VIII

VIIITHE NEW CONCEPTS OF TIME AND SPACEThe Essay in Behalf of Which the Greatest Number of Dissenting Opinions Have Been RecordedBY MONTGOMERY FRANCIS NEW YORKWe have all had experiences, on trains and boats, illustrating our inability to tell, without looking off to some external body, whether we are at rest or moving uniformly; and when we do so look, to tell, without reference to the ground or some other point external to both systems, whether ours or the other be the seat of motion. Uniform motion must be relative, because we find nowhere in the universe a body in the unique state of absolute rest from which alone absolute motion might be measured.True, the wave theory of light with its homogeneous space-filling ether seemed to provide a reference standard for the concept of absolute motion, and for its measurement by experiment with light rays. But whenMichelsonand Morley looked for this absolute motion they found no trace of it. To the physicist, observational student of the external world, nothing exists save observationally; what he can never observe is not there. So: I.By nomeans whatever may we regard uniform straight-line motion as other than relative.As a further direct consequence of theMichelson-Morleyexperiment we have: II.Light in a vacuum presents the same velocity,upper C equals 186 comma 330miles per second, to all observers whatever their velocity of relative motion.In addition to being experimentally established, this is necessary to support I, for if light will distinguish between our velocities, its medium is necessarily a universal standard for absolute motion. But it is contrary to common sense to suppose that if I pass you at 100 miles per hour, the same light impulse can pass us both at the same speed,C. We feel, instinctively, that space and time are not so constituted as to make this possible. But the fact has been repeatedly demonstrated. And when common sense and fundamental concepts clash with facts, it is not the facts that must yield. We have survived such crises, notably one where we had to change the fundamental concept of up-and-down; if another one is here, says Einstein, let us meet it.This the Special Theory of Relativity does. It accepts Postulates I and II above; their consequences it deduces and interprets. For extensive demonstration of these I lack space, and this has been satisfactorily done by others so it is not my chief duty; but clearly they will be startling. For the very ray of light which refuses to recognize our relative motion is the medium through which I must observe your system and you mine.It turns out that I get different values for lengths and time intervalsin your systemthan you get, andvice versa. And we are both right! For me to accept your “correction” were for me to admit that you are at absolute rest and I in absolute motion, that your measure of light velocity is right and mine wrong: admissions barred by the postulates. We have nothing to correct; we can only recognize the reason for the discrepancy; and knowing our relative velocity, each can calculate from his own results what the other’s will be. We find, of course, that at ordinary velocities the discrepancy is many times too small for detection; but at relative velocities at all comparable with that of light it rises above the observational horizon.To inquire the “true” length is meaningless. Chicago is east of Denver, west of Pittsburgh, south of Milwaukee; we do not consider this contradictory, or demand the “true” direction of Chicago. Einstein finds that the concept of length, between points in space or events in time, does not as we had supposed represent an intrinsic property of the points or the events. Like direction, it is merely a relation between these and the observer—a relation whose value changes with the observer’s velocity relative to the object. If our ideas of the part played in the world by time and space do not permit us to believe this, we must alter these ideas. Let us see how we may do this.A World of PointsTo deal with points in a plane the mathematician draws two perpendicular lines, and locates any point, asP, by measuring its distances,XandY,from these “coordinate axes.” The directions of his axes acquire for him a peculiar significance, standing out above other directions; he is apt to measure the distancesupper X minus xandupper Y minus ybetween the pointsPandQin these directions, instead of measuring the single distancePQ. We do the same thing when we say that the railroad station is five blocks north and two east.The mathematician visualizes himself as an observer, located on his coordinate framework. For another observer on another framework, the horizontal and vertical distancesupper X prime minus x primeandupper Y prime minus y primebetweenPandQare different. But for both, thedistance fromPdirect toQis the same. In each case the right triangle tells us that:upper P upper Q equals StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRoot equals StartRoot left-parenthesis upper X prime minus x prime right-parenthesis squared plus left-parenthesis upper Y prime minus y prime right-parenthesis squared EndRootImagine an observer so dominated by his coordinate system that he knows no way of relatingPwithQsave by their horizontal and vertical separation. His whole scheme of things would be shattered by the suggestion that other observers on other reference frames find different horizontal and vertical components. We have to show him the linePQ. We have to convince him that this length is the absolute property enjoyed by his pair of points; that horizontals and verticals are merely relations between the points and the observer, result of the observer’s having analyzed the distancePQinto two components; that different observers effect this decomposition differently; that this seems not to make sense to him only because of his erroneous concept of a fundamental difference between verticals and horizontals.The Four-Dimensional World of EventsWe too have created a distinction in our minds corresponding to no sufficient reality. Our minds seize on time as inherently separable from space. We see the world made up of things in a continuum of three space dimensions; to make this dead worldlive there runs through it a one-dimensional time continuum, imposed from without, unrelated.But did you ever observe anything suggesting the presence of time in the absence of space, or vice versa? No; these vessels of the universe always occur together. Association of the space dimensions into a manifold from which time is excluded is purely a phenomenon of the mind. The space continuum cannot begin to exist until the time dimension is supplied, nor can time exist without a place to exist in.The external world that we observe is composed, not of points, but of events. If a point lacks position in time it does not exist; give it this position and it becomes an event. This world of events is four-dimensional—which means nothing more terrifying than that you must make four measures to locate an event. It does not mean, at all, that you must visualize four mutually perpendicular lines in your accustomed three-space or in a four-space analogous to it. If this world of four dimensions seems to lack reality you will be able to exhibit no better reality for your old ideas. Time belongs, without question; and not as an afterthought, but as part of the world of events.To locate an event we use four measures:X,YandZfor space,Tfor time. Using the same reference frame for time and space, we locate a second event by the measuresx,y,z,t. Minkowski showed that the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared minus left-parenthesis upper C upper T minus upper C t right-parenthesis squared EndRootis the same for all observers, no matter how differenttheirx’s,y’s,z’s andt’s; just as in the plane the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRootis the same for all observers, no matter how different theirx’s andy’s.Such a quantity, having the same value forallobservers, is absolute. In the plane it represents the true, absolute distance between the points—their intrinsic property. In dealing with events it represents the true, absolute “interval,”in time and space togetherbetween the events. It is not space, nor time, but a combination of the two. We have always broken it down into separate space and time components. In this we are as naive as the plane observer who could not visualize the distancePQuntil it was split into separate horizontals and verticals. He understood with difficulty that another observer, employing a different reference frame because in different position, would make the decomposition differently. We understand with difficulty that another observer, employing a different reference frame becausein uniform motion relative to us, will decompose the “interval” between events into time and space components different from ours. Time and space are relative to the observer; only the interval representing space-time is absolute. So common sense stands reconciled to the Special Theory of Relativity.Successive Steps Toward GeneralityIs then our laboriously acquired geometry of points in a three-dimensional space to go into thediscard? By no means. Jeans, investigating the equilibrium of gaseous masses, found the general case too difficult for direct attack. So he considered the case where the masses involved are homogeneous and incompressible. This never occurs; but it throws such light on the general case as to point the way toward attack on it.Euclidean geometry excludes motion, save that engineered by the observer; and then the time is immaterial. Time does not enter at all; the three space dimensions suffice. This simple case never occurs where matter exists; but its conclusions are of value in dealing with more general cases.When we look into a world alleged to be that of Euclid and find motion, we may retain the Euclidean concept of what constitutes the world and invent a machinery to account for the motion; or we may abandon the Euclidean world, as inadequate, in favor of a more general one. We have adopted the second alternative.Newton’s laws tells us that a body free to move will do so, proceeding in a straight line at uniform velocity until interfered with. We do not ask, nor does the theory tell us, whence comes the initial motion. There is no machinery to produce it; it is an inherent property of Newton’s world—assured by the superposition of the time continuum upon Euclid’s world to make Newton’s, accepted without question along with that world itself.But Newton saw that his world of uniform motion, like Euclid’s, was never realized. In the neighborhood of one particle a second is interfered with, forced to give up its uniform motion and acquire aconstant acceleration. This Newton explained by employing the first of the alternatives mentioned above. He tells us that in connection with all matter there exists a force which acts on other matter in a certain way. He does not display the actual machinery through which this “force” works, because he could not discover any machinery; he had to stop with his brilliant generalization of the observed facts. And all his successors have failed to detect the slightest trace of a machinery of gravitation.Einstein asks whether this is not because the machinery is absent—because gravitation, like position in Euclid’s world and motion in Newton’s, is a fundamental property of the world in which it occurs. His point of attack here lay in precise formulation of certain familiar facts that had never been adequately appreciated. These facts indicate that even accelerated motion is relative, in spite of its apparently real and absolute effects.Gravitation and AccelerationAn observer in a closed compartment, moving with constant acceleration through empty space, finds that the “bottom” of his cage catches up with objects that he releases; that it presses on his feet to give him the sensation of weight, etc. It displays all the effects that he would expect if it were at rest in a gravitational field. On the other hand, if it were falling freely under gravitational influence, its occupant would sense no weight, objects released would not leave his hand, the reaction from his everymotion would change his every position in his cage, and he could equally well assume himself at rest in a region of space free from gravitational action. Accelerated motion mayalwaysbe interpreted, by the observer on the system, as ordinary force effects on his moving system, or as gravitational effects on his system at rest.An alternative statement of the Special Theory is that the observed phenomena of uniform motion may equally be accounted for by supposing the object in motion and the observer with his reference frame at rest, or vice versa. We may similarly state the General Theory: The observed phenomena of uniformlyacceleratedmotion may in every case be explained on a basis of stationary observer and accelerated objective, or of stationary objective with the observer and his reference system in accelerated motion. Gravitation is one of these phenomena. It follows that if the observer enjoy properly accelerated axes (in time-space, of course), the absolute character of the world about him must be such as to present to him the phenomenon of gravitation. It remains only to identify the sort of world, of which gravitationas it is observedwould be a fundamental characteristic.Euclid’s and Newton’s systems stand as first and second approximations to that world. The Special Relativity Theory constitutes a correction of Newton, presumably because it is a third approximation. We must seek in it those features which we may most hopefully carry along, into the still more general case.Newton’s system retained the geometry of Euclid.But Minkowski’s invariant expression tells us that Einstein has had to abandon this; for in Euclidean geometry of four dimensions the invariant takes the form:StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared plus left-parenthesis upper T minus t right-parenthesis squared EndRoot commaanalogous to that of two and three dimensions. It is not the presence of the constantCin Minkowski’s formula that counts; this is merely an adjustment so that we may measure space in miles and time in the unit that corresponds to a mile. It is the minus sign where Euclidean geometry demands a plus that makes Minkowski’s continuum non-Euclidean.The editor has told us what this statement means. I think he has made it clear that when we speak of the geometry of the four-dimensional world, we must not read into this term the restrictions surrounding the kind of geometry we are best acquainted with—that of the three-dimensional Euclidean continuum. So I need only point out that if we are to make a fourth (and we hope, final) approximation to the reality, its geometry must preserve the generality attained by that of the third step, if it goes no further.Einstein’s Time-Space WorldEinstein accordingly examined the possible non-Euclidean geometries of four dimensions, in search of one displaying fundamental characteristics which, interpreted in terms of space-time, would lead to the observed facts of gravitation. The mathematics of this investigation is that part of his work which, we are told, but twelve men can follow; so we may only outline his conclusions.If we assume that in the neighborhood of matter the world of space-time is non-Euclidean, and that its curvature or distortion or non-Euclideanism is of a certain type already known to mathematicians; that the curvature of this world in the neighborhood of matter increases with the mass, and decreases as the distance from the matter increases; and that every particle of matter that is not interfered with travels through space-time in the most direct path possiblein that continuum; then the observed facts of gravitation are accounted for as an inherent geometric property of this space-time world. We usually say that the presence of matter distorts this world, and that this distortion gives the track of particles through the region affected its non-uniform character.Gravitation then is not a force at all; it is the fundamental nature of things. A body free to move through the world must follow some definite path. Euclid says it will stand still; Newton that it will traverse a straight line in three-space at uniform time-rate; Einstein that it will move in a “geodesic” through time-space—in every-day language, that it will fall.The numerical consequences of Einstein’s theory are, within the limits of observation, the same as those of Newton’s for all bodies save one—Mercury. This planet shows a small deviation from the path predicted by Newton’s law; Einstein’s theory gives its motion exactly. Again, when modern research showed that light must be affected by gravitation, Einstein’s theory, because of the extreme velocity of light, deviates from Newton’s, where thespeed is less a determining factor; and observations of starlight deflected by the sun during the eclipse were in much better accord with Einstein’s theory than Newton’s. Moreover, the Special Theory predicts that mass is an observational variable like length and duration. Radioactive emanations have a velocity high enough to give appreciable results here, and the prediction is verified, tending to support the general theory by supporting its limiting case.We like always to unify our science; and seldom, after effecting a unification, are we forced to give it up. Einstein for the first time brings mechanical, electromagnetic and gravitational phenomena within one structure. This is one reason why physicists are so open minded toward his theory—they want it to be true.The Layman’s Last DoubtThe final answer to any series of questions is inevitably “because the world is so constructed.” The things we are content to leave on that basis are those to which we are accustomed, and which we therefore think we understand; those for which this explanation leaves us unsatisfied are those which are new and unfamiliar. Newton told us that the world of three-dimensional space with one-dimensional time superposed was so constructed that bodies left to themselves would go on forever in a straight line at constant speed. We think we understand this, but our understanding consists merely of the unspoken query, “Why, of course; what is there to prevent?”The Greeks, an intelligent people, looked at this differently; they would have met Newton with the unanimous demand “Why so; what is there to keep them going?” So if, in seeking an explanation of anything, we come sooner than we had expected to the finality “Because the world is so constructed,” let us not feel that we have been cheated.

