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Spacetime#Basic concepts

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**spacetime**is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams can be used to visualize relativistic effects such as why different observers perceive

*where*and

*when*events occur differently.Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. However, in 1905, Albert Einstein based (wikisource:Translation:On the Electrodynamics of Moving Bodies|his seminal work on special relativity) on two postulates: (1) The laws of physics are invariant (i.e., identical) in all inertial systems (i.e., non-accelerating frames of reference); (2) The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.The logical consequence of taking these postulates together is the inseparable joining together of the four dimensions, hitherto assumed as independent, of space and time. Many counterintuitive consequences emerge: in addition to being independent of the motion of the light source,

*the speed of light has the same speed regardless of the frame of reference in which it is measured*; the distances and even temporal ordering of pairs of events change when measured in different inertial frames of reference (this is the relativity of simultaneity); and the linear additivity of velocities no longer holds true.Einstein framed his theory in terms of kinematics (the study of moving bodies). His theory was a breakthrough advance over Lorentz's 1904 theory of electromagnetic phenomena and PoincarÃ©'s electrodynamic theory. Although these theories included equations identical to those that Einstein introduced (i.e. the Lorentz transformation), they were essentially ad hoc models proposed to explain the results of various experimentsâ€”including the famous Michelsonâ€“Morley interferometer experimentâ€”that were extremely difficult to fit into existing paradigms.In 1908 Hermann Minkowskiâ€”once one of the math professors of a young Einstein in ZÃ¼richâ€”presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. A key feature of this interpretation is the formal definition of the spacetime interval. Although measurements of distance and time between events differ for measurements made in different reference frames, the spacetime interval is independent of the inertial frame of reference in which they are recorded.Minkowski's geometric interpretation of relativity was to prove vital to Einstein's development of his 1915 general theory of relativity, wherein he showed how mass and energy

*curve*this

*flat*spacetime to a

*Pseudo Riemannian*manifold.{{anchor|Contents}}{{TOC limit|3}}

## Introduction

{{anchor|Introduction}}{{anchor|Definitions}}### Definitions

Non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space and which is separate from space. Classical mechanics assumes that time has a constant rate of passage that is independent of the state of motion of an observer, or indeed of anything external.WEB, Rynasiewicz, Robert, Newton's Views on Space, Time, and Motion,weblink Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University, 24 March 2017, Furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense.BOOK, Davis, Philip J., Mathematics & Common Sense: A Case of Creative Tension, 2006, A.K. Peters, Wellesley, Massachusetts, 9781439864326, 86, In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field.In ordinary space, a position is specified by three numbers, known as dimensions. In the Cartesian coordinate system, these are called x, y, and z. A position in spacetime is called an*event*, and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig. 1). Spacetime is thus four dimensional. An event is something that happens instantaneously at a single point in spacetime, represented by a set of coordinates

*x*,

*y*,

*z*and

*t*.The word "event" used in relativity should not be confused with the use of the word "event" in normal conversation, where it might refer to an "event" as something such as a concert, sporting event, or a battle. These are not mathematical "events" in the way the word is used in relativity, because they have finite durations and extents. Unlike the analogies used to explain events, such as firecrackers or lightning bolts, mathematical events have zero duration and represent a single point in spacetime.The path of a particle through spacetime can be considered to be a succession of events. The series of events can be linked together to form a line which represents a particle's progress through spacetime. That line is called the particle's

*world line*.{{rp|105}}Mathematically, spacetime is a

*manifold*, which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, a globe appears flat.WEB, Rowland, Todd, Manifold,weblink Wolfram Mathworld, Wolfram Research, 24 March 2017, An extremely large scale factor, c (conventionally called the

