The hoop conjecture, proposed by
Kip Thorne in 1972, states that an
imploding object forms a
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
when, and only when, a circular hoop with a specific critical circumference could be placed around the object and rotated about its diameter. In simpler terms, the entirety of the object's mass must be compressed to the point that it resides in a perfect sphere whose radius is equal to that object's
Schwarzschild radius, if this requirement is not met, then a black hole will not be formed. The critical circumference required for the imaginary hoop is given by the following equation listed below.
:
where
:
is the critical circumference;
:
is the object's Schwarzschild radius;
Thorne calculated the effects of
gravitation
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
on objects of different shapes (spheres, and cylinders that are infinite in one direction), and concluded that the object needed to be compressed in all three directions before gravity led to the formation of a black hole. With cylinders, the
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an outside observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive c ...
was formed when the object could fit inside the hoop described above. The mathematics to prove the same for objects of all shapes was too difficult for him at that time, but he formulated his hypothesis as the hoop conjecture.
By
Penrose singularity theorem of 1964 it is known that if there is
trapped null surface (and some other conditions) then a singularity must form, in 1983
Schoen and
Yau proved how much matter must be crammed into a given volume to create a closed trapped surface, sometimes referred as the Schoen-Yau black hole existence theorem
and more recently in 2023 using Gromov's "cube inequality"
some tori inequalities used in the results of 1983 have been generalized to cube ones which are more akin to Thorn's circular hoops.
See also
*
General relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
*
Bounding sphere
*
Black hole stability conjecture
References
*
Thorne, Kip, ''
Black Holes and Time Warps: Einstein's Outrageous Legacy'', W. W. Norton & Company; Reprint edition, January 1, 1995. .
General relativity
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