VIIITHE NEW CONCEPTS OF TIME AND SPACEThe Essay in Behalf of Which the Greatest Number of Dissenting Opinions Have Been RecordedBY MONTGOMERY FRANCIS NEW YORK

The Essay in Behalf of Which the Greatest Number of Dissenting Opinions Have Been Recorded

BY MONTGOMERY FRANCIS NEW YORK

We have all had experiences, on trains and boats, illustrating our inability to tell, without looking off to some external body, whether we are at rest or moving uniformly; and when we do so look, to tell, without reference to the ground or some other point external to both systems, whether ours or the other be the seat of motion. Uniform motion must be relative, because we find nowhere in the universe a body in the unique state of absolute rest from which alone absolute motion might be measured.True, the wave theory of light with its homogeneous space-filling ether seemed to provide a reference standard for the concept of absolute motion, and for its measurement by experiment with light rays. But whenMichelsonand Morley looked for this absolute motion they found no trace of it. To the physicist, observational student of the external world, nothing exists save observationally; what he can never observe is not there. So: I.By nomeans whatever may we regard uniform straight-line motion as other than relative.As a further direct consequence of theMichelson-Morleyexperiment we have: II.Light in a vacuum presents the same velocity,upper C equals 186 comma 330miles per second, to all observers whatever their velocity of relative motion.In addition to being experimentally established, this is necessary to support I, for if light will distinguish between our velocities, its medium is necessarily a universal standard for absolute motion. But it is contrary to common sense to suppose that if I pass you at 100 miles per hour, the same light impulse can pass us both at the same speed,C. We feel, instinctively, that space and time are not so constituted as to make this possible. But the fact has been repeatedly demonstrated. And when common sense and fundamental concepts clash with facts, it is not the facts that must yield. We have survived such crises, notably one where we had to change the fundamental concept of up-and-down; if another one is here, says Einstein, let us meet it.This the Special Theory of Relativity does. It accepts Postulates I and II above; their consequences it deduces and interprets. For extensive demonstration of these I lack space, and this has been satisfactorily done by others so it is not my chief duty; but clearly they will be startling. For the very ray of light which refuses to recognize our relative motion is the medium through which I must observe your system and you mine.It turns out that I get different values for lengths and time intervalsin your systemthan you get, andvice versa. And we are both right! For me to accept your “correction” were for me to admit that you are at absolute rest and I in absolute motion, that your measure of light velocity is right and mine wrong: admissions barred by the postulates. We have nothing to correct; we can only recognize the reason for the discrepancy; and knowing our relative velocity, each can calculate from his own results what the other’s will be. We find, of course, that at ordinary velocities the discrepancy is many times too small for detection; but at relative velocities at all comparable with that of light it rises above the observational horizon.To inquire the “true” length is meaningless. Chicago is east of Denver, west of Pittsburgh, south of Milwaukee; we do not consider this contradictory, or demand the “true” direction of Chicago. Einstein finds that the concept of length, between points in space or events in time, does not as we had supposed represent an intrinsic property of the points or the events. Like direction, it is merely a relation between these and the observer—a relation whose value changes with the observer’s velocity relative to the object. If our ideas of the part played in the world by time and space do not permit us to believe this, we must alter these ideas. Let us see how we may do this.A World of PointsTo deal with points in a plane the mathematician draws two perpendicular lines, and locates any point, asP, by measuring its distances,XandY,from these “coordinate axes.” The directions of his axes acquire for him a peculiar significance, standing out above other directions; he is apt to measure the distancesupper X minus xandupper Y minus ybetween the pointsPandQin these directions, instead of measuring the single distancePQ. We do the same thing when we say that the railroad station is five blocks north and two east.The mathematician visualizes himself as an observer, located on his coordinate framework. For another observer on another framework, the horizontal and vertical distancesupper X prime minus x primeandupper Y prime minus y primebetweenPandQare different. But for both, thedistance fromPdirect toQis the same. In each case the right triangle tells us that:upper P upper Q equals StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRoot equals StartRoot left-parenthesis upper X prime minus x prime right-parenthesis squared plus left-parenthesis upper Y prime minus y prime right-parenthesis squared EndRootImagine an observer so dominated by his coordinate system that he knows no way of relatingPwithQsave by their horizontal and vertical separation. His whole scheme of things would be shattered by the suggestion that other observers on other reference frames find different horizontal and vertical components. We have to show him the linePQ. We have to convince him that this length is the absolute property enjoyed by his pair of points; that horizontals and verticals are merely relations between the points and the observer, result of the observer’s having analyzed the distancePQinto two components; that different observers effect this decomposition differently; that this seems not to make sense to him only because of his erroneous concept of a fundamental difference between verticals and horizontals.The Four-Dimensional World of EventsWe too have created a distinction in our minds corresponding to no sufficient reality. Our minds seize on time as inherently separable from space. We see the world made up of things in a continuum of three space dimensions; to make this dead worldlive there runs through it a one-dimensional time continuum, imposed from without, unrelated.But did you ever observe anything suggesting the presence of time in the absence of space, or vice versa? No; these vessels of the universe always occur together. Association of the space dimensions into a manifold from which time is excluded is purely a phenomenon of the mind. The space continuum cannot begin to exist until the time dimension is supplied, nor can time exist without a place to exist in.The external world that we observe is composed, not of points, but of events. If a point lacks position in time it does not exist; give it this position and it becomes an event. This world of events is four-dimensional—which means nothing more terrifying than that you must make four measures to locate an event. It does not mean, at all, that you must visualize four mutually perpendicular lines in your accustomed three-space or in a four-space analogous to it. If this world of four dimensions seems to lack reality you will be able to exhibit no better reality for your old ideas. Time belongs, without question; and not as an afterthought, but as part of the world of events.To locate an event we use four measures:X,YandZfor space,Tfor time. Using the same reference frame for time and space, we locate a second event by the measuresx,y,z,t. Minkowski showed that the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared minus left-parenthesis upper C upper T minus upper C t right-parenthesis squared EndRootis the same for all observers, no matter how differenttheirx’s,y’s,z’s andt’s; just as in the plane the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRootis the same for all observers, no matter how different theirx’s andy’s.Such a quantity, having the same value forallobservers, is absolute. In the plane it represents the true, absolute distance between the points—their intrinsic property. In dealing with events it represents the true, absolute “interval,”in time and space togetherbetween the events. It is not space, nor time, but a combination of the two. We have always broken it down into separate space and time components. In this we are as naive as the plane observer who could not visualize the distancePQuntil it was split into separate horizontals and verticals. He understood with difficulty that another observer, employing a different reference frame because in different position, would make the decomposition differently. We understand with difficulty that another observer, employing a different reference frame becausein uniform motion relative to us, will decompose the “interval” between events into time and space components different from ours. Time and space are relative to the observer; only the interval representing space-time is absolute. So common sense stands reconciled to the Special Theory of Relativity.Successive Steps Toward GeneralityIs then our laboriously acquired geometry of points in a three-dimensional space to go into thediscard? By no means. Jeans, investigating the equilibrium of gaseous masses, found the general