4vectors

In physics, a 4-vector is a mathematical object that has four components and transforms in a specific way under Lorentz transformations. These transformations are fundamental to special relativity and describe how observations change between different inertial frames of reference. A 4-vector can be thought of as a generalization of a 3-vector from Euclidean space to spacetime.

The most common 4-vectors in relativity are the position 4-vector, momentum 4-vector, and velocity 4-vector. The position 4-vector is given by (ct, x, y, z), where c is the speed of light and t, x, y, z are the spacetime coordinates. The momentum 4-vector is (E/c, px, py, pz), where E is energy and p is the 3-momentum. The velocity 4-vector is related to the rate of change of the position 4-vector with respect to proper time.

A key property of 4-vectors is their invariance under Lorentz transformations. This means that the "length" of a 4-vector, defined using a specific metric (the Minkowski metric), remains the same regardless of the observer's inertial frame. This invariance is crucial for formulating physical laws in a way that is consistent with special relativity. The inner product of two 4-vectors is invariant, which allows for the formulation of scalar quantities that are independent of the observer's motion. The concept of 4-vectors simplifies the expression of many relativistic equations, making them more elegant and easier to manipulate.