Do rotating dust stars exist in general relativity?

Abstract

If rotating dust stars would exist in general relativity, they would represent examples of an improbable complete balance between the attractive quasi-Newtonian force (gravitoelectricity) and the repulsive gravitomagnetism. However, nonexistence proofs are available hitherto only for some dust "stars" extending to infinity, and for isolated dust stars of a very restricted class. By analyzing the lines of constant generalized Newtonian potential U in the interior and exterior of a large class of (hypothetical) stationary and axisymmetrically rotating dust stars in general relativity, we find that the existence of such stars can be disproved as soon as minima of the potential U in the exterior vacuum region can be excluded. We present some ideas how this minimum-problem could be attacked, and we summarize the present knowledge about Newtonian and Einsteinian rotating dust systems.

Source Integrals of Asymptotic Multipole Moments

“…This contradicts the presence of a dust distribution with positive mass density. Of course, this result is already known and more general non-existence results for dust including the rotating case can be found in [4,9] and references therein. Although the non-existence is proved here, this example shows in a concise way how the source integrals can be applied in more difficult physical situations like rotating stars.…”

Source integrals of multipole moments for static space–times

“…(iii) In [11] the author uses source integrals to recover easily an old result concerning non-existence results for dust configurations [26]. It is shown that static, axially symmetric and isolated dust configurations do not exist in general relativity.…”

“…The result in (68) is a consequence of the orthogonality relation (26) whereas for the integral (69) the inequality…”

A new and quite general existence proof for static and spherically symmetric perfect fluid stars in general relativity

“…This enforces p c to be strictly positive, because otherwise p(r) had to be identically zero. However, a static star of dust cannot exist in general relativity (see [40], and the literature quoted there). Equally, the constant c := (p c ) has to be strictly positive, because otherwise the non-increasing function (p(r)) had to be identically zero: no real star.…”

“…According to item (7) of section 3 we have d 2 (r) d 0 (r) = 0. If we could also prove the relation d 2 (r) d 1 (r), and therefore d 2 (r) d 1 (r) for all r, the extension of the inequalities (39)- (40) to the third iteration step would proceed in a similarly simple way as for the first two steps. However, hitherto we have not been able to prove the relation d 2 (r) d 1 (r), the main obstacle being the partly negative factor (3 − 4s/r) in front of the function p 1 (s) in equation (50).…”