title: Asana

text: Surya Namaskar, the Salute to the Sun, commonly practiced in most forms of modern yoga, links up to twelve asanas in a dynamically expressed yoga series. A full round consists of two sets of the series, the second set moving the opposing leg first. The asanas include Adho Mukha Svanasana (downward dog), the others differing from tradition to tradition with for instance a choice of Urdhva Mukha Svanasana (upward dog) or Bhujangasana (cobra) for one pose in the sequence. Schools, too, differ in their approaches to the sequence; for example, in Iyengar Yoga, variations such as inserting Maricyasana I and Pascimottanasana are suggested.


title: Asana

text: In the Western world, asanas are taught in differing styles by the various schools of yoga. Some poses like Trikonasana are common to many of them, but not always performed in the same way. Some independently documented approaches are described below.


title: Asana

text: Iyengar Yoga "emphasises precision and alignment", and prioritises correct movement over quantity, i.e. moving a little in the right direction is preferred to moving more but in a wrong direction. Postures are held for a relatively long period compared to other schools of yoga; this allows the muscles to relax and lengthen, and encourages awareness in the pose. Props including belts, blocks and blankets are freely used to assist students in correct working in the asanas. Beginners are introduced early on to standing poses, executed with careful attention to detail. For example, in Trikonasana, the feet are often jumped apart to a wide stance, the forward foot is turned out, and the centre of the forward heel is exactly aligned with the centre of the arch of the other foot.


title: Asana

text: Sivananda Yoga practices the asanas, hatha yoga, as part of raja yoga, with the goal of enabling practitioners ""to sit in meditation for a long time". There is little emphasis on the detail of individual poses; teachers rely on the basic instructions given in the books by Sivananda and Swami Vishnu-devananda. In Trikonasana, the top arm may be stretched forward parallel to the floor rather than straight up.


title: Asana

text: Sivananda Yoga identifies a group of 12 asanas as basic. These are not necessarily the easiest poses, nor those that every class would include. Trikonasana is the last of the 12, whereas in other schools it is one of the first and used to loosen the hips in preparation for other poses.


title: Asana

text: In Ashtanga Vinyasa Yoga, poses are executed differently from Iyengar Yoga. "Vinyasa" means flowing, and the poses are executed relatively rapidly, flowing continuously from one asana to the next using defined transitional movements. The asanas are grouped into six series, one Primary, one Intermediate, and four Advanced. Practice begins and ends with the chanting of mantras, followed by multiple cycles of the Sun Salutation, which "forms the foundation of Ashtanga Yoga practice", and then one of the series. Ashtanga Vinyasa practice emphasises aspects of yoga other than asanas, including drishti (focus points), bandhas (energy locks), and pranayama.


title: Asana

text: Kripalu Yoga uses teachers from other asana traditions, focussing on mindfulness rather than using effort in the poses. Teachers may say "allow your arms to float up" rather than "bring up your arms". The goal is to use the asanas "as a path of transformation." The approach is in three stages: firstly instruction in body alignment and awareness of the breath during the pose; secondly, holding the pose long enough to observe "unconscious patterns of tension in the body-mind"; and thirdly, through "deep concentration and total surrender", allowing oneself "to be moved by prana". In Trikonasana, the teacher may direct pupils' attention to pressing down with the outer edge of the back foot, lifting the arch of the foot, and then experimenting with "micro-movements", exploring where energy moves and how it feels.


title: Asana

text: In Bikram Yoga, as developed by Bikram Choudhury, there is a fixed sequence of 26 poses, in which Trikonasana is ninth, its task to focus on opening the hips. The Bikram version of Trikonasana is a different pose (Parsvakonasana) from that in Iyengar Yoga. The position of the feet is seen as critically important, along with proper breathing and the distribution of weight: about 30% on the back foot, 70% on the front foot.


title: Asana

text: Apart from the brands, many independent teachers, for example in Britain, offer an unbranded "hatha yoga".


