Elusive 'Buchdahl stars' are black holes without event horizons. But do they really exist?

Does the virial theorem apply to all massive objects? In other words, why doesn't the Earth, for example, collapse to a smaller volume, due to its self-gravitation? What force balances the gravitational force, to keep it the size that it is?

bolide2 said:

Does the virial theorem apply to all massive objects? In other words, why doesn't the Earth, for example, collapse to a smaller volume, due to its self-gravitation? What force balances the gravitational force, to keep it the size that it is?

Our G2 main sequence yellow dwarf star, a relatively small one that isn't even close to enough to the necessary mass to implode, which requires a nearly exponentially larger star, making our star's size look like earth's next to our star (planets, even massive gas giants are simply microscopic in comparison, with so much less energy they're insignificant versus blue giant.

Once a star has met the size threshold, it's guaranteed to go supernova, with such a vicious energetic explosion it collapses in on itself afterward tearing the very fabric of space and time itself, with a stellar mass singularity being the end product I'm but a very interested layman, I'm not familiar with virial theorem you speak of, but I can positively answer your question that no star, and certainly no planet or smaller stellar object, will ever have enough energetic force to collapse, and go supernova in the manner truly giant stars do (I'm also very confused why you wouldn't even already know this bringing up an astrophysics theorem concerning it). Massive stars are simply hard to even imagine their scale is so ridiculously huge, and as you probably know burn so brightly, putting out so much energy they live in the hundreds of millions, and die extremely young compared to our someday brown dwarf yellow sun, which is already nearing 5 billion years old, and should continue up to 9-10 billion years before it turns into a giant red star, swallowing up the entire inner solar system in the process.
And our final brown dwarf star system, could very well exist until the end of time measuring in the trillions of years I believe, and probably existing till the next Big Bounce, or until space's expansion makes the law of physics no longer feasible, as all lower mass stellar objects will that aren't destroyed by a collision, or consumed by their host star, or are actually lucky enough to cross paths with a stellar mass black hole, or even a galactic sizef super massive black hole, if they're actually within the very inner most orbits of their galaxy.

A Buchdahl star is a highly compact star for which the boundary radius R obeys R=9/4r+, where r+ is the gravitational radius of the star itself.

In 1959, Hans Adolf Buchdahl, a German-Australian physicist, studied the behavior of a highly idealized "star" represented as a perfectly spherical blob of matter, as it is compressed as much as possible. As the blob becomes smaller, its density increases, making its gravitational pull stronger. Using the principles of Einstein's general theory of relativity, Buchdahl determined an absolute lower limit for the size of the blob.

This special radius is calculated as 9/4 times the mass of the blob, multiplied by Newton's gravitational constant, divided by the speed of light squared. The Buchdahl limit is significant as it defines the densest possible object that can exist without ever becoming a black hole.

According to the theory of relativity, any object below this limit must always become a black hole. Naresh Dadhich, a physicist at the Inter-University Centre for Astronomy and Astrophysics in Pune, India, has discovered a new property held by Buchdahl stars. He calls Buchdahl stars "black hole mimics" as their observable properties would be nearly identical.

A quasiblack hole, a Buchdahl star, is a maximum compact star, or more generically a maximum compact object, for which the boundary radius R obeys R=r+. Quasiblack holes are objects on the verge of becoming black holes. Continued gravitational collapse ends in black holes and has to be handled with the Oppenheimer-Snyder formalism. Quasistatic contraction ends in a quasiblack hole and should be treated with appropriate techniques.

Quasiblack holes, not black holes, are the real descendants of Mitchell and Laplace dark stars. Quasiblack holes have many interesting properties. José P. S. Lemos, Oleg B. Zaslavskii develop the concept of a quasiblack hole, give several examples of such an object, define what it is, draw its Carter-Penrose diagram, study its pressure properties, obtain its mass formula, derive the entropy of a nonextremal quasiblack hole, and through an extremal quasiblack hole give a solution to the puzzling entropy of extremal black holes.

A quasiblack hole is an object in which its boundary is situated at a surface called the quasihorizon, defined by its own gravitational radius. Lemos and Zaslavskii elucidate under which conditions a quasiblack hole can form under the presence of matter with nonzero pressure. It is supposed that in the outer region an extremal quasihorizon forms, whereas inside, the quasihorizon can be either nonextremal or extremal. It is shown that in both cases, nonextremal or extremal inside, a well-defined quasiblack hole more always admits a continuous pressure at its own quasihorizon. Both the nonextremal and extremal cases inside can be divided into two situations, one in which there is no electromagnetic field, and the other in which there is an electromagnetic field. The situation with no electromagnetic field requires a negative matter pressure (tension) on the boundary.

On the other hand, the situation with an electromagnetic field demands zero matter pressure on the boundary. So in this situation an electrified quasiblack hole can be obtained by the gradual compactification of a relativistic star with the usual zero pressure boundary condition. For the nonextremal case inside the density necessarily acquires a jump on the boundary, a fact with no harmful consequences whatsoever, whereas for the extremal case the density is continuous at the boundary. For the extremal case inside we also state and prove the proposition that such a quasiblack hole cannot be made from phantom matter at the quasihorizon. The regularity condition for the extremal case, but not for the nonextremal one, can be obtained from the known regularity condition for usual black holes.

