New form for GR's gravitational red shift & a novel interpretation of the underlying physics
star...@sbcglobal.net
an enormous work to write it. LM]
I am posting an unusual version of the standard relation for
the gravitational red shift embodied in general relativity. This
new version uses the Klein-Gordon equation, sans psi notation,
and gives results which are exactly identical to those yielded
by the fully relativistic GRS equation in standard texts. The
proof of equivalence can be found in Part II below.
I emphasize here that my K-G approach is NOT 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.
This new viewpoint both supports and flows from considerations
of a larger toy model where 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_ DECREASES with increase in g-potential- - that
energy content going increasingly over into the gravitational field
itself, until at a black hole's event horizon the particle's own energy
is effectively zero. This is one more way of saying black holes
"have no hair". Also, for reasons given below, the gravitational
fine structure 'constant' should approach 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 / nuclear
transition Q, 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 r1 from the center of the mass and
traveling to a receiver at a greater distance r2, is
(1)
(1 - Rs/r1)^.5 - (1 - Rs/r2)^.5
= -------------------------------------- =
(1 - Rs/r1)^.5
(1 - Rs/r2)^.5
1 - ---------------
(1 - Rs/r1)^.5
Below I will show that
E2
1 - ------ = delta E'/E' =
E1
(1 - Rs/r2)^.5
1 - ---------------
(1 - Rs/r1)^.5
where E1 is the photon's energy measured at r2, and E2 is the
expected energy of an identical transition Q photon if it were both
emitted and then measured at r2. Rs is the mass's Schwarzschild
radius, = 2GM / Co^2 . For E' , see [B] below.
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 par-
ticles, eg: two hydrogen atoms, <at rest> at two different eleva-
tions in a gravitational field potential phi, E'1 associated with
particle 1 at lower elevation r1, and E'2 with particle 2 at higher
elevation r2. Each particle is in an inertial frame with respect
to the gravitational field.
(3)
E'1 - E'2 E'2
--------- = 1 - ------ = delta E'/E' =
E'1 E'1
[ (m2*C2^2)^2 + (M2v2C2)^2 ]^.5
1 - ----------------------------------------------
[ (m1*C1^2)^2 + (M1v1C1)^2 ]^.5
C1 and C2 are the local velocities of light, < Co, due to the
action of M's gravitational field at r1 and r2, where
(4)
C1,2 = Co[ 1 - 2GM/(Co^2*r1,2) , from general relativity,
and m1 and m2 are the 'rest'- or 'invariant'- masses of the two
particles, which will be seen to differ as a function of the
magnitude of the field at r1 and r2. Also
(5)
m1C1 = m2C2 ,
(6)
M1,2 = m1,2 / [1 - 2GM / (Co^2*r1,2) ]^.5 ,
and
(7)
v1,2 = [2GM / (Co^2*r1,2) ]^.5 * C1,2
and p = (M1,2) * (v1,2) , effectively a form of relativistic potential-
momentum, purely as a function of the local g-field.
[C]
Rel. (3) turns out to give identically the same prediction as
(1), and indeed both are shown below to be algebraically
identical (see Part II below), and seem to reflect different ways
of looking at the same phenomenon. The physical interpre-
tation of (3) is of course open, but I propose the following:
When mass particle 1 is at rest at r1 in the gravitational field of
M, its rest mass is greater than that of identical particle 2 at
rest at r2 (> r1), by a factor C2 / C1, or equivalently a factor of
(8)
1 - (2GM / Co^2*r2)
--------------------------- .
1 - (2GM / Co^2*r1)
On the other hand, the total rest mass energy E2 (=m2(C2)^2)
of particle 2 is -greater- than that of particle 1 by the same factor.
This means that when particle 1 emits photon 1 for a character-
istic emission line Q, photon 1 has proportionately LESS energy
than an equivalent transition Q photon 2, emitted and measured
entirely at r2 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'2 - E'1 to the gravitational field along the way
from r1 to r2.
As an example, this would mean that a Lyman-alpha photon
emitted by a hydrogen atom at r1 could be seen as having in-
trinsically less energy than a Lyman-alpha photon emitted by
an identical hydrogen atom higher up at r2. This leads immed-
iately to a further result...Since we know already from experi-
ments (eg: Pound, Rebka, Snider) that photon 1's wavelength
lambda1 is larger than that of photon 2 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 r1 is actually
(9)
h1 = (E1 lambda1) / C1 =
1 - (2GM / Co^2*r2)
-------------------------- * h2 .
