Basic Theoretical Physics: A Concise Overview P11 pptx

14.2 Coriolis Force and Weather Pattern 97
which has, as for “flat” Minkowski spaces, one eigenvalue of one sign and
three eigenvalues of the opposite sign.
The space of Minkowski four-vectors
˜x := (x, y, z, ict)
thus becomes a curved Minkowski manifold, where local coordinates of space
and time may be defined by means of radar signals.
Further details which would lead to Einstein’s general theory of relativity,
will not be treated here, although the effects of curvature of space and time
not only play a part in present-day astrophysics, e.g., in the neighborhood of
neutron stars and black holes, but have also entered our daily lives through
the Global Positioning System (GPS), a satellite navigation system, which is
presently used for many purposes.
14.2 Coriolis Force and Weather Pattern
Apart from elevator forces, Coriolis forces are perhaps the most important
inertial force. In contrast to centrifugal forces they are proportional to ω and
not ∝ ω
2
(i.e., not of second-order):
F

Coriolis
= −2m
t
· ω ×v

.
Particularly important are the consequences of this force in the weather
pattern, where the Coriolis force governs the deflection of wind currents from
regions of high atmospheric pressure to those of low pressure. For example,
if there were a high pressure region at the equator, with coordinates

x

= −H, y

=0,
and if the nearest low-pressure region were at
x

=+T,y

=0,
then without rotation of the earth, i.e., for ω = 0, the wind would only have
avelocitycomponentv
x
, i.e., it would directly move from west to east on the
shortest path from high pressure to low pressure.
However, due to the rotation of the earth,
ω =
2π
24h
ˆz ,
where ˆz denotes the axis of rotation, we have
F

Coriolis
= −2m
t
·ωv
x
e

y
,
98 14 Accelerated Reference Frames
i.e., a force directed from north to south. As a consequence, in the northern
hemisphere the wind flows out of the high-pressure region with a deflection
to the right and into the low-pressure region with a deflection to the left.
In addition, for the so-called trade winds (Passat wind) over the oceans,
which without rotation of the earth would blow directly towards the equator,
i.e., southwards (in the northern hemisphere), the rotation of the earth leads
to a shift of the direction. As a consequence the trade winds are directed from
north-east to south-west.
A further subject for which the rotation of the earth plays an essential
part concerns the Foucault pendulum, a very long pendulum swinging from
the ceiling of a large building, such that one can directly infer the rotation of
the earth from the varying plane of oscillations of the swings, c.f. subsection
14.4.
14.3 Newton’s “Bucket Experiment” and the Problem
of Inertial Frames
We return to the problem of ascertaining whether a frame of reference is
rotating, ω = 0, or whether, perhaps, it is an inertial frame. For this purpose,
we can return to the proposal already known by Newton, of observing the
surface of water in a bucket rotating with the reference frame.
Due to the internal friction of the liquid, its surface shows a profile given
by
z(r)=
r
2
ω
2
2g

,
where z(r) is the height of the liquid surface measured from the central
(lowest) point. In addition, the local slope of the surface of the liquid in the
bucket is given by
tan α =
dz
dr
=
rω
2
g
.
This relation is obtained by equating the centrifugal force (per unit mass of
liquid)
(F

x
)
centrifugal
= m
t
ω
2
r
with the gravitational force (per unit mass)
F
z
= −m
s
g.

In this way one can therefore ascertain whether one is in a rotating reference
frame, and here the third inertial force,thecentrifugal force, explicitly comes
into play.
In view of the difficulties in defining an inertial reference frame,towards
the end of the nineteenth century, the physicist and important Viennese
philosopher, Ernst Mach, postulated that a global inertial system can only
14.4 Application: Free Falling Bodies with Earth Rotation 99
be defined in a constructive way by the totality of all stars. But as mentioned
above it was only Einstein, who, some years later, cut the “Gordian knot” by
stating that only local inertial frames exist, thus solving in an elegant way
an essential problem left over from Newtonian mechanics. More details can
be found above in Sect. 14.1.
14.4 Application: Free Falling Bodies with Earth
Rotation; the Foucault Pendulum
In a cartesian coordinate system fixed to the moving surface of the earth, let
e
1
correspond to the direction “east”, e
2
to “north” and e
3
to the geometrical
vertical direction (e
3
≡ e
1
× e
2
).
Let the geographical latitude correspond to the angle ψ such that ψ =0

at the equator and ψ =
π
2
at the North Pole. Hence for the angular velocity
of the rotation of the earth we have
ω = ω cos ψe
2
+ ω sin ψe
3
.
The three equations of motion are, therefore:
m
t
¨x
1
= m
s
G
1
− 2m
t
ω ·(cos ψv
3
− sin ψv
2
) ,
m
t
¨x
2

