Minkowski spacetime is different from Newtonian space and time due
to the way one measures distance, four dimensional versus three. Spacetime
symmetry must be broken. The Minkowski interval is a consequence
of the relationship between time
and space
. For
classical physics, the functional relationship between time and space
must be severed. In the static field approximation, there is a scalar
distance
which is the same magnitude as the interval
.
If the interval
is replaced by the scalar distance
in
the relativistic 4-velocity, then that will sever the functional relationship
between time and space:
![]() |
(30) |
Substitute into weak-field gravitational Lorentz 4-force density equation (23) to create a classical 3-force equation:
![]() |
(31) |
This is not quite Newton's gravitational force density law. The reason
is that one must now consider the right-hand side of the force equation
carefully. According to the chain rule:
![]() |
(32) |
An open question is how should spacetime symmetry be broken for the
derivatives with respect to the interval ? An interval is composed
of both changes in time and space. For the acceleration term,
,
if the interval is only about time, then one gets back Newtonian acceleration,
a second derivative of time. One might be tempted to use time in the
mass distribution in spacetime term,
.
However, the system is presumed to be static, so this would be zero
by presumption. If this derivative is to have any chance at being
non-zero, it would have to change with respect to the scalar distance
as has been done earlier in the derivation. Note that this new
term will not point in a radial direction. Instead, the change in
mass with respect to space points in the direction of the velocity
of that mass. The classical 3-force law would look like so:
For a point source, the change in mass distribution term,
,
will not make a contribution, and one gets Newton's law of gravity.
It is only if the inertial mass is distributed over space as is the
case for galaxies will the new effect term come into play. If the
velocity is constant, then the acceleration is zero. The equation
describes the distribution in space of the inertial mass density
that contributes to the total gravitational source mass
. The
solution to 33 when there is no acceleration
has the inertial mass distribution that decays exponentially. There
is a problem with the rotation profile of thin disk galaxies.[7,8]
Once the maximum velocity is reached, the velocity stays constant
while the mass density declines exponentially. It has been shown that
galaxies should not be stable at all.[14] The new effect
term promises a stable exponential decay of the mass distribution
for large radii with constant velocity, which sounds like a fit. The
new effect has an inverse distance dependence for small accelerations
due to the factor of c required to get the units right, which matches
the MOND proposal that has been successfully applied to explain the
velocity profile of thin disk galaxies.[10]
doug 2005-11-18