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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 4×4 matrices, forms the Lorentz group. | In ], a '''four-vector''' is a ] in a four-dimensional real ], whose components transform like the ] and ] 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 ], forms the Lorentz group. | ||
| The basic vector of the ''']''' is the "event" |
The basic vector of the ''']''' is the "event" four-vector, defined as: | ||
| :<math> \mathbf{r} = \left( c \cdot t, x, y, z \right) </math> | :<math> \mathbf{r} = \left( c \cdot t, x, y, z \right) </math> | ||
| where ''c'' is the ]. | where ''c'' is the ]. | ||
| It is called an "event" vector because of ]'s interpretation which state that the only |
It is called an "event" vector because of ]'s interpretation which state that the only actual measurable ] entities are intersection of two events in space and time (i.e. Two bodies meet at point ''(x,y,z)'' in time ''t''). | ||
| Examples of four-vectors include the coordinates (''ct'', ''x'', ''y'', ''z'') themselves, the ] (''c''ρ, '''J''') formed from charge density ρ and current density '''J''', the ] (φ, '''A''') formed from the scalar potential φ and vector potential '''A''', and the ] (''E''/''c'', '''p''') formed from the (relativistic) energy ''E'' and momentum '''p'''. The ] (''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). | Examples of four-vectors include the coordinates (''ct'', ''x'', ''y'', ''z'') themselves, the ] (''c''ρ, '''J''') formed from charge density ρ and current density '''J''', the ] (φ, '''A''') formed from the scalar potential φ and vector potential '''A''', and the ] (''E''/''c'', '''p''') formed from the (relativistic) energy ''E'' and momentum '''p'''. The ] (''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). | ||
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| Strictly speaking, this is not a proper ] because ''x'' · ''x'' < 0 for some ''x''. 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 used 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 ]. | Strictly speaking, this is not a proper ] because ''x'' · ''x'' < 0 for some ''x''. 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 used 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 |
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== | ==See also== | ||
Revision as of 22:49, 26 January 2005
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.
The basic vector of the Minkowski space is the "event" four-vector, defined as:
where c is the speed of light. It is called an "event" vector because of Albert Einstein's interpretation which state that the only actual measurable physics entities are intersection of two events in space and time (i.e. Two bodies meet at point (x,y,z) in time t).
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:
Strictly speaking, this is not a proper inner product because x · x < 0 for some x. 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 used 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.