3 Black Holes3.1 Concept Questions1. What evidence do astronomers currently accept as indicating the presence of a blackhole in a system?2. Why can astronomers measure the masses of sup ermassive black holes only in relativelynearby galaxies?3. To what extent (with what accuracy) are real black holes in our Universe described bythe no-hair theorem?4. Does the no-hair theorem apply inside a black hole?5. Black holes lose their hair on a light-crossing time. How long is a light-crossing timefor a typical stellar-sized or supermassive astronomical black hole?6. Relativists say that the metric is gµν, but they also say that the metric is ds2=gµνdxµdxν. How can bo t h statements b e correct?7. The Schwarzschild geometry is said to describe the geometry of spacetime outside thesurface of the Sun or Earth. But the Schwarzschild geometry supposedly describesnon-rotating masses, whereas the Sun and Earth are rotating. If you collapsed the Sunor Earth to a black hole conserving their mass M and angular momentum L, whatwould the spin parameter a = L/M relative to the maximum possible for a Kerr blackhole?8. What happens at the horizon of a black hole?9. As cold matter becomes denser, it goes through the stages of being solid/liquid likea planet, then electron degenerate like a white dwarf, then neutron degenerate like aneutron star, then finally it collapses to a black hole. Why could there not be a denserstate of matter, denser than a neutron star, that brings a star to rest inside its horizon?10. How can an observer determine whether they are “at rest” in the Schwarzschild geom-etry?11. An observer outside the horizon of a black hole never sees anything pass through thehorizon, even to the end of the Universe. Does the black hole then ever actuallycollapse, if no one ever sees it do so?12. If nothing can ever get out of a black hole, how does its gravity get out?13. Why did Einstein believe that black holes could not exist in nature?14. In what sense is a rotating black hole “stationary”?115. What is a white hole? Do they exist?16. Could the expanding Universe be a white hole?17. Could the Universe be the interior of a black hole?18. You know the Schwarzschild metric for a black hole. What is the corresponding metricfor a white hole?19. What is the b est kind of black hole to fall into if you want to avoid being tidally tornapart?20. Why do astronomers o ften assume that the inner edge of an accretion disk around ablack hole occurs at the innermost stable orbit?21. A collapsing star of uniform density has the geometry of a collapsing Friedmann-Robertson-Walker cosmology. If a spatially flat FRW cosmology corresponds to astar that starts from zero velocity at infinity, then to what do open or closed FRWcosmologies corresp ond?22. Is the singularity of a Reissner-Nordstr¨om black hole gravitationally attractive or re-pulsive?23. If you are a charged particle, which dominates near the singularity of the Reissner-Nordstr¨om geometry, the electrical attraction/repulsion or the gravitational attrac-tion/repulsion?24. Is a white hole gravitationally attractive or repulsive?25. What happens if you fall into a white hole?26. Which way does time go in Parallel Universes in the Reissner-Nordstr¨om geometry?27. What does it mean that geodesics inside a black hole can have negative energy?28. Can geodesics have negative energy outside a black hole? How about inside the ergo -sphere?29. Physically, what causes mass inflation?30. Is mass inflation likely to occur inside real astronomical black holes?31. What happens at the X point, where the ingoing and outgoing inner horizons of theReissner-Nordstr¨om geometry intersect?32. Can a particle like an electron or proton, whose charge far exceeds its mass (in Planckunits), be modeled as Reissner-Nordstr¨om black hole?233. Does it makes sense that a person might be at rest in the Kerr-Newman geometry?How would the Boyer-Linquist coordinates of such a person vary along their worldline?34. In identifying M as the mass and a the a ngular momentum per unit mass of the blackhole in the Boyer-Linquist metric, why is it sufficient to consider the behavior of themetric at r → ∞?35. Does space move faster than light inside the ergosphere?36. If space moves faster than light inside the ergo sphere, why is the outer boundary ofthe ergosphere not a horizon?37. Do closed timelike curves make sense?38. What does Carter’s fourth integral of motion Q signify physically?39. What is special about a principal null congruence?40. Evaluated in the locally inertial frame of a principal null congruence, the spin-0 com-ponent of the Weyl scalar of the Kerr geometry is C = −M/(r − ia cos θ)3, whichlooks like the Weyl scalar C = −M/r3of the Schwarzschild geometry but with radiusr replaced by the complex radius r − ia cos θ. Is t here something deep here? Can theKerr geometry be constructed from the Schwarzschild geometry by complexifying theradial coordinate r?3.2 What’s important?1. Astronomical evidence suggests that stellar-sized and supermassive black holes existubiquitously in nature.2. The no-hair theorem, and when and why it applies.3. The physical picture of black holes as regions of spacetime where space is falling fasterthan light.4. A physical understanding of how the metric of a black hole relates to its physicalproperties.5. Penrose (conformal) diagr ams. In particular, the Penrose diagrams of the various kindsof vacuum black hole: Schwarzschild, Reissner-Nordstr¨om, Kerr-Newman.6. What really happens inside black holes. Collapse of a star. Mass inflation instability.33.3 Observational evidenceIt is beyond the scope of this course to discuss the observational evidence for black holes inany detail. However, it is useful to summarize a few facts.1. Observational evidence supports the idea that black holes occur ubiquitously in nature.They are not observed directly, but reveal themselves through their effects on theirsurroundings. Two kinds of black hole are observed: stellar-sized black holes in x-raybinary systems, mostly in our own Milky Way galaxy, and supermassive black holes inActive Galactic Nuclei (AGN) found at the centers of our own and other galaxies.2. The primary evidence that astronomers accept as indicating the presence of a blackhole is a lot of mass compacted into a tiny space.(a) In an x-ray binary system, if the mass of the compact object exceeds 3 M⊙, t hemaximum theoretical mass of a neutron star, then the object