case too difficult for direct attack. So he considered the case where the masses involved are homogeneous and incompressible. This never occurs; but it throws such light on the general case as to point the way toward attack on it.Euclidean geometry excludes motion, save that engineered by the observer; and then the time is immaterial. Time does not enter at all; the three space dimensions suffice. This simple case never occurs where matter exists; but its conclusions are of value in dealing with more general cases.When we look into a world alleged to be that of Euclid and find motion, we may retain the Euclidean concept of what constitutes the world and invent a machinery to account for the motion; or we may abandon the Euclidean world, as inadequate, in favor of a more general one. We have adopted the second alternative.Newton’s laws tells us that a body free to move will do so, proceeding in a straight line at uniform velocity until interfered with. We do not ask, nor does the theory tell us, whence comes the initial motion. There is no machinery to produce it; it is an inherent property of Newton’s world—assured by the superposition of the time continuum upon Euclid’s world to make Newton’s, accepted without question along with that world itself.But Newton saw that his world of uniform motion, like Euclid’s, was never realized. In the neighborhood of one particle a second is interfered with, forced to give up its uniform motion and acquire aconstant acceleration. This Newton explained by employing the first of the alternatives mentioned above. He tells us that in connection with all matter there exists a force which acts on other matter in a certain way. He does not display the actual machinery through which this “force” works, because he could not discover any machinery; he had to stop with his brilliant generalization of the observed facts. And all his successors have failed to detect the slightest trace of a machinery of gravitation.Einstein asks whether this is not because the machinery is absent—because gravitation, like position in Euclid’s world and motion in Newton’s, is a fundamental property of the world in which it occurs. His point of attack here lay in precise formulation of certain familiar facts that had never been adequately appreciated. These facts indicate that even accelerated motion is relative, in spite of its apparently real and absolute effects.Gravitation and AccelerationAn observer in a closed compartment, moving with constant acceleration through empty space, finds that the “bottom” of his cage catches up with objects that he releases; that it presses on his feet to give him the sensation of weight, etc. It displays all the effects that he would expect if it were at rest in a gravitational field. On the other hand, if it were falling freely under gravitational influence, its occupant would sense no weight, objects released would not leave his hand, the reaction from his everymotion would change his every position in his cage, and he could equally well assume himself at rest in a region of space free from gravitational action. Accelerated motion mayalwaysbe interpreted, by the observer on the system, as ordinary force effects on his moving system, or as gravitational effects on his system at rest.An alternative statement of the Special Theory is that the observed phenomena of uniform motion may equally be accounted for by supposing the object in motion and the observer with his reference frame at rest, or vice versa. We may similarly state the General Theory: The observed phenomena of uniformlyacceleratedmotion may in every case be explained on a basis of stationary observer and accelerated objective, or of stationary objective with the observer and his reference system in accelerated motion. Gravitation is one of these phenomena. It follows that if the observer enjoy properly accelerated axes (in time-space, of course), the absolute character of the world about him must be such as to present to him the phenomenon of gravitation. It remains only to identify the sort of world, of which gravitationas it is observedwould be a fundamental characteristic.Euclid’s and Newton’s systems stand as first and second approximations to that world. The Special Relativity Theory constitutes a correction of Newton, presumably because it is a third approximation. We must seek in it those features which we may most hopefully carry along, into the still more general case.Newton’s system retained the geometry of Euclid.But Minkowski’s invariant expression tells us that Einstein has had to abandon this; for in Euclidean geometry of four dimensions the invariant takes the form:StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared plus left-parenthesis upper T minus t right-parenthesis squared EndRoot commaanalogous to that of two and three dimensions. It is not the presence of the constantCin Minkowski’s formula that counts; this is merely an adjustment so that we may measure space in miles and time in the unit that corresponds to a mile. It is the minus sign where Euclidean geometry demands a plus that makes Minkowski’s continuum non-Euclidean.The editor has told us what this statement means. I think he has made it clear that when we speak of the geometry of the four-dimensional world, we must not read into this term the restrictions surrounding the kind of geometry we are best acquainted with—that of the three-dimensional Euclidean continuum. So I need only point out that if we are to make a fourth (and we hope, final) approximation to the reality, its geometry must preserve the generality attained by that of the third step, if it goes no further.Einstein’s Time-Space WorldEinstein accordingly examined the possible non-Euclidean geometries of four dimensions, in search of one displaying fundamental characteristics which, interpreted in terms of space-time, would lead to the observed facts of gravitation. The mathematics of this investigation is that part of his work which, we are told, but twelve men can follow; so we may only outline his conclusions.If we assume that in the neighborhood of matter the world of space-time is non-Euclidean, and that its curvature or distortion or non-Euclideanism is of a certain type already known to mathematicians; that the curvature of this world in the neighborhood of matter increases with the mass, and decreases as the distance from the matter increases; and that every particle of matter that is not interfered with travels through space-time in the most direct path possiblein that continuum; then the observed facts of gravitation are accounted for as an inherent geometric property of this space-time world. We usually say that the presence of matter distorts this world, and that this distortion gives the track of particles through the region affected its non-uniform character.Gravitation then is not a force at all; it is the fundamental nature of things. A body free to move through the world must follow some definite path. Euclid says it will stand still; Newton that it will traverse a straight line in three-space at uniform time-rate; Einstein that it will move in a “geodesic” through time-space—in every-day language, that it will fall.The numerical consequences of Einstein’s theory are, within the limits of observation, the same as those of Newton’s for all bodies save one—Mercury. This planet shows a small deviation from the path predicted by Newton’s law; Einstein’s theory gives its motion exactly. Again, when modern research showed that light must be affected by gravitation, Einstein’s theory, because of the extreme velocity of light, deviates from Newton’s, where thespeed is less a determining factor; and observations of starlight deflected by the sun during the eclipse were in much better accord with Einstein’s theory than Newton’s. Moreover, the Special Theory predicts that mass is an observational variable like length and duration. Radioactive emanations have a velocity high enough to give appreciable results here, and the prediction is verified, tending to support the general theory by supporting its limiting case.We like always to unify our science; and seldom, after effecting a unification, are we forced to give it up. Einstein for the first time brings mechanical, electromagnetic and gravitational phenomena within one structure. This is one reason why physicists are so open minded toward his theory—they want it to be true.The Layman’s Last DoubtThe final answer to any series of questions is inevitably “because the world is so constructed.” The things we are content to leave on that basis are those to which we are accustomed, and which we therefore think we understand; those for which this explanation leaves us unsatisfied are those which are new and unfamiliar. Newton told us that the world of three-dimensional space with one-dimensional time superposed was so constructed that bodies left to themselves would go on forever in a straight line at constant speed. We think we understand this, but our understanding consists merely of the unspoken query, “Why, of course; what is there to prevent?”The Greeks, an intelligent people, looked at this differently; they would have met Newton with the unanimous demand “Why so; what is there to keep them going?” So if, in seeking an explanation of anything, we come sooner than we had expected to the finality “Because the world is so constructed,” let us not feel that we have been cheated.

