Natural Science Forum / Physics / Particle Physics / March 2006

> >    See AE's GR1916 Chapter 2, "The Need for an Extension..."
> >       (pg 112 in Dover's Relativity) for more insight.
> >
> > The alternative solution to Kerr's is using the toroidal mass
> > placed in the field to eliminate rotations, and would arrive
> > at the same metric if the EP is applied by replacing the
> > centrifugal inertial effect with a purely gravitation energy
> > density distribution to cause the ellipsoid.
>
> Let's approach this in a slightly different way, by asking about the
> metric symmetries which one assumes precedent to deriving the Kerr
> solution:
>
> It seems to me that in Kerr one is assuming the symmetry of an
> "axisymmetric" and "stationary" metric and then coming upon spin as
> part of the solution to the Einstein field equations. (By Birkhoff,
> if we assume spherical symmetry then all solutions must be static
> and Schwarzschild, therefore, no spin.)
>
> It also seems that one could, alternatively, assume the symmetry
> of an "axisymmetric" (but not spherically symmetric) and "static"
> metric.
>
> In the former case, one starts off allowing the dtdx, dtdy, and /
> or dtdz terms to be non-zero, but requires the dxdy, dxdz and / or
> dydz to be zero.
>
> In the latter case, dxdy, dxdz and / or dydz are allowed to be
> non-zero, but dtdx, dtdy, and / or dtdz must be zero.
>
> For Kerr, with dtdr not= 0 and the mixed spatial terms all = 0,
> in the course of solving Einstein's second order nonlinear
> differential equations in vacuo for this assumed axisymmetric and
> stationary metric, we come across two constants of integration.
>  One of these constants is equated to "2GM" in order to reproduce
> Newton's law in the linear approximation / Schwarzchild's solution
> in the spherically symmetric approximation.  The other constant,
> often denoted "a", shows up in the form of "Ma" in the dtdr term
> of the metric, and is associated with spin angular momentum S=Ma.
>
> Wouldn't the ellipsoid that Ken talks about with no spin be a
> different metric with dtdr=0, but dxdy, dxdz and / or dydz mixing?
>  And, what solutions of this sort are known at this time, if any?
> If there are no solutions known of this sort, have they been ruled
> out, or just not yet solved?  (Just purchased Stephani, Kramer,
> MacCallum, Hoenselaers, and Herlt, "Exact Solutions of Einstein's
> Field Equations," following S. Carlip's suggestion on sci.physics,
> will poke around there to see.)
>
>     Wouldn't a comparison of "axisymmetric stationary" versus
> "axisymmetric static" solutions also be one way of approaching
> the questions in AE's GR1916 Chapter 2?       > > Best, > > Jay.