It's believed that a sufficiently quickly rotating planet-sized mass could be stable in a toroidal planet formation (though vanishingly unlikely to form naturally). However, assuming no cosmological constant, toroidal black holes can't exist long enough for light to traverse the centre. Does anyone have a good explanation of what, specifically, causes this difference? I'm curious as to where relativistic effects change the stability -- could a toroidal neutron star exist? Or, say that we have some mystery fluid material that can resist arbitrary gravitational pressure, but deforms to hydrostatic equilibrium. Could a toroid of this material exist when its minor circumference describes a photon orbit (sort of a photon "sphere")?
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... could a toroidal neutron star exist?
Yes, if we take exist to mean that there are toroidal solutions for a realistic equation of state for neutron matter [$1$].
No, in the sense that it is extremely unlikely that any realistic star would pass through the “toroidal neutron star” phase in the course of its normal evolution.
... say that we have some mystery fluid material that can resist arbitrary gravitational pressure ...
Actually, there is no need for some exotic equation of state, toroidal stars can exist for a large classes of matter EoS, rotation by itself is enough to prevent collapse. Such objects can exhibit continuous transition to the extreme Kerr black hole. A solution close to this limit has all the features of outer extreme Kerr geometry: ergosphere, photon surfaces etc. but sourced by a relativistically rotating ring.
Besides toroidal stars there are other interesting relativistic figures of rotation: multiple concentric rings, ring around a central spheroidal body, thin disk etc. Those solutions all have the same extreme Kerr black hole limit. As that limit is approached the “inner world” in essence is decoupled from the outer extreme Kerr metric. For more details see the monograph [$2$].
... where relativistic effects change the stability
One purely relativistic effect is an instability against non-axisymmetric perturbations that leads to angular momentum loss via gravitational radiation. This instability is present for all rotating perfect fluid equilibrium configurations.
Ansorg, M., Kleinwächter, A., & Meinel, R. (2002). Relativistic Dyson rings and their black hole limit. The Astrophysical Journal, 582(2), L87, doi:10.1086/367632.
Meinel, R., Ansorg, M., Kleinwächter, A., Neugebauer, G., & Petroff, D. (2008). Relativistic figures of equilibrium. Cambridge University Press, ISBN: 9780521863834, CUP.
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