True, the wave theory of light with its homogeneous space-filling ether seemed to provide a reference standard for the concept of absolute motion, and for its measurement by experiment with light rays. But whenMichelsonand Morley looked for this absolute motion they found no trace of it. To the physicist, observational student of the external world, nothing exists save observationally; what he can never observe is not there. So: I.By nomeans whatever may we regard uniform straight-line motion as other than relative.

As a further direct consequence of theMichelson-Morleyexperiment we have: II.Light in a vacuum presents the same velocity,upper C equals 186 comma 330miles per second, to all observers whatever their velocity of relative motion.In addition to being experimentally established, this is necessary to support I, for if light will distinguish between our velocities, its medium is necessarily a universal standard for absolute motion. But it is contrary to common sense to suppose that if I pass you at 100 miles per hour, the same light impulse can pass us both at the same speed,C. We feel, instinctively, that space and time are not so constituted as to make this possible. But the fact has been repeatedly demonstrated. And when common sense and fundamental concepts clash with facts, it is not the facts that must yield. We have survived such crises, notably one where we had to change the fundamental concept of up-and-down; if another one is here, says Einstein, let us meet it.

This the Special Theory of Relativity does. It accepts Postulates I and II above; their consequences it deduces and interprets. For extensive demonstration of these I lack space, and this has been satisfactorily done by others so it is not my chief duty; but clearly they will be startling. For the very ray of light which refuses to recognize our relative motion is the medium through which I must observe your system and you mine.

It turns out that I get different values for lengths and time intervalsin your systemthan you get, andvice versa. And we are both right! For me to accept your “correction” were for me to admit that you are at absolute rest and I in absolute motion, that your measure of light velocity is right and mine wrong: admissions barred by the postulates. We have nothing to correct; we can only recognize the reason for the discrepancy; and knowing our relative velocity, each can calculate from his own results what the other’s will be. We find, of course, that at ordinary velocities the discrepancy is many times too small for detection; but at relative velocities at all comparable with that of light it rises above the observational horizon.

To inquire the “true” length is meaningless. Chicago is east of Denver, west of Pittsburgh, south of Milwaukee; we do not consider this contradictory, or demand the “true” direction of Chicago. Einstein finds that the concept of length, between points in space or events in time, does not as we had supposed represent an intrinsic property of the points or the events. Like direction, it is merely a relation between these and the observer—a relation whose value changes with the observer’s velocity relative to the object. If our ideas of the part played in the world by time and space do not permit us to believe this, we must alter these ideas. Let us see how we may do this.

