Einstein's Theories of Relativity and Gravitation / A selection of material from the essays submitted in the competition for the Eugene Higgins prize of USD5,000

NEW CONCEPTS FOR OLD

What the World Looks Like After Einstein Has Had His Way With It

BY JOHN G. McHARDY, COMMANDER R.N.,
LONDON

“The new-created world, which fame in heaven
Long had foretold, a fabric wonderful,
Of absolute perfection.”

Einstein’s Theory of Relativity has led to determining a key law of nature—the law of gravitation—which is also the basic law of mechanics. Thus it embraces a whole realm of physics, and promises, through the researches of Professor Weyl, to embrace another realm—electro-dynamics. Its limitations are not yet reached, for Einstein has already postulated therefrom a theory of a finite, yet unbounded, universe. This essay, however, is mainly concerned with mechanics, and electrical forces are not considered.

To have synthesised Newton’s two great principles—his law of motion and law of gravitation—interpreting in the process the empirical law of equality of gravitational and inertial mass, is alone an immense achievement; but Einstein’s researches have opened up a new world to the physicist and philosopher which is of greater importance. He has given us a vision of the immaterial world, a geometrical or mathematical vision, which is more satisfying than the “ether” conceptions hitherto presented. The fabric of his vision is not baseless. It is this fabric we shall consider, touching on certain aspects of the Einstein theory in the endeavor to present an image in miniature of his edifice of thought and to show the firmness of its foundations. That they are well and truly laid was demonstrated by the verification, from observations made during the solar eclipse in 1919, of Einstein’s prediction of the displacement of a wave of light in a gravitational field, showing light to have the property of weight.

The physical world is shown by Einstein to be a world of “relations.” Underlying it there is an absolute world of which physical phenomena are the manifestation. “Give me matter and motion,” says Descartes, “and I will construct the world.” “Give me a world in which there are ordered relations,” says the Relativist, “and I will show you the behavior of matter therein” (mechanics). We first view this underlying world as an abstraction, abstracting energy (“bound” as in matter and electrons, “free” as in light), and its attribute force. This abstraction we will call the “World-Frame.” Later, we will study the underlying world in connection with energy, and will call this absolute world the “World-Fabric.” The connection between the geometrical character of the World-Frame and the geometrical characters of the World-Fabric is the key to the law of gravitation.

The World-Frame

This is our conception of a world, if such were possible, entirely free from the influence of energy. We may conceive of it as an amorphous immaterial something containing “point-events” (a point-event being an instant of time at a point in space—a conception, not a definition). These point-events have a fourfold order and definite relation in this Frame, i.e. they can be specified by four variables or coordinates in reference to some base called a reference system, with respect to which they are forward or backward, right or left, above or below, sooner or later. This shows the World-Frame to be four-dimensional. Thus an aggregate of point-events (or an “event,” which implies limited extension in space and limited duration in time)1 would have what we familiarly describe as length, breadth, height and time. To express these metrical properties most simply we must choose a four-dimensional reference system having a particular form—rectilinear axes (Cartesian coordinates), and a particular motion—uniform and rectilinear, i.e. unaccelerated, and non-rotating with respect to the path of a light ray. We call this an inertial system because Newton’s Law of Inertia holds for such a system alone. This system indicates how observers partition the World-Frame into space and time. It restricts observers to uniform rectilinear motion, and observations to bodies and light-pulses in such motion. Thus gravitational and other forces are discounted, and we obtain World-Frame conditions notwithstanding the fact that observers are in the presence of energy.

Now the separation between point-events which have a definite relation to each other must be absolute. The separation between two points in a plane is defined by the unique distance between them (the straight line joining them). Between point-events the analogue of this unique distance, which we call the “separation-interval” (to indicate its time-like and space-like nature), is also unique. Its unique and absolute character give it great importance as thereby it is the same for all observers regardless of their reference system.

