Environmental Engineering Reference
In-Depth Information
PROBLEMS 11
11.1 If a particle moves with speed
u
along the
x
-axis, show that if cosh
η
=
γ(u)
and tanh
η
=
u/c
then
2
ln
E
.
1
+
cp
=
η
−
E
cp
The variable
η
is known as the 'rapidity' of the particle.
11.2 Represent the Galilean transformation
t
=
t
x
=
x
−
ut
(11.41)
x
=
G
as a 2
×
2 matrix equation, i.e.
(u)
x
. Now consider a second transfor-
mation represented by
G
(v)
. Compute the matrix
G
(u)
G
(v)
and show that
it is also a Galilean transformation.
In the same co-ordinate basis, Lorentz transformations can be generated by
the 2
×
2 matrix (see Eq. (11.29)):
cosh
η
.
−
sinh
η
L
(η)
=
−
sinh
η
cosh
η
η
2
)
.
11.3 Prove Eq. (11.14). You may find the following identity useful:
Show that
L
(η
1
)
L
(η
2
)
=
L
(η
1
+
ε
ijk
=
R
ia
R
jb
R
kc
ε
abc
.
