From Galileo's Law of Inertia to Einstein's and Mach's Conjecture Principle of Inertia

"The function of science, as we take it, is to replace experience. Thus, on the one hand, science must remain in the province of experience, but, on the other, must hasten beyond it, constantly expecting confirmation, constantly expecting the reverse. Where neither confirmation nor refutation is possible, science is not concerned." - Ernst Mach ( 1838 - 1916 )

"It would be unsatisfactory, in my opinion, if a world without matter were possible" - Albert Einstein ( 1879 - 1955 )

" ... it is justified to consider Mach as the precursor of the general theory of relativity" ^[1] - Albert Einstein ( 1879 - 1955 )

^[1] Source: Abraham Pais ( 1918 - 2000 ): "Subtle is the Lord: The Science and Life of Albert Einstein", 1982 & 2005

A. Pais quoting famously "Subtle is the Lord, but malicious He is not!", by Albert Einstein

note: Abraham Pais ( 1918 - 2000, Dutch - born, Holocaust surviving Zionist Jew, American physicist and science historian of Niels Bohr and Albert Einstein prominently among others ) was taken under the wing of Albert Einstein at the Institute for Advanced Study, Princeton University after the war where an early cure for cancer died at Auschwitz at the butchering hands of Nazi Germans. However A. Pais is another example of enduring Jewish brilliance now thriving in science and medicine in modern Jewish Israel.

Ernst Mach - The Father of "Mach Speed"

( born 1838, Chirlitz-Turas, near Brno, Austrian Empire, now Czech Republic; died 1916, Haar, near Munich, German Empire )
Ernst Waldfried Josef Wenzel Mach, Father of "Mach Speed", born a Catholic but philosophically evolved into an atheist,
Logical Positivist Philosopher of Science, Physicist and Mathematician, forerunner of General Relativity by his critique
of Newton's Laws of Gravity and Inertia. And by Einstein's own attribution to him of the phrase "Mach Principle",
the physical origin of inertia and centrifugal force.
Author of six volumes: The Science of Mechanics (1883), The Analysis of Sensations (1897), Popular Scientific Lectures (1895),
The Principles of Physical Optics (1926), Knowledge and Error (1976), and Principles of the Theory of Heat (1986),
the latter three published posthumously.

Galileo's General Law of Inertia

Historically Galileo empirically demonstrated in a somewhat non - formalized mathematics that in a vacuum a falling body of matter possessing inertial mass would forever remain in whatever initial state it was originally given, either at rest or in relative ( Galilean - Cartesian ) motion.

However it took two years after Galileo's death in 1642 for René Descartes ( 1596 - 1650, French philosopher, mathematician ) in his most enduring legacy in the development of analytic geometry ( algebra combined with Euclid's congruent shapes and analytic forms ) to enable Isaac Newton and Gottfried Wilhelm von Leibnitz ( also: Leibniz ) both independently to develop 'the calculus' and for which in turn allowed Newton using Descartes' analytic geometry to explicity formulate Galileo's General Law of Inertia in a vacuum:

And hence the quantity of mass, , is operationally the numerical measurement of inertia!!

Theoretically, therefore, Newton's inertial mass is the necessary and measurable amount of coercive or enabling force needed to modify the motion and/or shape of an amount of a body of matter dislodged from its initial inert status otherwise signifying the condition of that body at rest, all of which begs some rather deep philopsophical questions.

And this concept of Galilean inertia is easily carried over into Newton's 2nd Law of Motion:

That is, 'inertial masses' are those masses for which Newton's 1st and 2nd Laws of Motion are valid.

Einstein's Equivalency Principle

Thus as to the General Relativity Equivalency Principle of Acceleration and Fields of Gravity, the entire Einstein general theory rests upon Galileo's empiricism that all falling bodies of matter possessing inertial mass fall in a vacuum with the same acceleration - viz.: a feather vs. a lead weighted ball arrive in a vacuum from a height to the earth's surface at the same time!