If, in place of the rather cumbersome expression upper X minus x to indicate the difference between the x-coordinates of two points, we employ the more compact expression dx; if for the benefit of readers who have a little algebra but no analysis we state explicitly that this expression is a single symbol for a single quantity, and has nothing to do with any product of two quantities d and x; and if we extend this notation to all our coordinates: then it is clear from previous essays that the distance S between two points in a plane referred to a rectilinear system OX, OY, is given by the simple equation upper S squared equals left-parenthesis d x right-parenthesis squared plus left-parenthesis d y right-parenthesis squared. Einstein and Minkowski show that the value for the separation interval normal upper Omega, the analogue of S, referred to an inertial system is given by the equation normal upper Omega squared equals left-parenthesis d x right-parenthesis squared plus left-parenthesis d y right-parenthesis squared plus left-parenthesis d z right-parenthesis squared minus left-parenthesis d t right-parenthesis squared comma

which is seen to be a modified extension to four dimensions of the equation for S. We must measure t in the same units as x, y, z. By taking the constant velocity of light (300,000 kilometres per second) as unit velocity, we can measure in length or time indiscriminately.2

We will analyse briefly this equation as it epitomizes the Special Theory of Relativity. If the World-Frame had been Euclidean the equation would have been normal upper Omega squared equals left-parenthesis d x right-parenthesis squared plus left-parenthesis d y right-parenthesis squared plus left-parenthesis d z right-parenthesis squared plus left-parenthesis d t right-parenthesis squared

but this would not satisfy the “transformation equations” which resulted from the Special Theory. These transformation equations arose directly from a reconciliation between two observed facts; (a) the observed agreement of all natural phenomena with the “Restricted Principle of Relativity”—a principle which shows that absolute rectilinear motion cannot be established—(as regards mechanics this was recognized by Newton; the Michelson-Morley and other experiments showed this principle also applied to optical and electro-dynamical phenomena); and (b) the observed disagreement of optical and electro-dynamical phenomena (notably the constancy of light velocity) with the laws of dynamics as given by classical mechanics, e.g., in regard to the compounding of relative velocities. Einstein effected this reconciliation by detecting a flaw in classical mechanics. He showed that by regarding space and time measurements as relative to the observer—not absolute as Newton defined them—there was nothing incompatible between the Principle of Relativity and the laws of dynamics so modified. Newton’s definitions were founded on conception. Einstein’s recognition of the relativity of space and time is based on observation.

Equation (1) shows that the geometry of the World-Frame referred to an inertial system is semi-Euclidean (hyperbolic), and that space and time measurements are relative to the observer’s inertial reference system. The equation shows that the World-Frame has a certain geometrical character which we distinguish as four-dimensional “flatness.” It is everywhere alike (homaloidal). Its flat character is shown by the straight line nature of the separation-interval and of the system to which it is most simply referred.

Thus we have found two absolute features in the World-Frame—(1) Its geometrical character—”flatness”; (2) The separation-interval—which can be expressed in terms of measurable variables called space and time partitions, this partitioning being dependent on the observer’s motion.

We are now in a position to explore the World-Fabric. Already we see that, studied under inertial conditions (free of force), it agrees with the World-Frame.

The World-Fabric

The General Theory of relativity is largely concerned with the investigation of the World-Fabric. Consider the World-Frame to be disturbed. We may regard this disturbance, which manifests itself in physical phenomena, as energy, or more correctly “action.”

When energy is thwarted in its natural flow, force is manifested, with which are associated non-uniform motions such as accelerations and rotations. This disturbed World-Frame we distinguish as the World-Fabric. It is found to have various non-Euclidean characters differing from the simple “flat” character of the World-Frame according to the degree of disturbance (action) in the region. Disturbance gives the fabric a geometrical character of “curvature”; the more considerable the disturbance, the greater the curvature. Thus an empty region (not containing energy, but under its influence) has less curvature than a region in which free energy abounds.

Our problem, after showing the relativity of force (especially gravitational force), is to determine the law underlying the fabric’s geometrical character; to ascertain how the degree of curvature is related to the energy influencing a region, and how the curvature of one region is linked by differential equations to that of neighboring regions. Such a law will be seen to be the law of gravitation.

We study the World-Fabric by considering tracks on which material particles and light-pulses progress; we find such tracks regulated and defined by the Fabric’s curvature, and not, as hitherto supposed, by attractive force inherent in matter. As a track is measurable by summing the separation-intervals between near-by point-events on it, all observers will agree which is the unique track between two distant point-events. Einstein postulates that freely progressing bodies will follow unique tracks, which are therefore called natural tracks (geodesics).

