The equation giving the distance between two points in a particular space is called** the metric**. Once we know the metric of a space, we know almost everything about the geometry of the space, which is why the metric is of fundamental importance.

We have already met the function that defines the distance between two points in Minkowski spacetime (see Minkowski's Four-Dimensional Space-Time article): it's the **spacetime interval** given by the formula

ds^{2} = c^{2}Δt^{2} - Δx^{2 } - Δy^{2 } - Δz^{2}

This can be written as ds^{2} = 1xc^{2}Δt^{2} - 1xΔx^{2 } - 1xΔy^{2 } -1xΔz^{2}

And +1, -1, -1, -1 can be defined as the diagonal elements of a 4x4 matrix, denoted by **η _{μν}**:

The indices μ,ν after the η symbol identify the elements of the matrix by reference to its rows (μ) and its columns (ν). The convention is that the metric coefficients run from 0 to 3, so η_{00}=1, η_{11}=-1, η_{22}=-1, η_{21}=0, etc.

This matrix simply tells us how to multiply the differentials cdt, dx, dy, dz to obtain the spacetime interval equation. We can see it by writing the matrix product as follow:

So finally, as only η_{00}, η_{11}, η_{22} and η_{33} are not null, we get the product equals to

1xc^{2}Δt^{2} - 1xΔx^{2 } - 1xΔy^{2 } -1xΔz^{2 } = c^{2}Δt^{2} - Δx^{2 } - Δy^{2 } -Δz^{2 }

which is the spacetime interval defining Minkoswki spacetime.

Finally, using the index notation, the Einstein convention (implicit summation on repetead indices) and the Minkowski metric **η _{μν}** , we can write the Minkowski line element in a more compact form:

Recall that the symbol **η _{μν }** irefers specifically to the Minkowski metric.

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