The concept of inertia spawned
Galileo’s concept of relativity: the __sensory__
and __empirical__ equivalence of the uniform rectilinear velocities of
terrestrial bodies, and the accelerated motions that occur on them. As we learned in Chapter 4, the magnitudes of
force, mass and acceleration vary in such a way (covariantly) that

**A. Galileo’s Relativity**

In Aristotle’s theory of the cosmos, the Earth was at rest at the center, and the Sun, Moon, planets, and stars all rotated around the stationary Earth once each day. (see Figure 2.1A) Aristotle attempted to justify his theory that the Earth was stationary in space with the following rationalization. When a rock is allowed to fall to the ground from a tall tower, it is observed to land at the foot of the tower. If the Earth was moving through space while the rock was falling, then it should land in a much different place. (Gamow, 1961, pp. 42 – 43)

On the other
hand, in the 16^{th} century Nicolas Copernicus theorized that the Sun
was at the center of the Cosmos, and that the Earth, Moon and planets all
rotated around it. (Figure 2.1B) This heliocentric concept was completely at
odds with Aristotle’s theory and Christian dogma, and was vehemently opposed by
the Catholic Church. In 1632, Galileo
wrote a book that supported the Copernican theory, entitled *Dialogues on the
Two Great World Systems*. (French, p.
67)

In his book,
Galileo “argued that the vertical path of a falling object does not compel one
to the conclusion that the earth is stationary.” (*Id*.

“He gave, by way of analogy, the example of a rock dropped from the top of the mast of a ship. Whether the ship is at rest or moving with a constant velocity, the rock always lands just at the foot of the mast. Thus an observation of the point of impact on a deck reveals nothing of the ship’s state of motion.” (French, p. 67; see Figure 5.1)

By analogy to the above empirical experiment, Galileo argued that the point of impact on the Earth of the rock that falls from a tower reveals nothing of the Earth’s state of motion.

Galileo also argued that the illusion of being at rest that we sense when we stand on the Earth does not compel the conclusion that the Earth is not moving. He then asserted, by way of analogy, the following thought experiment. If an observer is enclosed in a windowless cabin on a large ship, the observer cannot tell from the uniform rectilinear velocity of the ship and from observing the accelerated motions of the objects inside the cabin, whether he is sailing uniformly straight ahead at a constant speed on calm seas, or whether he is docked in port and at rest relative to the Earth’s surface. Galileo’s actual words were:

“so long as the vessel stands still [the motions of __material__
objects take place in a normal manner.
Then] make the ship move with __what velocity you please__, so long
as the motion is uniform and not fluctuating in this way and that. You shall not be able to discern the least
alteration in all the forenamed effects, nor can you gather by any of them
whether the ship moves or stands still...”
(Galileo, *Dialogues*, 1632 [Gamow, 1961, p. 45])

Ergo being on a body that exhibits uniform rectilinear
motion, or observing accelerated motions on such body, reveals nothing of the
state of motion of such body (i.e. Earth).
Why is this true? Because all
observers and all other material objects on __any__ body with any uniform
rectilinear velocity empirically have exactly the same __experience__…a
sensory illusion of rest. This concept
of the sensory equivalence (relativity) of all uniform rectilinear terrestrial
motions, and the empirical equivalence of all accelerated motions which occurs
on them, is sometimes referred to as Galileo’s principle of relativity.

Galileo’s Relativity demonstrated that all uniform rectilinear velocities of terrestrial bodies (including the Earth itself) are equivalent states of motion with respect to any mechanical events that occur on such bodies.[1] (Figure 5.2) This concept of mechanical relativity is nothing new. Everyone since the dawn of civilization has experienced this same phenomenon of empirical equivalence (relativity). (Bird, p. 51) But Galileo was possibly the first person to specifically describe it.

The __distance__
(vt) that the ship traveled from the port did not matter, nor did the __magnitude__
of its uniform velocity (v). [2]
The only thing that mattered was that in each position of the ship (in dock or
sailing away from the dock) the velocity of the ship was uniform and
rectilinear. (*Id*.

