Einstein 1920: The Structure of Space According to the General Theory of Relativity

Reference: Einstein’s 1920 Book

This paper presents Part III, Chapter 3 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 Structure of Space According to the General Theory of Relativity

According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known. We know from experience that, for a suitably chosen co-ordinate system, the velocities of the stars are small as compared with the velocity of transmission of light. We can thus as a rough approximation arrive at a conclusion as to the nature of the universe as a whole, if we treat the matter as being at rest.

Matter is a substance of infinite inertia. It occurs at the upper end of the dimension of inertia. The geometrical properties of space are not only dependent on matter, but also determined by the inertia of substance. This is because space is the extent of substance. As the inertia of the substance increases, its velocity decreases. This appears as a “whirlpool” of substance. At the center of the “whirlpool” we have substance as solid matter. Because of the “whirlpool” phenomenon, the matter at the center spins about an axis.

This whirlpool is elliptical that is almost flat as visible in the shape of a galaxy. The same whirlpool model applies to the solar system. It is very likely that this model applies also to the atom and to the whole universe too.

We already know from our previous discussion that the behaviour of measuring-rods and clocks is influenced by gravitational fields, i.e. by the distribution of matter. This in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe. But it is conceivable that our universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun. We might imagine that, as regards geometry, our universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake. Such a universe might fittingly be called a quasi-Euclidean universe. As regards its space it would be infinite. But calculation shows that in a quasi-Euclidean universe the average density of matter would necessarily be nil. Thus such a universe could not be inhabited by matter everywhere; it would present to us that unsatisfactory picture which we portrayed in Section XXX.

Einstein is operating on the model of matter and void. He is considering the dimension of inertia only indirectly. It is true that inertia takes a big jump at the interface between “void” and matter. It increases very slowly in the “void” domain, and again quite slowly in the material domain.

If we are to have in the universe an average density of matter which differs from zero, however small may be that difference, then the universe cannot be quasi-Euclidean. On the contrary, the results of calculation indicate that if matter be distributed uniformly, the universe would necessarily be spherical (or elliptical). Since in reality the detailed distribution of matter is not uniform, the real universe will deviate in individual parts from the spherical, i.e. the universe will be quasi-spherical. But it will be necessarily finite. In fact, the theory supplies us with a simple connection1 between the space-expanse of the universe and the average density of matter in it.

Each atom is an elliptical whirlpool at the center of which the nucleus exists. That means a solid body is made if an infinite number of atomic whirlpools. But that solid body itself forms the center of a much larger whirlpool. This same model then scales up to solar and star systems, the galaxies, and finally the universe. The universe shall exist much farther outwards than its relatively solid core.

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FINAL COMMENTS

The final comments are pretty much the comments above.

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Einstein 1920: The Possibility of a “Finite” and Yet “Unbounded” Universe

Reference: Einstein’s 1920 Book

This paper presents Part III, Chapter 2 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.

.

The Possibility of a “Finite” and Yet “Unbounded” Universe

But speculations on the structure of the universe also move in quite another direction. The development of non-Euclidean geometry led to the recognition of the fact, that we can cast doubt on the infiniteness of our space without coming into conflict with the laws of thought or with experience (Riemann, Helmholtz). These questions have already been treated in detail and with unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on them briefly here.

The development of non-Euclidean geometry led to the recognition of the fact, that we can cast doubt on the infiniteness of our space without coming into conflict with the laws of thought or with experience.

In the first place, we imagine an existence in two-dimensional space. Flat beings with flat implements, and in particular flat rigid measuring-rods, are free to move in a plane. For them nothing exists outside of this plane: that which they observe to happen to themselves and to their flat “things” is the all-inclusive reality of their plane. In particular, the constructions of plane Euclidean geometry can be carried out by means of the rods, e.g. the lattice construction, considered in Section XXIV. In contrast to ours, the universe of these beings is two-dimensional; but, like ours, it extends to infinity. In their universe there is room for an infinite number of identical squares made up of rods, i.e. its volume (surface) is infinite. If these beings say their universe is “plane,” there is sense in the statement, because they mean that they can perform the constructions of plane Euclidean geometry with their rods. In this connection the individual rods always represent the same distance, independently of their position.

Let’s consider the reality of flat beings existing in two-dimensional space. What they observe to happen to themselves and to their flat “things” in their plane is their reality. To them their space will extend to infinity. It will be consistent with Euclidian geometry.

