From Chirality to Helicity when our cack-handed universe goes massless.

Only when our Universe's last two, presumably rather large Black holes have collided and the last pop of Hawking radiation and gravitational waves is emanated as maximal Thermodynamic Entropy is reached will the universe's Chirality be one and the same as its Helicity, massless as it will be once again.

The stress-energy tensor is the source of space-time curvature and can be viewed as 4x4 matrix. The time–time component is the density of relativistic mass, e.g. the energy density. The time-space component is the density of the i-th component of Linear momentum or (relativistic) mass per unit volume.

From the diagram we see that Linear momentum is a vector invariant under translations. It arises from the assumed homogeneity of space: you move your experiment through space and unless the environment is permeated by markedly different fields or objects (!) it will deliver the same observations. Energy is the scalar invariant arising from the invariance of closed systems to translations in time. That is Energy is the conserved (Noether) current as time is deemed homogenous and indiscernibly the same so that an experiment done today then tomorrow (with all other things being equal) will deliver the same results .



Linear momentum and energy are formed by contractions of the energy-momentum tensor with the Killing field generator of their isometries. In the Lie vector field sense the local metric is preserved when Lie dragged along these tangent vector fields.

Similarly Angular momentum arises from the isotropy of space, it is a pseudo-vector (since it has a handedness -directionality) whereas rotational kinetic energy is a pseudo-scalar. Neither is elaborated on the diagram. These would arise from the spatial diagonal elements of the stress-energy tensor that represent normal stress or "pressure" and the off diagonal spatial components that are the shear stress.

Discrete Symmetries in field theories

In quantum field theories we assume fields exists in a exogenously defined flat space-time imbued with the symmetries of Poincare. At a high level we come across the following terminology:

1- "Handedness" is the "Parity", P of CPT.

2- Charge Parity, CP invariance is broken through Weak interaction decay channels so are not deemed exact symmetries.

3-CPT is deemed sacrosanct so that Poincare (Lorenz plus space-time translation) invariance is preserved.

As such broken CP implies Time (inversion), T symmetry is broken a problem given the even more sacrosanct Zeroth Law of Thermodynamics.

Field theorists distinguish two possible particle states:

• "chirality" - from the Greek, χειρ, ’hand’ 
• "helicity"-from the Greek, ελιξ, ’twisted’ 

The latter is defined via the direction of the momentum and angular momentum of a particle and is not a Lorentz-invariant property for massive particles (as you can attached an inertial frame to them). In the massless case (of photons, possibly gravitons and neutrinos) however, helicity can be linked directly to chirality.

As such we talk of two-spinor, (self dual) "chiral formulations of gravity, with their associated (linearised) quantised spin 2- helicity graviton states. Gravitational as well as electromagnetic waves were subject to polarisation at recombination and are imprinted on the CMBR. Traditionally neutrinos are represented by chiral Weyl-Spinors whether or not they are massless and subject to some mixing theorem.

The Relativity of Helicity

The particulates that give rise to the interaction between massive particles are the fields of force mediating bosons of Quantum Electrodynamics (QED), Quantum Chromodynamics and some quantum field theory of gravity. They are respectively the photon, gluon, and graviton each possessing an unambiguous chirality (‘handedness”) that is the same as their helicity (‘twistedness”).

These massless particles have spins that are in the same direction along its axis of motion regardless of the point of view of the observer. That their Chirality is absolute in this sense is due to both the invariance and finite speed of light. That such massless particles move at the speed of light, means that a massive observer (travelling at less than the speed of light) cannot travel in a faster reference frame in which the particle would appear to reverse its relative direction.

Their handedness is unambiguous in that all real observers see the same chirality. Accordingly we say that the direction of spin of the massless particles is not affected by a Lorentz boost (the relativistic equivalent of a Galilean change of reference frame) in the direction of motion of the particle. As such the sign of the projection (helicity) is fixed for all reference frames.

A photon’s “twistedness” has a sense in that helix described the rotation of its electric field (say) can be clockwise or anti-clockwise. By definition the helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion. It is left-handed if the directions of spin and motion are opposite.

Poincare Algebra of Helicity states

Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: left is negative, right is positive. This can be cast in the language of linear operators acting on a vector space. For our purposes think of a matrix acting on a column vector.