*speed-of-light*) relates distances measured in space with distances measured in time. The magnitude of this scale factor (nearly {{convert|300000|km|disp=or||}} in space being equivalent to one second in time), along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe which is noticeably different from what they might observe if the world were Euclidean. It was only with the advent of sensitive scientific measurements in the mid-1800s, such as the Fizeau experiment and the Michelsonâ€“Morley experiment, that puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space.BOOK, French, A.P., Special Relativity, 1968, CRC Press, Boca Raton, Florida, 0748764224, 35â€“60, {{anchor|Figure 1-1}}File:Observer in special relativity.svg|thumb|Figure 1-1. Each location in spacetime is marked by four numbers defined by a (frame of reference]]: the position in space, and the time (which can be visualized as the reading of a clock located at each position in space). The 'observer' synchronizes the clocks according to their own reference frame.)In special relativity, an observer will, in most cases, mean a frame of reference from which a set of objects or events are being measured. This usage differs significantly from the ordinary English meaning of the term. Reference frames are inherently nonlocal constructs, and according to this usage of the term, it does not make sense to speak of an observer as having a location. In Fig. 1‑1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout the three dimensions of space. Any specific location within the lattice is not important. The latticework of clocks is used to determine the time and position of events taking place within the whole frame. The term

*observer*refers to the entire ensemble of clocks associated with one inertial frame of reference.BOOK,weblink Spacetime Physics: Introduction to Special Relativity, Taylor, Edwin F., Wheeler, John Archibald, 1992, Freeman, 1992, 071670336X, 2nd, San Francisco, 14 April 2017, {{rp|17â€“22}} In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording. A real observer, however, will see a delay between the emission of a signal and its detection due to the speed of light. To synchronize the clocks, in the data reduction following an experiment, the time when a signal is received will be corrected to reflect its actual time were it to have been recorded by an idealized lattice of clocks.In many books on special relativity, especially older ones, the word "observer" is used in the more ordinary sense of the word. It is usually clear from context which meaning has been adopted.Physicists distinguish between what one

*measures*or

*observes*(after one has factored out signal propagation delays), versus what one visually sees without such corrections. Failure to understand the difference between what one measures/observes versus what one sees is the source of much error among beginning students of relativity.JOURNAL, Scherr, Rachel E., Shaffer, Peter S., Vokos, Stamatis, Student understanding of time in special relativity: Simultaneity and reference frames, American Journal of Physics, July 2001, 69, S1, S24â€“S35, 10.1119/1.1371254,weblink 11 April 2017, 2001AmJPh..69S..24S, physics/0207109,

### History

{{anchor|History}}{{multiple image
| direction = vertical

| width = 220

| image1 = Michelson-Morley experiment conducted with white light.png

| image2 = MichelsonMorleyAnimationDE.gif

| caption2 = Figure 1-2. Michelson and Morley expected that motion through the aether would cause a differential phase shift between light traversing the two arms of their apparatus. The most logical explanation of their negative result, aether dragging, was in conflict with the observation of stellar aberration.

}}By the mid-1800s, various experiments such as the observation of the Arago spot (a bright point at the center of a circular object's shadow due to diffraction) and differential measurements of the speed of light in air versus water were considered to have proven the wave nature of light as opposed to a corpuscular theory.BOOK, Hughes, Stefan, Catchers of the Light: Catching Space: Origins, Lunar, Solar, Solar System and Deep Space, 2013, ArtDeCiel Publishing, Paphos, Cyprus, 9781467579926, 202â€“233,weblink Propagation of waves was then assumed to require the existence of a medium which | width = 220

| image1 = Michelson-Morley experiment conducted with white light.png

| image2 = MichelsonMorleyAnimationDE.gif

| caption2 = Figure 1-2. Michelson and Morley expected that motion through the aether would cause a differential phase shift between light traversing the two arms of their apparatus. The most logical explanation of their negative result, aether dragging, was in conflict with the observation of stellar aberration.

*waved*: in the case of light waves, this was considered to be a hypothetical luminiferous aether.

