## Info

In this equation, E is energy measured at infinity, a constant of the motion, and Eshgii is the local energy measured by a shell observer. Equation 27 is general, it turns out. In the exercises of Chapter 5 you show that the same equation even applies to a flash of light fired outward from a fixed shell. In brief, Eshell is a local quantity measured by the shell observer, a quantity that he analyzes using the laws of special relativity. But Eshell is not a constant of the motion it is different...

## Solution

In Sample Problem 1 we found for this case that Um 5.775 Wand E m 1.033. Now, a particle launched radially outward (zero angular momentum ) with the lesser energy E m 1 000 will coast to rest at a very great distance from the black hole If the satellite of this problem were launched radially outward with the slightly greater energy E m 1.033, it would arrive at a great distance with some kinetic energy. In contrast, launching the particle radially inward will surely lead to capture by the black...

## Q Lu X

Figure 2 Three alternative cases of a stone moving along a straight line in space as it emits three flashes, 7, 2, and 3 The space locations of emissions are the same in all three cases, as are the times of first and last emissions and 3 But emission time for the middle flash 2 is different for the three cases We ask At what time will a free stone following a natural path pass the intermediate point and emit flash 2 > We answer this question by demanding that the total wristwatch time x from...

## Exploring Black Holes

General relativity flows from Einstein's field equations, which relate the mass and pressure in a region of spacetime to the warping of spacetime across that region. The field equattons reveal how this warping is experienced by any observer, whether she is moving, accelerating, rotating, stretching, twisting, or tumbling. The field equations are wonderfully general. But this generality has a price mathematical sophistication. The field equations speak the language of tensors or differential...

## Effective Potential in Schwarzschild Spacetime

The box on page 4-9 describes a method of using a computer to reckon the orbit of a stone. In this section we outline a much simpler method. By looking at the so-called effective potential, we can describe at a glance the major features of the trajectory of any particle. Clearer than our computer printouts, clearer even than our orbits, the effective potential lets us see at a glance the central features of the motion of a satellite. The basic idea is similar to the Newtonian analysis (Section...

## The Rain Frame

We dive to the center of a black hole riding in an unpowered spaceship that moves radially inward. To simplify further, we ride on a spaceship launched in a particular way, namely starting from rest at a great distance from the black hole. We call such a free-float frame a rain frame, because on Earth rain also starts from rest at a great height (even though near Earth the braking effect of air keeps rain from falling freely). The rain frame starts from rest at so great a height a radius so...

## Joyous Excitement

What discovery sent Einstein into joyous excitement in November of 1914 It was the calculation showing that his brand new (actually not quite completed) theory of general relativity gave the correct value for one detail of the orbit of the planet Mercury that had previously been unexplained. Mercury circulates around Sun in a not quite circular orbit The planet oscillates in and out radially while it circles tangentially. The result is an elliptic orbit. Newton tells us that if we consider only...

## Schwarzschild Map vs Shell View

Different maps different directions The observer on a given shell and the Schwarzschild bookkeeper both track the path of a light flash between two events, A and B, that lie near the shell observer. The shell observer tracks this light flash moving past him at an angle 0shell with respect to the radially outward direction (Figure 9). At what angle chw does the Schwarzschild bookkeeper record the light beam to be traveling Because of the way we define the reduced circumference r (Section 4 of...

## Seeing

Tell all the Truth but tell it slant Success in Circuit lies Too bright for our infirm Delight The Truth's superb surprise As Lightning to the Children eased With explanation kind The Truth must dazzle gradually Or every man be blind Emily Dickinson, about 1868 (poem 1129) What do we see when we (finally ) look around What can we say about the motion of light around, past, or into a spherically symmetric nonspinning black hole We ask here no small question. Almost everything that we learn about...

## Einstein Rings

Einstein was discussing some problems with me in his study when he suddenly interrupted his explanation and handed me a cable from the windowsill with the words, This may interest you. It was the news from Eddington confirming the deviation of light rays near the sun that had been observed during the eclipse. I exclaimed enthusiastically, How wonderful, this is almost what you calculated. He was quite unperturbed. I knew that the theory was correct Did you doubt it When I said, Of course not,...

