Chapter summary

In this chapter we saw how the general theory of relativity has changed our thinking about the nature of space and time.

We then saw how the ideas of space-time carry over to a theory of gravitation - general relativity. The interpretation of gravitational fields is that they alter the geometry of space-time, causing it to behave like that on a curved surface. The starting point for general relativity is the principle of equivalence, which tells us that inertial and gravitational masses are the same.

We saw that there are several effects of general relativity that can be tested. These include the advancement of the periastron of orbiting bodies, the bending of electromagnetic radiation, gravitational redshift and time dilation, and gravitational radiation.

We also saw how the gravitational redshift leads to a concept of black holes, objects from which nothing can escape.


8.1. What do we mean when we say that gravity alters the geometry of space-time?

8.2. Re-do the analysis of the person on a scale in the elevator (all three cases) explicitly noting the uses of inertial mass and gravitational mass.

8.3. Why don't you need a solar eclipse to measure the bending of radio waves past the edge of the Sun?

8.4. Briefly describe the tests of general relativity discussed in this chapter.

8.5. Since we cannot run a rule from the center of a black hole to the Schwarzschild radius, how would you "measure" the radius of a black hole? (Hint: Think in terms of a measurement that doesn't involve crossing the event horizon.)

8.6. (a) What is the exit cone? (b) When the exit cone closes, what happens to photons aimed straight up?

8.7. Is there a place near a black hole where you could look straight ahead and see the back of your head? Explain.

8.8. If the Schwarzschild radius of the Sun is 3 km, does that mean that the inner 3 km of the Sun is a black hole?


8.1. Consider an object with the same density as the Sun. Find an expression for the bending of starlight past the edge of this object as a function of the size of the object.

8.2. For a neutron star (to be discussed in Chapter 11) with 1 M0 and a radius of 10 km, through what angle is light bent as it passes the edge?

8.3. For a white dwarf with 1 M0 and a radius of

5 X 103 km, find the wavelength to which the Ha line will be shifted by the time it is seen by a distant observer.

8.4. (a) For the test of the gravitational redshift involving the Mossbauer effect, calculate the shift in going from the Earth's surface to 50 m above the Earth's surface. (b) How fast would the receiver have to move toward the source to compensate for the redshift? (c) Compare your answer in (b) with the speed that an object falling from the roof would acquire just before striking the ground.

8.5. Show that the Schwarzschild radius can also be found by taking the escape velocity from an object of mass M and radius R, and setting it equal to c.

8.6. (a) Compute your Schwarzschild radius.

(b) What would the density be for a black hole of your mass?

8.7. For what mass black hole does the density equal 1 g/cm3 ?

8.8. For what mass black hole does the difference between the acceleration of gravity at an astronaut's feet and head equal the acceleration of gravity on the Earth (1000 cm/s2 )?

8.9. Find an expression for dg/dr at the surface of a black hole as a function of the mass of a black hole. Your expression should be a scaling relationship as in equation (8.10).

8.10. How does the rate of a clock 1.5 RS from a 3M0 black hole compare with the rate of a clock far from the black hole?

8.11. How close must you be (in terms of RS) to a 3M0 black hole to find that a clock runs at 10% the rate it runs when it is far away?

8.12. If an electron-positron pair forms from the vacuum, how long can they live before they must annihilate?

Computer problems

8.1. For stars in the mid-range of each spectral type (O5, B5, . . .), make a table showing the angle of bending of starlight just passing the limb of that star. Also include in your table a white dwarf which has 1 solar mass in an object the size of the Earth.

8.2. For gravitational redshift, make a graph of Aa/a vs. r, for an interesting range of r (assuming 1 M0).

8.3. Make a table showing the Schwarzschild radii and average density for black holes of 10" M0 where n goes from 0 to 10. Also include a column showing the acceleration of gravity at the surface.

Part III

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