Galileo’s example also demonstrated that such uniform rectilinear velocities may either be ‘inertial’ (without applied force, like we perceive the motion of the Earth through space to be); or it can result from the application of a force (like the wind uniformly pushing the sailing ship forward). Therefore, Galileo’s concept of the empirical equivalence (or the ‘relativity’) of uniform velocities is not really dependent upon the concept of inertia…uniform rectilinear motion without applied force.[3]

Why is
Galileo’s concept of relativity true?
Because the uniform motion of the ship is __common__ to the observers
and the other material objects relatively at rest on it. [4] In other words, the ship and its contents
all __experience__ the same __common uniform velocity__. Empirically, the motions of such material
objects are the same as when the uniformly moving ship was lying in the harbor
at rest relative to the Earth’s surface.
(Figures 5.2A and
5.2C) [5]
The different magnitudes of uniform velocity of two moving bodies (the Earth
and the ship) are not sensed or perceived by an observer or by any other
material object in either situation. All that is perceived by an observer in
both equivalent situations is a __sensory illusion of rest__. Again, the observer and his material objects
have exactly the __same experience__ in both situations of perceived rest.[6]

To summarize,
the principle of Galileo’s Relativity was comprised of __two__ separate but
related concepts:

1. All uniform rectilinear (inertial) velocities
of terrestrial bodies are __sensorally__ and __empirically__ equivalent
states of motion; __and__

2. The (covariant) __accelerated__ motions of
terrestrial objects __on__ such inertially moving bodies demonstrate the
‘invariance’ of

Without __both__ concepts,
there is no principle of Galileo’s Relativity.[7]

One might ask
at this point: What relevance or
importance does Galileo’s sensory and empirical concept of relativity have with
respect to Einstein’s Special Theory of Relativity? The answer is that Einstein adopted a very
different mathematical variation of Galileo’s Relativity (which we shall call
“Galilean translational relativity’) as the basic mathematical framework for
his Special Theory, and he needed to characterize this very different mathematical
variation as merely an __extension__ of Galileo’s Relativity so that his *ad hoc* mathematical Special Theory might
__appear__ to have some semblance of an empirical foundation.[8] (see Chapters 13, 14, and 24) This subterfuge by Einstein was an attempt to
disguise the fact that his mathematical Special Theory had __no__ empirical
foundation at all and was in fact totally *ad* *hoc*.

**B. Mathematical Variations of Galileo’s
Relativity Concept**

Fifty-five
years after 1632, __portion__ of Galileo’s concept of relativity for his 1687 *Principia*:

“The motions of bodies included in a given space are the same among
themselves, whether that space is at rest, or moves uniformly forward in a
right [straight] line…A clear proof of this we have from the experiment of a
ship; where all motions happen after the same manner, whether the ship is at
rest, or carried uniformly forwards in a right [straight] line.” [9] (Newton, *Principia* [Motte, Vol. 1, p.
21])

Very importantly, Newton left
out *inter alia* the __sensory
illusion__ of rest which is perceived by the inertial observer, so that he
cannot determine whether he is moving or not.

During the 18^{th}
and 19^{th} centuries, the above empirical concept described by

This __mathematical
equivalence__ of every uniformly and rectilinearly moving frame on the Earth
for the same algebraic operation of __sensory__
or __empirical__ operation and equivalence thereof, it is a very different
concept than Galileo’s original concept of relativity. (see Chapters 13 and 14)

After

There never
was a question that Newton’s second law of motion (F = ma) worked the same way
and did not change its algebraic form where the uniform velocity of the
terrestrial laboratory was __common__ to the inertial motion of the Earth,
such as two different acceleration experiments conducted at different locations
on the surface of the inertially moving Earth.
As Galileo stated in 1632, “no
man doubts that, so long as the vessel stands still [on the surface of the
Earth, such accelerated motions of objects] ought to take place in [the normal]
manner.” (Galileo, *Dialogues*,
1632 [Gamow, 1961, p. 45])

