LANGE’S ABSRACT RELATIVITY & INERTIAL REFERENCE FRAMES

*During the period between 1885 and 1904, Ludwig
Lange and others abandoned *

**A.
Dissatisfaction with Absolute Space**

What is meant by the word __space__? ‘Space’ has at least three separate and
distinct meanings:

1.
__Void__:
That nothingness which is occupied by, and separates, material bodies.

2.
__Place__:
That certain part of a material body.

3.
__Distance__: The physical interval between two
material bodies or places.

With his concept of ‘immovable absolute space,’ __there__? To paraphrase Gertrude Stein: “With empty
space, there is no there, there.” As
stated by James Clerk Maxwell:

“The arrangement of the
parts of space can no more be altered than the order of the portions of
time. To conceive them to move from
their places is to conceive a __place to move away from itself__. But as there is nothing to distinguish one
portion of time from another except the different events which occur in them,
so there is nothing to __distinguish__ one part of space from another except
its relation to the place of material bodies.
We cannot describe the time of an event except by __reference to some
other event__, or the place of a body except by __reference to some other
body__. All our knowledge, both of
time and space, is essentially __relative__.”[1] (Maxwell, 1877, p. 12)

Thus, again it is obvious that

__relative__ positions of celestial bodies in the void of space and
measure the relative change of positions (distances and motions) between them. [2]

On the other hand, a stake in the
ground on Earth may be considered as __relatively__ at rest or relatively
immovable with respect to the Earth, so that the distances and positions of
material objects on Earth can be measured from such stake and can be defined as
relative places, relative motions and relative directions of motion. (Chapter 2)
Therefore, on Earth, some of

Newton attempted to justify his concept of an ‘immovable absolute space’
based on his concept of absolute motion,[3]
and on his assumption that it was necessary as a stationary reference system
for the description of his other laws and their time, place, and motion. For example,

With the gradual acceptance of ^{th}
century. By the latter part of the 19^{th}
century it was even suggested by many scientists that the concept of __stationary__
ether should be identified or equated with __immovable__ absolute
space. (Jammer, 1954, pp. 125, 141 –
142) Thus, the amorphous concept of
immovable absolute space became materialized with the mythical substance of
stationary ether.

During this same period of time,
however, more and more scientists in *Id*.*Id*.

**B. Lange’s Relative Inertial Frames**

In 1885,
German philosopher Ludwig Lange (1863 – 1936) suggested a hypothetical and
mathematical way out of this paradoxical dilemma. Since it is impossible to determine a fixed
spot in absolute space from which to measure, Lange proposed to eliminate

When two inertial reference frames (coordinate systems) were at rest together in Lange’s abstract model, their common position and common motion was geometrically illustrated as two inertial systems, S and S', each with x, y, z coordinate axes located at their mutual zero coordinate position. (see Figure 13.1A) Algebraically, this scenario can be relatively described as x = x', y = y', z = z', t = t'.

When the S'
system theoretically __translates__ at a uniform and rectilinear velocity
away from the S system in the same (x) direction, their different positions and
equivalent motions may be geometrically illustrated as shown on Figure 13.1B. [4]
This second scenario may also be algebraically described, as follows:

S system: x = x' + vt, y = y', z = z', t = t', and

S' system: x' = x – vt, y' = y, z' = z, t' = t,

where x indicates the __relative
position__ of inertial system S theoretically at rest at the zero coordinate
position; x' indicates the __relative position__ of inertial system S'
uniformly translating relative to x at the __relative velocity__ of v in the
same (x) direction; vt indicates the __distance traveled__ by S' at velocity
v relative to S during a time interval (t); and
t = t' indicated the same __absolute instant__ of time for the occurrence of
events on either frame of reference, as theoretically perceived by all 19^{th}
century observers on each frame. (see
Chapter 14)

Therefore, in place of a fixed spot in stationary absolute space, a mathematician substitutes the fiction that an inertial frame located at the origin point of a coordinate system in limited space is ‘stationary’ relative to another inertial frame (coordinate system) that translates uniformly and rectilinearly away from it. Thence, from this fictional ‘stationary’ inertial frame (like a stake in space), the mathematician can measure, compute and relate magnitudes of position and velocity between the two relatively moving inertial frames and within such limited space. These hypothetical coordinate frames became known as ‘inertial reference frames.’ (Rohrlich, p. 43) By means of this idealized fiction, the mathematician can create an abstract mathematical certainty and absoluteness with respect to such measurements of position and velocity that does not exist in the real world.[5]

In Lange’s
view, by means of this fictional substitution, “the essential…content of the
law of inertia, and with it the whole of mechanics, retains its full physical
meaning...An ‘inertial system’ is a coordinate system in respect to which
Newton’s law of inertia holds.”[6]
(Jammer, 1954, pp. 138 – 139) In effect,
“in order to give [mathematical] meaning to the first law of motion we do not
need absolute space and absolute motion but we can understand it entirely in
terms of __relative motion__” (Rohrlich, p. 43), and limited space.

