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{{short description|Mathematical model combining space and time}}{{other uses}}{{Spacetime|cTopic=Types}}In physics, spacetime is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects such as how different observers perceive where and when events occur.Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe). However, space and time took on new meanings with the Lorentz transformation and special theory of relativity.In 1908, Hermann Minkowski 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. This interpretation proved vital to the general theory of relativity, wherein spacetime is curved by mass and energy.{{anchor|Contents}}{{TOC limit|3}}

Fundamentals

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

Definitions

Non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space, and separate from space. Classical mechanics assumes that time has a constant rate of passage, independent of the observer's state of motion, or anything external.WEB, Rynasiewicz, Robert, Newton's Views on Space, Time, and Motion,weblink Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University, August 12, 2004, 24 March 2017, 16 July 2012,weblink" title="archive.today/20120716191122weblink">weblink live, It assumes that space is Euclidean: 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, 978-1-4398-6432-6, 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.{{rp|214–217}} General relativity 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). An event is represented by a set of coordinates x, y, z and t.BOOK, Fock, V., The Theory of Space, Time and Gravitation, 1966, Pergamon Press Ltd., New York, 0-08-010061-9, 33, 2nd,weblink 14 October 2023, Spacetime is thus four-dimensional.Unlike the analogies used in popular writings to explain events, such as firecrackers or sparks, mathematical events have zero duration and represent a single point in spacetime.BOOK, Lawden, D. F., Introduction to Tensor Calculus, Relativity and Cosmology, 1982, Dover Publications, Mineola, New York, 978-0-486-42540-5, 7, 3rd,weblink Although it is possible to be in motion relative to the popping of a firecracker or a spark, it is not possible for an observer to be in motion relative to an event.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, the surface of a globe appears flat.WEB, Rowland, Todd, Manifold,weblink Wolfram Mathworld, Wolfram Research, 24 March 2017, 13 March 2017,weblink" title="web.archive.org/web/20170313111306weblink">weblink live, A 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 Michelson–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, 0-7487-6422-4, Boca Raton, Florida, 35–60, en-us, {{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 is 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 whole 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, 0-7167-0336-X, 2nd, San Francisco, California, 14 April 2017, {{rp|17–22}}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, 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.{{rp|17–22}}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., Rachel Scherr, Shaffer, Peter S., Vokos, Stamatis, Student understanding of time in special relativity: Simultaneity and reference frames, American Journal of Physics, American Association of Physics Teachers, College Park, Maryland, July 2001, 69, S1, S24–S35, 10.1119/1.1371254,weblink 11 April 2017, 2001AmJPh..69S..24S, physics/0207109, 8146369, 28 September 2018,weblink live,

History

{{anchor|History}}{{multiple image }}By the mid-1800s, various experiments such as the observation of the Arago spot 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, 978-1-4675-7992-6, 202–233,weblink 7 April 2017, 17 January 2023,weblink live, Propagation of waves was then assumed to require the existence of a waving medium; 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 αἰθήρ (aithÄ“r), pure air, clear sky The various attempts to establish the properties of this hypothetical medium yielded contradictory results. For example, the Fizeau experiment of 1851, conducted by French physicist Hippolyte Fizeau, 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.WEB, Williams, Matt, 2022-01-28, What is Einstein's Theory of Relativity?,weblink 2022-08-13, Universe Today, en-US, 3 August 2022,weblink live, 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, The Universe of General Relativity, 2005, Birkhäuser, 0-8176-4380-X, Kox, A. J., Boston, Massachusetts, 1–13, en-us, Fresnel's (Dragging) Coefficient as a Challenge to 19th Century Optics of Moving Bodies., Eisenstaedt, Jean,weblinkweblink 13 April 2017, dead, The famous Michelson–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,WEB, George Francis FitzGerald,weblink 2022-08-13, The Linda Hall Library, en-US, 17 January 2023,weblink live, 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 was to derive later, i.e. the Lorentz transformation.WEB, The Nobel Prize in Physics 1902,weblink 2022-08-13, NobelPrize.org, en-US, 23 June 2017,weblink" title="web.archive.org/web/20170623231447weblink">weblink live, 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,weblink 'Subtle is the Lord–': The Science and the Life of Albert Einstein, 1982, Oxford University Press, 0-19-853907-X, 11th, Oxford, en, {{rp|163–174}} 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.{{multiple image|perrow = 2|total_width=300

width1=280|height1=396| caption1 = Hendrik Lorentz

width2=371|height2=500| caption2 = Henri Poincaré

width3=280|height3=396| caption3 = Albert Einstein

width4=813|height4=1093| caption4 = Hermann Minkowski| | footer = Figure 1-3.| footer_align = center}}Henri Poincaré was the first to combine space and time into spacetime.{{Citation |author=Darrigol, O. |title=The Genesis of the theory of relativity |year=2005 |journal=Séminaire Poincaré |volume=1 |pages=1–22 |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 |access-date=17 July 2017 |archive-date=28 February 2008 |archive-url=https://web.archive.org/web/20080228124558weblink |url-status=live }}BOOK, Miller, Arthur I., Albert Einstein's Special Theory of Relativity, 1998, Springer-Verlag, New York, 0-387-94870-8, {{rp|73–80,93–95}} He argued in 1898 that the simultaneity of two events is a matter of convention.BOOK, Galison, Peter, Einstein's Clocks, Poincaré's Maps: Empires of Time, 2003, W. W. Norton & Company, Inc., New York, 0-393-02001-0, 13–47,weblink {{refn|group=note|By stating that simultaneity is a matter of convention, Poincaré 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 Poincaré 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. In 1905/1906JOURNAL, Poincare, Henri, On the Dynamics of the Electron (Sur la dynamique de l'électron), Rendiconti del Circolo Matematico di Palermo, 1906, 21, 129–176,weblink 15 July 2017, 10.1007/bf03013466, 1906RCMP...21..129P, 2027/uiug.30112063899089, 120211823, free, 11 July 2017,weblink live, 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 spacetime by defining various four vectors, namely four-position, four-velocity, and four-force.{{Citation |author=Zahar |first=Elie |title=Einstein's Revolution: A Study in Heuristic |year=1989 |chapter=Poincaré's Independent Discovery of the relativity principle |location=Chicago, Illinois |publisher=Open Court Publishing Company |isbn=0-8126-9067-2 |orig-year=1983}} 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., The Genesis of General Relativity, Volume 3, 2007, Springer, Renn, Jürgen, Berlin, Germany, 193–252, Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910, 15 July 2017, Schemmel, Matthias,weblinkweblink 15 July 2017, dead, Even as late as 1909, Poincaré continued to describe the dynamical interpretation of the Lorentz transform.{{rp|163–174}}In 1905, Albert Einstein analyzed special relativity in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. His results were mathematically equivalent to those of Lorentz and Poincaré. He obtained them 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. His work 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 Körper), Annalen der Physik, 1905, 322, 10, 891–921,weblink 7 April 2018, 1905AnP...322..891E, 10.1002/andp.19053221004, free, 6 November 2018,weblink live, {{refn|group=note|A hallmark of Einstein's career, in fact, was his use of visualized thought experiments (Gedanken–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,weblink registration, 2007, Simon & Schuster, 978-0-7432-6473-0, 26–27;122–127;145–146;345–349;448–460, }}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 that the gravitational mass of a body is proportional to its energy content, which was one of the 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, 0-521-45506-5, Reprint,weblink 24 May 2017, en, 17 January 2023,weblink live, {{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. [...] 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, Poincaré et al. Minkowski saw Einstein's work as an extension of Lorentz's, and was most directly influenced by Poincaré.JOURNAL, Galison, Peter Louis, Minkowski's space–time: From visual thinking to the absolute world, Historical Studies in the Physical Sciences, 1979, 10, 85–121, 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 Göttingen Mathematical society with the title, The Relativity Principle (Das Relativitätsprinzip).