A four- vector is a quantity with four components which changes like spacetime coordinates under a coordinate transformation. We will write the displacement four- vector as:

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:

The Lorentz transformations can be written as

where is the Lorentz transformation matrix

and , can be regarded as column vectors.

The positioning of the indices is explained later but indicates that we can use the summation convention: sum over repeated indices, if one index is up and one index down. Thus we can write:

is a dummy- index which can be replaced by any other index; is a free- index, so the above equation is equivalent to four equations. For a general four- vector

we can write

If and are two four- vectors, clearly and are also four- vectors with obvious components

and

In any frame we can define a set of four- basis vectors:

and

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.

Writing:

where we have exchanged the dummy indices and and and , we see that this equals for all if and only if

This gives the transformation law for basis vectors:

Note that the basis transformation law is different from the transformation law for the components since takes one from frame to .

So in summary, for vector basis and vector components we have:

Since has a velocity relative to , we have:

So is the inverse of . Likewise

It follows that the Lorentz transformations with gives the components of a four- vector in from those in .

The magnitude of a four- vector is defined as :

in analogy with the line element

The sign on the will be explained later. This is a frame invariant scalar.

is spacelike if , timelike if and null if . The scalar product of two four- vectors and is:

Since

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.

Basis vectors form an orthonormal tetrad since they are orthogonal: if , normalized to unit magnitude: :

We will see later what the geometrical significance of is.