Spacetime

"Bolder even than Riemann, Clifford confessed his belief (1870) that matter is only a manifestation of curvature in a space-time manifold. This embryonic divination has been acclaimed as an anticipation of Einstein’s (1915–16) relativistic theory of the gravitational field. The actual theory, however, bears but slight resemblance to Clifford’s rather detailed creed. As a rule, those mathematical prophets who never descend to particulars make the top scores. Almost anyone can hit the side of a barn at forty yards with a charge of buckshot."

"Unfortunately, scientists adore the Minkowski theory because, secretly they detest the 3+1 formula as not entirely objective, since man is the one to add the time; and yet Einstein has shown that there is no universal, or god-given cosmic time, thus making man supreme in the determination of physical reality. So the Minkowski proposal that enables them to pretend (it is only a pretence because it is not true, even Eddington told us so), is preferred. 'S=ct...', as Russell has averred, makes things easy for the mathematicians, but it is not true of the physical world (the same Russell said it is an artefact); therefore the concept of curved space-time and time travel based on it are all bogus science."

"General relativity (GR) is a geometric theory of gravitation: a gravitational field is simply curved spacetime. The gravitational time dilation implied by the equivalence principle... can be interpreted as showing the warpage of spacetime in the time direction."

"In GR, the mass/energy source determines the metric function through the field equation. ...In this approach, gravity is the structure of spacetime and is not regarded as a force (that brings about acceleration). Thus a test-body will move in a force-free way in such a curved spacetime; it is natural to expect it to follow in this spacetime the shortest and straightest possible trajectory, the geodesic curve..."

"We may conceive our space to have everywhere a nearly uniform curvature, but that slight variations of the curvature may occur from point to point, and themselves vary with the time. These variations of the curvature with the time may produce effects which we not unnaturally attribute to physical causes independent of the geometry of our space. We might even go so far as to assign to this variation of the curvature of space 'what really happens in that phenomenon which we term the motion of matter.'"

"Every artist's strictly illimitable country is himself. An artist who plays that country false has committed suicide;and even a good lawyer cannot kill the dead. But a human being who's true to himself — whoever himself may be — is immortal;and all the atomic bombs of all the antiartists in spacetime will never civilize immortality."

"Mathematicians call the infinite curvature limit of spacetime a singularity. In this picture, then, the big bang emerges from a singularity. The best way to think about singularities is as boundaries or edges of spacetime. In this respect they are not, technically, part of spacetime itself... So the first moment of the universe — in this highly simplified picture — is not a moment or a place at all, but a boundary to moments and places. ...it signals a "no further" warning."

"The discovery of Minkowski... is to be found... in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, \sqrt -1\cdot ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. ...These inadequate remarks can give the reader only a vague notion of the important idea contributed by Minkowski. Without it the general theory of relativity... would perhaps have got no farther than its long clothes."

"Every physical description resolves itself into a number of statements, each of which refers to the space-time coincidence of two events A and B. In terms of Gaussian co-ordinates, every such statement is expressed by the agreement of their four co-ordinates x1, x2, x3, x4. Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference, without suffering from the defects of the latter mode of description; it is not tied down to the Euclidean character of the continuum which has to be represented."

"The use of rigid reference-bodies, in the sense of the method followed in the special theory of relativity, is in general not possible in space-time description. The Gauss co-ordinate system has to take the place of the body of reference. The following statement corresponds to the fundamental idea of the general principle of relativity: "All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.""

"To say that the four-dimensional continuum "exists now" implies that all cross sections "exist now" or, in other words, that the cross section t = t0 is identical with the cross-section t = t1. Otherwise, it could not exist "now." If we allow for this confusing way of thinking, the assertion that the "four-dimensional space-time continuum" has always existed and we are merely traveling through it asserts no more than the statement that the three-dimensional space continuum changes in time. ...if we call the ...space-time continuum a "reality," we are encouraged to adopt Lagrange's assertion that mechanics is a four-dimensional geometry, and to say that the four-dimensional continuum "exists now," and that therefore all future events exist now, and the "future" consists in our moving through the... continuum. But exactly as before Minkowski's formulation... we must also admit that the use of the word "now" in the formulation is rather misleading. By "now" we mean the cross section of the four-dimensional space-time continuum that is defined by t = t0. Therefore it is self-contradictory that any future instant of time t > t0 can exist "now." Use has often been made of this four-dimensional space-time continuum to "prove" that the future is "predetermined." ... The four-dimensional formulation is a useful instrument for the presentation of physical events, but it cannot be interpreted in our everyday language by simply speaking about the... space-time continuum as we have been accustomed to speak about our ordinary three-dimensional space."