A World of PointsTo deal with points in a plane the mathematician draws two perpendicular lines, and locates any point, asP, by measuring its distances,XandY,from these “coordinate axes.” The directions of his axes acquire for him a peculiar significance, standing out above other directions; he is apt to measure the distancesupper X minus xandupper Y minus ybetween the pointsPandQin these directions, instead of measuring the single distancePQ. We do the same thing when we say that the railroad station is five blocks north and two east.The mathematician visualizes himself as an observer, located on his coordinate framework. For another observer on another framework, the horizontal and vertical distancesupper X prime minus x primeandupper Y prime minus y primebetweenPandQare different. But for both, thedistance fromPdirect toQis the same. In each case the right triangle tells us that:upper P upper Q equals StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRoot equals StartRoot left-parenthesis upper X prime minus x prime right-parenthesis squared plus left-parenthesis upper Y prime minus y prime right-parenthesis squared EndRootImagine an observer so dominated by his coordinate system that he knows no way of relatingPwithQsave by their horizontal and vertical separation. His whole scheme of things would be shattered by the suggestion that other observers on other reference frames find different horizontal and vertical components. We have to show him the linePQ. We have to convince him that this length is the absolute property enjoyed by his pair of points; that horizontals and verticals are merely relations between the points and the observer, result of the observer’s having analyzed the distancePQinto two components; that different observers effect this decomposition differently; that this seems not to make sense to him only because of his erroneous concept of a fundamental difference between verticals and horizontals.

A World of Points

To deal with points in a plane the mathematician draws two perpendicular lines, and locates any point, asP, by measuring its distances,XandY,from these “coordinate axes.” The directions of his axes acquire for him a peculiar significance, standing out above other directions; he is apt to measure the distancesupper X minus xandupper Y minus ybetween the pointsPandQin these directions, instead of measuring the single distancePQ. We do the same thing when we say that the railroad station is five blocks north and two east.The mathematician visualizes himself as an observer, located on his coordinate framework. For another observer on another framework, the horizontal and vertical distancesupper X prime minus x primeandupper Y prime minus y primebetweenPandQare different. But for both, thedistance fromPdirect toQis the same. In each case the right triangle tells us that:upper P upper Q equals StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRoot equals StartRoot left-parenthesis upper X prime minus x prime right-parenthesis squared plus left-parenthesis upper Y prime minus y prime right-parenthesis squared EndRootImagine an observer so dominated by his coordinate system that he knows no way of relatingPwithQsave by their horizontal and vertical separation. His whole scheme of things would be shattered by the suggestion that other observers on other reference frames find different horizontal and vertical components. We have to show him the linePQ. We have to convince him that this length is the absolute property enjoyed by his pair of points; that horizontals and verticals are merely relations between the points and the observer, result of the observer’s having analyzed the distancePQinto two components; that different observers effect this decomposition differently; that this seems not to make sense to him only because of his erroneous concept of a fundamental difference between verticals and horizontals.

The mathematician visualizes himself as an observer, located on his coordinate framework. For another observer on another framework, the horizontal and vertical distancesupper X prime minus x primeandupper Y prime minus y primebetweenPandQare different. But for both, thedistance fromPdirect toQis the same. In each case the right triangle tells us that:upper P upper Q equals StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRoot equals StartRoot left-parenthesis upper X prime minus x prime right-parenthesis squared plus left-parenthesis upper Y prime minus y prime right-parenthesis squared EndRoot

Imagine an observer so dominated by his coordinate system that he knows no way of relatingPwithQsave by their horizontal and vertical separation. His whole scheme of things would be shattered by the suggestion that other observers on other reference frames find different horizontal and vertical components. We have to show him the linePQ. We have to convince him that this length is the absolute property enjoyed by his pair of points; that horizontals and verticals are merely relations between the points and the observer, result of the observer’s having analyzed the distancePQinto two components; that different observers effect this decomposition differently; that this seems not to make sense to him only because of his erroneous concept of a fundamental difference between verticals and horizontals.

The Four-Dimensional World of EventsWe too have created a distinction in our minds corresponding to no sufficient reality. Our minds seize on time as inherently separable from space. We see the world made up of things in a continuum of three space dimensions; to make this dead worldlive there runs through it a one-dimensional time continuum, imposed from without, unrelated.But did you ever observe anything suggesting the presence of time in the absence of space, or vice versa? No; these vessels of the universe always occur together. Association of the space dimensions into a manifold from which time is excluded is purely a phenomenon of the mind. The space continuum cannot begin to exist until the time dimension is supplied, nor can time exist without a place to exist in.The external world that we observe is composed, not of points, but of events. If a point lacks position in time it does not exist; give it this position and it becomes an event. This world of events is four-dimensional—which means nothing more terrifying than that you must make four measures to locate an event. It does not mean, at all, that you must visualize four mutually perpendicular lines in your accustomed three-space or in a four-space analogous to it. If this world of four dimensions seems to lack reality you will be able to exhibit no better reality for your old ideas. Time belongs, without question; and not as an afterthought, but as part of the world of events.To locate an event we use four measures:X,YandZfor space,Tfor time. Using the same reference frame for time and space, we locate a second event by the measuresx,y,z,t. Minkowski showed that the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared minus left-parenthesis upper C upper T minus upper C t right-parenthesis squared EndRootis the same for all observers, no matter how differenttheirx’s,y’s,z’s andt’s; just as in the plane the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRootis the same for all observers, no matter how different theirx’s andy’s.Such a quantity, having the same value forallobservers, is absolute. In the plane it represents the true, absolute distance between the points—their intrinsic property. In dealing with events it represents the true, absolute “interval,”in time and space togetherbetween the events. It is not space, nor time, but a combination of the two. We have always broken it down into separate space and time components. In this we are as naive as the plane observer who could not visualize the distancePQuntil it was split into separate horizontals and verticals. He understood with difficulty that another observer, employing a different reference frame because in different position, would make the decomposition differently. We understand with difficulty that another observer, employing a different reference frame becausein uniform motion relative to us, will decompose the “interval” between events into time and space components different from ours. Time and space are relative to the observer; only the interval representing space-time is absolute. So common sense stands reconciled to the Special Theory of Relativity.

The Four-Dimensional World of Events

We too have created a distinction in our minds corresponding to no sufficient reality. Our minds seize on time as inherently separable from space. We see the world made up of things in a continuum of three space dimensions; to make this dead worldlive there runs through it a one-dimensional time continuum, imposed from without, unrelated.But did you ever observe anything suggesting the presence of time in the absence of space, or vice versa? No; these vessels of the universe always occur together. Association of the space dimensions into a manifold from which time is excluded is purely a phenomenon of the mind. The space continuum cannot begin to exist until the time dimension is supplied, nor can time exist without a place to exist in.The external world that we observe is composed, not of points, but of events. If a point lacks position in time it does not exist; give it this position and it becomes an event. This world of events is four-dimensional—which means nothing more terrifying than that you must make four measures to locate an event. It does not mean, at all, that you must visualize four mutually perpendicular lines in your accustomed three-space or in a four-space analogous to it. If this world of four dimensions seems to lack reality you will be able to exhibit no better reality for your old ideas. Time belongs, without question; and not as an afterthought, but as part of the world of events.To locate an event we use four measures:X,YandZfor space,Tfor time. Using the same reference frame for time and space, we locate a second event by the measuresx,y,z,t. Minkowski showed that the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared minus left-parenthesis upper C upper T minus upper C t right-parenthesis squared EndRootis the same for all observers, no matter how differenttheirx’s,y’s,z’s andt’s; just as in the plane the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRootis the same for all observers, no matter how different theirx’s andy’s.Such a quantity, having the same value forallobservers, is absolute. In the plane it represents the true, absolute distance between the points—their intrinsic property. In dealing with events it represents the true, absolute “interval,”in time and space togetherbetween the events. It is not space, nor time, but a combination of the two. We have always broken it down into separate space and time components. In this we are as naive as the plane observer who could not visualize the distancePQuntil it was split into separate horizontals and verticals. He understood with difficulty that another observer, employing a different reference frame because in different position, would make the decomposition differently. We understand with difficulty that another observer, employing a different reference frame becausein uniform motion relative to us, will decompose the “interval” between events into time and space components different from ours. Time and space are relative to the observer; only the interval representing space-time is absolute. So common sense stands reconciled to the Special Theory of Relativity.

But did you ever observe anything suggesting the presence of time in the absence of space, or vice versa? No; these vessels of the universe always occur together. Association of the space dimensions into a manifold from which time is excluded is purely a phenomenon of the mind. The space continuum cannot begin to exist until the time dimension is supplied, nor can time exist without a place to exist in.

The external world that we observe is composed, not of points, but of events. If a point lacks position in time it does not exist; give it this position and it becomes an event. This world of events is four-dimensional—which means nothing more terrifying than that you must make four measures to locate an event. It does not mean, at all, that you must visualize four mutually perpendicular lines in your accustomed three-space or in a four-space analogous to it. If this world of four dimensions seems to lack reality you will be able to exhibit no better reality for your old ideas. Time belongs, without question; and not as an afterthought, but as part of the world of events.

To locate an event we use four measures:X,YandZfor space,Tfor time. Using the same reference frame for time and space, we locate a second event by the measuresx,y,z,t. Minkowski showed that the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared minus left-parenthesis upper C upper T minus upper C t right-parenthesis squared EndRootis the same for all observers, no matter how differenttheirx’s,y’s,z’s andt’s; just as in the plane the quantityStartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared EndRoot

is the same for all observers, no matter how different theirx’s andy’s.

Such a quantity, having the same value forallobservers, is absolute. In the plane it represents the true, absolute distance between the points—their intrinsic property. In dealing with events it represents the true, absolute “interval,”in time and space togetherbetween the events. It is not space, nor time, but a combination of the two. We have always broken it down into separate space and time components. In this we are as naive as the plane observer who could not visualize the distancePQuntil it was split into separate horizontals and verticals. He understood with difficulty that another observer, employing a different reference frame because in different position, would make the decomposition differently. We understand with difficulty that another observer, employing a different reference frame becausein uniform motion relative to us, will decompose the “interval” between events into time and space components different from ours. Time and space are relative to the observer; only the interval representing space-time is absolute. So common sense stands reconciled to the Special Theory of Relativity.

Successive Steps Toward GeneralityIs then our laboriously acquired geometry of points in a three-dimensional space to go into thediscard? By no means. Jeans, investigating the equilibrium of gaseous masses, found the general case too difficult for direct attack. So he considered the case where the masses involved are homogeneous and incompressible. This never occurs; but it throws such light on the general case as to point the way toward attack on it.Euclidean geometry excludes motion, save that engineered by the observer; and then the time is immaterial. Time does not enter at all; the three space dimensions suffice. This simple case never occurs where matter exists; but its conclusions are of value in dealing with more general cases.When we look into a world alleged to be that of Euclid and find motion, we may retain the Euclidean concept of what constitutes the world and invent a machinery to account for the motion; or we may abandon the Euclidean world, as inadequate, in favor of a more general one. We have adopted the second alternative.Newton’s laws tells us that a body free to move will do so, proceeding in a straight line at uniform velocity until interfered with. We do not ask, nor does the theory tell us, whence comes the initial motion. There is no machinery to produce it; it is an inherent property of Newton’s world—assured by the superposition of the time continuum upon Euclid’s world to make Newton’s, accepted without question along with that world itself.But Newton saw that his world of uniform motion, like Euclid’s, was never realized. In the neighborhood of one particle a second is interfered with, forced to give up its uniform motion and acquire aconstant acceleration. This Newton explained by employing the first of the alternatives mentioned above. He tells us that in connection with all matter there exists a force which acts on other matter in a certain way. He does not display the actual machinery through which this “force” works, because he could not discover any machinery; he had to stop with his brilliant generalization of the observed facts. And all his successors have failed to detect the slightest trace of a machinery of gravitation.Einstein asks whether this is not because the machinery is absent—because gravitation, like position in Euclid’s world and motion in Newton’s, is a fundamental property of the world in which it occurs. His point of attack here lay in precise formulation of certain familiar facts that had never been adequately appreciated. These facts indicate that even accelerated motion is relative, in spite of its apparently real and absolute effects.

Successive Steps Toward Generality

Is then our laboriously acquired geometry of points in a three-dimensional space to go into thediscard? By no means. Jeans, investigating the equilibrium of gaseous masses, found the general case too difficult for direct attack. So he considered the case where the masses involved are homogeneous and incompressible. This never occurs; but it throws such light on the general case as to point the way toward attack on it.Euclidean geometry excludes motion, save that engineered by the observer; and then the time is immaterial. Time does not enter at all; the three space dimensions suffice. This simple case never occurs where matter exists; but its conclusions are of value in dealing with more general cases.When we look into a world alleged to be that of Euclid and find motion, we may retain the Euclidean concept of what constitutes the world and invent a machinery to account for the motion; or we may abandon the Euclidean world, as inadequate, in favor of a more general one. We have adopted the second alternative.Newton’s laws tells us that a body free to move will do so, proceeding in a straight line at uniform velocity until interfered with. We do not ask, nor does the theory tell us, whence comes the initial motion. There is no machinery to produce it; it is an inherent property of Newton’s world—assured by the superposition of the time continuum upon Euclid’s world to make Newton’s, accepted without question along with that world itself.But Newton saw that his world of uniform motion, like Euclid’s, was never realized. In the neighborhood of one particle a second is interfered with, forced to give up its uniform motion and acquire aconstant acceleration. This Newton explained by employing the first of the alternatives mentioned above. He tells us that in connection with all matter there exists a force which acts on other matter in a certain way. He does not display the actual machinery through which this “force” works, because he could not discover any machinery; he had to stop with his brilliant generalization of the observed facts. And all his successors have failed to detect the slightest trace of a machinery of gravitation.Einstein asks whether this is not because the machinery is absent—because gravitation, like position in Euclid’s world and motion in Newton’s, is a fundamental property of the world in which it occurs. His point of attack here lay in precise formulation of certain familiar facts that had never been adequately appreciated. These facts indicate that even accelerated motion is relative, in spite of its apparently real and absolute effects.

Euclidean geometry excludes motion, save that engineered by the observer; and then the time is immaterial. Time does not enter at all; the three space dimensions suffice. This simple case never occurs where matter exists; but its conclusions are of value in dealing with more general cases.

When we look into a world alleged to be that of Euclid and find motion, we may retain the Euclidean concept of what constitutes the world and invent a machinery to account for the motion; or we may abandon the Euclidean world, as inadequate, in favor of a more general one. We have adopted the second alternative.

Newton’s laws tells us that a body free to move will do so, proceeding in a straight line at uniform velocity until interfered with. We do not ask, nor does the theory tell us, whence comes the initial motion. There is no machinery to produce it; it is an inherent property of Newton’s world—assured by the superposition of the time continuum upon Euclid’s world to make Newton’s, accepted without question along with that world itself.

