INERTIA,

Galileo discovered the empirical
relationships of gravitational acceleration and the phenomena of inertia. Later,

**A. Gravitational and Parabolic Motion, Inertia
and Inertial Motion**

Galileo Galilei (1561 – 1642) “was trained in the medieval Aristotelian tradition.” (Goldberg, p. 22) Aristotle had conjectured that “heavy objects fall down faster than light ones.” (Gamow, 1961, p. 35) In order to test Aristotle’s conjecture, Galileo (in 1589) took a light cannonball and a heavy cannonball to the top of the Leaning Tower of Pisa and simultaneously dropped them. To everyone’s amazement they appeared to hit the ground at the same instant. (Figure 4.1A)

Both of such objects appeared to go faster and faster as
they fell. The inquisitive Galileo
wanted to know what relationship governs this motion. So in order to dilute or slow down the fall
of a ball he rolled it down an inclined plane.
(Gamow, 1961, p. 35; Figure 4.1B) After several years of trial and error, in
about 1604, Galileo finally discovered the mathematical relationship of
gravitational acceleration near the surface of the Earth. The total distance covered by a free-falling terrestrial
object “during a certain period of time” is proportional to the square of that
time. (*Id*., p. 36; see Figure 4.2A)

As a by-product of these and similar experiments, Galileo
also discovered (in about 1608) the law that governs the path of a projectile
near the surface of the Earth. (Cohen,
1960, p. 212; Figure 4.1C) The trajectory of a projectile has two
independent components (Gamow, 1961, p. 39), “a vertical component that follows
the law of free fall (just as if there were no horizontal component) and a
horizontal component of forward motion that is uniform (just as if there were
no vertical component).” (Cohen, 1960,
p. 212) The distance of the horizontal
component is proportional to the time elapsed.[1]
(*Id*.

Thus, Galileo demonstrated and concluded that terrestrial objects (such as wagons) and projectiles (such as arrows and cannonballs) do tend to continue in motion through space or along the ground even after the force has been withdrawn.[3] He called this theoretical phenomenon of continuous uniform rectilinear motion of matter without applied force, ‘inertia’ or inertial motion.[4]

According to Aristotle’s point of view, a stone released
from the top of the mast of a sailing ship will fall vertically down and land
close to the stern of the ship. By the
end of the 16^{th} century, several people had actually tested
Aristotle’s conjectures and found that the stone instead falls to the base of
the mast. [5]
(see Figure 5.1) However, no one could explain why this
paradox occurs. No one, that is, except
Galileo after 1608.

The reason for the paradox was the physical phenomenon of
inertia and the resulting inertial motions of moving objects. [6]
Objects on the surface of the Earth which share a __common__ lateral
inertial motion, such as the ship, the man on the mast, and the stone, maintain
this common lateral inertial motion relative to the Earth, even when the force
is withdrawn and they become physically detached from one another, like the
falling stone. The detached stone also tends to accelerate
downward toward the Earth with a parabolic trajectory due to the forces of
gravity.

Why does the falling stone share the common lateral
inertial motion of the man on the mast and of the ship? One reason is that the lateral motion is
perpendicular to the force of gravity; therefore, there is no opposing force in
the lateral direction, so (ignoring the effects of air) inertial motion is
sustained. Another reason is that they
are all material bodies and have mass (m), similar velocity (v) and thus similar
__momentum__. After the force on the
stone is withdrawn (by the man dropping it), the stone continues to move
inertially in common with the ship because of its material ‘momentum:’ its mass times its velocity (mv). (Goldberg, p. 52) On the other hand, a photon or a ray of light
theoretically does not have any mass, and therefore it cannot exhibit inertia,
inertial motion, or lateral material momentum like the stone. Such material concepts of mass, velocity,
inertia, inertial motion, and momentum are all irrelevant to non-material
light.[7]

Descartes was one of the first philosophers to fully understand the concept of inertia and to extend it to the motions of heavenly bodies. He theorized that a planet that continuously moves uniformly and rectilinearly (substantially in a straight line) does not require a force to maintain such motion. (Goldberg, p. 46; Cohen, 1960, p. 210) This is the classical law of celestial inertial motion (the persistence of motion without apparent applied force), which we observe as the perpetual motions of the planets, the stars, and the galaxies.[8]

**B. Newton’s Three Laws of Motion**

Early in the *Principia*,

1.