Now, however, the numerical measure for inertia in General ( and Special! ) Relativity becomes

which is now also understood as the effective relativistic mass as well as the inertial mass. Furthermore the idea of inertia is broadened to the extent of now speaking of "the inertia of energy" instead of merely referring to some sort of static or constant "mass".

Mach's Conjecture Principle of Inertia and Centrifugal Force

The far distant distributions of masses of stars along with unseen dark matter and their fields of gravity determine the nearby conditions of localized inertia for bodies of matter possessing inertial mass. Thus nearby bodies of mass and their inertia contained within and connected to their own respective local inertial ( non - accelerating ) frames of reference are effected by the masses of distant stars and their fields of gravity. This is more a philosophical statement of the world of nature than anything else from which any direct physics equation can be derived.

However it is in the Machian sense of large - scale, distant structures of gravitational masses effecting nearby bodies possessing inertial mass in their own localized inertias, that one can philosophically imagine arising from General Relativity theory a sort of 'fabric of [ gravitational ] spacetime' permeating the cosmos.

From "Gravitation", by Misner, Thorne, Wheeler - §21.12 - Mach's Principle

"In my opinion the general theory of relativity can only solve this problem [ of inertia ] satisfactorily if it regards the world as spacially self - enclosed." - Albert Einstein ( 1934 )

1. "Einstein's theory says that inertia is a manifestation of the geometry of spacetime."


2. "Einstein's theory identifies gravitation as the mechanism by which matter there influences inertia here."


3. "... that matter there governs inertia here ..."

At a Deeper Machian Level of Philosophical Understanding

Already during this pre - relativity Machian state of physics we can discern the changing ideas of "matter" and "mass" whereby in altering the frame of inertial reference for a body of matter possessing inertial mass, it is therefore possible by differing Cartesian ( better: Riemannian! ) geometries to "create" or "destroy" fields of gravity as well as to effect differing amounts of inertial mass, , as later shown in Einstein's Special Relativity ( 1905 ) thusly:

And,

"Therefore the component of the momentum vector along the axis of , multiplied by , is to be defined as the kinetic energy of the point mass. The expression for this is

... It comes out very clearly in this way, how the energy depends on the system of reference. But as the axis of may be laid in the direction of any time - like vector, the law of energy, framed for all possible systems of reference, already contains, on the other hand, the whole system of the equations of motion." from "Space and Time", by H. Minkowski, Cologne, 1908.

Translate as follows:

§ References:

1. Mach's original paper as it affected Einstein's thinking regarding relative accelerations of bodies of mass in his 1916 seminal general relativity paper, "Die Mechanik In Ihrer Entwickelung" ( 'Mechanics and Its Evolution' or 'History of Mechanics' ), von Dr. Ernst Mach, Professor Der Physik An Der Deutschen Universität Zu Prag, published Leipzig: F.A. Brockhaus - 1883.


2. Historically seminal paper attempting to incorporate Mach's Principle into general relativity, "Mach's Principle and a Relativistic Theory of Gravitation", by C. Brans and R.H. Dicke, published November 1, 1961, Physical Review, Volume 124, Number 3.


3. Instead of WIMPS, weakly interacting massive particles, or axions, rather dark matter may be massive, according to physics professors Glenn Starkman of Case Western Reserve University and David Jacobs of University of Cape Town, in their joint paper "Macro Dark Matter", published October 13, 2014.


4. General Relativity theory is enlisted along with its concomitant inertial frame dragging to substantiate the general consensus that dark energy indeed comprises approximately 73% of the universe. But in doing so, Mach's Conjecture Principle for Inertia is also invoked, thereby lending credence to Mach's overall conjecture. "An interesting consequence of the general principle of relativity", authors Øyvind Grøn and Torkild Jemterud, Oslo and Akershus University College of Applied Sciences, Oslo, Norway, published April 12, 2016.

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