title: Asana

text: Asanas can be classified in different ways, which may overlap: for example, by the position of the head and feet (standing, sitting, reclining, inverted), by whether balancing is required, or by the effect on the spine (forward bend, backbend, twist), giving a set of asana types agreed by most authors. Mittra uses his own categories such as "Floor & Supine Poses". Darren Rhodes and others add "Core strength", while Yogapedia and "Yoga Journal" also add "Hip-opening" to that set.


title: Asana

text: The table shows an example of each of these types of asana, with the title and approximate date of the earliest document describing (not only naming) that asana.


title: Asana

text: Religious Indian art makes use of a variety of seated asanas for figures of Buddha, Shiva, and other gods and religious figures. Most are meditation seats, especially the lotus position, Padmasana, but Lalitasana and its "royal ease" variant are not. Jain tirthankaras are often shown seated in the meditation asanas Siddhasana and Padmasana.


title: Asana

text: The actress Mariel Hemingway's 2002 autobiography "Finding My Balance: A Memoir with Yoga" describes how she used yoga to recover balance in her life after a dysfunctional upbringing: among other things, her grandfather, the novelist Ernest Hemingway, killed himself shortly before she was born, and her sister Margaux killed herself with a drug overdose. Each chapter is titled after an asana, the first being "Mountain Pose, or Tadasana", the posture of standing in balance. Other chapters are titled after poses including Trikonasana, Virabhadrasana, Janusirsasana, Ustrasana, Sarvangasana, and finally Garudasana, in each case with some life lesson related to the pose. For example, Garudasana, "a balancing posture with the arms and legs intricately intertwined … requires some flexibility, a lot of trust, and most of all, balance"; the chapter recounts how she, her husband and her daughters all came close to drowning in canoes off Kauai, Hawaii.


title: Asana

text: Among yoga novels is the author and yoga teacher Edward Vilga's 2014 "Downward Dog", named for Adho Mukha Svanasana, which paints a humorously unflattering picture of a man of the world who decides to become a private yoga teacher in New York society. Ian Fleming's 1964 novel "You Only Live Twice" has the action hero James Bond visiting Japan, where he "assiduously practised sitting in the lotus position." The critic Lisa M. Dresner notes that Bond is mirroring Fleming's own struggles with the pose.


title: The Fifth Season (novel)

text: The Fifth Season is a 2015 science fantasy novel by N. K. Jemisin. It was awarded the Hugo Award for Best Novel in 2016. It is the first volume in the "Broken Earth" series and is followed by "The Obelisk Gate" and "The Stone Sky".


title: The Fifth Season (novel)

text: "The Fifth Season" takes place on a planet with a single supercontinent called the Stillness. Every few centuries, its inhabitants endure what they call a "Fifth Season" of catastrophic climate change.


title: The Fifth Season (novel)

text: In a prologue, an extraordinarily powerful orogene discusses the sad state of the world and laments the oppression of his race. He then uses his enormous power to fracture the entire continent across its length, threatening to cause the worst Fifth Season in recorded history. The story then follows three female orogenes (Essun, Damaya, and Syenite) across the Stillness from different time periods.


title: The Fifth Season (novel)

text: Essun is a middle-aged woman with two young children living in a small southern comm named Tirimo. Secretly, she is an orogene, a human with the ability to manipulate earth and stone by absorbing or redirecting heat and energy from elsewhere. Her children have orogenic abilities as well, which often manifest themselves subconsciously, requiring Essun to constantly work to avoid their discovery. One day, she arrives home to find her young son has been beaten to death by her husband after inadvertently revealing his orogenic abilities. Her husband has taken their daughter and left town. Numb with grief and rage, she instinctually shunts the massive earthquake from the events of the prologue, which has just arrived from up north, around the comm. This saves it from complete destruction, but alerts the townspeople an orogene is present.


title: The Fifth Season (novel)

text: Due to the massive earthquake and ominous signs coming from the North, the townspeople are aware that a Fifth Season is likely imminent. When Essun attempts to leave the village she is outed as an orogene, and an angry mob attempts to kill her. In a rage, she kills many townspeople by leeching the heat from their bodies, freezing them solid, accidentally destroying the comm's only water supply in the process. She runs away, intent on following her husband and reclaiming her daughter.