{{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, 1909, Raum und Zeit, Space and Time,weblink live, Jahresbericht der Deutschen Mathematiker-Vereinigung, B. G. Teubner, 1–14,weblink 28 July 2017, 17 July 2017, 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 G∞ is the Galilean group and the group Gc the Lorentz group.) "With respect to this it is clear that the group Gc in the limit for {{nowrap|1=c = ∞}}, i.e. as group G∞, exactly becomes the full group belonging to Newtonian Mechanics. In this state of affairs, and since Gc is mathematically more intelligible than G∞, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena actually possess an invariance, not for the group G∞, but rather for a group Gc, where c 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, É., Fano, G., 1955, 1915, Encyclopédie des Sciences Mathématiques Pures et Appliquées, 3, 1, La théorie des groupes continus et la géométrie, 39–43,weblink 6 April 2018, 23 March 2018,weblink" title="web.archive.org/web/20180323032943weblink">weblink live, (Only pages 1–21 were published in 1915, the entire article including pp. 39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the Encyclopédie in 1991.){{rp|41–42}}The Lorentz group is isomorphic to the Laguerre group transforming planes into planes,{{rp|39–42}}it is isomorphic to the Möbius 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, 9–10, 2008, 631–690, 0808.2730, 10.1002/andp.200810324, 2008AnP...520..631K, 12020510, {{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, 56–104, New York, 0-387-94348-X,weblink {{rp|3.2.3}} }}Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as überflüssige 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. In 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity.{{rp|151–152}} 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 interval {{See also|Causal structure}}In three dimensions, the distance Delta{d} between two points can be defined using the Pythagorean theorem: (Delta{d})^2 = (Delta{x})^2 + (Delta{y})^2 + (Delta{z})^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.BOOK, Kogut, John B., Introduction to Relativity, 2001, Harcourt/Academic Press, Massachusetts, 0-12-417561-9, {{rp|48–50;100–102}}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). Special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure the time and distance between any two events will end up computing the same spacetime interval. Suppose an observer measures two events as being separated in time by Delta t and a spatial distance Delta x. Then the squared spacetime interval (Delta{s})^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:BOOK, Introducing Einstein's Relativity: A Deeper Understanding, Ray d'Inverno, James Vickers, illustrated, Oxford University Press, 2022, 978-0-19-886202-4, 26–28,weblink Extract of page 27 (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)^2. The constant c, the speed of light, converts time units (like seconds) into space units (like meters). The squared interval Delta s^2 is a measure of separation between events A and B that are time separated and in addition space separated either because there are two separate objects undergoing events, or because a single object in space is moving inertially between its events. The separation interval is derived by squaring the spatial distance separating event B from event A and subtracting it from the square of the spatial distance traveled by a light signal in that same time interval Delta t. If the event separation is due to a light signal, then this difference vanishes and Delta s =0.When the event considered is infinitesimally close to each other, then we may write ds^2 = c^2dt^2 - dx^2-dy^2-dz^2. In a different inertial frame, say with coordinates (t',x',y',z'), the spacetime interval ds' can be written in a same form as above. Because of the constancy of speed of light, the light events in all inertial frames belong to zero interval, ds=ds'=0. For any other infinitesimal event where dsneq 0, one can prove that ds^2=ds'^2which in turn upon integration leads to s=s'.Landau, L. D., and Lifshitz, E. M. (2013). The classical theory of fields (Vol. 2).{{rp|2}} The invariance of interval of any event between all intertial frames of reference is one of the fundamental results of special theory of relativity.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 (ct)^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}} Note: There are two sign conventions in use in the relativity literature: s^2 = (ct)^2 - x^2 - y^2 - z^2 and s^2 = -(ct)^2 + x^2 + y^2 + z^2 These sign conventions are associated with the metric signatures {{nowrap|(+−−−)}} and {{nowrap|(−+++).}} A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study.BOOK, Carroll, Sean, The Biggest Ideas in the Universe, 2022, Penguin Random House LLC, New York, 9780593186589, 155–156, In the following discussion, we use the first convention. In general s^2 can assume any real number value. If s^2 is positive, the spacetime interval is referred to as timelike. Since spatial distance traversed by any massive object is always less than distance traveled by the light for the same time interval, positive intervals are always timelike. If s^2 is negative, the spacetime interval is said to be spacelike. Spacetime intervals are equal to zero when x = pm ct. 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.{{rp|48–50}}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 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 ±1.{{rp|23–25}} In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.{{anchor|Reference frames}} Reference frames {{More citations needed section|date=March 2024}}(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 a standard 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 S′ (pronounced "S prime") belongs to a second observer O′. The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′. Frame S′ moves in the x-direction of frame S with a constant velocity v as measured in frame S. The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.BOOK, A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity, Collier, Peter, Incomprehensible Books, 2017, 978-0-9573894-6-5, 3rd, {{rp|107}} Fig. 2-3a redraws Fig. 2-2 in a different orientation. Fig. 2-3b illustrates a relativistic spacetime diagram from the viewpoint of observer O. Since S and S′ are in standard configuration, their origins coincide at times t = 0 in frame S and t′ = 0 in frame S′. The ct′ axis passes through the events in frame S′ which have x′ = 0. But the points with x′ = 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 ct′ axis is tilted with respect to the ct axis by an angle θ given by{{rp|23–31}} tan(theta) = v/c. The x′ 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 O′. Event P represents the emission of a light pulse at x′ = 0, ct′ = 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 x′ = 0, ct′ = 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 with PQ and QR both at 45 degrees to the x and ct axes. Since OP = OQ = OR, the angle between x′ and x must also be θ.{{rp|113–118}}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.{{rp|23–31}} 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 the light 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 0, and thus the worldline (not shown in the pictures) of this clock intersects the worldline of the moving clock (the ct′-axis) in the event labelled A, where "two clocks are simultaneously at one place". In the lower picture the place for W′2 is taken to be Cx′ This introductory section has focused on the spacetime of special relativity, since it is the easiest to describe. Minkowski spacetime is flat, takes no account of gravity, is uniform throughout, and serves as nothing more than a static background for the events that take place in it. The presence of gravity greatly complicates the description of spacetime. In general relativity, spacetime is no longer a static background, but actively interacts with the physical systems that it contains. Spacetime curves in the presence of matter, can propagate waves, bends light, and exhibits a host of other phenomena.A basic goal is to be able to compare measurements made by observers in relative motion. If there is an observer O in frame S who has measured the time and space coordinates of an event, assigning this event three Cartesian coordinates and the time as measured on his lattice of synchronized clocks {{nowrap|1=(x, y, z, t)}} (see Fig. 1-1). A second observer O′ in a different frame S′ measures the same event in her coordinate system and her lattice of synchronized clocks {{nowrap|1=({{′|x}}, {{′|y}}, {{′|z}}, {{′|t}})}}. With inertial frames, neither observer is under acceleration, and a simple set of equations allows us to relate coordinates {{nowrap|1=(x, y, z, t)}} to {{nowrap|1=({{′|x}}, {{′|y}}, {{′|z}}, {{′|t}})}}. Given that the two coordinate systems are in standard configuration, meaning that they are aligned with parallel {{nowrap|1=(x, y, z)}} coordinates and that {{nowrap|1=t = 0}} when {{nowrap|1={{′|t}} = 0}}, the coordinate transformation is as follows:In scenario (a), the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time (i.e. dr/dt = 0 where r is the distance between receiver and source) and hence no longitudinal Doppler shift. The source observes the receiver as being illuminated by light of frequency {{′|f}}, but also observes the receiver as having a time-dilated clock. In frame S, the receiver is therefore illuminated by blueshifted light of frequency m_{1}+m_{2} }}.{{rp|94–97}} Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable elementary particle spontaneously decays into two lighter particles, total energy is conserved, but the mass is not. Part of the mass is converted into kinetic energy.{{rp|134–138}} Choice of reference frames {{More citations needed section|date=March 2024}}{{multiple image|align=right|image1=2-body Particle Decay-Lab.svg|width1=115|image2=2-body Particle Decay-CoM.svg|width2=105|caption1=Figure 3-11. (above) Lab Frame.