By Reinhard Meinel, Marcus Ansorg, Andreas Kleinwachter, Gernot Neugebauer, David Petroff
ISBN-10: 0511415141
ISBN-13: 9780511415142
ISBN-10: 052186383X
ISBN-13: 9780521863834
A worthy reference at the classical challenge of gravitational physics for researchers in most cases relativity, mathematical physics, and astrophysics.More studies and proposals
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Extra resources for Relativistic Figures of Equilibrium
Sample text
29) only. We denote the coordinate radius of the sphere by r0 . 30) eα = , eν = 3 2 2 M 2 1 + 2r0 1 + Mr3 1 + Mr3 2r0 and 2r0 p = µ(eν0 −ν − 1), eν0 ≡ eν(r0 ) . 31) The exterior solution (r > r0 ) is naturally the vacuum Schwarzschild metric given by 1− M M 2 2r eα = 1 + , eν = . 2 M is the gravitational mass. For a given value of the mass-density µ, the solution depends on only one parameter, say ν0 . M and r0 are related to ν0 according to M = 1 4 3 1 − e2ν0 2πµ 3/2 1 − eν0 M = . 35) 3 must hold.
The right solid line in Fig. 5) sweeps over the entire K-surface and puts ‘upper’ K values into the lower sheet and ‘lower’ K values into the ‘upper’ sheet. As a consequence, λ will change from ±1 to ∓1 between = 0, ζ = +∞ and = 0, ζ = −∞. This ‘exchange of sheets’ is important for the solution of the LP: The initial value ( 0 , ζ0 , λ) can (and must) be fixed in only one sheet of the K-surface. 41) along a suitable path. 41) along the closed dotted line of Fig. 4 into two steps: (i) Integrating along A+ CA− .
In other words, a cusp in the equatorial plane necessarily implies that the star is rotating at the massshedding limit. It can easily be verified that the same is true in Einsteinian theory. Moreover, numerical results suggest both in Newtonian and Einsteinian gravity that a cusp is a necessary and sufficient condition for the existence of a mass-shedding limit. The potentials and surface function describing a mass-shedding star are not analytic, which makes a highly accurate description of them particularly challenging.