Frequently Asked Questions
Series #10
Relativity and the GPS
Series #10
Relativity and the GPS
Source
of Energy Between the Pole and the Equator.
Let us examine how the principle of energy conservation is applied to
the two experiments above. In the case of the orbiting satellite,
it
is well known that the total energy of the satellite is conserved
because the sum of potential plus kinetic energy everywhere along the
orbit is equal to zero. There is no exchange of energy between
the
orbiting satellite and the Earth.
However, we have
seen above that when a mass moves from the pole to the rotating
equator, there is a net increase of energy (potential plus kinetic),
given to that moving clock. Since it requires no external energy
to
carry a mass slowly between the Earth pole and the equator (neglecting
friction), some people believe that there is no difference of
energy.
That is an error, as we will see now. Let us calculate
where that
energy comes from.
When a mass is at the
Earth
pole, its angular momentum around the Earth axis of rotation is
zero.
Let us slowly move the clock away from the Earth pole. When that
clock, moving along a meridian, gets to an increasing distance from the
pole, the rotating Earth makes it move faster and faster in a direction
perpendicular to the meridian. It is the inertia of the rotating
Earth, which accelerates the moving clock in a transverse direction,
until the clock reaches the equator and attains the transverse velocity
of the equator, which is 1670 Km/hr.
At the same
time the clock moves away from the Earth pole, another phenomenon takes
place. Since the equator is further away from the Earth center, a
radial force is required to maintain the effortless motion of the clock
along the meridian, which becomes larger near the equator. That
effortless motion is natural, since the Earth rotation produces a
centrifugal force away from the pole in the direction of the equator.
Those two forces compensates naturally. In fact, the exact shape
of
the Earth is the natural equilibrium between these forces.
Let us calculate first
the total increase of energy DE(clock)
given to the clock moving from the pole to the equator. It is
equal to the potential energy plus the kinetic energy.
DE(clock)
=
(kinetic
energy) + (potential
energy)
1
Taking
into account the flattened shape of the Earth and the centrifugal
force, it is well known that the potential energy is equal to the
kinetic energy. Therefore the total increase of energy of the
clock is:
DE(clock)
=2
(kinetic
energy)
2
The kinetic energy of the
clock at the equator is
K.E. (clock)=½mV2
3
Where m is the mass of
the clock and V is the velocity of the Earth equator.
Equation 3 in 2 gives
that the total increase of energy DE(clock)
of the clock moving from the Earth pole to the equator.
DE(clock)
=2
(½mV2)=mV2.
4
The clock at Earth pole has zero angular momentum “P” with respect to
the Earth axis (because R=0). At the equator, the increase of
angular
momentum DP of the clock around the Earth
axis is:
DP(clock)=mVR
5
Where R is the equatorial
radius of the Earth.
We know that the Earth is an inertial frame in space and its total
angular momentum remains constant even when a mass (like a clock) moves
on its surface. Since there is no external force acting on the
Earth,
its total angular momentum is constant. We know that the moment
of
inertia “I” of a spheroid of revolution (as the Earth) along the polar
axis is
6
We have M is the mass of
the Earth. The angular momentum “P” around its axis is:
P=Iw
7
The angular velocity w is defined as:
8
Using equations 6, 7 and
8, we find that the total Earth angular momentum “P” before the clock
moves to the equator is:
9
When the clock is initially at the Earth pole, its mass brings no
contribution to the Earth angular momentum (R=0). However, after the
mass of the clock reaches the equator, its mass gives a new
contribution to the angular momentum. However, since Newton’s
laws
requires that the total Earth angular momentum is constant, the new
angular momentum of the clock at the equator appears at the detriment
of the total initial Earth angular momentum.
Equation 5 gives the change of angular momentum due to the displacement
of the clock. The relative change of angular momentum of the
Earth
(minus the clock) due to the displacement of the clock is obtained
using equations 5 and 9. This gives:
10
Taking the derivative of
equation 9 for the same Earth mass M at the same distance R (equator)
we find:
11
Equation 11 in 10 gives
12
Let us calculate the
energy corresponding to that change of Earth velocity. The energy
of a rotating body is
E(Earth)=½ Iw2
13
Equations 6 and 7 in 13
give:
14
The variation of the
Earth rotational energy is given by the derivative of equation
14. This gives:
15
Using equation 15 let us calculate the total loss of the Earth
rotational energy due to the displacement of the clock from the pole to
the equator. Let us substitute equation 12 in 15. This
gives:
16
We find that equation 16 is identical to equation 4. Therefore
the
energy acquired by the clock at the equator is equal to the energy lost
due to the slowing down of the Earth rotation. Therefore,
17
Equation 17 shows that the increase of energy of the clock moving from
the pole to the equator is exactly equal to the lost of kinetic energy
of the rotating Earth. Therefore, as expected, supplementary
energy is
really needed to move the clock from the pole to the equator.
However,
that energy comes from the kinetic energy of the rotating Earth.
Therefore there is mass-energy conservation and also angular momentum
conservation in the Earth frame. Therefore is no need to use any
force
to carry the clock from the pole to the equator, because the internal
Earth energy of rotation transfers that energy naturally.
The whole Earth is slowed down when a mass is moved from the pole to
the equator as demonstrated here. For that reason, the height of
the
sea level is in a natural equilibrium, higher at the equator than at
the pole. Unfortunately, Hatch paper neglects the fact that the
mass
of the clock has absorbed energy when passing from the pole to the
equator while there is no change of energy between perigee and
apogee.
We have seen that the increase of kinetic energy slows down the clock
at the same time as an increase of potential energy speeds it up.
This
explains the fact that the clock rate does not change when we move on
the Earth surface remaining at sea level. No new theory is
required.
Conversely, in the case of an orbiting clock, there is no change of
total energy between the perigee and apogee so that this leads
naturally to a change of clock rate between perigee and apogee of an
orbiting body. Those two completely different experiments must
not be
confused.
References
1- Natural Length Contraction Mechanism Due to Kinetic Energy.
On the web at:
http://www.newtonphysics.on.ca/kinetic/index.html
2- Natural Physical Length Contraction Due to Gravity. On the Web at:
http://www.newtonphysics.on.ca/gravity/index.html
3 - A Detailed Classical Description of the Advance of the
Perihelion of Mercury. On the web at:
http://www.newtonphysics.on.ca/mercury/index.html
4 – Einstein’s Theory of Relativity Versus Classical Mechanics
P. Marmet, Book, (1997) Ed. Newton Physics Books, 2401 Ogilvie Rd.
Ottawa, Canada, K1J 7N4
On the Web at:
http://www.newtonphysics.on.ca/einstein/index.html
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Paul Marmet, April 4, 2002
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