Helicity is always rotationally invariant but for massless particles it is also Lorentz invariant: no boosts are possible that will reverse the sign of the 3-momentum, p of the particle. The sign of p though does reverse under a Parity transformation as although the momentum is a true (polar) vector changing sign with x, angular momentum vector, J is a pseudo-vector (axial vector) remaining unchanged with a reversal of x. Helicity which is the scalar product of the two J.p is thus a pseudo scalar changing sign under a parity transformation. In order to preserve parity both potential helictites of the particle need to be present and this is the case for the photon with spin ±1.

The spin, s of a massive particle means something different to the spin of a massless particle. We view the generators of the Proper Lorentz group as Angular Momentum (pseudo) vectors. They are generators, L in the sense that they infinitesimally create boosts by the exponential operation exp(𝟄L) between the algebra and group - think roughly here of a Taylor expansion of X around 0 for a small epsilon, 𝟄. They are (vector) operators in the sense that they operate linearly in a linear space so that compositions of them are additive. But they are special vectors being axial in nature with a definite orientation.a ll this means that they are closed under the [,] composition law of commutation:[J,J]=ieJ

In addition to the boost invariance we have considered so far, so-called Poincare invariance includes symmetry under translations of space-time. That is if we shift space or time incrementally along (generated by a vector, P) by say redefining the zero point of our co-ordinate axis we observe that our laws of motion do not change.

Casimir Operator

An operator that has two eigenvalues ±1 defines the Chirality for a massive Dirac fermion. Any Dirac field can therefore be projected into its left- or right-handed (chiral) Weyl components. The coupling of the weak interaction to fermions is proportional to such a projection operator, which is responsible for its parity symmetry violation.

What we look for in analysing the algebraic structure is a combination, “C" of operators (J and P), which commute with all the other generators. If that is the case we can say that their are states in the linear system that are simultaneous eigenstates of C(J,P). For the Poincare group there are two such "Casimir" combinations and as such a simultaneous state possesses two eigenvalues- one of mass and one of spin.

For massive particles it turns out that the sub-group of Lorentz boosts that leave 4-momentum untouched induces a representation of the Poincare group. This sub-group has the character of 3-dimensional rotations and is covered by the (symplectic) spin group.

It turns out then that the set of generators J and P are closed under [,] composition law and thus particles have two qualities: a definite mass and a spin. The mass is associated to the 4-momentum, P the spin to J. So in the massive case we have a SU(2) representation in which the Casimir operator J^2 has eigenvalues s(s+1) and J_3 values of s and -s. In the massless case only the angular momentum operator is specified by its Helicity. For a neutrino a massless fermion produced in beta decay the spin is -1/2.

Nature appears to treat (Charge Parity and Time) CPT invariance as a fundamental symmetry such that particles linked by them in unison are deemed indistinguishable from each other. CP transformations do however distinguish certain meson decays which express an orientation preference.

CPT Invariance

Three possible symmetries to consider as invariants of nature, which in fact only usually, but not always, hold, are those of charge conjugation (C), parity (P), and time reversal (T):
Charge conjugation(C): reversing the electric charge and all the internal quantum number
Parity (P): space inversion; reversal of the space coordinates, but not the time
Time reversal (T): replacing t by -t. This reverses time derivatives like momentum and angular momentum.

It is a reasonable presupposition that nature should not care whether its coordinate system is right-handed or left-handed, but surprisingly, that turns out not to be so. In a famous experiment by C. S. Wu, the non-conservation of parity in beta decay was demonstrated.

This and subsequent experiments have consistently shown that a neutrino always has its intrinsic angular momentum (spin) pointed in the direction opposite its velocity. It is called a left-handed particle as a result. Anti-neutrinos have their spins parallel to their velocity and are therefore right-handed particles. Therefore we say that the neutrino has an intrinsic parity.

Nature at a very fundamental level distinguishes "left-handed" and "right-handed" systems. The combination of the parity operation (=P) and "charge conjugation" (changing each particle into its antiparticle = C) thought initially to be an inviolate conservation law (CP invariance), through the study of Kaon decay in 1964 was shown to be only a partial law. Only upon adding time reversal (=T) to our arsenal of transformations will a system be deemed indistinguishable from its original state.

