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