Four-vector: Difference between revisions

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	In ], a '''four-vector''' is a ] in a four-dimensional real ] ~~with~~ a ] ~~between~~ ~~four-vectors~~ ''a'' and ''b'' ~~defined~~ as ~~follows:~~		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 <math>4 \times 4</math> 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).

			The ] between four-vectors ''a'' and ''b'' is defined as follows:

	:<math>		:<math>
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	a_0 b_0 - a_1 b_1 - a_2 b_2 - a_3 b_3		a_0 b_0 - a_1 b_1 - a_2 b_2 - a_3 b_3
	</math>		</math>
	Strictly speaking, the ] of four-vectors is not a proper inner product, since its value can be negative.		Strictly speaking, the ] of four-vectors is not a proper inner product, 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.)

	In the above, ''a''<sub>0</sub> would represent time, and ''a''<sub>1</sub>, ''a''<sub>2</sub> and ''a''<sub>3</sub> would represent space, and similarly for ''b''. The proportion between space coordinates and time coordinates should be ]; for example, if the time ] is given in ]s, the space coordinates should be given in ]s.

	The scalar product of two four-vectors is ] under any ]. The laws of ] are also supposed to be invariant under a Lorentz transformation.

	An object will perceive the universe as if the universe were lorentz-transformed so that the perceiving object is stationary. ''Stationary'' is defined only with respect to a chosen ].

	Often, ''ds'' or ''d''τ will be used, when talking about a four-vector; in that case, it will mean:

	:<math>ds=d\tau=\left\|da\right\|=\sqrt{(da)^2}.</math>

			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.
	''ds'' will be used when talking about distances, and ''d''τ used when talking about time. ∫dτ is the amount of time a particle will experience travelling between two points in ].

	See also: ], ], ], ].		See also: ], ], ], ].

Revision as of 23:31, 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 $4\times 4$ 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:

ab={\begin{vmatrix}a_{0}\\a_{1}\\a_{2}\\a_{3}\end{vmatrix}}.{\begin{vmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{vmatrix}}=a_{0}b_{0}-a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}

Strictly speaking, the inner product of four-vectors 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 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.