Environmental Engineering Reference
In-Depth Information
where
1
cosh
θ
=
γ(u)
=
1
u
2
/c
2
,
−
γ(u)
u
sinh
θ
=
c
,
(11.28)
i.e. tanh
θ
=
u/c
. In matrix form we can equivalently write
=
ct
x
y
z
cosh
θ
−
sinh
θ
00
ct
x
y
z
−
sinh
θ
cosh
θ
00
.
(11.29)
0
0
1 0
0
0
0 1
We have done nothing except to write the Lorentz transformations of Part II in
a different way, however the similarity to the formalism for rotations, embodied
in Eq. (11.5) and Eq. (11.6), is clearly striking. Roughly speaking, it seems
that the Lorentz transformations are something akin to rotating a vector in a
four-dimensional space through an imaginary angle. Furthermore, we also know
from Eqs. (7.33) and (7.34) in Part II that the energy and momentum of a particle
transform in precisely the same way:
=
E
/c
p
x
p
y
p
z
cosh
θ
−
sinh
θ
00
E/c
p
x
p
y
p
z
−
sinh
θ
cosh
θ
00
.
(11.30)
0
0
1 0
0
0
0 1
Having seen these results, we are encouraged to follow Minkowski
3
in supposing
that we really should think of space and time not as seperate entities but rather as
forming a unified four dimensional 'space-time' (often called 'Minkowski space')
and that the equations in physics should be built using vectors and scalars in this
space-time for they are the objects which do not vary as we move from one inertial
frame to another. Actually we should pause for a moment and admit that there may
be other types of object available to us, such as four-tensors or perhaps even more
exotic objects
4
but to admit such a possibility does not undermine the potential
value of four-vectors and four-scalars.
For example, an event in space-time would then have a position 'four-vector'
X
(ct,x,y,z),
and instead of seperately speaking of the energy and momentum
of a particle we should speak of its momentum four-vector
P
=
(E/c, p
x
,p
y
,p
z
)
.
Other directional quantities should likewise be described by an appropriate
four-vector
5
. We are laying claim to the idea that space and time form a four
dimensional space which supports the existence of scalars and vectors. However
if this space is to be useful to us it should possess a well defined scalar product.
=
3
Hermann Minkowski (1864-1909).
4
Such objects do actually exist. For example, to describe the relativistic motion of electrons we should
use objects known as 'spinors'.
5
From this point onwards we use upper case boldface characters to represent four-vectors.
