metric tensor inner product Minkowski metric Minkowski metric spacetime matrix tensor see
The metric tensor allows one to define the inner product of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the Minkowski metric η has components (valid with suitably chosen coordinates) which can be arranged in a 4 × 4 matrix:





η

α
β


=


(



−
1


0


0


0




0


1


0


0




0


0


1


0




0


0


0


1



)




{\displaystyle \eta _{\alpha \beta }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}


which is equal to its reciprocal, 




η

α
β




{\displaystyle \eta ^{\alpha \beta }}

, in those frames. Throughout we use the signs as above, different authors use different conventions – see Minkowski metric alternative signs.
