Einstein 1920: The Solution of the Problem of Gravitation

Reference: Einstein’s 1920 Book

This paper presents Part II, Chapter 12 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 Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity

If the reader has followed all our previous considerations, he will have no further difficulty in understanding the methods leading to the solution of the problem of gravitation.

We start off from a consideration of a Galileian domain, i.e. a domain in which there is no gravitational field relative to the Galileian reference-body K. The behaviour of measuring-rods and clocks with reference to K is known from the special theory of relativity, likewise the behaviour of “isolated” material points; the latter move uniformly and in straight lines.

In special relativity there is no gravitational field. Special relativity addresses particles in material domain moving uniformly in straight lines. It uses light as a reference frame to predict length contraction and time delay at higher velocities. This is saying that inertia increases with velocity. But this conclusion is just the opposite of what we observe happening in the field and material domains. Increase in inertia is accompanied by decrease in velocity.

The great contribution of special relativity is indicating that there is a dimension of inertia.

Now let us refer this domain to a random Gauss co-ordinate system or to a “mollusk” as reference-body K’. Then with respect to K’ there is a gravitational field G (of a particular kind). We learn the behaviour of measuring-rods and clocks and also of freely-moving material points with reference to K’ simply by mathematical transformation. We interpret this behaviour as the behaviour of measuring-rods, clocks and material points under the influence of the gravitational field G. Hereupon we introduce a hypothesis: that the influence of the gravitational field on measuring-rods, clocks and freely-moving material points continues to take place according to the same laws, even in the case when the prevailing gravitational field is not derivable from the Galileian special case, simply by means of a transformation of co-ordinates.

The great contribution of general relativity is to use a reference system that allows the dimension of inertia to be plotted. When we use the Gauss coordinate system of general relativity, it accounts for varying duration of substance, or the dimension of inertia. Einstein looks at this mathematically only, and does not interpret it in terms of inertia.

As inertia increases, the velocity decreases, and the path of quanta or particles starts to curve like in a whirlpool. The gravitational field comes about like a “whirlpool” to balance the acceleration.

The next step is to investigate the space-time behaviour of the gravitational field G, which was derived from the Galileian special case simply by transformation of the co-ordinates. This behaviour is formulated in a law, which is always valid, no matter how the reference-body (mollusk) used in the description may be chosen.

The law dictating the space-time behavior in a gravitational field shall, more properly, be the Law of Inertia.

This law is not yet the general law of the gravitational field, since the gravitational field under consideration is of a special kind. In order to find out the general law-of-field of gravitation we still require to obtain a generalisation of the law as found above. This can be obtained without caprice, however, by taking into consideration the following demands:

  • The required generalisation must likewise satisfy the general postulate of relativity.
  • If there is any matter in the domain under consideration, only its inertial mass, and thus according to Section XV only its energy is of importance for its effect in exciting a field.
  • Gravitational field and matter together must satisfy the law of the conservation of energy (and of impulse).

The general postulate of relativity requires consistency among all observations of the natural laws. Mass and energy are different only in terms of inertia. The conservation law that needs to be satisfied is the Conservation of Force as described by Faraday. Here “force” means the essence of substance, and “energy” means substance in motion. It is accounted for by the inertia-velocity relationship.

Finally, the general principle of relativity permits us to determine the influence of the gravitational field on the course of all those processes which take place according to known laws when a gravitational field is absent, i.e. which have already been fitted into the frame of the special theory of relativity. In this connection we proceed in principle according to the method which has already been explained for measuring-rods, clocks and freely-moving material points.

Special theory of relativity provides the relationship between inertia and velocity. The general theory adds curvature to the path and radial acceleration, which is then balanced by the gravitational field.

The theory of gravitation derived in this way from the general postulate of relativity excels not only in its beauty; nor in removing the defect attaching to classical mechanics which was brought to light in Section XXI; nor in interpreting the empirical law of the equality of inertial and gravitational mass; but it has also already explained a result of observation in astronomy, against which classical mechanics is powerless.

The equivalence between inertial and gravitational mass expands into the equivalence between inertia and gravity. The curved path of inertia provides acceleration one way. The gravity of gravitational field then provides the opposing acceleration. This general theory has been confirmed for consistency with observations to a greater degree than the classical mechanics.

If we confine the application of the theory to the case where the gravitational fields can be regarded as being weak, and in which all masses move with respect to the co-ordinate system with velocities which are small compared with the velocity of light, we then obtain as a first approximation the Newtonian theory. Thus the latter theory is obtained here without any particular assumption, whereas Newton had to introduce the hypothesis that the force of attraction between mutually attracting material points is inversely proportional to the square of the distance between them. If we increase the accuracy of the calculation, deviations from the theory of Newton make their appearance, practically all of which must nevertheless escape the test of observation owing to their smallness.