"Think, for a moment, of a cheetah, a sleek, beautiful animal, one of the fastest on earth, which roams freely on the savannas of Africa. In its natural habitat, it is a magnificent animal, almost a work of art, unsurpassed in speedor grace by any other animal. Now, think of a cheetah that has been captured and thrown into a miserable cage in a zoo. It has lost its original grace and beauty, and is put on display for our amusement. We see only the broken spirit of the cheetah in the cage, not its original power and elegance. The cheetah can be compared to the laws of physics, which are beautiful in their natural setting. The natural habitat of the laws of physics is higher-dimensional space-time. However, we can only measure the laws of physics when they have been broken and placed on display in a cage, which is our three-dimensional laboratory. We can only see the cheetah when its grace and beauty have been stripped away."

"Einstein proclaimed that all objects in the universe are always traveling through spacetime at one fixed speed—that of light. ...this one fixed speed can be shared between the different ...space and time dimensions ...If an object is sitting still ...all of the object's motion is used to travel through one dimension ...time ...all objects that are at rest ...age—at exactly the same rate or speed. If an object does move through space... some of the previous motion through time must be diverted. ...its clock will tick more slowly if it moves through space. ...The speed of an object through space is thus merely a reflection of how much of its motion through time is diverted. ...the maximum speed through space occurs if all of an object's motion through time is diverted to motion through space. But having used up all of its motion through time, this is the fastest speed through space ...something traveling at light speed through space will have no speed left for motion through time. Thus light does not get old; a photon that emerged from the big bang is the same age today as it was then. There is no passage of time at light speed."

"In a paper he sent to Einstein in 1919, Kaluza made an astounding suggestion. He proposed that the spatial fabric of the universe might possess more than the three dimensions... it provided an elegant and compelling framework for weaving together Einstein's general relativity and Maxwell's electromagnetic theory into a single, unified conceptual framework. ...implicit in Kaluza's work and subsequently made explicit and refined by... Oskar Klein in 1926... the spatial fabric of our universe may have both extended and curled-up dimensions. ... Einstein had formulated general relativity in the familiar setting of a universe with three spatial dimensions and one time dimension. The mathematical formalism... however, could be extended fairly directly to write down analogous equations for a universe with additional space dimensions. Under the "modest" assumption of one additional space dimension, Kaluza... derived the new equations. ...Kaluza found extra equations... those Maxwell had written down in the 1880s for deriving the electromagnetic force! ...Kaluza had united Einstein's theory of gravity with Maxwell's theory of light."

"Much as Kaluza found that a universe with five spacetime dimensions provided a framework for unifying electromagnetism and gravity, and much as string theorists found that a universe with ten spacetime dimensions provided a framework for unifying quantum mechanics and general relativity, Witten found that a universe with eleven spacetime dimensions provided a framework for unifying all string theories."

"The subject of this book is the structure of space-time on length-scales from 10-13 cm, the radius of an elementary particle, up to 1028 cm, the radius of the universe. ...we base our treatment on Einstein's General Theory of Relativity. This theory leads to two remarkable predictions about the universe: first, that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly, that there is a singularity in our past which constitutes, in some sense, a beginning to the universe."

"This picture would explain why we haven't been over run by tourists from the future."

"The conclusion of this lecture is that rapid space-travel, or travel back in time, can't be ruled out, according to our present understanding. They would cause great logical problems, so let's hope there's a Chronology Protection Law, to prevent people going back, and killing our parents. But science fiction fans need not lose heart. There's hope in string theory."

