Chapter VI: completes the preliminary course in the fundamentals of (11)

The way in which a curvature of space might appear to us as a force is made plainer by an example. Suppose that in a certain room a marble dropped anywhere on the floor always rolled to the center of the room; suppose the same thing happened to a baseball, a billiard ball, and a tennis ball. These results could be explained in two ways; we might assume that a mysterious force of attraction existed at the center of the floor, which affected all kinds of balls alike; or we might assume that the floor was curved. We naturally prefer the latter explanation. But when we find that in the neighborhood of a large material body all other bodies move toward it in exactly the same manner, regardless of their nature or their condition, we are accustomed to postulate a mysterious attractive force (gravitation); Einstein, on the contrary, adopts the other alternative, that the space around the body is curved.]223

In the ordinary "analytical geometry," the position and motion of all the points considered is referred to a rigid "body" or "frame of reference." This usually consists of an imaginary room of suitable size. The position of any point is then given by three numbers, i.e., its distances from one side wall and from the back wall and its height above the floor. These three numbers can only give one point, every other point having at least one number different. In four-dimensional geometry a fourth wall may be vaguely imagined as perpendicular to all three walls, and a fourth number added, giving the distance of the "point" from this wall also. Since "rigid" bodies do not exist in gravitational fields the "frame of reference" must be "non-rigid." The frame of reference in the Gaussian system need not be rigid, it can be of any shape and moving in any manner, in fact a kind of jelly. A "point" or "event" in the four dimensioned world is still given by four numbers but these numbers do not represent distances from anywhere; all that is necessary is that no two events shall have exactly the same four numbers to represent them, and that two events which are very close together shall be represented by numbers which differ only slightly from one another. This system assumes so little that it will be seen to be very wide in its scope; although to the ordinary mind, what is gained in scope seems more than that lost in concreteness. This does not concern the mathematician, however, and by using this system he gains his object, proving that the general laws of nature remain the same when expressed in any Gaussian coordinate system whatever.]220

Einstein enunciates a general principle that it is possible to find a transformation of coordinate axes which is exactly equivalent to any force, and in particular one which is equivalent to the force of gravitation. That is by concentrating our attention on the transformation which is a purely mathematical operation we can afford to neglect the force completely. To get a better idea of this principle of equivalence as it is called, let us consider a relatively simple example (which actually has nothing to do with gravitation, but which will serve to make our notions clearer.) A person on the earth unconsciously refers all his experiences, i.e., the motions of the objects around him to a set of axes fixed in the earth on which he stands. However, we know that the earth is rotating about its axis, and his axes of reference are also rotating with respect to the space about him. From the point of view of general relativity it is exactly because we do refer motions on the surface of the earth to axes rotating with the earth that we experience the so-called centrifugal force of the earth's rotation, with which everyone is familiar. If we could find it convenient to transform from moving axes to fixed axes, the force would vanish, since it is exactly equivalent to the transformation from one set of axes to the other. However, we find it unnatural to refer daily experiences to axes that are not placed where we happen to be, and so we prefer to take the force and rotating axes instead of no force and fixed axes.]272

We seem to have a direct experience of force in our muscular sensations. By pushing or pulling we can set bodies in motion. It is natural to assume, that something similar occurs, when Nature set bodies in motion. But is this not a relic of animism? The savage and the ancients peopled all the woods and skies with Gods and demons, who carries on the activities of nature by their own bodily efforts. Today we have dispossessed the demons, but the ghost of a muscular pull still holds the planets in place.]141

The general theory is an extension of the special theory which enables the law of gravitation to be deduced. Not in Newton's form, it is true, but in a better form, that is, one that accounts for two important facts otherwise not explained. But it is a far more general theory that indicated above. It is a complete study of the relations between laws expressed by means of any four coordinates (of which three space and one time is a special case), and the same laws expressed in the four coordinates of a system having any motion whatever with respect to the first system. By restricting this general study in accordance with certain postulates about the nature of the universe we live in, we arrive at a number of conclusions which fit more closely with observed facts that the conclusions drawn from Newton's theory.]221

NOTES

[1] It will be noted that Mr. Bolton pronounces the geometry of space to be Euclidean in the absence of gravitational fields, not that of space-time. This is in accord with what was pointed out on page 161.--Editor.

[2] The author here comes perilously close to ascribing to this "contraction" the sort of physical reality which it does not possess. See page 96.--Editor.

[3] Dr. Davis went rather fully into the algebra of the Michelson-Morley experiment. But Dr. Russell has covered the same ground in a form somewhat more advantageous from the typographical viewpoint, and the point is not one which it is profitable to discuss twice; so we eliminate this part of Dr. Davis' text.--Editor.

[4] This statement is objectionable, as explained in Chapter IV.--Editor.

[5] A. Einstein: Die Grundlage der allgemeinen Relativitätstheorie. Ann. d. Physik. 4, vol. 49, page 776.

[6] At this point we have again used the blue pencil on Dr. Davis' text, his discussion of the three observational tests of the General Theory adding nothing to Dr. Pickering's.--The Editor.

[7] 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.

[8] 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.

[9] 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.

[10] Not all gravitational fields may be transformed away by a proper choice of coordinates. If this were so, the space, whose nature is independent of any choice of coordinates, would always be Euclidean.--Author.

[11] Thus when it it said that a body contracts or that a clock runs slow when it is put in motion no actual physical change is implied. The judgment of different observers--one at rest with respect to the body and one not--are different.--Author.

[12] The balance of Dr. Royds' essay is given to a discussion of the phenomena of Mercury's perihelial advance, the deflection of light under the gravitational field of the sun, and the shift in spectral lines, in connection with which alone Einstein's theory makes predictions which are sufficiently at variance with those of Newtonian science to be of value in checking up the theory observationally. In the interest of space conservation and in the presence of Dr. Pickering's very complete discussion of these matters we omit Dr. Royds' statement.--Editor.

[13] There is, on any view, no difference as regards observation of position only.--Author.