Author: Mysid; 20 November 2015
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- The Relativity of motion
Modern science is based on a limited number of fundamental principles. One of these is the principle of relativity. This principle is associated in our minds with the name of Einstein and his relativity theories (Restricted and General). However the principle’s history started three centuries earlier in the 17th century with Galileo Galilei (1564-1642).
Galileo was the first to discuss the relativity of motion. In the “Dialogue Concerning the Two Chief World Systems” (1632), Galileo presented a thought experiment in which he described an observer watching various types of movements around him, in a ship’s cabin below deck with no means to see the outside world, He concludes: 1
“…You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving (“at constant velocity”) or standing still. In jumping, you will pass on the floor the same spaces as before …”.
Half a century later, in the Principia (1687), Newton embedded Galileo’s observation in his theory as Corollary V where he stated the principle of Relativity and deduced it from his laws of motion, as one would expect in an axiomatic system: 2
“The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.”
In the commentary on the Corollary Newton refers to Galileo’s “ship experiment” as a proof:
“…A clear proof of which we have from the experiment of a ship; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line” .
In clearer terms, the Galilean-Newtonian principle of Relativity states that the laws of motions i.e. laws of Mechanics remain unchanged when observed from the perspective of a reference object (frame of reference) moving “uniformly in a right line” or in modern terms, Uniform Rectilinear Motion (URM).
Note that, the use of the terms “at rest” and “moves uniformly in a right line” implies that motion is considered to be measured with respect to another object which is at absolute rest w.r.t. the Universe!
- Newtonian absolute space and absolute time
In the Newtonian and Galilean outlook which was the accepted view until the advent of Einstein’s special relativity in 1905, space, time and motion were considered to be absolute.
In the first Scholium, Newton distinguished clearly between absolute and relative motion. We can ascertain this fact by following Newton’s statements: 3
“Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies”.
Since space is absolute, then we can specify absolute places and positions:
“Place is a part of space which a body takes up, and is according to the space, either absolute or relative. […] Positions properly have no quantity, nor are they so much the places themselves, as the properties of places”.
If space is absolute, and we can specify absolute places and positions; then motion is also absolute:
“Absolute motion is the translation of a body from one absolute place into another and relative motion, the translation from one relative place into another” (Principles, p.83).
Therefore, if motion is absolute then “true” values (absolute) of the velocity v can be defined, i.e. absolute space can be used in principle to define the velocity v in an absolute manner.
Newton argued at length for the existence of this immutable absolute space, providing proofs that he considered definitive for its existence: the rotating water pail and the rotating tethered pair of globes 4.
However, Newton knew that in order to measure absolute motion, he required an object of reference which is at rest w.r.t to this “theoretical” absolute space, but in his own words, “there may be no body really at rest”, therefore we can measure only relative motion:
“And so, instead of absolute places and motions, we use relative ones; [….] for it may be that there is no body really at rest, to which the places and motions of others may be referred”.
Newton distinguished also between absolute time and relative measured time:
“Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration. Relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.” (Principles, p. 78)
- Frames of References
We know now, as Newton surmised, that everything is moving with respect to everything, that there is no object in absolute rest in the hypothetical absolute Space 5.
Subsequently the absolute “frame” was replaced by the set of frames with relative URM motion or “Inertial Frames of Reference”. However it remained in the background as a theoretical possibility for two and half centuries, in case that elusive object that is in absolute rest were found. This remained the case until the advent of Einstein’s theory of special relativity in 1905.
Newton did not use the expression “frame of reference”. This expression came into use much later during the 19th century. What he used instead is the term “relative spaces” and absolute space. He also explained how absolute motion (in absolute space) is decomposed into the sum of relative motions, using relative spaces.
Now, before going further, we must take a look at what is meant by frame of reference?
By definition, a frame of reference is a “material” body to which we attach a system of space coordinates (usually Cartesian, OXYZ), calibrated meter rods to measure position and synchronized clocks to measure durations.
Note that the fact that the frame of reference must be a material body is overlooked by many who discuss or teach Mechanics. For example Wikipedia defines a frame of reference as a system of coordinates without specifying that it has to be assigned to a material body. Most general Physics and Mechanics textbooks use the same definition 6.
