title: Assassin's Creed Origins

text: Bayek tracks Flavius to Cyrene, where he has used the activated Apple of Eden to enthrall the population. He kills Flavius, avenging Khemu's death and returns to Aya. Aya recruits Marcus Junius Brutus the Younger and Gaius Cassius Longinus to their cause and heads to Rome to assassinate Caesar and Septimius. Bayek and Aya part ways but form the Hidden Ones, the foundation of the Assassin Brotherhood, swearing to protect the world from the shadows. In Rome, Aya confronts and kills Septimius, who wields the Staff of Eden. She infiltrates the Roman Senate and assassinates Caesar. Later, she meets Cleopatra and warns her to be a fair ruler, or she will return to assassinate her. Afterward, Bayek and Aya, now calling herself Amunet, begin recruiting and training other Hidden Ones as they build the Brotherhood in Egypt and Rome, respectively.


title: Assassin's Creed Origins

text: In 2017, Layla Hassan, a researcher for Abstergo Industries, is tasked with retrieving an artifact in Egypt on their behalf. Instead, she finds a tomb containing the mummified bodies of Bayek and Aya. Hoping to find information that would secure her a position in the company's Animus Project, Layla covertly relives both Bayek and Aya's memories using a modified Animus. When Layla fails to report in, Abstergo sends a team to kill her, but their plan fails. Layla returns to the Animus but is later awoken by William Miles, an Assassin Mentor and the father of the deceased Desmond Miles. She accepts his offer to work with the Assassins, but does not join them. The two depart for modern-day Alexandria. In experiencing Bayek's memories, Layla may encounter a series of ancient structures built by the First Civilization. Each contains a message that alludes to Layla playing a pivotal role in an upcoming apocalyptic event.


title: Assassin's Creed Origins

text: Ubisoft Montreal led the game's development, with assistance provided by other Ubisoft studios in different parts of the world. The team in Montreal previously worked on "Assassin's Creed Revelations" and "". Development began after "Black Flag"s completion and lasted approximately four years. Ubisoft Sofia and Ubisoft Singapore played prominent roles in designing the game's map and quests. Sofia worked on the game's tomb and pyramids, while Singapore crafted the naval combat. Jean Guesdon and Ashraf Ismail, who both worked on "Black Flag", returned as the game's directors. Nearly 1,000 people worked on the game; only 300 came from the Montreal studio. Ubisoft described this model of development as "co-development" where, unlike previous installments in the series, the support studios had more freedom and their work was more integral to the whole. According to Ismail, the co-development model allows each area in the game's world to be unique. Despite this, the Montreal studio set some guidelines that all studios needed to follow. For instance, the distance that a quest requires the players to travel is limited. For the quests in a hub area, there can only be one "funny" mission and one eradication mission. Ubisoft's goal was to modernize the series.


title: Assassin's Creed Origins

text: Ancient Egypt was one of the most popular choices requested by fans of the series, but Ubisoft had declined the idea. Alex Hutchinson, the creative director of "Assassin's Creed III", called Ancient Egypt—along with the other two chosen settings, Feudal Japan and World War II—"the worst choices" for a setting. In a later interview, however, Ismail said Egypt was a setting the team wanted to explore and acknowledged fans' requests. He believed Ubisoft had not chosen Egypt for previous installments because of technological constraints. 49 BCE was chosen as the game's setting because it reflected an "impressive clash of civilizations". Egyptian culture was thriving but nearing its decline, while the Romans and the Greeks exerted a strong influence over Egypt, culminating in its annexation by the Roman Empire following the game's events. Ismail called this setting "epic" because it showed the "death of one world, [and] the birth of a new one". Initially, the team started with the world map from "Black Flag" and turned all its water bodies into landmasses. With such a large map, they needed to fill it with meaningful content, so they incorporated elements from role-playing video games and quests into the game. Puzzles, hunting, and military outposts were added to facilitate the player's exploration of the game's world. The game's combat was completely overhauled, as the developer wanted to give players more freedom of choice. For the first time, difficulty settings were introduced to the series to ensure the combat would be accessible.


title: Assassin's Creed Origins

text: The studio consulted Egyptologists and historians, and secured deals with universities to ensure that experts on the subject could provide the team with information and research. In addition to recruiting in-house historians, they also consulted academics like Jean-Claude Golvin to place landmarks in the game's world and recreate ancient paintings. The historians were involved in the game's creative process. For instance, Evelyne Ferron, one of the game's consultants, convinced Ubisoft to modify a scene featuring public mummification in a temple because the scene went against the "Egyptian mentality" at that time. Alan Gardiner's "" served as the foundation for the language spoken by the NPCs, though the team also drew from the works of James Allen and Raymond Faulkner. The goal was not to create a historically accurate version of Egypt but to make Egypt an authentic setting. The team watched films and TV series to see how ancient Egypt was portrayed in pop culture, and adjusted the game when they felt the focus on history had impeded gameplay and entertainment. Some cities featured in the game were larger than they were initially as the developer "wanted to create this sense of pharaonic scale in places like Alexandria and Memphis". Ismail added that if an event is well documented, the artists followed the historian's consensus. However, for parts that were not well-researched, they would design and recreate them on their own.


title: Assassin's Creed Origins

text: The team also endeavored to make the game world more immersive. To this end, the studio added sandstorms to the game, and players would experience a mirage if they remained in the heat too long. Cities have distinct designs which reflect their origins. While designing Alexandria, inspirations were taken from Pergamon, Turkey, whose design resembled that of ancient Alexandria. While most locations were handcrafted, artists and programmers made use of procedural generation to fill vast, open areas with rocks, grass and trees. Mini-map from the head-up display (HUD) were replaced with a compass, as they believed this made the HUD less obstructive and encouraged players to explore. To make the world more dynamic, the team invested a lot of time into designing the artificial intelligence of the non-playable characters, ensuring they would have a routine every day such as working during daytime and sleeping at night. Quest givers travel and do not stay in a fixed position, and players can choose the time of day to assassinate their targets. Factions also respond differently to players.


title: Assassin's Creed Origins

text: Alain Mercieca was invited to serve as the game's narrative director after a cinematic director saw one of his "punk plays" called "Squeegee Nights". The lead character, Bayek, was more experienced and mature than other protagonists in the series. Ismail described Bayek as a "reactive" protagonist, who could express various emotions. Bayek, as a Medjay from Siwa Oasis, a traditional village in Egypt, embodies an older way of life and Egyptian traditions. As Egypt is about to be annexed by the Roman Empire in the game, Bayek must find out "what he needs to do, what he needs to become". His personal quest would lead ultimately to the formation of the Assassin Brotherhood. Abubakar Salim, who was initially told that he was auditioning for an animated TV series that required motion capture, voiced Bayek. In 2020, a report from "Bloomberg" alleged that Ubisoft, in particular the marketing department and Chief Creative Officer Serge Hascoët, had tried to minimize the role of female assassins in a number of "Assassin's Creed" games. Initially, Bayek was expected to be injured or killed off early in the game, and the main playable character would become Aya. Actress Alix Wilton Regan voiced Aya.


