Environmental Engineering Reference
In-Depth Information
14.4 Eq. (14.6) relates the proper acceleration to the magnitude of the acceleration
four-vector, i.e. A
a 2 . Working in CERN's rest frame, the position
·
A
=−
four-vector
of
an
orbiting
electron
is X
=
(ct, R cos ωt, R sin ωt, 0 )
where
2 πR
=
27
km.
Differentiating
twice
with
respect
to
the
proper
time
and
using
d t/ d τ
=
γ
gives
an
acceleration
four-vector
equal
to
2 γ 2 ( 0 , cos ωt, sin ωt, 0 ) .
A
=−
Thus
the
proper
acceleration
has
a
of γ 2 ω 2 R (as
magnitude
one
might
have
anticipated
on
the
grounds
of time dilation). Finally we need ω
=
v/R where v is the electron's
E/(mc 2 )
10 3 ,i.e. v
speed. Can deduce v from γ
=
=
88
×
=
c to a good
10 23
ms 2 .
=
×
approximation and a
1 . 6
10 15 .
14.6 (a) τ is a proper time interval, i.e. is the invariant spacetime distance
between the events A and B where A is specified by providing the position
of the clock and its time initially and B is specified by providing the position
of the clock and its time at the end of the interval. In the vicinity of the
Earth's surface we might therefore venture to make use of Eq. (14.53) for a
uniform gravitational field, i.e.
gh/c 2
=
=
×
14.5 Expect z
2 . 46
( 1
c 2 ( d x ) 2 1 / 2
gh
c 2 ) 2 ( d t) 2
1
τ
+
1
2 1 / 2
c 2 d x
2 gh
c 2
1
+
d t,
d t
where h is the height above the Earth's surface (it is in general a function of
t ). It is to be understood that the time t refers to the time in an inertial frame
that is approximately at rest relative to the centre of the Earth (approximately
since we only need to be able to neglect length contraction effects in the spec-
ification of h ), e.g. when g
0 we regain the Minkowski interval expressed
in inertial co-ordinates. The above equation leads directly to the quoted result
upon expanding the square root. (b) Let v be the speed of a clock relative
to the ground, and let V be the speed of a point fixed on the Earth's surface
relative to the inertial frame from part (a), i.e. V
=
2 πR earth /( 1day ) .Now
work out the proper time elapsed on (i) the clock at rest in the airplane; (ii)
the clock at rest in the airport. For the clock on the airplane we have
=
t 1
v) 2
2 c 2
gh
c 2
(V
+
τ plane
+
(ignoring the variation of h as the airplane ascends and descends) and for
the clock on the ground
t 1
2 c 2 .
V 2
τ ground
Strictly speaking we should account for the fact that the clock on the
plane
must
actually
travel
slightly
faster
than V
+
v (by
an
amount
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