Spacetime

"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."

"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."

"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."

"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 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:dx^2 + dy^2 + dz^2 -c^2dt^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., \frac{\vartriangle l}{\vartriangle t} = c, or expressed as differentials, \frac{dl}{dt} = c, which implies \frac{dl^2}{dt^2} = c^2 and {dl^2} - c^2dt^2 = 0. But, by the Pythagorean theorem, {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 v existing between the two systems. These rules of transformation are given by the Lorentz-Einstein transformations."

"The discovery of the invariantdx^2 + dy^2 + dz^2 -c^2dt^2whose value we shall designate 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."

"Minkowski demonstrated the significance of the expression for ds^2 by taking the new variable T = ict, where i stands for \sqrt{-1}. With this change, ds^2 can be written: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 v are represented by two space-time Cartesian co-ordinate systems differing in orientation by the imaginary angle \theta, where \theta is connected with v by the formula tan\theta = \frac{iv}{c}."

"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 l^2 - c^2t^2 will remain invariant, and so we shall have l^2 - c^2t^2 = l'^2 - c^2t'^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."

"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."

"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 d\omega, we haved\omega = dxdydzdt,so that our principle of action, \partial A = 0, becomes\partial \int\,L\,d\omega = 0.Now there is a perfect symmetry between the rôles of space and time."

"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."

"In the Vortex that lies beyond time and space tumbled a police box that was not a police box."

"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."

""Imagination is creativity playing outside the realms of intelligence or knowledge. Imagination entangles within the quantum brain of the world and self." *"

"We seek ourselves in quests so deep,"