“Every body
continues in its state of rest, or of uniform motion in a right [straight]
line, unless it is compelled to change that state by forces impressed upon it.” [9]** **(Newton, *Principia* [Motte, Vol.
1, p. 13])

__innate force__ of matter…[its] power of __resisting__…” (*Id*., p. 2) In this regard, it can be asserted that the
inertia of a body (its power or __tendency to resist__ change in its __state__
of motion or of relative rest) is __proportional__ to its mass.[11] (see Cohen, 1960, p. 156) This concept is often referred to as the
‘inertial mass’ of an object.

However,
unlike Galileo, who merely described the orbital motions of the planets as
inertial, Newton asserted that such inertial motion must have constant __direction__ of motion
in a __straight__ line as well as a constant magnitude (speed).[12] In other words, __in a particular
direction__.[13] (see Goldberg, p. 33; and Figure 4.3A) Although orbital motion is not rigidly
inertial motion under __almost__ uniform straight-line motion of
certain heavenly bodies (such as the Sun and our Milky Way (MW) galaxy) can be
approximated to be inertial motion.[14]

German
astronomer Johannes Kepler (1570 – 1631), who described the three laws of
planetary motion in 1609 and 1618, also introduced the Latin word ‘inertia’
(meaning ‘laziness’) to physics. (Cohen,
1960, p. 210) Galileo described uniform
inertial motion as where the increments of distance, time and speed “repeat
itself always in the same manner.” (*Id*.,
pp. 88 – 89; Figure 4.3A)
French philosopher Rene Descartes (1596 – 1650) first clearly described the
phenomenon of inertia as a ‘state’ in his unpublished book, ‘Le Monde,’ (Cohen,
1960, p. 210; Goldberg, p. 53), and French scientist Pierre Gassendi (1592 –
1655) first published a description of the law and he also tested it with
experiments.[15]
(Cohen, 1960, p. 211; Harrison, pp. 125 – 126)
Thus,

2.

“The
change of motion [acceleration] is proportional to the motive force impressed;
and is made in the direction of the right [straight] line in which that force
is impressed.”[16] (Newton, *Principia* [Motte, Vol. 1, p.
13])

This law described the motion of a body that is not in equilibrium, vis.,
where a force is acting on the body and is not counterbalanced by another
force. *Id*, p. 2) Galileo mainly talked about accelerations, not
forces. On the other hand, __forces cause__ accelerations, and in which __direction__. Thus

__continuous__ net force (F) applied to it and is __proportional__
to such force.[17] (see Figure 4.3B) Therefore, F = ma.[18] (Young, p. 100)

__impact__ or __instantaneous__
(but not continuous) force, such as when a bat strikes a ball. Upon impact, the ball or other projectile is
initially accelerated in proportion to the motive force applied, and it gains
momentum. In space, far away from the
Earth, the projectile will maintain a constant uniform rectilinear velocity,
whereas near the surface of the Earth it soon succumbs to air resistance and
the downward force of gravity and it decelerates. According to

__mass__’
(m) of an object to mean its “quantity of matter…its density and bulk [volume]
conjointly.” (Newton, *Principia* [Motte, Vol. 1, p.
1]) The greater a body’s mass, the more
a body ‘resists’ being accelerated, vis. having its state or direction of
motion changed. Empirically, it is
observed that: “If a force causes a
large acceleration, the mass of the [accelerated] body is small; if the same
force causes only a small acceleration, the mass of the [accelerated] body is
large.” (Young, p. 100) Thus, the mass of a body is the quantitative
measure of its power or force of inertial resistance, and the magnitude of mass
is __inversely proportional__ to the force applied to cause its
acceleration: thus, m = F/a.[20] (*Id*.