Thus one
major reason why late 19^{th} century mathematicians liked Lange’s
abstract model was because it gave them a mathematical way to theorize about
celestial space and inertial motion in limited, specific and relative ways,
rather than as an uncertain absolute place or absolute motion. (Rohrlich, p. 43) Their equations would not work with amorphous
absolute space and absolute motions, any more than they work with the uncertain
and intangible concepts of infinity or eternity.

Why were __inertial__
frames (sometimes called ‘Galilean frames’) adopted as the standard frames of
reference in Lange’s model, rather than some other type of frame? Because if viewed from __accelerating or
arbitrarily__ moving bodies (frames), “the laws of mechanics would appear
hopelessly complex…[and the] study of mechanics would present tremendous
difficulties.”[7] (D’Abro, 1927, p. 107) On the other hand, when referred to inertial
or Galilean frames, “mechanical phenomena, including the planetary motions,
were susceptible of being formulated by very simple [mathematical] laws.”[8] (*Id*., p. 107) Thus, “it was essential to classical science
that the fundamental law of mechanics [vis., Newton’s second law of motion]
should remain the same in all Galilean [inertial] frames…[so that] similar
conclusions…[could] apply to all the mechanical laws.”[9] (D’Abro, 1950, p. 113)

Not only did
Lange’s relativistic model provide mathematicians with relative inertial
motions and with finite limited space intervals, it also provided them with
relative instants in __time__ and finite limited time intervals between
inertial reference frames. Thus, it
provided mathematicians with an idyllic framework for mathematically precise
theoretical __measurements__ between inertial frames.[10] “Lange’s suggestion…was hailed by his
contemporaries as an outstanding contribution to the foundation of physics.” (Jammer, 1954, p. 139)

When Newton’s
second law of motion (force and acceleration) was theoretically applied in
hypothetical experiments conducted upon each of Lange’s two imaginary inertial
reference frames, the resulting __algebraic__ laws became known as ‘__Newtonian
mechanics__’…in order to distinguish them from the prior __empirical__
laws of ‘Newton’s mechanics’ which were theoretically dependent upon absolute
space and absolute time. (Rohrlich, pp.
43 – 44) However, this so-called
distinction was also a myth, because relative motion “was used in actuality all
along by *Id*., p. 44) Thus, Lange’s relativistic concept merely
formalized the existing theoretical situation in abstract terms.[12]

It has been
suggested that Lange’s two abstract inertial coordinate frames simulate the
ship in port on the inertially moving Earth, and the __same__ ship later
moving uniformly and rectilinearly away from the relatively stationary port in
Galileo’s Relativity. (Figure 5.3) “They are the frames which are not subject to
acceleration, i.e. on which no forces are acting.” (Rohrlich, p. 43) The scientific conclusion was that Lange merely
reinterpreted Galileo’s Relativity and inertia “in a relativistic way.” (see Rohrlich, p. 43; Guilini, p. 14)

Did this historical
connection with Galileo’s Relativity mean that Lange’s relativistic model and
Galileo’s Relativity were basically the same concept? Hardly.
Lange’s new abstract model was __very different__ from the classical
concept of Galileo’s Relativity in many critical respects. First, in Lange’s model there were __two
co-moving inertial reference frames__ rather than just one inertial body (the
ship) in two equivalent states of uniform rectilinear motion. Second, in Lange’s idealized model both
reference frames were stipulated to be __inertial__, whereas in Galileo’s Relativity
the uniformly moving ship was non-inertial:
its uniform velocity was caused by a force (the wind). Nor was the motion of the Earth really
inertial, because of the forces of gravity impressed upon it.

