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ASTR 3740 Relativity & Cosmology Spring 2007. Problem Set 3.

Astronomy

• What observational evidence do astronomers use to infer the existence of black holes?
• If black holes emit no light, how are astronomers able to detect black holes?

  Physics

• If photons are massless, then is it true that they should be unaffected by gravity, and therefore unaffected by black holes?
• If gravity cannot escape from a black hole, how is it that the event horizon of a black hole manages to settle back to a spheroidal shape after something (a star, say) falls into the black hole, perturbing its horizon?
• Elements are created through fusion only when immense forces produce conditions of high density, pressure, and temperature, such as occur in the cores of stars. Tidal forces near a black hole can be powerful too. Might they be strong enough to fuse atoms?
• Black holes are distinguished by their different masses. What other parameters distinguish black holes?
• Is there a lower limit on the size of a black hole? Explain.
• Consider two black holes, one of mass 30 suns, the other of mass 3 million suns. Compare their statistics: their radii; how long it takes to fall in from horizon to singularity; the tidal force at the horizon; and their Hawking temperatures and luminosities.

  Near and inside black holes

• If you encountered a hypothetical mini-black hole in outer space, what would happen to your hand if you attempted to grab the black hole?
• If you are in the vicinity of a black hole, there is a direction in which you will be able to see the back of your head. Explain how this works.
• What is the photon sphere, and where is it?
• Describe all the things you would see if you watched a person fall through the horizon of a real black hole.
• If A remains stationary outside a black hole, and B falls into the black hole, what do each of A and B see on each other's clocks?
• You watch a person fall through the horizon of a black hole. As they fall through the horizon, they emit light in all directions. Since the only light that escapes from just above the horizon lies within a narrow cone about vertical, you will only see the person if you are within the cone of escaped photons.
• You watch another person fall through the horizon of a black hole. Can you see what happens to them inside the horizon?
• An astronaut is dropped off into a black hole by a spaceship. What does the astronaut see looking back out towards the spaceship?
• How does the tidal force on you vary as you approach a black hole? At what point will you be torn apart?
• What mass of black hole would cause you to blackout (and subsequently be torn apart) as you pass through its horizon? Assume that a tidal force of 10 gees between your head and toes is what it takes for you to blackout.
• Imagine that you survive the tidal forces of the black hole, and have enough time to consider the view. Describe what you experience as you approach the singularity of a Schwarzschild black hole.
• If you fall freely into a black hole, would the light from directly above you appear redshifted or blueshifted?
• Describe what you see just before hitting the singularity of a Schwarzschild black hole.
• A friend leaps into a black hole, and shortly afterwards you leap in after them. If you had stayed outside the black hole you would expect to see your friend freeze at the horizon. But what do you see of your friend if you followed them inside the horizon?
• The mathematical solution for the complete Schwarzschild geometry includes a white hole, two universes, and a black hole. The collapse of a real star to a black hole will not produce such a geometry, but mathematically such a geometry exists. Is contact between the two different universes possible through the Schwarzschild wormhole?

  Hawking radiation

• The Sun is a main sequence star, the most common type of star. For main sequence stars, more massive stars are both hotter and more luminous, and their main sequence lifetimes are shorter. Is it likewise true that more massive black holes are hotter and more luminous in Hawking radiation, and that they evaporate in shorter times?
• A black hole of 10¹¹kg (about the mass of a mountain) will evaporate by Hawking radiation in about the age of the Universe. Since more massive black holes emit more Hawking radiation, a 30 solar mass black hole will evaporate in much less than the age of the Universe. True or false?
• Is it possible to harness Hawking radiation as an energy source?
• An observer watches a person fall through the horizon of a black hole. The observer sees the person freeze at the horizon. If the black hole does not destroy information, then the observer sees the person eventually become encoded into thermal Hawking radiation that the black hole emits. On the other hand, from their own point of view, the person who falls in passes through the horizon without being destroyed. How can it be that the outside observer sees the person destroyed at the horizon, yet the person falling in does not experience being destroyed?