Einstein 1920: Exact Formulation of the General Principle of Relativity

Reference: Einstein’s 1920 Book

This paper presents Part II, Chapter 11 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).

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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Exact Formulation of the General Principle of Relativity

We are now in a position to replace the provisional formulation of the general principle of relativity given in Section XVIII by an exact formulation. The form there used, “All bodies of reference K, K’, etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion,” cannot be maintained, because the use of rigid reference-bodies, in the sense of the method followed in the special theory of relativity, is in general not possible in space-time description. The Gauss co-ordinate system has to take the place of the body of reference. The following statement corresponds to the fundamental idea of the general principle of relativity: “All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.”

The fundamental idea of the general principle of relativity is: “All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.”

The above is a mathematical device only. It lacks the explanation that substance has a dimension of inertia, and rigid reference bodies do not cover the whole range of inertia.

We can state this general principle of relativity in still another form, which renders it yet more clearly intelligible than it is when in the form of the natural extension of the special principle of relativity. According to the special theory of relativity, the equations which express the general laws of nature pass over into equations of the same form when, by making use of the Lorentz transformation, we replace the space-time variables x, y, z, t, of a (Galileian) reference-body K by the space-time variables x’, y’, z’, t’, of a new reference-body K’. According to the general theory of relativity, on the other hand, by application of arbitrary substitutions of the Gauss variables x1, x2, x3, x4, the equations must pass over into equations of the same form; for every transformation (not only the Lorentz transformation) corresponds to the transition of one Gauss co-ordinate system into another.

Lorenz transformations cover the inertia of the material domain only, using the reference-body of light. This works because, compared to material domain, light can be approximated as a reference-body of “zero inertia”. Gaussian transformation takes a different approach rather than just being an extension of special relativity. But there is no explanation to that mathematical approach.

If we desire to adhere to our “old-time” three-dimensional view of things, then we can characterize the development which is being undergone by the fundamental idea of the general theory of relativity as follows: The special theory of relativity has reference to Galileian domains, i.e. to those in which no gravitational field exists. In this connection a Galileian reference-body serves as body of reference, i.e. a rigid body the state of motion of which is so chosen that the Galileian law of the uniform rectilinear motion of “isolated” material points holds relatively to it.

What is missing is the connection between the “old-time” three-dimensional view of things, and the new abstraction of mathematics by Einstein. A Galileian body of reference is a rigid (high inertia) body. Therefore, it cannot serve as a reference-body for the dimension of inertia. Special relativity uses light as the reference-body. It works great for sorting out inertia in material domain. But light also has some inertia; therefore it cannot serve as the reference-body for the whole dimension of inertia. To handle the complete dimension of inertia, we need a reference-body of no inertia. Mathematics should explain this.

Certain considerations suggest that we should refer the same Galileian domains to non-Galileian reference-bodies also. A gravitational field of a special kind is then present with respect to these bodies (cf. Sections XX and XXIII).

Gaussian coordinates act mathematically as a reference-body of zero inertia. This needs to be explained and related to our material world of infinite inertia.

In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity. The motion of clocks is also influenced by gravitational fields, and in such a way that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility as in the special theory of relativity.

Gravitational fields have lesser inertia than rigid bodies. A gravitational field condenses into rigid matter at their center. A level of substance (inertia) has a level of space (extents) and time (duration).  The appropriate reference-body shall have no substance, space and time. This is possible mathematically only but its relationship to reality must be explained.

For this reason non-rigid reference-bodies are used which are as a whole not only moving in any way whatsoever, but which also suffer alterations in form ad lib. (at one’s pleasure) during their motion. Clocks, for which the law of motion is any kind, however irregular, serve for the definition of time. We have to imagine each of these clocks fixed at a point on the non-rigid reference-body. These clocks satisfy only the one condition, that the “readings” which are observed simultaneously on adjacent clocks (in space) differ from each other by an indefinitely small amount. This non-rigid reference-body, which might appropriately be termed a “reference-mollusk,” is in the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily. That which gives the “mollusk” a certain comprehensibleness as compared with the Gauss co-ordinate system is the (really unqualified) formal retention of the separate existence of the space co-ordinate. Every point on the mollusk is treated as a space-point, and every material point which is at rest relatively to it as at rest, so long as the mollusk is considered as reference-body. The general principle of relativity requires that all these mollusks can be used as reference-bodies with equal right and equal success in the formulation of the general laws of nature; the laws themselves must be quite independent of the choice of mollusk.

The non-rigid reference-body may be referred to as reference-quanta, which is expressed through the Gaussian four-dimensional co-ordinate system. I do not think that Gaussian coordinate system is chosen arbitrarily. It must follow the Law of Inertia.

The great power possessed by the general principle of relativity lies in the comprehensive limitation which is imposed on the laws of nature in consequence of what we have seen above.

Maybe we can use Einstein’s mathematical approach to derive a relationship between inertia and velocity. We may then study this relationship and improve upon it as the mathematical expression for the Law of Inertia. We may then use this Law to come up with simpler mathematics for Gravitational fields.

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FINAL COMMENTS

The obfuscation of science started when Newton simply presented a mathematical formula for gravitational attraction without explanation. This obfuscation still continues with general relativity, even though the math has gotten better.

Matter is only one phase of substance. Light forms the other phase. Together light and matter present a wide range of substance. This is the dimension of inertia.

Inertia may be defined as time-density of substance. The denser is the substance the more duration it has. In the field domain, the “frequency” may provide a good measure of inertia.

The more is the duration, the longer it takes to traverse across a location in space. This makes inertia inversely proportional to velocity. The proportionality constant appears to be “1/c2”. Large change in velocity produces infinitesimal change in inertia. The exact relationship between inertia and velocity may be called the Law of Inertia.

Einstein’s mathematical approach with Gaussian transformation may help derive the Law of Inertia. This law may then help derive simpler mathematics for the gravitational field.

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