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 ( 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, ChirlitzTuras, 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 )
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.
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