This omission leads to fundamental errors in solving problems and interpreting observation data especially in relativistic mechanics and Cosmology, as we shall see when I discuss (in a future blog) the use of “Cosmic Microwave Background Radiation CMBR as a (preferred) frame of reference to measure the “absolute velocity” of our Galaxy! 7,8.
Inertial Frames of reference
Frames of Reference are of two kinds: Inertial and non-Inertial. An inertial frame is, by definition, a frame in which Newton’s laws apply. This definition provides a means for identifying inertial frames by testing for Newton’s laws, in particular the Principle of Inertia.9
The principle of Inertia, also known as Newton’s first axiom or first law of motion, states the following: 10
“Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon”.
Any frame moving at constant velocity (uniform rectilinear motion) w.r.t. an inertial frame is itself an inertial frame. Thus there exists a set of inertial frames moving w.r.t. each other at constant velocity. Newton’s laws of motion and thus the whole of Classical Mechanics apply strictly in inertial frames of reference only. Note that in the Newtonian-Galilean conception of the universe, these Inertial Frames are supposed to be moving at constant velocities w.r.t. the hypothetical “Absolute Space”.
A non-Inertial Frame, is as the name indicates, any frame which is not moving at constant velocity with respect to an Inertial frame, that is when the material body used as reference frame possesses an accelerated motion, i.e. this body is involved in a mechanical interaction with other objects. Note that objects in uniform circular motion are subject to a centripetal acceleration and therefore are non-inertial frames.
- The Galilean Transformations
In Mechanics, the motion of a rigid object is studied by measuring its change in position as a function of time in a given frame of reference. In a frame equipped with Cartesian coordinates, the position P of the object at an instant t, is determined by 4 parameter: x, y, z, and t.
A point P (x, y, z, t) represents an observable “event” in the Universe. An event is by definition something that happens at a particular point in space and at a particular instant of time as measured w.r.t. a given frame of reference.
In the mechanical view of the Universe, a natural phenomenon is modeled as a series of successive events. In this manner, the stage is set for the study of natural phenomena.
The Galilean Transformations of coordinates are four equations which relate the space and time coordinates of event P in two inertial frames, O (x, y, z, t) and O’(x’, y’, z’, t’) moving with a relative velocity V.
To simplify the situation, we consider the following conditions (see figure):
- O’ is moving at velocity V w.r.t O, along the OX axis.
- The clocks and meters of O and O’ are identical.
- At time t = t’ = 0, O’ coincides with O and the clocks are synchronized.
The coordinates of event P in frames O and O’ are simply: P(x, y, z, t) and P(x’, y’, z’, t’).
We obtain the following 4 equations that relate the coordinates of the event P in both inertial frames:
x’ = x – Vt ; y’ = y ; z’ = z ; t’ = t [1]
Or for the general case:
r’ = r – Vt [2]
where r’ and r are the vectors position of P in O’ and O respectively.
The Galilean Transformation (GT) of the velocities is the derivative of [2] w.r.t. time:
v’ = v – V [3]
This is the well-known law of composition (and decomposition) of velocities that we learn in high school.
The Galilean Transformation (GT) of the accelerations and hence for the forces is similarly obtained by deriving [3] w.r.t. time:
a’ = a [4]
F’= ma’ = ma = F [5]
Assuming m = m’?
This verifies that the forces and the second Law of Newton, i.e. the laws of Newtonian Mechanics are “covariant” in a change of inertial frames of reference (IF) using Galilean transformations.
In scientific terminology, equations which conserve their form with respect to a given coordinates transformation are termed, “covariant” in this transformation. On the other hand, physical quantities which remain unchanged with respect to a given coordinates transformation are termed, “invariant” in this transformation
- The speed of light “dilemma”
In 1904, at the eve of Einstein’s new theory of relativity, Poincaré listed the Galilean-Newtonian principle of relativity as one of the 6 fundamental principles of physics. He generalized it to all physical (natural) phenomena and restated it in the following: 11
“The principle of relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for an observer carried along in a uniform motion of translation; so that we have not and can not have any means of discerning whether or not we are carried along in such a motion”.
Note that Poincaré used the epithet “stationary”, reflecting the implicit belief in an absolute space acting as a “rest” frame of reference.
