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:
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?
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
The solution to the Einstein equations for the spacetime around a planet
or star of mass M is called the Schwarzschild 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 R_{G} can be written
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>R_{G},
then we don't need to know about what happens at coordinate r=R_{G}
because this metric doesn't apply to r<R.
If R<R_{G}, we have to face the problem
of what happens when r=R_{G}. The metric looks singular there,
but actually the spacetime is smooth, so that an observer falling into
the body's gravitational pull from r>R_{G} to r<R_{G}
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>R_{G}.
In this case, this gravitating body is called
a black hole, and at the coordinate
value r=R_{G}, 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>R_{G} 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 lightyears in diameter. The mass of this black hole is about
300 million times the mass of our Sun.
