A Re-dimensionalization of Physics holding unified Physical Fields producing a purely gravitational Lorentz Force

(Response #1 by W.Kuijk-Grünbauer ¹ to Dr. L D Howe's paper on Symmetry between Gravitational and Electric forces ² )

Abstract

In this letter we follow a suggestion by R. Feynman to the effect that a re-dimensionalization of Physics may be helpful to solve problems that seem intractable with the physical notions within the framework of the mksC system of units and dimensions. A special re-dimensionalization is shown to reproduce the Coriolis force of Newtonian mechanics as a component of a novel "gravitational Lorentz force" in the context of a "unified field theory".

In a universe 'inhabited' only by pure Newtonian masses of dimension kg, no mechanics in the classical Newtonian vein can engender an entity the physical dimension of which contains the Coulomb unit C.

Pure and consequent application of the "Symmetry principle" between classical- and electric mechanics as suggested by L.Howe, means:

1. In a universe 'inhabited' only by pure Coulomb charges of dimension C, no "electric mechanics" as a theory that knows only charges, can bring forth an entity the physical dimension of which contains the Newtonian unit kg.
2. In our daily universe, with a view to deal with "charged masses", i.e. entities that in certain situations behave as Newtonian masses and in other situations as Coulomb charges, workers after Coulomb treated the H-atom in such a manner that the force (between electron and proton as charged particles) only counted the forces between the charges of these entities ³. It follows that the subsequent theory of the H-atom is not qualitatively symmetric with respect to masses and charges, and it is worthwhile to find out what could have been the consequences of the asymmetry entering the theory.

The "cause" of the asymmetry has been Coulomb's decision to accord, within the context of his "electric mechanics", the Newtonian physical unit kg a place that it did not necessarily deserve, namely in the dimension m^-3kg^-1s ²C of the well-known "dielectric constant", , that somewhat later on was experimentally found to have in vacuum the numerical value

In a few earlier proposals I have been arguing that one might for dielectric constant have better taken the expression

whose dimension takes a natural place in the purely electrical universe I. In terms of electric dimensions Coulomb's law would have then come out symmetric to the (for the occasion, algebraically rewritten) Newtonian law as follows:

Coulomb's law:

(in Nc = mCs-2)

(1)

(where q₁ and q₂ are like charges in C with separation r),

Newton's law:

(in Nc = mCkgs-2)

(2)

(where m₁ and m₂ are positive or negative masses in kg with separation r for , Newton's constant),

and the "diagravitational constant" of the same vacuum that thought attaches to a 'mass universe ' as well as to a 'universe of charges'. The beauty of this symmetry between definitions (1) and (2), and its implied re-dimensionalization of the fundamental laws is that they only affect the dimensions of concepts (but not their magnitudes and their algebraic relationships) in the first place, meaning that one can compare (quantities of) gravitational Newtons N that occur in classical mechanical formulas, with (quantities of) electrical Newtons N_C that occur in the new electrical mechanical formulas within the electric paradigm, in the same manner as one may compare quantities of apples and pears refraining from comparing an apple to a pear as such ⁴. As a further reward of the application of symmetry, the two new laws (1) and (2) reveal an inherent "cross-over force between mass and charge". For instance, for a mass m and a charge q, at arbitrary separation r, try to find an hybrid constant between the dielectric and diagravitational ones, such that

is a cross-force that 'builds from (1) and (2)', in this sense that from it one can formally recuperates (1) and (2). To that end, an inveterate algebraist is immediately led to put q = q₁ = q₂ in (1) and m = m₁ = m₂ in (2) and take the geometric average of the resulting two forces and find a new hybrid force formula

in

(1,2)

which does the job since, formally, for and one recovers (1) and (2). Calculation of the 'hybrid permittivity in vacuum' , alias "the diahybrid constant" that is given with this formula, leads to

and shape comparison of (1,2) with (1) and (2), counting that the entry must have dimension , we conclude that (1,2) is a third "inverse square law of force" holding for "charged masses", i.e. entities that are mathematically represented by expressions of the form with physical dimension . As an intertwinement of the fundamental laws (1) and (2), formula (1,2) hands us the following conclusions:

1. The full hybrid law reads as follows: the interactional force between two hybrid masses and , which are at separation r, is given by

   Hybrid's law:

   (3)

   and has dimension .
2. For arbitrary charge q and mass m at separation r, one infers from (1,2), (1) and (2) the identity

   (4a)

   with

   (4b)

This shows that is physically implied by the fundamental laws (1) and (2), and is not an artifice of imagination. So acts as soon as q and m are brought together into one medium with a permittivity, , that averages the permittivities that applies to any two charges and any to two masses in that same medium.

In what follows we will use the following terminology to discriminate between the three parts of Physics that we have distinguished in the above.

1. We speak of "the electric platform" as that part of existing Physics that can be formulated in terms of dimensions that only employ the three msC units,
2. The "gravitational platform" is the part that is in the three mks units only, and we add to this
3. 3. A new "hybrid platform" that uses only the three units, as defined and exhibited in the previous lines of text.

