## Einstein 1938: General Relativity

##### Reference: Evolution of Physics

This paper presents Chapter III, section 10 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

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## General Relativity

There still remains one point to be cleared up. One of the most fundamental questions has not been settled as yet: does an inertial system exist? We have learned something about the laws of nature, their invariance with respect to the Lorentz transformation, and their validity for all inertial systems moving uniformly, relative to each other. We have the laws but do not know the frame to which to refer them.

We are aware of the laws of nature but we do not have their frame of reference.

In order to be more aware of this difficulty, let us interview the classical physicist and ask him some simple questions:

“What is an inertial system?”

“It is a c.s. in which the laws of mechanics are valid. A body on which no external forces are acting moves uniformly in such a c.s. This property thus enables us to distinguish an inertial c.s. from any other.”

“But what does it mean to say that no forces are acting on a body?”

“It simply means that the body moves uniformly in an inertial c.s.”

Here we could once more put the question: “What is an inertial c.s.?” But since there is little hope of obtaining an answer differing from the above, let us try to gain some concrete information by changing the question:

“Is a c.s. rigidly connected with the earth an inertial one?”

“No, because the laws of mechanics are not rigorously valid on the earth, due to its rotation. A c.s. rigidly connected with the sun can be regarded for many problems as an inertial c.s. ; but when we speak of the rotating sun, we again understand that a c.s. connected with it cannot be regarded as strictly inertial.”

“Then what, concretely, is your inertial c.s., and how is its state of motion to be chosen?”

“It is merely a useful fiction and I have no idea how to realize it. If I could only get far away from all material bodies and free myself from all external influences, my c.s. would then be inertial.”

“But what do you mean by a c.s. free from all external influences?”

“I mean that the c.s. is inertial.”

Once more we are back at our initial question!

Our interview reveals a grave difficulty in classical physics. We have laws, but do not know what frame to refer them to, and our whole physical structure seems to be built on sand.

The classical frame of reference cannot account for rotation.

We can approach this same difficulty from a different point of view. Try to imagine that there is only one body, forming our c.s., in the entire universe. This body begins to rotate. According to classical mechanics, the physical laws for a rotating body are different from those for a non-rotating body. If the inertial principle is valid in one case, it is not valid in the other. But all this sounds very suspicious. Is it permissible to consider the motion of only one body in the entire universe? By the motion of a body we always mean its change of position in relation to a second body. It is, therefore, contrary to common sense to speak about the motion of only one body. Classical mechanics and common sense disagree violently on this point. Newton’s recipe is: if the inertial principle is valid, then the c.s. is either at rest or in uniform motion. If the inertial principle is invalid, then the body is in non-uniform motion. Thus, our verdict of motion or rest depends upon whether or not all the physical laws are applicable to a given c.s.

Take two bodies, the sun and the earth, for instance. The motion we observe is again relative. It can be described by connecting the c.s. with either the earth or the sun. From this point of view, Copernicus’ great achievement lies in transferring the c.s. from the earth to the sun. But as motion is relative and any frame of reference can be used, there seems to be no reason for favouring one c.s. rather than the other.

The inertial principle supports uniform motion and no acceleration. Visual sense of motion requires two bodies—one moving relative to another. Sense of motion with only one body requires the sense of acceleration. Motion as rotation provides that sense of acceleration in terms of being pulled by a centripetal force.

Physics again intervenes and changes our commonsense point of view. The c.s. connected with the sun resembles an inertial system more than that connected with the earth. The physical laws should be applied to Copernicus’ c.s. rather than to Ptolemy’s. The greatness of Copernicus’ discovery can be appreciated only from the physical point of view. It illustrates the great advantage of using a c.s. connected rigidly with the sun for describing the motion of planets.

No absolute uniform motion exists in classical physics. If two c.s. are moving uniformly, relative to each other, then there is no sense in saying, “This c.s. is at rest and the other is moving”. But if two c.s. are moving non-uniformly, relative to each other, then there is very good reason for saying, “This body moves and the other is at rest (or moves uniformly) “. Absolute motion has here a very definite meaning. There is, at this point, a wide gulf between common sense and classical physics. The difficulties mentioned, that of an inertial system and that of absolute motion, are strictly connected with each other. Absolute motion is made possible only by the idea of an inertial system, for which the laws of nature are valid.

Velocity refers to another body, but acceleration uses the single body itself as its reference. Therefore, acceleration is closer to being absolute motion. Here the body is its own reference.