__acceleration__ of an object is *inversely
proportional* to the mass of the accelerated object. (Zeilik, p. 68; Young, pp. 100 – 101) Feynman agreed: “a body reacts to a force by accelerating, or
by changing its velocity every second to an extent __inversely to its mass__.” (Feynman, 1965, pp. 4 – 5) Therefore, a = F/m. How do we know this to be true? Because (empirically) if you apply the __same__
continuous force to two balls (m_{1} and m_{5}) and one ball (m_{5})
accelerates only one-fifth as far and as fast as the other ball (m_{1})
during the same period of time (t), then *a priori* ball (m_{5})
must have 5 times as much mass as ball (m_{1}), and the magnitude of
acceleration (a) of each ball is __inversely proportional__ to the magnitude
of its own mass. (*Id*.

At the
beginning of the *Principia*, *Principia*
[Motte, Vol. 1, p. 1]) In other words,
the term “quantity of motion,” as defined by *Id*.

It might be claimed that Galileo anticipated Newton’s second law in his projectile experiments, because they combine two independent forces (one impact, i.e. the propulsion of a cannonball) and (one continuous, vis. gravity) in two different directions, into the combined (uniform velocity and uniform acceleration) motion of a mass (a projectile), which results in a parabolic trajectory in a third direction. (Figure 4.2) But Galileo never took the next step and synthesized these motions, forces, masses and accelerations into a generalized law.

3.

“To
every action there is always opposed an equal reaction; or, the mutual actions
of two bodies upon each other are always equal and directed to contrary parts.” (Newton, *Principia* [Motte, Vol. 1, p.
13])

__motions__ (caused by force) occur in equal (or
equivalent) and simultaneous pairs, and in opposite directions. When an impressed (applied) force accelerates
an object in one direction (the action motion), an equivalent __motion__ and
force of reaction is created in the opposite direction. This equivalent motion and force of reaction
is exemplified by the backward ‘g force’ that a passenger experiences during
the acceleration and take-off of an airplane, the recoil when a person fires a
rifle, or when a person on the ground pulls on a rope attached to a heavy
wagon, the person lurches (reacts) toward the wagon but the heavy wagon only
moves (reacts) slightly towards the puller.

The impact force and acceleration motion of the
relatively small mass of a bullet propelled out the barrel of a rifle is __equivalent__
to the force and acceleration motion of the more massive rifle in the opposite
direction. For example, when a bazooka
is fired, the explosion (force in the barrel) causes the __action__ motion
of the projectile out the front of the barrel and the equivalent __reaction__
motion and force of the exhaust out the rear of the barrel. (see Figure 4.3C) Because these opposite forces and motions are
equivalent they offset each other, and the person who holds the bazooka does
not feel any recoil; in effect, the bazooka remains in equilibrium.

In the third part of the Principia, entitled, “The
System of the World,”

**C. Covariance and Invariance**

__algebraic form__ of *Id*., p.
80) There is also a geometrical analogy
to classical ‘covariance’ that might help to explain the concept. (see Figure 4.5)

The
covariance of the interaction between the different variable quantities (F, m,
a) that invariantly results in the same fundamental law of motion (

There is also
another form or set of transformation equations that relate the motions of the __same__
mechanical experiment in two different __inertial__ frames of reference, in
order to demonstrate that __mechanics__
remain mathematically ‘invariant’ with respect to Galilean mathematical
transformations. (Goldberg, p. 80 – 81)

__inertial__ frames of
reference became so important to classical mechanics during the 19^{th}
century. (see Chapters 13 and 14)

Toward the
end of the 19^{th} century, the mathematical concept of covariance was
extended to the concept of ‘invariance’ for certain magnitudes and properties
of matter. An ‘invariant magnitude or property’ is one that does not change for
any observer regardless of his or the object’s velocity. [22]
(Goldberg, p. 81) The magnitudes and
properties of matter which were considered by late 19^{th} century
scientists to be invariant included: the
mass of an object, its length, its other dimensions, its shape and its color. (see Resnick, 1968, pp. 11, 15)