The greatest reward of Symmetry however is the unification of the electric, gravitational and hybrid fields that are in the usual fashion attached to the platforms of formulas (1), (2) and (3) respectively. They share the dimension of "acceleration", ms^-2 The fields are respectively

For q any charge;

(E1)

For m any mass;

(E2)

For any charged mass;

(E3)

where (E3) is the geometric average of (E1) and E2), so that for fixed charge q and mass m, these fields are scalar multiples of one another. Namely, one readily verifies the identities

(6)

where the bracketed expressions are (dimensionless) scalars. Recall from standard Maxwell theory, which concerns the electric platform and electric fields E(q), the deep result, that when q moves, say with velocity vector v, then there is a magnetic field that 'twines' with E(q) and writes

(7)

perpendicular to the electric field and v. Clearly, with the present dimensionalization, B(q) gets the dimension of (angular) frequency s^-1 instead of the "old" tesla ; this gives reason to rewrite it with the Greek symbol for 'angular frequency' as follows

Recall also that in the static case (v = 0) there is no such magnetic field. It is important that (7) multiplied with the scalar from (6), first formula, yields for an arbitrary moving mass m on the gravitational platform also an angular frequency field (of dimension s^-1 ),

which accompanies the (moving) gravitation field E(q) of the moving mass m.

It is noticeable that a given arbitrary angular frequency field need not, for a charge q or a mass m, be a twin field of the form or associated to an electric field E(q) or a gravitational field E(m). However, an equality of the form

is possible, namely if

and this is the case for and only for the charge to mass ratio

(cf (6))

Back on the electric platform, recall that the electric force (of dimension N_C), exerted on a charge q', which is at separation from q , is the well-known "Lorentz force"

(8)

where the summand

comes, thanks to the relative motion of q with respect to q', in addition to the re-dimensioned "centripetal" Coulomb force

This Coulomb force in turn, according to electric theory, may drive centripetally any possible motion between the two charges q and q'. More generally, it should be recalled that all the electromagnetic dynamical events that relate the charges q and q' and their fields, are described by the Lorentz force (8) and the four relativistically invariant Maxwell equations that connect the field pairs and . Changing to the gravitational platform, it follows that there is similarly a "gravitational Lorentz force"

(9)

between any moving mass m and any other mass that are at separation . Thus, since any circular kinematic motion of m' with respect to m can be simulated by a dynamic with central force and centripetal acceleration , the Lorentz invariant force (9) can be made to comprise what in Newtonian mechanics goes under the name "Coriolis acceleration" as follows. Let m' run in a plane circle of polar representation around m with central Newton force

and tangential velocity Its constant angular momentum writes and has magnitude . Now assume m' moves on the radius with speed , necessarily with constant , towards or away from m, then the angular momentum of m' changes. This change of momentum defines a torque , where F_C is by definition the Coriolis force. One has

and

(10)

The force F_C, which in this case is tangential to the circular motion in the same plane, can be added to the right hand side of (9) and tends to increase or decrease according as the radial velocity v' is pointing towards m () or away from it ().

Remark. One can, conversely, also get as a radial Coriolis force pointing towards m or away from m, by adding or subtracting v' to the tangential velocity v in (9), or make, more artificially and as it seems, contrary to the grain of experiment, the mass m to depend on time t. We leave this as an exercise to the reader.

The uniformity of the field laws (E1-3) for the three platforms suggest uniformity of the Maxwell theories for these platforms. Indeed, one can get get these uniform equations from the four standard electromagnetic Maxwell equations by redimensioning the entries of the physical constants that occur in these, and simultaneously redimension the field entries. Recall the four Maxwell equations for an arbitrary electromagnetic field pair (E,B), and the dimensions of their terms.

1. The "Magnetic Gauss Law": , with each term of dimension ,
2. The "Electric Gauss Law": , (with the "charge density' of dimension m^-3C and each term of dimension ),
3. The "Faraday-Henri Law"; , with dimension for each term,
4. The "Maxwell-Ampère Law": (with j the "electric current density" of dimension and each term of dimension ).

The proposed re-dimensionalization of these laws is obtained by subsituting in the dielectric constant , which then goes into , turning the pair (E,B) into a redimensioned pair, say on the electric platform, with laws:

1. The "First Gauss Law": , with each term of dimension ,
2. The "Second Gauss Law": , (with the "charge density' of dimension m^-3C and overall dimension of terms s^-1),
3. The "Faraday-Henri Law"; , with overall dimension s^-1,
4. The "Maxwell-Ampère Law": (with j' the "electric current density" of dimension each term of dimension ms^-3).

Note. Recall that E' has dimension ms^-2, and B' the dimension s^-1. Notice also that only the replacement of C by , respectively by kg, in the dimensions of these equations, which as before, carry into and into respectively, deliver, what should be called "the Maxwell equations for the hybrid platform", and "the Maxwell equations for the gravitational platform", respectively ⁵.

Footnotes

1. Former name: W.Kuyk. Present address: Science Department, Antwerp University, Groenenborgerlaan 171, 2020, Antwerpen.
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2. www.innovationgame.com
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3. This has been acceptable to all physicists, for reasons that the electric force between the charged parts q_e of two electrons amounts to being 10 ⁴⁰ times stronger than the Newtonian force of the interaction between their mass parts m_e . There is no hint in the literature of that time that anyone thought of considering the possibility of reckoning with the existence of 'cross-over forces' between q_e and m_e, i.e. between "charge and "mass".
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4. The conversion of the dimension in mksC units of an arbitrary valid formula in extant physics to the electric universe or "platform" as we shall say, is by substituting in the dimension, and thence to the purely gravitational platform by .
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5. In a second and third response paper to Howe's innovationgame.com paper on symmetry between gravitational and electric forces I will apply the above equation to the motion of the hydrogenic electron.
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www.innovationgame.com/physics/letters/resp1.htm May 2003