An unusual Klein-Gordon form for the static gravitational red shift

I am posting an unusual version of the standard relation for the
gravitational red shift (GRS) for photons traveling out of a grav-
itational well, predicted in general relativity. This new relation
involves an unusual use of the Klein-Gordon (K-G) equation,
sans psi notation, and gives results which are exactly identical
to those yielded by the fully relativistic GRS equation in standard
texts. The key assumptions enabling the discovery of this K-G
relation stem from a larger toy model showing possible connec-
tions between black holes, baby universes and dark matter. De-
tails of that model were posted previously on this newsgroup.
The proof of equivalence for the new relation can be found in
Part II below.

A Caveat:
I emphasize here that my K-G approach is NOT meant as an
alternative to Einstein's well confirmed general relativity or his
gravitational red shift result in particular. Indeed, his standard
result can be transformed algebraically (though tediously!) to
obtain this new K-G approach, and vice versa, without loss of
information. My equivalent rendering simply opens a window
on an unexpected and self-consistent reinterpretation of some
of the basic physics of objects in a gravitational field. The fact
that the well known standard GRS equation can be converted
into a relation explicitly showing specific variations of both the
particle's 'rest' mass and the Planck unit of angular momentum
with the gravitational potential gives support to a larger model
incorporating these specific changes. But whether this also
indicates the reality of the model, only time and observations
will tell.

This new viewpoint flows from considerations of a larger toy
model, not described here, where both particle rest mass and
h-bar increase directly with gravitational potential, with important
consequences for the uncertainty principle. However, the energy
content of a particle at rest will DECREASE with any increase
of the local gravity-potential - - that energy content going increas-
ingly over into the gravitational field itself, until at the surface of
a
black hole's event horizon, the particle's own energy is effectively
zero. This is compatible with, and is just one more way of stating,
the black holes "have no hair" theorems.

Also, for reasons given below, the gravitational fine structure 'con-
stant' (also called the gravitational coupling constant) should ap-
proach unity near the event horizon of a black hole.

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Part I

Below I will show a self consistent way of recasting the stan-
dard gravitational red shift (GRS) of general relativity which
gives identically the same results as GR:

[A]
Assume that a photon, characteristic of a specific atomic or nuclear
transition Q, (eg: the iron-57 gamma ray transition used by Pound
and Rebka) is moving upward in the gravitational well of a mass
M. The photon will experience a GRS in its wavelength, as cor-
rectly predicted by general relativity. The fully relativistic
relation
giving the fractional change in the energy of that photon, having
been emitted at distance Ra from the center of the mass and
traveling upward to a receiver at a greater distance Rb, is

(1)
(1 - Rs/Ra)^.5 - (1 - Rs/Rb)^.5
= -------------------------------------- =
(1 - Rs/Ra)^.5

(1 - Rs/Rb)^.5
1 - ------------------- .
(1 - Rs/Ra)^.5

Below I will show that

Eb
1 - ------ = (delta E')/E' =
Ea

(1 - Rs/Rb)^.5
1 - ------------------- ,
(1 - Rs/Ra)^.5

where Ea is the photon's energy measured at Rb, and Eb is the
expected energy of an identical transition Q photon if it were both
emitted and then measured at Rb. Rs is the mass's Schwarzschild
radius, = 2GM / Co^2 . For E' , see section [b] below.

It is important to note here that (A1) specifically describes the
static
situation with reference to the gravitational field, where both the
emitter and the detector/absorber are at rest, say at different
elevations on the surface of the earth. A different equation would
be required if either or both were in motion, eg: free-falling from in-
finity. From the fact that (A1) describes the static case by
definition,
it follows logically that its transformation into an algebraically
inden-
tical K-G form must also describe that same static case, as unlikely
as that must seem at first. This is central to its physical
interpretation.

Following is what I will show to be an exactly equivalent relation,
which gives predictions completely identical to those of (1). How-
ever, the nature of the second relation's variables seems to allow
an unusual interpretation of the physics involved, and suggests
that certain fundamental parameters may vary with gravitational
potential with no apparent contradiction implied for local physics.
This note will only touch on some of this.

The second relation (3 below) incorporates the Klein-Gordon eq.
for a particle's relativistic energy E where generally

(2)
E^2 = (mCo^2)^2 + (pCo)^2 ,

and p = relativistic momentum of a particle with rest mass m.
Co is velocity of light in field-free space.

[b]
Using (2) :
I consider the total K-G energy E' of each of two identical particles,
eg: two hydrogen atoms, at rest at two different elevations in a
gravitational field potential phi, E'a is associated with particle 'a'
at
lower elevation Ra, and E'b is associated with particle 'b' at higher
elevation Rb. Each particle is stationary and in an inertial frame
with respect to the gravitational field. The specific K-G equations
for each particle 'a' and 'b' are, respectively

(E'a)^2 = (ma*Ca^2)^2 + (Ma*va*Ca)^2 ,

and

(E'b)^2 = (mb*Cb^2)^2 + (Mb*vb*Cb)^2 .

(3)
E'a - E'b E'b
--------- = 1 - ------ = delta E'/E' =
E'a E'a

[ (mb*Cb^2)^2 + (Mb*vb*Cb)^2 ]^.5
1 - ----------------------------------------------
[ (ma*Ca^2)^2 + (Ma*va*Ca)^2 ]^.5

Ca and Cb are the local velocities of light, Co, due to the action
of M's gravitational field at Ra and Rb, respectively where

(4)
Ca,b = Co[ 1 - 2GM/(Co^2*Ra,b) , from general relativity,

and ma and mb are the 'rest' or 'invariant' masses of the two
identical particles which will be seen to differ as a function of the
magnitude of the field at Ra and Rb as required by the following
condition implicit in (B3):

(5)
ma*Ca = mb*Cb .

