# Spacetime

mathematical model combining space and time

In physics, spacetime (also space–time, space time or space–time continuum) is any mathematical model that combines space and time into a single interwoven continuum. By combining space and time into a single manifold called Minkowski space, physicists have significantly simplified a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.

Gravity Probe B circling earth

## Quotes

• 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.
• Samuel K. K. Blankson, Time and the Application of Time (2010).
• 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.
• Ta-Pei Cheng, Einstein's Physics: Atoms, Quanta, and Relativity Derived, Explained, and Appraised (2013).
• 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...
• Ta-Pei Cheng, Einstein's Physics: Atoms, Quanta, and Relativity Derived, Explained, and Appraised (2013).
• 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.
• Paul Davies, Cosmic Jackpot: Why Our Universe Is Just Right for Life (2007).
• 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, ${\displaystyle {\sqrt {-}}1\cdot ct}$ , 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.
• Phillipp Frank, Philosophy of Science: The Link Between Science and Philosophy (1957).
• 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.
• Brian Greene, The Elegant Universe (1999) Ch. 2 Space, Time, and the Eye of the Beholder.
• 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.
• Brian Greene, The Elegant Universe (1999) Ch. 8 More Dimensions Than Meet the Eye.
• 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.
• Stephen Hawking, with G.F.R. Ellis, The Large Scale Structure of Space-Time (1973) Preface.
• 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.
• James Jeans, Physics and Philosophy (1942)
• 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.
• Hans Kelsen, "Platonic Justice", Ethics, April 1938. Translated by Glenn Negley from "Die platonische Gerechtigkeit," Kantstudien, 1933. (The author corrected the translation in 1957), published in What is Justice? (1957).
• 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.
• 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.
• Jürgen Neffe, Einstein: A Biography (1956).
• 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.
• Daniel Peterson and Michael Silberstein, "Relativity of Simultaneity and Eternalism: In Defense of the Block Universe" Space, Time, and Spacetime: Physical and Philosophical Implications of Minkowski's Unification of Space and Time Vesselin Petkov, Ed. (2010).
• 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.
• Rudolf v. B. Rucker, Geometry, Relativity and the Fourth Dimension (1977).
• 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.
• Willem de Sitter, "Relativity and Modern Theories of the Universe," Kosmos (1932)
• 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.
• Lee Smolin, "Loop Quantum Gravity," The New Humanists: Science at the Edge (2003).
• 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.
• Lee Smolin, "Loop Quantum Gravity," The New Humanists: Science at the Edge (2003).
• 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.
• Lee Smolin, "A perspective on the landscape problem" arXiv (Feb 15, 2012).
• 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.
• Lee Smolin, "Positive energy in quantum gravity" arXiv (Jun 10, 2014).
• 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.
• Kip S. Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy (1994).
• 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?
• 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.
• Friedel Weinert, The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries (2005) see World line.
• 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.
• Hermann Weyl, Space—Time—Matter (1922) Ch. III Relativity of Space and Time, p. 217.
• 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.
• Gerald James Whitrow, The Structure of the Universe: An Introduction to Cosmology (1949).
• With regard to the Newtonian concept of absolute rotation, Eddington admitted that Einstein's plenum does in fact provide a world-wide inertial frame, with respect to which it can be measured. Nevertheless, Eddington believed that Einstein attributed to important a role to matter, for in his universe it appears that not only the metrical properties, as in General Relativity, but the very existence of space depends on the existence of matter. Eddington preferred to regard matter as a manifestation of the 'structure' of space-time.