title: The Fifth Season (novel)

text: Soon after leaving the comm, she encounters Hoa, a strange boy with ice-white skin and hair, who begins following her. Later she meets Tonkee, a commless and curious person. Together, they journey south, encountering the vast devastation caused by the "rifting" up north. Eventually, they arrive at a hidden comm called Castrima, built in a huge underground geode.


title: The Fifth Season (novel)

text: Damaya is a young girl in a northern comm, recently discovered by her parents to be an orogene. Unable to bring themselves to kill her, they summon Schaffa, a Guardian, to collect her. The Guardians are an ancient order of humans with supernatural abilities whose sole task is to manage and control orogenes. They control the Fulcrum, an organization that trains orogenes to use their abilities in a controlled fashion; nevertheless, orogenes remain a hated and feared sub-class with no rights of their own.


title: The Fifth Season (novel)

text: As the two travel from Damaya's home to the Fulcrum, Schaffa begins the training that Damaya will need in order to become a trained orogene. In his first lesson, Schaffa breaks Damaya's hand, challenging her to control her powers even when in great pain. His job, he tells her, is to keep the world safe from her. When she passes his test, they continue on their journey.


title: The Fifth Season (novel)

text: Damaya learns quickly and progresses through the ranks of young orogenes, learning to control her powers. One night, the young daughter of a wealthy and politically-connected family sneaks into the compound, curious about the interior of the largest building. Damaya reluctantly helps her enter, where they find a huge faceted pit, lined with sharp iron shards. Damaya is discovered, and one of the Guardians attempts to kill her; however, Schaffa steps in and spares Damaya's life. He also tells her that she will need to immediately take her first ring test—a test not normally given to orogenes until they have had much more training. In asking this of her, Schaffa is giving her a chance to save herself. She may have broken the rules of the Fulcrum, but if she can prove that she can control her powers and become a useful orogene, the Guardians will let her live. Damaya passes the test, is formally inducted into the Fulcrum, and is allowed to choose her new name.


title: The Fifth Season (novel)

text: Syenite, a rising orogene star in the Fulcrum, is forcibly partnered with Alabaster, the most powerful living orogene, in order to conceive a child with him on a business trip to the countryside. Though they loathe each other, they have no choice in the matter. As they travel to their destination, Alabaster frequently alludes to hidden knowledge about the obelisks, strange crystals the size of buildings that drift amongst the clouds. They are assumed by most to be inert leftovers from a long-dead civilization.


title: The Fifth Season (novel)

text: Along the way, Alabaster shows her one of the node stations that the Fulcrum has positioned around the Stillness. Officially, each contains an orogene whose job it is to constantly quell small earthquakes that could endanger the larger continent, and it is assumed to be dull work. Alabaster reveals the reality, which is that the orogenes in the nodes have all been mutilated and lobotomized from an early age, to allow them to quell quakes by instinct but subjecting them to constant agony and suffering. Alabaster reveals that the boy in this node is one of his offspring, that many of the orogenes in nodes are also his, all with the same strength he has, and yet there are almost no other at his level in the Fulcrum. Syenite is horrified by the discovery.


title: The Fifth Season (novel)

text: The two arrive at their destination, a coastal city that is losing business due to a large reef blocking the harbor; their task is to move the reef. This is far beneath Alabaster's abilities, but soon after their arrival he barely survives an assassination attempt, forcing Syenite to try to clear the harbor alone. To her shock, she discovers that the blockage is, in fact, an obelisk lying on its side underwater, which responds to her presence and rises up out of the water. Alarmed by this interaction between orogenes and the obelisk, the Fulcrum sends a Guardian to kill them both. The Guardian succeeds in incapacitating Alabaster, and is just about to kill Syenite when she instinctively reaches out to the obelisk with her power. The obelisk completely destroys the town and Syenite passes out.


