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Gravitational collapse

    Try to jump so high that you fly right off of the Earth into outer space. What happens? Why don't you get very far? You are essentially trapped on Earth, unless you can find a rocket that can travel at escape velocity away from the Earth.
    The escape velocity can be calculated in Newtonian gravity by using energy conservation of an object of mass m in the gravitational field of a planet of mass M in D space dimensions:

Escape velocity in Newtonian gravity

    The escape velocity for the surface of the Earth is about 11 km/sec. Notice that's only 37 millionths of the speed of light. Under what conditions would the escape velocity from the surface of some planet or star be equal to the speed of light?

Schwarzschild radius from Newtonian gravity

For a planet the mass of the Earth, this distance is only about a centimeter. So if the Earth were less than a centimeter in diameter, the escape velocity at the surface would be greater than the speed of light.
    But thanks to Einstein we learned that when any velocity in a gravitating system approaches the speed of light, the Newtonian theory of gravity has to be put aside for the relativistically invariance theory of Einstein. The relativistic formulation of gravity in General Relativity starts with the Einstein equation relating the curvature of the spacetime geometry to the energy of the matter and radiation in the spacetime

Einstein equation

The solution to the Einstein equations for the spacetime around a planet or star of mass M is called the Schwarzschild metric

Black hole metric

(This is for d=4 spacetime dimensions. Can you guess from the Newtonian limit for D space dimensions what the Schwarzschild metric looks like for d spacetime dimensions?) In units where Newton's constant and the speed of light are both set to unity, the gravitational radius RG can be written

Schwarzschild radius

    Note that an assumption has been made that we are outside the gravitating body in question. If we're outside the body, and the radial size R of the body satisfies R>RG, then we don't need to know about what happens at coordinate r=RG because this metric doesn't apply to r<R.
    If R<RG, we have to face the problem of what happens when r=RG. The metric looks singular there, but actually the spacetime is smooth, so that an observer falling into the body's gravitational pull from r>RG to r<RG won't feel anything special.
    But the problem is: such an observer will never, under any circumstances, not even with the most powerful rocket in the world, ever be able to cross back to r>RG.
    In this case, this gravitating body is called a black hole, and at the coordinate value r=RG, there exists something called a black hole event horizon. The event horizon is the relativistic geometric expression of the escape velocity becoming equal to the speed of light. Once anything, even light, crosses the event horizon, it can never escape back out to r>RG again.
    Black holes can be created by the gravitational collapse of large stars that are at least twice as massive as our Sun. Normally, stars balance the gravitational force with the pressure from the nuclear fusion reactions inside. When a star gets old and burns up all of its hydrogen into helium and then turns the helium into heavier elements like iron and nickel, it can have three fates. The first two fates occur for stars less than about twice the mass of our Sun (and one of them will be our Sun's eventual fate). These two fates both depend on the fermionic repulsion pressure described by quantum mechanics -- two fermions cannot be in the same quantum state at the same time. This means that the two stable destinies for a collapsing star will be:

1. a white dwarf supported by the fermionic repulsion pressure of the electrons in the heavy atoms in the core
2. a neutron star supported by the fermionic repulsion pressure of the neutrons in the nuclei of the heavy atoms in the core

    If the mass of the collapsing star is too large, bigger than twice the mass of our Sun, the fermionic repulsion pressure of either the electrons or the neutrons is not strong enough to prevent the ultimate gravitational collapse into a black hole.
    The estimated age of the Universe is several times the lifespan of an average star. This means there must have been a lot of stars bigger than twice the mass of our Sun that have burned their hydrogen and collapsed since the Universe began. Our Universe ought to contain many black holes, if the model that astrophysicists use to describe their formation is correct. Black holes created by the collapse of individual stars should only be about 2 to 100 times as massive as our Sun.
    Another way that black holes can be created is the gravitational collapse of the center of a large cluster of stars. These types of black holes can be very much more massive than our Sun. There may be one of them in the center of every galaxy, including our galaxy, the Milky Way. The black hole shown above sits in the middle of the galaxy called NGC 7052, surrounded by a bright cloud of dust 3,700 light-years in diameter. The mass of this black hole is about 300 million times the mass of our Sun.

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