*luminiferous*from the Latin

*lumen*, light, +

*ferens*, carrying;

*aether*from the Greek Î±á¼°Î¸Î®Ï (

*aithÄ“r*), pure air, clear sky However, the various attempts to establish the properties of this hypothetical medium yielded contradictory results. For example, the Fizeau experiment of 1851 demonstrated that the speed of light in flowing water was less than the sum of the speed of light in air plus the speed of the water by an amount dependent on the water's index of refraction. Among other issues, the dependence of the partial aether-dragging implied by this experiment on the index of refraction (which is dependent on wavelength) led to the unpalatable conclusion that aether

*simultaneously*flows at different speeds for different colors of light.BOOK, Stachel, John, Kox, A. J., Eisenstaedt, Jean, The Universe of General Relativity, BirkhÃ¤user, Boston, 2005, 1â€“13, Fresnelâ€™s (Dragging) Coefficient as a Challenge to 19th Century Optics of Moving Bodies., 081764380X,weblinkweblink 13 April 2017, yes, The famous Michelsonâ€“Morley experiment of 1887 (Fig. 1‑2) showed no differential influence of Earth's motions through the hypothetical aether on the speed of light, and the most likely explanation, complete aether dragging, was in conflict with the observation of stellar aberration.George Francis FitzGerald in 1889 and Hendrik Lorentz in 1892 independently proposed that material bodies traveling through the fixed aether were physically affected by their passage, contracting in the direction of motion by an amount that was exactly what was necessary to explain the negative results of the Michelson-Morley experiment. (No length changes occur in directions transverse to the direction of motion.)By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein were to derive later (i.e. the Lorentz transform), but with a fundamentally different interpretation. As a theory of dynamics (the study of forces and torques and their effect on motion), his theory assumed actual physical deformations of the physical constituents of matter.BOOK, Pais, Abraham, ""Subtle is the Lord-- ": The Science and the Life of Albert Einstein, 1982, Oxford University Press, Oxford, 019853907X, 11th,weblink {{rp|163â€“174}} Lorentz's equations predicted a quantity that he called

*local time*, with which he could explain the aberration of light, the Fizeau experiment and other phenomena. However, Lorentz considered local time to be only an auxiliary mathematical tool, a trick as it were, to simplify the transformation from one system into another.Other physicists and mathematicians at the turn of the century came close to arriving at what is currently known as spacetime. Einstein himself noted, that with so many people unraveling separate pieces of the puzzle, "the special theory of relativity, if we regard its development in retrospect, was ripe for discovery in 1905."BOOK, Born, Max, Physics in My Generation, 1956, Pergamon Press, London & New York, 194,weblink 10 July 2017, {{multiple image|perrow = 2|total_width=300