The only
possible remaining question was: Does
Newton’s second law of motion work the same way and does it retain the same
algebraic form on two different uniformly moving bodies where the __different
uniform rectilinear velocities__ are only __equivalent__? (see Figure 5.3) Thus, by the early 18^{th} century,
the __only__ possible purpose of Galileo’s Relativity was to theoretically
demonstrate the equivalence (empirical ‘covariance’) of accelerated motions on
two different and __spatially separated__ uniformly moving bodies, and thus
to theoretically demonstrate the empirical ‘invariance’ of Newton’s second law
of motion on all __equivalent__ terrestrial frames with different uniform
velocities anywhere on the planet.[11]
(Figures 5.2A and 5.2C)

By the latter
part of the 18th century there was no doubt that

As previously
mentioned, notably omitted from __observer__
located in an enclosed cabin on such ship cannot sensorally detect from the
uniform motion of the ship, nor empirically detect from the accelerated motions
of bodies in the cabin at either time “whether the ship moves or stands still.”[12] However, during the latter part of the 19^{th}
century, it was theorized that if stationary ether existed, then light
experiments conducted on inertial bodies (i.e. Earth) should be able to detect
the absolute velocity of the Earth relative to the theoretically stationary
ether. At this point, the Newtonians belatedly
adopted Galileo’s statement (concerning sensory and empirical detection) which
was omitted by __mechanical
experiment__ conducted in the cabin of the ship would allow any observer to
sense or empirically detect any difference in the two equivalent states of
uniform velocity (rest or uniform motion).[13] (Gamow, 1961, pp. 45, 46) It follows that if such observers could not
distinguish between two uniform motions, then they could not tell whether or
not they were moving through space. But
the famous Michelson and Morley light experiment (Chapter 9) and many other similar
light experiments paradoxically failed to detect any such absolute
velocity. This failure created a crisis
in physics and demanded an explanation.

For all of
the above reasons, during the score of years between 1885 and 1905, Galileo’s
simple sensory and empirical principle of relativity underwent a dramatic
theoretical and mathematical metamorphosis.
It was first modified into an abstract ‘__relativistic__’ concept of
co-moving inertial reference frames by Ludwig Lange in 1885 (Chapter 13), and
thereafter mathematically described and applied by what were called the
‘Galilean transformation equations.’[14]
(Chapter 14) In 1904, Lorentz
mathematically modified such Galilean transformation equations in a very
radical way so that they became *ad hoc* Lorentz transformation equations,
and Poincaré attempted *ad hoc* to generalize and apply Galileo’s complete
concept of Relativity (including the inability of inertial observers to tell
whether they were uniformly moving or at rest) to electromagnetics (the
constant velocity of light) and optics as well as mechanics.[15] (see Chapter 16)

In 1905,
Einstein adopted Lorentz’s *ad hoc*
transformation equations and Poincaré’s *ad hoc* concepts of relativity and
modified, reinterpreted and applied them so that *inter alia* they described
an impossible __absolute__ propagation velocity of light at *c*, an
impossible contraction of matter, an impossible dilation (expansion) of time,
an *ad hoc* increase in mass with
velocity, etc., etc. Einstein asserted
that he was merely reconciling Galileo’s mechanics concept of relativity with
Maxwell’s concept for the constant velocity of light at *c*, in order to
arrive at his Special Theory of Relativity.
However, as we will demonstrate in later chapters, the radical principle
of relativity that Einstein was referring to had nothing to do with Galileo’s
simple sensory and empirical concept of relativity, nor with the algebraic
version of relativity, nor with Maxwell’s concepts of light. By the time Einstein was through
mathematically fiddling with Galileo’s Relativity, Galileo most likely would
not have been able to recognize his own simplistic sensory and empirical
concept. Not surprisingly, Galileo’s
Relativity, the algebraic version of relativity, Lange’s Relativity, Lorentz’s
Relativity, Poincaré’s Relativity, and Einstein’s Special Theory of Relativity
are all very different concepts. (see Memo 24.3)