One could ask whether CPT invariance implied Lorentz or whether invariance under Lorentz transformations implies CPT invariance. Either way it is the latter that governs the following observations:

1. Integer spin particles obey Bose-Einstein statistics and half-integer spin particles obey Fermi-Dirac statistics. Operators with integer spins must be quantized using commutation relations, while anti-commutation relations must be used for operators with half integer spin. 
2. That particles and antiparticles have identical masses and their lifetimes arises from CPT invariance of physical theories. 
3. All the internal quantum numbers of antiparticles are opposite to those of the particles. 

Dirac Spinors

The notion of the chirality of a particle is both clarified and made more abstract by studying all the (Group of) possible transformations that do not materially affect the representative object in either real or mathematical abstract spin space.

The Chirality of particle is determined by whether the particle object transforms as either right or left-handed representations of the Poincare group: a connected set of orientation preserving Lorentz boosts and rotations with the addition of translational invariance. We note that different representative mathematical objects, despite being rooted from a common hierarchy, can represent certain particles consistently. Some representations, such as Dirac spinors used to describe massive fermions (like electrons), have both right and left-handed components, others such as the Weyl-Spinor describing (essentially) massless neutrinos have a handedness.

For massive fermion (spin half) particles (in which Lorentz boosts can operate at positive relative velocities) such as electrons, quarks, and neutrinos—chirality and helicity must be distinguished. As it is possible for an observer to change to a reference frame that overtakes these spinning particles, the particle will appear to move backwards, and its helicity (now only an 'apparent chirality') will be reversed. For massive particles therefore chirality is not the same as helicity.

Proper Orthochronous Lorentz Group

To finish a little let us review the structure of the full Lorentz group, O(1,3). It is relevant for the CPT theorem and has that has four disconnected, disjoint components according to the signs of the determinant of the Lorentz transformation, det(Λ)=±1.

For sets of boosts such that Λtt=γ>1, called "orthochronous proper(linear) transformations we form the restricted sub group SO+(1,3). When you see determinants, think of oriented parallel-pipeds, so this sub-group has already been quotiented out by P.

In increasing generality we have then the:

1. Proper orthochronous Lorentz group, SO+(1,3) considered to be the true (local) symmetry group of all physical laws governing quantum field theories in classical spacetimes. 
2. Special Orthogonal, SO(1, 3) group that includes spacetime reflections (PT) with γ ≤ −1, 
3. Full O(1, 3) group that contains parity (P) and time reversal (T) transformations not related to exact symmetries of the real world. 

The lowest-dimensional (non-trivial irreducible) representations of the proper orthochronous Lorentz group, are the relativistically generalized Pauli matrices.

The special linear group SL(2, C) in two complex dimensions is defined by:

SL(2, C) = {A∈GL(2, C)| detA = +1}. Here the SU(2) matrices (of Pauli) in the SL(2, C) generate spatial rotations; Hermitian matrices in the SL(2, C) generate the boosts.

Denoting the "fundamental" representation of the spinor Lorentz group SL(2, C) (by itself) as the (1/2, 0) representation and its complex conjugate, (1/2, 0)∗ by (0,1/2) we have: left "chiral" spinors: "chirality- Weyl spinors" that transform as (0,1/2)=(1/2, 0)∗

In most circumstances, two left-handed fermions interact more strongly than right-handed or opposite-handed fermions implying that the universe has a preference for left-handed chirality, which violates symmetry of the other forces of nature. Indeed only left-handed fermions interact with the weak interaction. There is no frame dependence of the weak interaction: a particle that interacts with the weak force does so in every frame.

A Morphology of Physical Descriptions

The hierarchy of conceptual frameworks we deploy to usefully explain phenomena, from the most  self-evident but restrictive to the myriad of the allowable reads as:

1. Axioms; 
2. (Theories of) Principle; 
3. (Constructivist) Laws;
4. Solutions (to the differential Field Equations that encode the laws).

As an example, we deploy the guiding principle that all objects imbued with the same "qualities" are indistinguishable from each other. Such a viewpoint has unified many an apparently disparate phenomena.