Let us consider now a second two-dimensional existence, but this time on a spherical surface instead of on a plane. The flat beings with their measuring-rods and other objects fit exactly on this surface and they are unable to leave it. Their whole universe of observation extends exclusively over the surface of the sphere. Are these beings able to regard the geometry of their universe as being plane geometry and their rods withal as the realisation of “distance”? They cannot do this. For if they attempt to realise a straight line, they will obtain a curve, which we “three-dimensional beings” designate as a great circle, i.e. a self-contained line of definite finite length, which can be measured up by means of a measuring-rod. Similarly, this universe has a finite area, that can be compared with the area of a square constructed with rods. The great charm resulting from this consideration lies in the recognition of the fact that the universe of these beings is finite and yet has no limits.

But if that two-dimensional existence is on a spherical surface instead of on a plane, then their geometry is no longer consistent with the Euclidean geometry. Their straight line will be a curve. Their universe will have a finite area. The universe of these beings is finite and yet has no limits.

But the spherical-surface beings do not need to go on a world-tour in order to perceive that they are not living in a Euclidean universe. They can convince themselves of this on every part of their “world,” provided they do not use too small a piece of it. Starting from a point, they draw “straight lines” (arcs of circles as judged in three-dimensional space) of equal length in all directions. They will call the line joining the free ends of these lines a “circle.” For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value, which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value

i.e. a smaller value than π, the difference being the more considerable, the greater is the radius of the circle in comparison with the radius R of the “world-sphere.” By means of this relation the spherical beings can determine the radius of their universe (“world”), even when only a relatively small part of their world-sphere is available for their measurements. But if this part is very small indeed, they will no longer be able to demonstrate that they are on a spherical “world” and not on a Euclidean plane, for a small part of a spherical surface differs only slightly from a piece of a plane of the same size.

These spherical-surface beings may not know that they are living on a spherical surface; but if they draw large enough circles, they will find the ratio of the circumference to its diameter is not constant. It becomes smaller as the radius increases. Thus, they can discover that they are not living in a Euclidean universe, and that their universe is finite.

Thus if the spherical-surface beings are living on a planet of which the solar system occupies only a negligibly small part of the spherical universe, they have no means of determining whether they are living in a finite or in an infinite universe, because the “piece of universe” to which they have access is in both cases practically plane, or Euclidean. It follows directly from this discussion, that for our sphere-beings the circumference of a circle first increases with the radius until the “circumference of the universe” is reached, and that it thenceforward gradually decreases to zero for still further increasing values of the radius. During this process the area of the circle continues to increase more and more, until finally it becomes equal to the total area of the whole “world-sphere.”

It may be difficult for them to discover this if they have access to only a very small part of their universe.

Perhaps the reader will wonder why we have placed our “beings” on a sphere rather than on another closed surface. But this choice has its justification in the fact that, of all closed surfaces, the sphere is unique in possessing the property that all points on it are equivalent. I admit that the ratio of the circumference c of a circle to its radius r depends on r, but for a given value of r it is the same for all points of the “world-sphere”; in other words, the “world-sphere” is a “surface of constant curvature.”

In this example, we assume the “world-sphere” to be a “surface of constant curvature because this makes all points equivalent.

To this two-dimensional sphere-universe there is a three-dimensional analogy, namely, the three-dimensional spherical space which was discovered by Riemann. Its points are likewise all equivalent. It possesses a finite volume, which is determined by its “radius” (2R3). Is it possible to imagine a spherical space? To imagine a space means nothing else than that we imagine an epitome of our “space” experience, i.e. of experience that we can have in the movement of “rigid” bodies. In this sense we can imagine a spherical space.

Riemann discovered a three-dimensional analogy of a spherical space. That means an object moving in a straight line in any direction will ultimately reach its starting point.

Suppose we draw lines or stretch strings in all directions from a point, and mark off from each of these the distance r with a measuring-rod. All the free end-points of these lengths lie on a spherical surface. We can specially measure up the area (F) of this surface by means of a square made up of measuring-rods. If the universe is Euclidean, then F = 4πr2; if it is spherical, then F is always less than 4πr2. With increasing values of r, F increases from zero up to a maximum value which is determined by the “world-radius,” but for still further increasing values of r, the area gradually diminishes to zero. At first, the straight lines which radiate from the starting point diverge farther and farther from one another, but later they approach each other, and finally they run together again at a “counter-point” to the starting point. Under such conditions they have traversed the whole spherical space. It is easily seen that the three-dimensional spherical space is quite analogous to the two-dimensional spherical surface. It is finite (i.e. of finite volume), and has no bounds.

It is easily seen that the three-dimensional spherical space is quite analogous to the two-dimensional spherical surface. It is finite (i.e. of finite volume), and has no bounds.

It may be mentioned that there is yet another kind of curved space: “elliptical space.” It can be regarded as a curved space in which the two “counter-points” are identical (indistinguishable from each other). An elliptical universe can thus be considered to some extent as a curved universe possessing central symmetry.

