Vierervektor

ViererVektor, in physics often called a four-vector, is a vector defined in four-dimensional spacetime that combines time and spatial coordinates into a single object. In special relativity a ViererVektor X^μ has components that depend on the chosen frame, for example X^μ = (ct, x, y, z) or, with natural units c = 1, X^μ = (t, x, y, z). It transforms under Lorentz transformations X'^μ = Λ^μ_ν X^ν, ensuring the form of physical laws is frame-independent.

Covariant and contravariant forms are related by the metric g_{μν}. With the common Minkowski metric g_{μν} = diag(1, -1, -1, -1) (or its opposite sign convention), the covariant components X_μ = g_{μν} X^ν are used to form invariant inner products X^μ X_μ = g_{μν} X^μ X^ν, which remain the same in all inertial frames.

Key examples include the four-position x^μ = (ct, x, y, z) and the four-momentum p^μ = (E/c, p_x, p_y, p_z). The four-momentum satisfies the invariant p^μ p_μ = m^2 c^2, yielding the energy–momentum relation E^2 = p^2 c^2 + m^2 c^4. The four-velocity u^μ = dx^μ/dτ, where τ is the proper time, satisfies u^μ u_μ = c^2.

Applications of ViererVektoren range from relativistic dynamics and collision kinematics to the formulation of field theories. In general relativity the concept generalizes to vectors in curved spacetime, transforming according to the local geometry with the metric g_{μν}(x).