Editing Four-vector (section)

=====Standard basis, (−+++) signature=====
The (-+++) [[metric signature]] is sometimes called the "east coast" convention.

Some authors define ''η'' with the opposite sign, in which case we have the (−+++) metric signature. Evaluating the summation with this signature:

<math display="block">\mathbf{A \cdot B} = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 </math>

while the matrix form is:

<math display="block">\mathbf{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) </math>

Note that in this case, in one frame:

<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>

while in another:

<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>

so that:

<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>

which is equivalent to the above expression for ''C'' in terms of '''A''' and '''B'''. Either convention will work. With the Minkowski metric defined in the two ways above, the only difference between covariant and contravariant four-vector components are signs, therefore the signs depend on which sign convention is used.

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]] if <math>\mathbf{A \cdot A} = 0</math>.