Very importantly, it should be pointed out at this early
juncture that Einstein drastically changed the above mechanical and empirical meaning
of algebraic ‘covariance’ for his Special Theory to mean the algebraic
‘invariance’ of physical laws and physical magnitudes with respect to the
Lorentz transformations. For Einstein,
the term ‘covariance’ meant the__ transformation__ (or translation) of __any__
algebraic law, equation, physical phenomenon or magnitude (including Maxwell’s
constant transmission velocity of light at *c*) from one inertial
reference frame to another inertial reference frame by means of his radical
Lorentz transformation equations or the equivalent. (see Chapters 21 and 27)

When so transformed by Lorentz transformations, the
classical laws of physics (including mechanics, electrodynamics and optics) would
be distorted and would dramatically change from their classical meaning or
magnitudes, but still they would remain __invariant__ “with respect to
Lorentz transformations.” (see Einstein,
*Relativity*, p. 48) In other
words, the distorted laws of physics and their magnitudes would be the same in
each inertial reference frame.

All of these arbitrary and radical mathematical
changes to physics were invented for one primary *ad hoc* purpose: to mathematically and artificially keep
Maxwell’s law for the constant transmission velocity of light at *c*, at the absolute magnitude of *c*
relative to all inertial reference frames, regardless of their linear motions, so
that the velocity of light relative to such reference frames could never
mathematically be *c* – v or *c* + v.
(see the Preamble) Rohrlich
referred to this artificial mathematical result as “Einstein’s fiat.” (Rohrlich, pp. 55 – 62)

All of the fundamental concepts described in this chapter will be important to a full understanding of the phenomena and theories discussed in the chapters to follow.

D. Failures of Classical Mechanics

__position__ in space with
time.” (Einstein, ‘*Relativity*,’
p. 10) “German mathematician Leonhard
Euler, who was only 20 when

“Many other
important developments of Newtonian mechanics[23]
and gravitation theory were made in the late eighteenth and early nineteenth
century. Such men as Lagrange, Laplace,
and Hamilton, who were both mathematicians, theoretical physicists and
astronomers, provided much more powerful mathematical techniques than were
available to Newton. They made it
possible to predict astronomical events with very great precision. But during this whole period of the
elaboration and extension of *relative motion*
and no difficulties were encountered.” (*Id*.

Since __time__’ (instant) of a distant event. Thus, classical mechanics also failed to
fully appreciate the intertwined relationship between position and time for the
algebraic description of motions.

During ^{th} century, the __mass__
of an object was always considered to be an inherent and invariant property of
the object. It never varied. Then during the period 1901 to 1904, Kaufmann
and several other scientists discovered by experiments, calculations and
theories that the ‘electromagnetic’ mass of an electron can vary depending upon
how much energy is applied to it. (see
Chapter 17) However, such variation can
depend upon how one defines the word mass. [24] (see Chapters 17 and 31) Nevertheless, classical mechanics failed to
realize these possibilities.

Classical mechanics also failed to recognize the correct relationship between matter, energy and mass. (Chapter 32) All of the above failings, in turn, may have affected the computation and mathematical description of positions, accelerations, momentum, resistance, time, and other values in classical mechanics and celestial mechanics.[25]

Such theoretical failings must of course be adequately remedied by
current physics. Nevertheless, *inter alia* to
radically modify *inter alia* to radically modify *ad hoc* theories of relativity was necessary or
even relevant for the correction of such failings, nor for almost any other
reason.

[1] Galileo demonstrated the path of an arrow, the path of a fired cannonball, and the path of any other spherical projectile on a wide slightly inclined plane, sometimes called a ‘wedge.’ (Cohen, 1960, p. 112; Figure 4.1C)

[2] By 1609,
Galileo had mathematically confirmed such parabolic motions. (Cohen, 1960, p. 212) Galileo’s mathematical language was
geometry. “He compared speed to speed,
position to position, time to time…”
Algebra did not become popular until the 18^{th} century. (Goldberg, p. 22)

[3] On Earth
such motion gradually slows to a stop because of the resistance of the air and
the friction of a surface. On the other
hand, in empty space the object will *a priori* coast at the same velocity
and in a straight line (although slightly curved due to the distant forces of
gravity), possibly forever.

