Galileo discovered the empirical relationships of gravitational acceleration and the phenomena of inertia. Later, Newton described his three laws of motion, and his law of gravitational attraction. When applied on any inertially moving body, Newton’s second law of motion (F = ma) is invariant, and the values of its variable components (F, m, a) are algebraically “covariant.”

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 (Ibid, p. 36; 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] (Ibid, Figure 4.2B). The combination of these components forms a parabola with yet a third distance traveled[2] (Figure 4.2C). These experimental conclusions also contradicted Aristotle who conjectured that an “object moves only as long as it is being pushed and will stop as soon as the force disappears” (Gamow, 1961, p. 42).

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 16th century, several people had actually tested Aristotle’s conjectures and found that the stone instead falls to the base of the mast[5] (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, Newton set forth three postulates that are commonly referred to as “Newton’s Laws of Motion.” They explain both the kinematics (the abstract description of motions) and the dynamics (the relationship of kinematic motions to the forces and masses which cause them) of terrestrial and celestial motions.

1. Newton’s First Law of Motion (the law of “inertia”) states:

“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]).

Newton described inertia as the theoretical “state” of a body in equilibrium (where no net external force is acting upon it in any direction).[10] Newton also defined inertia as the “innate force of matter… [its] power of resisting…” (Ibid, 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] (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, Newton’s inertial motion must maintain constant “velocity,” because “velocity” is defined as the speed (constant rate of motion) of an object in a particular direction[13] (Goldberg, p. 33; and Figure 4.3A). Although orbital motion is not rigidly inertial motion under Newton’s definition, the continuous 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]

Newton’s first law also tells us whether or not a net external force is acting upon a body. Why? Because: a) by Newton’s definition, “inertial” means “uniform motion in a straight-line,” and b) if a body (i.e. a planet) moves in a curved path, then according to Newton’s first and second laws of motion, there must be net external “forces impressed upon it” (Goldberg, pp. 52 – 53; Young, pp. 95 – 97).

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” (Ibid, 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, Newton’s fully developed law of inertia was really a group effort.

2. Newton’s Second Law of Motion (the law of “force and acceleration”) states:

“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. Newton defined “impressed force” as “an action exerted upon a body, in order to change its state…” (Ibid, p. 2). Galileo mainly talked about accelerations, not forces. On the other hand, Newton stated and demonstrated that forces cause accelerations and in which direction. Thus Newton explained why Galileo’s gravitational accelerations and Kepler’s planetary motions (accelerations) occur.

Newton intended the term “change of motion” to mean “change in velocity” or, in modern terminology, “acceleration” (Goldberg, p. 54). Newton’s second law, in effect, states that the acceleration (a) of a mass (m) is in the direction of the continuous net force (F) applied to it and is proportional to such force[17] (Figure 4.3B). Therefore, F = ma[18] (Young, p. 100).

Newton also described another form of his second law that involved an 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 Newton, his law of continuous force law (F = ma) was derived from this impact law [19] (Cohen, 1960, p. 184).

Newton defined the “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] (Ibid).

Newton’s second law also empirically implies that the magnitude of 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 (m1 and m5) and one ball (m5) accelerates only one-fifth as far and as fast as the other ball (m1) during the same period of time (t), then a priori ball (m5) must have 5 times as much mass as ball (m1), and the magnitude of acceleration (a) of each ball is inversely proportional to the magnitude of its own mass (Figure 4.3B).

At the beginning of the Principia, Newton defined the “quantity of motion [as] the measure of the…velocity and quantity of matter conjointly” (Newton, Principia [Motte, Vol. 1, p. 1]). In other words, the term “quantity of motion,” as defined by Newton, means the “mass (m) times the velocity (v)” of a body, which since the time of Newton has been referred to as “momentum,” or p (Goldberg, p. 52). Thus p = mv. The more mass and/or velocity an object has, the more momentum it has, and therefore the more force which must be applied to accelerate it, slow it down, or change its direction of motion (Ibid).

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. Newton’s Third Law of Motion (the law of “action and reaction”) states:

“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]).