## Strategy

From the initial position and directed velocity, find the value of angular momentum per unit mass L m and energy per unit mass E m. Step 1 Express dt and dty in terms of satellite wristwatch time dx Step 2 Substitute these results into the Schwarzschild metric to find dr as a function of dx. Result All bookkeeper increments dt, dt > , and dr are now locked to satellite time increment dx. Computer Starting with the initial position, let the computer advance satellite time x by increments as it...

## Roman Letters

A JIM Ratio of angular momentum to mass of a spinning center of attraction. Unit meter. F-2 b Impact parameter of an object near a center of attraction. Unit meters. 4-6, 5-6 c Speed of light. In flat spacetime defined to have the value given inside the back cover. Unit meters second. 1-2,1-11 dl Four-dimensional increment of distance used in the metric for the Universe in the Friedmann model. Unit meter. G-3 dr Increment of radial separation, where r may be the Euclidean radius or the reduced...

## Chapter

Plunging from Rest at Infinity Black Hole Alpha has a mass M 5 kilometers and a horizon at r 2M 10 kilometers. A stone starting from rest far away falls radially into Black Hole Alpha. A. At what velocity does a shell observer at r 35 kilometers measure the stone to be going as the stone passes him (Answer to nearest digit is -0.5. Supply three-digit accuracy.) What is the bookkeeper velocity dr dt of the stone as it passes r 35 kilometers (Answer to nearest digit is -0.4. Supply three-digit...

## Schwarzschild Maps of the Motion of Light

The larger view that no observer observes A light flash moving under the influence of a spherically symmetric center of attraction of given mass M has an orbit whose size and shape, praise be, depends on only a single quantity, the impact parameter b. The trajectory of a light flash near a black hole lends itself to a simple description using the effective potential. For example, Figure 6 is what we call a Schwarzschild map of the orbits of light for three sample values of the impact parameter...

## Knife Edge Orbit

Compute orbit around black We want to know whether that satellite whirring around the 2.6-million-hoie at center of galaxy. solar-mass black hole at the center of the Milky Way is going to fall in. Originally this satellite was orbiting at very high speed very close to the black hole. Suddenly a neutron star zoomed through the system and then disappeared, leaving our satellite in a perturbed state. For the perturbed satellite orbiting the black hole at the center of our galaxy, we feed the...

## The Plunging View

At the end of our chapters on the world of space and time, momentum Super cinema double feature and energy, planets and black holes, we celebrate with a final parade of an all-star cast. Let's follow Richard Matzner, Tony Rothman, and Bill Unruh looking at the starry heavens as we free float straight down into a black hole so massive, so large, that even after crossing the horizon at the Schwarzschild radius we have four hours of existence ahead of us the time of a super cinema double feature...

## Exercises

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....

## Rebel Stone and Obedient Computer

Try all possible worldlines between initial and final events. The free stone chooses the worldline of extremal aging. Suppose that the stone rebels let it disobey the command issued by space-time to follow the worldline of extremal aging. Or, more realistically, think of an external experimenter grasping the stone and forcing it to move along a worldline that it would not freely follow. This rebellion, this deviation from the natural, is partial the stone is present at the two obligatory...

## Summary

In general, the metric provides a complete description of spacetime the curvature of spacetime and the results of measurements carried out with rods and clocks. The metric for flat spacetime is the one that dominated our study of special relativity. However, special relativity cannot describe spacetime globally in the vicinity of a massive object. General relativity can do so, earning the name Theory of Gravitation. The Schwarzschild metric describes spacetime exterior to the surface of any...

## References and Acknowledgments

Initial quote from A Tale of Einstein, Congress and Safe Landings by Clifford Will, an online posting by the Editorial Services, Washington University, St. Louis. Carroll O. Alley provided much of the information used in this project. Carroll O. Alley, Proper Time Experiments in Gravitational Fields With Atomic Clocks, Aircraft, and Laser Light Pulses, in Quantum Optics, Experimental Gravity, and Measurement Theory, edited by Pierre Meystre and Marian O. Scully, Plenum Publishing, New York,...