But Newton saw that his world of uniform motion, like Euclid’s, was never realized. In the neighborhood of one particle a second is interfered with, forced to give up its uniform motion and acquire aconstant acceleration. This Newton explained by employing the first of the alternatives mentioned above. He tells us that in connection with all matter there exists a force which acts on other matter in a certain way. He does not display the actual machinery through which this “force” works, because he could not discover any machinery; he had to stop with his brilliant generalization of the observed facts. And all his successors have failed to detect the slightest trace of a machinery of gravitation.

Einstein asks whether this is not because the machinery is absent—because gravitation, like position in Euclid’s world and motion in Newton’s, is a fundamental property of the world in which it occurs. His point of attack here lay in precise formulation of certain familiar facts that had never been adequately appreciated. These facts indicate that even accelerated motion is relative, in spite of its apparently real and absolute effects.

Gravitation and AccelerationAn observer in a closed compartment, moving with constant acceleration through empty space, finds that the “bottom” of his cage catches up with objects that he releases; that it presses on his feet to give him the sensation of weight, etc. It displays all the effects that he would expect if it were at rest in a gravitational field. On the other hand, if it were falling freely under gravitational influence, its occupant would sense no weight, objects released would not leave his hand, the reaction from his everymotion would change his every position in his cage, and he could equally well assume himself at rest in a region of space free from gravitational action. Accelerated motion mayalwaysbe interpreted, by the observer on the system, as ordinary force effects on his moving system, or as gravitational effects on his system at rest.An alternative statement of the Special Theory is that the observed phenomena of uniform motion may equally be accounted for by supposing the object in motion and the observer with his reference frame at rest, or vice versa. We may similarly state the General Theory: The observed phenomena of uniformlyacceleratedmotion may in every case be explained on a basis of stationary observer and accelerated objective, or of stationary objective with the observer and his reference system in accelerated motion. Gravitation is one of these phenomena. It follows that if the observer enjoy properly accelerated axes (in time-space, of course), the absolute character of the world about him must be such as to present to him the phenomenon of gravitation. It remains only to identify the sort of world, of which gravitationas it is observedwould be a fundamental characteristic.Euclid’s and Newton’s systems stand as first and second approximations to that world. The Special Relativity Theory constitutes a correction of Newton, presumably because it is a third approximation. We must seek in it those features which we may most hopefully carry along, into the still more general case.Newton’s system retained the geometry of Euclid.But Minkowski’s invariant expression tells us that Einstein has had to abandon this; for in Euclidean geometry of four dimensions the invariant takes the form:StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared plus left-parenthesis upper T minus t right-parenthesis squared EndRoot commaanalogous to that of two and three dimensions. It is not the presence of the constantCin Minkowski’s formula that counts; this is merely an adjustment so that we may measure space in miles and time in the unit that corresponds to a mile. It is the minus sign where Euclidean geometry demands a plus that makes Minkowski’s continuum non-Euclidean.The editor has told us what this statement means. I think he has made it clear that when we speak of the geometry of the four-dimensional world, we must not read into this term the restrictions surrounding the kind of geometry we are best acquainted with—that of the three-dimensional Euclidean continuum. So I need only point out that if we are to make a fourth (and we hope, final) approximation to the reality, its geometry must preserve the generality attained by that of the third step, if it goes no further.

Gravitation and Acceleration

An observer in a closed compartment, moving with constant acceleration through empty space, finds that the “bottom” of his cage catches up with objects that he releases; that it presses on his feet to give him the sensation of weight, etc. It displays all the effects that he would expect if it were at rest in a gravitational field. On the other hand, if it were falling freely under gravitational influence, its occupant would sense no weight, objects released would not leave his hand, the reaction from his everymotion would change his every position in his cage, and he could equally well assume himself at rest in a region of space free from gravitational action. Accelerated motion mayalwaysbe interpreted, by the observer on the system, as ordinary force effects on his moving system, or as gravitational effects on his system at rest.An alternative statement of the Special Theory is that the observed phenomena of uniform motion may equally be accounted for by supposing the object in motion and the observer with his reference frame at rest, or vice versa. We may similarly state the General Theory: The observed phenomena of uniformlyacceleratedmotion may in every case be explained on a basis of stationary observer and accelerated objective, or of stationary objective with the observer and his reference system in accelerated motion. Gravitation is one of these phenomena. It follows that if the observer enjoy properly accelerated axes (in time-space, of course), the absolute character of the world about him must be such as to present to him the phenomenon of gravitation. It remains only to identify the sort of world, of which gravitationas it is observedwould be a fundamental characteristic.Euclid’s and Newton’s systems stand as first and second approximations to that world. The Special Relativity Theory constitutes a correction of Newton, presumably because it is a third approximation. We must seek in it those features which we may most hopefully carry along, into the still more general case.Newton’s system retained the geometry of Euclid.But Minkowski’s invariant expression tells us that Einstein has had to abandon this; for in Euclidean geometry of four dimensions the invariant takes the form:StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared plus left-parenthesis upper T minus t right-parenthesis squared EndRoot commaanalogous to that of two and three dimensions. It is not the presence of the constantCin Minkowski’s formula that counts; this is merely an adjustment so that we may measure space in miles and time in the unit that corresponds to a mile. It is the minus sign where Euclidean geometry demands a plus that makes Minkowski’s continuum non-Euclidean.The editor has told us what this statement means. I think he has made it clear that when we speak of the geometry of the four-dimensional world, we must not read into this term the restrictions surrounding the kind of geometry we are best acquainted with—that of the three-dimensional Euclidean continuum. So I need only point out that if we are to make a fourth (and we hope, final) approximation to the reality, its geometry must preserve the generality attained by that of the third step, if it goes no further.

An alternative statement of the Special Theory is that the observed phenomena of uniform motion may equally be accounted for by supposing the object in motion and the observer with his reference frame at rest, or vice versa. We may similarly state the General Theory: The observed phenomena of uniformlyacceleratedmotion may in every case be explained on a basis of stationary observer and accelerated objective, or of stationary objective with the observer and his reference system in accelerated motion. Gravitation is one of these phenomena. It follows that if the observer enjoy properly accelerated axes (in time-space, of course), the absolute character of the world about him must be such as to present to him the phenomenon of gravitation. It remains only to identify the sort of world, of which gravitationas it is observedwould be a fundamental characteristic.

Euclid’s and Newton’s systems stand as first and second approximations to that world. The Special Relativity Theory constitutes a correction of Newton, presumably because it is a third approximation. We must seek in it those features which we may most hopefully carry along, into the still more general case.

Newton’s system retained the geometry of Euclid.But Minkowski’s invariant expression tells us that Einstein has had to abandon this; for in Euclidean geometry of four dimensions the invariant takes the form:StartRoot left-parenthesis upper X minus x right-parenthesis squared plus left-parenthesis upper Y minus y right-parenthesis squared plus left-parenthesis upper Z minus z right-parenthesis squared plus left-parenthesis upper T minus t right-parenthesis squared EndRoot comma

analogous to that of two and three dimensions. It is not the presence of the constantCin Minkowski’s formula that counts; this is merely an adjustment so that we may measure space in miles and time in the unit that corresponds to a mile. It is the minus sign where Euclidean geometry demands a plus that makes Minkowski’s continuum non-Euclidean.

The editor has told us what this statement means. I think he has made it clear that when we speak of the geometry of the four-dimensional world, we must not read into this term the restrictions surrounding the kind of geometry we are best acquainted with—that of the three-dimensional Euclidean continuum. So I need only point out that if we are to make a fourth (and we hope, final) approximation to the reality, its geometry must preserve the generality attained by that of the third step, if it goes no further.

