## Step or Orbit

"Go straight!" implies extremal aging. Extremal aging implies that energy and angular momentum are constants of the motion.

A stone in orbit streaks around a black hole—or around Earth. What tells the stone how to move? Spacetime grips the stone, giving it the simplest possible command: "Follow a straight worldline in the local inertial frame." From instant to instant this directive is enough to tell the stone what to do next, what next step to take in its motion.

Knowledge of its next step is enough for the stone, but it is not enough for us. We want more. We seek a global description of the trajectory of the stone through spacetime, a trajectory that carries it far beyond the confines of a single local inertial frame. The Principle of Extremal Aging answers our need. It converts the local ordinance "Go straight!" into a general executive order spanning spacetime from fixed initial event to fixed final event. "Between initial and final events," the Principle decrees, "follow the worldline of extremal aging." This worldline—the straightest possible worldline through flat or curved spacetime—is called the geodesic.

In this chapter we win from the Principle of Extremal Aging a simplified account of the orbit of a stone around a spherically symmetric center of attraction. This simplification uses quantities that do not change as the motion progresses. These unchanging quantities—constants of the motion—are energy and angular momentum. The expression for energy for a general trajectory is the same as that derived in Chapter 3 for the

Nature shouts at the stone "Go straight!"

Constants of the motion energy and angular momentum

 What Then Is Time? What then is time? If no one asks me, 1 know what it is. Time is defined so that motion looks simple. If 1 wish to explain it to him who asks me, 1 do not know. —Misner, Thome, and Wheeler The world was made, not in time, but simultaneously with