## M

Substitute numbers into equation 7 and find the numerical value of b in the following equation The number represented by b in equation 8 is an estimate of the fractional difference in rates between stationary clocks at the position of the satellite and at Earth's surface. Is this difference negligible or important to the operation of the GPS Suppose the timing of a satellite clock is off by 1 nanosecond (10-9 second). In 1 nanosecond a light signal (or a radio...

## The Horizon as a One Way Barrier

A light flash is launched from the shell at r-coordi-nate rj. Its energy at launch is E(Vi)shell as measured by observers on the launching shell. Now the flash moves radially inward or outward and is received at a shell of different reduced circumference r2. A. What energy E shell does the light flash have as measured by the observers on the shell at r2 Hint Run equation 48 backward and forward, assuming constant energy E measured at infinity. B. Take the limit of your expression derived in...

## Keplers Laws of Planetary Motion

Johannes Kepler (1571-1630) provided a milestone in the history of mechanics with his Three Laws of Planetary Motion, deduced from a huge stack of planetary observations made by Tycho Brahe. 1. The planets circulate around Sun in elliptical orbits with Sun at one focus. 2. The radius vector from Sun to any planet sweeps out equal areas in equal times. 3. The square of the period of any planet is proportional to the cube of the planet's mean distance from Sun. A. Show by example that Kepler's...

## J

This integral is not in a common table of integrals So make the substitution r z2, from which dr 2zdz Then the integral and its solution become Which radius r do we use in the denominator of the right-hand expression' If we use r 4 kilometers, the result is On the other hand, if we use r 5 kilometers, the result is Here In is the natural logarithm (to the base e) and 11 stands for absolute value Substitute the values (units omitted) The trouble here is that the term 2M r changes significantly...

## Predicting Advance of the Perihelion

The advance of the perihelion of Mercury springs from the difference between the frequency at which the planet sweeps around in its orbit and the frequency at which it oscillates in and out radially. In the Newtonian analysis these two frequencies are equal if one considers only the interaction between planet and Sun. But Einstein's theory shows that these two frequencies are not quite equal, so Mercury reaches its maximum (or minimum) radius at a slightly different angular position in each...

## R 2rV dx2 fi mdt2[20

Divide through by dt1, solve for dr dt, and take the square root to obtain Bookkeeper measure of radial We take the minus square root because the expression describes a decreasing radius as the object falls toward the black hole outside the horizon. Equation 21 gives the bookkeeper velocity that describes the plunge as the rate of change of reduced circumference r with time t measured on faraway clocks. Why is this analysis so COMPLICATED How can a stone carry out all these calculations as it...

## The Spinning Black Hole

Black holes are macroscopic objects with masses varying from a few solar masses to millions of solar masses. To the extent they may be considered as stationary and isolated, to that extent, they are all, every single one of them, described exactly by the Kerr solution. This is the only instance we have of an exact description of a macroscopic object. Macroscopic objects, as we see them all around us, are governed by a variety of forces, derived from a variety of approximations to a variety of...

## Radial outward direction

Figure 3 Light velocity vector and its components as observed by the shell viewer located at radius r. The shell observer measures the speed of light to have the standard value unity Note that the sum of the squares of these components is equal to unity, as shown in Figure 3. Speed of light equal to unity is also measured by the free-float orbiting or plunging observer, safe inside her capsule of flat spacetime, an unpowered spaceship that orbits the black hole or hurtles radially inward. 5...

## M 2

(ignoring the fact that acceleration is in the negative radial direction). In these units, what is the value of gE, the acceleration of gravity at the surface of Earth LetgEconv be the acceleration of gravity in conventional units. Show that in geometric units has the units meter-1 and the approximate value gE cov 10 16 meter 1 51. Newton B. What is the corresponding prediction of general relativity First we need to decide which dr and dt we are talking about. The statement of the exercise...

## Sm

Figure 10 Pie charts of the view from the shell Each circle symbolizes the panoramic view from a point on a shell at that r-coordinate Angles listed, such as 2 x 54 , are the span of angles of the black portions, representing the directions in which the shell viewer sees the black hole or rather experiences the absence of light White portions represent the directions into which rays are compressed arriving from all the stars of the visible heavens. Each pie chart can be made three-dimensional...