C. Is Galileo’s Relativity a
fundamental law of nature?

Galileo’s analogy in favor of the Copernican theory, that the observers in a closed cabin on a uniformly moving ship could not physically sense such motion and that the objects on such uniformly moving ship empirically accelerated in the same manner as when the ship was at rest in port (relative to the Earth), helped Galileo in 1632 to persuade others that the apparently stationary Earth might actually be moving. It remains an interesting phenomenon and relationship for purposes of discussion. However, such analogy was irrelevant to the rest of Galileo’s concepts of relativity and to the relative motions that they described. If the observers went up on deck they would sense the wind in their faces and see the boat moving relative to the water. If they looked through a telescope they would realize that the Earth was moving relative to the other planets. If they constructed a Foucault pendulum on board, they could observe the motion of the Earth around its axis. Thus, we must ask the question: Is Galileo’s Relativity even a fundamental law of motion?

Because of the strict theoretical limitations (i.e. uniform rectilinear
velocity) of Galileo’s Relativity, and because it substantially ignores the
ubiquitous force of gravity, Galileo’s concept of relativity __in any form__
is really just an approximation of reality and an impossible idealization. Even the Earth does not meet these strict
requirements. Its surface is curved, its
elliptical path around the Sun is curved and results from gravitational forces,
and its elliptical velocity is not uniform.[16] On Earth, the uniform velocity of terrestrial
objects is only sustained by a continuous application of force; therefore such
velocity is not inertial. For these
reasons, Galileo’s Relativity most likely does not even qualify as a
fundamental law of mechanics, much less a fundamental law of physics or nature.

If two trains accelerate unequally over the same
circular track, are not their accelerated motions still covariant? Does not

From all of the above discussions,
it becomes apparent that the 17^{th} and 18^{th} century
mechanics concept of Galileo’s Relativity should not be characterized as a
fundamental law of motion, much less a fundamental law of physics or nature.[17] Nor should its mathematical counterparts,
such as the Galilean transformation equations, or Lange’s, Lorentz’s or
Einstein’s modified relativistic models of Galileo’s Relativity, be
characterized as fundamental laws of motion.
Although originally based on a fundamental law (inertia), Galileo’s
concept of relativity remains just a sensory and empirical illusion of rest and
a manmade convention that was __convenient__ for early scientists as an
intuitive theoretical demonstration of the empirical covariance and invariance
of

Whatever their current relevance, Galileo’s original principle of
relativity and its mathematical variations should be confined to the realm of
mechanics. Contrary to the assertions of
Poincaré and Einstein, the very different phenomena of electromagnetism,
radiation, and the constant velocity of light in a vacuum have absolutely __nothing__
to do with inertia, nor with the uniform rectilinear motions of matter, nor
with the accelerated motions of matter, nor with mechanics, nor with material
frames of reference, nor with coordinate transformation equations, nor with
Galileo’s Relativity. (see Chapters 23
and 24)

[1] For
example, the Earth uniformly rotates on its axis each day at a uniform velocity
of approximately 0.5 km/s. It more or
less uniformly orbits around the Sun at a uniform velocity of about 30 km/s
(Born, p. 67), and along with the Sun it more or less uniformly orbits around
the core of the Milky Way Galaxy at a uniform velocity of about 225 km/s. (see De Sitter, 1932, p. 95; Hubble, 1942, p.
103) Yet we on Earth do not notice nor
distinguish any of these great uniform velocities. (Gamow, 1961, p. 174) “All __mechanical__ events on the earth
occur as if this tremendous forward motion did not exist.” (Born, p. 68)

[2] The velocity of any moving ship (body) may be very different, as long as such velocity is uniform and rectilinear.

