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 around stars, white dwarfs, neutron stars, and black holes.

Throw a stone and let it fall back to Earth. The stone follows a parabolic path in space, the solid curve in the diagram to the left in Figure 1. At the beginning and end of this path, fix two events in space and time: Event 1, initial launch; Event 2, final impact. Why does the stone follow the particular path in space between Event 1 and Event 2, shown as a solid line in Figure 1? Why not hurry faster along a higher parabolic path, the upper dashed line in Figure 1, to get back in time for the appointed impact? Or move slower along a lower parabolic path, the lower dashed line? Why not some entirely different trajectory between these two events? What command does spacetime give to the stone, telling it how to move?

Spacetime shouts, "Go straight!" The free stone obeys. What does "straight" mean? Straight with respect to what? We know the answer: The path of the stone is straight in a free-float frame. Ride in a free-float frame that rises and falls vertically in concert with the stone, as shown in the right diagram of Figure 1. With respect to the free-float frame, the stone moves on a straight path during the entire trip between launch (Event 1) and impact (Event 2).

Not only must the trajectory of the stone be straight in an inertial frame, but the stone must also move with constant speed as measured in that frame. Figure 2 shows a plot of the position of the stone (horizontal axis)

Thrown stone follows straight path in free-float frame.

"Straight" means straight worldline.

Figure 1 Parabolic path of a stone (solid line, left diagram) connecting launch (Event 1) and impact (Event 2). Dashed lines show alternative spatial paths between these two events, alternatives that the stone does not take. (Why not?) On the right is a free-float frame that rises and falls with the stone. With respect to this free-float frame, the stone follows a straight path. Plotting its motion as a function of time yields a straight worldline (Figure 2).

Figure 1 Parabolic path of a stone (solid line, left diagram) connecting launch (Event 1) and impact (Event 2). Dashed lines show alternative spatial paths between these two events, alternatives that the stone does not take. (Why not?) On the right is a free-float frame that rises and falls with the stone. With respect to this free-float frame, the stone follows a straight path. Plotting its motion as a function of time yields a straight worldline (Figure 2).

Figure 2 Spacetime diagram of the stone's worldline in the free-float frame that rises and falls with the stone (right diagram of Figure 1). This worldline is straight between launch (Event 1) and impact (Event 2) Intermediate dock ticks are shown as event points along the worldline. Curved dashed lines between events 1 and 2 represent alternative worldlines of smaller aging, alternative worldlines that the stone does not take. (I said, Why not?!)

as time passes (vertical axis). The line traced out by the motion of the stone as it changes spatial location as a function of time is called a worldline. Constant velocity results in a straight worldline. Nature's command to the stone in its general form is "Follow a straight worldline in a local inertial frame." No description could be simpler.

"Follow a straight worldline!" is the command by which spacetime grips stone carries a mass, telling it how to move. The stone carries a wristwatch. During the wristwatch trip the stone's wristwatch ticks off the time lapse between events of launch and impact. Between Event 1 and Event 2 in Figure 2, the wristwatch ticks off intermediate events along the worldline of the particle in the spacetime diagram of the free-float frame (event points on the straight worldline of Figure 2).

More About the Black Hole

The term "black hole" was adopted in 1967 (by John Wheeler), but the concept is old As early as 1783, John Michell argued that light must "be attracted in the same manner as all other bodies" and therefore, if the attracting center is sufficiently massive and sufficiently compact, "all light emitted from such a body would be made to return toward it." Pierre-Simon Laplace came to the same conclusion in 1795, apparently independently, and went on to reason that "it is therefore possible that the greatest luminous bodies in the universe are on this very account invisible."

Michell and Laplace used Isaac Newton's "action at a distance" theory of gravitation in analyzing escape of light from or its capture by an already existing compact object. (See the box "Newton Predicts the Horizon of a Black Hole?" on page 2-22.) But is such a static compact object possible? In 1939, J Robert Oppenheimer and Hartland Snyder published the first detailed treatment of gravitational collapse within the framework of Einstein's theory of gravitation Their paper predicts the central features of nonspinning black holes described in this book.

Ongoing theoretical study has shown that the black hole is the result of natural physical processes. A nonsymmetric collapsing system is not necessarily blown apart by its instabilities but can quickly—in seconds!—radiate away its turbulence as gravitational waves and settle down into a stable structure. In its final form a black hole has three properties and three properties only: mass, charge, and angular momentum. No other property remains of anything that combined to form the black hole, from pins to palaces. This absence of all detail beyond these three properties has led to the saying (also by Wheeler) "The black hole has no hair"

An uncharged nonspinning black hole is completely described by the Schwarzschild metric (the generalization of equation [1] to three spatial dimensions) derived in 1915 by Karl Schwarzschild from Einstein's equations for general relativity. The energy of a nonspinning black hole is not available for use outside its horizon. For this reason, a nonspinning black hole is called a "dead black hole."