• Gerald James Whitrow, The Structure of the Universe: An Introduction to Cosmology (1949).
• Shortly after Einstein published his original memoir... de Sitter constructed an alternative static world-model... unlike Einstein's, space-time has an intrinsic structure of its own, independent of the presence of matter. ...there is, strictly speaking, no matter or radiation. ...whereas a test particle in Einstein's universe will remain at rest if it has no initial motion, a similar particle introduced in de Sitter's world will immediately acquire an ever-increasing velocity of recession from the observer. Moreover, in de Sitter's model, space-time is 'hyperbolic'. There is no absolute time, and each observer will perceive a horizon at which time will appear to him to stand still. ...This phenomenon. of course, is only apparent, like a rainbow. At any point on the (relative) horizon the time-flux experienced by an observer there will be the same as the original observer. Thus in de Sitter's world there will be an apparent slowing-down of distant atomic vibrations, if these keep standard time. Consequently the radiation from a distant nebula will appear to be shifted toward the red... This effect, of course, will be supplemented by the Doppler effect, due to the relative recession of the nebula regarded as a test particle.
• According to the special theory there is a finite limit to the speed of causal chains, whereas classical causality allowed arbitrarily fast signals. Foundational studies... soon revealed that this departure from classical causality in the special theory is intimately related to its most dramatic consequences: the relativity of simultaneity, time dilation, and length contraction. By now it had become clear that these kinematical effects are best seen as consequences of Minkowski space-time, which in turn incorporates a nonclassical theory of causal structure. However, it has not widely been recognized that the converse of this proposition is also true: the causal structure of Minkowski space-time contains within itself the entire geometry (topoligical and metrical structure) of Minkowski space-time. ...The problem of the independence of topological and metrical structures of space-time was clearly recognized by early writers on relativity such as Russell (1954) and, of course, Eddington...
• Replacing particles by strings is a naive-sounding step, from which many other things follow. In fact, replacing Feynman graphs by Riemann surfaces has numerous consequences: 1. It eliminates the infinities from the theory. ...2. It greatly reduces the number of possible theories. ...3. It gives the first hint that string theory will change our notions of spacetime. Just as in QCD, so also in gravity, many of the interesting questions cannot be answered in perturbation theory. In string theory, to understand the nature of the Big Bang, or the quantum fate of a black hole, or the nature of the vacuum state that determines the properties of the elementary particles, requires information beyond perturbation theory... Perturbation theory is not everything. It is just the way the [string] theory was discovered.
• Edward Witten, "The Past and Future of String Theory" in The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking's Contributions to Physics (2003) ed. G.W. Gibbons, E.P.S. Shellard & S.J. Rankin.
• We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations. ...the field view makes Einstein's theory of gravity look more like the other successful theories of fundamental physics, and so makes it easier to work toward a a fully integrated, unified description of all the laws. ...I'm a field man.
• Frank Wilczek, The Lightness of Being: Mass, Ether, and the Unification of Forces (2008).
• The traditional “cosmological” Multiverse considers that there might be physical realms inaccessible to us due to their separation in space-time. The quantum Multiverse arises from entities that occupy the same space-time, but are distant in Hilbert space – or in the jargon, decoherent.
• In an infinite universe, every point in space-time is the center.