Third, in
Galileo’s Relativity there was one observer in two different positions, whereas
in Lange’s model theoretically there were __two different observers__ in two
different positions (frames). Fourth, and
very importantly, the one observer in Galileo’s Relativity describes two
different events (accelerations) in two different locations, whereas the two
observers in Lange’s model each describe the __same event__ as viewed from
two different positions (perspectives). (see
Figure 14.1) Fifth, the observer in Galileo’s Relativity
describes his same physical __experiences__ at two different positions based
on his empirical perspective and __sensory__ illusion of rest, whereas the
two observers in Lange’s model __mathematically measure__ and describe the
positions of one event from two different frames (perspectives). These last three differences would later become
critical for Einstein’s Special Theory.

Sixth, in
Lange’s model one inertial system was assumed to be __stationary__ (at rest)
relative to the other. This was a
mathematical fiction for purposes of algebraic description, measurement and computation. Whereas, in Galileo’s Relativity, the only
purpose of the same ship being stationary or at ‘rest’ in port and then
uniformly moving away from the port was to empirically demonstrate that each
illusion of rest was an equivalent state of motion that produced visually
similar or covariant accelerations on it…so that the observer could not tell
from observing such accelerations whether or not he was moving. On the other hand, the observer’s ability to
tell whether or not he was moving played no part in Lange’s 1885 abstract
model. Theoretically and mathematically
it was irrelevant.[13]

Seventh, a
primary purpose of Galileo’s Relativity was to demonstrate the __physical and
empirical co-variance__ of different spatially separated accelerated motions
and thereby to demonstrate the invariance of __mathematical
measurement__,[14]
whereas measurement was __irrelevant__ to Galileo’s Relativity.

Eighth, the __relative
velocity__ of the ship at any position was __irrelevant__ to Galileo’s Relativity. The only requirement was that the motions of
the ship were uniform and rectilinear. On
the other hand, with Lange’s model the relative velocity of the two inertial frames,
v, was of primary importance because when v was combined with the time interval
traveled, t, it produced the all important distance traveled, vt, which was
necessary for mathematical measurement.
Ninth, the __distance traveled__ was __irrelevant__ to Galileo’s Relativity. The only thing that was relevant was that the
uniformly moving ship was __spatially separated__ from the port. On the other hand, the distance traveled by
the moving frame in Lange’s model was critical for purposes of measurement from
one frame to the other.

Tenth, the __time
interval traveled__ and the instant in time on the ship at either position
was __irrelevant__ to Galileo’s Relativity.
Whereas, with Lange’s mathematical model the time interval traveled was
important for purposes of measurement, and the instant in time at each
spatially separated position should have been important for measurement, as was
pointed out by Einstein in 1905.[15] Lastly, Galileo’s Relativity was totally
sensory and empirical in nature, whereas Lange’s model was totally abstract and
mathematical. Lange’s idealized model
was totally separated and isolated from the environment and empirical
influences of reality, such as gravity, pressure, temperature, wind, climate,
and friction. Without a sensory or
empirical foundation it cannot independently be stated that Lange’s model was
based upon experience. Nor can it
independently be stated that all inertial frames are equivalent states of
motion, or that one inertial observer cannot tell whether he is moving or at
rest. Nor can it independently be stated
that no mechanical experiment performed entirely on one inertial reference frame
cannot detect whether such frame is moving.[16]
All of these conclusions depend entirely upon the classical and very different sensory
and empirical concept of Galileo’s Relativity.

For all of
the above reasons, it becomes obvious that Lange’s idyllic abstract model was
designed to perform very different theoretical functions[17]
and was a completely __different concept__ than Galileo’s sensory and
empirical principle of relativity, which merely described reality for different
specific and limited purposes.

**C. Application
of Lange’s Inertial Reference Frames**

Why did we
just describe the many differences between Lange’s abstract model and Galileo’s
Relativity? Because in 1905, Einstein
adopted Lange’s very different abstract (coordinate) version of relativity as the
theoretical framework for his own mathematical Principle of Relativity. However, in the process Einstein (very
importantly) characterized his own Principle of Relativity as merely __an
extension__ of Galileo’s Relativity, even though it applied (not only to
mechanics, but) to all of physics as well…including the velocity *c* of light. (Einstein, 1905 [Dover, 1952, pp. 37 – 38];
Einstein, *Relativity*, pp. 15 – 18, 23)
Einstein then further radically modified Lange’s model of relativity by
applying the Lorentz transformation equations to it. He then claimed that his Special Theory was
just an attempt to reconcile the principle of relativity for mechanics with the
constant velocity of light at *c*.
All of these invalid rationalizations aside, Einstein was really
attempting to reconcile his own radically different mathematical version of
relativity with his own *ad hoc* concept for the absolute propagation
velocity of light. (see Chapters 19, 20,
21 and 23) In the final analysis, Galileo’s
simple sensory and empirical concept of relativity had absolutely nothing to do
with Einstein’s abstract mathematical Special Theory. (Chapters 19 and 24)

