Environmental Engineering Reference
In-Depth Information
imaginary, which means that it is not possible for an observer in any inertial frame
to be present at both events. Starting from this displacement four-vector we can
construct a number of other very useful four-vectors without too much hard work.
12.1 THE VELOCITY FOUR-VECTOR
The velocity four-vector is defined by
d
X
d
τ
,
U
=
(12.3)
c
d
t
.
d
τ
,
d
d
τ
=
(12.4)
It is a four-vector since the derivative is defined as
X
/τ
in the limit of vanish-
ingly small
τ
. The numerator is our prototypical four-vector and the denominator
is our prototypical four-scalar so the ratio must be a four-vector
1
. We can rewrite
the differential element of proper time using
(
d
t)
2
1
2
c
2
d
1
d
t
(
d
τ)
2
=
−
(12.5)
which implies that
d
τ
d
t
=
1
γ(u)
(12.6)
and therefore that
c,
d
,
d
t
d
τ
d
t
U
=
=
γ (u)(c,
u
).
(12.7)
u
Thus the velocity four-vector can be simply related to the velocity three-vector
.
Notice that the magnitude of the four-velocity is given by
c
2
U
·
U
=
(12.8)
and so everything moves through space-time with the same four-speed.
We can use the four-velocity to re-derive the formulae which relate the velocity
of a particle in one inertial frame of reference to that in another. As usual we focus
upon the particular case of inertial frames
S
and
S
. This is a very straightforward
task, for if we suppose that we know a four-velocity in
S
we can obtain the
corresponding four-velocity in
S
by applying the transformation specified by the
matrix in Eq. (11.29). In order to make precise contact with the results in Part II
we shall suppose that we are given a velocity four-vector in
S
and asked for its
1
This is a particular example of what is sometimes called the quotient theorem.
