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Casimir Potentials,
Electromagnetic Density Oscillations
and Gravitation

by Darrell Moffitt


       Among the less researched topics of classical and quantum physicists
       is that of  electromagnetic  density  oscillations  (e.d.o.s), i.e.,
       acoustic-form longitudinal waves in electromagnetic media.

       Though current literature  does  address  phonon  interactions,  and
       their influence on conductivity in solids, longitudinal plasma
       waves and their  quantum  relatives  are  largely   ignored.   While
       acoustic waves may    admit    transverse   polarities,   they   are
       fundamentally a longitudinal phenomenon.

       Plasma oscillations possess  up  to  four  longitudinal  modes;  the
       alternating compression and rarefaction zones comprising  a wave may
       consist of either  particles  or fields, and the field itself may be
       electric or magnetic.

       Still another form  of  e.d.o.  results   from   interacting  charge
       densities; the square  of charge density is dimensionally  identical
       to that of sound,

                                  (M/(R^3*t^2)),

       and van der  Waals  forces  both  derive  from  such  couplings  and
       contribute the restoring force responsible for ordinary sound.

       Quantum mechanics describes  van   der   Waals  forces  via  Casimir
       potentials, simple, non-relativistic equations which take their name
       from Hendrick B. G. Casimir.

       Casimir's polar-polar potential,  applied  to  any  two  polarizable
       bodies, is a  prototype  of  quantum-level  e.d.o.s.   This equation
       reads

                       E=((h/(2pi*c^5))(P1*P2)(w^6/6))(1/R)

             where (P1,P2) are the respective (volume) polarizations,
                       "w" is the characteristic frequency,
               and "R" is the separation of the bodies in question.

                      ("h" is, of course, Planck's constant,
                         and "c" is the speed of light..)

       The frequency is  evaluated over a cut-off determined by the size of
       the system, "r", and roughly equal to (c/r).

       Here, the factor (1/6) represents an integral over w^5, and r may be
       taken as an average atomic radius.  This equation is accurate over a
       wide range of scales, with corrections  on  the order of unity being
       found in most situations of interest.

       The factor

                                 (hw^6/(12pi*c^5))

       is the e.d.o.  term,  and  yields large exponents  in  the  case  of
       molecular or atomic  systems.   These  large exponents are deceptive
       however, as average polarization in an atomic system is on the order
       of

                              ((10^-27)cm^3) or less.

       Polar-polar potentials thus tend to produce energies which are small
       compared to the total energy of a  given  system.   This  low-energy
       behavior suggests some resemblance to gravitation,  which  a  sample
       calculation will make more evident.

       Consider first the physical dimensions of Newton's constant,

                                  (R^3/(M*t^2)).

       It can be written

                                     (w^2/d),

       where "d" is a volume mass density, and could as well be written

                              ((M/(R^3*t^2))(1/d)^2,

       the ratio of an e.d.o.divided by the square of a density.

       The Casimir interpretation of this expression takes the form

                             (hw^6/(2pi*c^5))(1/d)^2.

       To get an  estimate  of  the  frequency-to-density  ratios which are
       relevant, one should  look  at  a   simple  system  with  well-known
       parameters.

       Monoatomic hydrogen is just such a system.  Assume  that the atom in
       question is in  its ground state, with a minimal volume polarization
       equal roughly to one electron volume

                            (approx.(2.818*10^-13)^3).

       Then, the polar-polar  potential of this system will be proportional
       to the orbital frequency of the electron

                                  (a^3*c/4pi*r#)

          where "a" is the electromagnetic  coupling  constant (1/137.036),
                          and"r#" is the electron radius.

       The e.d.o. of this system

                         ((h/(2pi*c^5))(a^3*c/(4pi*r#)^6)

       is

                           5.52107*10^13(gm/(cm^3*t^2)),
                           (factor of (1/6) suppressed).

       When this quantity is multiplied by the electron volume squared, the
       result,

                           2.76449*10^-62(gm*cm^3/t^2),

       is a tiny value indeed.

       Herein lies a surprise.  Recall that Newton's constant  has multiple
       definitions in dimensional analysis.

       In this case, dividing the Casimir potential value just given by the
       electron mass squared creates a fascinating "coincidence", namely, a
       figure which is just shy of one-half Newton's constant.

       Properly reduced, and   doubled   to  account  for  two  interacting
       systems, the full expression will read

                           G=((hc/(pi*m#^2))(a^3/4pi)^6)

       which equals Newton's constant to within 99.85%("m#"=electron mass).

       A more accurate  derivation requires  evaluation  of  the  Lambshift
       contribution, and estimates of such factors as vacuum  polarization,
       charge screening, relativistic  corrections  (long-range) and higher
       order interactions.

       The model just presented is not, arguably,  an accurate "definition"
       of gravitation, as other models are much more precise and detailed.

       It does, however,    illustrate    the   utility   of    alternative
       conceptualization made feasible  by  the  use of e.d.o.s and Casimir
       potentials.

       The value of  this  approach lies  in  its  ability  to  reveal  new
       phenomena and relationships between seemingly well-known processes.

       A similar treatment  might  have  been  performed   on   quark-gluon
       coupling, say, where  longitudinal  virtuals have been found to play
       an active role.

       What is more important is that consideration  be  given to a broader
       range of qualitative issues in physics.  Quantitative methods are
       only as good  as  the  qualitative  concepts  they  address, for one
       cannot calculate our planet's circumference  without first asking if
       it is round.

       Add 1:  A good introduction to Casimir potentials will be found
               in
                    "Physics Today", 11/86, p.p.37-45,
                    titled "Retarded, or long-range, Casimir potentials",
                    by Larry  Spruch,  which  contains  a   very   complete
                    bibliography.


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