^\mu = (c\rho, J_x, J_y, J_z)$). The concept simplifies the formulation of many physical theories, such as electromagnetism and quantum field theory, by allowing relativistic equations to be written in a compact and covariant form. The "length" or norm of a 4-vector, defined using a specific metric, is an invariant quantity under Lorentz transformations, which has significant physical implications.