While Newton's (Laws) assumed implicitly the equality of inertia and gravitational mass, Einstein's General theory of Relativity (GR) made this presumption explicit by geometrising their equivalence. In effect Einstein, identified away the gravitational field as merely a kinematical illusion. At least that is in the locale of stronger gravitational fields, where a freely falling frame can always be identified rendering the gravitational field illusory. The gravitational field has been "socially constructed" and we need only consider as usefully "real" the inertial acceleration field "construct" sourced from the local inertial property of the matter. Such constructivism generally asserts that it is necessary to find (or "construct") a (mathematical) object to prove that it exists. Here it is not just semantics to say that we have deconstructed the gravitational field, supplanting it with the inertial acceleration one. That certain stuff has the quality that it resists a change to its motion in the spirit of Galileo or in Newton's reworking within his First law, is thus self-evidently, tautologically Axiomatised: Inertia is the quality that massive stuff possesses if nothing else (like charge, hypercharge, strangeness..).

The fibre bundle verbein formulation of GR in which a tetrad is tagged with Lorenzian, locally Minkowskian indices (of the inertial reference frame) as well as general co-ordinate ones (of the "inertial acceleration" field) makes the "Equivalence Principle" explicit (quote intended as a nod to the slight of hand being played out). Anyway for our purposes indistinguishability principle is more usually referred to as the Symmetry Principle.

That GR aspires to embrace Mach's Principle but ultimately comes up short rendering its Field Equations as mere purveyors of its solutions as Laws of motion.

Solutions are variously described with an increasing nod to their profundity as trajectories, equations of motion, or if delivered by a Principle as "On-Shell" solutions.  Field Equations as in the case of Maxwell to the left (written in Helmholtz form) can be described in either differential (at a 'point') or Integral form. The latter emphasises the full generality of the form of the equations. That integrals deliver general families of plausible solutions, the resulting trajectory of a system being governed by its particular initial conditions: to be fed exogenously as part of the solving process.

We live in a universe of accelerating matter whose equations of motion represent the (only) possible path that our universe could have taken given the initial condition (Big Bang) primal state and these laws running their course.We view our present epoch as the end of one trajectory, one chosen course from the myriad of otherwise pencilled-in trajectories that were initially possible. The Natural Laws (of GR and Quantum Mechanics) that guide the universe along its particular course delivering the peculiarity of the universe today are to be viewed as exogenously delivered. Themselves derived as solutions to some higher guiding principle. As we shall see it is the Stationary Action Principle that is the meta-principle from whence the field equations are derived.  That an Action principle comprises a Lagrangian object; the difference between the Kinetic and Potential energies of a system renders its definition just about in within the constructivist's scope.

In the following, we will try to unpack the status of assumptions and prescriptions that give
rise to useful explanations of the aggregate behaviour of our observed universe (which we model as an interacting system of material objects undergoing relative accelerated motions in a local inertial field rather rather (say) than due to their mutual attraction).

Newton’s Laws of trajectories on an Axiomatised Euclidean Geometry

Newton’s laws are a set of dynamical laws that only allow for motions in which particles react instantaneously to a change in the relative positioning of each other (and by extension react to any particle in the whole of the universe).

That such a law prescribes the motion means that in a hierarchical sense we view these laws as more fundamental than the trajectory the universe itself traces under its guidance. Omnipresent or omnipotent, either way Newton's laws are viewed as immutable and unchanging through a universal time and space. The laws stand beyond the acts and stage that is the unfolding universe.

We take as given (as Axiomatically the case), the infinitely non-converging parallel line geometry of Euclid to describe, by positional measurement, the motion of particles in terms of arcs and lines. We say that the kinematical description of the vector calculus of changes has itself been unchanging.

As such, the prescription is twofold immutable: the timeless laws prescribe the allowable trajectories from amongst the available omnipresent set of geometrical conic sections that emanate from a particle point. 

Our hierarchy is clear: the geometry as Axiomatic, is self-evidently the most presumptuous aspect of Newton's  model, Newton's Law play out on this fixed arena and its solutions are subservient derived trajectories.

Symmetry in the Laws not of the solutions

We need to be careful in distinguishing between a fundamental law and one of its solution. We need to also be careful to create a hierarchy "laws" based on their conceptual unifying character. To be sure, solutions arising from laws need not themselves look symmetric. 

The law dictates the trajectory; the resulting trajectory merely respects the law. There need be no intrinsic symmetry in the resulting trajectory, in the sense for example, that the family of trajectories from the central force law theories of Newton and Coulomb are not especially circular, merely but variations on conic sections as above. So in a two-body orbiting problem, we observe an elliptic trajectory around the common centre of mass of the two circulating bodies.

