• Suppose that we have an object of mass M and radius R, i.e.,

• Well, it is clear that the faster the rocket is propelled the higher up it will travel before it falls back to the planet. There will in fact be a critical speed at which the rocket will just escape the gravitational pull of the planet and so will escape to infinity.
• What is this critical velocity, i.e., the escape speed? Well, since Energy = 1/2 m v**2 - G M m / r, as the projectile moves upward, we see that G M m / r decreases since r is getting larger ===> in order to conserve energy, 1/2 m v**2 must get smaller.

  The key is now whether the projectile comes to a halt before it escapes the pull of gravity. Note that

  • if the energy at the surface of the object is Energy = 1/2 m V**2 - G M m / R less than 0 ===> v will go to 0 for some finite value of r and the object will fall back onto the object.
  • if the energy at the surface of the object is Energy greater than 0 ===> that v will never go to 0 and the object will escape to infinity.
• Technically, we say that if the energy greater than 0, then the projectile is unbound and can escape to infinity. If the energy is less than 0, then the projectile is bound and cannot escape to infinity. The critical speed which divides the world between these two possibilities is known as the escape speed for the object.
• The escape speed is given by

  • escape speed = sqrt [ 2 G M / R ]

More generally, since the speed of light is the maximum with which things can propogate through the Universe, this means that nothing can travel fast enough to escape from the surface of such an object.

Event Horizon

• the escape speed is sqrt[ 2 G M / R ]
• according to Newton and Einstein, it doesn't matter how the mass is distributed inside of the radius R as long as the star is spherically symmetric. This means that the surface of the star could be coincident with R or all of the mass could be contained at the very center of a sphere of radius R. It doesn't matter as the gravitational force will be the same for both situations. Hmmmm,

• The surface (where the escape speed is = c) is a boundary which separates the Universe into two parts which are unable to communicate with each other:
  • the inside is completely cut-off from the outside in the sense that an observer could only receive information, there is no way for the observer to communicate with someone on the outside
  • the outside can send in information to the inside but cannot ever hope to hear from someone who lives there

  The imaginary surface which separates the Universe into these regions is referred to as the Event Horizon.

  Note that since the escape speed is sqrt[ 2 G M / R ], we see that c = sqrt[ 2 G M / R ] ===> c**2 = 2 G M / R or that the radius of the event horizon is given by R = 2 G M / c**2.

  This is referred to as the Schwarzschild Radius in honor of Karl Schwarzschild who was the first person to work out the theory. The Scwarzschild radius is an exceedginly small number. The Schwarschild radius for a solar mass black hole is

  • Schwarzschild radius = 3 [M/M(Sun)] kilometers!

  That is, the Sun would have to collapse to a diameter of 6 kilometers in order to become a black hole. Comment -- A 10 M(Sun) black hole has a radius = 3 [10M(Sun)/M(Sun)] kilometers = 3 [10] kilometers = 30 kilometers.

  Some Comments

  Properties of black holes

• Photon Trajectories
• Cosmic Censorship "Theorem"
  • Nature always cloaks singularities inside of event horizons. So, whatever strange physics occurs near singularities cannot reach out back into our Universe.
• Black Holes Have No Hair
  • Black Holes are characterized by only their mass, spin, and electrical charge. This is odd, as it says that it doesn't matter if we make black holes out of elephants or gnats or old socks, it is only the mass, charge, and spin which are important. If I throw neutral material into a black hole, I do not know if I threw in electron-positron pairs, proton-anti-proton pairs, neutrons, neutrinos, ... . That is, we lose information about our Universe as it passes into a black hole. [Commet--if this information manages to sneak out into some other Universe through some quirk, then the other Universe would notice things which occurred randomly and without warning (the notion of cause and effect would break down). This is extremely bad.]

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