§7.9: Equivalence

Let's consider a thought experiment:

Suppose you are "standing" inside a closed room. You feel odd, however, because you are floating weightless inside the room. Looking out a window, you see absolutely nothing. You surmise that the room is floating off somewhere in space, without a planet nearby to provide your weight from gravity. Donning your handy spacesuit and opening the door, you discover that your featureless view is actually the interior of a dark elevator shaft. The elevator cables have snapped, the safety brake has failed, and you are now plunging to the bottom. The weightlessness you took to be a lack of gravity is really free fall! Thinking quickly, you pull out the steel safety cable and friction brake that you always carry with you. Snagging the side of the elevator shaft with the hook on the end of the cable, you safely bring the elevator to a stop.

Suppose you had dropped your handkerchief while you were in free fall. From your perspective, it would have just hung motionless in space. (Someone outside the elevator would see you, the handkerchief, and the elevator all falling with the same acceleration.) If you had "pushed" on your handkerchief, it would have moved off at constant speed in a straight line, as far as you could see. So, the room behaves like an inertial frame to you.

From your perspective, there is no discernible difference between an inertial system and free fall in gravity (so long as you stay within short enough distances to keep clear of effects from the planet's curvature).

Now, let's look into another thought experiment:

You are back in the closed elevator again. The elevator is not moving. You can tell because your weight feels normal, not greater like an elevator going upwards quickly or smaller like an elevator going down. Eventually, you become impatient with waiting for the elevator to move and open the door to step out. Much to your surprise, there is nothing but empty space outside! Rushing into your spacesuit, you carefully peer around outside. Attached to the outside of the elevator car are small rocket motors, causing it to accelerate uniformly upward at precisely the normal acceleration of gravity on the Earth. The accelerating reference frame of the elevator car was indistinguishable from a gravitational field.

Einstein's Principle of Equivalence says that no physical experiment can distinguish a uniformly accelerating reference frame (relative to an inertial frame) from an inertial reference frame with a uniform gravitational field.

But wait, you say. Here is an experiment that can distinguish between these cases:

Suppose a beam of light comes into the elevator car through its window. Since light is made up of photons, which we learned are massless particles, it will not be effected by gravity. Thus, a light beam going through the elevator in a uniform gravitational field will travel in a straight line across the elevator. If, instead, the elevator is accelerating, then the elevator car will move as the light beam crosses it (as seen by an outside, inertial observer) and its apparent path inside the elevator will be curved. The two paths are not the same to the elevator's passenger!

Einstein would disagree with you. While a photon has no rest mass, it certainly has energy as stated in the photoelectric effect, E = hc/l. Since mass and energy are equivalent in the concepts of special relativity, the photon's equivalent mass is m = E/c2 = h/lc. While this mass is very small, gravity will still effect it. The only difficulty is in finding a gravitational field strong enough (i.e., a mass large enough) for the change in the light beam's path to be observable.

One possibility is to compare the apparent position of a star in the background of other stars in two circumstances: as it is usually seen in the night sky compared to when its position in the sky is near to the Sun. Of course, the star would be impossible to see because of all the light we receive from our Sun - except on the occasion of a total solar eclipse. When this experiment was tried during the eclipse of 1919, the measured deflection in apparent position of the star agreed with Einstein's prediction.