In 1865, Maxwell unveiled his electromagnetic field theory. The theory, in 4 short differential equations encapsulated all the experimental studies characterizing electrical, magnetic and electromagnetic phenomena. When Maxwell merged the 4 equations together he obtained a single wave equation describing an electromagnetic wave travelling in vacuum at the speed of light c. 12
However, while the laws of classical mechanics obeyed the principle of relativity, Maxwell’s electromagnetic theory “apparently” did not. Electromagnetic equations and phenomena e.g. the electromagnetic wave equation, were not covariant in Galilean transformations between inertial frames.
In particular, the speed of light in vacuum, as it appears in the Maxwell’s wave equation, is a function of two constants ε0 and μ0 which measure the electrical and magnetic properties of space13. Therefore (c) is a property of space and thus possesses the same value regardless of the frame of reference, contrary to what one should expect by applying the Galilean Transformation of velocities (equation [3]).
The speed of light in vacuum, a central feature of Electromagnetic theory is not invariant in Galilean transformations between two inertial frames.
Of course in Science, it is the experiment which decides.
In fact, all the measurements of the speed of light showed that it is a constant c, unaffected by a change in the frame of reference. The crucial experiment was carried out by Michelson and Morley who failed to detect any variation of the speed of light due to Earth’s motion around the Sun 14.
It was clear that, the principle of relativity, Newtonian Mechanics and Maxwell’s electromagnetics theory were mutually incompatible. However all three were backed by experimental evidence showing that they correctly described the phenomena.
Nonetheless, one of the three had to be dropped or at the least revised.
The question was which one?
- The Universal principle of relativity
Let us recapitulate:
Classical mechanics has implicitly assumed a number of unproven hypotheses:
- Time is unaffected by a change in reference frame, i.e. durations are invariant dt =dt’ always.
- Mass is also unaffected by a change in reference frame m = m’ always.
- Length or distance is unaffected by a change in reference frame also d =d’, always.
Based on these assumptions Newton had deduced the laws of Mechanics and in particular the Galilean transformations. These assumptions were adopted ad hoc by Newton and his successors without sufficient experimental evidence. The classical experiments which demonstrated the invariance of m, dt and d, were carried out at low velocities, only. There was no evidence that as the velocities increased this would remain so.
On the other hand, there was unequivocal experimental and theoretical evidence that the speed of light is a constant independent of the frame of reference.
By the end of the 19th century a number of eminent scientists such as Lorentz, Poincaré, Sommerfeld, Langevin and others tried to reconcile electromagnetism with Galilean transformations and failed. However their efforts lead to the realization that a revision of the concepts of length, time, and simultaneity was needed. It was also clear that the Galilean transformations had to be replaced by a new transformation of the coordinates in which the theory of electromagnetism is covariant.
This new transformation was proposed by Poincaré who based it on the length contraction hypothesis advanced earlier by Lorentz and Fitzgerald to explain the negative results of the Michelson Morley experiment. He termed it the Lorentz transformations group.
By then, it had become also clear that the concepts of absolute immutable time and space were no longer adequate.
However none of these scientists was ready to take the crucial step of revising Newtonian mechanics and its premises.
Einstein’s “weird” Universe
In 1905, Einstein took this “crucial” step. He presented his Theory of Restricted Relativity in which he imbedded the principle of relativity as a first postulate and the principle of invariance of the speed of Light in free space or vacuum as second postulate.15
Postulate I or Principle of relativity states:
“The laws of nature are invariant w.r.t. inertial frames of reference”.
Postulate II states:
“The speed of light in free space is Invariant w.r.t. inertial Frames of reference.
Starting from these two postulates alone, Einstein derived the Lorentz transformation of coordinates, and constructed a completely new mechanics.
Einstein’s, theory of special Relativity, predicted 4 new “weird” phenomena .
- That for objects in motion, time is slowed down (time dilation) by a given factor γ (gamma) which increases with velocity (v).
- That for objects in motion, their length in the direction of motion is contracted by the same factor γ.
- That for objects in motion, their mass increases by the same factor γ.
- That matter is another form of energy with a conversion factor given by : E/m = c2(the famous E = mc2 equation).