= m
s
G
2
− 2m
t
ω sin ψv
1
+ m
t
ω
2
R cos ψ sin ψ,
m
t
¨x
3
= m
s
G
3
+2m
t
ω cos ψv
1
+ m
t
ω
2
R cos

2
ψ. (14.2)
If the earth had the form of an exact sphere, and if the mass distribution
were exactly spherical, then G
1
and G
2
would vanish, and G
3
≡−g.The
deviations correspond to gravitational anomalies, which are measured and
mapped by geophysicists, e.g., in mineral prospecting. As usual, for didactical
reasons we distinguish between the inertial mass m
t
and the gravitational
mass m
s
, although
m
s
= m
t
; R(= R(ψ))
is the radius of the earth at the latitude considered (where we average over
mountains and depressions at this value of ψ).
We shall now discuss the terms which are (a) ∝ ω
2
, i.e., the centrifugal
force, and (b) ∝|ωv|, i.e., the Coriolis force.
– (ai) The very last term in (14.2) leads to a weak, but significant flattening

of the sphere (→ “geoid” model of the earth = oblate spheroid), since
−m
s
g + m
t
ω
2
R cos
2
ψ =: −m
t
g
eff
(ψ)
depends on the geographical latitude. The gravitational weight of a kilo-
gram increases towards the poles.
100 14 Accelerated Reference Frames
– (aii) The terms ∝ ω
2
in the second equation and the north anomaly
G
2
4
of the gravitational force lead to a deviation in the direction of the
gravitational force both from the direction of e
3
and also from the G
direction. The deviations are described by the angles δ and δ

, respectively:

tan δ =
G
2
− ω
2
R cos ψ sin ψ
G
3
and
tan δ

=
−ω
2
R cos ψ sin ψ
G
3
.
For realistic values
G
3
≈ 981 cm/s
2
and ω
2
R ≈ 3.4cm/s
2
these effects are roughly of the relative order of magnitude 3 · 10
−3
.

With regard to the terms which are linear in ω (case b), we think of the
Foucault pendulum and assume therefore that v
3
≡ 0, whereas v
1
and v
2
are
not ≡ 0. The equations of motion are
m
t
¨x
1
= F
1
+2m
t
ωv
2
sin ψ and
m
t
¨x
2
= F
2
− 2m
t
ωv
1

sin ψ, i.e.,
˙
v =
F
m
t
+ ω
eff
(ψ) × v .
This can be formulated with an effective rotational velocity that depends on
the geographical latitude ψ, viz
ω
eff
(ψ)=−2ω sin ψe
3
, with ω =
2π
24h
.
A superb example of a Foucault pendulum can be seen in the Science Museum
in London.
4
The averaged east/west anomaly G
1
vanishes for reasons of symmetry, or with
perturbations of the east-west symmetry it is usually much smaller in magnitude
than |G
2
|.
15 Relativity II: E=mc

2
The Lorentz transformations,
x =
x

+
v
c
ct


1 −
v
2
c
2
,ct=
ct

+
v
c
x


1 −
v
2
c
2

,
have already been treated in Sect. 9.1, together with the difference between
the invariant proper time (eigenzeit) dτ – this is the time in the co-moving
system – and the noninvariant time dt in the so-called laboratory system.
Let us recall the invariant quantity
−(d˜x)
2
:= c
2
dt
2
− dx
2
− dy
2
− dz
2
,
which can be either positive, negative or zero. Positive results apply to time-
like Minkowski four-vectors
d˜x := (dx, dy, dz, icdt)
in all inertial frames. For space-like four-vectors the invariant expression on
the r.h.s. of the above equation would be negative and for light-like Minkowski
four-vectors it would be zero. These inequalities and this equality characterize
the above terms.
In fact, time-like, space-like and light-like Minkowski four-vectors exhaust
and distinguish all possible cases, and for time-like four-vectors one obtains
(with the proper time τ):
c
2

dτ
2
≡−d˜x
2
,
i.e., apart from the sign this is just the square of the invariant pseudo-length
in a pseudo-Euclidean Minkowski space. As mentioned above, Lorentz tran-
formations can be considered as pseudo-rotations in this space.
Since d˜x is an (infinitesimal) Minkowski four-vector and dτ an (infinites-
imal) four-scalar
1
, i.e., Lorentz invariant, the above-mentioned four-velocity,
˜v :=
d˜x
dτ
=
(v
x
,v
y
,v
z
, ic)

1 −
v
2
c
2
1

In contrast to dτ the differential dt is not a Minkowski scalar, i.e., not invariant
against Lorentz transforms.
102 15 Relativity II: E=mc
2
is a Minkowski four-vector. The factor
1

1 −
v
2
c
2
is thus compulsory ! In addition, if this result is multiplied by the rest mass
m
0
, one obtains a Minkowski four-vector, i.e.
˜p :=
m
0