In contrast to the spinlessness of a dead black hole, the typical black hole, like the typical star, has a spin, sometimes a great spin. The energy stored in this spin, moreover, is available for doing work for driving jets of matter and for propelling a spaceship. In consequence, the spinning black hole deserves and receives the name "live black hole " It has an angular momentum of its own

A spinning black hole—or any spinning mass, it turns out— drags around with it spacetime in its vicinity This "frame-dragging effect" is unquestionably measurable, even near our spinning Earth, by techniques now under development One technique employs a small gyroscope (a quartz ball 4 centimeters in diameter housed in Gravity Probe B, soon to be launched); the other uses a large "gyroscope" (a satellite orbiting Earth). In both cases the axis of the spinning gyroscope is dragged around by a fraction of a second of arc per year. To measure this tiny precession against background effects is the challenge. It will be exciting to see for the first time this new general relativistic effect. Theory predicts that near a rapidly spinning black hole, frame-dragging effects can be large, even inexorable, dragging along nearby spaceships no matter how strong their rockets

The metric for an uncharged spinning black hole was derived by Roy P. Kerr in 1963. (See Project F, The Spinning Black Hole.) In 1965 Ezra Theodore Newman and others solved the Einstein equations for the spacetime geometry around a charged spinning black hole. Subsequent theorems have proved that around a steady-state black hole of specified mass, charge, and angular momentum, Kerr-Newman geometry is the only solution to Einstein's field equations

Natural motion has maximum wristwatch time.

"Aging" measures total elapsed wristwatch time.

Natural motion implies extremal aging.

Einstein: There is no "gravitational force"!

How is the straight worldline different from all other possible worldlines that connect Event 1 and Event 2 (dashed lines in Figure 2)? We know the answer to that question too, from the Principle of Extremal Aging: The actual worldline has the longest wristwatch time of all possible worldlines between these two events. The free stone progresses uniformly from one event to the other, without jerks, jolts, or accelerations, thereby recording the longest possible time on its wristwatch between these two events. In contrast, a frantic traveler starting at the same Event 1 races at near-light speed to Moon, then streaks back in time for obligatory Event 2. The frantic traveler's wristwatch reads less elapsed time between Events 1 and 2 than does the wristwatch of the relaxed stone. The essential lesson of the Twin Paradox (Section 4 of Chapter 1) is that the natural motion between two events has maximum wristwatch time.

No frantic trip as far as Moon is necessary to demonstrate the basic principle: any deviation whatsoever from the straight worldline, no matter how small, leads to a shorter elapsed wristwatch time. The stone's wristwatch, accurate beyond all human timepieces, detects this difference and traces out the worldline of maximum wristwatch time. Wonder of wonders, the stone sniffs out and follows the worldline of maximum proper time without any wristwatch at all! How? Simply by going straight in local space time.

We use the word aging to describe the total elapsed proper time—the elapsed wristwatch time—along any worldline a particle takes from some initial event to another final event. Then the actual worldline the stone takes through spacetime is the worldline of maximal aging. Spacetime's command to the stone can be rephrased: "Follow the worldline of maximal aging!" From this simple command flows every description of motion in the remainder of this book. Amen.

Well, not quite "Amen." As described in Section 4 of Chapter 1, it is possible that the stone will follow a worldline not of maximal aging but of minimal aging. A noun that covers both cases is extremum. The corresponding adjective is extremal. The technical term for such a worldline is geodesic. To cover this unusual case, from now on we shift from maximum and maximal to the noun extremum and the adjective extremal. The Principle of Extremal Aging summarizes the result that a free particle follows a geodesic, a worldline of extremal aging. This fussy detail is presented for completeness. Tuck it away; all the worldlines we examine in this book, all our geodesies, result in maximal aging.

Principle of Extremal Aging (repeated): The path that a free particle takes between two events in spacetime is the path for which the time lapse between these events, recorded on its zoristwatch, is an extremum.

Figures 1 and 2 witness that, for slow speed and weak gravitational interaction, Newton's mechanics correctly describes the contrast between a straight worldline in spacetime and a curved path in space. So what is new about relativity? On the theory side, Einstein says that you can do away entirely with Newton's gravitational force, substituting instead the idea of geodesic: A free test particle moves along a worldline straight in spacetime as described with respect to every local free-float frame through which it passes. But the result may be a path curved in space as described by global Schwarzschild coordinates.

You claim that a worldline straight in every local free-float frame can nevertheless be curved In space, observed using the global Schwarzschild coordinates. You predict curved satellite orbits around a star. But your whole idea is obviously false; there is no way that tiny STRAIGHT motions can be added up to give overall CURVED motionI

Straight or curved? The description depends on the reference frame and on what kind of graph you draw. Figure 1 shows, in the left diagram, the curved path in space traced by a projectile observed with respect to Earth's surface and, in the right diagram, the straight path in space of the same projectile observed in a free-float frame. The projectile moves also with constant velocity in the free-float frame, a fact witnessed by the worldline of constant slope in Figure 2—the straight worldline in space-time The motion so described is as straight as it can possibly be—a geodesic. Yet for the Earth observer the path in space is curved.

Figure 1 describes motion confined to a local region of spacetime, where we can switch back and forth between a frame at rest with respect to Earth and a free-float frame in which spacetime is effectively flat for the flight time of the stone. In contrast, no free-float frame spans the entire orbit of Moon around Earth. Yet here too Moon moves straight in the spacetime of the local free-float frame. It follows a geodesic in spacetime, while its trajectory in space is curved.

General relativity stitches together the quilt squares of local free-float frames into a full quilt that covers wide regions of spacetime. Einstein predicts basically the same orbits as Newton does for motion near Earth and Sun. But even here Einstein corrects small discrepancies, while predicting motions different from Newton around compact stellar objects (and for the Universe as a whole!). In all known cases in which the two theories conflict, experiment verifies the predictions of general relativity. (See Projects A, C, D, and E.)

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