"Then the theory of relativity came and explained the cause of the failure. Electric action requires time to travel from one point of space to another, the simplest instance of this being the finite speed of travel of light... Thus electromagnetic action may be said to travel through space and time jointly. But by filling space and space alone [excluding time] with an ether, the pictorial representations had all supposed a clear-cut distinction between space and time."

"Space-Time. In 1908... Minkowski stated the whole content of the theory in a new and very elegant form. Hitherto the laws of nature had been thought of as describing phenomena which occurred in three-dimensional space, while time flowed on uniformly and imperturbably in another and quite distinct dimension of its own. Minkowski now supposed that this fourth dimension of time was not detached from and independent of the three dimensions of space. He introduced a new four-dimensional space to which ordinary space contributed three dimensions, and time one; we may call it 'space-time'. ...The succession of positions which a particle occupied in ordinary space at a succession of instants of time would be represented by a line in space-time; this he called the 'world-line' of the particle. ...Newton's absolute space and absolute time fell out of science, and they carried much with them in their fall. First to go was the concept of simultaneity. ...It now became necessary to find a way of treating gravitation which should not involve simultaneity. Einstein found this through the medium of his 'Principle of Equivalence'."

"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth... No force of gravitation is... needed to impress curvature on world-lines; the curvature is inherent in the space..."

"The mark of Platonic philosophy is a radical dualism. The Platonic world is not one of unity; and the abyss which in many ways results from this bifurcation appears in innumerable forms. It is not one, but two worlds, which Plato sees when with the eyes of his soul he envisages a transcendent, spaceless, and timeless realm of the Idea, the thing-in-itself, the true, absolute reality of tranquil being, and when to this transcendent realm he opposes the spacetime sphere of his sensuous perception—a sphere of becoming in motion, which he considers to be only a domain of illusory semblance, a realm which in reality is not-being."

"The uomo universale of the Renaissance, who was artist and craftsman, philosopher and inventor, humanist and scientist, astronomer and monk, all in one, split up into his component parts. Art lost its mythical, science its mystical inspiration; man became again deaf to the harmony of the spheres. The Philosophy of Nature became ethically neutral, and "blind" became the favourite adjective for the working of natural law. The space-spirit hierarchy was replaced by the space-time continuum. ...man's destiny was no longer determined from "above" by a super-human wisdom and will, but from "below" by the sub-human agencies of glands, genes, atoms, or waves of probability. ...they could determine his fate, but could provide him with no moral guidance, no values and meaning. A puppet of the Gods is a tragic figure, a puppet suspended on his chromosomes is merely grotesque."

"[By moving] I'm making sounds on the drums of spacetime"… "Space itself wobbles and rumbles like a drum... Black holes can bang on spacetime like mallets on a drum."

"Time exists not by itself; but simply from the things which happen, the sense apprehends what has been done in time past, as well as what is present, and what is to follow after."

"Descartes... fell back on his original confusion of matter with space—space being, according to him, the only form of substance, and all existing things but affections of space. This error... forms one of the ultimate foundations of the system of Spinoza."

"Minkowski's idea and the solution of the twin paradox can best be explained by means of an analogy between space and spacetime... Time as a fourth dimension rests vertically on the other three—just as in space the vertical juts out of the two-dimensional plane as a third dimension. Distances through spacetime comprise four dimensions, just as space has three. The more you go in one direction, the less is left for the others. When a rigid body is at rest and does not move in any of the three dimensions, all of its motion takes place on the time axis. It simply grows older. ...The faster he moves away from his frame of reference... and covers more distance in the three dimensions of space, the less of his motion through spacetime as a whole is left over for the dimension of time. ...Whatever goes into space is deducted from time. ...In comparison with the distances light travels, all distances in the dimensions of space, even those involving airplane travel, are so very small that we essentially move only along the time axis, and we age continually. Only if we are able to move away from our frame of reference very quickly, like the traveling twin... would the elapsed time shrink to near zero, as it approached the speed of light. Light itself... covers its entire distance through spacetime only in the three dimensions of space... Nothing remains for the additional dimension... the dimension of time... Because light particles do not move in time, but with time, it can be said that they do not age. For them "now" means the same thing as "forever." They always "live" in the moment. Since for all practical purposes we do not move in the dimensions of space, but are at rest in space, we move only along the time axis. This is precisely the reason we feel the passage of time. Time virtually attaches to us."

"Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be understood in the following discourse. I do not define time, space, place and motion, as being well known at all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. I. Absolute, true and mathematical time, of itself, and from its own nature, flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time is some sensible and external (whether accurate or unequable) measure of duration by means of motion, which is commonly used instead of true time; such as an hour, a day, a month or a year. II. Absolute space, in its own nature, without regard to any thing external, remains always similar and immovable. Relative space is some moveable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is vulgarly taken for immovable space; such is the dimension of a subterraneous, an æreal, or celestial space, determined by its position in respect of the earth. Absolute and relative space, are the same in figure and magnitude; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes, at another time it will be another part of the same, and so, absolutely understood, it will be perpetually mutable. III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space; not the situation, nor the external surface of the body. For the places of equal solids, are always equal; but their superficies, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places; and therefore the place of the whole, is the same thing with the sum of the places of the parts; and for that reason, it is internal, and in the whole body. IV. Absolute motion, is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses; or that part of its cavity which the body fills, and which therefore moves together with the ship: and relative rest, is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space; partly from the relative motion of the ship on the earth: and if the body moves also relatively in the ship; its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative motions as well of the ship on the earth, as of the body in the ship; and from these relative motions will arise the relative motion of the body on the earth."

"Guth... wanted to hear... Alex Vilenkin... describe a new theory of the origin of the universe, of how it could have emerged from nothing. Vilenkin's version of the infant universe... was a kind of metaphysical mole. ...a bubble of universe, space-time, had "tunneled" into a Wheeleresque superspace of possible space-times and then tunneled again into "real" space and time. ...But from where had the universe tunneled into this realm..? In Vilenkin's words, "from nothing." ...Vilenkin's tiny bubble... inflated and went through the standard expansion and evolution of the big bang. ...he, Guth, and Sidney Coleman sat and had a conversation that Lewis Carroll might have enjoyed, about nothing. ..."Nothing," answered Vilenkin... "is no time, no space." ..."There is an epoch without time," [Coleman] said finally as the shadows lengthened. "It is an enternity. So we make a quantum leap from eternity into time." Then, as good physicists did, they repaired to a Chinese restaurant."

"The idea of having an ambient space-time of some specific dimension seems to play less of a role in string theory than in conventional physics, and certainly less than the kind of role that I would myself feel comfortable with. It is particularly difficult to assess the functional freedom that is involved in a physical theory unless one has a clear idea of its actual space-time dimensionality."

"Ever since Hermann Minkowski's now infamous comments in 1908 concerning the proper way to view space-time, the debate has raged as to whether or not the universe should be viewed as a four-dimensional, unified whole wherein the past, present, and future are regarded as equally real or whether the views espoused by the possibilists, historicists, and presentests regarding the unreality of the future (and, for presentests, the past) are more accurate. Now, a century after Minkowski's proposed block universe first sparked debate, we present a new, more conclusive argument in favor of eternalism."

"The structure of space-time, taken as a whole, is the subject matter of the science called cosmology. Since you are asking about all space and all time in cosmology, you are interested in the entire universe, everywhere and everywhen, viewed as a static geometrical object."

"A person's lifeworm is a tangle of atomic worldlines. A braid. The dotty little atoms trace out smooth lines in spacetime: you are the pattern that these lines make up. There is no one single atom that is exclusively yours. I breathe an atom out, you breathe it in. Your garbage helps my tomatoes grow. And so the little spacetime threads weave us all together. The human race is a single vast tapestry, linked by our shared food and air. There are larger links as well: sperm, egg and umblilicus. Each family tree is an organic whole. Your spacetime body tapers back to the threads of mother's egg and father's sperm. And children, if you have them, are forever rooted in your flesh."