We may choose to be an observer fixed on the larger body of the system of two bodies and then observe that we are on a focus of an ellipse with the orbiting body tracing an ellipse around our centre of mass: quite a symmetrical pattern but not the most symmetrical of conic sections that we could envisage.

Rather than truly symmetrical circular trajectories being carved out in space, Nature codifies spherical symmetry at the level of the guiding law. The encoding of symmetry is announced mathematically by representing objects characterised as indistinguishable from each-other with respect to some quality (color, charge, "inertialness") as field or algebra-valued tensor fields.  Newton’s and Coulomb’s guiding inverse square laws (of mass and charge influence at a distance) are laws whose potentials satisfy Laplace's second order differential equation. They are tensor laws describing central conservative forces; vector derivatives of a potential that act in a way that does not distinguish any directionality. The laws inherit the isotropic nature of space with which the forces permeate in being spherically symmetric.

Einstein’s Laws and dynamic Kinematics of Riemannian Geometry

In Einstein's universe, the geometry of space is no longer axiomatically assumed as flat let alone hyperbolic. Rather the Geometry of the stage (of space) itself is demoted to but a solution of the field equations of General relativity.

The space as merely a solution is now viewed

 as having an "historical" character in the sense that the geometries that preside at any moment for an inertial observer are an echo of its local past history. To the right are the principal components of the Kerr metric tensor describing the external space-time intervals resulting from a rotating Black hole.

Parenthetically, it may be helpful to think in this context of the Einstein's inertial geometrisation of a non-linear gravity, in the sense that it remembers its past, as a Martingale: a stochastic process in which the conditional expectation of the next metic interval value is given a prior (value) that is just its current value. The dynamical playing field of rulers and stop-clocks, in themselves being path dependent are a product of the Field equations. The kinematical framework of tensor calculus evolves. Einstein's revolution of the invariant was to demote an (Euclid's fifth -parallel) axiom to the status of but a 'manifold of solutions' to a governing law.

Stationarity as the Unifying Principle of all Action

We have seen how through the evolution of ideas that we must take care to define what is the reach of our laws. [Smolin, Problem with Physics], in his critique of String theory cites as one of the five problems for a Theory of Principle to resolve as the merger of General Relativity and quantum theory into one overarching theory of quantum gravity. Such a Theory of Principle with its universal description of nature is to be distinguished from a lesser constructivist theory. 

For the latter, the archetype we return to is electromagnetic field theory; merely a description of  phenomenon in terms of models or equations. It is a theory of more limited scope in that it can co-exist consistently with other such constructivist theories such as theory of dark matter with which it plainly does not interact.

In contrast the Action Principle has the required unifying character. There are an infinite set of possible paths between two points in space and the guiding hand does not insist on the simple simplicity of linearity of lines or symmetry of circles to traverse that interval. Rather, that the law guides the system to a particular trajectory from its possible set choosing that solution that solves a minimisation problem. The chosen trajectory is in a sense the efficient solution to the law.

The Natural Law will state that the system of interacting particles from its initial configuration to its end configuration will minimise the difference between the system's Kinetic Energy, T and it's potential energy, V.

The latter energy form is of gravitational (or electromagnetic if particles in system are additionally charged) type. Either way the laws prescribe energy due to the relative positioning of the system of particles (with respect to each other) and energy due to their relative speeds with respect to a distant idealised inertial observer outside the system. The law, given the status of Principle: the "Action Principle", dictates this need for the distribution of total energy between these two types to be optimally constrained. The constraint is such that the chosen trajectory is the one in which the sum of these kinetic and positional energies is the minimum. In evolving through time, the path taken at every step is the one that minimises this.

Axioms by Rights Principles as Directives

 In looking at the progressive structure of beliefs that sit at the heart of Theoretical Physics we see that Nature’s adherence to fundamental tenets sometimes mirrors the best of man’s: from those Rights we hold so implicit and inalienable (so as to be presumed to be as god given) to those we see merely as Federal Directives that require enactment at local state level. The European self-evident "Human Rights" are the Axioms of Euclidean Geometry and Grassmanian Algebra and the likes while European "Directives" such as freedom to travel across borders are Physics’ founding Principles. Laws have Solutions that are akin to the various visa entry levels of the nation states. As with the European project there has been, and will continue to be, a need for an overhaul of various tranches of this hierarchy as more abuses and inconsistencies of treatment by participants become apparent.

