Questions
Review Questions
You drop a ball inside a car traveling at a steady 50 km/h in a straight line on a smooth road. Does it fall in the same way as it does inside a stationary car? How does this question relate to Einstein’s special theory of relativity?
In Einstein’s special theory of relativity, two different observers moving at different speeds will measure the same value of the speed of light. Will these same observers measure the same value of, say, the speed of an airplane? Explain your answer.
A friend summarizes the special theory of relativity by saying “Everything is relative.” Explain why this statement is inaccurate.
Serena flies past Michael in her spaceship at nearly the speed of light. According to Michael, Serena’s clock runs slow. According to Serena, does Michael’s clock run slow, fast, or at the normal rate? Explain your answer.
Ole flies past Lena in a spherical spaceship at nearly the speed of light. According to Lena, how does the distance from the front to the back of Ole’s spaceship (that is, measured along the direction of motion) compare to the distance from the top to the bottom (that is, measured perpendicular to the direction of motion)? Explain your answer.
Why does the speed of light represent an ultimate speed limit?
Why is Einstein’s general theory of relativity a better description of gravity than Newton’s universal law of gravitation? Under what circumstances is Newton’s description of gravity adequate?Describe two different predictions of the general theory of relativity and how these predictions were tested experimentally. Do the results of the experiments agree with the theory?
How does a gravitational redshift differ from a Doppler shift?
In what circumstances are degenerate electron pressure and degenerate neutron pressure incapable of preventing the complete gravitational collapse of a dead star?
Should we worry about Earth being pulled into a black hole? Why or why not?
How does rapid flickering of an X-ray source provide evidence that the source is small (see discussion of Cygnus X-1)?
All the stellar-mass black hole candidates mentioned in the text are members of very short-period binary systems. Explain how this makes it possible to detect the presence of the black hole.Astronomers cannot actually see the black hole candidates in close binary systems. How, then, do they know that these candidates are not white dwarfs or neutron stars?
Describe two ways in which a member of a binary star system could become a black hole.
What is a gamma-ray burst? What is the evidence that gamma-ray bursts are not located in the disk of our Galaxy or in a halo surrounding our Galaxy?
Summarize the evidence that gamma-ray bursts result from a process involving a star in a distant galaxy.
What is a collapsar? How does the collapsar model account for the existence of long-duration gamma-ray bursts?
How do astronomers locate supermassive black holes in galaxies?
What is an intermediate-mass black hole? How are such objects thought to form?
When we say that the Moon has a radius of 1738 km, we mean that this is the smallest radius that encloses all of the Moon’s material. In this sense, is it correct to think of the Schwarzschild radius as the radius of a black hole? Why or why not?
A twenty-third-century instructor at Starfleet Academy tells her students, “If someday your starship falls into a black hole, it’ll be your own fault.” Explain why it would require careful piloting to direct a spacecraft into a black hole.
In what way is a black hole blacker than black ink or a black piece of paper?
If the Sun suddenly became a black hole, how would Earth’s orbit be affected? Explain your answer.
According to the general theory of relativity, why can’t some sort of yet-undiscovered degenerate pressure prevent the matter inside a black hole from collapsing all the way down to a singularity?
Is it possible to tell which chemical elements went into a black hole? Why or why not?
Why is it unlikely that a black hole has a large electric charge?
What kind of black hole is surrounded by an ergoregion? What happens inside the ergoregion?
What is the no-hair theorem?
As seen by a distant observer, how long does it take an object dropped from a great distance to fall through the event horizon of a black hole? Explain your answer.
Summarize what leads to Hawking radiation.
If even light cannot escape from a black hole, how is it possible for black holes to evaporate?
Why do smaller black holes evaporate more quickly than larger black holes?
Advanced Questions
Questions preceded by an asterisk (*) involve topics discussed in the Boxes.
Problem-solving tips and tools
Remember that the time to travel a certain distance is equal to the distance divided by the speed, and the density of an object is its mass divided by its volume. The volume of a sphere of radius r is 4πr3/3. Section 4-7 describes Newton’s law of universal gravitation. Box 4-4 shows how to use Newton’s formulation of Kepler’s third law, which explicitly includes masses; when using this formula, note that the period P must be expressed in seconds, the semimajor axis a in meters, and the masses in kilograms. For another version of this formula, in which period is in years, semimajor axis in AU, and masses in solar masses, see Section 17-9.
*A spaceship flies from Earth to a distant star at a constant speed. Upon arrival, a clock on board the spaceship shows a total elapsed time of 8 years for the trip. An identical clock on Earth shows that the total elapsed time for the trip was 10 years. What was the speed of the spaceship relative to Earth?
*An unstable particle called a positive pion (pronounced “pieon”) decays in an average time of 2.6 × 10−8 s. On average, how long will a positive pion last if it is traveling at 95% of the speed of light?
*How fast should a meter stick be moving in order to appear to be only 60 cm long?