(right) Center of Momentum Frame.| }}The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For analysis of momentum and energy problems, the most convenient frame is usually the "center-of-momentum frame" (also called the zero-momentum frame, or COM frame). This is the frame in which the space component of the system's total momentum is zero. Fig. 3-11 illustrates the breakup of a high speed particle into two daughter particles. In the lab frame, the daughter particles are preferentially emitted in a direction oriented along the original particle's trajectory. In the COM frame, however, the two daughter particles are emitted in opposite directions, although their masses and the magnitude of their velocities are generally not the same. Energy and momentum conservation In a Newtonian analysis of interacting particles, transformation between frames is simple because all that is necessary is to apply the Galilean transformation to all velocities. Since {{tmath|1=v' = v - u}}, the momentum {{tmath|1=p' = p - mu}}. If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame.{{rp|241–245}}Conservation of momentum in the COM frame amounts to the requirement that {{math|1=p = 0}} both before and after collision. In the Newtonian analysis, conservation of mass dictates that {{tmath|1=m=m_{1}+m_{2} }}. In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition. In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities. In the case of a completely inelastic collision with total loss of kinetic energy, the outgoing velocities of the rebounding particles will be zero.{{rp|241–245}}Newtonian momenta, calculated as {{tmath|1=p = mv}}, fail to behave properly under Lorentzian transformation. The linear transformation of velocities {{tmath|1=v' = v - u}} is replaced by the highly nonlinear{{tmath|1=v^{prime}=(v-u) Big/left(1- frac{v u} { c^{2} }right)}} so that a calculation demonstrating conservation of momentum in one frame will be invalid in other frames. Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum. This second option was what he chose.{{rp|104}}{{multiple image | align = right | direction = vertical | width = 220 | image1 = Energy-momentum diagram for pion decay (A).png | width1 = | alt1 = | caption1 = Figure 3-12a. Energy–momentum diagram for decay of a charged pion. | image2 = Energy-momentum diagram for pion decay (B).png | width2 = | alt2 = | caption2 = Figure 3-12b. Graphing calculator analysis of charged pion decay. }}The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass.{{rp|127}}{{smalldiv|1=Example: Because of the equivalence of mass and energy, elementary particle masses are customarily stated in energy units, where {{nowrap|1=1 MeV = 106}} electron volts. A charged pion is a particle of mass 139.57 MeV (approx. 273 times the electron mass). It is unstable, and decays into a muon of mass 105.66 MeV (approx. 207 times the electron mass) and an antineutrino, which has an almost negligible mass. The difference between the pion mass and the muon mass is 33.91 MeV. {{SubatomicParticle|Pion-}} → {{SubatomicParticle|link=yes|Muon-}} + {{SubatomicParticle|link=yes|Muon antineutrino}} Fig. 3-12a illustrates the energy–momentum diagram for this decay reaction in the rest frame of the pion. Because of its negligible mass, a neutrino travels at very nearly the speed of light. The relativistic expression for its energy, like that of the photon, is {{tmath|1=E_{v}=p c,}} which is also the value of the space component of its momentum. To conserve momentum, the muon has the same value of the space component of the neutrino's momentum, but in the opposite direction.Algebraic analyses of the energetics of this decay reaction are available online,WEB, Nave, R., Energetics of Charged Pion Decay,weblink Hyperphysics, Department of Physics and Astronomy, Georgia State University, 27 May 2017, 21 May 2017,weblink" title="web.archive.org/web/20170521075304weblink">weblink live, so Fig. 3-12b presents instead a graphing calculator solution. The energy of the neutrino is 29.79 MeV, and the energy of the muon is {{nowrap|1=33.91 MeV − 29.79 MeV = 4.12 MeV}}. Most of the energy is carried off by the near-zero-mass neutrino.}} Beyond the basics {{anchor|Rapidity}}The topics in this section are of significantly greater technical difficulty than those in the preceding sections and are not essential for understanding Introduction to curved spacetime. Rapidity {{multiple image | direction = horizontal | width1 = 135 | image1 = Trig functions (sine and cosine).svg | caption1 = Figure 4-1a. A ray through the unit circle {{nowrap|1=x2 + y2 = 1}} in the point {{nowrap|1=(cos a, sin a)}}, where a is twice the area between the ray, the circle, and the x-axis. | width2 = 190 | image2 = Hyperbolic functions-2.svg | caption2 = Figure 4-1b. A ray through the unit hyperbola {{nowrap|1=x2 − y2 = 1}} in the point {{nowrap|1=(cosh a, sinh a)}}, where a is twice the area between the ray, the hyperbola, and the x-axis. }}File:Sinh+cosh+tanh.svg|thumb|180px|Figure 4–2. Plot of the three basic Hyperbolic functions: hyperbolic sine ((:File:Hyperbolic Sine.svg|sinh)), hyperbolic cosine ((:File:Hyperbolic Cosine.svg|cosh)) and hyperbolic tangent ((:File:Hyperbolic Tangent.svg|tanh)). Sinh is red, cosh is blue and tanh is green.]]Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas.This nonlinearity is an artifact of our choice of parameters.{{rp|47–59}} We have previously noted that in an {{nowrap|1=x–ct}} spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other.The natural functions for expressing these relationships are the hyperbolic analogs of the trigonometric functions. Fig. 4-1a shows a unit circle with sin(a) and cos(a), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that a is interpreted, not as the angle between the ray and the {{nowrap|1=x-axis}}, but as twice the area of the sector swept out by the ray from the {{nowrap|1=x-axis}}. Numerically, the angle and {{nowrap|1=2 × area}} measures for the unit circle are identical. Fig. 4-1b shows a unit hyperbola with sinh(a) and cosh(a), where a is likewise interpreted as twice the tinted area.BOOK, Thomas, George B., Weir, Maurice D., Hass, Joel, Giordano, Frank R., Thomas' Calculus: Early Transcendentals, 2008, Pearson Education, Inc., Boston, 978-0-321-49575-4, 533, Eleventh, Fig. 4-2 presents plots of the sinh, cosh, and tanh functions.For the unit circle, the slope of the ray is given by text{slope} = tan a = frac{sin a }{cos a }. In the Cartesian plane, rotation of point {{nowrap|1=(x, y)}} into point {{nowrap|1=(x{{'}}, y{{'}})}} by angle θ is given by begin{pmatrix}x' y' end{pmatrix} = begin{pmatrix}cos theta & -sin theta sin theta & cos theta end{pmatrix}begin{pmatrix}x y end{pmatrix}.In a spacetime diagram, the velocity parameter beta is the analog of slope. The rapidity, φ, is defined by{{rp|96–99}} beta equiv tanh phi equiv frac{v}{c}, where tanh phi = frac{sinh phi}{cosh phi} = frac{e^phi-e^{-phi}}{e^phi+e^{-phi}}. The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;{{rp|47–59}} beta = frac{beta_1 + beta_2}{1 + beta_1 beta_2} = frac{tanh phi_1 + tanh phi_2}{1 + tanh phi_1 tanh phi_2} = tanh(phi_1 + phi_2), or in other words, phi = phi_1 + phi_2.The Lorentz transformations take a simple form when expressed in terms of rapidity. The γ factor can be written as gamma = frac{1}{sqrt{1 - beta^2}} = frac{1}{sqrt{1 - tanh^2 phi}} = cosh phi, gamma beta = frac{beta}{sqrt{1 - beta^2}} = frac{tanh phi}{sqrt{1 - tanh^2 phi}} = sinh phi. Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called boosts.Substituting γ and γβ into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the {{nowrap|1=x-direction}} may be written as begin{pmatrix} c t' x' end{pmatrix} = begin{pmatrix} cosh phi & -sinh phi -sinh phi & cosh phi end{pmatrix} begin{pmatrix} ct x end{pmatrix}, and the inverse Lorentz boost in the {{nowrap|1=x-direction}} may be written as begin{pmatrix} c t x end{pmatrix} = begin{pmatrix} cosh phi & sinh phi sinh phi & cosh phi end{pmatrix} begin{pmatrix} c t' x' end{pmatrix}. In other words, Lorentz boosts represent hyperbolic rotations in Minkowski spacetime.{{rp|96–99}}The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage.BOOK,weblink Spacetime Physics, Taylor, Edwin F., Wheeler, John Archibald, 1992, W. H. Freeman, 0-7167-2327-1, 2nd, Rapidity arises naturally as a coordinates on the pure boost generators inside the Lie algebra algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo {{nowrap|2{{pi}}}}) on the pure rotation generators in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal resemblance.{{anchor|4‑vectors}} 4‑vectors Four‑vectors have been mentioned above in context of the energy–momentum {{nowrap|1=4‑vector}}, but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, {{nowrap|1=4‑vectors}}, and more generally tensors, greatly simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are manifestly relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of Maxwell's equations in their usual form is not trivial, while it is merely a routine calculation, really no more than an observation, using the field strength tensor formulation.BOOK, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, 1962, Dover Publications Inc., 978-0-486-65427-0, E. J. Post, On the other hand, general relativity, from the outset, relies heavily on {{nowrap|1=4‑vectors}}, and more generally tensors, representing physically relevant entities. Relating these via equations that do not rely on specific coordinates requires tensors, capable of connecting such {{nowrap|1=4‑vectors}} even within a curved spacetime, and not just within a flat one as in special relativity. The study of tensors is outside the scope of this article, which provides only a basic discussion of spacetime. Definition of 4-vectors A 4-tuple, {{tmath|1=A=left(A_{0}, A_{1}, A_{2}, A_{3}right)}} is a "4-vector" if its component Ai transform between frames according to the Lorentz transformation.If using {{tmath|1=(ct, x, y, z)}} coordinates, A is a {{nowrap|1=4–vector}} if it transforms (in the {{nowrap|1=x-direction}}) according to begin{align} A_0' &= gamma left( A_0 - (v/c) A_1 right) A_1' &= gamma left( A_1 - (v/c) A_0 right) A_2' &= A_2 A_3' &= A_3 end{align}which comes from simply replacing ct with A0 and x with A1 in the earlier presentation of the Lorentz transformation.As usual, when we write x, t, etc. we generally mean Δx, Δt etc.The last three components of a {{nowrap|1=4–vector}} must be a standard vector in three-dimensional space. Therefore, a {{nowrap|1=4–vector}} must transform like {{tmath|1=(c Delta t, Delta x, Delta y, Delta z)}} under Lorentz transformations as well as rotations.