All four predictions have been confirmed experimentally (in thousands of experiments) and the measured value of the γ-factor was found to agree with the expression predicted by Einstein with high precision.
Note that γ = (1 –v2/c2)-1/2 is termed the Lorentz factor , where c is the speed of light in vacuum.
for v= 30,000km/s (one tenth of the speed of light), γ = 1.005.
Note that the first three phenomena become appreciable (observable) at very high speed close to the speed of light (v > 30,000 km/s or one tenth the light speed or greater), which explains why they are undetectable at the low speeds of everyday activities.
The fourth prediction provided of course the foundation of the atomic bomb and of nuclear power technologies.
Later on, in his theory of General Relativity (1915), Einstein extended the scope of the principle of Relativity to all (Non-Inertial) Frames of reference, turning it into a universal principle that applies anywhere and at any time for all natural Phenomena.
- The Universe according to the principle of relativity
The two theories of relativity changed forever our view of the Universe. The new universe does not resemble the “quaint” static and immutable universe experienced by our ancestors throughout humanity’s history.
First, the theory of relativity eliminated space and time as two distinct and immutable entities and replaced them with a new entity of space-time whose properties are defined by its matter and energy content. Matter and energy create deformations (kinks) in an otherwise smooth space-time.
It also eliminated the matter-energy dichotomy by stating that mass is another form of energy. Matter became the last of a long list of phenomena that the concept of energy had gradually absorbed: kinetic energy or the energy of motion, potential energy with all its manifestations and “heat” or thermal energy.
Relativity also removed the distinction between inertia and gravity, showing, in the principle of equivalence of general relativity, that both inertia and gravity are manifestations of the same property intrinsic to matter.
Relativity also showed that the asymmetry between electrical and magnetic phenomena was an artifact of the frame of reference: the EM theory was originally formulated for sources at rest in a particular frame. In a relativistic approach the electric and magnetic fields observed from moving frames are completely symmetrical forming one entity termed electromagnetic field.
To be clear, unlike Newtonian mechanics electromagnetism was a relativistic theory, covariant with the Lorentz transformations, and did not require revision of its premises.
A new counter intuitive and strange universe has emerged from the fray at the end of the 19th century. Science in its attempt at objectivizing our knowledge about the universe succeeded very rapidly in stripping it from colors, sounds, tastes and other qualities such as sensations of softness, warmth, smoothness and so on. The concept of a space time whose properties and “shape” change and adapt in response to its matter and field constituents is a consequence of the further objectivizing of our experience of the world. 16
Space-time has always been there. Its discovery by science is another confirmation of the power of the scientific method to disentangle objective reality from subjective experience.
Images
a- Lattice analogy of the deformation of space-time caused by a planetary mass.
Author: Mysid; 20 November 2015
https://en.wikipedia.org/wiki/File:Spacetime_lattice_analogy.svg
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References
1– Galilei, Galileo (1632). “Dialogue Concerning the Two Chief World Systems”, 2nd day, pp. 186 – 187, translated by Stillman Drake, University of California Press, (1953), (as quoted in Wikipedia, “Galileo’s ship”).
2- Newton, Isaac (1687). “The Mathematical Principles of Natural Philosophy”, Book I, Corollary V, page 89, translated by Andrew Motte, (1846), MacMillan, On Line Library.
Newton’s full commentary is as follows:
“For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by supposition), in both cases the same; and it is from those sums and differences that the collisions and impulses do arise with which the bodies mutually impinge one upon another. Wherefore (by Law II), the effects of those collisions will be equal in both cases; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the mutual motions of the bodies among themselves in the other. A clear proof of which we have from the experiment of a ship; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line”.
3- Newton, Principia, Book I, Scholium following the Definitions, pp. 77-82:
In this Scholium, Newton discusses in detail his concepts of Space, Time and Motion.