*An astronaut flies from Earth to a distant star at 80% of the speed of light. As measured by the astronaut, the one-way trip takes 15 years. (a) How long does the trip take as measured by an observer on Earth? (b) What is the distance from Earth to the star (in light-years) as measured by an Earth observer? As measured by the astronaut?
In the binary system of two neutron stars discovered by Hulse and Taylor (Section 21-2), one of the neutron stars is a pulsar. The distance between the two stars varies between 1.1 and 4.8 times the radius of the Sun. The time interval between pulses from the pulsar is not constant: It is greatest when the two stars are closest to each other and least when the two stars are farthest apart. Explain why this is consistent with the gravitational slowing of time (Figure 21-7a).
Find the total mass of the neutron star binary system discovered by Hulse and Taylor (Section 21-2), for which the orbital period is 7.75 hours and the average distance between the neutron stars is 2.8 solar radii. Is your result reasonable for a pair of neutron stars? Explain your answer.
Estimate how long it will be until the two neutron stars that make up the binary system discovered by Hulse and Taylor collide with each other. Assume that the distance between the two stars will continue to decrease at its present rate of 3 mm every 7.75 hours, and use the data given in Question 39. (You can assume that the two stars are very small, so they will collide when the distance between them is equal to zero.)
The orbital period of the binary system containing A0620-00 is 0.32 day, and Doppler shift measurements reveal that the radial velocity of the X-ray source peaks at 457 km/s (about 1 million miles per hour). (a) Assuming that the orbit of the X-ray source is a circle, find the radius of its orbit in kilometers. (This is actually an estimate of the semimajor axis of the orbit.) (b) By using Newton’s form of Kepler’s third law, prove that the mass of the X-ray source must be at least 3.1 times the mass of the Sun. (Hint: Assume that the mass of the K5V visible star—about 0.5 M⊙ from the mass-luminosity relationship—is negligible compared to that of the invisible companion.)
Contrast gamma-ray bursts with X-ray bursts (discussed in Section 20-12). From our models of what causes these energetic phenomena, explain why X-ray bursts emit repeated pulses but gamma-ray bursts apparently emit just once.
Long-duration gamma-ray bursts are only observed in galaxies where there is ongoing star formation. Explain how this is consistent with the collapsar model of how these bursts occur.
The spectrum of a Type Ic supernova lacks absorption lines of hydrogen and helium. This means that when a black hole formed at the center of the progenitor star, the resulting jets were more easily able to escape into space. Explain why this is so.
Find the orbital period of a star moving in a circular orbit of radius 500 AU around the supermassive black hole in the galaxy NGC 4261 (Section 21-5).
*Find the Schwarzschild radius for an object having a mass equal to that of the planet Saturn.
*What is the Schwarzschild radius of a black hole whose mass is that of (a) Earth, (b) the Sun, (c) the supermassive black hole in NGC 4261 (Section 21-5)? In each case, also calculate what the density would be if the matter were spread uniformly throughout the volume of the event horizon.
*What is the mass in kilograms of a black hole whose Schwarzschild radius is 11 km?
*To what density must the matter of a dead 8-M⊙ star be compressed in order for the star to disappear inside its event horizon? How does this compare with the density at the center of a neutron star, about 3 × 1018 kg/m3?
*Prove that the density of matter needed to produce a black hole is inversely proportional to the square of the mass of the hole. If you wanted to make a black hole from matter compressed to the density of water (1000 kg/m3), how much mass would you need?
Discussion Questions
The speed of light is the same for all observers, regardless of their motion. Discuss why this requires us to abandon the Newtonian view of space and time.
Describe the kinds of observations you might make in order to locate and identify black holes.
Speculate on the effects you might encounter on a trip to the center of a black hole (assuming that you could survive the journey).
Web/eBook Questions
Search the World Wide Web for information about a stellar-mass black hole candidate named V4641 Sgr. In what ways does it resemble other black hole candidates such as Cygnus X-1 and V404 Cygni? In what ways is it different and more dramatic? How do astronomers explain why V4641 Sgr is different?
Search the World Wide Web for information about supernova SN 2006aj, which was associated with gamma-ray burst GRB 060218. In what ways were this supernova and gamma-ray burst unusual? Are the observations of these objects consistent with the collapsar model?
Search the World Wide Web for information about the intermediate-mass black hole candidate in M82. Is this still thought to be an intermediate-mass black hole? What new evidence has been used to either support or oppose the idea that this object is an intermediate-mass black hole?
The Equivalence Principle. Access the animation “The Equivalence Principle” in Chapter 21 of the Universe Web site or eBook. View the animation and answer the following questions. (a) Describe what is happening as viewed from the frame of reference of the elevator. What causes the apple to fall to the floor of each elevator? (b) Describe what is happening as viewed from the frame of reference of the stars. What causes the apple to fall to the floor of each elevator? (c) Think of another experiment you could perform with the apple. Describe what would happen during this experiment as seen by Newton (in the left-hand box) and by Einstein (in the right-hand box).