Four-vector: Difference between revisions

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Revision as of 18:57, 18 July 2003 editStevenj (talk \| contribs)Extended confirmed users14,829 edits a four-vector in physics is a more specific kind of object than a generic 4-tuple← Previous edit		Revision as of 00:26, 11 August 2003 edit undoPhys (talk \| contribs)3,350 edits Added link to sign conventionNext edit →
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	Strictly speaking, this is not a proper ], since its value can be negative. Like the ordinary ] of three-vectors, however, the result of this scalar product is a ]: it is ] 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.		Strictly speaking, this is not a proper ], since its value can be negative. Like the ordinary ] of three-vectors, however, the result of this scalar product is a ]: it is ] 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. See ].

	The laws of ] 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 Lorentz-transformed so that the perceiving object is stationary.		The laws of ] 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 Lorentz-transformed so that the perceiving object is stationary.

Revision as of 00:26, 11 August 2003

In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). The set of all such rotations and boosts, called Lorentz transformations and described by 4×4 matrices, forms the Lorentz group.

Examples of four-vectors include the coordinates (ct, x, y, z) themselves, the four-current (cρ, J) formed from charge density ρ and current density J, the electromagnetic four-potential (φ, A) formed from the scalar potential φ and vector potential A, and the four-momentum (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 (time-like, labeled by index 0) has the same units as the following three coordinates (space-like, labeled by indices 1,..,3).

The scalar product between four-vectors a and b is defined as follows:

a\cdot b=\left({\begin{matrix}a_{0}&a_{1}&a_{2}&a_{3}\end{matrix}}\right)\left({\begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)\left({\begin{matrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{matrix}}\right)=-a_{0}b_{0}+a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}

Strictly speaking, this is not a proper inner product, since its value can be negative. Like the ordinary dot product of three-vectors, 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. See Sign convention.

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 Lorentz-transformed so that the perceiving object is stationary.