title: Assassin's Creed Origins

text: Following the release of "Assassin's Creed Syndicate" (2015), which sold fewer copies than its predecessors, Ubisoft decided against releasing an "Assassin's Creed" game in 2016. Ubisoft allowed one year of additional development time so they could "evolve the game mechanics" and reposition the series as a "premier open-world franchise". Ubisoft CEO Yves Guillemot later added that this also gave the Montreal studio additional time to polish the game and learn from "Assassin's Creed Unity" (2014), whose troubled launch alienated the series' fans. Ubisoft had released new installments in the franchise every year since 2009's "Assassin's Creed II". "Assassin's Creed Origins" was first leaked in January 2016, with "Kotaku" sources suggesting the game was codenamed "Empire" and set in ancient Egypt. Ubisoft announced it officially at Microsoft's press conference during E3 2017. The game was released for Windows, PlayStation 4 and Xbox One worldwide on October 27, 2017. There were five special editions of the game available when it launched. The Stadia version released on December 15, 2020.


title: Assassin's Creed Origins

text: Ubisoft supported the game extensively following its official launch. Several gear packs were released for the game, and Ubisoft partnered with Square Enix, the developers of "Final Fantasy XV", to create crossover content. All season pass holders have access to two pieces of story-driven downloadable content (DLC) packs. The first pack, "The Hidden Ones", was released on January 23, 2018, and is set five years after the game's main narrative, focusing on Bayek and Amunet's efforts to expand the Hidden Ones' influence and liberate the Sinai Peninsula from Roman occupation. The second pack, "The Curse of the Pharaohs", was set to be released on March 6 but was delayed to March 13. This DLC focuses on Egyptian mythology, as Bayek travels to Thebes to investigate a curse that has apparently brought several ancient pharaohs back from the dead. Both packs raise the maximum players' level and introduce new outfits, mounts and gears.


title: Assassin's Creed Origins

text: Ubisoft also released several free updates for the game. In December 2017, they added a new mission named "Here Comes a New Challenger", introducing a new gladiatorial arena in Cyrene. The update also includes a new difficulty mode named Nightmare, and enemy scaling, where low-level enemies will have their level increased to match the player's level. Ubisoft also released a series of missions named "Trials of the Gods", which allows players to fight high-level bosses inspired by Egyptian deities. A New Game Plus mode was introduced in February 2018. An update in April introduced the Animus Control Panel for PC players, allowing them to adjust gameplay parameters such as increasing the movement speed, possessing infinite health, and increasing the number of tame animals.


title: Assassin's Creed Origins

text: Ubisoft released "Discovery Tour" in February 2018. It consists of 75 guided tours, each lasting from five to 25 minutes. They focus on the landmarks featured in the game and the social traditions and the way of life of ancient Egyptians. The tours feature "academic information curated by historians and Egyptologists" based on research from universities and institutions like the British Museum. "Discovery Tour" removes combat from the game entirely, allowing teachers to show this portion of the game to schoolchildren directly. The team received feedback from people in the education field and designed the tour and its control to be as accessible as possible, and allow players to explore at their own pace. In an article on history-themed video games, "Damals" magazine wrote the "Discovery Tour" mode presented a transformation from an entertainment product to an interactive learning aid. In its 2018 exhibition on the "Queens of Egypt", the Pointe-à-Callière Museum in Montreal, Canada, incorporated images and video sequences from "Discovery Tour". A standalone free version of "Discovery Tour" was released in May 2020. A similar "Discovery Tour" mode would later be released for "Assassin's Creed Odyssey" and "Assassin's Creed Valhalla".


title: Assassin's Creed Origins

text: In June 2022, Ubisoft released a patch for "Origins", which boosted the framerate to 60fps on Xbox Series X/S and PlayStation 5.


title: Assassin's Creed Origins

text: "Assassin's Creed Origins" received "generally favorable" reviews from critics, according to review aggregator Metacritic.


title: Assassin's Creed Origins

text: Louise Blain of "GamesRadar" praised the varied map design writing that each region felt unique. IGN's Alanah Pearce agreed, adding each city had its own unique culture and architecture. She added the game offered a "delightful sense of discovery" as it allowed players to come upon locations of interest organically. Chris Carter of "Destructoid" wrote that "at no point did "Origins" feel inauthentic", and applauded Egypt as a unique setting for a video game. Like Pearce, he explored the game's world more than previous "Assassin's Creed" games. "VentureBeat"s Stephanie Chan described Egypt as "a spectacle to behold", praising its varied map design and the puzzles inside tombs and temples. Alessandro Fillari of "GameSpot" also liked Egypt as a setting, calling it "vibrant and lush". He believed it had a "strong sense of life", though he noted some regions are too sparse. He praised the dense towns and cities in the world, and commended Ubisoft's attention to detail. Writing for "PC Gamer", Christopher Livingston noted the recommended level of an area made "Origins" similar to a massively multiplayer online game (MMO) and remarked it created "a feeling of artificial difficulty".


title: Assassin's Creed Origins

text: Blain praised the game's design and applauded the team for ditching the franchise's staple gameplay features such as paired animation combat, tailing missions, and an excessive amount of collectibles. Pearce liked the elimination of forced stealth missions and the implementation of a redesigned parkour system, which removed the frustration she had with earlier installments. She also applauded the new progression system, which gave players more choices than previous games, and the action-focused combat, which she described as "tense". Suriel Vasquez of "Game Informer" praised the game's openness, since it gave players freedom to approach objectives in their own way. He described the combat as being more active and involving. Chan described the combat as "flawed" but "generally solid", though she disliked segments featuring Aya and Layla as she felt they interrupted the gameplay. Christian Donlan, writing for "Eurogamer", believed that the combat was a significant improvement over its predecessors and felt it reflected the franchise's new direction. He believed that the combat was more engaging, commenting it required players to use strategy, especially when they were facing different enemy archetypes together. However, he felt that the combat and the gameplay were unoriginal, and added that "everywhere you look in "Origins" you'll find things that you have done in other games". Both Pearce and Fillari complained the lock-on system did not work properly. Fillari added it turned "battles that could be tactical and fierce into disorienting and clumsy encounters". Fillari also noted that the control for stealth was less responsive than the previous games and considered it one of "Origins" weakest aspects.