title: The Fifth Season (novel)

text: Syenite awakens much later on a populated island off the coast, having been rescued along with Alabaster by a Stone Eater called Antimony. The island's leader is Innon, a very rare example of an untrained orogene living openly in society. She and Alabaster are accepted there, and both eventually enter into a loving polyamorous relationship with Innon. Syenite conceives a child with Alabaster, and raises him in peace on the island for a few years.


title: The Fifth Season (novel)

text: However, they are eventually discovered by the Fulcrum, who sends multiple ships with Guardians to retrieve them. Alabaster is abducted by Stone Eaters before he can mount a successful defense. Innon is killed in front of Syenite by one of the Guardians, and Schaffa approaches to take her child from her. She smothers her child rather than allow it to suffer in the Fulcrum or be lobotomized for the node stations. In her grief and anger, she draws power from a nearby obelisk, destroying the ship and killing most of the Guardians present as well as many fleeing islanders.


title: The Fifth Season (novel)

text: Reaching Castrima, Essun realizes that an old acquaintance has been waiting for her: Alabaster. Here it is revealed that Damaya, Syenite, and Essun are all the same woman at different points in her life; Syenite (the older Damaya) and Alabaster survived the attack on the island, and she went into hiding as Essun to try to start a new life while Alabaster has been living in enforced isolation amongst the Stone Eaters. Essun realizes that it was Alabaster who cracked the continent in half, triggering the present Fifth Season and likely dooming civilization in the process. He states that it was necessary to accomplish his plan to finally end the Fifth Seasons, and concludes with: "Have you ever heard of something called a "moon"?"


title: The Fifth Season (novel)

text: Names in the Stillness are denoted by given name followed by use-caste (profession) followed by comm (community, town, or city). For instance, Schaffa Guardian Warrant is a Guardian named Schaffa from the comm Warrant. Orogenes go by a (usually) geological-themed first name, the use-caste Orogene, and their comm, the Fulcrum.


title: The Fifth Season (novel)

text: "The New York Times" review stated ""The Fifth Season" invites us to imagine a dismantling of the earth in both the literal and the metaphorical sense, and suggests the possibility of a richer and more fundamental escape. The end of the world becomes a triumph when the world is monstrous, even if what lies beyond is difficult to conceive for those who are trapped inside it." NPR wrote that "Jemisin brilliantly illustrates the belief that, yes, imaginative world-building is a vital element of fantasy—but also that every character is a world unto herself."


title: The Fifth Season (novel)

text: In August 2017 it was announced that "The Fifth Season" is being adapted for television by TNT with rapper and actor Daveed Diggs attached as an executive producer.


title: The Fifth Season (novel)

text: "The Fifth Season" was awarded the Hugo Award for Best Novel at the 74th World Science Fiction Convention on August 20, 2016, making Jemisin the first black person to win the award. It also won the Sputnik Award, and was nominated for the Nebula Award and World Fantasy Award for Best Novel.


title: The Fifth Season (novel)

text: This is the first of three books in the Broken Earth series. The second novel in the trilogy, "The Obelisk Gate", was published on August 16, 2016, and won a Hugo Award for Best Novel in 2017. The third book, "The Stone Sky", was published in August 2017 and won the 2018 Hugo Award for Best Novel. These consecutive wins made Jemisin the first person to win the Hugo Award three years in a row or for all three books in a trilogy.


title: Ü

text: Ü (lowercase ü) is a Latin script character composed of the letter U and the diaeresis diacritical mark. In some alphabets such as those of a number of Romance languages or Guarani it denotes an instance of regular U to be construed in isolation from adjacent characters which it would usually form a larger unit with; other alphabets like the Azerbaijani, Estonian, German, Hungarian and Turkish ones treat it as a letter in its own right. In those cases it typically represents a close front rounded vowel .


title: Ü

text: Although not a part of their alphabet, Ü also appears in languages such as Finnish and Swedish when retained in foreign proper names like "München" ("Munich"). A small number of Dutch and Afrikaans words employ the character to mark vowel hiatus (e.g. "reünie" /reːyˈni/ ("reunion"), a loanword marked with diaeresis to suppress the native reading of "eu" as a digraph pronounced /øː/).