{{rp|36–59}} Properties of 4-vectors Closure under linear combination: If A and B are {{nowrap|1=4-vectors}}, then {{tmath|1=C = aA + aB}} is also a {{nowrap|1=4-vector}}. Inner-product invariance: If A and B are {{nowrap|1=4-vectors}}, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a {{nowrap|1=3-vector}}. In the following, vec{A} and vec{B} are {{nowrap|1=3-vectors}}: A cdot B equiv A_0 B_0 - A_1 B_1 - A_2 B_2 - A_3 B_3 equiv A_0 B_0 - vec{A} cdot vec{B} In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in {{nowrap|1=3-space}}. Two vectors are said to be orthogonal if A cdot B = 0. Unlike the case with {{nowrap|1=3-vectors,}} orthogonal {{nowrap|1=4-vectors}} are not necessarily at right angles with each other. The rule is that two {{nowrap|1=4-vectors}} are orthogonal if they are offset by equal and opposite angles from the 45° line which is the world line of a light ray. This implies that a lightlike {{nowrap|1=4-vector}} is orthogonal with itself. Invariance of the magnitude of a vector: The magnitude of a vector is the inner product of a {{nowrap|1=4-vector}} with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which A cdot A = 0 , while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval c^2 t^2 - x^2 and the invariant length of the relativistic momentum vector E^2 - p^2 c^2 .{{rp|178–181}}{{rp|36–59}} Examples of 4-vectors Displacement 4-vector: Otherwise known as the spacetime separation, this is {{nowrap|1=(Δt, Δx, Δy, Δz),}} or for infinitesimal separations, {{nowrap|1=(dt, dx, dy, dz)}}. dS equiv (dt, dx, dy, dz) Velocity 4-vector: This results when the displacement {{nowrap|1=4-vector}} is divided by d tau, where d tau is the proper time between the two events that yield dt, dx, dy, and dz. V equiv frac{dS}{d tau} = frac{(dt, dx, dy, dz)}{dt/gamma} = gamma left(1, frac{dx}{dt}, frac{dy}{dt}, frac{dz}{dt} right) = (gamma, gamma vec{v} ) {{multiple image | direction = horizontal | width1 = 150 | image1 = Momentarily Comoving Reference Frame.gif | caption1 = Figure 4-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame. | width2 = 150 | image2 = Lorentz transform of world line.gif | caption2 = Figure 4-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center). }} The {{nowrap|1=4-velocity}} is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle. An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found which is momentarily comoving with the particle. This frame, the momentarily comoving reference frame (MCRF), enables application of special relativity to the analysis of accelerated particles. Since photons move on null lines, d tau = 0 for a photon, and a {{nowrap|1=4-velocity}} cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path. Energy–momentum 4-vector: P equiv (E/c, vec{p}) = (E/c, p_x, p_y, p_z) As indicated before, there are varying treatments for the energy-momentum {{nowrap|1=4-vector}} so that one may also see it expressed as (E, vec{p}) or (E, vec{p}c) . The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy-momentum {{nowrap|1=4-vector}} is a conserved quantity. Acceleration 4-vector: This results from taking the derivative of the velocity {{nowrap|1=4-vector}} with respect to tau . A equiv frac{dV}{d tau} = frac{d}{d tau} (gamma, gamma vec{v}) = gamma left( frac{d gamma}{dt}, frac{d(gamma vec{v})}{dt} right) Force 4-vector: This is the derivative of the momentum {{nowrap|1=4-vector}} with respect to tau . F equiv frac{dP}{d tau} = gamma left(frac{dE}{dt}, frac{d vec{p}}{dt} right) = gamma left( frac{dE}{dt},vec{f} right) As expected, the final components of the above {{nowrap|1=4-vectors}} are all standard {{nowrap|1=3-vectors}} corresponding to spatial {{nowrap|1=3-momentum}}, {{nowrap|1=3-force}} etc.{{rp|178–181}}{{rp|36–59}} 4-vectors and physical law The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving {{nowrap|1=4-vectors}} rather than give up on conservation of momentum.Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving {{nowrap|1=4-vectors}} require the use of tensors with appropriate rank, which themselves can be thought of as being built up from {{nowrap|1=4-vectors}}.{{rp|186}}{{anchor|Acceleration}} Acceleration {{Further|Acceleration (special relativity)}}It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is only when gravitation is significant that general relativity is required.WEB, Gibbs, Philip, Can Special Relativity Handle Acceleration?,weblink The Physics and Relativity FAQ, math.ucr.edu, 28 May 2017, 7 June 2017,weblink" title="web.archive.org/web/20170607102331weblink">weblink live, Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating, or rotating. To accommodate this difference, general relativity uses curved spacetime.In this section, we analyze several scenarios involving accelerated reference frames.{{anchor|Dewan–Beran–Bell spaceship paradox}} Dewan–Beran–Bell spaceship paradox The Dewan–Beran–Bell spaceship paradox (Bell's spaceship paradox) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.(File:Bell's spaceship paradox - two spaceships - initial setup.png|thumb|Figure 4-4. Dewan–Beran–Bell spaceship paradox)In Fig. 4-4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string which is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration.In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Will the string break?When the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.{{rp|106,120–122}} To observers in the rest frame, the spaceships start a distance L apart and remain the same distance apart during acceleration. During acceleration, L is a length contracted distance of the distance {{nowrap|1=L{{'}} = γL}} in the frame of the accelerating spaceships. After a sufficiently long time, γ will increase to a sufficiently large factor that the string must break. Let A and B be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. A says that B has the same acceleration that he has, and B sees that A matches her every move. So the spaceships stay the same distance apart, and the string does not break.{{rp|106,120–122}} The problem with the first argument is that there is no "frame of the spaceships." There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.{{rp|106,120–122}}(File:Bell spaceship paradox.svg|thumb|Figure 4–5. The curved lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dashed lines are lines of simultaneity for either observer before acceleration begins and after acceleration stops.)A spacetime diagram (Fig. 4-5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude k acceleration for proper time sigma (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length along the line of simultaneity A'B'' turns out to be greater than the length along the line of simultaneity AB.The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. 4-5, the acceleration is finished, the ships will remain at a constant offset in some frame S'. If x_{A} and x_{B}=x_{A}+L are the ships' positions in S, the positions in frame S' are:JOURNAL, Franklin, Jerrold, Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity, European Journal of Physics, 31, 2010, 291–298, 10.1088/0143-0807/31/2/006, 2010EJPh...31..291F, 2, 0906.1919, 18059490, begin{align} x'_{A}& = gammaleft(x_{A}-vtright)x'_{B}& = gammaleft(x_{A}+L-vtright)L'& = x'_{B}-x'_{A} =gamma Lend{align}The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame S. As shown in Fig. 4-5, Bell's example asserts the moving lengths AB and A'B' measured in frame S to be fixed, thereby forcing the rest frame length A'B'' in frame S' to increase.{{anchor|Accelerated observer with horizon}} Accelerated observer with horizon Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as event horizons. In the text accompanying Fig. 2-7, the magenta hyperbolae represented actual paths that are tracked by a constantly accelerating traveler in spacetime. During periods of positive acceleration, the traveler's velocity just approaches the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.File:Accelerated relativistic observer with horizon.png|thumb|Figure 4–6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed (:File:ConstantAcceleration02.jpg|here). ]]Fig. 4-6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity parameter beta approaches a limit of one as ct increases. Likewise, gamma approaches infinity.The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows: We remember that beta = ct/x. Since c^2 t^2 - x^2 = s^2, we conclude that beta (ct) = ct/ sqrt{c^2 t^2 - s^2}. gamma = 1/sqrt{1 - beta ^2} = sqrt{c^2 t^2 - s^2}/s From the relativistic force law, F = dp/dt = dpc/d(ct) = d(beta gamma m c^2)/d(ct). Substituting beta(ct) from step 2 and the expression for gamma from step 3 yields F = mc^2 / s , which is a constant expression.