For smaller velocities and weak gravitational fields we obtain the Newton’s theory from general theory as the first approximation. The concept of distance does not come into play.

We must draw attention here to one of these deviations. According to Newton’s theory, a planet moves round the sun in an ellipse, which would permanently maintain its position with respect to the fixed stars, if we could disregard the motion of the fixed stars, themselves and the action of the other planets under consideration. Thus, if we correct the observed motion of the planets for these two influences, and if Newton’s theory be strictly correct, we ought to obtain for the orbit of the planet an ellipse, which is fixed with reference to the fixed stars. This deduction, which can be tested with great accuracy, has been confirmed for all the planets save one, with the precision that is capable of being obtained by the delicacy of observation attainable at the present time. The sole exception is Mercury, the planet which lies nearest the sun. Since the time Leverrier, it has been known that the ellipse corresponding to the orbit of Mercury, after it has been corrected for the influences mentioned above, is not stationary with respect to the fixed stars, but that it rotates exceedingly slowly in the plane of the orbit and in the sense of the orbital motion. The value obtained for this rotary movement of the orbital ellipse was 43 seconds of arc per century, an amount ensured to be correct to within a few seconds of arc. This effect can be explained by means of classical mechanics only on the assumption of hypotheses which have little probability, and which were devised solely for this purpose.

On the basis of the general theory of relativity, it is found that the ellipse of every planet round the sun must necessarily rotate in the manner indicated above; that for all the planets, with the exception of Mercury, this rotation is too small to be detected with the delicacy of observation possible at the present time; but that in the case of Mercury it must amount to 43 seconds of arc per century, a result which is strictly in agreement with observation.

The general theory accurately predicts the delicate rotation of the ellipse of mercury.

Apart from this one, it has hitherto been possible to make only two deductions from the theory which admit of being tested by observation, to wit, the curvature of light rays by the gravitational field of the sun, [Observed by Eddington and others in 1919. (Cf. Appendix III.)] and a displacement of the spectral lines of light reaching us from large stars, as compared with the corresponding lines for light produced in an analogous manner terrestrially (i.e. by the same kind of molecule). I do not doubt that these deductions from the theory will be confirmed also.

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FINAL COMMENTS

Einstein’s theory of relativity is not only accurate, but also seminal. But Einstein’s interpretation is mathematical and very abstract. So improvement in this theory can be made in terms of explaining it better.

The special theory of relativity predicted changes in the characteristics of space and time with velocity. The general theory predicted further changes in space-time with acceleration. Astronomical phenomena, which classical mechanics was unable to explain, could now be explained by the theory of relativity.

Descartes had declared space and time to be the characteristics of substance, According to him, if there were no substance, there was neither space nor time. Einstein did not agree with Descartes. But Einstein’s other discovery of quanta shows the field to be a much “diluted” form of substance, and this seems to confirm Descartes’ intuition.

We may then look at space as the “extent of substance,” time as the “duration of substance,” and inertia as the “innate force (substantive-ness) of substance.”  In this sense, changes in space and time characteristics shall imply changes in the inertia of substance.

Therefore, we may say that the great contribution of special relativity is indicating that there is a dimension of inertia. Light as a substance has very high velocity but extremely small inertia. On the other hand, matter has very high inertia but very low velocity.

Understandably, there is an inverse relationship between inertia and velocity. Inertia is how substantive (dense) the substance is; velocity is how rapidly substance moves across a location. The denser is the substance, the longer it shall take to move across a location. In other words, higher is the inertia, the lower shall be the velocity. This corrects the misinterpretation of special theory that length contraction and time delay occurs with increase in velocity.

The great contribution of general relativity is to use a reference system that allows the duration of matter particle to be plotted. When we use the Gauss coordinate system of general relativity, it is essentially accounting for the dimension of inertia. But Einstein interpreted it mathematically in a deeply abstract fashion.

The mathematics of General Theory of Relativity is very accurate, but it has lacked real explanation. That explanation may now be provided with the concept of inertia as time-density of substance. Inertia is inversely proportional to the natural velocity of the substance as described above.

As inertia increases, the velocity decreases, and the path of quanta or particles starts to curve like in a whirlpool. The curve exists due to a centripetal acceleration balanced by the “whirlpool” type structure of the gravitational field.

The law dictating the space-time behavior in a gravitational field shall be the conservation of Inertia. Mass and energy are different only in terms of inertia. The conservation law was described by Faraday as Conservation of Force.

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