1 −
v
2
c
2
(v
x
,v
y
,v

z
, ic) .
This four-vector is the relativistic generalization of the momentum, and
the dependence of the first factor on the velocity, i.e., the velocity dependence
of the mass,
m =
m
0

1 −
v
2
c
2
,
is also as compulsory as previously that of ˜v.
The imaginary component of the four-momentum is thus
i ·
m
0
c

1 −
v
2
c
2
.
On the other hand, in the co-moving system (v = 0), a force vector can
be uniquely defined by the enforced velocity change, i.e.,

d(m
0
v)
dτ
=: F ,
analogously to Newton’s 2nd law.
The work done by this force, δ
ˆ
A, serves to enhance the kinetic energy
E
kin
: δ
ˆ
A −
˙
E
kin
dt = F · dr −
l
F
c
cdt ≡ 0 ,
with the so-called power l
F
:=
˙
E
kin
.
Formally this means the following:

The force-power four-vector
˜
F :=

F
x
,F
y
,F
z
, i
l
F
c

is pseudo-perpendicular to
d˜x := (dx, dy, dz, icdt):
˜
F · d˜x ≡ 0 .
This is valid in all inertial frames (Lorentz frames).
15 Relativity II: E=mc
2
103
We now define
E := E
kin
+ m
0
c
2

.
It is then natural to supplement the relation
l
F
=
˙
E by F = ˙p
and in this way to define the energy-momentum four-vector
˜p =
m
0

1 −
v
2
c
2
(v
x
,v
y
,v
z
, ic) ,
i.e.,
˜p =

p
x
,p

y
,p
z
, i
E
c

with the following two relations, which belong together, equivalent to
˜
F · d˜x ≡ 0:
p =
m
0
v

1 −
v
2
c
2
and E =
m
0
c
2

1 −
v
2
c

2
. (15.1)
In fact, one can explicitly evaluate that in this way the equation
˜
F · d˜x ≡ 0
is satisfied.
The invariant pseudo-length of the energy-momentum four-vector is thus
˜p
2
= −m
2
0
c
2
; the energy E = mc
2
is the sum of the rest energy m
0
c
2
and the kinetic energy
T :=
m
0
c
2

1 −
v
2

c
2
− m
0
c
2
,
which in a first approximation leads to the usual nonrelativistic expression,
T≈
m
0
v
2
2
.
In addition we have the relativistic form of the equation of motion, which
is that in every inertial system:
dp
dt
= F , (15.2)
104 15 Relativity II: E=mc
2
with
p =
m
0
v

1 −
v2

c
2
.
By integrating this differential equation for the velocity at constant force
and with the initial values
v(t =0)≡ 0 ,
one obtains
p
2
= F
2
t
2
,
or
v
2
c
2
=
ˆ
F
2
t
2
1+
ˆ
F
2
t

2
,
with
ˆ
F :=
F
m
0
c
.
For t →∞the r.h.s. of
v
2
c
2
converges to 1 (monotonically from below); c is
thus the upper limit of the particle velocity v.
Our derivation of the above relations was admittedly rather formal, but
Einstein was bold enough, and gave reasons (see below) for proposing the
even more important interpretation of his famous relation,
E = mc
2
,
which not only implies the above decomposition of the energy into a rest
energy and a kinetic part, but that the formula should additionally be inter-
preted as the equivalence of mass and energy; i.e., he suggested that mass
differences (×c
2
) can be transformed quantitatively into energy differences
2

.
This is the essential basis, e.g., of nuclear energetics; nuclei such as He, which
have a strong binding energy δE
B
, also have a measurable mass defect
δm
B
= δE
B
/c
2
.
One could give many more examples.
The above reasons can, in fact, be based on considerations of the impact
between different particles. For example, viewed from the rest frame of a very
heavy point mass, which forms the target of other particles moving towards
it with very high velocity (almost c) the kinetic energy of any of the moving
particles is essentially identical to its (velocity dependent) mass times c
2
.
Thus in this case the rest mass is negligible.
All these considerations, which in our presentation have been partly stringent
and partly more or less heuristic, have proved valid over decades, without
2
Here one should not forget the fact that m
0
c
2
(in contrast to E) is relativistically
invariant.

15 Relativity II: E=mc
2
105
any restriction, not only by thought experiments and mathematical deriva-
tions, but also by a wealth of nontrivial empirical results. As a consequence,
for decades they have belonged to the canonical wisdom not only of university
physics, but also of school physics and gradually also in daily life, although
there will always be some people who doubt the above reasoning and experi-
ence. We shall therefore conclude Part I of the book by restating Einstein’s
relation
E = mc
2
, where m = m
0
/

1 −
v
2
c
2
,
almost exactly one century after it was originally conceived in the “miraculous
year” of 1905.
Part II
Electrodynamics and Aspects of Optics