# New Generally Covariant First Order pde by 6M3r37qE

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```									       New Generally Covariant First
Order pde
Here we show the consequences of a new generally
covariant partial differential equation generalization of the
Dirac equation:

(/x) -=0                                     (1)
Note the new 00=1-rH/r , rr =1/oo diagonalized coefficients giving the
general covariance.
Note substitution of rH=2e2/mec2 gives the standard Dirac equation result
/x+i=0 as r gets large

See http://davidmaker.com for derivation of this new pde
Next we Solve /x-=0                       (1)

Solve equation 1 directly (see similar methods of solving Schrodinger equation
using separation of variables =R(r)yL,m½(,), spherical harmonics (eg.,S, P
states), etc., given oo=1-rH/r=1-2e2/rmec2). For example there is stability at
rrH, since the metric time component oo (=1-rH/r) is zero so clocks slow down
and we get (baryon) stability. Also at this radius rrH
1)     2P3/2 states fill ahead of the 2P½ as noted in Alberto (Alberto, 1998), in the
hard shell ultra-relativistic approximation giving his L-S coupling. Thus the 2P3/2
state(and its sp2 hybrid) for this new electron Dirac equation gives an azimuthal
trifolium, 3 lobe shape, so this ONE charge e (color not needed to
guarantee this) spends 1/3 of its time in each lobe (1/3e fractionally charged
lobes).
2)    Lobe structure is locked into the center of mass (asymptotic freedom), can‟t
leave, just as with the Schrodinger equation spherical harmonics (here as
Dirac half integer spherical harmonics).
3)    There are six 2P states (corresponding to the 6 flavors) (Halzen, quarks); the
P wave scattering produces jets; giving us all the main properties of
quarks! (strong force) from first principles (eq.1). Note that the 2S½ state
solution is twice the energy of the 2P3/2 of equation 1 (Alberto, 1998).
4)   We can identify the 2S½ then with the tauon and the 1S½ with the muon
respectively with the 2P3/2 at r=rH then the baryons.
5) From equ.1 the small electron proper mass and these large baryon 2P3/2 ,
lepton 2S½ state energy eigenvalues imply ultrarelativistic electron motion.
There is straight line ultrarelativistic electron motion for the S state (since no
rotation) so Fitzgerald contraction rH(1-v2/c2) is to Fermi/2000, a “point”. But
2P states (L=1) are rotational and so even for the same degree of
ultrarelativistic motion have the small Fitzgerald contraction=
rHcos(1-cos2)d=½rH=1/2 Fermi, not a point.
This explains why leptons are “point” particles and baryons are not.

6) Frobenius series numerical solution for rrH of equation 1 done in ch.19 of
http://davidmaker.com. For example the J=1 state 2P solution for rrH is mass
eigenvalue 2mP forming the two 2P z direction lobes giving the properties of the
deuteron and thereby deriving the core concept of nuclear physics.

7). Again the rH boundary provides a hard shell potential producing a Van der
Waals equation of state thereby implying the liquid equation of state observed
in 100GeV gold-gold collisions at BNL.
8) The eq.1 energy E=1/√oo =1/√ (1-rH/r) has singularities in it and so
when its associated potential is substituted into the S matrix we find the
W and Z as resonances.
This new pde can be used to derive the weak interaction.
 1  rH / 2r  3 / 8rH / r   ..
1           1
Energy            
2
For r>>rH
 oo            rH
1
r
9) The first term is the mass energy, the second term is the Coulomb
potential (energy) and the third term gives a small addition to the Coulomb
potential that allows us to drop the higher order diagrams and
renormalization in QED. This third term then provides the Lamb shift.
(i.e.,Energy perturbation expectation value of hydrogen atom 2,0,0 state).
The precision of the old renormalization QED is still maintained.