"When, in youth, I learned what was called "philosophy" … no one ever mentioned to me the question of "meaning." Later, I became acquainted with Lady Welby's work on the subject, but failed to take it seriously. I imagined that logic could be pursued by taking it for granted that symbols were always, so to speak, transparent, and in no way distorted the objects they were supposed to "mean." Purely logical problems have gradually led me further and further from this point of view. Beginning with the question whether the class of all those classes which are not members of themselves is, or is not, a member of itself; continuing with the problem whether the man who says "I am lying" is lying or speaking the truth; passing through the riddle "is the present King of France bald or not bald, or is the law of excluded middle false?" I have now come to believe that the order of words in time or space is an ineradicable part of much of their significance – in fact, that the reason they can express space-time occurrences is that they are space-time occurrences, so that a logic independent of the accidental nature of spacetime becomes an idle dream. These conclusions are unpleasant to my vanity, but pleasant to my love of philosophical activity: until vitality fails, there is no reason to be wedded to one's past theories."

"It ought to arouse our suspicions that people who spend enormous efforts on interpreting [Martin Heidegger's] work disagree on the fundamental question whether he was an idealist. For the purposes of this discussion, his lack of a resolute commitment to the basic facts is enough. Suppose you took the notion of Dasein seriously, in the sense that you thought it referred to a real phenomenon in the real world. Your first question would be: How does the brain cause Dasein and how does Dasein exist in the brain? Or if you thought the brain was not the right explanatory level you would have to say exactly how and where Dasein is located in the space time trajectory of the organism and you would have to locate the right causes, both the micro causes that are causing Dasein and its causal effects on the organic processes of the organism. There is no escaping the fact that we all live in one space-time continuum, and if Dasein exists it has to be located and causally situated in that continuum. Furthermore, if you took Dasein seriously you would then have to ask how does Dasein fit into the biological evolutionary scheme? Do other primates have it? Other mammals? What is its evolutionary function? I can’t find an answer to these questions in Heidegger or even a sense that he is aware of them or takes them seriously. But taking these questions seriously is the price of taking Dasein seriously, unless of course you are denying the primordiality of the basic facts."

"In Newton's system of mechanics... there is an absolute space and an absolute time. In Einstein's theory time and space are interwoven, and the way in which they are interwoven depends on the observer. Instead of three plus one we have four dimensions."

"It may be helpful to a good understanding of the conception of the physical universe implied by the general theory of relativity, to consider the different definitions of a straight line. ...In the old mechanics, there are four of these, viz.: (1) ray of light, (2) the track of a material particle not subject to any forces, (3) a stretched cord, (4) an axis of rotation. The fourth definition is the one favored by the great mathematician Henri Poincaré. ...Are they still identical in the theory of relativity? The definitions 1 and 2 define the straight line as a projection on the three-dimensional space x, y, z of a geodesic in the four-dimensional space-time continuum. This projection will be a geodesic in three-dimensional space only under very special conditions. In the general case the two projections will differ from each other, and neither of them will be a geodesic. Also the projection may be a geodesic in one system of coordinates but not in another. The stretched cord is by definition a geodesic in the three-dimensional space. As a rule, this will not be a geodesic in the four-dimensional continuum. The rotation axis is also by definition a line in three-dimensional space. The definition, however, presupposes the possibility of the rotation of a rigid body, which would be possible only in a homogeneous, isotropic, and statical field, i.e., in a world without any material bodies... in it, which by their gravitational field would upset the isotropy. The definition is thus meaningless in the general theory of relativity."

"Spacetime... turns out to be discrete, described by a structure called spin foam."

"In string theory one studies strings moving in a fixed classical spacetime. ...what we call a background-dependent approach. ...One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves—at the rate predicted by general relativity to the... accuracy of eleven decimal places—tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up."

"The hypothesis underlying all approaches to the landscape is that there is a cosmological setting in which different regions or epochs of the universe can have different effective laws. This implies the existence of spacetime regions not directly observable... These regions must either be in the past of our big bang, or far enough away from us to be causally unrelated."

