## G2    The Equivalence Principle

What does Einstein do, when no logical explanation can be found for an experimental result? He makes that result the starting point for a new theory! He simply elevated the unexplained constancy of the speed of light (with Maxwell as his rear guard) to a basic principle and thereby founded the STR. In a similar manner he tackles gravity - he makes the equivalency of inertial and gravitational mass an axiom.

This equivalence principle of Einstein is so important that we will present various formulations of it (considering [29] we should call it the ‘strong equivalence principle’):

1. In principal, it is not possible in a local experiment to determine whether a laboratory is suspended in the gravitational field of a large body causing a gravitational acceleration g or whether it is gravitationally free and being subjected to a constant acceleration g.
2. There are no local experiments that can distinguish whether a laboratory is free falling in a gravitational field or whether it is resting unaccelerated in gravity-free space.
3. In a homogeneous gravitational field all operations run in exactly the same way as in a uniformly accelerated, but gravity-free reference frame.
4. A small laboratory in a gravitational field, falling freely and not rotating, is an inertial frame in the sense of the STR.
5. The effect of gravity can be locally produced (or reversed) by a suitable acceleration.

In the third formulation, the claim that the experiments should be ‘local’, that is, they should not stretch out over a ‘large’ area of space, is replaced by the requirement that the gravitational field should be ‘homogeneous’, which of course in all cases is valid only in a small area to a very good approximation. The third formulation is so vividly presented by Einstein himself in his popular presentation [30] of the relativity theories that it must have been a conscious reference to the description of phenomena in the ship's belly by Galileo (see quote in A2):

“We imagine a large portion of empty space, so far removed from the stars and other appreciable masses, that we have before us approximately the conditions required by the fundamental law of Galilei. lt is then possible to choose a Galileian reference-body for this part of space (world), relative to which points at rest remain at rest and points in motion continue permanently in uniform rectilinear motion. As reference-body let us imagine a spacious chest resembling a room with an observer inside who is equipped with apparatus. Gravitation naturally does not exist for this observer. He must fasten himself with strings to the floor, otherwise the slightest impact against the floor will cause him to rise slowly towards the ceiling of the room.

To the middle of the lid of the chest is fixed externally a hook with rope attached, and now a 'being' (what kind of a being is immaterial to us) begins pulling at this with a constant force. The chest together with the observer then begins to move 'upwards' with a uniformly accelerated motion. In course of time their velocity will reach unheard-of values - provided that we are viewing all this from another reference-body which is not being pulled with a rope.

But how does the man in the chest regard the process? The acceleration of the chest will be transmitted to him by the reaction of the floor of the chest. He must therefore take up this pressure by means of his legs lf he does not wish to be laid out full length on the floor. He is then standing in the chest in exactly the same way as anyone stands in a room of a house on our earth. If he releases a body which he previously had in his hand, the acceleration of the chest will no longer be transmitted to this body, and for this reason the body will approach the floor of the chest with an accelerated relative motion. The observer will further convince himself that the acceleration of the body, towards the floor of the chest is always of the same magnitude, whatever kind of body he may happen to use for the experiment.

Relying on his knowledge of the gravitational field (as it was discussed in the preceding section), the man in the chest will thus come to the conclusion that he and the chest are in a gravitational field which is constant with regard to time. Of course he will be puzzled for a moment as to why the chest does not fall in this gravitational field. Just then, however, he discovers the hook in the middle of the lid of the chest and the rope which is attached to it, and he consequently comes to the conclusion that the chest is suspended at rest in the gravitational field.

Should we smile at the man and say that he errs in his conclusion? I do not believe we ought to if we wish to remain consistent; we must rather admit that his mode of grasping the situation violates neither reason nor known mechanical laws. Even though it is being accelerated with respect to the 'Galileian space' first considered, we can nevertheless regard the chest as being at rest. We have thus good grounds for extending the principle of relativity to include bodies of reference which are accelerated with respect to each other, and as a result we have gained a powerful argument for a generalised postulate of relativity.

We must note carefully that the possibility of this mode of interpretation rests on the fundamental property of the gravitational field of giving all bodies the same acceleration, or, what comes to the same thing, on the law of the equality of inertial and gravitational mass. If this natural law did not exist, the man in the accelerated chest would not be able to interpret the behaviour of the bodies around him on the supposition of a gravitational field, and he would not be justified on the grounds of experience in supposing his reference-body to be 'at rest'.

Suppose that the man in the chest fixes a rope to the inner side of the lid, and that he attaches a body to the free end of the rope. The result of this will be to stretch the rope so that it will hang 'vertically' downwards. If we ask for an opinion of the cause of tension in the rope, the man in the chest will say: “The suspended body experiences a downward force in the gravitational field, and this is neutralised by the tension of the rope; what determines the magnitude of the tension of the rope is the gravitational mass of the suspended body.” On the other hand, an observer who is poised freely in space will interpret the condition of things thus: “The rope must perforce take part in the accelerated motion of the chest, and it transmits this motion to the body attached to it. The tension of the rope is just large enough to effect the acceleration of the body. That which determines the magnitude of the tension of the rope is the inertial mass of the body." Guided by this example, we see that our extension of the principle of relativity implies the necessity of the law of the equality of inertial and gravitational mass. Thus we have obtained a physical interpretation of this law.”  [30-75ff]

Maybe this long quotation will incite you to give one of the generally comprehensible texts from Einstein a try. Of course, there are also many drawings, animations, videos and DVDs, which portray these ideas for those not inclined themselves to do the reading. They are all nice and even funny – have a look at your leisure. It is amusing to watch Professor Albert together with his elevator comfortably bolting in free fall down the elevator shaft. I myself am much more interested in the drop tower in Bremen, where one can actually perform such free fall experiments for time intervals up to 9 seconds, than in these humorous drawings meant to illustrate a thought experiment.