is a frame independent vector joining near by points in spacetime. means: has components in frame . are the coordinates themselves [ which are coordinate dependent ].
In frame , the coordinates are so:
where is the Lorentz transformation matrix
and , can be regarded as column vectors.
we can write
If and are two four- vectors, clearly and are also four- vectors with obvious components
In general we can write
where labels the basis vector and labels the coordinate.
Any four- vector can be expressed as a sum of four- vectors parallel to the basis vectors i.e.
The last equality reflects the fact that four- vectors are frame independent.
where we have exchanged the dummy indices and and and , we see that this equals for all if and only if
Note that the basis transformation law is different from the transformation law for the components since takes one from frame to .
Since has a velocity relative to , we have:
It follows that the Lorentz transformations with gives the components of a four- vector in from those in .
in analogy with the line element
The sign on the will be explained later. This is a frame invariant scalar.
is frame independent.
and are orthogonal if ; they are not necessarily perpendicular in the spacetime diagram [ for example a null vector is orthogonal to itself ], but must make equal angles with the line.
We will see later what the geometrical significance of is.