Exercises

1. Proper Distance Between Spherical Shells

A black hole has mass M = 5 kilometers, a little more than three times that of our Sun. Two concentric spherical shells surround this black hole. The inner shell has r-coordinate (reduced circumference) r; the outer one has r-coordinate r + dr, where dr = 1 meter. What is the radial separation do = drshe|[ between these spherical shells as measured directly by an observer on one of these shells? Treat three cases of the reduced circumference r of the inner shell.

A. r = 50 kilometers

B. r = 15 kilometers

2. Grazing the Sun

Verify the statement in Section 4 (top of page 2-11) concerning two spherical shells around our Sun. The inner shell, of reduced circumference r\ = 695 980 kilometers, just grazes the surface of Sun. The outer shell is of reduced circumference one kilometer greater, namely r2 = 695 981 kilometers. Verify the prediction that the directly measured distance between these shells will be 2 millimeters more than 1 kilometer. (Outbursts and flares leap thousands of kilometers up from Sun's roiling surface, so this exercise is a bit unrealistic, even if we could build these shells!) Hint: Use the approximation

The exponent n can be a positive or negative integer or a positive or negative fraction.

3. Gravitational Red Shift

Consider a black hole with M = 1.5 kilometers, approximately equal to that of our Sun. An observer standing on a spherical shell of reduced circumference r shines a steady laser beam of wavelength 400 nanometers (4 x 10 meters: violet light) radially outward. This light is received by a remote observer at a radius very much greater than 2M. What is the wavelength of the light received by this remote observer in each of the following cases? Note that red light has wavelength 700 nanometers and that, in conventional units,

Treat three cases: The person shining the laser outward stands on a spherical shell of reduced circumference r with the value

A. r = 20 kilometers

D. Guess: Suppose the source is aimed in some other direction than the outward radial one, but the laser beam still arrives at a distant observer. Will this distant observer measure the same wavelength as computed in cases A, B, and C, or will the wavelength be different for a non-radial initial direction?

4. How Many Shells?

The President of the Black Hole Construction Company is waiting in your office when you arrive. He is waxing wroth. ("Let Roth wax him for a while."— Groucho Marx)

"You are bankrupting me!" he shouts. "We signed a contract that I would build spherical shells centered on Black Hole Alpha, the shells to be 1 meter apart extending down to the horizon. Now my staff relativist tells me that, starting at any radius whatever outside the horizon, I have to build an infinite number of these shells between that radius and the horizon. We do not have materials for that many!"

"Calm down a minute," you reply. "Black Hole Alpha has a horizon radius r = 2M = 10 kilometers = 10 000 meters. You agreed to build 1000 spherical shells starting at reduced circumference r = 10 001 meters, then r = 10 002 meters, then r - 10 003 meters, and so forth, ending at r = 11 000 meters. So what is the problem?"

"I don't know. Maybe we can figure it out if I describe our construction method. My worker robots hang 1-meter rods down vertically (radially) from each completed shell, measure them in place to be sure they are exactly 1 meter long, then weld to the ends of these rods the horizontal (tangential) beams of the next spherical shell of smaller radius."

"Ah, then your Black Hole Construction Company is indeed facing a large unnecessary expense," you conclude. "But I think I can help you."

A. Explain to the President of the Black Hole Construction Company how to alter his construction method in order to complete his obligation to build 1000 correctly spaced spherical shells. Be specific, but do not be a fussbudget.

B. Using the radius of the innermost shell in the relevant equation, make a first estimate of the directly measured separation between the innermost shell and the second shell, the one with the next-larger radius.

C. Using the radius of the second shell, the one just outside the innermost shell, make a second estimate of the directly measured separation between the innermost shell and the second shell.

D. Optional. If you are unhappy with the estimates of parts B and C, you may use calculus to make a correct calculation of the directly measured separation between the innermost shell and the one just outside it.

E. Was the contractor's staff relativist correct in predicting an infinite number of shells for this contract, even using the method described in the fourth paragraph of this exercise?

5. A Dilute Black Hole

Most descriptions of black holes are so apocalyptic that one gets the impression that black holes are extremely dense objects. Of course, a black hole is not dense throughout, because all matter quickly plunges to the central crunch point. Still, one can speak of an artificial "average density," defined, say, by the total mass M divided by a spherical Euclidean volume of radius r = 2M. In terms of this definition, general relativity does not require that a black hole have a large average density. In this exercise you design a black hole with average density equal to that of the atmosphere you breathe on Earth, approximately 1 kilogram per cubic meter. Do all calculations to one-digit accuracy—we want an estimate!

A. From the Euclidean equation for the volume of a sphere find an equation for the mass M of air contained in a sphere of radius r, in terms of the density p kilograms/meter3. Use the conversion factor G/c2 = 7 x 10"28 meter/kilogram (Section 6) to express this mass in meters. (The volume formula used here is for Euclidean geometry, and we are applying it to curved space geometry—so this exercise is only the first step in a more sophisticated analysis.)