4- For a discussion of Newton’s arguments and proofs, see for example:
Robert Rynasiewicz, Newton’s Scholium on Time, Space, Place and Motion,
(2011, revised 2020), Stanford Encyclopedia of Philosophy
(2018), https://www.researchgate.net/publication/)
5- see my article : “The cosmic Ferris wheels”
6- A quick word search on Google using “frame of reference definition” revealed that only the “Encyclopedia Britannica” used the term body in the definition:
“Reference frame, also called frame of reference, in dynamics, system of graduated lines symbolically attached to a body , that serve to describe the position of points relative to the body”
However, conceptually, it is more correct to say that it is a body to which are attached…etc…
7- Consoli, M. and Pluchino, A. (2018). “Cosmic Microwave Background and the issue of a fundamental preferred frame”.
8- Koberlein, Brian (2018), “A New Way To Measure The Speed Of Our Galaxy Through The Cosmos”.
https://www.forbes.com/sites/briankoberlein/2018/07/19/a-new-way…
9- see my article: “The Foucault pendulum or how to identify inertial frames of reference”
10- Newton, Principia, Book I, Axioms, p. 83.
11- Poincare, Henri (1904) ‘The Principles of Mathematical Physics”. Translated by George Bruce Halsted.
https://en.wikisource.org/wiki/The_Principles_of_Mathematical_Physics
Original French version: Poincaré, Henri (1904), “L’état actuel et l’avenir de la physique mathématique”, Bulletin des sciences mathématiques 28 (2): 302-324
12- The wave equation is a second order homogeneous partial derivative equation. The 1-D form was discovered by d’Alembert in the mid-18th century and has the form:
utt = v2uxx
where u is the amplitude of the wave and (v) its speed; the subscripts denote the second derivative with respect to (t) and (x) respectively.
The wave equation describes a travelling wave motion and applies to mechanical waves such as, sound and seismic waves.
In the case of an electromagnetic wave, (u) represents the amplitude of the electric field E or of the magnetic field H, and v the speed of light c (c= (ε0μ0)-1 ) in vacuum:
Ett = (ε0μ0)-1 Exx
Htt = (ε0μ0)-1 Hxx
The particular solution of the 1-D equation is easier to follow in its sine form:
E(x, t) = Em sin (kx – ω t)
H(x, t) = Hm sin (kx – ω t)
where, Em or Hm is the maximum amplitude of electric or magnetic field, k the wave vector (k= 2π/λ) and ω , the angular frequency (ω = 2πf) and c = ω/k.
13- c = (ε0μ0) -1/2
ε0 = (36πx109)-1S.I.units, the dielectric permittivity of free space, measured electrically using a vacuum gap capacitor
μ0 = 4π x10-7 S.I. units, the magnetic permeability of free space
which gives c = 3×108 m/s
The speed c is a function of two universal constants describing the electrical and magnetic properties of free space. Therefore Maxwell’s theory already states that the speed of light in vacuum must be a constant.
14- The Michelson – Morley experiment in 1887 showed that the speed of light w.r.t. to the “ether”, supposedly at rest w.r.t. absolute space, was not affected by the earth’s motion around the Sun (~ 30 km/s).
https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment
Today’s definition of the standard meter is based on the fact that the speed of light in vacuum is a universal constant and has been measured with increased accuracy and precision to be c = 299 792 458 metres per second.
https://en.wikipedia.org/wiki/Speed_of_light
15– Einstein, Albert (1905). “Zur Elektrodynamik bewegter Körper“. Annalen der Physik. 17 (10): 891–921, translated from German by Meghnad Saha and Wikisource.
I Quote:
“no properties of the phenomena correspond to the concept of absolute rest, but rather that for all coordinate systems for which the mechanical equations hold, the equivalent electrodynamical and optical equations hold also, as has already been shown for magnitudes of the first order. In the following we will elevate this guess to a presupposition (whose content we shall subsequently call the “Principle of Relativity”) and introduce the further assumption, — an assumption which is only apparently irreconcilable with the former one — that light in empty space always propagates with a velocity V which is independent of the state of motion of the emitting body“
16- This process of objectivization of our subjective perception of natural phenomena was carried even further by the Theory of Quantum Mechanics. The “particle-wave duality” removed the apparent distinction between the concepts of particle and wave created by our brain’s interpretation of the images of reality received through our senses, and merged them in a new entity , “the wavicle” . All quantum particles such as photons, electrons, nucleons, quarks and others behave as if they possess the properties of a wave and of a particle at the same time.
The God particle and other mysteries 
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