By 1887, an algebraically modified version of Lange’s relativistic model
was being used by German mathematician Woldemar Voigt (1850 – 1919) for his
paper on the Doppler theory. (Pais, p.
121) In 1892, Dutch mathematician H. A.
Lorentz used Lange’s abstract model for his relativistic contraction of matter
theory. (Miller, pp. 25 – 27) In 1895, Lorentz used the classical equation
x' = x – vt as part of a set of radical transformation equations for his
contraction of matter theory, but he then regarded such transformation
equations “only as a convenient mathematical tool for proving a physical
theorem…” (Pais, pp. 124 – 125) In 1898, Irish mathematician Joseph Larmor
(1857 – 1942) used a radically modified algebraic version of Lange’s model for
a set of transformation equations in an attempt to confirm Lorentz’s 1895
contraction theory. (*Id*., p. 126) In 1899, Lorentz used Lange’s model to devise
yet another radical set of transformation equations for his contraction theory,
but for several years he did nothing with them.
(*Id*., p. 125)

During 1902, French mathematician Henrí Poincaré (1854 – 1912) published
a book, entitled *Science and Hypothesis*,
wherein he talked about ‘relative motion,’ ‘relative positions,’ ‘relative
velocity,’ ‘the principle of relative motion,’ ‘the relativity of space,’ and
‘the principle of relativity.’[18] (Poincaré, 1902, pp. 90, 112 – 114, 243 – 244) Then, in 1904, at the

As previously explained, in June 1905, Einstein adopted for his own Special Theory of Relativity, the basic concepts of Lorentz’s 1895 relativistic contraction of matter theory, Lange’s abstract version of relativity as his framework, and Lorentz’s radical 1904 transformation equations. By this time, June 1905, Galileo probably would not have recognized his own simple sensory and empirical concept of relativity.[19]

If it had not been for Lange’s modified abstract model of Galileo’s Relativity, Lorentz probably would not have been able to theoretically construct his relativistic contraction theory and his relativistic transformation equations in 1904. In this event, Einstein could never have adopted Lange’s, Lorentz’s and Poincaré’s relativistic concepts, and probably never would have invented his own Special Theory of Relativity in 1905.

What __is__ the current relevance of Lange’s 1885 abstract
model of Galileo’s Relativity? Its sole
relevance is that it was adopted by Einstein in 1905 as the abstract framework
for the construction of his Special Theory.
Other than this, Lange’s abstract model of Galileo’s 17^{th}
century relativity concept does not appear to retain much (if any) current or
independent relevance.

Why is it necessary for us to learn
and understand the exact criteria for, and the difference between, Galileo’s Relativity
and Lange’s modified abstract model thereof in the 21^{st}
century? Because Einstein theories of
Special Relativity and General Relativity evolved from them and were entwined
with their concepts, and because Einstein’s relativistic theories were
predicated upon the applicability of Galileo’s Relativity to electromagnetics,
optics, electrodynamics and the velocity of light. Why was Galileo’s empirical concept of
relativity so important to Einstein?
Because Einstein needed some __empirical__ basis for his mathematical
Special Theory so that it would not appear to be completely *ad hoc*.

If Einstein’s relativistic theories were not based on Galileo’s empirical
concept of relativity, and if Galileo’s Relativity has no relevance to
electromagnetics, optics, electrodynamics and the velocity of light (which it does
not), then where does this leave Einstein’s relativistic theories? The answer is: completely without any __empirical__
foundation whatsoever. Einstein’s
relativistic theories are therefore totally *ad
hoc*. [20]

At this point it is recommended that the reader turn to Chart 24.1 in order to study and understand the various different concepts and definitions of ‘Relativity.’ We shall repeatedly refer to all of these concepts in great detail in later chapters.

[1] It should be noted that Maxwell made these comments 25 years before Poincaré, and 28 years before Einstein made similar comments.

