| Revision as of 23:31, 11 June 2003 editStevenj (talk | contribs)Extended confirmed users14,829 edits revisions, corrected definition (not defined by inner product, but by how it transforms under Lorentz group)← Previous edit | Revision as of 23:38, 11 June 2003 edit undoStevenj (talk | contribs)Extended confirmed users14,829 edits gave metric tensorNext edit → | ||
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| In ], a '''four-vector''' is a ] in a four-dimensional real ], 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 ]). The set of all such rotations and boosts, called ] and described by |
In ], a '''four-vector''' is a ] in a four-dimensional real ], 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 ]). The set of all such rotations and boosts, called ] 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 (''c''&rho, '''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 energy ''E'' and momentum '''p'''. In all of these cases, the ] (''c'') is 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). | Examples of four-vectors include the coordinates (''ct'', ''x'', ''y'', ''z'') themselves, the four-current (''c''&rho, '''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 energy ''E'' and momentum '''p'''. In all of these cases, the ] (''c'') is 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). | ||
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| :<math> | :<math> | ||
| (a,b) | |||
| ab | |||
| = | = | ||
| \begin{ |
\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) | |||
| . | |||
| \begin{ |
\left( \begin{matrix}b_0 \\ b_1 \\ b_2 \\ b_3 \end{matrix} \right) | ||
| = | = | ||
| a_0 b_0 |
-a_0 b_0 + a_1 b_1 + a_2 b_2 + a_3 b_3 | ||
| </math> | </math> | ||
| Strictly speaking, |
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. | ||
| 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 23:38, 11 June 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 (c&rho, 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 energy E and momentum p. In all of these cases, the speed of light (c) is 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:
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.
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.
See also: four-velocity, four-acceleration, four-momentum, four-force.