title: Assassin's Creed Origins

text: Blain liked the game's quests noting that starting one side quest might initiate a chain of additional narrative side missions that players can complete. Pearce called the story "delightfully mystical" and "elaborate", while Fillari applauded the narrative for successfully balancing "moments of heartbreak and earnestness". Pearce also praised the side-quests for being memorable and featuring "multi-faceted, interesting characters with believable motives". While Carter noted that the game had a rough start, the tale involving the cult and the Order of the Ancients slowly became more intriguing and interesting. Chan criticised some campaign missions, especially those involving assassinating members of the cult, for being anti-climactic. She praised the side-quests for reflecting the everyday life of Egypt, but she lamented they were repetitive. She compared completing locations of interest to running errands. Andy Kelly, also from "PC Gamer", added that the game was "guilty of a particularly egregious example of padding", and wrote that during his 28 hours play time to finish "Origins", at least eight hours were spent on completing side content "against [his] will". Livingston remarked the player was often forced to grind for experience points before they could complete the main campaign missions, which broke the flow of the main campaign. Bayek is generally well-liked by critics. Donlan called him "charming", while "Polygon"s Colin Campbell described him as "kind" and "devout".


title: Assassin's Creed Origins

text: Chris Naunton of Southampton University created "Playing in the Past", a T.tv series, using "Origins" to teach the history of ancient Egypt. He and other Egyptologists described the game's depiction of the era as "the best visualization of Ancient Egypt ... amazing".


title: Assassin's Creed Origins

text: In November 2017, Ubisoft announced sales of "Assassin's Creed Origins" during its first 10 days were double those of "Assassin's Creed Syndicate", while player engagement increased. 35% of the sales were digital download, compared to just 12% for "Syndicate". It was the best-selling retail game in the UK in its first week of release, beating competitors including "Super Mario Odyssey" and "", which were released on the same day as "Origins". It was the second best-selling retail game in the US in October 2017, behind "", and the third best-selling game in November, behind "" and "Star Wars Battlefront II". The game sold more than 10 million copies during life of eighth generation of video game consoles.


title: Assassin's Creed Origins

text: The game was followed by "Assassin's Creed Odyssey", which takes place in Ancient Greece during the Peloponnesian War. Developed by Ubisoft Quebec, the game was released on October 5, 2018 for Windows, PlayStation 4 and Xbox One.


title: Spacetime

text: In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur.


title: Spacetime

text: Until the 20th century, it was assumed that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. The physicist Albert Einstein helped develop the idea of spacetime as part of his theory of relativity. Prior to his pioneering work, scientists had two separate theories to explain physical phenomena: Isaac Newton's laws of physics described the motion of massive objects, while James Clerk Maxwell's electromagnetic models explained the properties of light. However, in 1905, Einstein based on two postulates:


title: Spacetime

text: The logical consequence of taking these postulates together is the inseparable joining of the four dimensions—hitherto assumed as independent—of space and time. Many counterintuitive consequences emerge: in addition to being independent of the motion of the light source, the speed of light is constant regardless of the frame of reference in which it is measured; the distances and even temporal ordering of pairs of events change when measured in different inertial frames of reference (this is the relativity of simultaneity); and the linear additivity of velocities no longer holds true.


title: Spacetime

text: Einstein framed his theory in terms of kinematics (the study of moving bodies). His theory was an advance over Lorentz's 1904 theory of electromagnetic phenomena and Poincaré's electrodynamic theory. Although these theories included equations identical to those that Einstein introduced (i.e., the Lorentz transformation), they were essentially ad hoc models proposed to explain the results of various experiments—including the famous Michelson–Morley interferometer experiment—that were extremely difficult to fit into existing paradigms.


title: Spacetime

text: In 1908, Hermann Minkowski—once one of the math professors of a young Einstein in Zürich—presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. A key feature of this interpretation is the formal definition of the spacetime interval. Although measurements of distance and time between events differ for measurements made in different reference frames, the spacetime interval is independent of the inertial frame of reference in which they are recorded.


title: Spacetime

text: Minkowski's geometric interpretation of relativity was to prove vital to Einstein's development of his 1915 general theory of relativity, wherein he showed how mass and energy curve flat spacetime into a pseudo-Riemannian manifold.


title: Spacetime

text: 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. Furthermore, it assumes that space is Euclidean; it assumes that space follows the geometry of common sense.


title: Spacetime

text: In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer. General relativity also provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field.


title: Spacetime

text: 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". Space time is thus four dimensional. Mathematical events have zero duration and represent a single point in spacetime.


title: Spacetime

text: 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".


title: Spacetime

text: Mathematically, spacetime is a "manifold", which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, a globe appears flat. A scale factor, formula_1 (conventionally called the "speed-of-light") relates distances measured in space with distances measured in time. The magnitude of this scale factor (nearly 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.


title: Spacetime

text: 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 entire ensemble of clocks associated with one inertial frame of reference. In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording. A real observer, however, will see a delay between the emission of a signal and its detection due to the speed of light. To synchronize the clocks, in the data reduction following an experiment, the time when a signal is received will be corrected to reflect its actual time were it to have been recorded by an idealized lattice of clocks.


title: Spacetime

text: 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.


title: Spacetime

text: 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.


title: Spacetime

text: 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. 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. However, 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. 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. 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.


title: Spacetime

text: George Francis FitzGerald in 1889, and Hendrik Lorentz in 1892, independently proposed that material bodies traveling through the fixed aether were physically affected by their passage, contracting in the direction of motion by an amount that was exactly what was necessary to explain the negative results of the Michelson–Morley experiment. (No length changes occur in directions transverse to the direction of motion.)


title: Spacetime

text: 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), but with a fundamentally different interpretation. As a theory of dynamics (the study of forces and torques and their effect on motion), his theory assumed actual physical deformations of the physical constituents of matter. Lorentz's equations predicted a quantity that he called "local time", with which he could explain the aberration of light, the Fizeau experiment and other phenomena. However, Lorentz considered local time to be only an auxiliary mathematical tool, a trick as it were, to simplify the transformation from one system into another.


title: Spacetime

text: Other physicists and mathematicians at the turn of the century came close to arriving at what is currently known as spacetime. Einstein himself noted, that with so many people unraveling separate pieces of the puzzle, "the special theory of relativity, if we regard its development in retrospect, was ripe for discovery in 1905."


