Indistinguishability as the Guiding Principle to Unification stumbles at Inertia

Weight-fulness is a (socially) constructed concept. Only our inertia is real and unexplained. Einstein rendered inertial mass indistinguishable from gravitational mass through his equivalence principle. He then spirited away the gravitational field altogether in teasing out a freely falling inertial frame leaving, by transitivity the purely kinematical acceleration field. Through (electrostatic) contact with a macroscopic body we feel the illusion of weight due to "gravity". How liberating to think that modulo the electron-electron hovering sensation of "contact" we are just freely accelerating (falling) with the rest of the universe?

The indistinguishability principle traces through the unifying programs of Newton via Maxwell to Einstein. By "describing", with his Equivalence Principle" rather than assuming (as Newton did) the indistinguishability of "inertial" and "gravitational" mass, Einstein eschewed Newton's static, globally-defined, specially ordained inertial reference frame. Einstein's prior unification of electrodynamics with mechanics came from his observation that the vacuum speed of light is indistinguishably different to all inertial observers. Recognising the "invariance" of light speed meant embracing Lorenzian over Galilean Relativity as he further eschewed the aether of Maxwell.

His General Theory of Relativity in essence eschews the gravitational field itself, so to deem the identification of inertial and "gravitational" mass as an "equivalence" is almost erroneous. A "test" object is only ever apparently "immersed" in the "gravitational' field that surrounds some larger (accreted) macroscopic body. Indeed if placed on the surface of that body its trajectory would trace as a freely falling frame geodesic but for the (electrostatic) contact forces that maintain it in equilibrium. That gravitation is explained away as geometry, did not wholly satisfy Einstein who wanted to root his General theory of macroscopic objects in Mach's Principle. That the origin of the now unidentified inertial "quality" of matter of which Mach sources to the mass of the rest of the universe renders the theory incomplete. Einstein never liked the "Relativity" of his Special Relativity theory, preferring the "invariance" moniker. Given General Relativity's (minor!) failure to capture the source of the inertia, leaving absolute acceleration causally sourceless of explanation suggest an alternative name would be more appropriate. A further indistinguishability argument suggests that the scale of subsuming deeper gravitational field associated to macroscopic body versus microscopic test particle in its midst is irrelevant is illusory. Both objects can serve as inertial frames freely falling in a localised inertial acceleration field.

Einstein's Relatively Principled Revolution

Einstein elevated the invariance and constancy of the speed of light to a fundamental tenet. In augmenting Galileo's Principle by identifying those observers as indistinguishable rather by the Lorentz transformations of his Special (Principle of) Relativity he deemed electricity and magnetism as indistinguishably complimentary phenomena. That is in the eyes of (inertial frame) beholder, being merely a question of observer perspective. As a result, absolute (universal) time (and space) was lost and the notion of simultaneity (of multiple events) became relative as time was put on equal footing with space (in a 4-dimensional space-time) and observers tick off a private as well as public time of their clocks.

Einstein further sheds the intermediary space between particles of its mechanical aether property which was deployed even by Maxwell to mediate interaction. Space is not a vessel defined by the configuration of massive particles rather the causal relation between events are to be considered our elemental constructs. In his General Theory (of Relativity) he rendered gravity not a force, rather just the (non-inertial) effects of taking an accelerated point of view of an experiment, so supplanting the Axioms of Euclidean geometry with Riemann's (4-d) curved hyperbolic geometry.

A perspective on the conceptual unification of gravity and electromagnetism through field theory is presented in the figure below as three strands of innovation.

That there is this extraneously to be defined inertia, the resistance of any object to acceleration appearing in his equations of motion still begs the question, "acceleration with respect to what?" Is the (rest or otherwise) mass of a particle intrinsically and arbitrarily defined or exogenously determined by its environment? Ernst Mach, argues that both a body's acceleration and its inertia are relative. Relative that is to all that external matter in the universe that exerts an influence on all instrumentation. The disregarded estranged relatives are thus inertia and its second cousin, acceleration.