Newton’s third law asserts that action and reaction 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 (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,” Newton described his law of the mutual gravitational attraction of all material objects:all masses attract each other with a force that is proportional to each mass, and which is inversely proportional to the square of the distance between their centers. Newton’s law of gravitational attraction is basically an application of his three laws of motion working together with respect to the different masses of opposing bodies.[21]

C. Covariance and Invariance

Newton’s second law of motion led to the classical mathematical concepts of “covariance” and “invariance” (Goldberg, pp. 80 – 81). By the classical term “covariance,” we refer to a comparison between two different accelerating events, where the magnitudes of mass, force and/or acceleration are different, but the interaction between such different magnitudes demonstrates that each acceleration event is the result of Newton’s second law: a = F/m (Figure 4.4). In other words, in mathematical terms, the magnitudes of the variables (F, a, m change in such a way (“covariantly”) so that the algebraic form of Newton’s second law of motion (F = ma) remains “invariant” (it does not change) with respect to each different acceleration event. As Goldberg expressed it,“the quantities in the laws vary, each so that they leave the [algebraic] form of the law unvaried” (Ibid, p. 80). There is also a geometrical analogy to classical “covariance” that might help to explain the concept (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 (Newton’s second law) is somewhat analogous to the various different visual perspectives of numerous observers with respect to the same motion of an object (Chapter 3). Algebraic transformation equations must be devised in order to relate such very different visual perspectives (or coordinate measurements) of the same motion or event, and likewise algebraic transformation equations have been devised to relate the interaction of such variable quantities which result in the same fundamental law of motion. The algebraic transformation equations for such variables are: F = ma, a = F/m and m = F/a. When any two of these variables or magnitudes are ascertained, the third variable can automatically be computed (Figure 4.4).

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 Newton’s second law of acceleration is independent of the velocity of the inertial frame in which it is applied. These transformation equations are called “Galilean transformations” and they are discussed in detail in Chapter 14. Newton’s second law of motion remains algebraically “covariant” and the laws of mechanics remain mathematically “invariant” with respect to Galilean mathematical transformations (Goldberg, p. 80 – 81).

Newton’s second law only appears to retain the form F = ma (and it is only obviously and intuitively covariant) in inertial reference systems (Lindsay, p. 290). In accelerated reference systems, or reference systems with arbitrary motions (like a roller coaster), Newton’s second law appears to take different algebraic forms and becomes very confusing. These functions of simplicity and intuitiveness are among the main reasons why inertial frames of reference became so important to classical mechanics during the 19th century (Chapters 13 and 14).

Toward the end of the 19th 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 19th century scientists to be invariant included: the mass of an object, its length, its other dimensions, its shape and its color (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 (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” (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

Newton’s three laws of motion, along with his laws of gravitation and Galileo’s theories of inertia, inertial motion, and projectiles (and related laws) are today referred to as “classical mechanics” (Young, p. 92). Einstein stated that:“ the purpose of mechanics is to describe how bodies change their position in space with time” (Einstein, Relativity, p. 10). German mathematician Leonhard Euler, who was only 20 when Newton died, contributed much to making Newton’s work accessible to a wider audience.He cast Newton’s calculus and the mathematical formulation of his mechanics into the [algebraic] form which we use today…F = ma. (Rohrlich, p. 42).

“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 Newton’s work, absolute space and time seemed to play no role at all.Everyone worked with relative motion and no difficulties were encountered” (Ibid).

Since Newton’s era, we have come to realize that classical mechanics had certain theoretical failings. We shall briefly describe three of the most important failings. As Einstein pointed out in 1905, classical mechanics failed to appreciate the importance of the distance/time interval delay of the light signal with respect to a local observer’s perception of the “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 Newton’s time, and throughout history until the dawn of the 20th 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 (Chapter 17). However, such variation can depend upon how one defines the word mass[24] (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, Newton’s three laws of motion along with his law of gravitational attraction have governed most areas of mechanics and astronomy for over 300 years. Einstein’s Special Theory of Relativity claims inter alia to radically modify Newton’s laws of motion well beyond that which is necessary for the correction of the previously described theoretical failings.[26] Einstein’s General Theory of Relativity also claims inter alia to radically modify Newton’s laws of gravity. We will demonstrate in this treatise that neither of Einstein’s ad hoc theories of relativity was necessary or even relevant for the correction of such failings, nor for almost any other reason.