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!

One man's heap of Recombination embers is another's countable set of photons with traceable histories.

We often read claims that such and such an observational anomaly cannot be plausibly explained by our current models of the universe so that the explanatory models needs tweaking. Here is an example, ras org :

"The voids we have detected cannot explain the Cold Spot under standard cosmology. There is the possibility that some non-standard model could be proposed to link the two in the future but our data place powerful constraints on any attempt to do that. If there really is no supervoid that can explain the Cold Spot, simulations of the standard model of the universe give odds of 1 in 50 that the Cold Spot arose by chance"

Should we care any more about an "unlikely" 2% than a 50-50 when it comes to a sample of one (universe)? I feel that we are being persuaded by an irrelevant "probability measure" argument which I am unable to debunk cogently but will try in the following. One first turns Anthropically to the following kind of argument. That it did happen (even under the guiding hand of Standard Model of Cosmology) seems as unremarkable as reflecting that there has been at least one instance of a single winner of two lotteries. carolina lottery winner. From the perspective of the winner, indeed a remarkable coincidence, one which they wouldn't have appreciated given their participation in the mad odds in the first place. To us, given all the irrational hopefuls in the world it was just a matter of time.

Moving away from such Bayesian prior arguments we can rather ruminate about what might constitute outrageous odds anyhow. That is how to ascribe some plausibility measure to a probability claim about the state of our universe given our model for it.

The following amounts to little more than a coarse-grained view of sand heaps, the Cantor's number line and an introduction to the probability measure: that one man's heap is another's countable set of grains, that one man's unlikely aberration in the CMB is another's man's noise.

I will mix metaphors and mince the maths. We start with the paradox of the heap that reads: "you have a heap of sand, and take a single grain away, you still have a heap of sand. But if you keep removing one grain at a time, eventually you will only have a single grain remaining, and that’s clearly not a heap. So when did it stop being a heap?" Sorites_paradox

Compare now probability theory, in which we have a function called a "probability measure" if it assigns to each event in the collection of all events a non-negative real number. The "measure" is a scale telling you the weight of any subset (of the collection) where the total weight of everything is one. Whether you put a couple items on the scale separately one by one or you weigh them together at once, the sum of their weights will be the same.
Consider again the grains of sand that individually weigh zero, but collectively (as an uncountably large aggregate) in a jar have a weight bigger than zero.

Similarly there are uncountably many rationals within a number line interval.

If we cover all rational numbers between 0 and 1 by rational sized intervals the total length of the covering is an infinite series that sums to only a half. Despite the rational numbers being everywhere in between 0 and 1, and all being covered with its own interval we fill only half the number interval [0 ,1]. There are uncountably many real (rational and irrational pi, e and phi and the likes) numbers within a number line interval, so the probability of choosing an irrational number in the interval has a probability measure of zero even though the whole interval has some positive measure.

What we discern as distinguishable light brightness and differentiation in colours adheres to (log) power laws.

Colours are indiscernibly the same over a small enough wavelength interval.

What we can distinguish across the COBE pictures of the CMB as more than noise we would not deem as anything but noise across small enough neighbourhoods.

COBE ring data

So to a self-selecting universe adhering to its own laws from an uncountably infinite set of possibilities, what is the appropriate "measure" to ascribe a probability "odds" to an observed behaviour?

Eotvos for a Ferromagnet falling as a dielectric in vaccuum

An Eotvos experiment to include paramagnetic effects?

Galileo considered the necessary equivalent rate of acceleration of a rubix cube of material not in "contact" versus a "whole" cube of the same material (as such slightly denser) falling in a gravitational field. His conclusion was that because the constituent cubes must all fall at g=9.81ms^-2 just as the whole cube then g is universal for all bodies immersed in the field regardless of their mass (indeed density).

Image credit: (weisshouse.com). See discussion googlePlus-science on what is contact.

Eotvos has affirmed the indifference of the gravitational field to the materiality of a body freely falling through to one part in 10^17. Two test masses effectively follow the same geodesic as long as they have the same bulk mass; the precise make-up of the constituent stuff bound as they are by differing electrostatic and nuclear forces is irrelevant to the external acceleration field. One may ask whether it is useful to augment the Eotvos experiment by choosing one material that possesses an additional bulk property of paramagnetism.

