Einstein 1920: The Space-Time Continuum of the General Theory

Reference: Einstein’s 1920 Book

This paper presents Part II, Chapter 10 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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The Space-Time Continuum of the General Theory of Relativity Is not a Euclidean Continuum

In the first part of this book we were able to make use of space-time co-ordinates which allowed of a simple and direct physical interpretation, and which, according to Section XXVI, can be regarded as four-dimensional Cartesian co-ordinates. This was possible on the basis of the law of the constancy of the velocity of light. But according to Section XXI, the general theory of relativity cannot retain this law. On the contrary, we arrived at the result that according to this latter theory the velocity of light must always depend on the coordinates when a gravitational field is present. In connection with a specific illustration in Section XXIII, we found that the presence of a gravitational field invalidates the definition of the co-ordinates and the time, which led us to our objective in the special theory of relativity.

The Law of “constancy of light” has simply been hiding the Law of Inertia that inertia varies inversely with velocity. The Law of Inertia is consistent throughout. We can retain the Cartesian coordinates in General relativity if we use inertia as a dimension and apply the Law of Inertia in that dimension.

In view of the results of these considerations we are led to the conviction that, according to the general principle of relativity, the space-time continuum cannot be regarded as a Euclidean one, but that here we have the general case, corresponding to the marble slab with local variations of temperature, and with which we made acquaintance as an example of a two-dimensional continuum. Just as it was there impossible to construct a Cartesian co-ordinate system from equal rods, so here it is impossible to build up a system (reference-body) from rigid bodies and clocks, which shall be of such a nature that measuring-rods and clocks, arranged rigidly with respect to one another, shall indicate position and time directly. Such was the essence of the difficulty with which we were confronted in Section XXIII.

Einstein is using rigid space-time (rods and clocks) as a substitute for material inertia.

But the considerations of Sections XXV and XXVI show us the way to surmount this difficulty. We refer the four-dimensional space-time continuum in an arbitrary manner to Gauss co-ordinates. We assign to every point of the continuum (event) four numbers, x1, x2, x3, x4 (co-ordinates), which have not the least direct physical significance, but only serve the purpose of numbering the points of the continuum in a definite but arbitrary manner. This arrangement does not even need to be of such a kind that we must regard x1, x2, x3 as “space” co-ordinates and x4 as a “time” co-ordinate.

Einstein is actually using the general coordinates x1, x2, x3, x4 to somehow incorporate the inertial dimension of the substance.

The reader may think that such a description of the world would be quite inadequate. What does it mean to assign to an event the particular co-ordinates x1, x2, x3, x4, if in themselves these co-ordinates have no significance? More careful consideration shows, however, that this anxiety is unfounded. Let us consider, for instance, a material point with any kind of motion. If this point had only a momentary existence without duration, then it would be described in space-time by a single system of values x1, x2, x3, x4. Thus its permanent existence must be characterised by an infinitely large number of such systems of values, the co-ordinate values of which are so close together as to give continuity; corresponding to the material point, we thus have a (uni-dimensional) line in the four-dimensional continuum. In the same way, any such lines in our continuum correspond to many points in motion. The only statements having regard to these points which can claim a physical existence are in reality the statements about their encounters. In our mathematical treatment, such an encounter is expressed in the fact that the two lines which represent the motions of the points in question have a particular system of co-ordinate values, x1, x2, x3, x4, in common. After mature consideration the reader will doubtless admit that in reality such encounters constitute the only actual evidence of a time-space nature with which we meet in physical statements.

Here the variable duration is expressed in terms of length of lines in this coordinate system. The length provides the measure of inertia. Intersection of these lines provides claim to physical existence.

When we were describing the motion of a material point relative to a body of reference, we stated nothing more than the encounters of this point with particular points of the reference-body. We can also determine the corresponding values of the time by the observation of encounters of the body with clocks, in conjunction with the observation of the encounter of the hands of clocks with particular points on the dials. It is just the same in the case of space-measurements by means of measuring-rods, as a little consideration will show.

Corresponding values of space and time can then be determined.

The following statements hold generally: Every physical description resolves itself into a number of statements, each of which refers to the space-time coincidence of two events A and B. In terms of Gaussian co-ordinates, every such statement is expressed by the agreement of their four co-ordinates x1, x2, x3, x4. Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference, without suffering from the defects of the latter mode of description; it is not tied down to the Euclidean character of the continuum which has to be represented.

Agreement of the Gaussian four coordinates is equivalent to space-time coordinates.

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FINAL COMMENTS

The Law of “constancy of light” has simply been hiding the Law of Inertia that inertia varies inversely with velocity. The Law of Inertia is consistent throughout. We can retain the Cartesian coordinates in General relativity if we use inertia as a dimension and apply the Law of Inertia in that dimension.

Instead of using the inertia-velocity relationship of Law of Inertia, Einstein uses the Gaussian coordinates to establish the space-time relationship with inertia..

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