1.2 The Puzzle of Black Hole Remnants

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From stars to black holes

Gravity is an attractive force that causes matter to collapse into itself.
Stars do not collapse into a point because they have nuclear fusion that creates heat and pressure which balance the attraction of gravity.
The Sun is in fact not even particularly dense, with an average density only slightly larger than water.
At some point in a star’s lifespan the fuel for fusion gets exhausted, so the heat and pressure cannot be maintained and the star will start collapsing.

Electron degeneracy pressure prevents further gravitational collapse

One major tenet of quantum mechanics is the Pauli exclusion principle, which tells us that certain particles (including electrons) cannot overlap.
Electrons have very low mass, so according to quantum mechanics they have relatively large wavelengths.
As large objects, like stars, collapse under their own gravitational force, the electrons in their atoms eventually get so close together that their wave functions will be forced to overlap should the process of collapse continue. The pressure that resists this overlapping of electron quantum states is called electron degeneracy pressure.
Objects that are stabilized by electron degeneracy pressure are called white dwarfs. White dwarfs have a density of about one metric ton per cubic centimeter.

Neutron degeneracy pressure is the last defense against gravitational collapse

If the mass of the object is too large, electron degeneracy pressure cannot hold off further compression. At this point electrons and protons will get pushed into each other, creating neutrons and releasing anti-neutrinos in a process called electron capture.
These neutrons will also exhibit degeneracy pressure as they get compressed, preventing further collapse and creating a neutron star, which is about 1 billion times more dense than a white dwarf.
Neutrons are more massive than electrons and therefore their wavefunctions are smaller, so they can undergo far more compression before stabilizing the neutron star.

Black holes form when neutron degeneracy pressure cannot resist gravity

When gravity overcomes neutron degeneracy pressure, objects shrink into more and more compact volumes until all the mass disappears into one point with infinite density. This is a black hole.
Every object has an escape velocity, the speed at which a projectile launched from the surface would be able to fly away and never fall back down.
Black holes do not have true surfaces, but they do have a volume around them in which the escape velocity is equal to the speed of light. The edge of this volume is called the event horizon. Once something crosses the event horizon, it can never escape.

The gravitational attraction between two objects can be recast in terms of the potential energy between them

The gravitational potential energy between two masses is described by the following equation: $$PE = -\frac{GMm}{r}$$
If you make the distance $r$ between the two masses go all the way to 0, the gravitational potential energy of each object becomes infinitely negative. Most objects in nature have finite size, so a distance of 0 is impossible. Only black holes allow for infinite closeness.
If we add a test mass in the vicinity of the black black hole, we know from relativity that its intrinsic energy is $E=mc^2$. The closer its distance to the singularity, the greater the negative potential energy between the test mass and the black hole. In fact, at a critical distance from the singularity, the negative potential energy overcomes the test mass’s positive intrinsic energy and its overall energy contribution becomes negative.
According to general relativity, the critical radius where a test mass’s total energy goes from positive to negative is $r = \frac{2GM}{c^2}$ where $M$ is the mass of the black hole. It turns out that this is the same as the event horizon.
This creates a very unusual situation: A test mass inside a black hole’s event horizon contributes net negative energy to the black hole, which decreases the black hole’s mass. Simply put, we add something to the black hole and its mass goes down.
If there are enough objects inside a black hole, the black hole will be massless. These massless objects are called remnants. According to Professor Mathur, remnants pose a problem for physics because, if they exist, they should dominate every physical process.”