Editing Four-vector (section)

=====Standard basis, (+−−−) signature=====

The (+−−−) [[metric signature]] is sometimes called the "mostly minus" convention, or the "west coast" convention.

In the (+−−−) [[metric signature]], evaluating the [[Einstein notation|summation over indices]] gives:
<math display="block">\mathbf{A} \cdot \mathbf{B} =  A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3  </math>
while in matrix form:
<math display="block">\mathbf{A \cdot B}
  = \begin{pmatrix} A^0 & A^1 & A^2 & A^3 \end{pmatrix}
    \begin{pmatrix}
      1 &  0 &  0 &  0 \\
      0 & -1 &  0 &  0 \\
      0 &  0 & -1 &  0 \\
      0 &  0 &  0 & -1
    \end{pmatrix} \begin{pmatrix} B^0 \\ B^1 \\ B^2 \\ B^3 \end{pmatrix}
</math>

It is a recurring theme in special relativity to take the expression
<math display="block"> \mathbf{A}\cdot\mathbf{B} = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = C</math>
in one [[Frame of reference|reference frame]], where ''C'' is the value of the inner product in this frame, and:
<math display="block"> \mathbf{A}'\cdot\mathbf{B}' = {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3 {B'}^3 = C' </math>
in another frame, in which ''C''&prime; is the value of the inner product in this frame. Then since the inner product is an invariant, these must be equal:
<math display="block"> \mathbf{A}\cdot\mathbf{B} = \mathbf{A}'\cdot\mathbf{B}' </math>
that is:
<math display="block"> \begin{align}
C &= A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 \\[2pt]
&= {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3{B'}^3
\end{align} </math>

Considering that physical quantities in relativity are four-vectors, this equation has the appearance of a "[[Conservation law (physics)|conservation law]]", but there is no "conservation" involved. The primary significance of the Minkowski inner product is that for any two four-vectors, its value is [[invariant (physics)|invariant]] for all observers; a change of coordinates does not result in a change in value of the inner product. The components of the four-vectors change from one frame to another; '''A''' and '''A'''&prime; are connected by a [[Lorentz transformation]], and similarly for '''B''' and '''B'''&prime;, although the inner products are the same in all frames. Nevertheless, this type of expression is exploited in relativistic calculations on a par with conservation laws, since the magnitudes of components can be determined without explicitly performing any Lorentz transformations. A particular example is with energy and momentum in the [[energy-momentum relation]] derived from the [[four-momentum]] vector (see also below).

In this signature we have:
<math display="block"> \mathbf{A \cdot A} = \left(A^0\right)^2 - \left(A^1\right)^2 - \left(A^2\right)^2 - \left(A^3\right)^2 </math>

With the signature (+−−−), four-vectors may be classified as either [[Minkowski space#Causal structure|spacelike]] if <math>\mathbf{A \cdot A} < 0</math>, [[Minkowski space#Causal structure|timelike]] if <math>\mathbf{A \cdot A} > 0</math>, and [[Minkowski space#Causal structure|null vector]]s if <math>\mathbf{A \cdot A} = 0</math>.