Four-vector

Four-vector

A four-vector is a mathematical object consisting of four components, typically representing the coordinates of an event in a four-dimensional Minkowski space. Vectors and tensors that transform according to the Lorentz transformation are called four-vectors and four-tensors. Momentum and energy combine as a four-vector in special relativity.From: Quantum Mechanics [2001], Classical Mechanics [2019], Light - The Physics of the Photon [2019], Vacuum friction [2018]

Geometrical optics

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Published in Timothy R. Groves, Charged Particle Optics Theory, 2017

Timothy R. Groves

An arbitrary four-vector Aμ = (A1, A2, A3, A4) is defined in terms of its four components. For two reference frames in relative uniform motion with velocity v along the z-direction, the components of Aμ are related in the two frames by A1′=A1A2′=A2A3′=γ(A3+iβA4)A4′=γ(−iβA3+A4) where γ is given by (2.9) and β = υ/c. This is known as a Lorentz transformation. Show that the inner product of any two four-vectors Aμ and Bμ satisfies ∑μ=14Aμ′Bμ′=∑μ=14AμBμ. An inner product of two four-vectors is thus said to be invariant under a Lorentz transformation.

Vacuum friction

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Published in Journal of Modern Optics, 2018

Stephen M. Barnett, Matthias Sonnleitner

From the viewpoint of those schooled in special relativity, this makes perfect sense: the transition has lowered the total energy of the atom by and using the most famous equation in physics, , we are led to conclude that the ground state atom is indeed lighter than the excited state atom by precisely . Indeed very early in the development of relativity, Einstein established this relationship between a change in internal energy and a change in inertia [14]. From the viewpoint of special relativity this is entirely unsurprising, of course, given the close relationship between momentum and energy, which combine as a four-vector.

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Geometrical optics

Vacuum friction