### The Evolution of Scientific Thought from Newton to Einstein (1927)

A. D'Abro, Book @archive.org
• The principle of the invariant velocity of light states that in whatever Galilean system we might have operated, the measured velocity of light in vacuo would always be the same. ...The mathematical translation of this principle of physics yields us the following equation, which must remain invariably zero in value for all Galilean frames:
${\displaystyle dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}=0}$  (using differentials)
[ Note: the above is derived from the velocity of light c being equal to the change in length divided by the change in time, i.e., ${\displaystyle {\frac {\vartriangle l}{\vartriangle t}}=c}$ , or expressed as differentials, ${\displaystyle {\frac {dl}{dt}}=c,}$  which implies ${\displaystyle {\frac {dl^{2}}{dt^{2}}}=c^{2}}$  and ${\displaystyle {dl^{2}}-c^{2}dt^{2}=0}$ . But, by the Pythagorean theorem, ${\displaystyle {dl^{2}}={dx^{2}}+{dy^{2}}+{dz^{2}}}$  ].
From a purely mathematical standpoint problems of this type form a branch of mathematics known as the theory of invariants. ...the transformations to which it was necessary to subject these variables (in order to satisfy the condition of invariance...), were given by a wide group of transformations known as conformal transformations. Conformal transformations are those which vary the shape of the lines while leaving the values of their angles of intersections unaltered. They are of wide use in maps, e.g., in Mercator's projection or in the stereographic projection.
But when, in addition, the relative velocity is taken into consideration it is seen that conformal transformations are far too general. ...when the required restrictions are imposed we find that the rules of transformation according to which the space and time co-ordinates of one Galilean observer are connected with those of another depend in a very simple way on the relative velocity ${\displaystyle v}$  existing between the two systems. These rules of transformation are given by the Lorentz-Einstein transformations.
 ${\displaystyle dx'={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}(dx-vdt)}$ ${\displaystyle dt'={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}(dt-{\frac {vdx}{c^{2}}})}$
...the lesser generality of the Lorentz-Einstein transformations has ...the further restriction of the conditions of invariance ...Not only will this expression have a zero value for all Galilean frames when it has a zero value for one particular Galilean frame, but in addition, if it does not happen to have a zero value in one frame but has some definite non-vanishing numerical value, it will still maintain this same definite non-vanishing value in all other Galilean frames. In other words, Einstein's premises are represented mathematically by the invariance of the total value of ${\displaystyle dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}}$  for all Galilean frames, regardless of whether this value happens to be zero or non-vanishing.
The deep significance of this condition of invariance was first noted by Minkowski, and it led... to the discovery of four-dimensional space-time.
• pp. 193-195
• The discovery of the invariant
${\displaystyle dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}}$
whose value we shall designate ${\displaystyle ds^{2}}$  marks the date of immense importance in the history of natural philosophy. ...It mattered not whether we were situated in this frame or in that one; in every case ...it still maintained the same value when referred to any other frame. ...we were in the presence of something which, contrary to a distance in space or a duration in time, transcended our variable points of view ...a common absolute world underlying the relativity of physical space and time.
Minkowski immediately recognised in the mathematical form of this invariant the expression of the square of the distance in a four-demensional continuum. This distance was termed the Einsteinian interval, or, more simply, the interval. ...The continuum was neither space nor time, but it pertained to both ...it may appear strange that measurements with clocks can be co-ordinated with measurements with rods or scales. This difficulty, however, need not arrest us; for although dt is a time which can only be measured with a clock, yet cdt, being the product of a velocity by a time, is a spatial length since it represents the distance covered by light in the time dt.
• p. 195
• Minkowski demonstrated the significance of the expression for ${\displaystyle ds^{2}}$  by taking the new variable ${\displaystyle T=ict}$ , where ${\displaystyle i}$  stands for ${\displaystyle {\sqrt {-1}}.}$  With this change, ${\displaystyle ds^{2}}$  can be written:
${\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2}+dT^{2},}$
which is the expression of the square of a distance in a four-dimensional Euclidean space when a Cartesian co-ordinate system is taken. Since this expression is to remain unmodified in value and form in all Galilean frames, we must conclude that in a space-time representation a passage from one Galilean frame to another is given by a rotation of our four-dimensional Cartesian space-time mesh-system. Now rotation constitutes... a variation in the co-ordinates of the points of the continuum. In other words, they correspond to mathematical transformations. The transformations which accompany a rotation of a Cartesian co-ordinate system are of a particularly simple nature; they are called "orthogonal transformations." It follows that if we write out the orthogonal transformations for Minkowski's four-dimensional Euclidean space-time, we should obtain ipso facto the celebrated Lorentz-Einstein transformations which represent the passage from one Galilean system to another. ...we obtain the following result: Two Galilean systems moving with a relative velocity ${\displaystyle v}$  are represented by two space-time Cartesian co-ordinate systems differing in orientation by the imaginary angle ${\displaystyle \theta }$ , where ${\displaystyle \theta }$  is connected with ${\displaystyle v}$  by the formula ${\displaystyle tan\theta ={\frac {iv}{c}}.}$
• footnote, pp. 196-197
• With the rejection of such classical absolutes as length and duration, our ability to conceive of an objective impersonal world, independent of the presence of an observer, seems to be imperiled. The great merit of Minkowski was to show that an absolute world could nevertheless be imagined, although it was a far different world from that of classical physics. In Minkowski's world the absolute which supersedes the absolute length and duration of classical physics is the Einsteinian interval. ...
Thus suppose that, as measured in our Galilean frame of reference, two flashes occur at points A and B, situated at a distance l apart, and suppose the flashes are separated in time by an interval t. If we change our frame of reference, both l and t will change in value, becoming l' and t' respectively, exhibiting by their changes the relativity of length and duration. In Minkowski's words, "Henceforth space and time themselves are mere shadows." On the other hand, the mathematical construct ${\displaystyle l^{2}-c^{2}t^{2}}$  will remain invariant, and so we shall have ${\displaystyle l^{2}-c^{2}t^{2}=l'^{2}-c^{2}t'^{2}.}$  It is this invariant expression, which involves both length and duration, or both space and time, which constitutes the Einsteinian interval; and the objective world which it cannotes is the world of four-dimensional space-time. The Einsteinian interval... remains the same for all observers, just as distance alone or duration alone were mistakenly believed to remain the same for all observers in classical physics. ...the Einsteinian interval still remains an invariant as measured for all frames of reference, whether accelerated or not. In the case of accelerated frames, however, we must restrict our attention to Einsteinan intervals of infinitesimal magnitude, and then add up the intervals when finite magnitudes are involved.
• pp. 210-211
• In the study of electricity and magnetism we may consider phenomena in which conditions do not vary as time passes by; the electric charges and the magnets remain at rest, and the currents flowing in fixed wires do not vary in intensity. Conditions are then termed stationary [static]; it is as though time played no part. The laws which govern this type of phenomena were discovered empirically over a century ago, and were expressed mathematically in terms of spatial vectors. The problem of ascertaining how electric and magnetic phenomena would behave when conditions ceased to be stationary was one that could not be predicted; further experimental research was necessary before the general laws could be obtained. Even so, the difficulties were considerable, and it needed Maxwell's genius to establish the laws from the incomplete array of experimental evidence then at hand. All this work extended over nearly a century; it was slow and laborious.
Yet, had men realised that our world was one of four-dimensional Minkowskian space-time, and not one of separate space and time, things would have been different. By extending the well-known stationary laws to four-dimensional space-time, through the mere addition of time components to the various trios of space ones, we should have written out inadvertently the laws governing varying fields, or, in other words, we should have constructed Maxwell's celebrated equations. Electromagnetic induction, discovered experimentally by Faraday, the additional electrical term introduced tentatively by Maxwell, radio waves, everything in the electromagnetics of the field, could have been foreseen at one stroke of the pen. A century of painstaking effort could have been saved. We are assuming that a four-dimensional vector calculus would have been in existence; but this is purely a mathematical question.
• pp. 319-320
• In classical science, it was strange to find that action... should yet present the artificial aspect of an energy in space multiplied by a duration. As soon, however, as we realise that the fundamental continuum of the universe is one of space-time and not one of separate space and time, the reason for the importance of the seemingly artificial combination of space with time in the expression for the action receives a very simple explanation. Henceforth, action is no longer energy in a volume of space multiplied by a duration; it is simply energy in a volume of the world, that is to say, in a volume of four-dimensional space-time. Designating a volume of space-time by ${\displaystyle d\omega }$ , we have
${\displaystyle d\omega =dxdydzdt,}$
so that our principle of action, ${\displaystyle \partial A=0,}$  becomes
${\displaystyle \partial \int \,L\,d\omega =0.}$
Now there is a perfect symmetry between the rôles of space and time.
• p. 323
• When we realize the important rôle played by space-time in our attempts to avoid a belief in absolute rotation, we can well understand that the doctrine of the relativity of all motion would have been absurd in Newton's day. ...any speaker prior to, say, the year 1900 could never have anticipated the discovery of space-time, for its sole justification arose from the negative experiments in optics and electrodynamics attempted at about that time. As for Newton, not only did he know nothing of the non-mechanical negative experiments, but in addition, the equations of electrodynamics had not been discovered... even if he had conceived of space-time through some divine inspiration, he could never have utilised it for the purpose of establishing the relativity of all motion. His ignorance of non-Euclidean geometry would have rendered the task impossible. In fact, space-time, in the seventeenth century, would have been a hindrance, and the sole result of its premature introduction into science would have been to muddle everything up and render the discovery of Newton's law of gravitation well-nigh impossible.
• pp. 428-429

### Quotes from fiction

• In the Vortex that lies beyond time and space tumbled a police box that was not a police box.
• Stephen Baxter, Doctor Who - The Wheel of Ice
• Doc Brown:  I foresee two possibilities.  One: coming face-to-face with herself thirty years older would put her into shock and she'd simply pass out.  Or two: the encounter could create a time paradox, the result of which could cause a chain reaction that would unravel the very fabric of the spacetime continuum and destroy the entire universe!  Granted, that's a worst-case scenario.  The destruction might in fact be very localised, limited to merely our own galaxy.
Marty McFly:  Well, that's a relief.