title: Spacetime

text: An important example is Henri Poincaré, who in 1898 argued that the simultaneity of two events is a matter of convention. 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. In 1900 and 1904, he suggested the inherent undetectability of the aether by emphasizing the validity of what he called the principle of relativity, and in 1905/1906 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. 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". Furthermore, even as late as 1909, Poincaré continued to believe in the dynamical interpretation of the Lorentz transform. For these and other reasons, most historians of science argue that Poincaré did not invent what is now called special relativity.


title: Spacetime

text: In 1905, Einstein introduced special relativity (even though without using the techniques of the spacetime formalism) in its modern understanding as a theory of space and time. While his results are mathematically equivalent to those of Lorentz and Poincaré, Einstein showed that the Lorentz transformations are not the result of interactions between matter and aether, but rather concern the nature of space and time itself. He obtained all of his results by recognizing that the entire theory can be built upon two postulates: The principle of relativity and the principle of the constancy of light speed.


title: Spacetime

text: Einstein performed his analysis in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. His work introducing the subject was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples.


title: Spacetime

text: In addition, Einstein in 1905 superseded previous attempts of an electromagnetic mass–energy relation by introducing the general equivalence of mass and energy, which was instrumental for his subsequent formulation of the equivalence principle in 1907, which declares the equivalence of inertial and gravitational mass. By using the mass–energy equivalence, Einstein showed, in addition, that the gravitational mass of a body is proportional to its energy content, which was one of the early results in developing general relativity. While it would appear that he did not at first think geometrically about spacetime, in the further development of general relativity Einstein fully incorporated the spacetime formalism.


title: Spacetime

text: 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:


title: Spacetime

text: 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. However, it is not at all clear when Minkowski began to formulate the geometric formulation of special relativity that was to bear his name, or to which extent he was influenced by Poincaré's four-dimensional interpretation of the Lorentz transformation. Nor is it clear if he ever fully appreciated Einstein's critical contribution to the understanding of the Lorentz transformations, thinking of Einstein's work as being an extension of Lorentz's work.


title: Spacetime

text: 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"). On 21 September 1908, Minkowski presented his famous talk, "Space and Time" ("Raum und Zeit"), 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.


title: Spacetime

text: 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.


title: Spacetime

text: 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, and in 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity. 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."


title: Spacetime

text: In three dimensions, the "distance" formula_2 between two points can be defined using the Pythagorean theorem:


title: Spacetime

text: 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".


title: Spacetime

text: 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.


title: Spacetime

text: In four-dimensional spacetime, the analog to distance is the interval. Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from four-dimensional Euclidean space. The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events (because of time dilation) or the distance between the two events (because of length contraction). But special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure 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 formula_4 and a spatial distance formula_5 Then the spacetime interval formula_6 between the two events that are separated by a distance formula_7 in space and by formula_8 in the formula_9-coordinate is:


title: Spacetime

text: The constant formula_12 the speed of light, converts time units (like seconds) into space units (like meters). The squared interval formula_13 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 formula_4. If the event separation is due to a light signal, then this difference vanishes and formula_15.


title: Spacetime

text: In a different inertial frame, say with coordinates formula_17, the spacetime interval formula_18 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, formula_19. For any other infinitesimal event where formula_20, one can prove that formula_21


title: Spacetime

text: which in turn upon integration leads to formula_22. The invariance of interval of any event between all intertial frames of reference is one of the fundamental results of special theory of relativity.


title: Spacetime

text: 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, formula_23 means formula_7, 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.


title: Spacetime

text: The equation above is similar to the Pythagorean theorem, except with a minus sign between the formula_25 and the formula_26 terms. The spacetime interval is the quantity formula_27 not formula_28 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 formula_29 as a distinct symbol in itself, rather than the square of something.


title: Spacetime

text: In general formula_29 can assume any values of the real number. If formula_29 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, real intervals are always timelike. If formula_29 is negative, the spacetime interval is said to be spacelike, where the spacetime interval is imaginary. Spacetime intervals are equal to zero when formula_33 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.


title: Spacetime

text: 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 formula_1 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. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.


title: Spacetime

text: These sign conventions are associated with the metric signatures and A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study.


title: Spacetime

text: 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′.


title: Spacetime

text: Fig. 2-3a redraws Fig. 2-2 in a different orientation. Fig. 2-3b illustrates a 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


title: Spacetime

text: 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 ±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"′ = −"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).


title: Spacetime

text: 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 −1, so that △PQR 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".


title: Spacetime

text: While the rest frame has space and time axes that meet at right angles, the moving frame is drawn with axes that meet at an acute angle. The frames are actually equivalent. The asymmetry is due to unavoidable distortions in how spacetime coordinates can map onto a Cartesian plane, and should be considered no stranger than the manner in which, on a Mercator projection of the Earth, the relative sizes of land masses near the poles (Greenland and Antarctica) are highly exaggerated relative to land masses near the Equator.


title: Spacetime

text: 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).


title: Spacetime

text: A light (double) cone divides spacetime into separate regions with respect to its apex. The interior of the future light cone consists of all events that are separated from the apex by more "time" (temporal distance) than necessary to cross their "spatial distance" at lightspeed; these events comprise the "timelike future" of the event O. Likewise, the "timelike past" comprises the interior events of the past light cone. So in "timelike intervals" Δ"ct" is greater than Δ"x", making timelike intervals positive. The region exterior to the light cone consists of events that are separated from the event O by more "space" than can be crossed at lightspeed in the given "time". These events comprise the so-called "spacelike" region of the event O, denoted "Elsewhere" in Fig. 2-4. Events on the light cone itself are said to be "lightlike" (or "null separated") from O. Because of the invariance of the spacetime interval, all observers will assign the same light cone to any given event, and thus will agree on this division of spacetime.


title: Spacetime

text: The light cone has an essential role within the concept of causality. It is possible for a not-faster-than-light-speed signal to travel from the position and time of O to the position and time of D (Fig. 2-4). It is hence possible for event O to have a causal influence on event D. The future light cone contains all the events that could be causally influenced by O. Likewise, it is possible for a not-faster-than-light-speed signal to travel from the position and time of A, to the position and time of O. The past light cone contains all the events that could have a causal influence on O. In contrast, assuming that signals cannot travel faster than the speed of light, any event, like e.g. B or C, in the spacelike region (Elsewhere), cannot either affect event O, nor can they be affected by event O employing such signalling. Under this assumption any causal relationship between event O and any events in the spacelike region of a light cone is excluded.