A quantum equivalent of Maxwell's displacement ("current") field should add drag even though the quantum electrodynamic vaccuum that it is falling though is electrically neutral. Would a ferromagnet decelerate while falling through free-space (permeated only by a uniform gravitational field) more than a non-ferrous metal due to the retarding action described by Lenz's law as the creation of "eddy-displacement fields" in the "dielectric" of the (quantum electrodynamic) vacuum resist its fall?

To answer this we will have to consider the standard Eotvos experiments that look to confirm the degree to which the so-called Weak Equivalence Principle of Einstein applies across matter types and the nature of the displacement ("current") field in Ampere's Law. First, lets make reference to Maxwell's equations in describing the classic demonstration of Lenz's law and the appearance of his mysteriously misnamed "displacement current".

Maxwell's "Displacement field" and Lenz's Law

A popular high school demonstration is that of dropping a neodymium magnet through a (conducting but non -significantly-ferromagnetic) tube

and observing its rate of descent as slower than if a non-magnet metal the same size was dropped. The movement of circulating magnet flux lines during the magnet's descent induces a displacement field through the copper tube. According to Maxwell-Lorentz-Einstein electrodynamics this bona fide electric field then accelerates the free electrons in the copper tubing creating an "eddy' current in the copper. The magnetic part of the Lorentz force, F=qE + qvB associated to the movement of electric charge induces a Magnetic force that acts in the opposite direction to the gravitational force thus slowing its descent according to what is called "Lenz's law". Without a conducting material you just have a latent time varying Electric field.

The name given to the field of "displacement current" is a misnomer as it is an electric current only when there are moving charges. It is not the J in Ampere's law but the dD/dt. Without a conductor for a tube (or indeed some dielectric material) the displacement field exists.

The induced electric field (just as with a "normal" displaced electric charge induced electric field) in the presence of a copper conductor (with its free electrons) enables an "induced current" to flow. This induced electric field is there whether or not there is a conductor present though, just as the gravitational field is there whether there is test matter to freely fall in it. Below are Maxwell's equations using Helmholtz divergence and curl notation.

You read the "point form" as right hand side is source of left. But as with any equality it goes both ways! So that the latter two divergence laws (Nabla dot) mirror Newtonian gravity's inverse square law (from a massive point source). That is, they can all be described by Laplace's equation for a scalar potential field that falls off as 1/r.

For Maxwell, instead of the bulk gravitational mass quality of matter generating a potential that gives rise to a conservative (path independent) force, in his equations the surface electric and magnetic (monopole) charges are the sources.
D is the "displacement" current that Maxwell added to give the equations full symmetry in space and time and J is an electric current source which we don't have in our copper tube so the Eddy current is the time changing D. Curl H (Nabla cross) is the circulating Magnetic field source. A picture used in the context of polarization modes imprinted on the CMBR sums up the difference in the divergence and circulatory properties of the Divergence and Curl operations.

Typically, because no magnetic monopoles have been discovered the Electric field is traditionally defined by Gauss's law with its potential field diverging (E Mode) from a point (at least spherical) source and magnetic dipoles to the circulatory field ("B mode') lines.


Magnetic fields can though, as Faraday's Law shows us can be generated from a circulating (centripetally accelerating) electric current.

Eddy, Vortex or Circulating Current?

One might as why this Magnetic sourced current is not called a "circulation current". It is described by the Ampere's law of Maxwell's four equations and shows that the Eddy current is sourced from a circulatory Magnetic field, H (note use of the Helmholtz's "Curl" notation "Nabla x", an axial rotating tangential vector field!). On the etymology even Maxwell believed in an mechanical (aether-based) mechanism for the propagation of his light in a vacuum of intertwining electric and magnetic fields.

Within the fluid dynamics community there is little difference between the terms "vortex" and an "eddy" just their contextual usage. That is within turbulence in all fluid types, be they liquids, gases or plasma you usually speak of eddies with these eddies decaying into to smaller eddies to form a turbulent cascade. Vortices are deemed more stable structures and the description of their physics (e.g. geostrophic winds) does not necessarily include the formation of turbulent cascades.

