Examples of fourvectors include the coordinates (ct, x, y, z) themselves, the fourcurrent (cρ, J) formed from charge density ρ and current density J, the electromagnetic fourpotential (φ, A) formed from the scalar potential φ and vector potential A, and the fourmomentum (E/c, p) formed from the (relativistic) energy E and momentum p. The speed of light (c) is often used to ensure that the first coordinate (timelike, labeled by index 0) has the same units as the following three coordinates (spacelike, labeled by indices 1,..,3).
The scalar product between fourvectors a and b is defined as follows:
Strictly speaking, this is not a proper inner product, since its value can be negative. Like the ordinary dot product of threevectors, however, the result of this scalar product is a scalar: it is invariant under any Lorentz transformation. (This property is sometimes use to define the Lorentz group.) The 4×4 matrix in the above definition is called the metric tensor, sometimes denoted by g; its sign is a matter of convention, and some authors multiply it by 1.
The laws of physics are also postulated to be invariant under Lorentz transformations. An object in an inertial reference frame will perceive the universe as if the universe were Lorentztransformed so that the perceiving object is stationary.
See also: fourvelocity, fouracceleration, fourmomentum, fourforce.
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