If material bodies are prevented from following natural tracks by contact with matter or other causes, the phenomenon of gravitational force is manifested relative to them. Whenever the natural flow of energy is interrupted force is born. For example, when the piston interrupts the flow of steam, or golf ball flow of club, force results—the interruption is mutual, and the force relative to both. Likewise when the earth interrupts the natural track of a particle (or observer) gravitational force is manifested relative to both.

So long as a body moves freely no force is appreciated by it. A falling aviator (neglecting air resistance) will not appreciate any gravitational force. He follows a natural track, thereby freeing himself from the force experienced in contact with matter. He acquires an accelerating motion with respect to an inertial system. By acquiring a particular accelerating motion an observer can annul any force experienced in any small region where the field of force can be considered constant.

Thus Einstein, interpreting the equality of gravitational and inertial mass, showed that the same quality manifests itself according to circumstances as “weight” or as inertia, and that all force is purely relative and may be treated as one phenomenon (an interruption in energy flow). This “Principle of Equivalence” shows that small portions of the World-Fabric, observed from a freely moving particle (free of force), could be treated as small portions of the World-Frame.3

If such observations were practicable, we could determine the Fabric curvature by referring point-event measurements to equation (1). We cannot observe from unique tracks but we can observe them from our restrained situation. Their importance is now apparent, because, by tracing them over a region, we are tracing something absolute in the Fabric—its geometrical character. We study this curvature by exploring separation-intervals on the tracks of freely moving bodies, relating these separation-intervals to actual measurements in terms of space and time components depending on the observer’s reference system. The law of curvature must be the law of gravitation. To illustrate the lines on which Einstein proceeded to survey the World-Fabric from the earth we will consider a similar but more simple problem—the survey of the sea-surface curvature from an airship. We study this curvature by exploring small distances on the tracks of ships (which we must suppose can only move uniformly on unique tracks—arcs of great circles), relating such distances to actual measurements in terms of length and breadth components depending on the observer’s reference system. This two-dimensional surface problem can be extended to the four-dimensional Fabric one.

We consider the surface to be covered by two arbitrarily drawn intersecting series of curves: curves in one series not intersecting each other, vide figure. This Gaussian system of coordinates is appropriate only when the smaller the surface considered, the more nearly it approximates to Euclidean conditions. It admits of defining any point on the surface by two numbers indicating the curves intersecting at that point. P is defined by x 1, x 2. upper P 1 (very near P) is defined by x 1 plus d x 1, x 2 plus d x 2. The equation for the minute distance s between two adjacent points in such a system is given by the general formula s squared equals g 11 d x 1 squared plus g 12 d x 1 d x 2 plus g 22 d x 2 squared period The g’s may be constants or functions of x 1, x 2. Their value is dependent on the observer’s reference system and on the geometrical character of the surface observed. The curves being arbitrary, the formula is appropriate for any reference system, or even if the observer does not know exactly what his reference system is. (The Fabric observer does not know what his space and time partitioning actually is because he is in a gravitational field). It is the g’s which disclose the geometry of an observer’s partitions, and their values also contain a reflection of the character of the region observed.

We find s by direct exploration with a moving ship (normal upper Omega is found by direct exploration with a freely moving particle); d x 1, d x 2 are the observed length and breadth measurement differences which we have to relate to s. By making sufficient observations in a small area and referring them to the general formula we can find the values of the g’s for the observer’s particular reference system. Different values for g’s will be found if the observer changes his reference system, but there is a limitation to the values so obtainable owing to the part played by the surface itself, which is diffidently expressing its intrinsic geometrical character in the g’s in each observation.

