Environmental Engineering Reference
In-Depth Information
In the
(ct,x,y,z)
basis the metric for this system of co-ordinates is
ω
2
(x
2
y
2
)/c
2
1
−
+
ωy/c
−
ωx/c
0
ωy/c
−
1
0
0
g
=
.
(14.34)
−
ωx/c
0
−
10
0
0
0
−
1
Now the motion of a free particle through space-time is some unique trajectory,
which cannot depend upon how we choose our co-ordinates, and that means even if
we choose a non-inertial co-ordinate system. There must therefore be a genuinely
co-ordinate independent way to write the equations of motion of this free particle.
Here we quote the answer and defer the proof to Appendix A. The co-ordinate
independent way to write the equation of motion of a particle not acted upon by
any force is given by
∂g
ik
∂x
l
+
u
k
u
l
g
ij
d
u
j
d
τ
1
2
∂g
il
∂x
k
−
∂g
kl
∂x
i
+
=
0
,
(14.35)
where
u
i
d
x
i
/
d
τ
is the four-velocity of the particle. This is an equation that treats
all frames (inertial and non-inertial) on an equal footing. Notice that for inertial
co-ordinates all of the derivatives of the metric vanish and we are left with the
expected statement that all components of the four-acceleration are constant for a
free particle (i.e. d
=
0).
Given Eq. (14.35) we can go ahead and check to see that it gives the expected
answer for the motion of a non-relativistic free particle in the rotating frame. We
will assume that
u
/
d
τ
=
d
d
τ
=
(
0
,
x,
¨
y,
¨
z),
¨
(14.36)
where the dots indicate differentiation with respect to
t
, which will be fine in the
non-relativistic limit. Setting
i
=
2 will then give us the equation of motion in
x
.
We need to evaluate
g
2
j
d
u
j
d
τ
=−¨
x,
(14.37)
∂g
2
k
∂x
l
∂g
2
l
∂x
k
u
k
u
l
=
u
k
u
l
=
ω
y
˙
(14.38)
and
∂g
kl
∂x
2
u
k
u
l
=−
2
ω
2
x
−
2
ω
y.
˙
(14.39)
Hence Eq. (14.35) reduces to
ω
2
x.
x
¨
=
2
ω
y
˙
+
(14.40)
Similarly with
i
=
3 we obtain
ω
2
y.
y
¨
=−
2
ω
x
˙
+
(14.41)