{{rp|110–113}} Fig. 4-6 illustrates a specific calculated scenario. Terence (A) and Stella (B) initially stand together 100 light hours from the origin. Stella lifts off at time 0, her spacecraft accelerating at 0.01 c per hour. Every twenty hours, Terence radios updates to Stella about the situation at home (solid green lines). Stella receives these regular transmissions, but the increasing distance (offset in part by time dilation) causes her to receive Terence's communications later and later as measured on her clock, and she never receives any communications from Terence after 100 hours on his clock (dashed green lines).{{rp|110–113}}After 100 hours according to Terence's clock, Stella enters a dark region. She has traveled outside Terence's timelike future. On the other hand, Terence can continue to receive Stella's messages to him indefinitely. He just has to wait long enough. Spacetime has been divided into distinct regions separated by an apparent event horizon. So long as Stella continues to accelerate, she can never know what takes place behind this horizon.{{rp|110–113}} Introduction to curved spacetime {{anchor|Basic propositions}} Basic propositions Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean reference frame that extends throughout all space and all time. Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space.Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley: "That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it." In contrast, Einstein denied that there is any background Euclidean reference frame that extends throughout space. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself.{{rp|175–190}}File:Principle of the tidal force.svg|thumb|Figure 5–1. Tidal effects.Different reporters viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first reporter, at the center of mass of particles {{nowrap|1=2 and 3}} but unaware of the large mass 1, concludes that a force of repulsion exists between the particles in scenario A while a force of attraction exists between the particles in scenario B. (ii) A second reporter, aware of the large mass 1, smiles at the first reporter's naiveté. This second reporter knows that in reality, the apparent forces between particles {{nowrap|1=2 and 3}} really represent tidal effects resulting from their differential attraction by mass 1. (iii) A third reporter, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along geodesicsgeodesicsIn spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun. Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a geodesic. No evidence of gravitation can be discovered following alongside the motions of a single particle.{{rp|175–190}}In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of two bodies or two separated particles. In Fig. 5-1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The tidal accelerations that these particles exhibit with respect to each other do not require forces for their explanation. Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime. These tidal accelerations are strictly local. It is the cumulative total effect of many local manifestations of curvature that result in the appearance of a gravitational force acting at a long range from Earth.{{rp|175–190}}Two central propositions underlie general relativity. The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion."BOOK, Lorentz, H. A., Einstein, A., Minkowski, H., Weyl, H., The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity,weblink registration, 1952, Dover Publications, 0-486-60081-5, {{rp|113}} This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.BOOK, Mook, Delo E., Vargish, Thoma s, Inside Relativity, 1987, Princeton University Press, Princeton, New Jersey, 0-691-08472-6,weblink {{rp|137–149}} (File:Elevator gravity.svg|thumb|Figure 5–2. Equivalence principle) The equivalence principle states that in any sufficiently small region of space, the effects of gravitation are the same as those from acceleration. In Fig. 5-2, person A is in a spaceship, far from any massive objects, that undergoes a uniform acceleration of g. Person B is in a box resting on Earth. Provided that the spaceship is sufficiently small so that tidal effects are non-measurable (given the sensitivity of current gravity measurement instrumentation, A and B presumably should be Lilliputians), there are no experiments that A and B can perform which will enable them to tell which setting they are in.{{rp|141–149}} An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation, {{nowrap|1=F = GMmg/r2 = }} mgg and in Newton's second law, {{nowrap|1=F = mia,}} there is no a priori reason why the gravitational mass mg should be equal to the inertial mass mi. The equivalence principle states that these two masses are identical.{{rp|141–149}} To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study. Without these mathematical tools, it is possible to write about general relativity, but it is not possible to demonstrate any non-trivial derivations.{{further|Introduction to general relativity|General relativity}}{{anchor|Curvature of time}} Curvature of time (File:Einstein's argument suggesting gravitational redshift.svg|thumb|Figure 5–3. Einstein's argument suggesting gravitational redshift)In the discussion of special relativity, forces played no more than a background role. Special relativity assumes the ability to define inertial frames that fill all of spacetime, all of whose clocks run at the same rate as the clock at the origin. Is this really possible? In a nonuniform gravitational field, experiment dictates that the answer is no. Gravitational fields make it impossible to construct a global inertial frame. In small enough regions of spacetime, local inertial frames are still possible. General relativity involves the systematic stitching together of these local frames into a more general picture of spacetime.{{rp|118–126}}Years before publication of the general theory in 1916, Einstein used the equivalence principle to predict the existence of gravitational redshift in the following thought experiment: (i) Assume that a tower of height h (Fig. 5-3) has been constructed. (ii) Drop a particle of rest mass m from the top of the tower. It falls freely with acceleration g, reaching the ground with velocity {{math|1=v = (2gh)1/2}}, so that its total energy E, as measured by an observer on the ground, is {{tmath|1=m + frac{frac 1 2 m v^2} {c^2} = m + frac{m g h} {c^2} }} (iii) A mass-energy converter transforms the total energy of the particle into a single high energy photon, which it directs upward. (iv) At the top of the tower, an energy-mass converter transforms the energy of the photon E{{'}} back into a particle of rest mass m{{'}}.{{rp|118–126}}It must be that {{math|1=m = m{{'}}}}, since otherwise one would be able to construct a perpetual motion device. We therefore predict that {{math|1=E{{'}} = m}}, so that frac{E'}{E} = frac{h nu , '}{ h nu} = frac{m}{m + frac{mgh}{c^2}} = 1 - frac{gh}{c^2} A photon climbing in Earth's gravitational field loses energy and is redshifted. Early attempts to measure this redshift through astronomical observations were somewhat inconclusive, but definitive laboratory observations were performed by Pound & Rebka (1959) and later by Pound & Snider (1964).WEB, Mester, John, Experimental Tests of General Relativity,weblink Laboratoire Univers et Théories, 9 June 2017,weblink 18 March 2017, dead, Light has an associated frequency, and this frequency may be used to drive the workings of a clock. The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower. Suppose we build two identical clocks whose rates are controlled by some stable atomic transition. Place one clock on top of the tower, while the other clock remains on the ground. An experimenter on top of the tower observes that signals from the ground clock are lower in frequency than those of the clock next to her on the tower. Light going up the tower is just a wave, and it is impossible for wave crests to disappear on the way up. Exactly as many oscillations of light arrive at the top of the tower as were emitted at the bottom. The experimenter concludes that the ground clock is running slow, and can confirm this by bringing the tower clock down to compare side by side with the ground clock.{{rp|16–18}} For a 1 km tower, the discrepancy would amount to about 9.4 nanoseconds per day, easily measurable with modern instrumentation.Clocks in a gravitational field do not all run at the same rate. Experiments such as the Pound–Rebka experiment have firmly established curvature of the time component of spacetime. The Pound–Rebka experiment says nothing about curvature of the space component of spacetime. But the theoretical arguments predicting gravitational time dilation do not depend on the details of general relativity at all. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence.{{rp|16}} This includes Newtonian gravitation. A standard demonstration in general relativity is to show how, in the "Newtonian limit" (i.e. the particles are moving slowly, the gravitational field is weak, and the field is static), curvature of time alone is sufficient to derive Newton's law of gravity.ARXIV, Carroll, Sean M., Lecture Notes on General Relativity, 2 December 1997, gr-qc/9712019, {{rp|101–106}}Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time and curved space. Given G as the gravitational constant, M as the mass of a Newtonian star, and orbiting bodies of insignificant mass at distance r from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable:{{rp|229–232}} Delta s^2 = left( 1 - frac{2GM}{c^2 r} right) (c Delta t)^2 - , (Delta x)^2 - (Delta y)^2 - (Delta z)^2 {{anchor|Curvature of space}} Curvature of space The (1 - 2GM/(c^2 r) ) coefficient in front of (c Delta t)^2 describes the curvature of time in Newtonian gravitation, and this curvature completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to G and M, and because of the r in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is curved.But general relativity is a theory of curved space and curved time, so if there are terms modifying the spatial components of the spacetime interval presented above, shouldn't their effects be seen on, say, planetary and satellite orbits due to curvature correction factors applied to the spatial terms?The answer is that they are seen, but the effects are tiny. The reason is that planetary velocities are extremely small compared to the speed of light, so that for planets and satellites of the solar system, the (c Delta t)^2 term dwarfs the spatial terms.