7Transformation groups

IA Vectors and Matrices


7.3 Lorentz transformations
Consider the Minkowski 1 + 1 dimension spacetime (i.e. 1 space dimension and
1 time dimension)
Definition
(Minkowski inner product)
.
The Minkowski inner product of 2
vectors x and y is
hx | yi = x
T
Jy,
where
J =
1 0
0 −1
Then hx | yi = x
1
y
1
− x
2
y
2
.
This is to be compared to the usual Euclidean inner product of
x, y ∈ R
2
,
given by
hx | yi = x
T
y = x
T
Iy = x
1
y
1
+ x
2
y
2
.
Definition
(Preservation of inner product)
.
A transformation matrix
M
pre-
serves the Minkowski inner product if
hx|yi = hMx|Myi
for all x, y.
We know that
x
T
Jy
= (
Mx
)
T
JMy
=
x
T
M
T
JMy
. Since this has to be
true for all x and y , we must have
J = M
T
JM.
We can show that M takes the form of
H
α
=
cosh α sinh α
sinh α cosh α
or K
α/2
=
cosh α −sinh α
sinh α −cosh α
where H
α
is a hyperbolic rotation, and K
α/2
is a hyperbolic reflection.
This is technically all matrices that preserve the metric, since these only
include matrices with
M
11
>
0. In physics, these are the matrices we want, since
M
11
< 0 corresponds to inverting time, which is frowned upon.
Definition
(Lorentz matrix)
.
A Lorentz matrix or a Lorentz boost is a matrix
in the form
B
v
=
1
√
1 − v
2
1 v
v 1
.
Here
|v| <
1, where we have chosen units in which the speed of light is equal to
1. We have B
v
= H
tanh
−1
v
Definition
(Lorentz group)
.
The Lorentz group is a group of all Lorentz matrices
under matrix multiplication.
It is easy to prove that this is a group. For the closure axiom, we have
B
v
1
B
v
2
= B
v
3
, where
v
3
= tanh(tanh
−1
v
1
+ tanh
−1
v
2
) =
v
1
+ v
2
1 + v
1
v
2
The set of all
B
v
is a group of transformations which preserve the Minkowski
inner product.