## John Archibald Wheeler

An imprint of Addison Wesley Longman San Francisco Boston New York Capetown Hong Kong London Madrid Mexico City Montreal Munich Paris Singapore Sydney Tokyo Toronto For more than 12 years, students in many classes have read through sequential versions of this text, shared with us their detailed difficulties, and given us advice. (See Only the Student Knows, American Journal of Physics, Volume 60, Number 3, March 1992, pages 201-202.) Many of the thinker objections in the text come from these...

## Equation [31 defines the effective potential Vr in its dependence on r by what we have to take away from the squared

Figure 8 Computer plot Effective potential curve versus radius for an object orbiting a black hole for angular momentum L m 3.75M. Both effective potential and particle energy are measured along the vertical axis. Particle energy does not vary with radius it is a constant of the motion. As a result the energy-versus-position plot is a horizontal line. Upper diagram When the particle has energy corresponding to the minimum of the potential (point indicated by open circle), the particle remains...

## The Principle of Extremal Aging

Go straight spacetime shouts at the stone. The stone's wristwatch verifies that its path is straight. All the exotic talk about curved spacetime geometry near stars and black holes leaves us unprepared for a revelation about motion right at home Schwarzschild geometry correctly describes the motions of baseballs and stones near the surface of Earth. Even more surprising Analyzing trajectories of near-Earth objects using Schwarzschild geometry prepares us to go back and describe trajectories...

## Right

Figure 8 Top view of Saturn and its ring as if looking down from above on Saturn displayed in the lower image on the cover. Not to scale Query 19 asks questions about the angles at which the observer at the right sees points A, B, and C when the black hole M is in place. Figure 8 Top view of Saturn and its ring as if looking down from above on Saturn displayed in the lower image on the cover. Not to scale Query 19 asks questions about the angles at which the observer at the right sees points A,...

## Advance of the Perihelion of Mercury

This discovery was, I believe, by far the strongest emotional experience in Einstein's scientific life, perhaps in all his life. Nature had spoken to him. He had to be right. For a few days, I was beside myself with joyous excitement. Later, he told Fokker that his discovery had given him palpitations of the heart. What he told de Haas is even more profoundly significant when he saw that his calculations agreed with the unexplained astronomical observations, he had the feeling that something...

## Energy Measured by Shell Observer

The shell observer uses special relativity to analyze motion locally. The farther toward the center a stone falls, the faster it goes as observed by a sequence of local shell observers (equation 24 ). But a shell observer sees a faster-moving stone as having more energy the faster the stone moves past him, the more energy he can in principle extract from it by slowing it down to rest. In this sense a shell observer at smaller radius attributes more energy to the faster-moving stone as it passes...

## Metric for the Rain Frame

Up until now we have examined only events in the immediate neighborhood of a single in-falling rain observer. In this near-surround, one can construct a local free-float frame consisting of a limited latticework of clocks. Now we want to describe the motion of light from distant stars and the motion of resupply packages hurled radially inward for our use. These trajectories span large regions of spacetime. To describe them we need a global reference frame. For the region outside the black hole...

## Radii of Circular Orbits Around a Black Hole

Note Exercises 2 through 8 build on one another. Later ones in the sequence require answers from earlier ones. Find the radii of circular orbits around a black hole as a function of the angular momentum of the satellite, using the following outline or some other method. An object has known angular momentum per unit mass, L m. This value of L m fixes the effective potential (equation 32 ) for all r-coordinate values. The particle moves in a stable circular orbit if its energy is equal to the...

## The Friedmann Universe

Will we ever penetrate the mystery of creation There is no more inspiring evidence that the answer will someday be yes than our power to predict, and predict correctly, and predict against all expectation, so fantastic a phenomenon as the expansion of the universe. 1 'The Biggest Blunder of My Life The Friedmann model Universe is the simplest cosmological model based on Einstein's field equations. In 1922 Alexander A. Friedmann idealized the sprinkling of stars through space as a cloud of dust...

## Falling from Rest at Infinity

Drift slowly inward, then plunge toward the center. Special case stone starts The fact that E m is constant for a free particle yields great simplification in from rest at r > . describing the motion of a radially plunging particle. As an example, think of a stone originally at rest a very great distance from a spherically symmetric nonspinning black hole. Over the eons this stone moves gradually toward the center of attraction, finally plunging radially inward to oblivion. Formally we say...