width1=280|height1=396| caption1 = Hendrik Lorentz | width2=371|height2=500| caption2 = Henri PoincarÃ© | width3=280|height3=396| caption3 = Albert Einstein | width4=813|height4=1093| caption4 = Hermann Minkowski | | footer = Figure 1-3.}}An important example is Henri PoincarÃ©,{{Citation|author=Darrigol, O. |title=The Genesis of the theory of relativity |year=2005 |journal=SÃ©minaire PoincarÃ©|volume=1|pages=1â€“22|url=http://www.bourbaphy.fr/darrigol2.pdf|doi=10.1007/3-7643-7436-5_1|isbn=978-3-7643-7435-8 |bibcode=2006eins.book....1D }}BOOK, Miller, Arthur I., Albert Einstein's Special Theory of Relativity, 1998, Springer-Verlag, New York, 0387948708, {{rp|73â€“80,93â€“95}} who in 1898 argued that the simultaneity of two events is a matter of convention.BOOK, Galison, Peter, Einstein's Clocks, PoincarÃ©'s Maps: Empires of Time, 2003, W. W. Norton & Company, Inc., New York, 0393020010, 13â€“47,weblink {{refn|group=note|By stating that simultaneity is a matter of convention, PoincarÃ© meant that to talk about time at all, one must have synchronized clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention). This stance represented a fundamental philosophical break from Newton, who conceived of an absolute, true time that was independent of the workings of the inaccurate clocks of his day. This stance also represented a direct attack against the influential philosopher Henri Bergson, who argued that time, simultaneity, and duration were matters of intuitive understanding.}} In 1900, he recognized that Lorentz's "local time" is actually what is indicated by moving clocks by applying an explicitly operational definition of clock synchronization assuming constant light speed.{{refn|group=note|The operational procedure adopted by PoincarÃ© was essentially identical to what is known as Einstein synchronization, even though a variant of it was already a widely used procedure by telegraphers in the middle 19th century. Basically, to synchronize two clocks, one flashes a light signal from one to the other, and adjusts for the time that the flash takes to arrive.}} In 1900 and 1904, he suggested the inherent undetectability of the aether by emphasizing the validity of what he called the principle of relativity, and in 1905/1906JOURNAL, Poincare, Henri, On the Dynamics of the Electron (Sur la dynamique de l'Ã©lectron), Rendiconti del Circolo Matematico di Palermo, 1906, 21, 129â€“176,weblink 15 July 2017, 10.1007/bf03013466, 1906RCMP...21..129P, 2027/uiug.30112063899089, he mathematically perfected Lorentz's theory of electrons in order to bring it into accordance with the postulate of relativity. While discussing various hypotheses on Lorentz invariant gravitation, he introduced the innovative concept of a 4-dimensional space-time by defining various four vectors, namely four-position, four-velocity, and four-force.{{Citation|author=Zahar, Elie|year=1989|orig-year=1983|title=Einstein's Revolution: A Study in Heuristic|chapter=PoincarÃ©'s Independent Discovery of the relativity principle |publisher=Open Court Publishing Company|location=Chicago|isbn=0-8126-9067-2}} He did not pursue the 4-dimensional formalism in subsequent papers, however, stating that this line of research seemed to "entail great pain for limited profit", ultimately concluding "that three-dimensional language seems the best suited to the description of our world".BOOK, Walter, Scott A., Renn, JÃ¼rgen, Schemmel, Matthias, The Genesis of General Relativity, Volume 3, Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905â€“1910, 2007, Springer, Berlin, 193â€“252,weblink 15 July 2017,weblink 15 July 2017, yes, Furthermore, even as late as 1909, PoincarÃ© continued to believe in the dynamical interpretation of the Lorentz transform.{{rp|163â€“174}} For these and other reasons, most historians of science argue that PoincarÃ© did not invent what is now called special relativity.In 1905, Einstein introduced special relativity (even though without using the techniques of the spacetime formalism) in its modern understanding as a theory of space and time. While his results are mathematically equivalent to those of Lorentz and PoincarÃ©, it was Einstein who showed that the Lorentz transformations are not the result of interactions between matter and aether, but rather concern the nature of space and time itself. He obtained all of his results by recognizing that the entire theory can be built upon two postulates: The principle of relativity and the principle of the constancy of light speed.Einstein performed his analysis in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. His seminal work introducing the subject was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples.JOURNAL, Einstein, Albert, On the Electrodynamics of Moving Bodies ( Zur Elektrodynamik bewegter KÃ¶rper), Annalen der Physik, 1905, 322, 10, 891â€“921,weblink 7 April 2018, 1905AnP...322..891E, 10.1002/andp.19053221004, {{refn|group=note|A hallmark of Einstein's career, in fact, was his use of visualized thought experiments (Gedankenâ€“Experimente) as a fundamental tool for understanding physical