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where ''c'' is the ], and where ''c'' is the ], and


Other examples of four-vectors include the ] <math>J^a= \left( j, \rho c \right)</math> formed from the current and charge densities (<math>j</math> and <math>\rho</math>, respectively), the ] <math>\left( A, \phi \left)</math> formed from the vector and scalar potentials (<math>A</math> and <math>\phi</math>, respectively) and the ] (''E''/''c'', '''p''') formed from the (relativistic) energy ''E'' and momentum '''p'''. The ] (''c'') is often used to ensure that the last coordinate (''time-like'', labeled by index 4) has the same units as the first three coordinates (''space-like'', labeled by indices 1,..,3). Other examples of four-vectors include the ] <math>J^a= \left( j, \rho c \right)</math> formed from the current and charge densities (<math>j</math> and <math>\rho</math>, respectively), the ] <math>\left( A, \phi \right)</math> formed from the vector and scalar potentials (<math>A</math> and <math>\phi</math>, respectively) and the ] (''E''/''c'', '''p''') formed from the (relativistic) energy ''E'' and momentum '''p'''. The ] (''c'') is often used to ensure that the last coordinate (''time-like'', labeled by index 4) has the same units as the first three coordinates (''space-like'', labeled by indices 1,..,3).


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

Revision as of 12:41, 6 April 2005

In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski 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.

A point in Minkowski space is called an "event" and is described by the position four-vector defined as:

x a = ( x , y , z , c t ) {\displaystyle x^{a}=\left(x,y,z,ct\right)}

where c is the speed of light, and

Other examples of four-vectors include the four-current J a = ( j , ρ c ) {\displaystyle J^{a}=\left(j,\rho c\right)} formed from the current and charge densities ( j {\displaystyle j} and ρ {\displaystyle \rho } , respectively), the electromagnetic four-potential ( A , ϕ ) {\displaystyle \left(A,\phi \right)} formed from the vector and scalar potentials ( A {\displaystyle A} and ϕ {\displaystyle \phi } , respectively) 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 last coordinate (time-like, labeled by index 4) has the same units as the first three coordinates (space-like, labeled by indices 1,..,3).

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

a ⋅ b = ( a 0 a 1 a 2 a 3 ) ( − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( b 0 b 1 b 2 b 3 ) = − a 0 b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 {\displaystyle 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 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.

See also

Category: