LECTURE IV THE CHARACTERISTICS OF SPACE A. THE GEOMETRY OF THE GRAVITATIONAL FIELD The starting point of the relativity theory is that the laws of nature, including the velocity of light in empty space, are the same everywhere and with regard to any system to which they may be referred — whether on the revolving platform of the earth or in the speeding railway train or in the space between the fixed stars. From this it follows that the length of a body is not a fixed property of it, but is relative, depending on the conditions of obser- vation— the relative velocity of the observer with regard to the body. It also is shown that the laws of motion of bodies in a gravitational field are identical with the laws of inertial motion with regard to an accelerating system (as exemplified by the billiard ball in the speeding railway train, Lecture I). From these two conclusions it follows that in the gravitational field the circumference of a circle is not equal to tt times its diameter, as we have learned to prove in our school geometry, but it is less than w times the diameter. As the theorems of mathematics depend upon each other, a change in one theorem involves a change in others. Thus from the theorem which we found to apply in a gravitational field, that ''the circumference of the circle is less than t times the diameter, and this the more so the larger the diameter," it follows that the ''sum of the angles in a triangle is greater than 180 degrees, and this the more so the larger the sides of the triangle." It also follows that any two lines in a plane intersect each other, 69 70 RELATIVITY AND SPACE that there exist no parallel lines, and that there exists no infinitely distant point on a straight line, but that all the points of a straight line are at finite distance and the total length of the straight line therefore is finite. Thus, going along a straight line in one direction, we come back to our starting point from the opposite direction, after going a finite distance — just as is the case in describing a circle. Just as the straight line is finite in length in this geometry of the gravitational field, so the plane is finite in area, though unlimited, that is, without boundary^ — like the surface of a sphere^ — and the volume of space is finite, though unlimited, and the conception of infinite distance or length or area or volume does not exist. By mathematical deduction from the relativity theory we thus derive the conclusion that our three-dimensional universe is not infinite, but finite, though inconceivably large. Although finite, it is limitless, just as the surface of a sphere is a two-dimensional space which is finite but limitless. We have always understood that mathematics is the most exact of all sciences and its theorems capable of abso- lute proof, and yet here in the gravitational field we find a space in which the proven theorems of our school geometry do not hold good any more. Mathematics is the most exact science, and its conclu- sions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical conclusions are relative, conditional. It is not correct to say, ''The sum of the angles in a triangle is 180 degrees," but the correct state- ment of the mathematical theorem is: "If certain premises or assumptions (the 'axioms') are chosen as valid, then the sum of the angles in a triangle equals 180 degrees." But whether these premises or axioms are "true" — that is, whether they are in agreement with physical experience — or not is no part of mathematics. The only requirements are that the number of axioms be sufficiently large to build conclusions or "theorems" on them and that they be THE CHARACTERISTICS OF SPACE 71 consistent with each other, that is, that one does not con- tradict the other. Thus we could well imagine the selection of another set of axioms, different from that of our school geometry, so as to lead to the theorem that ''the sum of the angles in the triangle is less than 180 degrees," or "is more than 180 degrees," etc. Our school geometry is built on a set of axioms which was selected by Euclid 2000 years ago, and therefore it is often called Euclidean geometry. Undoubtedly, Euclid was led by experience when selecting the axioms which he chose, and therefore the theorems of Euclidean mathe- matics have been in good agreement with physical experi- ence. This, however, is no part of mathematics. It would be a problem of mathematical physics to determine which set of axioms gives closest agreement between physical space and the mathematical theorems, in other words, what are the mathematical characteristics of physical space. If, then, experience shows that under certain conditions (in a gravitational field or in a centrifugal field) the characteristics of physical space do not agree with the theorems of Euclidean mathematics, it merely means that the set of axioms on which Euclid based his geometry does not apply to this physical space, and a different set of axioms, leading to a different, non-Euclidean geometry, has to be selected. In the realm of pure mathe- matics there is nothing new in this. Nearly a hundred years ago the great mathematicians of the nineteenth century, the Germans Gauss and Riemann, the Russian Lobatschewsky, the Hungarian Bolyai and others, investi- gated the foundations of geometry, which led them to the development of systems different from the Euclidean and based on different sets of axioms. Therefore, when finally, in the relativity theory, physics advanced beyond the range of Euclidean geometry, the mathematics of the new space characteristics was already fully developed. 72 RELATIVITY AND SPACE B. EUCLIDEAN, ELLIPTIC, HYPERBOLIC AND PROJECTIVE GEOMETRY Among the axioms of Euclidean geometry, such as: *'Two points determine a straight Une," "If two straight lines have two points in common, they have all points in common," "There are points outside of a straight line," *' There are points outside of a plane," etc., there is one axiom which appears less obvious, the so-called "parallel axiom." It is: "Through a point outside of a given straight line one, and only one, parallel line can be drawn" (the parallel line being defined as a line in the same plane which, no matter how far prolonged, never intersects the given line). The legitimacy of this axiom has always been doubted, and throughout all the centuries since Euclid numerous attempts have been made to "prove" this parallel law; that is, to show that it is not an axiom but a theorem, a conclusion from the other axioms. All these attempts failed, and finally the great mathematicians of the nine- teenth century attacked the problem from another side. Assuming that the parallel law is not an axiom, but a conclusion from other axioms, then we should be led to contradictions by choosing a different parallel law- — -for instance, assuming that there exist no parallels or that more than one parallel exists- — and developing the conclusions from this new assumption. On the other hand, if the par- allel law is a real axiom, then by assuming a different paral- lel law and developing the conclusions from it, we should get just as consistent a system of geometry as Euclid did, but one different from the Euclidean. Such systems were derived from the assumption stated, thus proving that the parallel law is a real axiom and not a conclusion from other axioms. In addition to the Euclid- ean geometry, which is based on the axiom of one parallel, a complete geometry (called the "hyperbolic geometry") was developed on the axiom that there exist more than one THE CHARACTERISTICS OF SPACE 73 parallel, and a complete geometry (the ''elliptic geometry") on the axiom that no parallel exists. The latter is the geometry shown by the relativity theory to apply in a gravitational field; the former applies in a centrifugal field. Furthermore, by leaving out the parallel axiom altogether and using only the remaining axioms of Euclid, a consistent geometry was developed, the ''geometry of position" (Geo- metrie der Lage) or "projective geometry," which is more general than the hyperbolic, elliptic and Euclidean geom- etries and includes these three as special cases. It deals exclusively with the relative positions of points, lines, fig- ures, etc., but not with size and measurement. Obviously it must do thus, since it must simultaneously fit all three conditions: C = ird, > 7r(iand<7rd. In the following tabulation I give some of the main char- acteristics of the four geometries:^ When the mathematicians of the nineteenth century had shown that Euclid's geometry is not the only possible one, but that two other geometries existed, the elliptic and the hyperbolic, fully as consistent as Euclid's, the question arose which of the three geometries completely represents the space of physical nature. The exact measurement of the angles in a triangle would determine this. If the physical space is non-Euclidean, the sum of the angles of the triangle would differ from 180 degrees, the more the larger the triangle. But there may be a slight departure of our space from Euclidean, which escapes notice, as the size of the triangle which we can meas- ure is limited to a few hundred million miles. ^ The mathe- maticians therefore used to speculate whether such a departure would be discovered if we could measure a tri- angle between some distant fixed stars with some hundred light-years as sides. The answer has now been given indirectly by the rela- tivity theory, showing that physical space varies between ^ Some of these properties will be explained later on. ^ The diameter of the orbit of the earth. 74 RELATIVITY AND SPACE General Euclidean Elliptic Hyperbolic geometry of or or or pseudo- position, or parabolic spherical spherical projective geometry Physical exist- Fieldless space. Gravitational Centrifugal field. General. ence. field. Characteristic of space .iC. Zero. Positive. Negative. None. Parallel axiom Through a point Any two straight Through a point The conception of of plane ge- outside of a lines in a plane outside of a infinity does not ometry. straight line intersect each straight line two exist, therefore one, and only other at finite straight lines also not the con- one, parallel line distance; that can be drawn ception of paral- can be drawn. is, parallels do not exist. which intersect the given straight line at infinite distance, and between these two lines exist an infinite number of straight lines through the point which do not intersect the given line at all. lels. Triangles The sum of the The sum of the The sum of the Comparison of angles in a tri- angles in a tri- angles in a tri- size, and meas- angle equals ISO angle is greater angle is less ure m e n t does degrees. than 180 de- than 180 de- not exist, and grees, the more grees, the more metric relations so, the larger so, the larger do not enter this the sides of the the sides of the geometry. triangle. triangle. Circles The circumfer- The circumfer- The circumfer- Metric relations ence of a circle ence of a circle ence of a circle do not exist. equals ir times is less than tt is more than ir The circle does the diameter. times the diam- times the diam- not exist as a eter, the more eter, the more distinct figure. so the larger the so the larger the but merely the diameter. diameter. general conic or curve of second order. Lines The straight line The straight line is finite in The straight line is infinite in is infinite in length and un- length, but un- length and un- limited. It has limited. It has limited. It has one infinitely no infinitely two infinitely distant point. distant point. distant points. Planes The plane is in- The plane is fi- The plane is in- finite in area nite in area, but finite in area and unbounded. unbounded. and unbounded. The conception of infinity does Space Space is infinite in volume and Space is finite in volume, but un- Space is infinite in volume and not exist. unlimited. limited. unlimited. Visual concep- Space visually Space visually Space visually tion. appears infinite appears infinite. appears finite. and is infinite. though it is finite. though it is in- finite. 2-d i mensional Plane (cylinder. Sphere (spin-die, Pseudo-sphere. ('(|uivalcnt. cone, etc.). etc.). THE CHARACTERISTICS OF SPACE 75 Euclidean at great distance from masses and elliptic space at masses, and that the average space characteristic thus is not zero, but has a slight positive value, is elliptic. It is curious that in speculating on the possible departure of space from Euclidean, in the last century, we expected to find it slightly hyperbolic. Why, I cannot remember. C. THE EARTH AS ELLIPTIC 2-SPACE Living in our physical space, which as far as our sense perceptions extend is Euclidean or zero-space, we cannot get outside of it to see how the world would look in an elliptic or hyperbolic space, and it therefore is difficult to get a conception of the two non-Euclidean forms of space, the positive or elliptic and the negative or hyperbolic. The only way in which we can get a partial conception is by analogy with two-dimensional spaces, or 2-spaces. We can produce the two-dimensional analogy of the two non- Euclidean spaces as surfaces, and, as three-dimensional beings, looking at these two-dimensional spaces from the outside, from a higher dimension we can see and compare their properties and characteristics with those of the Euclidean 2-space, that is, the plane. To illustrate: Suppose our earth were surrounded by a dense mass of clouds through which sun and moon could never be seen — about as it seems to be on the planet Venus. There would then be nothing to draw our attention to the earth being a sphere floating in three-dimensional space, and our world would practically be two-dimensional, lim- ited to the surface of the earth. The third dimension, the vertical, is accessible to us to a very limited extent only, so that we might forget it and for the moment think our- selves two-dimensional beings, limited to the surface of the earth. There would be no evidence to show us that the earth is not a flat plane. Indeed, in spite of all the evidence given by our view of the universe around it, by the sun and moon and stars and their motion, it took man many thou- 76 RELATIVITY AND SPACE sand years to come to the conclusion that the earth is round. On such an earth, cut off by clouds from any view of the universe, a Euclid might develop a geometry and would produce the same axioms and theorems as we have learned in school in plane geometry. That is, the straight line would still be defined in the same way, as the shortest dis- tance between two points, the sum of the angles in the triangle would be proven as 180 degrees, etc. Suppose now with the passing of time commerce extended on earth and ships traveled long distances. This would enable us to measure large triangles, for instance, that between New York, Rio de Janeiro and Liverpool. We would find that the sum of the angles of such a large triangle is not 180 degrees, as Euclid proved, but is materially larger. All the places lying 1000 miles from New York City we would find lying on a circle with 2000 miles diameter. But when measuring we would find the circumference of this circle materially shorter than r times 2000 miles. If we prolong a straight line on this two-dimensional surface of the earth, it does not extend into infinity, as Euclid claims, but has a finite length of 25,000 miles (the circumference of the earth), and then returns into itself. The surface of the earth is not infinite, as is that of the Euclidean plane, but is finite, though it has no limit. ^ If we prolong two paral- lel lines, we find that they approach nearer and nearer together and finally, after 6250 miles, intersect each other. 1 When the relativity theory leads to the conclusion that the extent of our universe is not infinite, but that the volume of the universe is finite, most people will ask, "What is beyond the finite extent of the universe?" and find it difficult to conceive that there is no "beyond," but that the universe, though finite, has no limits, or, in other words, that a finite volume can be all the universe. Just as for ages it was difficult for man to conceive that the earth on which we live should not be infinite, but finite in area and still have no limit or "edge," and kept asking for an "edge" of the world. We got over this by familiarity and have no difficulty to conceive a surface like that of the earth, which is finite in area but without boundary. Anal- ogous thereto is the universe of the relativity theory — finite in volume, but without limit or boundary. THE CHARACTERISTICS OF SPACE 11 In short, we find that the geometry on the surface of the earth is not the Euclidean geometry, but is the elliptic geometry of the mathematicians of the nineteenth century, the same which the relativity theory shows to apply in the gravitational field. When measuring the surface of the earth, we therefore cannot use the Euclidean geometry and the trigonometry corresponding thereto, the plane trigonometry of our school days, except on a small scale, as when surveying the lots of a city. When surveying countries and continents, we have to use the trigonometric formulas of the elliptic geometry, in which the sum of the angles of the triangle is greater than 180 degrees. Thus the non-Euclidean geometry which the relativity theory introduced into physics, after all, is nothing so very new. A part of the plane or two-dimensional elliptic geometry, its trigonometry, is in every day use in sur- veying the earth and is quite familiar. We all had it in school under the name of ''spherical trigonometry." Spherical trigonometry thus is the trigonometry of the eUiptic plane geometry. However, when studying spherical trigonometry, we usually take advantage of our being three-dimensional beings, and so do not limit ourselves to the surface of the sphere, but project outside of it into the third dimension. Naturally, in three-dimensional geom- etry we could not do this, as we have no fourth dimension beyond it. But, rigidly, we could develop the spherical trigonometry, and the geometry of the spherical surface, perfectly well without ever going outside of the surface into a higher or third dimension. So we have to do in three- dimensional geometry, where we have no higher dimension to go to. After all, we apply the same looseness of usage in plane geometry, for instance, when proving two figures as congruent by bringing them to coincidence. Sometimes we can do this by moving the one figure in the plane into coincidence with the other, but sometimes we cannot do this without turning the one figure over in the third dimen- sion— when two figures are symmetrical like the impres- 78 RELATIVITY AND SPACE sions of the right and the left hand. But, strictly, this is not permissible in plane geometry, and we should distinguish between congruent alike and symmetrical. In solid geom- etry we have to do so. A right-handed and a left-handed screw may be identical in every part, but still are not alike, as they cannot be brought into coincidence, because we cannot turn one over through a fourth dimension.^ The plane or two-dimensional elliptic geometry, therefore, is the geometry in the surface of a sphere, while the plane or two-dimensional Euclidean geometry is the geometry in a plane, and the plane or two-dimensional hyperbolic geom- etry is the geometry in a so-called pseudo-spherical surface. (Fig. 24.) This is rather disappointing. We were led to a new non- Euclidean geometry, in which the straight line has only a finite length, the plane a finite area, the angles of the tri- angles are more than 180 degrees, and other strange features exist, and then find that this is merely the geometry of the surface of a sphere, and the straight line of finite length is merely a largest circle, etc. We spoke of a straight line on the earth, and large tri- angles, like that between New York, Liverpool and Rio de Janeiro, in which the sum of the angles is greater than 180 degrees. But you will say that the path in which a ship travels from New York to Liverpool is not a straight line, but is a circle, a ''largest circle" of the earth, and the straight line between New York and Liverpool passes through the interior of the earth. The straight line between New York and Liverpool, which goes through the interior of the earth, is not a part of the earth's surface, but belongs to a higher dimensional space. What, then, is the straight line of the earth's surface? A straight line has been defined physically as the shortest distance between two points. In the two- 1 Thus, if a magician or spiritist claims the existence of a fovu'th dimension, ask him to prove it by taking a right-handed glove from some one in the audience and returning it as a left-handed glove. He would have to turn it over in the fourth dimension. THE CHARACTERISTICS OF SPACE 79 dimensional geometry of the earth's surface the shortest distance between two points is the largest circle, and this therefore is the straight line of the spherical surface. Or, a straight line has been defined as the path of a body mov- ing without any force acting on it. Such a body in a spherical surface moves in the largest circle. The mathe- matical definition of the straight line is 'Hhe line deter- mined by two points." This probably is the best definition. Analytically it means that the straight line is the line defined by an equation of the first degree. In the two- dimensional elliptic space which we call a spherical surface the largest circle therefore is the straight line and as seen fulfills all the characteristics of the straight line, and within the spherical surface the largest circle has no curva- ture ; that is, it is the same as our straight line in three- dimensional space is to us. Nevertheless, looking at the "straight line" of the spherical surface from the outside, from higher space, we see it curved, and therefore the mathematicians often use the term ''the straightest line." But in this respect it is in no way different from the straight line of our three-dimensional space which we get, as the shortest distance, by stretching a string between two points. We do not know whether what we call a straight line in our three-dimensional space would still be a straight line in a four-dimensional space of which our three-dimensional space was a part (and mathematically we can conceive such a space). Or, rather, when we speak of a straight line in our three-dimensional space, because it is determined by two points and is the shortest distance between these two points, we know that this line, from a four-dimensional Euclidean space of which our space is a part, is not always straight, is not always the shortest distance between two points, but may be curved, just as the straight line of the spherical 2-space is curved seen from 3-space. For instance, when the earth and the planet Mars are nearly at opposite sides of the sun, the straight line between the earth and Mars, which passes close to the sun, is straight 80 RELATIVITY AND SPACE only in our space, but as a line in a Euclidean 4-space, which contains our 3-space, it is curved^ and another line is a shorter distance between earth and Mars, but this other line passes out of our 3-space into 4-space and there- fore does not exist for us. As seen then, the ''straight line" has a meaning only with reference to the space in which it is defined, but a straight line of one space may not be a straight line for a higher space. Furthermore, while from and as a part of Euclidean 3-space, the elliptic 2-space appears curved, as a sphere, if we could look at it as a part of an elliptic 3-space of the same characteristic constant as the elliptic 2-space, the latter would not appear curved, as sphere, but flat, and the straight line of this elliptic 2-space would not appear curved, as circle, but straight. But an Euclidean plane and a straight line in it, as a part of elliptic 3-space, would appear curved (or as much of the plane or line as can be obtained in elliptic 3-space). Broadly then, any space, straight line or plane seen from and as part of a higher space of different characteristic appears curved, and it appears plane or straight only as part of a higher space of the same characteristic constant, as will be seen later. D. THE CHARACTERISTIC OR CURVATURE OF SPACE There is only one physical space for us, the space in which we are living, which is practically a Euclidean 3-space. We cannot go beyond this space, and therefore find it difficult to get a conception of the two non-Euclidean spaces, the elliptic and the hyperbolic, as we cannot look at them from the outside and see their characteristics and properties. The only way to get a partial conception of these non-Euclidean spaces is by analogy with the corre- sponding two-dimensional spaces, that is, surfaces. Just as the plane is the 2-space corresponding to our physical * Because it passes through the gravitational field of the sun, that is, a region of elliptic 3-space. THE CHARACTERISTICS OF SPACE 81 3-space, and as in school we first studied the geometry of the 2-space, or plane geometry, and then advanced to the geometry of the 3-space, or solid geometry, so we can construct and produce the elliptic and the hyperbolic 2-space, which correspond to the elliptic and hyperbolic 3-space in the same way as the Euclidean plane corresponds to the Euclidean or physical 3-space. These Euclidean, elhptic and hyperbolic 2-spaces we can, as three-dimensional beings, see from the outside and so get a complete conception of their characteristics and of the elliptic and hyperbolic plane geometry which corre- sponds to the Euclidean plane geometry. It is characteristic of the three types of 2-space or surface that the circumference of the circle is equal, larger or smaller than T times the diameter. That is, the quantity C jl-rr (where C is the circumference of a circle with radius r) equals one in Euclidean space, is less than one in elliptic and more than one in hyperbolic space. Thus the quantity 1 - C/27rr (1) equals zero in Euclidean space, is positive in elliptic and negative in hyperbolic space, and therefore to a certain extent characterizes the space. However, as the circum- ference of the circle differs from li^r the more the larger the diameter, the quantity (1) is not constant, but depends on the radius r. It can be shown, however, that the quantity^ - - K^ - 2^) (2) is independent of the numerical value of the radius r and is constant for each kind of space. It therefore is called the characteristic constant of the space, or sometimes, for reasons which we will see later, the curvature of the space. In Euclidean space the characteristic constant K is zero, and the Euclidean space therefore also is called zero space, or plane space, or parabolic space. 1 In this expression higher terms have been neglected, and it therefore is an approximation only, which holds when (1 — C/2irr) is a small quantity. 6 82 RELATIVITY AND SPACE In elliptic space the characteristic constant K is positive, and the larger the more the elliptic space differs from Eucli- dean. Elliptic space therefore may also be called positive space; that is, space with positive constant. In hyperbolic space the characteristic constant K is negative, and the larger the more the hyperbolic space differs from Euclidean. Hyperbolic space, therefore, may also be called negative space; that is, space with negative constant. Let a straight line be drawn through two points of the curve shown in Fig. 18 and these two points be brought Fig. 18. Fig. 19. infinitely near together. The line then becomes the tangent T of the curve and represents the direction of the curve at the point p where it touches. Let now a circle be drawn through three points of the curve and these three points be brought infinitely near together. The circle then osculates the curve, as seen in Fig. 19- — that is, touches it at three successive points — and therefore has at these points the same curvature as the curve. That is, it represents the curvature of the curve at its contact point p, and the radius R of this circle is the radius of curvature of the curve, and its reciprocal. /vi = 1/R, (3) is called the curvature of the curve. (The reciprocal of the radius is called the curvature, as a curve is called the more THE CHARACTERISTICS OF SPACE 83 curved the smaller its radius. Radius R = co gives zero curvature or the straight line.) When we come to surfaces, that is, two-dimensional spaces, we find that we have at every point of the surface an infinite number of tangents, in different directions, which all lie in a tangent plane. We also find that in every point we have an infinite number of osculating circles and there- fore radii of curvatures, which all lie in the normal to the surface at the point. Among this infinite number of radii are two radii at right angles to each other, Ri and R2, which are extremes, either one the shortest and the other the longest or both the shortest, but in opposite directions. The curvature of the surface at the point is measured by the product of the two main radii of curva- ture, and as the curvature of the surface is denoted the value: A% = I/R1R2. (4) It can be shown that the characteristic constant (2) of the 2-space, K, is equal to Ko (4) ; that is, is the curvature of the 2-space. That is: V " 2Tj)^RJf.^R'' ^^^ is the characteristic constant of the 2-space or the curvature of the surface which represents this 2-space. Instead of speaking of the characteristic constant of the space, we therefore often speak of the "curvature of space.'^ The Euclidean space thus is a space of zero curvature. The elliptic space is a space of positive curvature. The hyperbolic space is a space of negative curvature. Characteristic of all three spaces is constancy of curvature. R = VR1R2 = -^ (6) then may be called the radius of the space. In elliptic space, I = 27rR (7) 7 84 RELATIVITY AND SPACE is the length of the straight line. A = 4x7^2 (8) is the area or surface of the plane. V = 4.ir'R'/S ' (9) is the volume of the 3-space. In Euclidean space, K = 0, thus R = co ] that is, the radius of the Euclidean space is infinity. In the hyperbolic space K is negative, and R thus becomes imaginary; that is, the radius of the hyperbolic space is imaginary. Elliptic 2-space, therefore, is a surface having constant curvature of radius R; that is, it is a sphere, and the elliptic geometry is the geometry on a sphere of radius R = l/VK. (10) m In Euclidean 3-space the equation of a 2-space of con- stant curvature K is given by ;, |,. + ,. + (,_ _l.y 1 = 1. („) For K = 0 this gives : z = 0; (12) that is, the xy plane, or a Euclidean 2-space. For K differing from zero, shifting the coordinate center by l/^K gives: K {x' + y' + 2^) = 1. (13) For positive value of K this is a sphere of radius : R = l/VK. (14) For negative value of K, K = -l/R\ it is: x2 H- ^2 _^ ^2 - -R^; that is, all points x, y, z are imaginary. Thus in Euclidean 3-space no real (complete) hyperbolic THE CHARACTERISTICS OF SPACE 85 2-space exists, but the hyperbolic 2-space appears as a sphere with imaginary radius: R = j/VK. (15) Fig. 20. E. THE STRAIGHT LINE AND THE ELLIPTIC 2-SPACE In a surface or 2-space, as a plane, a line may be bent to the right, as a in Fig. 20, or to the left, as z. We can imag- ine the line gradually changed from a to b, c, etc., to z. In a it is bent to the right, in z to the left, and when changing from a to s it therefore must sometime pass through a position s, where it is not bent to the right any more, nor yet bent to the left — that is, where it is straight ; in other words, where it has no bend in the 2-space or as line or element of the 2-space and hence is a straight line of the 2-space, whatever the 2-space may be, whether a zero- space, that is, a Euclidean plane, or an elliptic 2-space, which seen from our Euclidean 3-space appears as a sphere etc. This straight line may be bent into a direction at right angles to the 2-space, out of the 2-space into a third dimension, for instance into the Euclidean 3-space (our physical space), from which we see the 2-space, and then would appear as a circle. Or it may appear straight even from the 3-space, this depending on whether the character- istic of the 3-space is the same or different from that of the 2-space. I may have a straight line L in an elliptic 2-space S, which is contained in our Euclidean 3-space (that is, a largest circle L on sphere S in Euclidean 3-space). I can put a Euclidean 2-space — that is, a Euclidean plane P— through the line L. For the elliptic 2-space >S, L is a straight line. For the Euclidean 2-space P, L is a circle. I may put another elliptic 2-space S\ of lesser characteristic 86 RELATIVITY AND SPACE constant^ — ^that is, a sphere of larger radius — through L. On S\ L will be curved, but less than on P. (But I could not put an elliptic 2-space of greater curvature through L; that is, seen from Euclidean 3-space, put through a circle a sphere of smaller radius than that of the circle, so that the circle lies on the sphere.) The elliptic 2-space S, as part of the Euclidean 3-space, appears curved, as sphere. But I could, at least mathe- matically, consider S as an elliptic 2-space contained in an elliptic 3-space of the same constant, and seen from this elliptic 3-space, S would not be curved, but would appear plane, as a flat plane, of finite area. The elliptic 2-space S would then be in common to the Euclidean 3-space and the eUiptic 3-space, and in the Euclidean 3-space S would be a sphere, but in the elliptic 3-space S would be a flat plane of finite area. Any straight line on S would also be a straight line for the elliptic 3-space, but would be a circle for the Euclidean 3-space, a "largest circle" of the sphere S. I may consider S as contained in an elliptic 3-space of lesser curvature than that of S. In this 3-space S would still appear curved, but less so than it appears in the Euclidean 3-space; that is, it would appear as a sphere of larger radius. But we could not consider the elliptic 2-space *S as a part of an elliptic 3-space of greater curvature than S. If we could con- sider the elliptic 2-space >S as a part of an elliptic 3-space of greater curvature than that of ^S, S would again appear curved, but not as sphere, but now with hyperbolic or negative curvature, and that would make it an imaginary sphere, as we have seen above. S could not as a whole be contained in the elhptic 3-space of greater curvature. The reason is that the straight lines in the 3-space are finite in length and shorter than the straight lines on the 2-space S, when the latter has a lesser curvature, and the latter therefore cannot be contained in the 3-space. Thus only a distorted part of S could find room in the 3-space, as pseudo-sphere. THE CHARACTERISTICS OF SPACE 87 In other words, a 2-space can be contained in a 3-space of lesser curvature, but not in a 3-space of greater curvature. Thus a hyperbohc 3-space can contain eUiptic 2-spaces and EucUdean 2-spaces and hyperbohc 2-spaces of lesser negative curvature; a Euclidean 3-space can contain only Euclidean and elliptic 2-spaces, but no complete hyperbolic 2-space, and an elliptic 3-space can contain only elliptic 2-spaces of the same or greater curvature, the former appearing as planes, the latter as spheres. It thus follows that, absolutely, there exists no such thing as a ''straight line," but "straight line" is relative only, with reference to the space in which it is defined. Any straight line with regard to a space of higher dimension than the space in which it is a straight line may not be a straight line; it is a straight line if the higher space has the same curvature, but is curved in a higher space of different curvature. This is exactly the case in our physical space, which as the relativity theory shows, has a slight positive curvature. If we could imagine our three-dimensional space as con- tained in and as a part of a four-dimensional Euclidean space (and mathematically there is no difficulty in this), then from this four-dimensional Euclidean space we would see that the straight lines of our space are really circles with about 100,000,000 light-years' radius. But the center of the circle and its curvature are outside of our 3-space, in the fourth dimension, exactly as the straight line of the elliptic 2-space is a circle seen from the Euclidean 3-space containing the elliptic 2-space as sphere, but a circle of which the center and the curvature are outside of the 2- space, and within the 2-space it has no curvature. Thus, also, what we see as a plane in our space, from the four-dimensional Euclidean space in which our space is contained, would be seen as a sphere with 100,000,000 light- years' radius, and our entire three-dimensional space would be a three-dimensional hypersurface, finite but unlimited, in the Euclidean 4-space. But while we mathematically 88 RELATIVITY AND SPACE can conceive of such a "hyper surface," physically we cannot. F. BENDING OF SPACE Suppose in a Euclidean 2-space — that is, a plane like the sheet of paper on which is printed Fig. 21 — we have two points, Pi and P^. Through these two points we can put one, and only one, straight line, Lq. This is the shortest distance between the two points Pi and Po, and any other line between them, as Li or L2 — shown dotted in Fig. 21 — Fig. 21. is longer. Suppose we have a straight line L in the plane Fig. 21 and a point P outside of L. Any line drawn in the plane through point P, as Li, Lo, L3, etc. (shown dotted), intersects the line L, except one line Lo, which no matter how far it is produced does not intersect L and is called the parallel. From this it follows that in a triangle ABC the sum of the angles equals 180 degrees, and that in any circle the circumference C is tt times the diameter DOE of the circle. THE CHARACTERISTICS OF SPACE 89 Suppose now we take the plane of Fig. 21- — -the sheet of paper — and bend it in any desired manner, but without stretching it. We may bend it into a cylinder or a cornu- copia—that is, a cone — ^or into a corrugated sheet as shown in Fig. 22.1 Looking at it now, from outside space, the lines L, Lo, Li, the sides of the triangle ABC, etc., are not straight any more, but curved. But within the plane of the paper the dimensions all remain the same, and if we measure the sides of the triangle or the circumference of the circle, the angles, areas, etc., with a rule or measure con- tained within the plane of the paper (that is, within the 2-space, the rule therefore in the 3-space bending with the paper) , we then find exactly the same measurements as before we bent the plane into a cylinder or corrugated shape. The lineLo, though not appearing straight anymore from the out- side, still is, within the sheet of paper, shorter than the lines Li and L2 and still is the shortest path between the points Pi and Pi — that is, still is the ''straight line" of the 2- space- — ^and Lo still never intersects L — ^that is, remains the parallel. Thus all the geometry which we derived and proved in the Euclidean plane (Fig. 21) still holds just the same in a cylinder, a cone, a corrugated sheet or any other 2-space which we may produce by bending this Euclidean plane, and there is no way and no possibility to show and prove from the inside of the 2-space (Figs. 21 and 22) whether we have bent it out of shape or not. Indeed, there is no such thing as bending the 2-space per se out of shape, but we have bent it only with regard to its location with respect to the 3-space from which we look at it. The same thing applies to an elliptic or hyperbolic 2- space. We may take a piece of a sphere (Fig. 23 or 27) and bend it — always without stretching — into a spindle, as in * This figure and some of the following ones are printed stereoscopically, so that the reader may take them out and look at them through a stereo- scope to see the curvature. With a little practice it is possible to see stereoscopically without a stereoscope. 90 RELATIVITY AND SPACE Fig. 28. All dimensions measured within the surface have remained the same, and a two-dimensional being living within the surface could never find out whether he was in the surface of a sphere, or of a spindle, or of some other shape produced by bending. The straight lines of the sphere — that is, seen from the outside, its largest circles — also are the straight lines of the spindle — that is, the short- est lines between two points; the angles are the same, etc. Thus the characteristic constant, or the curvature of the space, remains unchanged by the bending of the space. Euclidean 2-space thus is the plane and any surface made by bending it or a part of it in any desired manner — • into cylinder, cone, wave surface, etc; elliptic 2-space is the sphere and any surface made by bending a piece of the sphere into some other shape, as a spindle; hyperbolic 2-space is the pseudo-sphere — ^not existing in Euclidean 3-space — or any surface which can be considered as made by bending a piece of the pseudo-sphere into some other shape as shown in Fig. 24. But it must be bending without stretching. If, for instance, in bending the plane Fig. 21 I stretch it at some places, contract it at others, the line Lq between P1P2 of Fig. 21 may be stretched and so may become longer than lines Li or L2, and therefore will cease to be the shortest line — that is, the straight line of the bent space^ — ^and the geometry of this space would not remain the same. This illustrates the relativity of the conception ''straight line." The shortest path between two points Pi and P2 of the cylinder or cone looks very different to us from a straight line, and still it is geometrically identical with the straight line of our plane geometry; and it becomes like it in looks on simply um-olling the cone or cylinder — an opera- tion which makes no change whatever within the surface, but merely changes its relation to a higher dimensional space. However, when we bend a 2-spacc into some other shape we may get ''kinks" or "singular points" into it, and at THE CHARACTERISTICS OF SPACE 91 these singular points conditions may become indefinite; and in the geometry on a surface with such singular points we have to take care to arrange our figures so that they keep away from the singular points. For instance, w^e can bend a plane into a cone, and the geometry on the cone therefore is identical with the plane geometry of our school days, but when drawing figures on the cone — ^straight lines like the lines traced by a string stretched on the cone — we must keep away from the point of the cone. So the geometry on the spindle is the geometry on the sphere, but while on the sphere we may draw the figures anyw-here, on the spindle we have to keep the lines of the figure away from the two points of the spindle, as singular points. This possibility of bending a space into some other form without change of its constant permits us to illustrate hyperbolic geometry on a real surface. As has been seen, a complete hyperbolic 2-space cannot exist in Euclidean 3-space. But in Fig. 24 is shown a picture of a hyperbolic 2-space which bears to the complete hyperbolic 2-space about the same relation as the cone in Euclidean 2-space bears to the plane. It is Beltrami's pseudo-sphere, the rotation surface of the tractrix^ as meridian curve. As seen, Beltrami's pseudo-sphere has a singular point at infinity- — that is, the surface points into infinity in two opposite directions (the lower side is cut shorter in the model) — and a singular line, a circular knife edge. Taking the precaution to draw the figures on the pseudo-sphere so that they do not run into the singular line and the singular point, we can study the hyperbolic geometry on it. 1 The tractrix is the curve described by a weight at one end of a rod when the other end of the rod is dragged along a straight line. That is, it is the curve given by the condition, that tlie length of the tangent of the curve is constant. Its equation is: , , fc + V/c- - a;2 ,- y = k log V fC^ — x^ 92 RELATIVITY AND SPACE G. MATHEMATICAL SPACE AND PHYSICAL SPACE We must sharply distinguish between physical space and mathematical space. Mathematical space is the concep- tion of a dimensional continuous manifold, and an infinite number of different mathematical spaces, of any number of dimensions, can be conceived and have the same reality in the science of mathematics as philosophical conceptions. Physical space is the form of conception in which our mind clothes the (supposed) extraneous cause of our sense per- ceptions. There is therefore only one physical space, and it exists only as a form of something acting on our senses ; that is, exists only as far as there is something filling space, and ''empty space" in this respect has no meaning. As physical space is a dimensional continuous manifold, it is of interest thus to ask which of the innumerable con- ceivable mathematical spaces agrees best with the proper- ties of physical space. Mathematical space itself has nothing to do with nature and things in nature, but is entirely conceptional. The only characteristic required and assumed for the "point" as the element of mathematical space is the property of continuity. That is, to any point you can get other points infinitely close and can thus go con- tinuously from one point to another. Such is for instance, the case with the instances of time, with the temperatures, colors, etc. The "straight line" then is defined by the axiom: "Two points a and b determine one and only one straight line L." The straight line contains an infinite number of points, which are given by: p, ^ ^^. (1) Each point pi is determined by a ratio x -f- y, and this ratio is called its coordinate. THE CHARACTERISTICS OF SPACE 93 We extend beyond the straight line by the axiom: ''There are points outside of the straight line." Any such point c with any point pi of the line L gives again a line, and any point on this line is given by : ^ x'c + tj'yi ^ xa -\- yb + zc . . ^' x' + y' x + y + z' ^^^ The ratios x ^ y -^ z, then, are called the coordinates of the point p^- The infinite number of points pi of the first straight line L give an infinite number of straight lines, and as each of these straight lines has an infinite number of points p2, the number of points p^ thus is infinitely larger than that of the points pi. The totality of the points po, therefore, is called a two-dimensional manifold, or a mathe- matical 2-space, or a mathematical plane. We extend beyond the plane by the axiom: "There are points outside of the plane." Any such point d with any of the 0° ^ points p^ gives oo 2 lines, and each of these lines contains an infinite number of points ps, given by: _ x"d + y"p2 ^ xa + yh -{- zc + ud ,^. P'- x" -f- y" xTT-f-2 + u ^^ so that infinite times as many points pz exist as points p2, and the points pz thus constitute a three-dimensional mani- fold or mathematical 3-space. Each of these points pz is given by its coordinates, the three ratios : x ^ y -^ z -^ u. Mathematically, we extend in the same manner beyond the 3-space by the axiom: "There are points outside of the 3-space."^ Any such point e again gives a straight line with any of the 00 3 points, and so leads to 0° ^ points, xa -\- yh -\- cz -[- ud -\- ve ,.. P4 = r — I 1 1 ' (4) x-{-y-\-z + u-\'V given by the coordinates x^y-^z-iru^v and constitut- ing a four-dimensional manifold or mathematical 4-space. ' This axiom disagrees with physical experience, therefore all the mathe- matical spaces from here on have no physical representation. 94 RELATIVITY AND SPACE In this manner we can build up to any number of dimen- sions, and as a mathematical conception the n-dimensional manifold, or n-space, is just as real as the 3-space or 2-space. The mathematical n-space merely is the continuous mani- FiG. 25. fold of oo« elements which are given by the n ratios: x : y : z : u . . . Xn as coordinates. As the physical space is a three-dimensional manifold of physical points^ — that is, positions in space — ^it could be represented by some mathematical 3-space, while the surface of a physical body could be represented by some mathematical 2-space, etc. As in these mathematical spaces we have defined points, lines and planes, etc., we can deal with triangles, polygons, space figures, etc., and so can construct a geometry of THE CHARACTERISTICS OF SPACE 95 mathematical 2-space, or plane geometry, a geometry of mathematical 3-space, or solid geometry, of mathematical n-space, etc. These geometries, however, would be entirely geometries of position, or projective geometries (the case 4 of page 74) ; that is, they would deal only with the location of points and lines, etc., to each other, but not with the sizes and metric relations, with equality, etc. Thus, they contain theorems like Fig. 25. ''If the three lines connecting corresponding corners of two triangles meet in one point, the three points of inter- section of corresponding sides lie in a straight line." But we could not have a theorem reading : "Two triangles are congruent if their corresponding sides are equal," or: ''The sum of the squares of the two sides of a right-angled triangle equals the square of the hypotenuse." All metric relations, comparison of sizes, measurements, etc., are based on bringing figures into coincidence, for instance, the measuring rod with the measured length, etc. We prove the congruence of two triangles by moving one into coincidence with the other. Before this is possible in the mathematical spaces defined above we must add an axiom specifying that: "A figure can be moved in space without changing." This is by no means obvious. It is the case, for instance, in the surface of a sphere, but it is not the case in the surface of an egg-shaped figure. We can thus have a theorem of two congruent triangles in the sphere and prove it by moving the one triangle into coincidence with the other. But we could not have two congruent triangles in the sur- face of an egg, because we could not move one triangle from a part of the egg's surface to another one, where the curvature is different, without distorting it, stretching some dimensions and contracting others, and thereby changing the dimensions. Thus measurements on the 96 RELATIVITY AND SPACE surface of the egg and theorems on metric relations, as the Phythagorean theorem, are impossible. It is not possible to move figures from one space to another space of different curvature, for instance, from a sphere into a plane, without stretching and thereby chang- ing and distorting the dimensional relations. Thus figures in one space cannot be represented in correct dimension in another space. (Hence the difficulty in map making: to represent a part of the earth's surface, of a sphere, on a plane map with the least possible distortion of dimensions.) This axiom of metric relations expressed mathematically means that the characteristic or curvature of space K is constant. Thus, not all spaces are metric spaces^ — that is, spaces in which measurements are possible and in which we can speak of and compare the sizes of figures, deal with equality, congruence, infinity, etc. The general space is projective — that is, merely positional relations exist in it — and a special condition or axiom is required, that of con- stancy of curvature, to establish metric relations. Positionally, there is no difference between finite and infinitely distant elements of space, and the geometry of position, or projective geometry, thus does not contain the conception of the infinitely distant. The distinctions based on the relation to the infinitely distant — for instance, the differences between the ellipsis, which is all finite, and the hyperbola, which runs into infinity — are thus absent in it. This does not mean that infinitely distant elements may not exist in the geometry of position and that all the elements of projective geometry are finite (as all the ele- ments of the elliptic geometry are finite). But it means that in their projective properties infinitely distant ele- ments differ in no way from finite elements. All the differ- ence between finite and infinitely distant elements is metric, and the conception of infinitely distant elements as differ- ent from finite elements is introduced only by the metric axioms. THE CHARACTERISTICS OF SPACE 97 While mathematical spaces may be conceived as of any number of dimensions and have equal reality as mathe- matical conceptions, physical space is limited by experience to three dimensions- — that is, any point in physical space is determined by three data or coordinates, for instance, the three distances x, y, z from some arbitrarily chosen coor- dinate axes. It must be realized, however, that the three-dimensional character of physical space is in reality not entirely empirical, but also is conceptional. Physical space is three- dimensional when considering the point as the element of physical space; that is, any point in physical space is determined by three coordinates, x, y, z. But in the real physical space of our experience there are no points, but bodies, and the point is a mere mathematical conception, an abstraction, but not a physically existing thing. Thus the three-dimensional physical point-space also is a mere abstraction and is no more a physical reality than its element, the mathematical point, is. What is real is the physical body. But the location of a (rigid) body in physical space is not fixed by three coordinates, but requires six coordinates. Three coordinates, Xi, yi, Zi, would fix one point. Pi, of the body, for instance, its center of mass. This would not fix the body, for the body could still have an infinite number of positions in turning around its center point. Pi. We must thus fix a second point, P2. With point Pi fixed, P2 can move anywhere on the surface of a sphere with Pi as center and P1P2 as radius. As the surface of the sphere is two-dimensional, two data or coordinates, X2, ^2, thus are necessary to fix point P2 on this sphere. Fixing the two points Pi andP2, however, does not yet locate the body ; it still can turn around the axis P1P2. A third point, P3, thus has still to be fixed. As P3 could move on a circle around P1P2 as axis, one coordinate, X3, is sufficient to fix point P3 and thereby locate the body in space. Thus six data or coordinates, Xi, yi, Zi, Xo, yo, X3, are required to locate the position of the rigid body in physical xi, yi, zi, S12' = xi, 2/2, Z'l, Sis" — Xz, 2/3, Zz, S23" = 98 RELATIVITY AND SPACE space, and as empirically the body is the element of physical space, we might speak of the physical space of our experi- ence as six-dimensional and not three-dimensional. This is not a mere quibbling, but is very real, and all through physics and mechanics we find that not three but six data or coordinates are required to locate an element in physical space, whether a position, a motion or the cause of a motion, that is, a force. Thus : The location of a body in space is given by six coordi- nates, as above shown, or by nine coordinates, those of three points of the body, with three equations between them (which represent the constancy of the distances between the points) : (xi — X2)- + (2/1 — 2/2)^ + (zi — z^y. (xi - XzY -\- {ijx — yzY + {zi — ZzY. {x, -xzy + (2/2 - yzY + {Z2 - ZzY. The motion of a body from one position in physical space to another position is given by six independent components, of which for convenience three are usually chosen as translations, in the direction of the coordinate axes: Vi, Vo, Vs, and the other three as rotations around the coordinate axes : Ti, To, Vs. The total force acting upon a body is expressed by six independent components, of which usually three are chosen as vector forces along the coordinate axes: Fi, Fo, F3, and the other three as couples or torques around the axes: Tr, T„ T3. As in physical and mechanical calculations we almost always have to come to the abstraction of the ''point" as an element of our calculation, it is more convenient to con- sider the mathematical point as space element and treat the six coordinates of the physical body not as dimensions but as ''degrees of freedom." We must realize, however, that THE CHARACTERISTICS OF SPACE 99 the statement of the three dimensions of physical space is not direct experience but already includes a mathematical abstraction. Thus by choosing something else than the mathematical point as space element we can consider physical space as of a different number of dimensions. For instance, any point in physical space can be con- sidered as center of an infinite number of spheres, of differ- ent radii r. As there are oo ^ points in space, given by their coordinates x, y, z, and each as center gives rise to oo spheres, given by their radii r, there are therefore existing in our physical space » ^ spheres, represented by their coordinates X, y, z, r. That is, with the sphere as element, physical space is four-dimensional, and a four-dimensional geometry — at least a projective or non-metric geometry — ^can be constructed in physical space with the sphere as element or ''point." This has been done.^ Following our outline as above: Let Si, S2, S3, S4, Ss be five spheres in space. Then any of the CO 4 spheres is given by : with four coordinates, the four ratios: Xi "T" X2 "^ X^ ~v X4 ~r' Xq Three linear equations between the x then give a ''line," just as two equations in the three Cartesian coordinates give a line in three-dimensional point space. This "line" in 4-space consists of all the spheres which intersect in a circle, etc. As has been seen, therefore, in spite of all our conviction of the three-dimensionality of space, in reality space and the number of dimensions of space are relative, dependent on the condition of the observer^ — that is, on the thing chosen as element of the space^ — and physical space with the point as element may be three-dimensional. With the sphere as element it is four-dimensional, with the rigid 1 Reye, "Geometrie der Kugelii." 100 RELATIVITY AND SPACE body as element six-dimensional. Thus space and time ^,nd dimension are relative.^ H. THE BUNDLE AS ELLIPTIC 2-SPACE We try to get a conception of non-Euclidean space, as the elliptic space of the gravitational field and the hyper- bolic space of the centrifugal field, by considering the corresponding two-dimensional spaces. These we can construct as surfaces in our physical 3-space, and so can study them, by looking at them from a higher dimension as part of our 3-space, The disadvantage, however, is that we can look at these various 2-spaces from one 3-space only, our physical space, which is essentially Euclidean, and then it is difficult to abstract and realize which of the properties and characteristics of the 2-space are inherent in the 2-space and which are merely incidental to the relation of the 2-space with our particular 3-space and would not be the same when seen from some other 3-space, and therefore are not essentially characteristic of the 2- space. Thus we have seen that the elliptic 2-space, or ''elliptic plane," appears from our Euclidean 3-space as a spherical surface, and the straight line of the elliptic 2-space appears as a circle in the Euclidean 3-space ; that is, as curved. We realize that this line has no curvature, is straight, in the elliptic 2-space. But still, it is curved into the Euclidean 3-space, and as we look at it from the Euclidean 3-space, it is difficult to avoid the feeling that, after all, it is not a straight line but a curve, and only appears straight in the elliptic 2-space because from this space we cannot see the curvature. The same applies to 1 The fallacy of the magician's or spiritist's four-dimensional space is not in the conception of a four-dimensional space in general, but in the conception of a four-dimensional 'point space — in claiming a dimension higher than shown by experience. Physical space is a three-dimensional point space, and attributing to it a higher dimension with the point as space element, therefore, is against experience. But not so with something else as element. On the contrary, with something else chosen as ele- ment of physical space, we may expect to find it with a different number of dimensions. THE CHARACTERISTICS OF SPACE 101 the elliptic 2-space as a whole; from within we can notice no curvature, and it is an elliptic plane, but from Euclidean 3-space we plainly see it curved, as sphere, and so cannot entirely avoid the feeling that after all it is not plane but is curved, and that it appears plane from within merely because from within we cannot see the curvature. We may explain that from an elliptic 3-space of the same curvature the elliptic 2-space would not appear curved but as a plane of finite area; but as we cannot look at it from an elliptic 3-space, this is not entirely convincing — ■ the less so as it has to some extent become customary to speak of the characteristic constant of the space as its "curvature," because of its appearance from a Euclidean higher space. In reality this is incorrect, as the elliptic 2-space and the lines in it are not curved inherently and their appearance to us as sphere and as circle is merely the appearance of their relation to the Euclidean 3-space from which we view them. As already brought out, the same 2-space and its lines would appear differently curved from a different 3-space, and would not appear curved from a 3-space of the same characteristic constant as that of the 2-space. Unfortunately we have no such elliptic 3-space. However, the idea can be grasped by viewing an elliptic 2-space under conditions where we do not have a Euclidean 3-space from which to look at the 2-space, but where we view the elliptic 2-space by itself, in comparison with a Euclidean 2-space. We see then that there is noth- ing curved about the former any more than about the latter. Mathematical space is a dimensional continuous mani- fold. The elements of it, while we may call them points, have nothing to do with the physical point^ — ^that is, the position in physical space — but may be any continuously changeable quantity, for instance, color, composition of a gas mixture, forces in physical space. ^ Physical space is a 1 Thus for instance, all the forces in physical space form a six-dimen- sional manifold. 102 RELATIVITY AND SPACE three-dimensional manifold with the point^ — ^that is, the location in space^ — as element, and therefore when com- paring it with mathematical space we generally choose the point as the element of mathematical space. This, how- ever, is not essential, and we could just as well conceive of a mathematical space with the line as element or the plane as element, etc. This is commonly done in projective geometry, and it leads to valuable results. Suppose, then, we have a Euclidean plane or 2-space of zero characteristic E; that is, the two-dimensional manifold of points p and lines L contained in the plane E. From a point 0 outside of E we can project E. Every point p of E then is projected by a line through 0 (usually called a "ray"). I = Op, and every line L of £' is projected from 0 by a plane P = OL. To the points p of E thus corre- spond the lines or rays I through 0, and to the lines L of E correspond the planes P through 0. All these lines I and planes P through 0 thus form a two-dimensional con- tinuous manifold- — that is, a 2-space — which corresponds element for element to the points and lines p and L of the Euclidean 2-space E. Such a 2-space, consisting of all the lines and planes through the point 0, is called a "bundle." Every figure consisting of points and lines in the Euclidean plane E is projected by and gives a figure consisting of lines and planes in the bundle 0 ; every angle between two lines in E is projected by an angle between the two corresponding planes in 0; every triangle in E is projected by a triside in 0, that is, a three-sided pyramid; every curve in £' by a cone in 0, etc., and to the geometry of the plane E thus corresponds a geometry of the bundle 0, by projection. We can directly read off the theorems of the geometry of the bundle 0 from the theorems of the geometry of the plane E by saying, in the bundle, "line" or "ray" for "point" in the plane, and saying in the bundle 0 "plane" for "line" in plane E. Thus, for instance, the theorem of plane geometry of Fig. 25: THE CHARACTERISTICS OF SPACE 103 "If the three lines connecting corresponding corners of two triangles meet in one point, the three points of inter- section of corresponding sides lie in a straight line," would be in the geometry of the bundle: "If the three planes connecting corresponding edges of two trisides intersect in a line, the three lines of intersection of corresponding side planes lie in a plane." The latter is derived from the former by projecting it from point 0. Two congruent triangles in E are projected by two tri- sides in 0, but these two trisides are not congruent. Two congruent trisides in 0 (that is, two trisides of which the one can be moved into coincidence with the other) give two triangles in E, but these two triangles are not congru- ent. Thus, while all the theorems dealing with the relative position of the elements- — points and lines in E, lines and planes in 0- — remain the same in the two 2-spaces, the Euclidean plane E and the bundle 0, the theorems dealing with metric relations, such as congruent, equal, etc., do not transfer. That is, the geometry which E and 0 have in common is the geometry of position, or projective geometry (column 4 on page 74), but not the metric geometry. In other words, the characteristic constant of the 2-space 0 is different from that of the plane E. "Length" or "distance" we call the part of the line between two points. To the line L oi E corresponds the plane P projecting the line L from 0, and to the two points J) I and p2 on the line L in E correspond two lines ^i and U in the plane P (the lines projecting the two points from 0). To the distance, as the part jpipi of the line L in E, thus corresponds in 0 the part of the plane P between the lines liU] that is, the angle ^1^2. Thus "distance" or "length" in our new 2-space, the bundle 0, is the angle between the two lines or rays, which are the "points" or elements of the 2-space 0. The length of the straight line L in the Euclidean plane E is infinite. That is, starting from a point p of this line L 104 RELATIVITY AND SPACE and moving along this line in one direction, I have to go into infinity and then come back from infinity on the other side of the line before I again reach point p. If in the ''line" of the bundle 0 — that is, the plane P which corresponds to the line L of E — I start from a "point"- — that is, the ray or line I corresponding to the point p of E — and follow the motion of p along line L, the corresponding ray I moves through the plane P; that is, turns around the point 0. But while the point p on L in the plane E traverses an infinite distance before it returns, the corresponding ray I on P in 0 returns after one complete revolution; that is, after traversing the angle 27r, Since the "length" in our 2-space 0 is measured by the angle, it thus follows that the "length" of the straight line P in the 2-space 0 is 27r. In other words, the length of the straight line in 2-space 0 is finite, and this 2-space, the bundle, thus is an elliptic 2-space. The length of a straight line in an elliptic 2-space is 2TrR, where R is the "radius" of the 2-space, and is given by the characteristic constant K = 1/R^. As in the bundle 0 the length of the straight line is 27r, it follows that in this bundle it is i^ = 1; thus K = 1. That is, the bundle 0 is an elliptic 2-space of characteristic constant or curvature 1. An area in E, like that of a triangle, is projected from 0 by a spherical angle. ' ' Area " or " volume ' ' in the 2-space 0 thus is measured as a spherical angle. The total area of E is projected by the total spherical angle at 0. But while the area of E is infinite, the total spherical angle of 0 — that is, the total area or "volume" of the 2-space — is finite. Every angle between two lines LiLo in E is projected by an angle between the two planes Pi and Po, which corre- spond to the lines Li and Lo. The angle between the planes of the bundle 0 differs from the angle between the lines of the plane E, to which it corresponds. It is larger if the perpendicular form 0 on to E falls inside of the angle, smaller if the perpendicular falls outside of the angle. THE CHARACTERISTICS OF SPACE 105 Every triangle in E is projected by a triside in 0 — that is, a three-sided pyramid, which is the ''triangle" of the 2-space called the bundle. If the triside and the corre- sponding triangle are situated so that the perpendicular from 0 onE falls inside of the triside, then each angle of the triside is larger than the corresponding angle of the triangle, as seen above, and as the sum of the angles in the triangle in E equals 180 degrees, the sum of the angles in the triside is thus larger than 180 degrees. If the triside is situated so that the perpendicular from 0 on E falls outside of the triside, the bundle 0 and with it the triside can be turned so that the perpendicular falls inside of it. (The correspond- ing triangle obviously then has changed in shape and posi- tion on E, but the sum of its angles is still 180 degrees.) Thus in any triside the sum of the angles is greater than 180 degrees, and the bundle thus is an elliptic 2-space. A curve in the Euclidean plane E gives a cone in the bundle 0, and inversely. Suppose we have a circle in the 2-space 0; that is, a circular cone. We may move it so that its center line is perpendicular to E. It then will project on £" by a circle, and if C is the circumference, r the radius of the circular cone (both in angular measure, as ''length" in 2-space 0 is measured as angle), then from the relation between the circular cone and its projecting circle it follows: C = 2t sin r, and as sin r always is less than r, it is : C<27rr, and the circumference of the circle is less than tt times the diameter, the more so the larger the circle. The largest circle which can exist in the 2-space 0 has the radius 7r/2, or diameter x, and its circumference is 27r. It is the straight line of the 2-space 0. The straight line of the elliptic 2-space 0 thus can be considered as a circle with radius ir/2, just as the straight line of the Euclidean 2-space can be considered as a circle with radius 0° . 106 RELATIVITY AND SPACE Thus we have here, by projection from a Euclidean plane E, produced an elliptic 2-space of unit curvature, in the bundle 0, by using the ray — that is, the line- — -as element instead of the point, and thus have a chance to study the characteristics of the elliptic 2-space on the bundle 0, with the advantage that now we do not look at it from a higher space of different characteristic constant, but from an entirely unrelated space; that is, a space of different elements. We see now that in this 2-space 0, while it is elliptic, of characteristic constant or ''curvature" 1, there is nothing curved about it. Its straight line is the plane P. It has a finite length 27r, but there is nothing curved about P, and we see thus that while the elliptic 2-space in the Euclidean 3-space appears curved, as sphere, and the straight lines in it appear curved, as circles, this is not a feature of the elliptic 2-space, but of its relation to the Euclidean 3-space, since when unrelated to an Euclidean 3-space there is nothing curved in the elliptic 2-space, as the bundle 0. In the Euclidean plane E we have parallels. But two parallels of E project from 0 by two planes as ''lines" of the bundle, which intersect in a line or ray — that is, a "point" of 0 — and this is just as finite and real as any other. Any two planes of 0 intersect; thus there are no parallels in 0. The infinitely distant points of E project from 0 by rays, which are parallel to the plane E, but in 0 are just like any other rays, are finite. Thus there is no infinity in the elliptic 2-space 0. The elliptic 2-space 0 is finite; the length of any straight line in it is 2t; its total volume or area is 47r. And still there is nothing outside or beyond it in 2-space, but it is all the space, and so shows how a finite volume — -or "area" in 2-space, here measured by the spherical angle from 0 — can fill all the space. In many respects this elliptic 2-space, derived by using the line or "ray" as element instead of the point, is more convenient and illustrative than the point space on the THE CHARACTERISTICS OF SPACE 107 spherical surface for realizing the characteristics of elliptic space. I. PROJECTIVE GEOMETRY By eliminating the metric axioms we limit the general geometry, or projective geometry, to mere relations of position, but exclude all theorems dealing with equality, congruence, proportionality and similarity, with numerical relations and measurement, with parallels and the effect of infinitely distant elements, as the distinction between ellipsis, hyperbola, circle and parabola, etc. It might be thought that very little is left then for consideration, and a few theorems of the general or projective geometry may therefore be given to illustrate the wide and interesting field of geometrical research outside of the metric axioms. Theorems of geometry of position, or projective geom- etry, which hold on the sphere and pseudo-sphere, and in the bundle as well as in the Euclidean plane, are : 1 . If the three lines connecting corresponding corners of two triangles meet in a point, then the three points of intersection of corresponding sides of these two triangles lie on a straight line. This theorem is shown in Fig. 25 on a Euclidean plane, in Fig. 26 on corrugation as Euclidean 2-space, in Fig. 27 on a sphere, and in Fig. 28 on a spindle as elliptic 2-space. We have mentioned it already in the bundle as elliptic 2-space with a different element. 2. In a plane (Fig. 29) are given two lines, Li andLa, and three points, po, pi, p^, outside of these two lines. From the point Po lines are drawn intersecting the lines Li and L2 in the respective points ai and a2, &i and 62, Ci and C2, etc. Then the points of intersection, a of the lines piai and p2a2, b of pihi and pobo, c of piCi and P2C2, etc., lie on a conic sec- tion. On this conic also lie the points pi and p^, the point of intersection q of the lines Li and L2 with each other, and the points of intersection Si of the line Li with pop2 and S2 of L2 with popi. 108 RELATIVITY AND SPACE It is interesting to note that the conic is an element of projective geometry, defined as a curve of second order; that is, by the feature that a straight fine can intersect it in a maximum of two points. The classifications of the conic, however, are no part of projective geometry, as they are Fig. 29. made by its relation to infinity and therefore are metric in character : The hyperbola has two infinitely distant points, the parabola has one infinitely distant point, the ellipsis has no infinitely distant point; that is, its two infinitely distant points are imaginary, and the circle is the ellipsis in which the two imaginary infinitely distant points are at right angles to each other. This classification of conies applies only in the Euclidean plane, but not on the sphere or in the bundle, as in the elliptic space there is no infinity. 3. If from a point p (Fig. 30) two lines are drawn inter- secting a conic (as, for instance, a circle) in the points ai and a2 and 6i and 62 respectively, and the points of inter- section, p' of aihi and 0062 and p" of ^162 and a-^bi, are con- nected by a straight line P, and pi and p2 are the points of THE CHARACTERISTICS OF SPACE 109 intersection of this line P with the conic, then ppi and pp2 are tangents of the conic at the point pi and p2 respectively. The line P is called the polar of the point p. Any line through p intersects the polar F in a point po, which is the fourth harmonic point to p with respect to the two intersections of ppo with the conic. If Pi and P2 are the polars of the two points pi and p^ with regard to a conic, then the line connecting the points Fig. 30. pi and P2 is the polar of the point of intersection of Pi and P2, and inversely. This gives a means to construct the tangents on a conic; for instance, a circle. As has been seen, the harmonic relation between four points exists in projective geometry. This shows that this relation, while usually considered as metric and introduced as such, in its nature is not a metric relation but a positional relation. 110 RELATIVITY AND SPACE 4. If we have, in Fig. 31, four points a, b, c, d, in a plane, and draw the six Unes through them, ah, ac, ad, he, bd, cd, and denote the three additional points of intersection of these six lines by e = ah, cd;f = ac, hd; g = ad, he, and Fig. 31. draw the three additional lines ef, eg and fg, we get a total of nine lines and four points on each of these nine lines. Each of these nine groups of points is composed of four harmonic points. This shows the positional nature of the four harmonic points. J. THE METRIC AXIOM AND THE LAW OF GRAVITATION In Lecture I, and more completely in Lecture III, we have seen that the laws of the motion in a gravitational field are identical with the laws of inertial motion in an accelerated system, and that the former — that is, the law of gravitation^ — can be derived as the equations of mathe- matical transformation to an accelerated system without THE CHARACTERISTICS OF SPACE 111 making any assumption on the physical nature of the gravi- tational force. In this manner Einstein has derived his law of gravitation. This transformation, however, which leads to the law of gravitation, is not entirely without arbitrary assumption, as it would appear to be at first. The law of gravitation is a metric relation. We have seen, however, in the preceding that metric relations do not exist in general space, but space in general is projective, involv- ing only relations of position, and that a particular char- acteristic of space, a metric axiom, is necessary to give relations of size and measurement between figures in a space, or between two spaces, such as the elliptic space of the gravitational field and the Euclidean characteristic of fieldless space. Einstein's derivation of the law of gravita- tion, however, assumes the possibility of a mathematical transformation to the accelerated system, which is of metric nature; that is, assumes the existence of metric rela- tions and therefore requires the selection of a metric axiom. Any axiom which establishes metric relations would obvi- ously fulfill the conditions of permitting the transformation to the accelerated system, and thus give a law of gravita- tion. Thus many different forms of the law of gravitation could be derived, since different forms of metric axioms could be chosen and the form of the metric axiom is arbi- trary. Einstein chose the metric axiom which gives the simplest formulation of the physical and mathematical laws. This is it: ''Whatever may be the characteristic or curvature of the space, and however it may vary from point to point or remain constant, an element of space — that is, an infinitely small part of it — is plane or Euclidean." The general geometry of space (whether 2-space, that is, plane geometry, or 3-space, that is, solid geometry) is pro- jective or non-metrical; that is, it deals with problems of position only, and its space is merely a dimensional con- tinuous manifold, as discussed before. Metric relations, such as equality of length, area, volume, congruence, the relation of smaller or larger, proportionality, the distinction 112 RELATIVITY AND SPACE between finite and infinitely distant, etc., are foreign to general space, but require an additional property of space beyond that of being a continuous manifold, namely, the property, that measurement^ — that is, comparison of sizes of figures — can be carried out. Measurement is carried out by bringing figures into coincidence — that is, the measuring rule with the length to be measured, the area to be measured with the unit area used as measure, the triangles which are to be proven as congruent with each other, etc. To make this possible it is necessary that figures can be moved in space without changing. Obviously, if moving a figure, as a triangle, would change its size, we could not prove or discuss the equality or congruence of two figures by moving one into coincidence with the other, since when we moved it it would not be the same figure any more. Metric space is, therefore, characterized, in distinction to general space, by the additional axiom: "Figures can be moved in space without change of their size or of the size of their component parts." This metric axiom leads to the condition that the charac- teristic, or curvature, of the space must be constant. Thus there are only three spaces in which complete metric rela- tions are possible, the Euclidean or plane, the elliptic and the hyperbolic space. Complete metric relations, such as congruence of figures, etc., exist only within a space of constant curvature or between different spaces of the same curvature, as between the plane and the conical surface (zero 2-space) or between the sphere and the surface of a spindle of the same curvature as the sphere. They do not exist, for instance, between the plane and the sphere — 2-spaces of different curvature. A triangle in the plane and a triangle on a sphere can never be congruent and can never be made to coincide. If the sides of the two triangles were equal and I try to make them coincide, I have to stretch or contract the area of one to make it fit the other, and this changes its area, its angles, etc., so that it is not the same triangle any more. The same thing is true with two spheri- THE CHARACTERISTICS OF SPACE 113 cal triangles of different curvature, as, for instance, located at two different parts of an egg-shaped surface, a 2-space of varying positive curvature. The metric relation of con- gruence, or proportionality, etc., does not exist in a space of varying curvature, as figures cannot be moved to bring them into coincidence without changing their shape by stretching or contracting. It is not possible, therefore, to represent an elliptic 2- space— like the surface of the earth — on a plane 2-space — as in a plane map^ — without distorting the dimensions, getting the different parts out of proportion with each other. Nevertheless, a partial metric relation is possible in spaces of varying curvature or between spaces of different curva- ture. For instance, we measure the length of a curve, like the circumference of a circle, by comparing it with a straight line.