{{rp|234–238}}Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In 1859, Urbain Le Verrier, in an analysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848, reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury. The perihelion of Mercury's orbit exhibited an excess rate of precession over that which could be explained by the tugs of the other planets.JOURNAL, Le Verrier, Urbain, Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 1859, 49, 379–383,weblink The ability to detect and accurately measure the minute value of this anomalous precession (only 43 arc seconds per tropical century) is testimony to the sophistication of 19th century astrometry.File:General relativity time and space distortion frame 1.png|thumb|Figure 5–4. General relativity is a theory of curved time and curved space. (:File:General relativity time and space distortion extract.gif|Click here to animate.) ]]As the famous astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing wobbles in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed.WEB, Worrall, Simon, The Hunt for Vulcan, the Planet That Wasn't There,weblink National Geographic, 4 November 2015,weblink" title="web.archive.org/web/20170524004444weblink">weblink dead, 24 May 2017, In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct.The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of Â±1 on a spacetime diagram. Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components.{{rp|234–238}} Delta s^2 = left( 1 - frac{2GM}{c^2 r} right) (c Delta t)^2- , left( 1 + frac{2GM}{c^2 r} right) left[ (Delta x)^2 + (Delta y)^2 + (Delta z)^2 right] In Newton's gravitation, the (1 - 2GM/(c^2 r) ) coefficient in front of (c Delta t)^2 predicts bending of light around a star. In general relativity, the (1 + 2GM/(c^2 r) ) coefficient in front of left[ (Delta x)^2 + (Delta y)^2 + (Delta z)^2 right] predicts a doubling of the total bending.{{rp|234–238}}The story of the 1919 Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere.WEB, Levine, Alaina G., May 29, 1919: Eddington Observes Solar Eclipse to Test General Relativity,weblink APS News, This Month in Physics History, American Physical Society, May 2016, 25, 5,weblink" title="web.archive.org/web/20170602134913weblink">weblink live, 2 June 2017, {{anchor|Sources of spacetime curvature}} Sources of spacetime curvature (File:StressEnergyTensor contravariant.svg|thumb|250px|Figure 5-5. Contravariant components of the stress–energy tensor)In Newton's theory of gravitation, the only source of gravitational force is mass.In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the Einstein field equations,the sources of gravity are presented on the right-hand side in T_{mu nu}, the stress–energy tensor.Fig. 5-5 classifies the various sources of gravity in the stress–energy tensor: T^{00} (red): The total mass–energy density, including any contributions to the potential energy from forces between the particles, as well as kinetic energy from random thermal motions. T^{0i} and T^{i0} (orange): These are momentum density terms. Even if there is no bulk motion, energy may be transmitted by heat conduction, and the conducted energy will carry momentum. T^{ij} are the rates of flow of the {{nowrap|1=i-component}} of momentum per unit area in the {{nowrap|1=j-direction}}. Even if there is no bulk motion, random thermal motions of the particles will give rise to momentum flow, so the {{math|1=i = j}} terms (green) represent isotropic pressure, and the {{math|1=i ≠ j}} terms (blue) represent shear stresses.BOOK, Hobson, M. P., Efstathiou, G., Lasenby, A. N., General Relativity, 2006, Cambridge University Press, Cambridge, 978-0-521-82951-9, 176–179, One important conclusion to be derived from the equations is that, colloquially speaking, gravity itself creates gravity.{{refn|group=note|More precisely, the gravitational field couples to itself. In Newtonian gravity, the potential due to two point masses is simply the sum of the potentials of the two masses, but this does not apply to GR. This can be thought of as the result of the equivalence principle: If gravitation did not couple to itself, two particles bound by their mutual gravitational attraction would not have the same inertial mass (due to negative binding energy) as their gravitational mass.{{rp|112–113}} }} Energy has mass. Even in Newtonian gravity, the gravitational field is associated with an energy, {{tmath|1=E = mgh,}} called the gravitational potential energy. In general relativity, the energy of the gravitational field feeds back into creation of the gravitational field. This makes the equations nonlinear and hard to solve in anything other than weak field cases.{{rp|240}} Numerical relativity is a branch of general relativity using numerical methods to solve and analyze problems, often employing supercomputers to study black holes, gravitational waves, neutron stars and other phenomena in the strong field regime. Energy-momentum {{multiple image | total_width = 330 | image1 = Gravitation space source.svg | width1 = 869 | height1 = 580 | image2 = Gravitomagnetic field due to angular momentum.svg | width2 = 261 | height2 = 200 | footer_background = | footer_align = | footer = Figure 5-6. (left) Mass-energy warps spacetime. (right) Rotating mass–energy distributions with angular momentum J generate gravitomagnetic fields H.}} In special relativity, mass-energy is closely connected to momentum. Just as space and time are different aspects of a more comprehensive entity called spacetime, mass–energy and momentum are merely different aspects of a unified, four-dimensional quantity called four-momentum. In consequence, if mass–energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as gravitomagnetism.BOOK, Thorne, Kip S., Fairbank, J. D., Deaver, B. S. Jr., Everitt, W. F., Michelson, P. F., Near zero: New Frontiers of Physics, 1988, W. H. Freeman and Company, 573–586, 12925169,weblinkweblink 28 July 2017, dead, (File:Special relativistic explanation of gravitomagnetism.svg|250px|thumb|Figure 5–7. Origin of gravitomagnetism)It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II, {{nowrap|1=chapter 13–6}} of his Lectures on Physics, available online.)BOOK, Feynman, R. P., Leighton, R. B., Sands, M., The Feynman Lectures on Physics, vol. 2, 1964, Basic Books, 978-0-465-02416-2, 13–6 to 13–11, New Millenium,weblink 1 July 2017, 17 January 2023,weblink live, Analogous logic can be used to demonstrate the origin of gravitomagnetism.{{rp|245–253}}In Fig. 5-7a, two parallel, infinitely long streams of massive particles have equal and opposite velocities −v and +v relative to a test particle at rest and centered between the two. Because of the symmetry of the setup, the net force on the central particle is zero. Assume {{tmath|1=v ll c}} so that velocities are simply additive. Fig. 5-7b shows exactly the same setup, but in the frame of the upper stream. The test particle has a velocity of +v, and the bottom stream has a velocity of +2v. Since the physical situation has not changed, only the frame in which things are observed, the test particle should not be attracted towards either stream.{{rp|245–253}}It is not at all clear that the forces exerted on the test particle are equal. (1) Since the bottom stream is moving faster than the top, each particle in the bottom stream has a larger mass energy than a particle in the top. (2) Because of Lorentz contraction, there are more particles per unit length in the bottom stream than in the top stream. (3) Another contribution to the active gravitational mass of the bottom stream comes from an additional pressure term which, at this point, we do not have sufficient background to discuss. All of these effects together would seemingly demand that the test particle be drawn towards the bottom stream.{{rp|245–253}}The test particle is not drawn to the bottom stream because of a velocity-dependent force that serves to repel a particle that is moving in the same direction as the bottom stream. This velocity-dependent gravitational effect is gravitomagnetism.