It may seem as though there is no way out of these difficulties, as though no physical theory can avoid them. Their root lies in the validity of the laws of nature for a special class of c.s. only, the inertial. The possibility of solving these difficulties depends on the answer to the following question. Can we formulate physical laws so that they are valid for all c.s., not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over. We shall then be able to apply the laws of nature to any c.s. The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either c.s. could be used with equal justification. The two sentences, “the sun is at rest and the earth moves”, or “the sun moves and the earth is at rest”, would simply mean two different conventions concerning two different c.s.

Could we build a real relativistic physics valid in all c.s.; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!

But the theory of special relativity makes the motion absolute by using velocity of light as its reference point. What is missing is the relationship between motion (as acceleration) and inertia.

We have at least one indication, though a very weak one, of how to build the new physics. Really relativistic physics must apply to all c.s. and, therefore, also to the special case of the inertial c.s. We already know the laws for this inertial c.s. The new general laws valid for all c.s. must, in the special case of the inertial system, reduce to the old, known laws.

The problem of formulating physical laws for every c.s. was solved by the so-called general relativity theory; the previous theory, applying only to inertial systems, is called the special relativity theory. The two theories cannot, of course, contradict each other, since we must always include the old laws of the special relativity theory in the general laws for an inertial system. But just as the inertial c.s. was previously the only one for which physical laws were formulated, so now it will form the special limiting case, as all c.s. moving arbitrarily, relative to each other, are permissible.

The physical laws under general relativity reduce to those under special relativity for inertial systems.

This is the programme for the general theory of relativity. But in sketching the way in which it was accomplished we must be even vaguer than we have been so far. New difficulties arising in the development of science force our theory to become more and more abstract. Unexpected adventures still await us. But our final aim is always a better understanding of reality. Links are added to the chain of logic connecting theory and observation. To clear the way leading from theory to experiment of unnecessary and artificial assumptions, to embrace an ever-wider region of facts, we must make the chain longer and longer. The simpler and more fundamental our assumptions become, the more intricate is our mathematical tool of reasoning; the way from theory to observation becomes longer, more subtle, and more complicated. Although it sounds paradoxical, we could say: Modern physics is simpler than the old physics and seems, therefore, more difficult and intricate. The simpler our picture of the external world and the more facts it embraces, the more strongly it reflects in our minds the harmony of the universe.

The general relativity has, however, become mathematical and abstract and has moved farther away from reality.

Our new idea is simple: to build a physics valid for all c.s. Its fulfilment brings formal complications and forces us to use mathematical tools different from those so far employed in physics. We shall show here only the connection between the fulfilment of this programme and two principal problems: gravitation and geometry.

Formal complications arise in general relativity as it is not explained in terms of the reality we are familiar with.

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Classical mechanics applies to inertial systems. Inertial systems comply with Galileo’s law of inertia. This law treats inertia qualitatively and not quantitatively. These inertial systems are limited to the familiar material domain. Motion in different inertial systems is related according to the Galilean transformation. Such motion is perceived in a relative sense only. It has no absolute basis.

Special relativity replaces the Galilean transformation of classical mechanics by Lorentz transformation. The Lorentz transformation uses the velocity of light as the reference point for all motion. This makes measurement of motion closer to being absolute and more consistent. Hence special relativity gives better results; but it is still limited to inertial frames that deal only with non-varying inertia and uniform velocity. Inertial frames do not account for rotation and acceleration.

A universal theory of “relativity” is expected to work not only for inertial frames but also for the frames containing rotation, acceleration and variations of inertia. It will work not only in the material domain but also in the electromagnetic and the field domains. It will measure motion on an absolute basis. It shall not require external reference and will work with internal reference like inertia.

The last point about internal reference requires that we measure the velocity of a body based on its internal characteristics and not relative to other bodies. The acceleration of a body is based on its internal characteristics and so does its inertia. The condition of uniform velocity comes about when the acceleration of a body is balanced by its inertia. Considering the broad range of a universal theory from field to matter, it is obvious that motion is reciprocal of inertia. Therefore, the natural uniform velocity of a body may be described in terms of its inertia. This will truly be a universal theory because it does not require an external reference point.

The universe has no external reference. It acts as the reference to everything within the universe. It can provide an absolute basis to all motions including rotations within the universe. It provides an absolute scale from infinite velocity of zero inertia to zero velocity of infinite inertia.

We are aware of the laws of nature. We are now aware that their ultimate frame of reference is the universe itself. Here we may have the universal observer of general relativity. We hope that this universal observer can help relate the abstract mathematics of general relativity to a reality that we can grasp.

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