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.
The heading below is linked to the original materials.
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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.
An inertial system is a coordinate system where objects not subject to forces move in straight lines at constant speed. This aligns with Newton’s first law and forms the foundation of special relativity. In the theory of relativity, the laws of nature are consistent with Lorentz transformation. This means that neither space nor time are absolute. And that points to changes in both velocity and inertia according to some law that we need to discover.
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 CS in which the laws of mechanics are valid. A body on which no external forces are acting moves uniformly in such a CS This property thus enables us to distinguish an inertial CS 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 CS“
The body moves uniformly only when its velocity is balanced by its inertia. In a system of bodies, the velocity and inertia of all bodies are in a dynamic equilibrium. This further expands the definition of inertial system. The only forces acting on the bodies are inertial forces. The gravitational forces come into play only when the dynamic equilibrium of a system is disturbed. Such gravitational forces act only to restore the equilibrium.
Here we could once more put the question: “What is an inertial CS?” 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 CS 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 CS rigidly connected with the sun can be regarded for many problems as an inertial CS; but when we speak of the rotating sun, we again understand that a CS connected with it cannot be regarded as strictly inertial.”
If a CS is rigidly connected with the earth, then it is part of earth’s inertial system. Earth is part of the larger inertial system of the solar system. As long as every particle is in equilibrium it is part of some inertial system.
“Then what, concretely, is your inertial CS, 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 CS would then be inertial.”
“But what do you mean by a CS free from all external influences?”
“I mean that the CS 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.
This interview reveals that the definition of inertial system needs to be expanded for a system of bodies, because no body is totally isolated.
We can approach this same difficulty from a different point of view. Try to imagine that there is only one body, forming our CS, 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 CS 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 CS.
The rotation of a single body in the entire universe shall contribute to its inertia (centeredness). The degree of centeredness shall determine its forward velocity on an absolute basis; because the forward velocity of a body with infinite inertia would be zero. Einstein thinks that it is not permissible to consider the motion of only one body in the entire universe; but the motion of a body can be considered relative to itself as in acceleration.
Take two bodies, the sun and the earth, for instance. The motion we observe is again relative. It can be described by connecting the CS with either the earth or the sun. From this point of view, Copernicus’ great achievement lies in transferring the CS 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 CS rather than the other.
We can differentiate the CS of sun from the CS of earth on the basis of their respective inertia. The forward velocity of earth shall be greater than the forward velocity of sun on an absolute basis because the inertia of sun is obviously greater than the inertia of earth. Einstein falsely believes, “But as motion is relative and any frame of reference can be used, there seems to be no reason for favouring one CS rather than the other.”
Physics again intervenes and changes our commonsense point of view. The CS connected with the sun resembles an inertial system more than that connected with the earth. The physical laws should be applied to Copernicus’ CS 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 CS connected rigidly with the sun for describing the motion of planets.
No absolute uniform motion exists in classical physics. If two CS are moving uniformly, relative to each other, then there is no sense in saying, “This CS is at rest and the other is moving”. But if two CS 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.
The common sense fourth coordinate of a CS shall be it rigidity or inertia by which the CS is centered in space. This is represented by time. Absolute uniform motion shall be based on zero motion for infinite inertia. Unfortunately, this is not directly recognized by the theory of relativity. It may be hidden under the math of relativity.
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 CS 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 CS, 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 CS. The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either CS 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 CS.
Could we build a real relativistic physics valid in all CS; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!
Einstein seems to think that an absolute measure of velocity is not possible; but, the theory of relativity does make it possible by using the velocity of light as an absolute reference point in reverse. A more straightforward absolute reference point shall be zero velocity at infinite inertia. Such a reference point shall be a black hole at the center of a galaxy.
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 CS and, therefore, also to the special case of the inertial CS. We already know the laws for this inertial CS. The new general laws valid for all CS must, in the special case of the inertial system, reduce to the old, known laws.
The problem of formulating physical laws for every CS 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 CS was previously the only one for which physical laws were formulated, so now it will form the special limiting case, as all CS moving arbitrarily, relative to each other, are permissible.
The special theory of relativity applies to inertial systems only; but the general theory of relativity formulates physical laws for all CS moving arbitrarily, relative to each other.
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 way the general theory of relativity is accomplished is quite vague and abstract. Einstein says, “The simpler and more fundamental our assumptions become, the more intricate is our mathematical tool of reasoning.” But our final aim is always a better understanding of reality.
Our new idea is simple: to build a physics valid for all CS. 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.
The new idea in general theory of relativity is to build a physics valid for all CS. But the way it is achieved is very abstract at the moment.
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Final Comment
Classical mechanics applies to inertial systems that are limited to the familiar material domain. Inertial systems comply with Galileo’s law of inertia that treats inertia as constant and velocity as uniform in all inertial systems. Motion in different inertial systems is related by addition of velocities according to the Galilean transformations. Such motion is perceived in a relative sense only. It has no absolute basis.
In his Special theory of relativity, Einstein introduces Lorentz transformations. His inertial frames are now constrained by the limit of a constant velocity of light. The special relativity gives somewhat 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. Non-inertial frames shall include rotation that will add to inertia, and acceleration that will overcome inertia. The proper physics that applies to all CS, shall quantify inertia and provide a relationship between velocity and inertia.
The general theory of relativity appears to develop an absolute law that accounts for varying inertia and velocity. That law appears to be based on a series of assumptions and logic that is hidden under abstract mathematical reasoning. The law is described only through a complex mathematical expression.
Any presence of “external force” would imply a system of at least two bodies that influence each other through a field that occupies the space between them. General relativity deals with this in terms of gravitation and geometry.
Postulate Mechanics looks at it as a dynamic equilibrium of inertia among the bodies that results in an ensemble of motion. This is our universe.
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