In general relativity, a compact object is a body whose radius R is not much larger than its own gravitational radius r+. Compact objects are realized in compact stars. The concept of a compact object within general relativity achieved full form with the work of Buchdahl1 where it was proved on quite general premises that for any nonsingular static and spherically symmetric perfect fluid body configuration of radius R with a Schwarzschild exterior, the radius R of the configuration is bounded by R ≥ 89 r+, with r+ = 2m in this case, m being the spacetime mass, and we use units in which the constant of gravitation and the velocity of light are set equal to one. Objects with R = 89 r+ are called Buchdahl stars, and are highly compact stars. A Schwarzschild star, i.e., what is called the Schwarzschild interior solution,2 with energy density ρ equal to a constant, is a realization of
this bound. Schwarzschild stars can have any relatively large radius R compared to their gravitational radius r+, but when the star has radius R = 9/8 r+, i.e., it is a Buchdahl star, the inner pressure goes to infinity and the solution becomes singular at the center, solutions with smaller radii R being even more singular.

From here, one can infer that when the star becomes a Buchdahl star, i.e., its radius R, by a quasistatic process say, achieves R = 9/8 r+, it surely collapses. A neutron star, of radius of the order R = 3r+, although above the Buchdahl limit, iscertainly a compact star, and its apparent existence in nature to Oppenheimer and others, led Oppenheimer himself and Snyder to deduce that complete gravitational collapse should ensue. By putting some interior matter to collapse, matched to a Schwarzschild exterior, it was found by them that the radius of the star crosses its own gravitational radius and an event horizon forms with radius r+, thus discovering Schwarzschild black holes in particular and the black hole concept in general.

Note that when there is a star r+ is the gravitational radius of the star, whereas in vacuum r+ is the horizon radius of the spacetime, so that when the star collapses, the gravitational radius of the star gives place to the horizon radius of the spacetime. In its full vacuum form, the Schwarzschild solution represents a wormhole, with its two
phases, the expanding white hole and the collapsing black hole phase, connecting two belonging to the Kerr-Newman family, having as particular cases, the Reissner-Nordström solution with mass and electric charge, and the Kerr solution with mass I.e., are there black hole mimickers?

Unquestionably, it is of great interest to conjecture on the existence of maximum compact objects that might obey R = r+. Speculations include gravastars, highly compact boson stars, wormholes, and quasiblack holes. Here we advocate the quasiblack hole. It has two payoffs. First, it shows the behavior of maximum compact objects and second, it allows a different point of view to better understand a black hole, both the outside and the inside stories. To bypass the Buchdahl bound and go up to the stronger limit R ≥ r+, that excludes trapped surfaces within matter, one has to put some form of charge. Then a new world of objects and states opens up, which have R = r+. The charge can be electrical, or angular momentum, or other charge. Indeed, by putting electric charge into the gravitational system, Andr ́easson7 generalized the Buchdahl bound and found that for those systems the bound is R ≥ r+. Thus, systems with R = r+ are indeed possible, see8 for a realization of this bound, and for some physical asymptotically flat universes.

Classically, black holes are well understood from the outside. For their inside, however, it is under debate whether they harbor spacetime singularities or have a regular core. Clearly, the understanding of the black hole inside is an outstanding problem in gravitational theory. Quantifiabally, black holes still pose problems related to the Hawking radiation and entropy. Both are low energy quantum gravity phenomena, whereas the singularity itself, if it exists, is a full quantum gravity problem. Black holes form quite naturally from collapsing matter, and the uniqueness theorems are quite powerful, but a time immemorial question is: Can there be matter objects with radius R obeying R = r+?

Are there black hole mimickers? Unquestionably, it is of great interest to conjecture on the existence of maximum compact objects that might obey R =r+. Speculations include gravastars, highly compact boson stars, wormholes, and quasiblack holes. The quasiblack hole has two payoffs. First, it shows the behavior of maximum compact objects and second, it allows a different point of view to better understand a black hole, both the outside and the inside stories. To bypass the Buchdahl bound and go up to the stronger limit R ≥ r+, that excludes trapped surfaces within matter, one has to put some form of charge. Then a new world of objects and states opens up, which have R = r+. The charge can be electrical, or angular momentum, or other charge. Indeed, by putting electric charge into the gravitational system, Andreasson generalized the Buchdahl bound and found that for those systems the bound is R ≥ r+. Thus, systems with R = r+ are indeed possible and there are other black holes in general relativity,

Scientists are puzzled by a strange object in the cosmos that appears to be a black hole, behaves like a black hole, and may even have similar characteristics to a black hole, but it has a key difference: there is no event horizon, meaning it is possible to escape its gravitational pull if enough effort is made.

This object, known as a Buchdahl star, is the densest object that can exist in the universe without turning into a black hole. Despite its theoretical existence, no one has ever observed one, sparking debate on whether these objects exist. A physicist may have recently discovered a new property of Buchdahl stars that could provide answers.

The existence of black holes is widely accepted by astronomers due to various forms of evidence, such as the detection of gravitational waves during collisions and the distinct shadows they cast on surrounding matter. It is also understood that black holes form from the catastrophic collapse of massive stars at the end of their life, following a Verve Times report.

See: https://www.sciencetimes.com/articles/41868/20230116/stars-exist-forever-buchdahl-wont-turn-black-holes.htm
See the paper:
Quasiblack holes with pressure: General exact results
José P. S. Lemos, Oleg B. Zaslavskii
2010 Physical Review D
See: https://scholar.archive.org/work/uicyzqjgvvd6ppankzh5vdjolq
There is still a lack of understanding of the limit of compression an object can endure before collapsing into a black hole. White dwarfs, containing the sun's mass in the Earth's volume, and neutron stars, which compress even further to the size of a city, are known to exist. But it remains unclear if other smaller objects can actually exist without becoming black holes. In the latter case, Buchdahl stars are offered as quasi-black holes.
Hartmann352