On the Coupling Constants in the Linear Approximation of Gravity
And its Generalization
William R. Koepke
Kremmling, Colorado
e-mail:w_koepke@yahoo.com
July 26,2002
Abstract
Recent discoveries such as the accelerated expansion of the Universe in conjunction
with the distribution of gravitational lenses, the anomalous velocity curves of galaxies,
and the anomalous acceleration of Pioneer 10 and 11, behests one to an explanation. Here
we find a new solution to Einstein’s field equations, show that because the gravitational
field is not a gauge field, gravity is a continuum at all scales, and derive the conditions to
uphold Galilean equivalence.
Introduction
When deriving the coupling constants in the linear approximation of gravity, I find a
generalization that renders the field equations non-gauge transformable.This property
implies that gravity is a continuum at all scales. Also a new external scalar field or
cosmic potential is derived modifying the general field equations. Upholding the weak
principle of equivalence, an auxiliary equation for the behavior of the cosmic potential is
derived.
The “weak” principle of equivalence or from now on referred to as “Galilean
Equivalence” is that the translational motion of point particles is equivalent between
gravitation and acceleration. I hold this in my derivation. The “strong” principle of
equivalence, that a reference frame is indistinguishable between gravitation and
acceleration, must be rejected because of the ability to detect tidal forces.
General Invariance, that all laws of physics must be invariant, i.e. generally covariant
and constants and anything independent of the state of matter remain unchanged, under
general coordinate transformations, I use in my derivation to show that the cosmic
potential has an absolute character. Spacetime is quasi-absolute.
As far as the properties of mass being intrinsic or extrinsic, I leave unanswered. Mass is
just the proportionality constant between compared accelerations of different objects.
Here we find the general solution to the field equations. In the linear approximation, the
most general that is linear in h  , is of second differential order, and contains T  as
source is of the form
Page 1 of 5
    h   a   h

 a ' (   h       h )
 b      h  b'      h    T 
it is convenient to make a shorthand
h  h
where a, a’, b, b’, and  are constants. If we operate with   on both sides of this
equation we obtain
      h   a     h  a' (     h         h )
 b      h  b'       h     T 
The right side of this equation vanishes because of the general conservation law for the
energy-momentum tensor. We demand for minimal coupling, that the left side vanishes
identically. Therefore the constants should satisfy
 a 'b'  0
1  a'  0
ab0
Hence a’=1, b’=1, and b=-a. And so the field equation reduces to
(1)
    h   a   h  (   h       h )
 a      h       h   T 
The gauge transformation for this equation is
h   h         
1  a 
   
2a  1
As “a” approaches ½ the gauge transformation becomes undefined. So at a=1/2 the
linear approximation is not a gauge field. This implies that the gravitational field is not
quantized. There is not a spin 2 massless boson, a graviton, mediating gravity. Gravity is
a distentional-temporal manifestation of matter-energy. Spacetime is a continuum at all
scales. Gravity is not a “force” but just a four dimensional Reimannian manifold.
Page 2 of 5
The value of the constant “a” has physical significance. If we vary “a”, we come up with
different tensors that describe the same gravitational field. But the tensors are related with
one caveat. If instead of using the tensor h  to describe the gravitational field we use a
new tensor h  where
h   h   C  h
Where C is a constant, then the field equation for
Has exactly the same form as (1) but “a” is replaced by a ;
a  a(1  4C )  2C
If we choose
C
1 a
2  4a
C is indeterminate when a=1/2 and h      is a solution where  is arbitrary.
a=1/2 is justified because of the proposition:
All consistent axiomatic formulations of physical phenomena include phenomena that are
inconsistent with that formulation. Any well-formed formulation will necessarily be
incomplete.
Hence  is an external scalar field, or a cosmic potential, of some form, but is in the
large-scale structure of the Universe, a function of time only to preserve isotropy and
homogeneity. In the solar system being of only a radial function could approximate it.
Returning to the linear approximation with a=1/2;
    h  
1  
  h  (   h       h )
2
1
       h        h   T 
2
Page 3 of 5
g       ( h     )
   (1   )  h
1
g  
1
1
 
g 
1
    

 
In local geodesic coordinates and using general invariance, we substitute;
h 
    (

 [   (
1

1
1
1
     )     ( g      )
 
2


 
g 
1
1
1
      )]
     )      ( g
 


1
1
1
1
1
        )
       ( g      )        ( g 
2





 T
Distributing the partials and collecting;
2 1 
     g 
      g
     g         g 
[ (    g 
 2
1
1
 )     g         g ]
       g 
4
4
1
                    
 T
2
1
               
2
but
 
R
1 
     g 
      g
     g )
(    g 
2
And
1
1
    R   (    g 
     g 
      g
     g        g        g 
 )
R
2
2
So the field equations in local geodesic coordinates are
1
1
1
    R      g         g 
R
2
4
4
2


1
1
  [                                   ]

T
2
2
2
2
Page 4 of 5
Now we must see if the right hand side of the equation is a tensor. The coordinate
transformation from x   to x  gives for non-inertial frames;
1
1
1
g  R     g  g      g
2
4
4
2


1
1

  T   g        g     g       g  g      g  g       
2
2
2
2

R 
for inertial frames g=4, hence;
R 

2
2
With
2 
1
g  R 
2
T 

2
[2     g        g       g  g       ]
16G
c4
To preserve Galilean equivalence;
1
( R    R ) ;  0
2
(Bianchi identity)
(T );  0
for Galilean equivalence. Then that implies
[ g       2     
1 
g g       2 g      ];  0
2
This last condition puts constraints on the behavior of the metric.
Page 5 of 5