Einstein’s Time-Space WorldEinstein accordingly examined the possible non-Euclidean geometries of four dimensions, in search of one displaying fundamental characteristics which, interpreted in terms of space-time, would lead to the observed facts of gravitation. The mathematics of this investigation is that part of his work which, we are told, but twelve men can follow; so we may only outline his conclusions.If we assume that in the neighborhood of matter the world of space-time is non-Euclidean, and that its curvature or distortion or non-Euclideanism is of a certain type already known to mathematicians; that the curvature of this world in the neighborhood of matter increases with the mass, and decreases as the distance from the matter increases; and that every particle of matter that is not interfered with travels through space-time in the most direct path possiblein that continuum; then the observed facts of gravitation are accounted for as an inherent geometric property of this space-time world. We usually say that the presence of matter distorts this world, and that this distortion gives the track of particles through the region affected its non-uniform character.Gravitation then is not a force at all; it is the fundamental nature of things. A body free to move through the world must follow some definite path. Euclid says it will stand still; Newton that it will traverse a straight line in three-space at uniform time-rate; Einstein that it will move in a “geodesic” through time-space—in every-day language, that it will fall.The numerical consequences of Einstein’s theory are, within the limits of observation, the same as those of Newton’s for all bodies save one—Mercury. This planet shows a small deviation from the path predicted by Newton’s law; Einstein’s theory gives its motion exactly. Again, when modern research showed that light must be affected by gravitation, Einstein’s theory, because of the extreme velocity of light, deviates from Newton’s, where thespeed is less a determining factor; and observations of starlight deflected by the sun during the eclipse were in much better accord with Einstein’s theory than Newton’s. Moreover, the Special Theory predicts that mass is an observational variable like length and duration. Radioactive emanations have a velocity high enough to give appreciable results here, and the prediction is verified, tending to support the general theory by supporting its limiting case.We like always to unify our science; and seldom, after effecting a unification, are we forced to give it up. Einstein for the first time brings mechanical, electromagnetic and gravitational phenomena within one structure. This is one reason why physicists are so open minded toward his theory—they want it to be true.

Einstein’s Time-Space World

Einstein accordingly examined the possible non-Euclidean geometries of four dimensions, in search of one displaying fundamental characteristics which, interpreted in terms of space-time, would lead to the observed facts of gravitation. The mathematics of this investigation is that part of his work which, we are told, but twelve men can follow; so we may only outline his conclusions.If we assume that in the neighborhood of matter the world of space-time is non-Euclidean, and that its curvature or distortion or non-Euclideanism is of a certain type already known to mathematicians; that the curvature of this world in the neighborhood of matter increases with the mass, and decreases as the distance from the matter increases; and that every particle of matter that is not interfered with travels through space-time in the most direct path possiblein that continuum; then the observed facts of gravitation are accounted for as an inherent geometric property of this space-time world. We usually say that the presence of matter distorts this world, and that this distortion gives the track of particles through the region affected its non-uniform character.Gravitation then is not a force at all; it is the fundamental nature of things. A body free to move through the world must follow some definite path. Euclid says it will stand still; Newton that it will traverse a straight line in three-space at uniform time-rate; Einstein that it will move in a “geodesic” through time-space—in every-day language, that it will fall.The numerical consequences of Einstein’s theory are, within the limits of observation, the same as those of Newton’s for all bodies save one—Mercury. This planet shows a small deviation from the path predicted by Newton’s law; Einstein’s theory gives its motion exactly. Again, when modern research showed that light must be affected by gravitation, Einstein’s theory, because of the extreme velocity of light, deviates from Newton’s, where thespeed is less a determining factor; and observations of starlight deflected by the sun during the eclipse were in much better accord with Einstein’s theory than Newton’s. Moreover, the Special Theory predicts that mass is an observational variable like length and duration. Radioactive emanations have a velocity high enough to give appreciable results here, and the prediction is verified, tending to support the general theory by supporting its limiting case.We like always to unify our science; and seldom, after effecting a unification, are we forced to give it up. Einstein for the first time brings mechanical, electromagnetic and gravitational phenomena within one structure. This is one reason why physicists are so open minded toward his theory—they want it to be true.

If we assume that in the neighborhood of matter the world of space-time is non-Euclidean, and that its curvature or distortion or non-Euclideanism is of a certain type already known to mathematicians; that the curvature of this world in the neighborhood of matter increases with the mass, and decreases as the distance from the matter increases; and that every particle of matter that is not interfered with travels through space-time in the most direct path possiblein that continuum; then the observed facts of gravitation are accounted for as an inherent geometric property of this space-time world. We usually say that the presence of matter distorts this world, and that this distortion gives the track of particles through the region affected its non-uniform character.

Gravitation then is not a force at all; it is the fundamental nature of things. A body free to move through the world must follow some definite path. Euclid says it will stand still; Newton that it will traverse a straight line in three-space at uniform time-rate; Einstein that it will move in a “geodesic” through time-space—in every-day language, that it will fall.

The numerical consequences of Einstein’s theory are, within the limits of observation, the same as those of Newton’s for all bodies save one—Mercury. This planet shows a small deviation from the path predicted by Newton’s law; Einstein’s theory gives its motion exactly. Again, when modern research showed that light must be affected by gravitation, Einstein’s theory, because of the extreme velocity of light, deviates from Newton’s, where thespeed is less a determining factor; and observations of starlight deflected by the sun during the eclipse were in much better accord with Einstein’s theory than Newton’s. Moreover, the Special Theory predicts that mass is an observational variable like length and duration. Radioactive emanations have a velocity high enough to give appreciable results here, and the prediction is verified, tending to support the general theory by supporting its limiting case.

We like always to unify our science; and seldom, after effecting a unification, are we forced to give it up. Einstein for the first time brings mechanical, electromagnetic and gravitational phenomena within one structure. This is one reason why physicists are so open minded toward his theory—they want it to be true.

The Layman’s Last DoubtThe final answer to any series of questions is inevitably “because the world is so constructed.” The things we are content to leave on that basis are those to which we are accustomed, and which we therefore think we understand; those for which this explanation leaves us unsatisfied are those which are new and unfamiliar. Newton told us that the world of three-dimensional space with one-dimensional time superposed was so constructed that bodies left to themselves would go on forever in a straight line at constant speed. We think we understand this, but our understanding consists merely of the unspoken query, “Why, of course; what is there to prevent?”The Greeks, an intelligent people, looked at this differently; they would have met Newton with the unanimous demand “Why so; what is there to keep them going?” So if, in seeking an explanation of anything, we come sooner than we had expected to the finality “Because the world is so constructed,” let us not feel that we have been cheated.

The Layman’s Last Doubt

The final answer to any series of questions is inevitably “because the world is so constructed.” The things we are content to leave on that basis are those to which we are accustomed, and which we therefore think we understand; those for which this explanation leaves us unsatisfied are those which are new and unfamiliar. Newton told us that the world of three-dimensional space with one-dimensional time superposed was so constructed that bodies left to themselves would go on forever in a straight line at constant speed. We think we understand this, but our understanding consists merely of the unspoken query, “Why, of course; what is there to prevent?”The Greeks, an intelligent people, looked at this differently; they would have met Newton with the unanimous demand “Why so; what is there to keep them going?” So if, in seeking an explanation of anything, we come sooner than we had expected to the finality “Because the world is so constructed,” let us not feel that we have been cheated.

Back to Index Next