Using index notation, the time and space coordinates can be combined into a single "4-vector", that has units of length(i.e. ct is a distance).
We can express the space time interval using the index notation
Sinceis nothing more than a dot product of a vector with itself, we should expect the components of the indices to follow a similar relationship. Following the rules of matrix multiplication, the dot product of two 4-vectors should follow the form:
This gives the desired results as expected.
The change in signs in the covariant term,
from the contravarient term
Comes from the Minkowski metric
Similarly, for two different 4-vectors,
This is useful in that it shows that the scalar product of two 4-vectors is an invariant since the time-space interval is an invariant.