title: Spacetime

text: All observers will agree that for any given event, an event within the given event's future light cone occurs "after" the given event. Likewise, for any given event, an event within the given event's past light cone occurs "before" the given event. The before–after relationship observed for timelike-separated events remains unchanged no matter what the reference frame of the observer, i.e. no matter how the observer may be moving. The situation is quite different for spacelike-separated events. Fig. 2-4 was drawn from the reference frame of an observer moving at From this reference frame, event C is observed to occur after event O, and event B is observed to occur before event O. From a different reference frame, the orderings of these non-causally-related events can be reversed. In particular, one notes that if two events are simultaneous in a particular reference frame, they are "necessarily" separated by a spacelike interval and thus are noncausally related. The observation that simultaneity is not absolute, but depends on the observer's reference frame, is termed the relativity of simultaneity.


title: Spacetime

text: Fig. 2-6 illustrates the use of spacetime diagrams in the analysis of the relativity of simultaneity. The events in spacetime are invariant, but the coordinate frames transform as discussed above for Fig. 2-3. The three events are simultaneous from the reference frame of an observer moving at From the reference frame of an observer moving at the events appear to occur in the order From the reference frame of an observer moving at , the events appear to occur in the order . The white line represents a "plane of simultaneity" being moved from the past of the observer to the future of the observer, highlighting events residing on it. The gray area is the light cone of the observer, which remains invariant.


title: Spacetime

text: A spacelike spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. A spacelike spacetime interval hence provides a measure of "proper distance", i.e. the true distance = formula_38 Likewise, a timelike spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given world line. A timelike spacetime interval hence provides a measure of the "proper time" = formula_39


title: Spacetime

text: In Euclidean space (having spatial dimensions only), the set of points equidistant (using the Euclidean metric) from some point form a circle (in two dimensions) or a sphere (in three dimensions). In Minkowski spacetime (having one temporal and one spatial dimension), the points at some constant spacetime interval away from the origin (using the Minkowski metric) form curves given by the two equations


title: Spacetime

text: with formula_41some positive real constant. These equations describe two families of hyperbolae in an "x"–"ct" spacetime diagram, which are termed "invariant hyperbolae".


title: Spacetime

text: In Fig. 2-7a, each magenta hyperbola connects all events having some fixed spacelike separation from the origin, while the green hyperbolae connect events of equal timelike separation.


title: Spacetime

text: The magenta hyperbolae, which cross the "x" axis, are timelike curves, which is to say that these hyperbolae represent actual paths that can be traversed by (constantly accelerating) particles in spacetime: Between any two events on one hyperbola a causality relation is possible, because the inverse of the slope—representing the necessary speed—for all secants is less than formula_1. On the other hand, the green hyperbolae, which cross the "ct" axis, are spacelike curves because all intervals "along" these hyperbolae are spacelike intervals: No causality is possible between any two points on one of these hyperbolae, because all secants represent speeds larger than formula_1.


title: Spacetime

text: Fig. 2-7b reflects the situation in Minkowski spacetime (one temporal and two spatial dimensions) with the corresponding hyperboloids. The invariant hyperbolae displaced by spacelike intervals from the origin generate hyperboloids of one sheet, while the invariant hyperbolae displaced by timelike intervals from the origin generate hyperboloids of two sheets.


title: Spacetime

text: The (1+2)-dimensional boundary between space- and timelike hyperboloids, established by the events forming a zero spacetime interval to the origin, is made up by degenerating the hyperboloids to the light cone. In (1+1)-dimensions the hyperbolae degenerate to the two grey 45°-lines depicted in Fig. 2-7a.


title: Spacetime

text: Fig. 2-8 illustrates the invariant hyperbola for all events that can be reached from the origin in a proper time of 5 meters (approximately ). Different world lines represent clocks moving at different speeds. A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0.3 "c", the elapsed time measured by the observer is 5.24 meters (), while for a clock traveling at 0.7 "c", the elapsed time measured by the observer is 7.00 meters (). This illustrates the phenomenon known as time dilation. Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x–axis within that proper time than they would have without time dilation. The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O measures the clocks of observer O′ as running slower in his frame, observer O′ in turn will measure the clocks of observer O as running slower.


title: Spacetime

text: Length contraction, like time dilation, is a manifestation of the relativity of simultaneity. Measurement of length requires measurement of the spacetime interval between two events that are simultaneous in one's frame of reference. But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference.


title: Spacetime

text: Fig. 2-9 illustrates the motions of a 1 m rod that is traveling at 0.5 "c" along the "x" axis. The edges of the blue band represent the world lines of the rod's two endpoints. The invariant hyperbola illustrates events separated from the origin by a spacelike interval of 1 m. The endpoints O and B measured when  = 0 are simultaneous events in the S′ frame. But to an observer in frame S, events O and B are not simultaneous. To measure length, the observer in frame S measures the endpoints of the rod as projected onto the "x"-axis along their world lines. The projection of the rod's "world sheet" onto the "x" axis yields the foreshortened length OC.


title: Spacetime

text: (not illustrated) Drawing a vertical line through A so that it intersects the "x"′ axis demonstrates that, even as OB is foreshortened from the point of view of observer O, OA is likewise foreshortened from the point of view of observer O′. In the same way that each observer measures the other's clocks as running slow, each observer measures the other's rulers as being contracted.


title: Spacetime

text: In regards to mutual length contraction, Fig. 2-9 illustrates that the primed and unprimed frames are mutually rotated by a hyperbolic angle (analogous to ordinary angles in Euclidean geometry). Because of this rotation, the projection of a primed meter-stick onto the unprimed "x"-axis is foreshortened, while the projection of an unprimed meter-stick onto the primed x′-axis is likewise foreshortened.


title: Spacetime

text: Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. If an observer in frame S measures a clock, at rest in frame S', as running slower than his', while S' is moving at speed "v" in S, then the principle of relativity requires that an observer in frame S' likewise measures a clock in frame S, moving at speed −"v" in S', as running slower than hers. How two clocks can run "both slower" than the other, is an important question that "goes to the heart of understanding special relativity."


title: Spacetime

text: This apparent contradiction stems from not correctly taking into account the different settings of the necessary, related measurements. These settings allow for a consistent explanation of the "only apparent" contradiction. It is not about the abstract ticking of two identical clocks, but about how to measure in one frame the temporal distance of two ticks of a moving clock. It turns out that in mutually observing the duration between ticks of clocks, each moving in the respective frame, different sets of clocks must be involved. In order to measure in frame S the tick duration of a moving clock W′ (at rest in S′), one uses "two" additional, synchronized clocks W and W at rest in two arbitrarily fixed points in S with the spatial distance "d".


title: Spacetime

text: Conversely, for judging in frame S′ the temporal distance of two events on a moving clock W (at rest in S), one needs two clocks at rest in S′.


