Gravitational Field of Shells and Disks in General Relativity

Details Gravitational Field of Shells and Disks in General Relativity Gravitational Field of Shells and Disks in General Relativity

Dec 1970

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1970phrvd...2.2756m&link_type=abstract

Physical Review D, vol. 2, Issue 12, pp. 2756-2761

Physics

39

Scientific paper

The problem of obtaining the gravitational field of static, axially symmetric, thin shells is elucidated. In particular, a clear distinction between global and local frames is made. An algorithm is given for obtaining the fields of disks. There are two significant gravitational potentials λ and φ. The potential λ is straightforwardly determined from the radial stresses by solving a two-dimensional potential problem. This potential is analytic everywhere except on the disk and, together with its stream function z¯, can be used to generate a conformal transformation which brings the equation for φ into the form of Laplace's equation. This potential can then be found by solving a Neumann boundary-value problem. However, the surface in the new coordinate system is not a disk since z¯ is discontinuous across the disk. This is due to the fact that the Cauchy-Riemann equations imply that if the normal derivative of ρ¯ is discontinuous, then the tangential derivative of z¯ will be discontinuous.

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