We can have the curved space to be “elliptical” in form too.  This shall look like a galaxy.

It follows from what has been said, that closed spaces without limits are conceivable. From amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it are equivalent. As a result of this discussion, a most interesting question arises for astronomers and physicists, and that is whether the universe in which we live is infinite, or whether it is finite in the manner of the spherical universe. Our experience is far from being sufficient to enable us to answer this question. But the general theory of relativity permits of our answering it with a moderate degree of certainty, and in this connection the difficulty mentioned in Section XXX finds its solution.

Thus, close spaces without limits are conceivable. From amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it are equivalent.

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FINAL COMMENTS

We as three-dimensional beings live in a material universe. We find it hard to imagine the fourth dimension of inertia in which the substance of variable durations has existence. This fourth dimension is not some abstract “time”. It is a dimension that accounts for variable duration characteristic (inertia) of substance. 

The fourth dimension is INERTIA and not some abstract notion of TIME.

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Einstein 1920: Cosmological Difficulties of Newton’s Theory

Reference: Einstein’s 1920 Book

This paper presents Part III, Chapter 1 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.

.

Cosmological Difficulties of Newton’s Theory

Apart from the difficulty discussed in Section XXI, there is a second fundamental difficulty attending classical celestial mechanics, which, to the best of my knowledge, was first discussed in detail by the astronomer Seeliger. If we ponder over the question as to how the universe, considered as a whole, is to be regarded, the first answer that suggests itself to us is surely this: As regards space (and time) the universe is infinite. There are stars everywhere, so that the density of matter, although very variable in detail, is nevertheless on the average everywhere the same. In other words: However far we might travel through space, we should find everywhere an attenuated swarm of fixed stars of approximately the same kind and density.

We may assume the whole universe to be flat like a galaxy rotating whirlpool like. It extends outwards beyond it relatively solid core. Therefore, it is unlikely to have a uniform density of matter.

This view is not in harmony with the theory of Newton. The latter theory rather requires that the universe should have a kind of centre in which the density of the stars is a maximum, and that as we proceed outwards from this centre the group-density of the stars should diminish, until finally, at great distances, it is succeeded by an infinite region of emptiness. The stellar universe ought to be a finite island in the infinite ocean of space. [Proof.—According to the theory of Newton, the number of “lines of force” which come from infinity and terminate in a mass m is proportional to the mass m. If, on the average, the mass-density P0 is constant throughout the universe, then a sphere of volume V will enclose the average mass P0V. Thus the number of lines of force passing through the surface F of the sphere into its interior is proportional to P0V. For unit area of the surface of the sphere the number of lines of force which enters the sphere is thus proportional to P0 · v/F} or P0R. Hence the intensity of the field at the surface would ultimately become infinite with increasing radius R of the sphere, which is impossible.]

The universe as a whirlpool is in harmony with both the theory of Newton and the general theory of relativity.

This conception is in itself not very satisfactory. It is still less satisfactory because it leads to the result that the light emitted by the stars and also individual stars of the stellar system are perpetually passing out into infinite space, never to return, and without ever again coming into interaction with other objects of nature. Such a finite material universe would be destined to become gradually but systematically impoverished.

On an infinite scale, light shall not be traveling in a straight line. It has some amount of inertia, so its path shall be curved. It will be part of an infinite whirlpool.

In order to escape this dilemma, Seeliger suggested a modification of Newton’s law, in which he assumes that for great distances the force of attraction between two masses diminishes more rapidly than would result from the inverse square law. In this way it is possible for the mean density of matter to be constant everywhere, even to infinity, without infinitely large gravitational fields being produced. We thus free ourselves from the distasteful conception that the material universe ought to possess something of the nature of centre. Of course we purchase our emancipation from the fundamental difficulties mentioned, at the cost of a modification and complication of Newton’s law which has neither empirical nor theoretical foundation. We can imagine innumerable laws which would serve the same purpose, without our being able to state a reason why one of them is to be preferred to the others; for any one of these laws would be founded just as little on more general theoretical principles as is the law of Newton.

Seeliger’s conception of the situation is not very satisfactory. Therefore, his recommended solution is not very satisfactory either.

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FINAL COMMENTS

Einstein is considering Seeliger’s paradox in this section. He finds Seeliger’s conception of the problem to be unsatisfactory.

Interestingly enough the general relativity provides a solution in terms of the universe being an infinite whirlpool-like field. On an infinite scale, The path of light shall be curved because it has finite amount of inertia. It will form a part of the infinite whirlpool.

Newton’s theory is modified in the sense that the path of light also has infinitesimal curvature.

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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.

.

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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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.

.

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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