Inertial Mass and Elevated status of Accelerated motion

An object's mass is a numerical measure of its inertia; being the amount of matter in the object and as such defines the strength of the acceleration field that it experiences for a given applied force. [D W Sciama, Unity of the Universe]. Once we can measure inertia by measuring the force required to change its velocity we can ask the question: is the inertia of a given body always the same or does it change when other bodies are brought near it? No such experimental change has been detected which means that inertia is an intrinsic property of matter according to Newton. Mach rather argues that a body has inertia because it interacts in some way with all the matter in the universe.

Newton’s view is that the constant of proportionality is the inertial mass of the body. There is a problem in Newton's second law in that the value of the acceleration depends on how it is measured. That is on which body is the standard of rest. Therefore, in a rest frame on earth the Sun's gravitational force is producing no acceleration of the earth at all. The force rather it exerts is objectively determined. Accelerations called absolute are measured in a special way. Those bodies on which no forces act will not possess absolute acceleration and only those bodies are said
to constitute an inertial frame of reference. As such, Newton's second law should read force equals mass times absolute acceleration.

A force may produce no acceleration at all in non inertial frames. By postulating the existence of

additional compensatory " inertial" (centripetal or centrifugal) forces which do not have a physical origin in material objects we can see that earth is non inertial because it is absolutely rotating. On the reference frame of an object falling in the gravitational field of earth an inertial force must act on that body to counteract the gravitational force of the earth.

According to Newton the only way rotation relative to absolute space can be detected is from the existence of Centrifugal and Coriolis forces. However, absolute space was precisely invented in order to account for these forces. Electric and magnetic forces do not induce the same acceleration on all bodies as in for example, neutral bodies.

Inertial Motions

Motions in which particles can accelerate indefinitely are allowed. Motions that are constant in both space and time, so called inertial motions are the idealised motions from which we measure relative changes in space (velocity) and changes in velocity (acceleration). Only accelerated motion reflects the influence of a force (in the vicinity so as to be significantly different than zero). Non-inertial motion is a codification of the law of forces governing possible motions.

We will not observe trajectories that are not the result of an acting net force. These will trace out either arcs reflecting changing directions in space or straight lines in space reflecting accelerated rectilinear motion. In the latter only when we move to the unified view of space-time are such rectilinear motions revealed as arcs in a Minkowskian space-time.

Einstein's Laws and Dynamic Kinematics

When Einstein was unable to reconcile the constancy of the speed of light (the relativity of magnetism and electrostatics) from Maxwell's field equations with the Galileo's Principle of Relativity (that declares all inertial frames as equally agreeable impartial observers) it was the Principle that was overhauled. The status of what is a given, that is what is assumed axiomatically, is altered under the resulting governing laws of Einstein’s Special Theory of Relativity.

Einstein in his General Theory of Relativity further relieved accelerated motions from their distinguishable status by identifying, through the Equivalence Principle, such changes in velocity with the pull of gravity. The Principle in asserting that the effects of acceleration and gravitation are physically indistinguishable identifies inertial with passive gravitational mass. There are so-called 'strong' and 'weak' forms claims:

-In the 'strong' form the claim is that, at least in a sufficiently small region of space-time (to remove Distinguishability brought about by tidal effects due to a gravitationally field being isotropically sourced from spherical bound matter), a gravitational field is in all respects identical to and indistinguishable from an accelerated frame of reference.

-In the 'weak' form the assertion is that passive gravitational and inertial rest-mass are quantitatively equal and it is this form, which is tested by Eotvos-type experiments.

Whereas the weak form of the Equivalence Principle involves only rest-mass, the strong form extends this statement to bodies with zero rest-mass for example the electromagnetic field which constitutes an independent assumption as Einstein explicitly included electromagnetic radiation in his definition of matter. To this end it can be instructive to put the strength of the gravitational attraction of all mass on an equal footing to the coupling strengths of the three standard model forces and refer to it as the 'inertial charge' of the body. Just as the electric or strange charges define the strength of the electromagnetic or strong nuclear force coupling between like charged bodies, the inertial mass defines the strength of the gravitational interaction between (as it happens all) matter.