{{rp|245–253}}Matter in motion through a gravitomagnetic field is hence subject to so-called frame-dragging effects analogous to electromagnetic induction. It has been proposed that such gravitomagnetic forces underlie the generation of the relativistic jets (Fig. 5-8) ejected by some rotating supermassive black holes.JOURNAL, Williams, R. K., 1995, Extracting X rays, ÎŽ rays, and relativistic e−–e+ pairs from supermassive Kerr black holes using the Penrose mechanism, Physical Review D, 51, 10, 5387–5427, 10.1103/PhysRevD.51.5387, 1995PhRvD..51.5387W, 10018300, JOURNAL, Williams, R. K., 2004, Collimated escaping vortical polar e−–e+ jets intrinsically produced by rotating black holes and Penrose processes, The Astrophysical Journal, 611, 2, 952–963, 10.1086/422304, 2004ApJ...611..952W, astro-ph/0404135, 1350543, Pressure and stress Quantities that are directly related to energy and momentum should be sources of gravity as well, namely internal pressure and stress. Taken together, {{nowrap|1=mass-energy}}, momentum, pressure and stress all serve as sources of gravity: Collectively, they are what tells spacetime how to curve.General relativity predicts that pressure acts as a gravitational source with exactly the same strength as mass–energy density. The inclusion of pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation. For example, the pressure term sets a maximum limit to the mass of a neutron star. The more massive a neutron star, the more pressure is required to support its weight against gravity. The increased pressure, however, adds to the gravity acting on the star's mass. Above a certain mass determined by the Tolman–Oppenheimer–Volkoff limit, the process becomes runaway and the neutron star collapses to a black hole.{{rp|243,280}}The stress terms become highly significant when performing calculations such as hydrodynamic simulations of core-collapse supernovae.JOURNAL, Kuroda, Takami, Kotake, Kei, Takiwaki, Tomoya, Fully General Relativistic Simulations of Core-Collapse Supernovae with An Approximate Neutrino Transport, The Astrophysical Journal, 755, 1, 11, 1202.2487, 2012, 10.1088/0004-637X/755/1/11, 2012ApJ...755...11K, 119179339, These predictions for the roles of pressure, momentum and stress as sources of spacetime curvature are elegant and play an important role in theory. In regards to pressure, the early universe was radiation dominated,WEB, Wollack, Edward J., Cosmology: The Study of the Universe, Universe 101: Big Bang Theory, NASA, 10 December 2010,weblink 15 April 2017,weblink" title="web.archive.org/web/20110514230003weblink">weblink 14 May 2011, dead, and it is highly unlikely that any of the relevant cosmological data (e.g. nucleosynthesis abundances, etc.) could be reproduced if pressure did not contribute to gravity, or if it did not have the same strength as a source of gravity as mass–energy. Likewise, the mathematical consistency of the Einstein field equations would be broken if the stress terms did not contribute as a source of gravity. Experimental test of the sources of spacetime curvature Definitions: Active, passive, and inertial mass Bondi distinguishes between different possible types of mass: (1) {{nowrap|1=active mass (m_a)}} is the mass which acts as the source of a gravitational field; (2){{nowrap|1=passive mass (m_p)}} is the mass which reacts to a gravitational field; (3) {{nowrap|1=inertial mass (m_i)}} is the mass which reacts to acceleration. m_p is the same as {{nowrap|1=gravitational mass (m_g)}} in the discussion of the equivalence principle. In Newtonian theory, The third law of action and reaction dictates that m_a and m_p must be the same. On the other hand, whether m_p and m_i are equal is an empirical result. In general relativity, The equality of m_p and m_i is dictated by the equivalence principle. There is no "action and reaction" principle dictating any necessary relationship between m_a and m_p.BOOK, Bondi, Hermann,weblink The Role of Gravitation in Physics: Report from the 1957 Chapel Hill Conference, 1957, Max Planck Research Library, 978-3-86931-963-6, DeWitt, Cecile M., Berlin, Germany, 159–162, 1 July 2017, Rickles, Dean,weblink" title="web.archive.org/web/20170728121700weblink">weblink 28 July 2017, live, Pressure as a gravitational source (File:Cavendish and Kreuzer Torsion Balance Diagrams.svg|thumb|330px|Figure 5–9. (A) Cavendish experiment, (B) Kreuzer experiment)The classic experiment to measure the strength of a gravitational source (i.e. its active mass) was first conducted in 1797 by Henry Cavendish (Fig. 5-9a). Two small but dense balls are suspended on a fine wire, making a torsion balance. Bringing two large test masses close to the balls introduces a detectable torque. Given the dimensions of the apparatus and the measurable spring constant of the torsion wire, the gravitational constant G can be determined.To study pressure effects by compressing the test masses is hopeless, because attainable laboratory pressures are insignificant in comparison with the {{nowrap|1=mass-energy}} of a metal ball.However, the repulsive electromagnetic pressures resulting from protons being tightly squeezed inside atomic nuclei are typically on the order of 1028 atm ≈ 1033 Pa ≈ 1033 kg·s−2m−1. This amounts to about 1% of the nuclear mass density of approximately 1018kg/m3 (after factoring in c2 ≈ 9×1016m2s−2).BOOK, Crowell, Benjamin, General Relativity, 2000, Light and Matter, Fullerton, CA, 241–258,weblink 30 June 2017, 18 June 2017,weblink" title="web.archive.org/web/20170618200413weblink">weblink live, {{multiple image | total_width = 330 | image1 = Apollo 11 Lunar Laser Ranging Experiment.jpg | width1 = 2349 | height1 = 2373 | image2 = LunarLaserMLRS.gif | width2 = 473 | height2 = 729 | footer_background = | footer_align = | footer = Figure 5-10. Lunar laser ranging experiment. (left) This retroreflector was left on the Moon by astronauts on the Apollo 11 mission. (right) Astronomers all over the world have bounced laser light off the retroreflectors left by Apollo astronauts and Russian lunar rovers to measure precisely the Earth-Moon distance. }}If pressure does not act as a gravitational source, then the ratio m_a/m_p should be lower for nuclei with higher atomic number Z, in which the electrostatic pressures are higher. {{nowrap|1=L. B. Kreuzer}} (1968) did a Cavendish experiment using a Teflon mass suspended in a mixture of the liquids trichloroethylene and dibromoethane having the same buoyant density as the Teflon (Fig. 5-9b). Fluorine has atomic number {{math|1=Z = 9}}, while bromine has {{math|1=Z = 35}}. Kreuzer found that repositioning the Teflon mass caused no differential deflection of the torsion bar, hence establishing active mass and passive mass to be equivalent to a precision of 5×10−5.JOURNAL, Kreuzer, L. B., Experimental measurement of the equivalence of active and passive gravitational mass, Physical Review, 1968, 169, 5, 1007–1011, 1968PhRv..169.1007K, 10.1103/PhysRev.169.1007, Although Kreuzer originally considered this experiment merely to be a test of the ratio of active mass to passive mass, Clifford Will (1976) reinterpreted the experiment as a fundamental test of the coupling of sources to gravitational fields.JOURNAL, Will, C. M., Active mass in relativistic gravity-Theoretical interpretation of the Kreuzer experiment, The Astrophysical Journal, 1976, 204, 224–234,weblink 1976ApJ...204..224W, 10.1086/154164, 2 July 2017, 28 September 2018,weblink" title="web.archive.org/web/20180928123722weblink">weblink live, free, In 1986, Bartlett and Van Buren noted that lunar laser ranging had detected a 2 km offset between the moon's center of figure and its center of mass. This indicates an asymmetry in the distribution of Fe (abundant in the Moon's core) and Al (abundant in its crust and mantle). If pressure did not contribute equally to spacetime curvature as does mass–energy, the moon would not be in the orbit predicted by classical mechanics. They used their measurements to tighten the limits on any discrepancies between active and passive mass to about 10−12.JOURNAL, Bartlett, D. F., Van Buren, Dave, 1986, Equivalence of active and passive gravitational mass using the moon, Physical Review Letters, 57, 1, 21–24, 1986PhRvL..57...21B, 10.1103/PhysRevLett.57.21, 10033347, With decades of additional lunar laser ranging data, Singh et al. (2023) reported improvement on these limits by a factor of about 100.JOURNAL, Singh, Vishwa Vijay, Müller, Jürgen, Biskupek, Liliane, Hackmann, Eva, Lämmerzahl, Claus, Equivalence of Active and Passive Gravitational Mass Tested with Lunar Laser Ranging, Physical Review Letters, 2023, 131, 2, 021401, 10.1103/PhysRevLett.131.021401, 37505941, 7 March 2024,weblink 2212.09407, 2023PhRvL.131b1401S, Gravitomagnetism (File:Gravity Probe B Confirms the Existence of Gravitomagnetism.jpg|330px|thumb|Figure 5–11. Gravity Probe B confirmed the existence of gravitomagnetism)The existence of gravitomagnetism was proven by Gravity Probe B {{nowrap|1=(GP-B)}}, a satellite-based mission which launched on 20 April 2004.WEB ,weblink , Gravity Probe B: FAQ , 2 July 2017 , 2 June 2018 ,weblink" title="web.archive.org/web/20180602231753weblink">weblink , live , The spaceflight phase lasted until 2005. The mission aim was to measure spacetime curvature near Earth, with particular emphasis on gravitomagnetism. Initial results confirmed the relatively large geodetic effect (which is due to simple spacetime curvature, and is also known as de Sitter precession) to an accuracy of about 1%. The much smaller frame-dragging effect (which is due to gravitomagnetism, and is also known as Lense–Thirring precession) was difficult to measure because of unexpected charge effects causing variable drift in the gyroscopes. Nevertheless, by August 2008, the frame-dragging effect had been confirmed to within 15% of the expected result,NEWS , Gugliotta , G. , Perseverance Is Paying Off for a Test of Relativity in Space , New York Times , 16 February 2009 ,weblink , 2 July 2017 , 3 September 2018 ,weblink , live , while the geodetic effect was confirmed to better than 0.5%.WEB, Everitt, C. W. F., Parkinson, B. W., 2009, Gravity Probe B Science Results—NASA Final Report,weblink live,weblink" title="web.archive.org/web/20121023062122weblink">weblink 23 October 2012, 2 July 2017, JOURNAL, Everitt, etal, 2011, Gravity Probe B: Final Results of a Space Experiment to Test General Relativity, Physical Review Letters, 106, 22, 221101, 10.1103/PhysRevLett.106.221101, 1105.3456, 2011PhRvL.106v1101E, 21702590, 11878715, Subsequent measurements of frame dragging by laser-ranging observations of the LARES, {{nowrap|1=LAGEOS-1}} and {{nowrap|1=LAGEOS-2}} satellites has improved on the {{nowrap|1=GP-B}} measurement, with results (as of 2016) demonstrating the effect to within 5% of its theoretical value,JOURNAL, Ciufolini, Ignazio, Paolozzi, Antonio Rolf Koenig, Pavlis, Erricos C., Koenig, Rolf, 2016, A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model, European Physical Journal C, 76, 3, 120, 1603.09674, 2016EPJC...76..120C, 10.1140/epjc/s10052-016-3961-8, 4946852, 27471430, although there has been some disagreement on the accuracy of this result.JOURNAL, Iorio, L., A comment on "A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model. Measurement of Earth's dragging of inertial frames," by I. Ciufolini et al, The European Physical Journal C, February 2017, 77, 2, 73, 10.1140/epjc/s10052-017-4607-1, 2017EPJC...77...73I, 1701.06474, 118945777, Another effort, the Gyroscopes in General Relativity (GINGER) experiment, seeks to use three 6 m ring lasers mounted at right angles to each other 1400 m below the Earth's surface to measure this effect.WEB, Cartlidge, Edwin, Underground ring lasers will put general relativity to the test,weblink physicsworld.com, 20 January 2016, Institute of Physics, 2 July 2017, 12 July 2017,weblink" title="web.archive.org/web/20170712144159weblink">weblink live, WEB, Einstein right using the most sensitive Earth rotation sensors ever made,weblink Phys.org, Science X network, 2 July 2017, 10 May 2017,weblink live, Technical topics {{anchor|Technical topics}}{{anchor|Is spacetime really curved}} Is spacetime really curved? In Poincaré's conventionalist views, the essential criteria according to which one should select a Euclidean versus non-Euclidean geometry would be economy and simplicity. A realist would say that Einstein discovered spacetime to be non-Euclidean. A conventionalist would say that Einstein merely found it more convenient to use non-Euclidean geometry. The conventionalist would maintain that Einstein's analysis said nothing about what the geometry of spacetime really is.WEB, Murzi, Mauro, Jules Henri Poincaré (1854–1912),weblink Internet Encyclopedia of Philosophy (ISSN 2161-0002), 9 April 2018, 23 December 2020,weblink" title="web.archive.org/web/20201223123326weblink">weblink live, Such being said, 1. Is it possible to represent general relativity in terms of flat spacetime? 2. Are there any situations where a flat spacetime interpretation of general relativity may be more convenient than the usual curved spacetime interpretation? In response to the first question, a number of authors including Deser, Grishchuk, Rosen, Weinberg, etc. have provided various formulations of gravitation as a field in a flat manifold. Those theories are variously called "bimetric gravity", the "field-theoretical approach to general relativity", and so forth.JOURNAL, Deser, S., Self-Interaction and Gauge Invariance, General Relativity and Gravitation, 1970, 1, 18, 9–8, gr-qc/0411023, 1970GReGr...1....9D, 10.1007/BF00759198, 14295121, JOURNAL, Grishchuk, L. P., Petrov, A. N., Popova, A. D., Exact Theory of the (Einstein) Gravitational Field in an Arbitrary Background Space–Time, Communications in Mathematical Physics, 1984, 94, 3, 379–396,weblink 9 April 2018, 1984CMaPh..94..379G, 10.1007/BF01224832, 120021772, 25 February 2021,weblink live, JOURNAL, Rosen, N., General Relativity and Flat Space I, Physical Review, 1940, 57, 2, 147–150, 10.1103/PhysRev.57.147, 1940PhRv...57..147R, JOURNAL, Weinberg, S., Derivation of Gauge Invariance and the Equivalence Principle from Lorentz Invariance of the S-Matrix, Physics Letters, 1964, 9, 4, 357–359, 10.1016/0031-9163(64)90396-8, 1964PhL.....9..357W, Kip Thorne has provided a popular review of these theories.BOOK, Thorne, Kip, Black Holes & Time Warps: Einstein's Outrageous Legacy, 1995, W. W. Norton & Company, 978-0-393-31276-8, {{rp|397–403}}The flat spacetime paradigm posits that matter creates a gravitational field that causes rulers to shrink when they are turned from circumferential orientation to radial, and that causes the ticking rates of clocks to dilate. The flat spacetime paradigm is fully equivalent to the curved spacetime paradigm in that they both represent the same physical phenomena. However, their mathematical formulations are entirely different. Working physicists routinely switch between using curved and flat spacetime techniques depending on the requirements of the problem. The flat spacetime paradigm is convenient when performing approximate calculations in weak fields. Hence, flat spacetime techniques tend be used when solving gravitational wave problems, while curved spacetime techniques tend be used in the analysis of black holes.{{rp|397–403}} Asymptotic symmetries The spacetime symmetry group for Special Relativity is the Poincaré group, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, viz., the Poincaré group.In 1962 Hermann Bondi, M. G. van der Burg, A. W. MetznerJOURNAL, Bondi, H., Van der Burg, M. G. J., Metzner, A., 1962, Gravitational waves in general relativity: VII. Waves from axisymmetric isolated systems, Proceedings of the Royal Society of London A, A269, 1336, 21–52, 1962RSPSA.269...21B, 10.1098/rspa.1962.0161, 120125096, and Rainer K. SachsJOURNAL, Sachs, Rainer K., 1962, Asymptotic symmetries in gravitational theory, Physical Review, 128, 6, 2851–2864, 1962PhRv..128.2851S, 10.1103/PhysRev.128.2851, addressed this asymptotic symmetry problem in order to investigate the flow of energy at infinity due to propagating gravitational waves. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at lightlike infinity to characterize what it means to say a metric is asymptotically flat, making no a priori assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields.ARXIV, Lectures on the Infrared Structure of Gravity and Gauge Theory, 1703.05448, 2017, Strominger, Andrew, hep-th, ...redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. To be published Princeton University Press, 158 pages., {{rp|35}}What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as supertranslations. This implies the conclusion that General Relativity (GR) does not reduce to special relativity in the case of weak fields at long distances.{{rp|35}}{{anchor|Riemannian geometry}} Riemannian geometry {{Excerpt|Riemannian geometry|template=-General geometry}}{{anchor|Curved manifolds}} Curved manifolds For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold (M, g). This means the smooth Lorentz metric g has signature (3,1). The metric determines the {{vanchor|geometry of spacetime|SPACETIME_GEOMETRY}}, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates (x, y, z, t) are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light c is equal to 1.BOOK, Bär, Christian, Fredenhagen, Klaus, Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations, 2009, Springer, Dordrecht, 978-3-642-02779-6, 39–58,weblink 14 April 2017,weblink 13 April 2017, Lorentzian Manifolds, dead, A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event p. Another reference frame may be identified by a second coordinate chart about p. Two observers (one in each reference frame) may describe the same event p but obtain different descriptions.Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing p (representing an observer) and another containing q (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event p). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples (x, y, z, t) (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces tensors into relativity, by which all physical quantities are represented.Geodesics are said to be timelike, null, or spacelike if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by timelike and null (lightlike) geodesics, respectively.{{anchor|Privileged character of 3+1 spacetime}} Privileged character of 3+1 spacetime {{Excerpt|Anthropic principle|Dimensions of spacetime}} See also {{cols}} Basic introduction to the mathematics of curved spacetime Complex spacetime Einstein's thought experiments Four-dimensionalism Geography Global spacetime structure List of spacetimes Metric space Philosophy of space and time Present Time geography {{colend}} Notes {{reflist|group=note|30em}} Additional details {{reflist|group="Click here for additional details"|30em}} References {{Reflist}} Further reading {{BarrowTipler1986}} George F. Ellis and Ruth M. Williams (1992) Flat and curved space–times. Oxford University Press. {{isbn|0-19-851164-7}} Lorentz, H. A., Einstein, Albert, Minkowski, Hermann, and Weyl, Hermann (1952) The Principle of Relativity: A Collection of Original Memoirs. Dover. Lucas, John Randolph (1973) A Treatise on Time and Space. London: Methuen. BOOK, Penrose, Roger, Roger Penrose, The Road to Reality, Oxford University Press, 2004, Oxford, 0-679-45443-8, The Road to Reality, Chapters 17–18. BOOK, Taylor, E. F., Wheeler, John A., John A. Wheeler, Spacetime Physics, Second Edition, W. H. Freeman, 1992, Internet Archive,weblink 0-7167-2327-1, SPEECH, The Doom of Spacetime: Why It Must Dissolve Into More Fundamental Structures, Nima, Arkani-Hamed, Nima Arkani-Hamed, The 2,384th Meeting Of The Society, Washington, D.C., December 1, 2017,weblink July 16, 2022, External links {{Commons category-inline}} Albert Einstein on space–time 13th edition Encyclopædia Britannica Historical: Albert Einstein's 1926 article Encyclopedia of Space–time and gravitation Scholarpedia Expert articles Stanford Encyclopedia of Philosophy: "Space and Time: Inertial Frames" by Robert DiSalle. {{Dimension topics}}{{Time Topics}}{{Time measurement and standards}}{{Relativity}}{{Authority control}}{{Use dmy dates|date=April 2019}}