*inertial*mass (a relationship embodied in his famous formula

*E = mc*

^{2}). In 1907, Max Planck added the supposition that energy is also related to

*gravitational*mass. Perhaps, said Planck, pure energy, even without inertial mass, should experience gravitational force and exert gravitational force. In the same year, as a stepping stone toward general relativity, Einstein formulated similar ideas in a bolder way. Inertial effects and gravitational effects, he said, are not merely closely related, they are identical and indistinguishable.

Imagine passengers in two closed cars with the windows painted over. The first car, after moving forward with constant velocity, abruptly slows. The passengers are “thrown” forward. This is an inertial effect. In front of the second car, which is stationary, is suddenly placed a gigantic boulder of enormous mass (lets say it’s made of kryptonite). The passengers are “yanked” forward. This is a gravitational effect. Einstein hypothesized that the inertial and gravitational effects are indistinguishable, that no experiment performed within either car could reveal whether the passengers were being accelerated forward (relative to their vehicles) by inertia or by gravity. This is the Principle of Equivalence, stated first by Einstein in 1907. Elaborated over the next eight years, it formed a cornerstone of the theory of general relativity. An immediate consequence of this principle is the equality of inertial and gravitational masses, for otherwise the difference in these two kinds of mass would afford a means of distinguishing gravitational and inertial forces.

Let me add two more examples to help clarify the meaning of the Principle of Equivalence. Consider first an elevator falling freely down a very long elevator shaft without friction. To the passengers in the elevator (capable of careful observation) there are no external forces at all. A ball released from the hand will not fall to the floor of the elevator, for it will fall toward the earth just as the elevator itself and the passengers are falling, and to the passengers it will appear to float in air. Relative to the elevator, the ball satisfies Newtons first law, and the passengers, pursuing this and other experiments as they fall, will conclude that they are in an inertial frame of reference. Not at all, say the ground-based observers. The elevator is an accelerated system. It just happens that upward-acting apparent inertial forces cancel out the downward-acting real gravitational forces, and the passengers are deluded into believing themselves to be in an inertial frame with no forces acting. The weightless condition of astronauts in orbit about the Earth arises exactly from this cancellation of inertial and gravitational forces.

Or consider a spaceship, far from any sources of gravitational force, being uniformly accelerated by its rocket engine. This time a ball dropped from the hand will fall with uniform acceleration to the floor of the cabin. From the point of view of an outside observer, the ball remains motionless in space, but the cabin floor is accelerated upward to meet it. The outside observer recognizes that there are no real forces acting, only apparent forces arising from the acceleration of the spaceship. To observers within the spaceship, however, these forces seem quite real. The crew might decide that they are motionless, parked on some planet. According to Einsteins Principle of Equivalence, there is no experiment that the occupants of the falling elevator or the accelerating rocket could carry out that would unambiguously decide in favor of their point of view or the viewpoint of the outside observers. That gravity and acceleration through space produce identical physical effects is the startling assertion of the Principle of Equivalence.

To return to the passengers in the two cars: Those in the first car are thrown forward by an inertial force. Those in the second car are being tugged forward by the gravitational force of a hypothetical huge boulder. We “know” that inertial forces are unreal, that the passengers in the first car only think they are experiencing a force. The inertial “force” they feel is not a force at all, but an effect associated with the behavior of space and time in their frame of reference. If we accept the Principle of Equivalence, we must accept the idea that gravitational forces also are “unreal,” and are in some way merely properties of space and time. This is Einsteins powerful point of view, which led, in due course, to the “dynamics” of spacetime, to black holes, and to gravitational radiation.^{1}

1^{} There is a sense, too, in which the Principle of Equivalence led to the stunning achievement of the GPS navigational system. That system works only because it incorporates the effects of general relativity as well as special relativity.