Four- vectors
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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
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.
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