Fulfilling Mach

According to [Woodward, 72] Mach's Principle says that the 'fixed stars' causally determines by itself the properties of the local inertial field and thus the inertial properties of local bodies and inertial frames of reference. Mach effectively asserts that there exists only one frame of reference at any given point in the universe in which the cosmic blackbody radiation appears isotropic.

Fulfilment of Mach's Principle [Woodward 75] comes down to all of the following criteria being met in your physical theory:

1 The Einstein criteria -:
A The inertia of a body must increase when ponderable masses are piled up in its neighbourhood.
B A body must experience an accelerating force when neighbouring masses are accelerated, and, in fact, the force must be in the same direction as that acceleration.
C A rotating hollow body must generate inside of itself a 'Coriolis field', which deflects moving bodies in the sense of the rotation, and a radial centrifugal field as well.
D The gravitational/inertial field equations must yield no solution for an empty universe.

2 The Hdnl criterion: The role of Mach's Principle is to act as a selection principle in the determination of physical and non-physical global solutions of the gravitational/inertial field equations.

3 The Sciama criterion : The inertial properties of a small, neutral body are almost totally induced by the gravitational interaction of the remainder of the matter in the universe with the body; and that the inertia of a body is a reaction against accelerations of the body relative to the background gravitational field produced by the rest of the matter in the universe.

4 The Pauli criterion: The inertia, and thus the gravitational field, of a single body in an otherwise empty universe must be null.

Consider isolating a body from the influence of all physical fields (inertial, gravitational, electromagnetic, etc.) produced by all of the external matter in the universe. By Mach it will possess no inertia as the local inertial field produced by the rest of the matter in the universe can permeate the locale of this isolated body.The body now being inertialess, means relative to an external observer, it may possess any arbitrary state of motion, even accelerated without an Impulse. It is now just equivalent to a body in an otherwise empty universe. Both scenarios are identical in as much as both bodies may have any arbitrary state of motion but Pauli would not approve.

Brans-Dicke and the Rifling Bullet

Brans-Dicke articulate the problem of inertia in their classic paper Brans-Dicke. What is being described in the following is more nearly an absolute space in the sense of Newton rather than a physical space in the sense of Berkeley and Mach:

"According to the ideas of Mach, the inertial forces observed locally in an accelerated laboratory may be interpreted as gravitational effects having their origin in distant matter accelerated relative to the laboratory. The imperfect expression of this idea in general relativity can be seen by considering the case of a space empty except for a lone experimenter in his laboratory. Using the traditional, asymptotically Minkowskian coordinate system fixed relative to the laboratory, and assuming a normal laboratory of small mass, its effect on the metric is minor and can be considered in the weak-field approximation.

The observer would, according to general relativity, observe normal behaviour of his apparatus in accordance with the usual laws of physics. However, also according to general relativity, the experimenter could set his laboratory rotating by leaning out a window and firing his 22-caliber rifle tangentially. Thereafter the delicate gyroscope in the laboratory would continue to point in a direction nearly fixed relative to the direction of motion of the rapidly receding bullet. The gyroscope would rotate relative to the walls of the laboratory.

Thus, from the point of view of Mach, the tiny, almost massless, very distant bullet seems to be more important that the massive, nearby walls of the laboratory in determining inertial coordinate frames and the orientation of the gyroscope. Apparently, we may assume one of at least three things:

1.that physical space has intrinsic geometrical and inertial properties beyond those derived from the matter contained therein;

2. that the above example may be excluded as non-physical by some presently unknown boundary condition on the equations of general relativity.

3. that the above physical situation is not correctly described by the equations of general relativity."

This seems too subtle for the Lord let alone me!