B. Set r = 2M for the Schwarzschild radius of the horizon of this black hole. What is the numerical value of 2M in meters? (Hint: Carry all units along to be sure you have not made a minor error somewhere.)

C. Compare your answer to the radius of our solar system. The mean radius of the orbit of the planet Pluto is approximately 6 x 1012 meters.

D. How many times the mass of our Sun is the mass of your designer black hole?

6. Astronaut Stretching According to Newton

As you plunge feet first radially inward toward the center of a black hole, you are not physically stress-free and comfortable! True, you detect no overall "force of gravity" accelerating you inward. But you do feel a tidal force pulling your feet and head apart and additional forces squeezing your middle inward from the sides like a high-quality corset. When do these tidal forces become uncomfortable? We cannot yet answer this question using general relativity, but Newton is available for consultation, so let's ask him. One-digit accuracy is plenty for numerical estimates in this exercise.

A. Cleaning up the formula. Let gconv be the local acceleration of gravity in conventional units (meters per second squared). (Here and hereafter "conv" is always a subscript.) Set m times Sconv equal to the gravitational force in New ton's law of gravitation (equation [2], page 2-13). In Newton's law use a subscript M^g to remind yourself that the mass of the center of attraction is in kilograms. Now cancel m (mass of your own body), divide through by c2, convert units, and show that the resulting formula for the local acceleration of gravity is g = M/r2. Here M is in meters (equation [5], page 2-14) and g = gconv/ c2 is also in geometric units: meter/meter2 = meter-1. (Note: Exponent is minus one).

B. Convert gconv at Earth's surface to geometric units to show that it has the approximate value g Earth = 1CT16 meter-1. Use this value in the remaining parts of this exercise.

What does "uncomfortable" mean? So that we all concur, let us say that you are uncomfortable when your head is pulled upward with half the usual force of gravity on Earth, your middle is in free-float and comfortable (except for being squeezed from the sides), and your feet are pulled downward with half Earth's usual force of gravity on them. In other words, the difference in gravitational acceleration between your head and feet as you fall is equal to the acceleration of gravity as it would be measured at Earth's surface: dg = gEarth.

C. Take the derivative with respect to r of the local acceleration g found in part A to obtain an expression dg/dr in terms of M and r.

D. How massive a black hole do you want to fall into? Suppose M = 10 kilometers = 10 000 meters, or about seven times the mass of our Sun. Assume your head and feet are 2 meters apart. Find the radius rouch, in meters, at which you become uncomfortable according to our criterion. Compare this radius with that of Earth, namely 6.4 x 106 meters.

E. Will your discomfort increase or decrease or stay the same as you continue to fall inward toward the center from this radius?

F. Suppose you fall from rest at infinity. How fast are you going when you reach the radius of discomfort rouch according to Newton? Express this speed as a fraction of the speed of light.

G. Taking the velocity in part F to be constant from that radius to the center, find the corre sponding (maximum) time in meters to travel from rouch to the center, according to Newton. This will be the maximum time lapse during which you will be—er—uncomfortable.

H. What is the maximum time of discomfort, according to Newton, expressed in seconds?

Note 1: If you carried the symbol M for the black hole mass through these equations, you found that it canceled out in expressions for the maximum time lapse of discomfort in parts G and H. In other words, your discomfort time is the same for a black hole of any mass when you fall from rest at infinity—according to Newton. This equality of discomfort time for all M is also true for the general relativistic analysis.

Note 2: The wristwatch time lapse from any radius to the center according to general relativity is analyzed in Chapter 3 and in an exercise of that chapter. The "ouch time" is examined more thoroughly in Section 7 of Project B, Inside the Black Hole.

7. General Relativity over Chesapeake Bay

On November 22,1975, a U.S. Navy P3C antisubmarine patrol plane flew back and forth for 15 hours at an altitude of 25 000 to 30 000 feet (7600 to 10 700 meters) over Chesapeake Bay in an experiment organized by Carroll Alley and collaborators. The plane carried atomic clocks that were compared by laser pulse with identical clocks on the ground. During the period of flight, the plane's clock gained 47.2 nanoseconds (47.2 x 10"9 seconds) compared with the ground clock. You have the tools to analyze this time difference. Take 9000 meters as an average altitude. Assume that the plane flew very slowly, just above stalling speed, so that time dilation due to relative speed can be neglected. (In fact, time stretching accounted for about 10 percent of the general relativity effect.) Call the clock on the surface of Earth the shell clock. Let fshell be the time the airplane has been "on station" at 9000 meters altitude and let t be the corresponding far-away time. From equation [19],

where t is far-away time.

A. Take the derivative of the expression for fshen with respect to r. (The plane's altitude h is much smaller than the radius r of Earth, so ignore h in places where it is added to r.) Use equations [12] and [19] to convert the resulting dr to drshen and the resulting t to fsheii- Show that the result gives the following relation between dfsheU and drsheil: Mdr.

dt shell shell

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