Equivalence Principle

Relativity, The Special and The General Relativity

Part: THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS AS AN ARGUMENT FOR THE GENERAL POSTULATE OF RELATIVITY
(The Equality of Inertial and Gravitational Mass as a Proof for the General Postulate of Relativity)

BY
ALBERT EINSTEIN, Ph.D.
PROFESSOR OF PHYSICS IN THE UNIVERSITY OF BERLIN

TRANSLATION
Prof. Gülen Aktaş
Boğaziçi University, Istanbul

Illustrative Flash Images
Han Erim

Equivalence Principle

Let us consider a region of empty space so far removed from stars and other large masses that the conditions required by the fundamental Galilean law are approximately fulfilled. Thus, for this region of space (of the world) it is possible to choose a Galilean reference-body for which points that are at rest remain at rest, and points that are in motion continue in a state of uniform rectilinear motion.

As a reference-body let us imagine a box resembling a room, inside which an observer equipped with instruments is located. Naturally there is no such thing as gravitation for this observer; he must fasten himself securely to the floor with straps, otherwise the slightest collision with the floor would cause him to drift slowly up towards the ceiling.

Let there be affixed to the middle of the ceiling of the box, from the outside, a hook with a rope attached to it, and now some “being” (it is immaterial of what kind) begins to pull it with a constant force. The box together with the observer then enters a state of uniformly accelerated motion “upwards”. In the course of time their speed will reach unheard-of values – of course, all this being judged from another reference-body which itself is not being pulled by the rope.

But how will the man inside the box interpret this process? The acceleration will be communicated to him by the reaction of the floor of the box. In order not to lie full length on the floor he must counteract this pressure with his legs. Thus he will stand in the box exactly as he would in a room in a house on the Earth.

If the observer lets go of a body he is holding in his hand, since the acceleration of the box is no longer imparted to the body, it will approach the floor of the box in a relative accelerated motion. Whatever body he chooses for his experiment, he will convince himself that the acceleration of the body towards the floor of the box always has the same magnitude.

On the basis of his knowledge of gravitational fields, the man inside the box will infer that he and the box are situated in a gravitational field which is constant in time. For a moment he may hesitate over the question of why the box does not fall in this field. But then he will notice the rope which holds the box, and he will conclude that the box is suspended at rest in the gravitational field.

Should we laugh at this man and tell him that he has drawn a wrong conclusion? If we do not wish to fall into contradictions, I do not think we are entitled to do so. We must admit that his way of interpreting the situation conflicts neither with logic nor with the known laws of mechanics.

In spite of the fact that, with respect to the “Galilean space” initially assumed, the box is in a state of accelerated motion, we may nevertheless regard it as being at rest. Thus we have sufficient grounds for extending the principle of relativity so that it shall also apply to reference-bodies in accelerated motion with respect to one another. As a result, we have obtained a strong argument in favour of the generalized postulate of relativity.

We must carefully emphasize that such an interpretation is based on a fundamental property of a gravitational field which imparts the same acceleration to all bodies, or, which comes to the same thing, on the law expressing the equality of inertial and gravitational mass.

Let us now suppose that the man inside the box fastens a rope to the inner side of the ceiling and attaches a body to the free end of the rope. As a result, the rope will hang “vertically” downwards. When we ask about the cause of the tension in the rope, the man inside the box will say:

“The suspended body senses a force acting downwards in the gravitational field, and this force is balanced by the tension in the rope. The magnitude of the tension in the rope is determined by the gravitational mass of the suspended body.”

On the other hand, an observer freely floating in space will interpret the situation as follows: “The rope must necessarily play a role in the accelerated motion of the box, and transmits this motion to the body attached to it. The tension in the rope is sufficiently great to produce acceleration of the body. The magnitude of the tension in the rope is determined by the inertial mass of the body.”

From this example we see that extending the principle of relativity in this way forces upon us the law of the equality of inertial and gravitational mass. In this manner we have obtained a physical interpretation of this law.

From our discussion of the accelerated box we see that the general theory of relativity must have important consequences for the laws of gravitation. In fact, a systematic development of the general relativity idea has yielded the laws satisfied by the gravitational field.

Before going any further, however, I must warn the reader against a misunderstanding that might easily arise from these ideas. For the man inside the box there exists a gravitational field, although, relative to the coordinate system originally chosen, no such field is present.

It is therefore easy to imagine that the existence of a gravitational field might always be merely apparent. One might suppose that whatever the nature of the gravitational field, we could always choose another reference-body relative to which no gravitational field would exist at all. However, this is not true for all gravitational fields, but only for very special ones.

For example, it is impossible to choose a reference-body with respect to which the gravitational field of the Earth would be completely eliminated.

We can now see why the objection raised at the end of Section 18 against the general principle of relativity is not convincing.

It is true that a passenger in a train will be thrown forward when the brakes are suddenly applied, and that he will interpret this event as due to the non-uniform motion of the train. But no one is compelled to ascribe this forward fall to a “real” acceleration of the train.

He may also interpret his experience in the following way: “My reference-body (the train) remains continuously at rest. However, there exists, relative to it, a gravitational field directed forwards and varying with time (at the instant when the brakes are applied). Under the influence of this field the ground together with the Earth moves in such a way that its original backward-directed velocity continually decreases.”