How does the Choice of Boundary Conditions at Infinity Affect Frame Dragging Inside Rotating Shells?.
This paper matches four different classes of axisymmetric solutions to Einstein's Field equations (Weyl and Levi -Civita static solutions, Kerr solution, Papapetrou solutions or Lewis and Van Stockum solutions) across a thin axisymmetric rotating shell to a flat interior metric. No assumptions about the behavior of the gravitational field at infinity are imposed, including the usual assumptions of asymptotic flatness or compactness. The purpose is to see how changing the boundary conditions at infinity affects frame dragging inside the shell. The first case discussed is that of a spherical rigidly rotating shell of uniform density. Excluding the limit of the shell radius equal to the gravitational radius, the only matching exterior solutions finite on the rotation axis are trivial flat and Schwarzschild metrics. The freedom gained by not imposing boundary conditions at infinity is not sufficient to allow other matchings across such a shell to a flat interior metric. Allowing the shell to be any of a class of axisymmetric prolate spheroidal shells with rotation and density dependent on latitude, I find matchings in cases other than the gravitational limit for both the Weyl and Levi-Civita static case and the Papapetrou case. Both contain asymptotically flat solutions as well as solutions with other behavior at infinity. The component of the stress energy tensor for time and angle of rotation as seen in an inertial frame is zero in the Weyl and Levi-Civita static case for all boundary conditions. It may however be nonzero in the Papapetrou case depending upon the conditions at infinity imposed. The interpretation is that changing the boundary conditions in this case changes the frame dragging of the interior inertial frames.
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- Physics: General