title: Ü

text: A glyph, U with umlaut, appears in the German alphabet. It represents the umlauted form of "u", which results in when long and when short. The letter is collated together with "U", or as "UE". In languages that have adopted German names or spellings, such as Swedish, the letter also occurs. It is however not a part of these languages' alphabets. In Swedish the letter is called "tyskt y" which means "German y".


title: Ü

text: In other languages that do not have the letter as part of the regular alphabet or in limited character sets such as ASCII, U-umlaut is frequently replaced with the two-letter combination "ue". Software for optical character recognition sometimes sees it falsely as "ii".


title: Ü

text: The letter Ü is present in the Hungarian, Turkish, Uyghur Latin, Estonian, Azeri, Turkmen, Crimean Tatar, Kazakh Latin and Tatar Latin alphabets, where it represents a close front rounded vowel . It is considered a distinct letter, collated separately, not a simple modification of U or Y, and is distinct from "UE".


title: Ü

text: This same letter appears in the Chinese Romanisations pinyin, Wade–Giles, and the German-based Lessing-Othmer, where it represents the same sound : 綠/lǜ (green) or 女/nǚ (female). Standard Mandarin Chinese pronunciation has both the sounds and . Pinyin only uses "Ü" to represent after the letters "L" or "N" to avoid confusion with words such as 路/lù (road) and 怒/nù (anger). Words such as 玉/yù (jade) or 句/jù (sentence) are pronounced with , but are not spelled with "Ü", although Wade–Giles and Lessing use "Ü" in all situations. As the letter "Ü" is missing on most keyboards and the letter "V" is not present in standard Mandarin pinyin, the letter "V" is used on most computer Chinese input methods to enter the letter "Ü". As a result, romanisation of Chinese with the letter "V" representing the Ü sound is sometimes found. However, Ü sound should be officially represented by "yu" in Pinyin when it is difficult to enter Ü. For example, the surname Lü (吕) would be written as "Lyu" in passports. Four extra tones for the letter "ü", which are "ǖ, ǘ, ǚ, ǜ", is added in Unicode as per GB/T 2312.


title: Ü

text: Several languages use diaeresis over the letter U to show that the letter is pronounced in its regular way, without dropping out or building diphthongs with neighboring letters.


title: Ü

text: In Spanish, it is used to distinguish between "gue"/"güe" / and "gui"/"güi" /: "nicaragüense" ("Nicaraguan"), "pingüino" ("penguin").


title: Ü

text: as in "aigües", "pingüins", "qüestió", "adeqüi". Also, "ü" is used to mark that vowel pairs that normally would form a diphthong must be pronounced as separate syllables, examples: "Raül", "diürn".


title: Ü

text: In French, the diaeresis appears over the "u" only very rarely, in some uncommon words, "capharnaüm" ('shambles'), "Capharnaüm/Capernaüm" or "Emmaüs" . After the 1990 spelling reforms, it is applied to a few more words, like "aigüe" (formerly "aiguë), "ambigüe" (formerly "ambiguë) and "argüer" (formerly without the diaeresis).


title: Ü

text: In the Rheinische Dokumenta, a phonetic alphabet for many West Central German, Low Rhenish, and related vernacular languages, "ü" represents a range from to .


title: Ü

text: Historically the unique letter "Ü" and U-diaeresis were written as a "U" with two dots above the letter.


title: Ü

text: U-umlaut was written as a "U" with a small "e" written above (Uͤ uͤ): this minute "e" degenerated to two vertical bars in medieval handwritings. In most later handwritings these bars in turn nearly became dots.


title: Ü

text: In modern typography there was insufficient space on typewriters and later computer keyboards to allow for both a U-with-dots (also representing "Ü") and a U-with-bars. Since they looked near-identical the two glyphs were combined, which was also done in computer character encodings such as ISO 8859-1. As a result, there was no way to differentiate between the three different characters. While the distinction can be recreated in modern Unicode using combining diacritics, modern typographic standards do not recommend doing so. In the Hungarian alphabet, double acute U (Ű) is a distinct letter representing a long "Ü".