^ We measure the area of a sphere or a part of it (as the surface of the earth or that of a country) by comparing it with a flat or plane unit area- — for instance, a square mile — though this can be done only indirectly and there is no possibility of physically carrying out such measurement, as we can never make a flat unit area coincide with a curved surface or a part of it. We can do it, however, by compar- ing the two, the curved line and the straight line, or the curved area and the flat unit area, element by element. If we desired to measure the circumference of a circle three feet in diameter by a straight rule one foot in length, we should get very inaccurate results, as a rule one foot in length cannot be made to coincide with the curved peri- phery of the circle. If, however, we were to use a straight rule one inch in length, we should get much more accurate results in measuring a circle three feet in diameter, as the ^ It appears to us obvious now that the length of a curved line can be measured, that is, compared with the length of a straight measuring rod; but this is not at all so obvious, and it was a matter of serious discussion by the great mathematicians of ancient time whether there could be such a thing as a "measurement" of the circumference of a circle. 8 114 RELATIVITY AND SPACE one-inch straight rule could be made almost to coincide with a part of the periphery. The shorter the straight rule we use, the more accurately it will coincide with the curved periphery of the circle and the more accurate will be the measurement, and if it were possible to use an infinitely short rule in measuring the length of a curved line, we should get perfect accuracy. This means, in other words, that the shorter a part of a curved line is the straighter it is, and that an infinitely short part of a curved line is straight. The same applies with areas or surfaces. The smaller a part of a curved surface is the planer it is, and an infinitely small part of a curved surface is plane. This makes it possible to measure the length of curved lines, the area of curved surfaces, the volume of curved spaces, etc. The characteristics are: "A piece of a curved line is the straighter the shorter it is, and an infinitely short piece of a curved line is straight." "A piece of a curved surface is the planer the smaller it is, and an infinitely small piece of a curved surface is plane." ''A piece of a curved space is the planer the smaller it is, and an infinitely small piece of a curved (or non-Euclidean) space is plane, that is, Euclidean." This, however, is not an inherent property of space. It does not hold for the general or non-metric space, but it is a special condition imposed upon space. It is a metric axiom, the adoption of which gives to space characteristics beyond those of general space, namely, the property that measurements can be made in space. For instance, at the point of the cone, or at the singular line of the pseudo-sphere, this axiom does not hold, and no matter how small a piece at the point of the cone or the singular line of the pseudo-sphere we take, it never is plane or Euclidean. This condition that the space element is Euclidean applies to all spaces of constant curvature, and therefore, in describing the characteristics of these spaces, we said, for instance, referring to elliptic space: ''The sum THE CHARACTERISTICS OF SPACE 115 of the angles in a triangle is greater than 180 degrees, the more so the larger the triangle." In an infinitely small triangle of elliptic space the sum of the angles is 180 degrees, because an infinitely small piece of elliptic space is Euclidean. However, the spaces of constant curvature are more special than the space with the Euclidean element. For instance, an egg-shaped surface does not have constant curvature, but its element is Euclidean. We thus have here three successive gradations of the conception of mathematical space: 1. The general space, as a continuous manifold, in which a geometry of position exists, but no metric geometry. 2. The differential metric space, that is, the general space with the differential metric axiom added, the axiom that an element of the space is Euclidean. This permits those metric relations which can be derived by segregation into elements (line elements, surface elements, etc.) and permits measure- ment of lengths and areas, but not comparison of figures, such as congruence, as motion of figures in the space is not possible without change. 3. The completely metric space — that is, the differential metric space with the integral metric axiom added, the axiom that figures can be moved in space without change of shape. This gives the space of constant curvature, Euclidean, elliptic or hyperbolic. The differentially metric space, established by the axiom that the element of the space is plane or Euclidean, is not the most general metric space, but the most general metric space would be established by the differential metric axiom that all the elements of the space are of the same character- istic. But whether this characteristic of the space element is Euclidean or of any other form is arbitrary. That is : 2o. The general differential metric space is the general space with the axiom added that all the elements of the space have the same characteristic. That is, if ds is a line element and dxi an element of the coordinate Xi (where 116 RELATIVITY AND SPACE i = 1 and 2 for 2-space, 1, 2 and 3 for 3-space, 1.2 — n for 7i-space), it is: ^§2 _ p{dx^ = independent of Xi] that is, dsMs a constant quadratic function of the coordinate element dXi. For the special differential metric space, in which the ele- ments are Euclidean, it is: ds"^- = Hdxi"^, while in general, for differential metric 3-space, it is: ds^ = Qidxi^ + g2dx2^ + gsdx^^ + giidxidx^ + g^zdxidxz + As the elliptic space of the gravitational field and the Euclidean space of fieldless regions have a different curva- ture, complete metric relations, such as congruence of fig- ures in the two spaces, cannot exist between them. Since, however, both spaces are differentially metric spaces — that is, their elements are Euclidean- — such metric relations as exist in differentially metric spaces- — that is, comparison of the length of lines, areas of surfaces, volumes of bodies — exist between elliptic and Euclidean space. In the trans- formation to an accelerated system, which leads to his formulation of the law of gravitation, Einstein therefore makes the assumption of the existence of the differentially metric axiom. This is necessary, as without it no metric relations could be established, but nevertheless it is an arbitrary feature, justified only by the simplicity of the results. K. VISUAL APPEARANCE OF CURVED SPACE The distance of an object we estimate by the difference in the direction in which we see it from the two eyes. Thus, if, in Fig. 32, 0 is an object and Ai and A2 the two eyes, the difference in the two lines of sight AiO and AoO gives us the distance of 0. If 0 is further away, the lines AiO and A2O differ less in direction, and finally, when 0 is THE CHARACTERISTICS OF SPACE 117 infinitely far away, the two lines of sight A^O and AiO have the same direction. That an object is infinitely dis- tant (that is, very far distant) we thus recognize by the Fig. 32. two lines of sight from our eyes to the object having the same direction. This is in Euclidean space. How would elliptic and hyperbolic space look to us? In elliptic space there is no infinite distance, and the straight line is finite in length. Thus, if there were no obstructions and no light-absorbing medium, the line of sight from the eye should return into itself; that is, every- where where the view is free we should see the back of our head covering all open space^ — because all lines of vision, in whatever direction we look, return into themselves through the back of our head. However, this is meaning- less in our universe, since, though finite, it is so enormously large — the length of the straight line being equal to about 400,000,000 light-years^ — that light would have been absorbed long before it had completed the closed path. In elliptic space there is no infinite distance and the straight line is finite in length. We would estimate the distance of an object 0 in the same manner as in Fig. 32, by the difference in the direction of the two lines of vision from the two eyes to the object, and the further the object is away the less is the difference in this direction, until finally, when the object is at the distance of one-quarter the length of the line (one quadrant of the circle as which the straight line of the elliptic space appears to us from a higher Euclidean space), the two lines of vision AiO and A2O have the same direction; that is, the object 0 appears at infinite distance. Thus, though elliptic space is finite in extent, every object at a quarter-line length from our 118 RELATIVITY AND SPACE eyes appears infinitely distant. When we move toward it, it comes into finite distance, and finally we reach it in a finite distance, while other objects beyond it now have come into quarter-line distance and appear infinitely distant. Thus, visually, infinity exists in the finite elliptic space, and we see objects apparently at infinite distance, but can reach them in finite distance. In the hyperbolic space each line L (Fig. 33) has two parallels Li and Lo through a point P — that is, two lines Fig. 33. which intersect L at infinity — and these tvv^o parallels Li and L2 make an angle L1PL2 with each other. Thus L] differs in direction from L at the point P, though parallel, that is, intersecting it at infinity. Thus, if I look at a receding object 0 in hyperbolic space, the difference in the direction of the two lines of vision from the two eyes, AiO and A2O (Fig. 32) gets the less the further the object is away. But even when the object 0 is infinitely far away, the two lines of sight toward it still differ in direction by the angle by which two parallels differ in hyperbolic space, and the object therefore appears to be at finite distance. Visually there is no infinite distance in hyperbolic space, but all objects appear at finite distance, even infinitely dis- tant objects. But such infinitely distant objects, though appearing at finite distance to the view, never get any nearer, no matter how far we move toward them. Wc may estimate their (apparent) distance and move toward them by this distance and more, and still they appear just as far, at the same apparently finite distance. (This reminds us of the similar characteristic of the velocity of light c in the relativity theory, which is finite, c = 3 X 10^'' cm., but still inapproachable, as no combination or addi- THE CHARACTERISTICS OF SPACE 119 tion of lesser velocities can ever add to a sum equal to c.) Helmholtz has shown that we can get a view of hyper- bolic space by looking at our space through a large, slightly concave lens which covers both eyes. We thus find: Euclidean space is infinite in extent, appears infinite visually and is unbounded. Elliptic space is finite in extent, but appears infinite visually and is unbounded. Objects at a quarter-line distance appear infinitely distant. Hyperbolic space is infinite in extent, but appears finite visually and is unbounded. L. THE TWO-DIMENSIONAL ANALOGUE OF THE UNI- VERSE, AND THE MATHEMATICAL CONCEPTION OF IT The relativity theory has reconfirmed the law of con- servation of energy, but has denied the law of con- servation of matter by showing matter as kinetic energy, moc^, where c = velocity of light and mo is a constant. Mass then is represented by: moC^ + E m _ v^ where v is the relative velocity and E the non-kinetic energy of the body. The constancy of the mass then is approximate only as long as V is small compared with c and E small compared with WqC^. This is the case at all but ionic velocities and energies. The gravitational field of matter, then, is of the character of an accelerated system, and in the gravitational field the laws of the Euclidean geometry cease to hold and space shows a positive or elliptic curvature. Space, then, and its characteristic or curvature are functions of matter and thus of energy. That is, space is 120 RELATIVITY AND SPACE plane or Euclidean in the absence of matter and becomes elliptic or positively curved in the presence of matter, the more so the larger the amount of matter present ; that is, the greater the gravitational field, as we say. The universe, as we know, consists of isolated huge masses, the fixed stars, which are surrounded by their gravitational fields, but separated from each other by enormous distances of empty or practically empty and therefore essentially Euclidean space. At the masses of the fixed stars the curvature of the universe thus is positive; in their gravitational fields it gradually tapers down to zero in the fieldless space between the fixed stars. Geometric- ally, we may thus picture for ourselves a two-dimensional analogy of our universe by replacing the fixed stars with very shallow spherical segments — ^positive 2-space — which continue as very shallow conical surfaces^ — ^the gravitational fields- — ^into the flat Euclidean planes of fieldless space, as shown in the reproduction of such a model in Fig. 34.