[3] Born and
other scientists have asserted that:
‘the root of this [relativity] law is clearly the law of inertia.” (Born, p. 69)
But these assertions are not rigorously correct as we have seen from
Galileo’s example, __and__ also because inertial motion without applied
force is only an idealization (due to the ubiquitous __force__ of
gravity). A more correct statement might
be that the root of Galileo’s Relativity is uniform rectilinear velocity (v).

[4]
Galileo’s actual words were: “…the cause
is that the ship’s __motion is common__ to all the things contained in it
and to the air also…” (Galileo, *Dialogues*,
1632 [Gamow, 1961, p. 45])

[5] The fact that all inertially (or uniformly and rectilinearly) moving bodies on Earth are equivalent is an absolute terrestrial concept; a constant phenomenon that is the same for every terrestrial observer at sea level. (Goldberg, p. 80) The observer is therefore irrelevant with regard to any attempts to distinguish one uniform terrestrial velocity from another.

[6] We on
Earth share these common uniform velocities with the uniformly moving mother
ship Earth. Relative to the Earth and to
each other, when we stand on the Earth we are all __relatively__ at rest. As
Born put it: “A system of bodies all of
which travel through space with the same constant velocity is…at rest as
regards their mutual position…” (Born,
p. 69)

[7] Nevertheless, in his 1905 Special Theory, Einstein substituted the constant speed of a ray of light with respect to inertial bodies for the accelerated motions of material objects on inertial bodies, and incorrectly characterized this scenario as Galileo’s Relativity, which of course it was not. (see Chapter 19)

[8] The reasons why ‘Galilean Translational Relativity’ (usually called the ‘Galilean transformation equations’) is so different from Galileo’s Relativity are described in Chapters 13 and 14.

[9] For example, “the billiard player in a closed box car of a train moving uniformly along a straight track cannot tell from the behavior of the balls what the motion [uniform velocity] of the train is with respect to the ground.” (Resnick, 1968, p. 13) The train may be going east or west, north or south, at any uniform velocity; or it may be stopped and just sitting on the tracks. The billiard balls exhibit exactly the same laws of mechanics (i.e. force, acceleration, momentum, conservation of momentum, inertial resistance and equal and opposite actions) in all of these equivalent situations.

[10] Born referred to Galilean Relativity as “the law of the relativity of mechanical events…” (Born, p. 69)

[11] These
theoretical demonstrations also applied to

[12] By
1687, the Copernican heliocentric theory of the Solar System was widely
accepted. Therefore, Newton (unlike
Galileo) did not need to use the __sensory and empirical equivalence__ of
uniform rectilinear moving bodies for an observer in order to demonstrate and
justify the idea that the apparently stationary Earth could be moving, and
thereby to advance, defend and justify the Copernican theory.

[13] A
modern statement of this additional concept “is that no __mechanical__
experiments carried out [or deduced] entirely in one inertial frame can tell
the observer [in that frame] what the motion [magnitude of velocity] of that
frame is with respect to any other inertial frame.” (Resnick, 1968, p. 13)

[14] The
term ‘relativistic’ means that there are two abstract __frames__ (i.e.
ships) in uniform rectilinear motion __relative__ to one another, rather
than Galileo’s empirical concept of two __positions__ of the same ship: one position in port and one position moving
uniformly away from port.

[15] Of course the reader should ask the question: What relevance does the constant velocity of light have with respect to the accelerated motions of matter or with respect to inertial motions? The obvious answer is: none.

[16] Kepler discovered in 1609 that the Earth speeds up when it is near to the Sun, and slows down when it is further from the Sun.

[17] On the
other hand, Einstein attempted to characterize Galileo’s Relativity as a
fundamental empirical law of physics and nature, because he needed its
empirical qualities in order to justify his Special Theory, and to give it an
empirical foundation. (see Einstein, *Relativity*,
pp. 15 – 18; Chapters 23 and 24)