Re: Questions on the Kerr metric cont'n

Ken S. Tucker wrote:

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.

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