[2] In this regard, Born also stated: “It is not space that is there and that impresses its form on things, but the things and their physical laws that determine space.” (Born, p. 71) Likewise, “…German philosopher Gottfried Wilhelm Leibniz…argued that space is nothing but the relationship of the location of objects.” (Rohrlich, p. 41)

[3] *Id*. pp. 107, 139 – 141)

[4] This translatory motion when applied to Lange’s relative inertial frames may also be referred to as an inertial ‘boost motion.’ In addition, Lange’s concepts may be applied to ‘displaced’ or ‘rotated’ inertial motion. (see Figures 13.1C and 13.1D)

[5] For example, idealized relative velocities measured in Lange’s abstract model never consider or experience the forces of gravity, the resistance of air and friction, temperatures, pressures, wind, weather, the myriad of other velocities of the Earth relative to other celestial objects, nor the direction of such relative velocities.

[6] These, of course, are idealistic conclusions.

[7] Laws formulated from these non-inertial bodies would be compounded and complicated with accelerations, arbitrary herky-jerky motions, as well as the phenomena of centrifugal (outward) forces and Coriolis (sideways) forces. (D’Abro, 1927, p. 107)

[8] This was
another reason why late 19^{th} century scientists liked Lange’s
inertial reference frames: it was
because of their simplicity and intuitiveness.
There were no sensors or computers during these periods that could
sense, sort out and compute the magnitudes of accelerating or arbitrary
motions. Uniform rectilinear velocities
allowed the accelerating motions and magnitudes of

[9]
Remember, we pointed out in Chapter 5 that Galileo’s Relativity was not really
a fundamental law of physics or motion, but rather only a convention of
convenience. After 1885, the inertial
motions of Galileo’s Relativity (in the form of ‘inertial frames’) also became
a __mathematical necessity__.

[10]
However, late 19^{th} century mathematicians did not realize the full
physical and mathematical significance of these relative times for such
measurements until Einstein explained them in 1905. (see Chapter 25)

[11] “People were simply not fully aware of this situation.” (Rohrlich, p. 44)

[12]
Therefore, the rationalization that Lange’s relativistic model was invented “to
rid

[13] The paradoxical 1887 M & M null result stimulated the attempt by scientists to discover the absolute motion (velocity) of the Earth relative to the stationary ether. Therefore, after 1887, a terrestrial observer’s ability to detect or prove (theoretically or by experiment) that the Earth was moving relative to the ether at an absolute velocity became a cause celeb for theoreticians and mathematicians and their mathematical applications of Lange’s reference frames, which strangely enough were irrelevant for this purpose.

[14] One
primary purpose of such mathematical measurement was to algebraically
demonstrate that the same laws of __mechanics__ were valid and invariant in
different inertial reference frames.
Einstein would later attempt to expand this mechanics concept to include
electrodynamics (the velocity of light), and he then characterized this __mathematical__
invariance of physics laws as mathematical ‘covariance.’ (see Einstein, *Relativity*, p. 48)

[15] There was another difference between Lange’s
Relativity and Galileo’s Relativity.
With Galileo’s Relativity the inertial bodies did not have to be moving __linearly__
relative to each other. Rather, such
inertial bodies could even be moving perpendicularly relative to one another.

[16] This statement, incorrectly generalized to include EM experiments, became the foundation for Poincaré’s and Einstein’s principles of relativity. (see Chapter 21)

[17] In Lange’s model, an inertial reference frame served at least three separate functions: it simulated a reference body with respect to which a distant observer might measure; it was the place where a local observer and his system of coordinates resided; and it was a uniformly moving platform upon which an accelerated event might occur, but it could not demonstrate the empirical covariance of Newton’s second law. Galileo’s Relativity only included the last function, and it did demonstrate such empirical covariance.

[18] However, in 1902, Poincaré did not specifically define such concepts.

[19] The
modifications to, and transformation equations for, Galileo’s Relativity
(invented by Lange, Voigt, Larmor, Poincaré, Lorentz, Einstein and others) were
completely new *ad hoc* mathematical concepts, with very different
purposes than Galileo’s Relativity, all without any direct empirical
support. In the process, the original
purposes for Galileo’s simple sensory and empirical concept were totally
abandoned.

[20] When
critically analyzed and scrutinized in later chapters, Einstein’s Special
Theory appears to be nothing more than an arbitrarily constructed __mathematical
exercise__, with no application whatsoever to reality.