Einstein’s Results

Thus we approach the absolute character of the surface through the relative nature of the observer’s reference system. There is a relationship common to all values of the g’s that belong to the same curvature. This relationship is expressed by a differential equation. It is this equation of curvature that the airship’s observer must find. Einstein’s problem was similar, but he was concerned with four dimensions, which entailed a general formula with ten g’s, and he had to find a set of differential equations of the second order to determine the law of Fabric curvature. He divided the Fabric into regions: I. World-Frame—beyond influence of energy. II. Empty region—free of energy, but under its influence. III. Region containing free energy only. Each region has a characteristic curvature. By means of an absolute differential calculus—a wonderful mathematical scaffolding erected by Riemann, Christoffel and others—involving the theory of tensors, he succeeded in finding such a set of equations. He kept the following points in view: (1) The equations must not only give the character of region II, but must satisfy the special case of region I; (2) They must be independent of any partitioning system, because the General Theory of Relativity demands that a law of nature be in a form appropriate for all observers whatever their position and motion; (3) They must be concerned with energy which is conserved, not mass which the Special Theory showed dependent on velocity. This set of differential equations which shows how the curvature of the Fabric at any point links to the curvature at neighboring points is the law of gravitation, a law which has been severely tested by the practical observation of the solar eclipse already referred to. At a first approximation these equations degenerate into Newton’s Law. At a second approximation they account for the motion of the perihelion of Mercury, which had hitherto baffled astronomers. All the laws of mechanics are deducible from this law of World-Fabric curvature, i.e. conservation of energy (which includes conservation of mass since we re-define mass as energy) and conservation of momentum (re-defined by a relativist). It must be noted that this law and the General Theory show that the velocity of light is not absolutely constant, but, like everything else, a light-pulse is affected by the Fabric curvature in a gravitational field. In conclusion we will contrast some conspicuous differences in the old world view of classical mechanics and the new view presented by Einstein.

1. A three-dimensional ether medium with variously conceived properties which communicated the supposed inherent attractive force in matter in some unexplained way, and transmitted electromagnetic waves, has been replaced by a four-dimensional external World-Fabric, the geometrical character of which controls the motion of matter (energy) and accounts for all mechanical laws.

2. After separating the observer’s subjective share in definitions from nature’s share in the things defined, space, time, and force, hitherto regarded as absolute, have been shown to be purely relative and dependent on the observer’s track. Mass has also proved to be relative to velocity unless re-defined as energy. As classical mechanics bases all definitions on space, time, and mass units, the relativity of such defined quantities is now apparent.

3. Newton’s laws of motion, his law of gravitation, and the laws of conservation, hitherto regarded as unrelated, are now synthesised in a basic law of mechanics.

Einstein has not disturbed the electric theory of matter, and both the old and new physics have in common the “Principle of Least Action.” We obtain a glimpse of this principle in the unique tracks pursued by freely moving bodies, which may be regarded as tracks of least effort, force only being manifested as an expression of the Fabric’s resentment when bodies depart from these natural tracks. Einstein has approached nearer to the truth in regard to the laws underlying nature, and, as always, this means a simplification. His theory, which entails a readjustment of such fundamental conceptions as space and time, opens up fresh fields to scientific investigation and to philosophic thought. It reveals a bridge uniting the domains of physics and philosophy, and it heralds a new era in the history of science.

1 Commander McHardy uses the term “event” in a sense somewhat different from that seen in a majority of the essays. He reserves for the four-dimensional element—the instant of time at a point in space—the name “point-event”; and the term “event” he applies to a collection of these forming, together, an observable whole. An actual physical happening, like a railroad wreck or a laboratory experiment, it will be realized is of the latter sort, occupying an appreciable region of space rather than a single point, and an appreciable interval of time rather than a single second. To the element, the “point-event” of Commander McHardy’s essay, this bears the same relation that the geometer’s solid bears to his point. This comment is in no sense to be taken as criticism of Commander McHardy’s terminology, which rather appeals to us; we make it merely to guard against confusion in the reader’s mind.—Editor.?

2 This paragraph is the result of an editorial revision of the author’s text, designed to retain the substance of his presentation, while tying up what he has to say more definitely with the preceding essays, and eliminating the distinction between finite and infinitesimal intervals, which we believe to be out of place in an essay of this character. We will not apologize to our mathematical readers for having used finite and differential notation in the same equation, in violation of mathematical convention.—Editor.?

3 Although gravitational force in a small region can be imitated or annulled by accelerating motion, there remains the disturbing influence of gravitational matter already referred to and expressed in the fabric curvature. It is this that defines how unique tracks run, or rather, how bodies progress.—Author.?

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