title: Spacetime

text: The necessary recordings for the two judgements, with "one moving clock" and "two clocks at rest" in respectively S or S′, involves two different sets, each with three clocks. Since there are different sets of clocks involved in the measurements, there is no inherent necessity that the measurements be reciprocally "consistent" such that, if one observer measures the moving clock to be slow, the other observer measures the one's clock to be fast.


title: Spacetime

text: Fig. 2-10 illustrates the previous discussion of mutual time dilation with Minkowski diagrams. The upper picture reflects the measurements as seen from frame S "at rest" with unprimed, rectangular axes, and frame S′ "moving with "v" > 0", coordinatized by primed, oblique axes, slanted to the right; the lower picture shows frame S′ "at rest" with primed, rectangular coordinates, and frame S "moving with −"v" < 0", with unprimed, oblique axes, slanted to the left.


title: Spacetime

text: Each line drawn parallel to a spatial axis ("x", "x"′) represents a line of simultaneity. All events on such a line have the same time value ("ct", "ct"′). Likewise, each line drawn parallel to a temporal axis ("ct", "ct′") represents a line of equal spatial coordinate values ("x", "x"′).


title: Spacetime

text: To show the mutual time dilation immediately in the upper picture, the event "D" may be constructed as the event at "x"′ = 0 (the location of clock W′ in S′), that is simultaneous to "C" ("OC" has equal spacetime interval as "OA") in S′. This shows that the time interval "OD" is longer than "OA", showing that the "moving" clock runs slower.


title: Spacetime

text: In the lower picture the frame S is moving with velocity −"v" in the frame S′ at rest. The worldline of clock W is the "ct"-axis (slanted to the left), the worldline of W′ is the vertical "ct"′-axis, and the worldline of W′ is the vertical through event "C", with "ct"′-coordinate "D". The invariant hyperbola through event "C" scales the time interval "OC" to "OA", which is shorter than "OD"; also, "B" is constructed (similar to "D" in the upper pictures) as simultaneous to "A" in S, at "x" = 0. The result "OB" > "OC" corresponds again to above.


title: Spacetime

text: The word "measure" is important. In classical physics an observer cannot affect an observed object, but the object's state of motion "can" affect the observer's "observations" of the object.


title: Spacetime

text: Many introductions to special relativity illustrate the differences between Galilean relativity and special relativity by posing a series of "paradoxes". These paradoxes are, in fact, ill-posed problems, resulting from our unfamiliarity with velocities comparable to the speed of light. The remedy is to solve many problems in special relativity and to become familiar with its so-called counter-intuitive predictions. The geometrical approach to studying spacetime is considered one of the best methods for developing a modern intuition.


title: Spacetime

text: The twin paradox is a thought experiment involving identical twins, one of whom makes a journey into space in a high-speed rocket, returning home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each should find the other to have aged less. The twin paradox sidesteps the justification for mutual time dilation presented above by avoiding the requirement for a third clock. Nevertheless, the "twin paradox" is not a true paradox because it is easily understood within the context of special relativity.


title: Spacetime

text: The impression that a paradox exists stems from a misunderstanding of what special relativity states. Special relativity does not declare all frames of reference to be equivalent, only inertial frames. The traveling twin's frame is not inertial during periods when she is accelerating. Furthermore, the difference between the twins is observationally detectable: the traveling twin needs to fire her rockets to be able to return home, while the stay-at-home twin does not.


title: Spacetime

text: These distinctions should result in a difference in the twins' ages. The spacetime diagram of Fig. 2-11 presents the simple case of a twin going straight out along the x axis and immediately turning back. From the standpoint of the stay-at-home twin, there is nothing puzzling about the twin paradox at all. The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B. More complex trajectories require integrating the proper time between the respective events along the curve (i.e. the path integral) to calculate the total amount of proper time experienced by the traveling twin.


title: Spacetime

text: Weiss's nomenclature, designating the stay-at-home twin as Terence and the traveling twin as Stella, is hereafter used.


title: Spacetime

text: Stella is not in an inertial frame. Given this fact, it is sometimes incorrectly stated that full resolution of the twin paradox requires general relativity:


title: Spacetime

text: Although general relativity is not required to analyze the twin paradox, application of the Equivalence Principle of general relativity does provide some additional insight into the subject. Stella is not stationary in an inertial frame. Analyzed in Stella's rest frame, she is motionless for the entire trip. When she is coasting her rest frame is inertial, and Terence's clock will appear to run slow. But when she fires her rockets for the turnaround, her rest frame is an accelerated frame and she experiences a force which is pushing her as if she were in a gravitational field. Terence will appear to be high up in that field and because of gravitational time dilation, his clock will appear to run fast, so much so that the net result will be that Terence has aged more than Stella when they are back together. The theoretical arguments predicting gravitational time dilation are not exclusive to general relativity. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence, including Newton's theory.


title: Spacetime

text: 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 few of these phenomena are described in the later sections of this article.


title: Spacetime

text: 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 (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 . With inertial frames, neither observer is under acceleration, and a simple set of equations allows us to relate coordinates to . Given that the two coordinate systems are in standard configuration, meaning that they are aligned with parallel coordinates and that when , the coordinate transformation is as follows:


title: Spacetime

text: Fig. 3-1 illustrates that in Newton's theory, time is universal, not the velocity of light. Consider the following thought experiment: The red arrow illustrates a train that is moving at 0.4 c with respect to the platform. Within the train, a passenger shoots a bullet with a speed of 0.4 c in the frame of the train. The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8 c. This is in accordance with our naive expectations.


title: Spacetime

text: More generally, assuming that frame S′ is moving at velocity "v" with respect to frame S, then within frame S′, observer O′ measures an object moving with velocity . Velocity "u" with respect to frame S, since , , and , can be written as = = . This leads to and ultimately


title: Spacetime

text: The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,


title: Spacetime

text: Fig. 3-2a illustrates a red train that is moving forward at a speed given by . From the primed frame of the train, a passenger shoots a bullet with a speed given by , where the distance is measured along a line parallel to the red axis rather than parallel to the black "x" axis. What is the composite velocity "u" of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig. 3-2b:


title: Spacetime

text: The relativistic formula for addition of velocities presented above exhibits several important features:


title: Spacetime

text: It is straightforward to obtain quantitative expressions for time dilation and length contraction. Fig. 3-3 is a composite image containing individual frames taken from two previous animations, simplified and relabeled for the purposes of this section.