issues. For special relativity, he employed moving trains and flashes of lightning for his most penetrating insights. For curved spacetime, he considered a painter falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his great Solvay Debates with Bohr on the nature of reality (1927 and 1930), he devised multiple imaginary contraptions intended to show, at least in concept, means whereby the Heisenberg uncertainty principle might be evaded. Finally, in a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as quantum entanglement. BOOK, Isaacson, Walter, Einstein: His Life and Universe, 2007, Simon & Schuster, 978-0-7432-6473-0, {{rp|26â€“27;122â€“127;145â€“146;345â€“349;448â€“460}} }}In addition, Einstein in 1905 superseded previous attempts of an electromagnetic mass-energy relation by introducing the general equivalence of mass and energy, which was instrumental for his subsequent formulation of the equivalence principle in 1907, which declares the equivalence of inertial and gravitational mass. By using the mass-energy equivalence, Einstein showed, in addition, that the gravitational mass of a body is proportional to its energy content, which was one of early results in developing general relativity. While it would appear that he did not at first think geometrically about spacetime,BOOK, Schutz, Bernard, Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity, 2004, Cambridge University Press, Cambridge, 0521455065, Reprint,weblink 24 May 2017, en, {{rp|219}} in the further development of general relativity Einstein fully incorporated the spacetime formalism.When Einstein published in 1905, another of his competitors, his former mathematics professor Hermann Minkowski, had also arrived at most of the basic elements of special relativity. Max Born recounted a meeting he had made with Minkowski, seeking to be Minkowski's student/collaborator:ARXIV, Weinstein, Galina, Max Born, Albert Einstein and Hermann Minkowski's Space-Time Formalism of Special Relativity, 1210.6929, physics.hist-ph, 2012, {{cquote|I went to Cologne, met Minkowski and heard his celebrated lecture 'Space and Time' delivered on 2 September 1908. [â€¦] He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other was pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendor. He never made a priority claim and always gave Einstein his full share in the great discovery.}}Minkowski had been concerned with the state of electrodynamics after Michelson's disruptive experiments at least since the summer of 1905, when Minkowski and David Hilbert led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz, PoincarÃ© et al. However, it is not at all clear when Minkowski began to formulate the geometric formulation of special relativity that was to bear his name, or to which extent he was influenced by PoincarÃ©'s four-dimensional interpretation of the Lorentz transformation. Nor is it clear if he ever fully appreciated Einstein's critical contribution to the understanding of the Lorentz transformations, thinking of Einstein's work as being an extension of Lorentz's work.JOURNAL, Galison, Peter Louis, Minkowski's space-time: From visual thinking to the absolute world, Historical Studies in the Physical Sciences, 1979, 10, 85â€“121, 10.2307/27757388, 27757388, (File:Minkowski Diagram from 1908 'Raum und Zeit' lecture.jpg|thumb|330px|Figure 1-4. Hand-colored transparency presented by Minkowski in his 1908 Raum und Zeit lecture)On 5 November 1907 (a little more than a year before his death), Minkowski introduced his geometric interpretation of spacetime in a lecture to the GÃ¶ttingen Mathematical society with the title, The Relativity Principle (Das RelativitÃ¤tsprinzip).{{refn|group=note|In the original version of this lecture, Minkowski continued to use such obsolescent terms as the ether, but the posthumous publication in 1915 of this lecture in the Annals of Physics (Annalen der Physik) was edited by Sommerfeld to remove this term. Sommerfeld also edited the published form of this lecture to revise Minkowski's judgement of Einstein from being a mere clarifier of the principle of relativity, to being its chief expositor.}} On 21 September 1908, Minkowski presented his famous talk, Space and Time (Raum und Zeit),JOURNAL, Minkowski, Hermann, Raum und Zeit, Jahresbericht der Deutschen Mathematiker-Vereinigung, 1909, 1â€“14,weblink Space and Time, B.G. Teubner, to the German Society of Scientists and Physicians. The opening words of Space and Time include Minkowski's famous statement that "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." Space and Time included the first public presentation of spacetime diagrams (Fig. 1‑4), and included a remarkable demonstration that the concept of the invariant interval (discussed below), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity.{{refn|group=note|(In the following, the group Gcâˆž is the Galilean group and the groupG the Lorentz group.) "With respect to this it is clear that the group }}, i.e. as group G = ∞cc in the limit for {{nowrap|1= is mathematically more intelligible than GcâˆžG, exactly becomes the full group belonging to Newtonian Mechanics. In this state of affairs, and since , but rather for a group GâˆžâˆžG, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena actually possess an invariance, not for the group is definitely finite, and only exceedingly large using the ordinary measuring units."}}The spacetime concept and the Lorentz group are closely connected to certain types of sphere, hyperbolic, or conformal geometries and their transformation groups already developed in the 19th century, in which invariant intervals analogous to the spacetime interval are used.{{refn|group=note|For instance, the Lorentz group is a subgroup of the conformal group in four dimensions.JOURNAL, Cartan, Ã‰., Fano, G., 1955, 1915, EncyclopÃ©die des Sciences MathÃ©matiques Pures et AppliquÃ©es, 3, 1, La thÃ©orie des groupes continus et la gÃ©omÃ©trie, 39â€“43,weblink (Only pages 1â€“21 were published in 1915, the entire article including pp. 39â€“43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the EncyclopÃ©die in 1991.){{rp|41â€“42}}The Lorentz group is isomorphic to the Laguerre group transforming planes into planes,{{rp|39â€“42}}it is isomorphic to the MÃ¶bius group of the plane,JOURNAL, Kastrup, H. A., On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics, Annalen der Physik, 520, 9â€“10, 2008, 631â€“690, 0808.2730, 10.1002/andp.200810324, 2008AnP...520..631K, {{rp|22}}and is isomorphic to the group of isometries in hyperbolic space which is often expressed in terms of the hyperboloid model.BOOK, Ratcliffe, J. G., 1994, Foundations of Hyperbolic Manifolds, Hyperbolic geometry, 56â€“104, New York, 038794348X, {{rp|3.2.3}} }}Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as Gcc, where Ã¼berflÃ¼ssige Gelehrsamkeit (superfluous learnedness). However, in order to complete his search for general relativity that started in 1907, the geometric interpretation of relativity proved to be vital, and in 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity.{{rp|151â€“152}} Since there are other types of spacetime, such as the curved spacetime of general relativity, the spacetime of special relativity is today known as Minkowski spacetime.## Spacetime in special relativity{{anchor|Spacetime interval}}## Spacetime intervalIn three-dimensions, thedistance Delta{d} between two points can be defined using the Pythagorean theorem:
{left(Delta{d}right)}^2 = {left(Delta{x}right)}^2 + {left(Delta{y}right)}^2 + {left(Delta{z}right)}^2
Although two viewers may measure the x,y, and z position of the two points using different coordinate systems, the distance between the points will be the same for both (assuming that they are measuring using the same units). The distance is "invariant".In special relativity, however, the distance between two points is no longer the same if measured by two different observers when one of the observers is moving, because of Lorentz contraction. The situation is even more complicated if the two points are separated in time as well as in space. For example, if one observer sees two events occur at the same place, but at different times, a person moving with respect to the first observer will see the two events occurring at different places, because (from their point of view) they are stationary, and the position of the event is receding or approaching. Thus, a different measure must be used to measure the effective "distance" between two events.In four-dimensional spacetime, the analog to distance is the interval. Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from four-dimensional Euclidean space. The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events (because of time dilation) or the distance between the two events (because of length contraction). But special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure time and distance carefully will find the same spacetime interval between any two events. Suppose an observer measures two events as being separated in time by Delta t and a spatial distance Delta x. Then the spacetime interval {left(Delta{s}right)}^2 between the two events that are separated by a distance Delta{x} in space and by Delta{ct}= cDelta t in the ct-coordinate is:
(Delta s)^2 = (Delta ct)^2 - (Delta x)^2 , or for three space dimensions, (Delta s)^2 = (Delta ct)^2 - (Delta x)^2 - (Delta y)^2 - (Delta z)^2BOOK, Differential Manifolds and Theoretical Physics, W. D., Curtis, F. R., Miller, Academic Press, 1985, 978-0-08-087435-7, 223,weblink The constant textrm{c}, the speed of light, converts the units used to measure time (seconds) into units used to measure distance (meters).Note on nomenclature: Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, x means Delta{x}, etc. We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning.(File:Spacetime Diagram of Two Photons and a Slower than Light Object.png|thumb|Figure 2-1. Spacetime diagram illustrating two photons, A and B, originating at the same event, and a slower-than-light-speed object, C) The equation above is similar to the Pythagorean theorem, except with a minus sign between the (textrm{c} , t)^2 and the x^2 terms. The spacetime interval is the quantity s^2, not s itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard s^2 as a distinct symbol in itself, rather than the square of something.