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of other expressions can be used because the metric tensor can raise and lower the components of or . For contra/co-variant components of and co/contra-variant components of , we have:

so in the matrix notation:

while for and each in covariant components:

with a similar matrix expression to the above.

Applying the Minkowski tensor to a four-vector with itself we get:

which, depending on the case, may be considered the square, or its negative, of the length of the vector.

Following are two common

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