[Condition (B5) is a core assumption carried over from the larger
cosmological model-in-progress. It is in part this assumption
which makes (B3) work at all and equal to the standard GRS eq.]

When the two particles are at the same elevation R, Ca = Cb and
their masses are then equal by identity.

(6)
Ma,b = ma,b / [1 - 2GM / (Co^2*Ra,b) ]^.5 ,

and
(7)
va,b = [2GM / (Co^2*Ra,b) ]^.5 * Ca,b

and here p = (Ma,b) * (va,b) , effectively a form of relativistic pot-
ential-momentum, purely as a function of the local g-field. Thus,
even though the particles in question are AT REST and in a
static situation, the local gravitational field has a relativistic
effect
on their masses -as if- they were in motion with velocity va,b. I
conjecture that the local potential may set up something like a
'virtual velocity' acting on their masses, as contrasted with a
scenario where they have real, kinematic velocities. By 'virtual'
I am suggesting its use in the sense of quantum field theory's
virtual processes paradigm. Whether this might reflect a point
of contact between GR and QFT, I cannot guess.

[C]
Rel. (3) turns out to give identically the same prediction as (1),
and indeed both are shown below to be algebraically identical,
and seem to reflect equally valid but different ways of looking
at the same phenomenon. The physical interpretation of (3) is
of course open, but I propose the following:

When mass particle 'a' is at rest at Ra in the gravitational field of
M, its rest mass is greater than that of identical particle 'b' at rest
at Rb ( Ra), by a factor Cb / Ca, or, equivalently, a factor of

(8)
1 - (2GM / Co^2*Rb)
--------------------------- .
1 - (2GM / Co^2*Ra)

On the other hand, the total rest mass energy Eb (=mb*(Cb)^2)
of particle 'b' is -greater- than that of particle 'a' by the same
factor.
This means that when particle 'a' emits photon 'a' associated with
a characteristic emission line Q, photon 'a' has proportionately
LESS energy than an equivalent transition Q photon 'b', emitted
and measured entirely at Rb higher up in the gravitational well.
This also means that the photon itself can be seen as having
-constant- energy throughout its trajectory. This is of course quite
different from the usual interpretation whereby the photon loses
an amount of energy, equal to E'b - E'a, to the gravitational field
along the way from Ra to Rb.

As an example, this would mean that a Lyman-alpha photon
emitted by a hydrogen atom at Ra could be seen as having in-
trinsically less energy than a Lyman-alpha photon emitted by
an identical hydrogen atom higher up at Rb. This leads immed-
iately to a further result...Since we know already from experi-
ments (eg: Pound, Rebka, Snider) that photon 'a's wavelength
lambda 'a' is larger than that of photon 'b' by the same factor
given by (8), we can also say that since generally

lambda = h / mC^2 = h C / E , for the photon,

then Planck's constant h at Ra is actually

(9)
h_a = (Ea * lambda 'a') / Ca =

1 - (2GM / Co^2*Rb)
-------------------------- * h_b .
1 - (2GM / Co^2*Ra)

That is, the value of Planck's constant would then vary directly
with the local g-field. It needs to be emphasized that, with one
exception, none of these dimensional parameter variations with
gravity can be observed locally in a lab, even in principle, since
all of the measuring apparatus at any level in a g-field is also
changed commensurately (local measuring rods, clocks, etc.),
along with the quantity being measured. Thus the simultan-
eously changed apparatus will be blind to the changes in val-
ues of fundamental parameters. The key exception is the pho-
ton since, from this new viewpoint, both its energy E and wave-
length lambda are truly constant and are preserved over its path
as long as it interacts only with the gravitational field. A point
needing further exploration: near the event horizon of a black
hole, the hugely increased value of Planck's 'constant' would
greatly enhance the local importance of the Heisenberg un-
certainty principle, where

(delta X) (delta momentum_x) = or ihbar .

Also, with the plausible argument that G is truly constant, the
local value of the gravitational fine structure 'constant', often
called the gravitational coupling constant

GFS = (GMp^2) / hbarCo

for a particle Mp should _increase_ to somewhere near unity at
the event horizon. This would result in near equality in the magni-
tudes of the electromagnetic and gravitational interactions at the
event horizon. What interesting effects might we expect from such
changes on all local classical and quantum physics?

- - - - - - - - - - - - - - - - - - - - - - - - -
Part II

Demonstration of the formal equivalence of (1) and (3) :

From

Ea-Eb (1 - Rs/Rb)^.5
------ = 1 - ------------------
Ea (1 - Rs/Ra)^.5

and Rs = 2GM/Co^2 , we have

(10)
Rs/Ra = 2GM/(Ra*Co^2) and let this equal A. Similarly, Let Rs/Rb
= 2GM/(Rb*Co^2) and let this equal B. Thus (1) is

(11)
(1 - Rs/Rb)^.5 (1 - B)^.5
1 - --------------- = 1 - ---------- .
(1 - Rs/Ra)^.5 (1 - A)^.5

Adding -1, then multiplying by -1 and squaring gives

(1 - B) (1 - A)^2 (1 - B)^3
------- = ------------ x ----------- =
(1 - A) (1 - B)^2 (1 - A)^3

(1 - A)^2 (1 -3B + 3B^2 - B^3)
---------------------------------------- .
(1 - B)^2 (1 -3A + 3A^2 - A^3)

And since "-3B" = - 4B + B, and "3B^2" = +6B^2 - 3B^2 ,
and "-B^3" = -4B^3 + 3B^3 (& similarly for A), by substituting
we get

(1 - A)^2 (1-4B+6B^2-4B^3+B^4+B-3B^2+3B^3-B^4)
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