[4] But
Galileo’s concept of inertial motion did not continue in a straight line
forever. Rather, it was limited to
straight segments and segments that __curved__ at great distances, because
Galileo could not grasp the concept of spatial infinitely. (Cohen, 1960, pp. 117 – 119, 122)

[5] We must assume that the effects of the air and wind on the stone are negligible.

[6] On or near the surface of the Earth, terrestrial inertial motion is usually lateral relative to the Earth’s surface, because of the downward force of gravity. However, in the vacuum of space, far from gravitational influences, celestial inertial motion may occur in any direction.

[7] We shall refer to this subject again in various Chapters.

[8] However, the concept of celestial inertial motion without applied force is also an impossible idealization, because the gravitational forces of other celestial objects that produce orbital or curved motions are not taken into account.

[9]

[10] The idea that inertial motion can be a ‘state’ of motion or a ‘state’ of rest was asserted by both Galileo and Descartes. (Cohen, 1960, p. 216)

[11] It also follows that bodies with the same quantity of matter (mass) have the same inertia. (Cohen, 1960, p. 157)

[12]
Although Galileo postulated that uniform motion on a “plane would be perpetual
if the plane were of infinite extent” (Cohen, 1960, p. 117), he could not
imagine this happening. At great
distances Galileo imagined inertial motion to be curvilinear. (*Id*., pp. 119, 112, 124) In effect, Galileo was really describing
uniform momentum.

[13]

[14]
Is the motion of celestial bodies ever rigorously uniform
straight-line inertial motion? The answer is probably no. A galaxy’s motion through space is about as close as one can get to idealized
inertial motion, because its motion appears to be uniform and random and it
doesn’t seem to orbit anything. The Sun’s combined galactic motion
and slightly orbital motion is also close to straight-line inertial
motion, and yet it is ever so slightly curved, because each 200 million
years or so the Sun circumnavigates the Milky Way Galaxy.
The Earth shares the Sun’s almost uniform straight-line motion at about
225 km/s relative to the core of the galaxy, but because of the Earth’s close
proximity to the Sun, the Earth also orbits the Sun at 30 km/s every 365 Earth
days. *A priori*, there can be no
perfectly uniform and straight-line inertial motion, because the trajectories
of all bodies are affected, more or less, by the gravitational attraction of
other objects in the universe.

[15] Gassendi dropped rocks from the mast of a moving ship, and because of their inertia the rocks landed near the base of the mast, rather than toward the stern. (Harrison, pp. 125 – 126)

[16] It
turns out that this second law was possibly first conceived and written down by

[17] The
standard unit of measure for any force is now called a ‘

[18] This
modern algebraic form of

[19] Newton asserted that a given force (F) results in a certain acceleration (a) of a body (m), but in order to determine the velocity v of mass m at any instant (during such acceleration), we must also know the duration of time (t) which such force has been applied. Therefore, v = at. (Cohen, 1960, p. 155)

[20] Again,
since ‘mass’ is the quantitative measure of matter’s __inertial__ force of
resistance, the magnitude of a body’s mass is often referred to as its
‘inertial mass.’

[21] In a
treatise to follow this one, entitled the *Relativity of Gravity*, we
shall fully discuss Galileo’s, Kepler’s and Newton’s laws of gravity, and
Einstein’s attempt to supplant Newton’s law with a radical new *ad hoc*
mathematical theory of gravity (curved spacetime), which he called ‘General
Relativity.’

[22] The concepts of ‘property’ and magnitude ‘invariance’ were absolute concepts, not relative ones. (Goldberg, p. 80)

[23] “

[24] For
example, it turned out that the term ‘electromagnetic mass’ was actually a
misnomer. In reality, electromagnetic
mass was just an electromagnetic __resistance__.

[25] The
term ‘celestial mechanics’ refers to the application of

[26] The
Latin phrase *inter* *alia* means “among other things.”