"The positive energy theorem was for half a century or more an open challenge to relativists. Many attempts were made to prove flat spacetime was stable, but none completely succeeded completely until a majestic tour de force of geometric reasoning of Shoen and Yau. This was followed two years later by a proof of Witten, which was as elegant as it was short. It is this proof of Witten’s that we take as a template here for the quantum theory."

"Minkowski, building on Einstein's work, had now discovered that the Universe is made of a four-dimensional "spacetime" fabric that is absolute, not relative."

"Einstein was guided by a principle he had inferred from the known properties of gravitation, the principle of the equivalence of gravitational forces to inertial effects such as centrifugal force. The development of the Standard Model was guided by a principle called gauge symmetry, a generalization of the well-known property of electricity that it is only differences of voltages that matter, not voltages themselves. But we have not discovered any fundamental principle that governs M-theory. The various approximations to this theory look like string or field theories in spacetimes of different dimensionalities, but it seems probable that the fundamental theory is not to be formulated in spacetime at all. Quantum field theory is powerfully constrained by principles concerning the nature of four-dimensional spacetime that are incorporated in the special theory of relativity. How can we get the ideas we need to formulate a truly fundamental theory, when this theory is meant to describe a realm where all intuitions derived from life in spacetime become inapplicable?"

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event."

"There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives. ...Really this is what is meant by the Fourth Dimension, though some people who talk about the Fourth Dimension do not know they mean it. It is only another way of looking at Time. There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it. ...space, as our mathematicians have it, is spoken of as having three dimensions, which one may call Length, Breadth, and Thickness, and is always definable by reference to these planes, each at right angle to the others. But some philosophical people have been asking why three dimensions particularly—why not another direction at right angles to the other three?—and have even tried to construct a Four Dimensional geometry. Professor Simon Newcomb was expounding this to the New York Mathematical Society only a month or so ago. You know how on a flat surface, which has only two dimensions, we can represent a figure of a Three-Dimensional solid, and similarly they think that by models of three dimensions they could represent one of four—if they could master the perspective of the thing. See?"

"And now, in our time, there has been unloosed a cataclysm which has swept away space, time, and matter hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope, and entailing a deeper vision."

"The scene of action of reality is not a three-dimensional Euclidean space but rather a four-dimensional world, in which space and time are linked together indissolubly. However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavors to crystallize out of direct experience. It is a four-dimensional continuum, which is neither "time" nor "space". Only the consciousness that passes on in one portion of this world experiences the detached piece which comes to meet it and passes behind it as history, that is, as a process that is going forward in time and takes place in space. ... It is remarkable that the three-dimensional geometry of the statical world that was put into a complete axiomatic system by Euclid has such a translucent character, whereas we have been able to assume command over the four-dimensional geometry only after a prolonged struggle and by referring to an extensive set of physical phenomena and empirical data. Only now the theory of relativity has succeeded in enabling our knowledge of physical nature to get a full grasp of the fact of motion, of change in the world."

"Spacetime tells matter how to move; matter tells spacetime how to curve."

"In 1908 the famous mathematician Minkowski made a remarkable discovery concerning the Lorentz formulae. He showed that, although each observer had his own private space and private time, a public concept which is the same for all observers can be formed by combining space and time in a particular way. If we regard an inverval of time as a kind of 'distance' in the time dimension, we can convert it into a true distance by multiplying it by the velocity of light, c; in other words, with any time interval we can associate a definite spatial interval, namely the distance which light can travel in empty space in that period. If, according to a particular observer, the difference in time between any two events is T, this associated spatial interval is cT. Then, if R is the space-distance between these two events, Minkowski showed that the difference of the squares of cT and R has the same value for all observers in uniform relative motion. The square root of this quantity is called the space-time interval between the two events. Hence, although time and three-dimensional space depend on the observer, this new concept of space-time is the same for all observers."

"Space-time is curved in the neighborhood of material masses, but it is not clear whether the presence of matter causes the curvature of space-time or whether this curvature is itself responsible for the existence of matter."