Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.5 Lorentz group property derivation
This note verifies the group property of the Lorentz transformation.
It is not recommended unless you have had a solid course in linear
algebra.
Note first that a much more simple argument can be given by defining
the Lorentz transformation more abstractly, {A.4}
(A.13). But that is cheating. Then you have to prove that
these Lorentz transform are always the same as the physical ones.
For simplicity it will be assumed that the observers still use a
common origin of space and time coordinates.
The group property is easy to verify if the observers B and C are
going in the same direction compared to A. Just multiply two matrices
of the form (1.13) together and apply the condition that
1 for each.
It gets much messier if the observers move in different directions.
In that case the only immediate simplification that can be made is to
align the coordinate systems so that both relative velocities are in
the planes. Then the transformations only involve in a
trivial way and the combined transformation takes the generic form
It needs to be shown that this is a Lorentz transformation from A
directly to C.
Now the spatial, , coordinate system of observer C can be
rotated to eliminate and the spatial coordinate system
of observer A can be rotated to eliminate . Next
both Lorentz transformations preserve the inner products. Therefore
the dot product between the fourvectors and
in the A system must be the same as the dot product between columns 1
and 3 in the matrix above. And that means that must
be zero, because will not be zero except in the
trivial case that systems A and C are at rest compared to each other.
Next since the proper length of the vector equals one in
the A system, it does so in the C system, so must be
one. (Or minus one, but a 180 rotation of the spatial
coordinate system around the axis can take care of that.) Next,
since the dot product of the vectors and is
zero, so is .
That leaves the four values relating the time and components.
From the fact that the dot product of the vectors and
is zero,
where is some constant. Also, since the proper lengths of
these vectors are minus one, respectively one,
or substituting in for and from the above
It follows that and must be equal, (or
opposite, but since both Lorentz transformations have unit
determinant, so must their combination), so call them .
The transformation is then a Lorentz transformation of the usual form
(1.13). (Since the spatial coordinate system cannot just
flip over from left handed to right handed at some point,
will have to be positive.) Examining the transformation of the origin
0 identifies
as , with the relative velocity of system A
compared to B, and then the above two equations identify as
the Lorentz factor.
Obviously, if any two Lorentz transformations are equivalent to a
single one, then by repeated application any arbitrary number of them
are equivalent to a single one.