The task is to find a physically relevant solution to the GEM field
equations, 5. The Poisson field equation of classical
Newtonian gravity can be solved by an inverse distance potential,
. The potential has a point singularity where
. The GEM
field equations are relativistic, so time must be incorporated. An
inverse distance potential does not solve the field equations in four
dimensions. The potential
solves the field equations, where
is the Lorentz invariant
distance squared, the negative of the Lorentz invariant interval squared,
. Distance is used instead of the interval because
classical gravity depends on distance, not time. The potential has
as a singularity that is the entire lightcone, where
.
The potential is not relevant in the classical domain since its derivative
will not be an inverse square as required for a classical gravitational
force.
Gravity is a weak effect. It is common in quantum mechanics to normalize a potential and study linear perturbations of weak fields, an approach that will be followed here. Assume spherical symmetry. Form a normalized potential with a linear perturbation:
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(17) |
Take the covariant derivative with respect to and
,
keeping only terms to first order in the spring constant
:
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(18) |
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The change in the potential is a function of a spring constant
over sigma squared. The classical Newtonian dependence on distance
is an inverse square, so this is promising. A potential that applies
exclusively to gravity is sought, yet the non-zero gradient of
indicates an electric field. The sign of the spring constant
does not effect the solution to the four dimensional wave field equations
but does change the derivative of the potential. A potential that
only has derivatives along the diagonal of the field strength tensor
can be constructed from two potentials
that differ by spring constants that either constructively interfere
to create non-zero derivatives, or destructively interfere to eliminate
derivatives. With this in mind, construct a potential that will have
no electric field:
Take the covariant derivative of this potential, keeping only the
terms to first order in the spring constant .
doug 2005-11-18