New form for GR's gravitational red shift & a novel interpretationof the underlying physics

  1. [Moderator's note: the message is posted because it had to be
    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.]

    - - - - - - - - - - - - - - - - - - - - - - - - -
    << 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 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.


    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?

    - - - - - - - - - - - - - - - - - - - - - - - - -
    << 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)
    -----------------------------------------------------------------------
     
  2. jcsd
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