## Three Coordinate Systems

(1) Free-float, (2) Spherical shell, (3) Schwarzschild bookkeeping. Live locally in the first two span spacetime with the third. Ride in an unpowered satellite as you fall toward a black hole. Or stand on Mar> y possible reference the scaffolding of a spherical shell and observe this satellite up close as it frames streaks past. Or analyze the satellite motion using the reduced circumference r, angle 0, and far-away time t. Each of these observations requires a different set of spacetime...

## Readings in General Relativity

Note ISBNs are for paperback editions, when available. Popular Books Black Holes and Time Warps Einstein's Outrageous Legacy, Kip S. Thorne, W. W. Norton & Co., New York, 1994, ISBN 0-393-31276-3. Our favorite popular treatment, as described in the front matter of this book. Black Holes, J.-P. Luminet, Cambridge University Press, 1992, ISBN 0-521 -40906-3. Superb, brief book. Similar to Thorne without describing the personalities. Relativity The Special and General Theory, Albert Einstein,...

## Three Views of a Circular Orbit

A shell observer on the shell of radius r compares measurements with a free-float observer moving past with speed i> sheii- Assume that these comparisons can be made using the laws of special relativity. That is, assume that locally we can use the usual Lorentz transformations, calling the shell frame laboratory, the free-float frame rocket, and the relative speed i> sheii- Be careful to have the x-axis of relative motion for the Lorentz transformation lie along the direction of motion of...

## Turning Around Using the Black Hole

The starship Enterprise is headed toward a black hole of known mass M. As captain, you want to use this black hole to reverse your direction of motion with a minimum expenditure of rocket fuel. You look at Figure 11, page 4-23 and decide to use Case 3, but with energy E m smaller than the peak of effective potential so that you will return outward immediately rather than enter the risky knife-edge orbit that has claimed the lives of so many graduates of Starfleet Academy. A. Will the value of E...

## Time Travel Using the Black Hole Unstable Circular Orbits

The proposal for travel forward in time (Exercise 7) is modified to use an unstable circular orbit, with the assumption that an automatic device controls correcting rockets to keep the satellite safely on its knife-edge orbit. A. What is the value of the ratio dx dt for the unstable circular orbit of smallest possible radius Based on this result, would you recommend in favor of using an unstable orbit around a black hole for time travel B. With a glance at effective potential curves for...

## Time Travel Using the Black Hole Stable Circular Orbits

You are on a panel of experts called together to evaluate a proposal to travel forward in time using the difference in rates between a clock in a stable circular orbit around a black hole and our far-away clocks remote from the black hole. Give your advice about the feasibility of the scheme, based on the following analysis or some other that you devise. A. Consider two sequential ticks of the clock of a satellite in a stable circular orbit around a black hole. We want to find the ratio dx dt....

## Energy Production by a Quasar

Note Some results of exercises in Chapter 4, as indicated, are used in solving the following exercise. A quasar (contraction of the name quasi-stellar object) is an astronomical object that pours out a prodigious amount of energy. Because they are so bright, quasars are the most distant visible objects, some of them as much as 14 x 109 light-years distant. A single bright quasar can give off energy at a rate greater than that of all the stars in our galaxy, though most quasars emit energy at a...

## Firing a Laser Pulse Outward

The shell commander on the shell at r-coordinate r fires a laser pulse of energy Esheii radially outward. A. What will be the energy of the laser pulse when it reaches a great distance Answer this question using the following outline or some other method. (Do not assume the answer that is asserted for equation 27 , page 3-17.) Redder color, lower frequency, smaller quantum energy characterizes the pulse received by the remote observer compared with the pulse launched by the shell observer at...

## Justifying the Approximations

We calculated the speed of a satellite in circular orbit and the speed of the clock on Earth's surface using Euclidean geometry and Newtonian mechanics with its universal time. Now, the numerator in each expression for speed, namely rdty, is the same for Euclidean geometry as for Schwarzschild geometry because of the way we defined r in Schwarz-schild spacetime. However, the time dt in the denominator of the speed is not the same for Newton as for Schwarzschild. In particular, the derivation of...