title: Spacetime

text: To reduce the complexity of the equations slightly, there are a variety of different shorthand notations for "ct":


title: Spacetime

text: In Fig. 3-3a, segments "OA" and "OK" represent equal spacetime intervals. Time dilation is represented by the ratio "OB"/"OK". The invariant hyperbola has the equation where "k" = "OK", and the red line representing the world line of a particle in motion has the equation "w" = "x"/"β" = "xc"/"v". A bit of algebraic manipulation yields formula_55


title: Spacetime

text: The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma formula_56:


title: Spacetime

text: If "v" is greater than or equal to "c", the expression for formula_56 becomes physically meaningless, implying that "c" is the maximum possible speed in nature. For any "v" greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one.


title: Spacetime

text: In Fig. 3-3b, segments "OA" and "OK" represent equal spacetime intervals. Length contraction is represented by the ratio "OB"/"OK". The invariant hyperbola has the equation , where "k" = "OK", and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/"β" = "c"/"v". Event A has coordinates


title: Spacetime

text: ("x", "w") = ("γk", "γβk"). Since the tangent line through A and B has the equation "w" = ("x" − "OB")/"β", we have "γβk" = ("γk" − "OB")/"β" and


title: Spacetime

text: The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities.


title: Spacetime

text: Lorentz transformations are used to transform the coordinates of an event from one frame to another in special relativity.


title: Spacetime

text: When "v" ≪ "c" and "x" is small enough, the "v"/"c" and "vx"/"c" terms approach zero, and the Lorentz transformations approximate to the Galilean transformations.


title: Spacetime

text: formula_62 formula_63 etc., most often really mean formula_64 formula_65 etc. Although for brevity the Lorentz transformation equations are written without deltas, "x" means Δ"x", etc. We are, in general, always concerned with the space and time "differences" between events.


title: Spacetime

text: Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse transformations are trivially related to each other, since the "S" frame can only be moving forwards or reverse with respect to . So inverting the equations simply entails switching the primed and unprimed variables and replacing "v" with −"v".


title: Spacetime

text: There have been many dozens of derivations of the Lorentz transformations since Einstein's original work in 1905, each with its particular focus. Although Einstein's derivation was based on the invariance of the speed of light, there are other physical principles that may serve as starting points. Ultimately, these alternative starting points can be considered different expressions of the underlying principle of locality, which states that the influence that one particle exerts on another can not be transmitted instantaneously.


title: Spacetime

text: The derivation given here and illustrated in Fig. 3-5 is based on one presented by Bais and makes use of previous results from the Relativistic Composition of Velocities, Time Dilation, and Length Contraction sections. Event P has coordinates ("w", "x") in the black "rest system" and coordinates in the red frame that is moving with velocity parameter . To determine and in terms of "w" and "x" (or the other way around) it is easier at first to derive the "inverse" Lorentz transformation.


title: Spacetime

text: The above equations are alternate expressions for the t and x equations of the inverse Lorentz transformation, as can be seen by substituting "ct" for "w", for , and "v"/"c" for "β". From the inverse transformation, the equations of the forwards transformation can be derived by solving for and .


title: Spacetime

text: The Lorentz transformations have a mathematical property called linearity, since and are obtained as linear combinations of "x" and "t", with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that was tacitly assumed in the derivation, namely, that the properties of inertial frames of reference are independent of location and time. In the absence of gravity, spacetime looks the same everywhere. All inertial observers will agree on what constitutes accelerating and non-accelerating motion. Any one observer can use her own measurements of space and time, but there is nothing absolute about them. Another observer's conventions will do just as well.


title: Spacetime

text: A result of linearity is that if two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation.


title: Spacetime

text: The Doppler effect is the change in frequency or wavelength of a wave for a receiver and source in relative motion. For simplicity, we consider here two basic scenarios: (1) The motions of the source and/or receiver are exactly along the line connecting them (longitudinal Doppler effect), and (2) the motions are at right angles to the said line (transverse Doppler effect). We are ignoring scenarios where they move along intermediate angles.


title: Spacetime

text: The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other. The analysis of such waves depends on whether the source, the receiver, or both are moving relative to the medium. Given the scenario where the receiver is stationary with respect to the medium, and the source is moving directly away from the receiver at a speed of "v" for a velocity parameter of "β", the wavelength is increased, and the observed frequency "f" is given by


title: Spacetime

text: On the other hand, given the scenario where source is stationary, and the receiver is moving directly away from the source at a speed of "v" for a velocity parameter of "β", the wavelength is "not" changed, but the transmission velocity of the waves relative to the receiver is decreased, and the observed frequency "f" is given by


title: Spacetime

text: Light, unlike sound or water ripples, does not propagate through a medium, and there is no distinction between a source moving away from the receiver or a receiver moving away from the source. Fig. 3-6 illustrates a relativistic spacetime diagram showing a source separating from the receiver with a velocity parameter "β", so that the separation between source and receiver at time "w" is "βw". Because of time dilation, . Since the slope of the green light ray is −1, . Hence, the relativistic Doppler effect is given by


title: Spacetime

text: Suppose that a source and a receiver, both approaching each other in uniform inertial motion along non-intersecting lines, are at their closest approach to each other. It would appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis, that expectation is not necessarily true. Nevertheless, when appropriately defined, transverse Doppler shift is a relativistic effect that has no classical analog. The subtleties are these:


title: Spacetime

text: 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 , but also observes the receiver as having a time-dilated clock. In frame S, the receiver is therefore illuminated by blueshifted light of frequency


title: Spacetime

text: In scenario (b) the illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clocks are time dilated as measured in frame S, and since dr/dt was equal to zero at this point, the light from the source, emitted from this closest point, is redshifted with frequency


title: Spacetime

text: Scenarios (c) and (d) can be analyzed by simple time dilation arguments. In (c), the receiver observes light from the source as being blueshifted by a factor of formula_56, and in (d), the light is redshifted. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. (The converse, however, is not true.) Most reports of transverse Doppler shift refer to the effect as a redshift and analyze the effect in terms of scenarios (b) or (d).


title: Spacetime

text: In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity. Linear momentum, the product of a particle's mass and velocity, is a vector quantity, possessing the same direction as the velocity: . It is a "conserved" quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change.


title: Spacetime

text: In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector . In exploring the properties of the spacetime momentum, we start, in Fig. 3-8a, by examining what a particle looks like at rest. In the rest frame, the spatial component of the momentum is zero, i.e. , but the time component equals "mc".


title: Spacetime

text: We can obtain the transformed components of this vector in the moving frame by using the Lorentz transformations, or we can read it directly from the figure because we know that and , since the red axes are rescaled by gamma. Fig. 3-8b illustrates the situation as it appears in the moving frame. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches "c".


title: Spacetime

text: Light particles, or photons, travel at the speed of "c", the constant that is conventionally known as the "speed of light". This statement is not a tautology, since many modern formulations of relativity do not start with constant speed of light as a postulate. Photons therefore propagate along a lightlike world line and, in appropriate units, have equal space and time components for every observer.


