Dark star Hawking radiation
Dark star Hawking radiation
If we consider Eighteenth-century Newtonian physics, with light treated as "corpuscles", then John Michell's 1983 equation tells us that if a star is dense enough or large enough for its mass-to-radius ratio to equal r=2M, then the surface gravitation at that radius will be such that the escape velocity equals the speed of light. if the star is any denser, then nothign emitted by the star on a simple ballistic trajectory at less than the speed of light will be able to fully escape, and even light itself will be unable to escape "to infinity".
Particles and light will still be able to brifly visit the region outside r=2M, but their ballistic paths will inevitably lead them back towards the r=2M radius ... unless something intervenes.
This may make us think that a distant observer, who cannot be reached directly by any of the star's radiation, will see the star as being perfectly black. However, although a single test particle approach might seem to support this conclusion, in actuality, the statistical behaviour of the rarefied atmosphere of visiting particles outside r=2M is different - since the particles are outside r=2M, they can collide with passing bodies and be knocked free, collide with each other with one paticipant in the collisionbeing knocked freee, or (in the case of an unstable particle) can undrgo radiactive decay and disintegrate into atwo or more daughter particles, one or mor eof which might be able to escape and reach a distant observer.
These "visiting particles" would seem to play the same role as "virtual particles" under quantum mechanics – they can not reach the distant observer directly, but their existence has observable consequences for the distant observer.
Applying SR conventions
If the distant observer applies SR-style definitions for observed causality, they will tend to say that for a particle emitted at or below r=2M the emission event takes place in the more-than-infinite future. Since subsequent signals from the particle (at this position) cannot reach the observer directly, the observer might reason that no consequences of the particle's existence can ever reach them, without involving reverse causality.
However, in the dark star situation, this logic would be wrong - light radiated at r=2M would never reach the distant observer directly along an unaccelerated path, but is a spaceship were to skim the surface of the star, its collision with trapped, "visiting" light-corpuscles would accelerate some of the light away from the star, and the distant observer would see the spaceship to be mysteriously illuminated by light from the star that supposedly "shouldn't exist". In modern QM terminology, the accelerations caused by interactions and momentum exchange with the spaceship's mass would be converting "virtual particles" into "real particles". Although the star's atmosphere would be "virtual" for the distant observer, its particles would be "real" for the inhabitants of the spaceship flying through the region.
The dark star model also produces a similar (or identical?) temperature relationship to quantum mechanics. For an arbitrarily-small "Newtonian black hole", the critical horizon is arbitrarily close to the supposed suppsosed central singularity, the drop-off in gravitational field strength per unit distance is greater, a visiting particle only has to be "bumped" a comparatively short distance in order to escape, and it's comparatively easy for the hole to radiate. The micro-dark-star is arbitrarily "hot", and the comparative thinness of its gravitational "buffer zone" means that the object is liable to blow itself apart unless it has some other effect holding it together (or unless the object is so small that quantum effects or wavelength-radius issues limit radiation). For a "full size" dark star, the concentration of visiting particles is lower, the collisions that lead to particles being accelerated out of the star's gravitational grip are correspondingly rarer, and if we model the star's atmosphere as extending down through the horizon as a perfect gas, for particles to escape even as far as the horizon might require such a sequence or chain of multiple convenient collisions, for particles at that height to be very few and far between – limiting the number of collisions between them that might lift a "potential escapee" particle any higher.
So, like a QM black hole, the indirect radiation temperature of a dark star is lower when the star's critical radius is larger.