^ As has been seen, owing to the curvature of the spherical segments (the masses), if we traverse the model in a straight line, we gradually curve in a direction at right angles to the plane of the model^ — that is, into the third dimension- — • so that we return to our starting point after passing about twelve masses. This shows how the existence of positively curved regions (masses) makes the total universe of finite extent. However, this analogy of a two-dimensional model with our universe must not be carried too far, as it is deficient in an essential feature. In the 2-space we have either positive or negative or zero curvature, depending on whether the two radii of curvature of the surface are in the same direc- tion or in opposite direction, or whether one is zero. In ' This Fig. 34 shows the two-dimensional analogy of our universe about one million times exaggerated. That is, the actual curvature is about one- millionth as much as shown, or, instead of twelve, about twelve million masses are passed before a straight line returns into itself in the physical universe. THE CHARACTERISTICS OF SPACE 121 3-space, however, there are three radii of curvature, and combinations exist which have no analogy in 2-space and therefore cannot be illustrated by analogy with it. This limits the completeness of the 2-space analogies with 3- space relations, so that the preceding can be considered only as giving a rather crude conception of the 3-space characteristics. Physical space, we have seen, is not a space of constant curvature, but of a curvature which varies from point to point with the distribution of matter. From what we have discussed, therefore, it follows that the metric axioms do not rigidly hold in physical space, and figures cannot be moved in space without stretching or contracting when passing from a point of space to a point of different curva- ture. Measurements and dimensional relations, therefore, are not rigidly possible in physical space, and strictly, we cannot speak of the length or the size of a body, as we can- not measure it by bringing the measure to it, because the length and shape of the measure change when it is moved through space. Thus length and size are not fixed proper- ties of a body, but depend on the conditions under which they are determined, and this brings us back to the conclu- sions of the first lecture on the relativity of length and time. Instead of the physical conception of mass as a kinetic energy which causes a curvature, a deformation or ''kink" in space, we may start from the mathematical side and consider mass and its gravitational field as the manifesta- tion or physical representation of a curvature of space. We may consider physical space as varying in curvature between zero and positive values. In other words, space, though in general of zero curvature, or Euclidean, contains singular points, or rather regions of positive curvature, which we call ''masses." In a region surrounding a region of positive curvature or ''mass" the curvature of the space gradually changes from the positive values in the mass to zero at distance from the mass, and such a region we call the gravitational field of the mass. Energy then becomes 122 RELATIVITY AND SPACE space curvature, and the characteristic constant of space is the measure of energy. The hmitations of this mathematical conception of the physical universe are that the electromagnetic energy and the electromagnetic field do not yet satisfactorily fit into it. INDEX Aberration of light, 15 Absolute number, meaning, 38 Accelerated motion, and gravitation, 52 Acceleration, 9, 47 Action at distance, 19 Alternating current, 14 dielectric field, 20 Analogue, 2 dimensional, of uni- verse, 119 Axioms of mathematics, 70 metric, of space, 95, 110 B Beltrami's pseudosphere, 91 Bending of space, 88 Betel geuse, 67 Bolyai, 71 Bullet velocity, 13 C Capacity and wave velocity, 23 Centrifugal field, 47 force and inertia, 49 mass, 47 Characteristic of space, 69 constant of space, 81 Charge, electrostatic, 47 Circle, in centrifugal and gravita- tional field, 62 circumference and diameter, 61 Color, relatively, 7 Combination of velocities in rela- tivity, 42 Comet, orbits, 60 velocity, 13 Completely metric space, 115 Cone, as Euclidean 2-space, 90 Conic in projective geometry, 107 Constant, characteristic, of space, 81 Coordinates of space elements, 92 Corpuscular theory of light, 13 Curvature of bundle as 2-space, 102 of curve, 82 of space, 80, 81, 83 Cylinder, as Euclidean 2-space, 90 D Deflection of light in gravitational field, 55 angle and equation, 59 Detonation velocity, 13 Dielectric field, 18 intensity, 47 Differential metric space, 115 Dimensions of physical space, 97 Direction of curve, 82 Distance between two events, 32 measure of time, 33 E Earth as elliptic 2-space, 75 Einstein, law of gravitation, 11 Electric field, 47 quantity, 47 Electricity, constancy of speed, 4 Electromagnet, 20 Electromagnetic field, 21 wave, 17, 21 frequency, 22 Electron velocity, 8 Electrostatic charge, 47 field, 18 ElUptic geometry, 64, 72, 74 trigonometry, 77 Energy equivalent of mass, 44 field, 22, 46 kinetic, 47 and mass, 41 of wave, 22 123 124 INDEX Entity energy, 24 Equations of transformation to moving system, 25, 27 Ether, 12, 14 as solid, 14 drift, 14 fallacy of conception, 16 illogical, 18 unnecessary, 17 waves, 18 Euclid, 71 Euclidean geometry, 64, 72, 74 F Fallacy of ether conception, 16 Faraday, 12, 17 Field, centrifugal, 47 dielectric, 18 electromagnetic, 21 electrostatic, 18 gravitational, 18, 46, 47 magnetic, 18 of energy, 22 of force, 12, 18 Finite volume of universe, 63 Force, magnetic, 46 Four-dimensional space with sphere as element, 99 Fraction, meaning of, 38 Frequencies of electromagnetic waves, 22 Friction of ether, 14 G Gauss, 71 General differential space, 115 geometry, 64 or projective geometry space, 115 Geometry, 64 of gravitational field, 69 Gravitation, 46 as accelerated motion, 52 as centrifugal force of radial acceleration, 55 as inertia of accelerated system, 53 laws of, 9, 11, 50 and metric axiom, 110 Gravitational field, 18, 21, 47 and deflection of light, 55, 59 as space curvature, 121 its geometry, 69 intensity, 47 shifting of spectrum lines, 68 Gravitational force as centrifugal force of imaginary velocity, 55 and inertia, 53 Gravitational mass, 47 H in projective Harmonic relation geometry, 108 as non-metric, 110 Hertz, 17, 21 Hyperbolic geometry, 64, 72, 74 Hypersurface, 88 Hypothesis of ether abandoned, 16 Imaginary number, meaning, 38 rotation, meaning, 39 representing relativity, 35 Inductance and wave velocity, 23 Inertial mass, 47 Infinitely distant elements in geom- etry, 96 Intensity of dielectric field, 47 of gravitational field, 47 of magnetic field, 47 Interference of light, 13 K Kinetic energy, 44, 47 Kinks, in space, 90 Law of gravitation, 50 Length, relativity, 6 of straight line, 87 shortening by motion, 5, 28 transformation by motion, 26 INDEX 125 Light, constancy of speed, 4 as wave, 13 deflection in gravitational field, 55, 59 orbit of beam, 59, 60 -year, 34 Limit velocity, that of light, 42 Lines of force, 18 as space element in bundle, 100 Lobatschewsky, 71 Lorentz transformation, 26 M Magnet, permanent, 18 Magnetic field, 18, 48 intensity, 47 Magnetic pole strength, 47 Manifold, continuous dimensional, 92 Mass, centrifugal, 47 and energy, 41 as space curvature, 121 gravitational, 47 inertial, 47 relativity, 8 Mathematical conception of mass, 121 space and physical space, 92 Mathematics, relative and not abso- lute, 70 Matter, law of conservation, 8 Maxwell, 12, 17, 21 Mercury, planet, 13 Metric axiom, differential, 115 and law of gravitation, 110 of space, 95, 110 Metric relations in space, 95 partial, 113 Minkowski's four-dimensional space, 24 N Negative number, meaning, 38 Newton, law of gravitation, 10 theory of light, 13 7i-space, 93 O Orbit of beam of light in gravita- tional field, 59 cosmic, 59 Parabolic geometry {See Euclidean geometry) Parallels, 64 axiom, 72 Partial metric relations, 113 Pathcurve of train, 30 Periodic phenomenon or wave, 20 Phase of wave, 14 Physical space, dimensions, 97 and mathematical, 92 Plane geometry (See Euclidean geometry). Planet, orbit, 59 Point as element of space, 92 Polar, 108 Polarization, 14 Pole and polar, 108 strength, magnetic, 47 Position geometry (See General geometry or Projective geom- etry). Projective geometry, 72, 74, 107 Pseudosphere, Beltrami's, 91 Pseudospherical or hyperbolic geom- etry Q Quantity, electric, 47 R Radius of curv^ature, 82 of space, 83 of world, 65 Ray as space element in bundle, 100 Riemann, 71 Rigidity of ether, 14 Rotation, imaginary, meaning, 39 representing relativity, 35 Rotation of pathcurve of train, 31 QC 6.S8 126 Sense perceptions as primary, 23 Simultaneousness of events, 28 Singular points and lines in space, 90 Six-dimensional physical space, 97 Space, bending of, 88 characteristic, 69 curved, visional appearance, 116 general, 95 mathematical and physical, 92 Spectrum lines, shift in gravitational field, 68 Speed, length and time, 8 (See velocity). Sphere as element of four-dimen- sional space, 99 Spherical or elhptic geometry, 74 trigonometry as plane trigonom- etry of elliptic 2-space, 77 Spindle as elliptic 2-space, 89 Steam velocity, 13 Straightest line, 79 Straight line in curved space, 85 definition, 79 length, 87 on earth, 76, 78 relativity of, 79, 87 3 9358 00103908 7 | Time effects of gravitational field, 67 measure of length, 33 relativity, 6 slowing down by motion, 5, 28 transformation by motion, 26 Track coordinates and train coordi- nates, 26 Tractrix, 91 Train coordinates and track coordi- nates, 26 pathcurve, 30 Transverse wave motion of light, 14 Triangle, sum of angles, 63 U Universe, finite volume, 63 two-dimensional analogue of, 119 V Velocity combination in relativity theory, 42 factor of shortening of length and slowing down of time, 29 of electromagnetic wave, 23 of light, 23 Visual appearance of curved space, 116 Volume, finite, of universe, 63, 65 W Tenuity of ether, 14 Theorems in bundle as elliptic Wave, electromagnetic, 17 2-space, 103 - and wave motion, 17 of mathematics, 70 theory of light, 13, 17 QC6 S8 SteinntetZf Charles Four lectures on space* 1st ed« New 1923. Xf 126 p» dlagrs* Proteus, 1865-1923. relativity and York, McGraw-Hill, , plates. 24 cm. 103908 ( MENU 07 SEP 78 992066 NEDDbp 23-6805 ^atf '^vp ( INSTRUCTIONS FOR OBSERVING THE ILLUS- TRATIONS STEREOSCOPICALLY By a little practice, one can throw these illustrations into stereoscopic vision without the use of a stereoscope. This is done by the use of a black cardboard barrier to constrain the left eye to see only the left image, and the right eye, the right. Hold a piece of black cardboard, 3>^ by 8 inches, perpendicular to the page, with the Z}4 inch end resting on the page midway between the two images. Hold the page so that both images are equally illuminated and no shadows on them, and with the end of the cardboard touching the nose and forehead. You will thus see only one image, which at first may appear unsteady and somewhat blurred, both because the image is too close for the eyes properly to adjust themselves in focus, and also the angle at which the eyes are now required to set themselves is unnatural. Slowly move the page and cardboard (the latter held always perpendicular to the page) away from the face, concentrating persistently on this image and on some single feature of the image, su:h as a line intersection, until the page is at the ordinary reading distance from the eyes. The lines will then be in focus, the single image will stand out in perfect stereoscopic effect, and attention can now be turned from the particular feature to the image as a whole. The surface of the ball and the lines on it will appear curved, as in an actual sphere, and the corrugations of the paper surfaces will stand out in relie^, so that the proper curvature of the lines can be observed. However, to get stereoscopic vision, it is necessary that both eyes have the same focal length, or to wear glasses correcting for the same focal length.