Editing Four-vector (section)

====Minkowski tensor====

{{See also|spacetime interval}}

Applying the [[Minkowski tensor]] {{math|''η<sub>μν</sub>''}} to two four-vectors {{math|'''A'''}} and {{math|'''B'''}}, writing the result in [[dot product]] notation, we have, using [[Einstein notation]]:
<math display="block">\mathbf{A} \cdot \mathbf{B} = A^{\mu} B^{\nu}  \mathbf{E}_{\mu} \cdot \mathbf{E}_{\nu} = A^{\mu} \eta_{\mu \nu} B^{\nu} </math>

in special relativity. The dot product of the basis vectors is the Minkowski metric, as opposed to the Kronecker delta as in Euclidean space. It is convenient to rewrite the definition in [[matrix (mathematics)|matrix]] form:
<math display="block">\mathbf{A \cdot B} = \begin{pmatrix} A^0 & A^1 & A^2 & A^3 \end{pmatrix} \begin{pmatrix} \eta_{00} & \eta_{01} & \eta_{02} & \eta_{03} \\ \eta_{10} & \eta_{11} & \eta_{12} & \eta_{13} \\ \eta_{20} & \eta_{21} & \eta_{22} & \eta_{23} \\ \eta_{30} & \eta_{31} & \eta_{32} & \eta_{33} \end{pmatrix} \begin{pmatrix} B^0 \\ B^1 \\ B^2 \\ B^3 \end{pmatrix} </math>
in which case {{math|''η<sub>μν</sub>''}} above is the entry in row {{math|''μ''}} and column {{math|''ν''}} of the Minkowski metric as a square matrix. The Minkowski metric is not a [[Euclidean metric]], because it is indefinite (see [[metric signature]]). A number of other expressions can be used because the metric tensor can raise and lower the components of {{math|'''A'''}} or {{math|'''B'''}}. For contra/co-variant components of {{math|'''A'''}} and co/contra-variant components of {{math|'''B'''}}, we have:
<math display="block">\mathbf{A} \cdot \mathbf{B} = A^{\mu} \eta_{\mu \nu} B^{\nu} = A_{\nu} B^{\nu} = A^{\mu} B_{\mu} </math>
so in the matrix notation:
<math display="block">\begin{align}
\mathbf{A} \cdot \mathbf{B}
  &= \begin{pmatrix} A_0 & A_1 & A_2 & A_3 \end{pmatrix} \begin{pmatrix} B^0 \\ B^1 \\ B^2 \\ B^3 \end{pmatrix} \\[1ex]
  &= \begin{pmatrix} B_0 & B_1 & B_2 & B_3 \end{pmatrix} \begin{pmatrix} A^0 \\ A^1 \\ A^2 \\ A^3 \end{pmatrix}
\end{align}
</math>
while for {{math|'''A'''}} and {{math|'''B'''}} each in covariant components:
<math display="block">\mathbf{A} \cdot \mathbf{B} = A_{\mu} \eta^{\mu \nu} B_{\nu}</math>
with a similar matrix expression to the above.

Applying the Minkowski tensor to a four-vector '''A''' with itself we get:
<math display="block">\mathbf{A \cdot A} = A^\mu \eta_{\mu\nu} A^\nu </math>
which, depending on the case, may be considered the square, or its negative, of the length of the vector.

Following are two common choices for the metric tensor in the [[Minkowski space#Standard basis|standard basis]] (essentially Cartesian coordinates). If orthogonal coordinates are used, there would be scale factors along the diagonal part of the spacelike part of the metric, while for general curvilinear coordinates the entire spacelike part of the metric would have components dependent on the curvilinear basis used.

=====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>.

=====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>.

=====Dual vectors=====

Applying the Minkowski tensor is often expressed as the effect of the [[dual space#Ailinear products and dual spaces|dual vector]] of one vector on the other:

<math display="block">\mathbf{A \cdot B} = A^*(\mathbf{B}) = A{_\nu}B^{\nu}. </math>

Here the ''A<sub>ν</sub>''s are the components of the dual vector '''A'''* of '''A''' in the [[dual basis]] and called the [[Covariance and contravariance of vectors|covariant]] coordinates of '''A''', while the original ''A<sup>ν</sup>'' components are called the [[Covariance and contravariance of vectors|contravariant]] coordinates.