{{rp|217}}Because of the minus sign, the spacetime interval between two distinct events can be zero. If s^2 is positive, the spacetime interval is timelike, meaning that two events are separated by more time than space. If s^2 is negative, the spacetime interval is spacelike, meaning that two events are separated by more space than time. Spacetime intervals are zero when x = pm textrm{c} , t. In other words, the spacetime interval between two events on the world line of something moving at the speed of light is zero. Such an interval is termed lightlike or null. A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage.A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig. 2‑1 presents a spacetime diagram illustrating the world lines (i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light-speed object. The vertical time coordinate is scaled by textrm{c} so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at the speed of light, their world lines have a slope of Â±1. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.Note on nomenclature: There are two sign conventions in use in the relativity literature:
s^2 = (textrm{c} t)^2 - x^2 - y^2 - z^2
and
s^2 = - (textrm{c} t)^2 + x^2 + y^2 + z^2
These sign conventions are associated with the metric signatures {{nowrap|(+ âˆ’ âˆ’ âˆ’)}} and {{nowrap|(âˆ’ + + +).}} A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study.{{anchor|Reference frames}}## Reference frames(File:Standard configuration of coordinate systems.svg|thumb|Figure 2-2. Galilean diagram of two frames of reference in standard configuration)(File:Galilean and Spacetime coordinate transformations.png|thumb|330px|Figure 2-3. (a) Galilean diagram of two frames of reference in standard configuration, (b) spacetime diagram of two frames of reference, (c) spacetime diagram showing the path of a reflected light pulse)To gain insight in how spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in astandard configuration. With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2‑2, two Galilean reference frames (i.e. conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame Sâ€² (pronounced "S prime") belongs to a second observer Oâ€². - The
*x*,*y*,*z*axes of frame S are oriented parallel to the respective primed axes of frame Sâ€². - Frame Sâ€² moves in the
*x*-direction of frame S with a constant velocity*v*as measured in frame S. - The origins of frames S and Sâ€² are coincident when time
*t*= 0 for frame S and*t*â€² = 0 for frame Sâ€².BOOK, A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity, Collier, Peter, Incomprehensible Books, 2017, 9780957389465, 3rd, {{rp|107}}
t = 0 in frame S and tâ€² = 0 in frame S'. The ctâ€² axis passes through the events in frame Sâ€² which have xâ€² = 0. But the points with xâ€² = 0 are moving in the x-direction of frame S with velocity v, so that they are not coincident with the ct axis at any time other than zero. Therefore, the ctâ€² axis is tilted with respect to the ct axis by an angle Î¸ given by
tan theta = v/c .
The xâ€² axis is also tilted with respect to the x axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always Â±1. Fig. 2‑3c presents a spacetime diagram from the viewpoint of observer Oâ€². Event P represents the emission of a light pulse at xâ€² = 0, ctâ€² = âˆ’a. The pulse is reflected from a mirror situated a distance a from the light source (event Q), and returns to the light source at xâ€² = 0, ctâ€² = a (event R).The same events P, Q, R are plotted in Fig. 2‑3b in the frame of observer O. The light paths have slopes = 1 and âˆ’1 so that â–³PQR forms a right triangle. Since OP = OQ = OR, the angle between xâ€² and x must also be Î¸.{{rp|113â€“118}}While the rest frame has space and time axes that meet at right angles, the moving frame is drawn with axes that meet at an acute angle. The frames are actually equivalent. The asymmetry is due to unavoidable distortions in how spacetime coordinates can map onto a Cartesian plane, and should be considered no stranger than the manner in which, on a Mercator projection of the Earth, the relative sizes of land masses near the poles (Greenland and Antarctica) are highly exaggerated relative to land masses near the Equator.{{anchor|Light cone}}## Light cone{{anchor|Figure 2-4}}(File:ModernPhysicsSpaceTimeA.png|thumb|Figure 2-4. The light cone centered on an event divides the rest of spacetime into the future, the past, and "elsewhere")In Fig. 2-4, event O is at the origin of a spacetime diagram, and the two diagonal lines represent all events that have zero spacetime interval with respect to the origin event. These two lines form what is called thelight cone of the event O, since adding a second spatial dimension (Fig. 2‑5) makes the appearance that of two right circular cones meeting with their apices at O. One cone extends into the future (t>0), the other into the past (t |

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