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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