Four-vector: Difference between revisions

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

		⚫	:<math>
	{\|
			a \cdot b
	\|         <math> a \cdot b \,</math>
			=
⚫	\| <math> =
	\left( \begin{matrix}a_0 & a_1 & a_2 & a_3 \end{matrix} \right)\,		\left( \begin{matrix}a_0 & a_1 & a_2 & a_3 \end{matrix} \right)
	\left( \begin{matrix} -1 & 0 & 0 & 0 \\		\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)
	0 & 1 & 0 & 0 \\
			=
	0 & 0 & 1 & 0 \\
		⚫	-a_0 b_0 + a_1 b_1 + a_2 b_2 + a_3 b_3
	0 & 0 & 0 & 1 \end{matrix} \right)\,
			</math>
⚫	\left( \begin{matrix}b_0 \\ b_1 \\ b_2 \\ b_3 \end{matrix} \right) ~~\; </math>~~
	\|-
	\|
	\|-
	\|
⚫	~~\| <math> =~~ -a_0 b_0 + a_1 b_1 + a_2 b_2 + a_3 b_3 ~~\,</math>~~
	\|}

	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 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 ].		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 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 ].

Revision as of 00:36, 1 June 2004

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:

\mathbf {r} =\left(c\cdot t,x,y,z\right)

where c is the velocity 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:

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 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 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.