Environmental Engineering Reference
In-Depth Information
two events satisfies (see Eq. (14.21))
(
d
s)
2
(c
d
t)
2
(
d
x)
2
(
d
y)
2
(
d
z)
2
=
−
−
−
c
2
g
+
x
cosh
gt
c
g
c
d
t
+
sinh
gt
c
d
x
2
=
c
2
g
x
sinh
gt
c
g
c
d
t
+
cosh
gt
c
d
x
2
−
+
(
d
y
)
2
(
d
z
)
2
−
−
1
c
2
2
gx
(c
d
t
)
2
(
d
x
)
2
(
d
y
)
2
(
d
z
)
2
.
=
+
−
−
−
(14.29)
Thus there is some warping of time in the accelerating frame but space is Euclidean
(which is not surprising since we constructed it that way).
Let us now turn to another accelerating frame of reference. This time our goal
will be to make contact with Chapter 8. Let us consider a non-inertial frame rotating
with angular speed
ω
about the
z
-axis. It is convenient first to work in cylindrical
polar co-ordinates. The space-time interval between two neighbouring events in an
inertial frame (i.e. one for which
ω
=
0) is
(
d
s)
2
(c
d
t)
2
(r
d
φ)
2
(
d
r)
2
(
d
z)
2
.
=
−
−
−
(14.30)
A most important property of space-time physics is that this interval must be the
same in any other system of co-ordinates, even an accelerating system. This is
nothing more than the statement that the distance between any two points on a
general manifold should be independent of the way we choose to parameterize the
manifold. Hence we can choose to work in a non-inertial frame with
φ
=
φ
−
ωt
,
t
=
t
,
r
=
r
and
z
=
z
such that
(
d
s)
2
(c
d
t
)
2
r
2
(
d
φ
+
ω
d
t
)
2
(
d
r
)
2
(
d
z
)
2
.
=
−
−
−
(14.31)
From now on we will only be interested in the rotating co-ordinates and will
subsequently drop the primes. Moreover, we shall find it more convenient to now
switch back to Cartesian co-ordinates, i.e.
x
=
r
cos
φ,
y
=
r
sin
φ.
(14.32)
In which case and after a little algebra it follows that
(
d
s)
2
(c
d
t)
2
(
1
ω
2
(x
2
y
2
)/c
2
)
(
d
x)
2
=
−
+
+
2
(ω/c)(y
d
x
−
x
d
y)(c
d
t)
−
(
d
y)
2
(
d
z)
2
.
−
−
(14.33)