## Newton Approximates Plunging Energy

Show that the general relativistic expressions for energy of a plunging particle reduce (sort of ) to the Newtonian result for small velocities and small values of 2M r. Use the following outline or your own method. A. Set up the Newtonian expression for the total energy of a particle in free fall around a center of gravitational attraction. Convert to geometric units and show that the result can be written and use the approximation (for I d I 1 and I nd I 1) several times to show that,...

## 1

Two firecrackers explode at the same place in the laboratory and are separated by a time of 3 years as measured on a laboratory clock. A. What is the spatial distance between these two events in a rocket in which the events are separated in time by 5 years as measured on rocket clocks B. What is the relative speed of the rocket and laboratory frames Two firecrackers explode in a laboratory with a time difference of 4 years and a space separation of 5 light-years, both space and time measured...

## Timetable to the Center

An astronaut drops from rest off a shell of radius r0. How long a time elapses, as measured on her wrist-watch, between letting go and arriving at the center of the black hole If she jumps off the shell just outside the horizon, what is her horizon-to-crunch time (the maximum possible horizon-to-crunch time, see equations 30 and 31 on page 3-21). Several hints The first goal is to find dr dx, the rate of change of r-coordinate with wristwatch time X, in terms of r and r0. Then form an integral...

## N

Of, 3-32 deflection of light by, D-8 kinetic energy hitting 3-28 pulsar, F-l circular orbits, 4-30-31 gravitation theory, 2-13 predicts horizon of black hole, 2-22 far-away, 1-18,2-35,2-38, F-3 in special relativity, 1-16-18 Schwarzschiid, 2-35 circular, 4-20,4-25, 4-28-32 computing, 4-9, F-30 forecasting, 4-8,4-23 knife-edge, 4-8-10, 4-19, 5-12-14 Newtonian, 4-30-31, C-4-6

## Selected Physical Constants

Speed of light in a vacuum Gravitational constant Planck's constant Electron charge Electron mass Radius of sphere having the same volume as Earth Mean distance of Earth from Sun 1 astronomical unit AU Mean speed of Earth in its orbit around Sun G 6.6726 x lCT11 meter3 kilogram-second2 h 6.6261 x 10-34 kilogram-meter2 second me 9.1094 x 10 31 kilogram 0.510999 MeV mp 1.67262 x 1027 kilogram 938.272 MeV Me 5.9742 x 1024 kilograms 0.444 centimeter rE 6.3710 x 106 meters 1 AU 1.495978 x 1011...

## Bnw

Note that when the viewing angle is 90 degrees for the shell observer, the Schwarzschild angle is also 90 degrees, since the tangent of 90 degrees is infinite for both. In the remainder of this chapter we deal primarily with the shell angle 0sheii of light propagation. Plotting the results on Schwarzschild maps such as Figures 7 and 8, however, requires the transformation of angles given by equation 29 , Relation between angles is Note that the derivation of equation 29 does not depend on a...

## Query

The period of Mercury's orbit is 7.602 x 106 seconds and that of Earth is 3.157 x 107 seconds. What is the value of Mercury's period in Earth-years Mercury's revolution in one century. How many revolutions around Sun does Mercury make in one century 100 Earth-years How many degrees of angle are traced out by Mercury in one century Correction factor. The mass M of Sun is 1.477 x 103 meters and the radius rQ of Mercury's orbit is 5.80 x 1010 meters. Calculate the value...

## Horizon Alarm for Your Spaceship

The Space Agency is anxious about the fate of their rocket ship and you and requests that you carry an automatic alarm designed to warn you when you are in danger of an irretrievable plunge into a black hole. The alarm should warn you when you are approaching the horizon. As a first cut at the design, assume that you are dropping from rest at a great distance and that the device has a register that allows you to enter the known value of the mass M of the black hole....

## Energy and Angular Momentum from Extremal Aging

Fix the end events of a short path vary the middle event to find constants of motion. When studying special relativity Chapter 1 , we were forced to use rela-tivistic expressions for energy and linear momentum, expressions different from those of Newton. Why did we accept the unfamiliar formulas Because only relativistic expressions satisfy laws of conservation of both total energy and total linear momentum in high-speed collisions and other interactions in an isolated system. What consolation...