1 - (2GM / Co^2*r1)
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'
GFS = hbar*C / (GMp^2)
for a particle Mp should _decline_ to somewhere near unity at
the event horizon. What interesting effects might we expect
from these changes on all local classical and quantum physics?
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<< Part II >>
Demonstration of the formal equivalence of (1) and (3) :
From
E1-E2 (1 - Rs/r2)^.5
------ = 1 - ------------------
E1 (1 - Rs/r1)^.5
and Rs = 2GM/Co^2 , we have
(10)
Rs/r1 = 2GM/r1Co^2 and let this equal A. Similarly, Let Rs/r2
= 2GM/r2Co^2 and let this equal B. Thus (1) is
(11)
(1 - Rs/r2)^.5 (1 - B)^.5
1 - --------------- = 1 - ---------- .
(1 - Rs/r1)^.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)
-----------------------------------------------------------------------
.
(1 - B)^2 (1-4A+6A^2-4A^3+A^4+A-3A^2+3A^3-A^4)
This simplifies so that we have
(1 - A)^2 [(1- B)^4 + B(1 - B)^3] (1 - B)
----------------------------------------- = -------- .
(1 - B)^2 [(1- A)^4 + A(1 - A)^3] (1 - A)
Taking the square root of both sides gives
(1 - A) [(1- B)^4 + B(1 - B)^3]^.5 (1 - B)^.5
------------------------------------------ = ----------- .
(1 - B) [(1- A)^4 + A(1 - A)^3]^.5 (1 - A)^.5
We can now recover the original eq. (1) in its (A,B) form by
multiplying both sides by -1, then adding 1:
(1 - B)^.5
1 - ------------- =
(1 - A).^5
(12)
(1 - A) [(1- B)^4 + B(1 - B)^3]^.5
= 1 - ------------------------------------------ .
(1 - B) [(1- A)^4 + A(1 - A)^3]^.5
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
Now it needs to be shown that (3) can be cast into a form in (A,B)
identical to (12), and thus being equal to (1)....
>From (4) and (5) we get
(13)
(m1*C1^2)^2 = {m1*[Co(1 - A)]^2}^2 =
m1^2*Co^4[1 - A]^4 ,
which is the first term of the denominator inside the radical of
(3), recast in (A,B) form.
>From (4), (6) and (7) we get
(14)
(M1*v1*C1) =
m1
---------- x [A^.5 Co (1 - A)] x Co*(1 - A) .
(1 - A)^.5
Squaring and simplifying gives
(15)
(M1*v1*C1)^2 = M1^2*Co^4*[A(1 - A)^3] ,
which is the second term of the denominator inside the rad-
ical of (3), in (A,B) form. (3)'s numerator can be recast ident-
ically, using B, so that (E'1-E'2)/E'1 , which equals
(16)
[(m2*C2^2)^2+(M2v2C2)^2]^.5
1 - ------------------------------------------ ,
[(m1*C1^2)^2+(M1v1C1)^2]^.5
is now
(17)
m2*Co^2[(1 - B)^4 + B(1 - B)^3]^.5
1 - ---------------------------------------------- .
m1*Co^2[(1 - A)^4 + A(1 - A)^3]^.5
Co^2 cancels out, and from (4) and (5) we know that
(18)
m2 (1 - A)
---- = ------- .
m1 (1 - B)
This gives (E'1-E'2)/E'1 a form in A,B equal to
(19)
(1 - A) [(1- B)^4 + B(1 - B)^3]^.5
= 1 - ------------------------------------------ .
(1 - B) [(1- A)^4 + A(1 - A)^3]^.5
But this is identical to (12), the (A,B) equivalent for (1), which
is general relativity's relation for the fully relativistic GRS.
Thus (1) is algebraically exactly equivalent to (3). The fact
that (1), the standard form of the GRS, turns out to have an
equivalent, though unusually formatted form (3), may be
understandably dismissed as simply the result of an incon-
sequential algebraic re-juggling of the original terms in (1),
having no useful physical significance or interpretation at
all. Yet (3) explicitly contains the Klein-Gordon eq., which
historically was found in the course of searching for a rela-
tion correctly combining the quantum mechanics of a par-
ticle and the requirements of special relativity. Does this
hint that (1) and (3)'s equivalence might mean something
more?
[Of course the above might be more persuasive if an equiva-
lence were found between (1) and the Dirac eq., which is
usually the tool of choice for finessing a particle's quantum
mechanics with special relativity.]
Cheers