The Equivalence Principle

We know that electric charge obeys a conservation law. That law tells us that the phenomena of Nature can neither create nor destroy electric charge; they can only rearrange it. Thus, when we put an electric charge on some object, we must of necessity put an equal amount of the opposite charge on some other object. When we describe that law mathematically we say that the amount of electric charge at some point cannot change as an inherent function of time, but can only change as an accidental function of time due to transportation of the charge. No phenomenon of Nature can change that fact, not even motion of the charged body.

We can reasonably expect that analysis to apply to any property of matter that generates a forcefield. But we seem to have a dilemma if we include gravitation in that statement. We know that changing the speed at which a body moves changes that body=s inertial mass. Now we need to ask whether the change in speed also changes the body=s gravitational mass (that aspect of the body=s mass that generates the body=s gravitational field) and, if so, how we can reconcile that fact with the statement that the source properties of forcefields do not change with changes in their motions.

Albert Einstein gained part of the information that we have thus asked for when he read a newspaper story about a carpenter falling off a roof. The carpenter had said that in the fraction of a second when he was falling he gained the impression of his tools floating weightless before him. That story almost certainly reminded Einstein of an experiment usually attributed to Galileo, in which experiment Galileo allegedly dropped balls made of different substances off the Leaning Tower of Pisa to demonstrate that the rate at which objects fall does not depend upon their composition.

We can interpret the Galileo experiment through two of Newton=s laws, his second law of motion and his law of gravitation. By Newton=s second law of motion we know that a body=s acceleration in response to an applied force depends upon that body=s inertial mass (through the statement that force equals mass multiplied by acceleration). And by Newton=s law of gravity we know that the force that Earth exerts upon any body depends directly upon that body=s gravitational mass, which also acts as the gravitational Acharge@ by which the body produces the gravitational field that attracts Earth in accordance with Newton=s third law of motion. The observation that bodies made of different substances fall at the same rate on Earth indicates an equivalence between a body=s gravitational mass and its inertial mass. Today physicists and engineers exploit that equivalence in describing the paths that bodies, such as artificial satellites and interplanetary probes, follow in a gravitational field: they don=t even bother with the forces, but calculate the accelerations directly by putting only the mass of the body (or bodies) producing the field into the inverse-square law.

But now we come to an important question. We know from Special Relativity that adding energy to a body increases that body=s inertial mass. Does it also increase the body=s gravitational mass? Does energy, in whatever form it may take, gravitate? To ask that question is to ask whether light and forcefields generate their own gravitational fields. How could we possibly hope to answer that question?

Whatever answer we come up with must conform to the requirements of the laws of physics, so if we can devise an answer through an imaginary experiment that obeys all of the relevant laws, we can accept it as valid. If light generates a gravitational field, then conservation of linear momentum necessitates that it respond to a gravitational field, that in some sense light must fall toward any gravitating body. So, in order to answer the above question, we need only prove and verify the proposition that gravity affects light. Since light consists of forcefields, that answer will apply to forcefields as well as to light.

Look again into the laboratory of your imagination and put the sun on the workbench. We know that our sun glows four megatonnes of light into space every second from its photosphere, but we only want to use a small part of that output. Assume that we have trapped a certain amount of sunlight at a specially-built helium cracking plant floating some distance above the photosphere. Now imagine taking the helium made in the fusions that created that light and lifting it out of the sun. We use the energy in that light to split the helium back into hydrogen. Finally we lower the hydrogen back down onto the sun, harnessing its gravitational energy, the work that its weight does upon our hoist, as we do so. In accordance with the law of conservation of energy, the energy in the hydrogen that we lower onto the photosphere must equal the total of the energy in the helium and the light at our helium cracking plant plus the energy that we gain from our hoist as we lower the hydrogen minus the energy that the hoist had to put into the helium when it raised it to our cracking plant.

Again we see how a conservation law allows to use rather simple bookkeeping techniques to solve a problem in physics. To simplify the analysis we will assume that the hydrogen originally underwent fusion into helium at the photosphere, rather than at the sun= s core. Now we see how the energy balance adds up:

1. At the cracking plant we have the mass-energy of the helium (A), minus the energy we put into our hoist to lift the helium out of the photosphere (B), and the energy in the light that we trap at our cracking plant (C).

2. At the photosphere we have the mass-energy of the hydrogen (D) minus the work done on the hoist in lowering the hydrogen (E).

The hydrogen that we lower onto the photosphere has slightly more mass than does the helium that it yields when it undergoes fusion. That difference in mass corresponds to the energy in the radiation emitted in the fusion (F). It also makes B smaller than E. Thus, the total energy we would measure at our cracking plant equals A-B+C and the energy that we would measure at the photosphere equals D-E=A+F-E. Those two numbers must equal each other (A-B+C=A+F-E), so, because B is smaller than E, we must infer that C is smaller than F; that is, the light that reached the cracking plant it must have carried less energy than it had when it left the sun. Therefore, the light must have done work against the sun=s gravitational field, which means that light responds to the gravitational force; that is, light has something equivalent to gravitational mass. Because the energy that light carries stands in direct proportion to the frequency of the radiation, physicists call this effect the gravitational redshift.

Now we know that we cannot discern a difference between inertial mass and gravitational mass in any phenomenon that carries energy. Indeed, now we can say that energy, rather than mass, generates a gravitational field and thus brings the production of gravity under the control of the conservation law pertaining to energy. That fact provides us with one of the foundations of the theory of General Relativity, the proposition that we call the Equivalence Principle. And that brings us back to the hapless carpenter, because his mishap gave the Equivalence Principle a special meaning for Einstein.

A man made of flesh, blood, and bone sees his wood-and-steel tools floating weightless before his eyes because both he and the tools accelerate at the same rate in Earth=s gravitational field. We now see that event as a manifestation of the Equivalence Principle. But Einstein also knew that weightlessness denotes a property that bodies possess in an inertial frame of reference. Here we see Einstein=s act of genius, in his reconception of the meaning of the Equivalence Principle. In Einstein=s imagination the carpenter and his tools did, indeed, occupy and mark an inertial frame of reference, but one that Earth=s gravity had warped out of true. That=s how Einstein reconceived gravity as a warping of space and time. Einstein then used that new concept as a stepping stone to the complete theory of General Relativity.

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