Superconducting Eddy Currents

Either way the eddy or vortex currents in the copper tube if the etymology is correct should not exhibited laminar (continuous non crossing field) flow characteristics. So does our solid conducting metal tube carrying circulating charges driven by a displacement field constitute the name an eddy flow? The argument goes that we have a "cascade of ever more smaller self-similar eddy currents circulating in the copper tube when a magnet is dropped down inside it. Maxwell tells us that as an eddy current flows a secondary magnetic field is generated which would then create a secondary eddy current, a tertiary magnetic field...

If somehow there is thermal dissipation of energy that makes this an energy dissipating cascade what then would be the case if the tube was a superconductor? No cascading eddies then? Then that induced current would be just a plain old circulatory current?

Eotvos and the Weak Equivalence Principle

The weak equivalence principle (WEP) states that in a uniform gravitational field all objects, regardless of their

composition, fall with precisely the same acceleration.

The principle asserts the exact identity of inertial mass mi (the mass appearing in Newton’s second law) and gravitational mass mg (the mass appearing in Newton’s law of gravity).


This was the first great unification program of Newton, relating the dynamics of the Heavens to the inertial trajectories of earth bound objects.

The presumption of Equivalence of inertial and gravitational masses was subsequently unpicked by Einstein in formulating his Equivalence principle of which his elevator experiments encapsulate his thinking nicely if not fully.

eotvos latest

Note that WEP implicitly assumes that the falling objects are bound by non gravitational (Nuclear-Yukawa type binding) forces. Departures from the exact equality of mg/mi for objects 1 and 2 as specified by the Eotvos parameter have not been detected to orders of 1 in 10^17

η1,2 = a1 − a2 = (mg/mi)1 − (mg/mi)2 , (1) (a1 + a2)/2 [(mg/mi)1 + (mg/mi)2 ]/2 


Our question then can be framed as to whether there is an equivalent Eotvos experiment for magnetic (ferromagnetic) versus non magnetic materials falling through an effective vaccuum dielectric?

Its effects would either mask or be masked by the different materiality of the freely falling bodies as captured in a pure Eotvos experiment.

To get a feel for the possible quantum effects that would have to be considered in motivating better such an experiment (our consideration to date have been classical). The related perhaps unobserved Unruh and observed Casimir effects are two phenemena that are predicted by the virtual particle froth that emmanates from the quantum electrodynamic vacuum.

Unruh effect

The Unruh effect is the prediction that an accelerating observer will observe blackbody radiation where an inertial observer would observe none. The vacuum background appears to be warm from an accelerating reference frame so a thermometer waved around in empty space will (by virtue of its accelerated motion) record a non-zero temperature. wiki Unruh effect

Casimir Effect

When two uncharged objects are placed in a vacuum with no external fields, we wouldn't expect them to have any force between them other than gravity. Quantum electrodynamics says otherwise. In a classical description, the lack of an external field means that there is no field between the plates, and no force would be measured between them. When this field is instead studied using the quantum electrodynamic vacuum, it is seen that the plates do affect the virtual photons which constitute the field, and generate a net force– either an attraction or a repulsion depending on the specific arrangement of the two plates.

Multi Graphene-layered Detector

Consider finally layers of graphene separated by a little more than their van der Waals bond length of 0.3nm. "Not in contact" That is, they have the electrical conductivity properties of graphene (graphineReview) sheets and not graphite.

If suitably separated vertically parallel sheets are dropped in a gravitational field they will at some point become graphite pulled in by tidal forces of the gravitational field. If rather separated horizontally they will separate further on falling, becoming more distended in their sheet formation.

As such the respective electrical properties of the two configurations will be different depending on "point of contact" reached through orientation in the external field. So random question/thoughts:

1-Is there some version of the strong equivalence principle that could be usefully investigated (an Eotvos experiment) here?

2-could one envision a Lenz law induced differential acceleration across the two orientations: eddy currents decelerating the material vertically aligned greater than horizontally by Faraday's law?

Given that graphene is a thin layer allotrope of pure carbon 12 ("in the structure of a plane of sp2 bonded atoms with a molecule bond length of 0.142 nm") one could consider a more traditional Eotvos in which an alternative set of graphene sheets doped with greater percentage of isotopes of carbon 13: arxiv.org - arxiv.org/pdf/1112.5752.pdf

as such denser are also dropped.

This would confirm the acceleration dependence, given density difference and materiality of object of free fall.