title: Spacetime

text: A consequence of Maxwell's theory of electromagnetism is that light carries energy and momentum, and that their ratio is a constant: . Rearranging, , and since for photons, the space and time components are equal, "E/c" must therefore be equated with the time component of the spacetime momentum vector.


title: Spacetime

text: Photons travel at the speed of light, yet have finite momentum and energy. For this to be so, the mass term in "γmc" must be zero, meaning that photons are massless particles. Infinity times zero is an ill-defined quantity, but "E/c" is well-defined.


title: Spacetime

text: By this analysis, if the energy of a photon equals "E" in the rest frame, it equals in a moving frame. This result can be derived by inspection of Fig. 3-9 or by application of the Lorentz transformations, and is consistent with the analysis of Doppler effect given previously.


title: Spacetime

text: Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to several famous conclusions.


title: Spacetime

text: Another way of looking at the relationship between mass and energy is to consider a series expansion of at low velocity:


title: Spacetime

text: The second term is just an expression for the kinetic energy of the particle. Mass indeed appears to be another form of energy.


title: Spacetime

text: The concept of relativistic mass that Einstein introduced in 1905, "m", although amply validated every day in particle accelerators around the globe (or indeed in any instrumentation whose use depends on high velocity particles, such as electron microscopes, old-fashioned color television sets, etc.), has nevertheless not proven to be a "fruitful" concept in physics in the sense that it is not a concept that has served as a basis for other theoretical development. Relativistic mass, for instance, plays no role in general relativity.


title: Spacetime

text: For this reason, as well as for pedagogical concerns, most physicists currently prefer a different terminology when referring to the relationship between mass and energy. "Relativistic mass" is a deprecated term. The term "mass" by itself refers to the rest mass or invariant mass, and is equal to the invariant length of the relativistic momentum vector. Expressed as a formula,


title: Spacetime

text: This formula applies to all particles, massless as well as massive. For photons where m equals zero, it yields, .


title: Spacetime

text: Because of the close relationship between mass and energy, the four-momentum (also called 4-momentum) is also called the energy–momentum 4-vector. Using an uppercase "P" to represent the four-momentum and a lowercase p to denote the spatial momentum, the four-momentum may be written as


title: Spacetime

text: In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time. In 1915, Emmy Noether discovered that underlying each conservation law is a fundamental symmetry of nature. The fact that physical processes don't care "where" in space they take place (space translation symmetry) yields conservation of momentum, the fact that such processes don't care "when" they take place (time translation symmetry) yields conservation of energy, and so on. In this section, we examine the Newtonian views of conservation of mass, momentum and energy from a relativistic perspective.


title: Spacetime

text: To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension.


title: Spacetime

text: In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity:


title: Spacetime

text: For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat.


title: Spacetime

text: and collide to produce a single particle of conserved mass traveling at the center of mass velocity of the original system, formula_80. The total momentum is conserved.


title: Spacetime

text: Fig. 3-10 illustrates the inelastic collision of two particles from a relativistic perspective. The time components and add up to total "E/c" of the resultant vector, meaning that energy is conserved. Likewise, the space components and add up to form "p" of the resultant vector. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual particles that collided. Indeed, it is larger than the sum of the individual masses: .


title: Spacetime

text: 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.


title: Spacetime

text: 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.


title: Spacetime

text: 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 , the momentum . 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.


title: Spacetime

text: Conservation of momentum in the COM frame amounts to the requirement that both before and after collision. In the Newtonian analysis, conservation of mass dictates that . 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.


title: Spacetime

text: Newtonian momenta, calculated as , fail to behave properly under Lorentzian transformation. The linear transformation of velocities is replaced by the highly nonlinear


title: Spacetime

text: 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.


title: Spacetime

text: 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."


title: Spacetime

text: 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.


title: Spacetime

text: This nonlinearity is an artifact of our choice of parameters. We have previously noted that in an 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.


title: Spacetime

text: 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 , but as twice the area of the sector swept out by the ray from the . (Numerically, the angle and 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. Fig. 4-2 presents plots of the sinh, cosh, and tanh functions.


title: Spacetime

text: In a spacetime diagram, the velocity parameter formula_83 is the analog of slope. The "rapidity", "φ", is defined by


title: Spacetime

text: 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;


title: Spacetime

text: The Lorentz transformations take a simple form when expressed in terms of rapidity. The "γ" factor can be written as


title: Spacetime

text: Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called "boosts".


title: Spacetime

text: Substituting "γ" and "γβ" into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the may be written as


title: Spacetime

text: 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.


title: Spacetime

text: Four‑vectors have been mentioned above in context of the energy–momentum , but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, , 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. On the other hand, general relativity, from the outset, relies heavily on , 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 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.


title: Spacetime

text: A 4-tuple, is a "4-vector" if its component "A" transform between frames according to the Lorentz transformation.


title: Spacetime

text: which comes from simply replacing "ct" with "A" and "x" with "A" in the earlier presentation of the Lorentz transformation.


title: Spacetime

text: The last three components of a must be a standard vector in three-dimensional space. Therefore, a must transform like under Lorentz transformations as well as rotations.


title: Spacetime

text: 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 rather than give up on conservation of momentum.


title: Spacetime

text: 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 require the use of tensors with appropriate rank, which themselves can be thought of as being built up from .


title: Spacetime

text: 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.


title: Spacetime

text: 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.


title: Spacetime

text: 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.


title: Spacetime

text: 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. Will the string break?


title: Spacetime

text: 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.


title: Spacetime

text: 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.


title: Spacetime

text: 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 formula_124 acceleration for proper time formula_125 (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 formula_126 turns out to be greater than the length along the line of simultaneity formula_127.


title: Spacetime

text: 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 formula_128 If formula_129 and formula_130 are the ships' positions in formula_131 the positions in frame formula_132 are:


title: Spacetime

text: 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 formula_134. As shown in Fig. 4-5, Bell's example asserts the moving lengths formula_127 and formula_136 measured in frame formula_134 to be fixed, thereby forcing the rest frame length formula_126 in frame formula_132 to increase.


title: Spacetime

text: 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.


title: Spacetime

text: 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 formula_83 approaches a limit of one as formula_9 increases. Likewise, formula_56 approaches infinity.


title: Spacetime

text: The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:


title: Spacetime

text: 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).


title: Spacetime

text: 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.


title: Spacetime

text: 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. 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.


title: Spacetime

text: In 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.


title: Spacetime

text: 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.


title: Spacetime

text: 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.


title: Spacetime

text: 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.