Summary:
Note these are ONE step derivations from eq. 1 of these important
results, not added assumptions as in the mainstream approach.
Note these successes are due to rH not being zero; thereby keeping
this theory generally covariant if nonzero forces are present.
Comparison with Standard Method

The alternative approach is to set rH=0 in equation 1. Setting rH=0 removes
the general covariance when nonzero force is present, which is a mistake
that people have compensated for by fudging in adhoc pathologies such as
many gauges, renormalization and counter-terms, higher dimensions, 19
free parameters that you can adjust any way you want, turning theoretical
physics into a veritable junk pile. The result is a confusion that has
stopped the progress of theoretical physics for the past 30 years.
Conclusion
This new generally covariant generalization of the Dirac
equation gives many results that are otherwise merely
assumptions added to the old Dirac equation theory. It solves
the many left over problems the old Dirac equation cannot
help us solve, gets new valid physics beyond the standard
model and thereby gets theoretical physics moving again.
Backup Slide A

Equation 1 Is Generally Covariant By Construction

New generally covariant (Dirac) PDE
The spherical symmetry background metric coefficient (44) 00=1-rH/r can be inserted into a Dirac equation
by starting with
4     4
ds     dx  dx
2
(2)
 1  1

which is a generally covariant expression. In the spherically symmetric case one can diagonalize to
ds 2  11dx12   22 dx2   33dx3   44 dx4
2         2          2

Define px from px = dx/ds and define  from:
d
p x  i
dx

( h 1, mo=1) , linearize to get the Dirac  matrices and multiply both sides of the resulting equation by /ds and
get
d            d           d             d          or
  11 1         1
  22  2 2   33  3 3  i  44  4 4
dx            dx           dx             dx

(/x)- = 0                                (1)
Note only multiplications, redefining, and at the end, the standard Dirac equation linearization was used
to modify equation 1. Thus we have not compromised the general covariance of equation 2
in deriving equation 1. Thus equation 1 by construction is generally covariant.
Backup Slide B

How Equivalence Principle Can Be Applied to E&M

Recall that the electrostatic force Eq=F=ma so E(q/m)=a. Thus there are
different accelerations „a‟ for different charges „q‟ in an ambient electrostatic
field „E‟. In contrast with gravity there is a single acceleration for two different
masses as Galileo discovered in his tower of Pisa experiment. Thus gravity
(mass) obeys the equivalence principle and so (in the standard result) the
metric formalism gij can apply to gravity.

Single Charge e Approach Allows Use of Metric Coefficients gij
Note that E&M can also obey the equivalence principle but in only one case: if
there is a single e and Dirac particle me in Eq=ma and therefore (to get the
correct geodesics):
oo = goo=1-2e2/rmec2 =1-rH/r
(with rr=1/oo) and so then trivially all charges will have the same acceleration
in the same E field. This then allows us to insert this metric gij formalism into the
standard Dirac equation derivation instead of the usual Minkowski flat space-
time gijs (below). Thus by making E&M obey the equivalence principle you
force it to have ONE nonzero mass with charge.
Thus you force a unified field theory on theoretical physics.
Backup Slide C
Summary of Applications

Fractalness: Note dr2=rrdr‟2 observer (i.e.,dr‟) near rH (most likely position)
also sees a huge selfsimilar universe
dr‟2/(1-rH/r‟)=dr2=dr‟2/(1-rH/rH)=dr‟2/(0)=
and his own high frequency dirac zitterbewegung of equation 1:
(/2)2 =(1/dT)2 =1/dt‟2(1-rH/r‟) = 1/(dt‟2(1-rH/rH)) = 1/0 = .

Solve equation 1 for r>rH, r=rH and r<rH to get the physics results. For
example:
rrH Frobenius series solution for  near rrH gives baryon properties:
as a spherical harmonic 2P3/2 without any of the QCD postulates and free
parameters
r<rH Note fractalness so that observer r<rH in huge universe:
cosmology, radial coordinate expansion due to next higher fractal scale
implies gravity, and eq 1 written between two fractal scales implies the left
handed Dirac doublet core of Standard Model.
r>rH QED             (without the higher order diagrams)

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