title: Ü

text: The methods available for entering and from the keyboard depend on the operating system, the keyboard layout, and the application.


title: Lorentz transformation

text: In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.


title: Lorentz transformation

text: The most common form of the transformation, parametrized by the real constant formula_1 representing a velocity confined to the -direction, is expressed as


title: Lorentz transformation

text: where and are the coordinates of an event in two frames with the origins coinciding at ==0, where the primed frame is seen from the unprimed frame as moving with speed along the -axis, where is the speed of light, and formula_3 is the Lorentz factor. When speed is much smaller than , the Lorentz factor is negligibly different from 1, but as approaches , formula_4 grows without bound. The value of must be smaller than for the transformation to make sense.


title: Lorentz transformation

text: Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating, moving in curved paths, rotational motion with constant angular velocity, etc.). The term "Lorentz transformations" only refers to transformations between "inertial" frames, usually in the context of special relativity.


title: Lorentz transformation

text: In each reference frame, an observer can use a local coordinate system (usually Cartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. An event is something that happens at a point in space at an instant of time, or more formally a point in spacetime. The transformations connect the space and time coordinates of an event as measured by an observer in each frame.


title: Lorentz transformation

text: They supersede the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). The Galilean transformation is a good approximation only at relative speeds much less than the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames. The invariance of light speed is one of the postulates of special relativity.


title: Lorentz transformation

text: Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The Lorentz transformation is in accordance with Albert Einstein's special relativity, but was derived first.


title: Lorentz transformation

text: The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.


title: Lorentz transformation

text: Many physicists—including Woldemar Voigt, George FitzGerald, Joseph Larmor, and Hendrik Lorentz himself—had been discussing the physics implied by these equations since 1887. Early in 1889, Oliver Heaviside had shown from Maxwell's equations that the electric field surrounding a spherical distribution of charge should cease to have spherical symmetry once the charge is in motion relative to the luminiferous aether. FitzGerald then conjectured that Heaviside's distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published the conjecture that bodies in motion are being contracted, in order to explain the baffling outcome of the 1887 aether-wind experiment of Michelson and Morley. In 1892, Lorentz independently presented the same idea in a more detailed manner, which was subsequently called FitzGerald–Lorentz contraction hypothesis. Their explanation was widely known before 1905.


title: Lorentz transformation

text: Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous aether hypothesis, also looked for the transformation under which Maxwell's equations are invariant when transformed from the aether to a moving frame. They extended the FitzGerald–Lorentz contraction hypothesis and found out that the time coordinate has to be modified as well ("local time"). Henri Poincaré gave a physical interpretation to local time (to first order in "v"/"c", the relative velocity of the two reference frames normalized to the speed of light) as the consequence of clock synchronization, under the assumption that the speed of light is constant in moving frames. Larmor is credited to have been the first to understand the crucial time dilation property inherent in his equations.


title: Lorentz transformation

text: In 1905, Poincaré was the first to recognize that the transformation has the properties of a mathematical group,


title: Lorentz transformation

text: Later in the same year Albert Einstein published what is now called special relativity, by deriving the Lorentz transformation under the assumptions of the principle of relativity and the constancy of the speed of light in any inertial reference frame, and by abandoning the mechanistic aether as unnecessary.


title: Lorentz transformation

text: An "event" is something that happens at a certain point in spacetime, or more generally, the point in spacetime itself. In any inertial frame an event is specified by a time coordinate "ct" and a set of Cartesian coordinates to specify position in space in that frame. Subscripts label individual events.


title: Lorentz transformation

text: in all inertial frames for events connected by "light signals". The quantity on the left is called the "spacetime interval" between events and . The interval between "any two" events, not necessarily separated by light signals, is in fact invariant, i.e., independent of the state of relative motion of observers in different inertial frames, as is shown using homogeneity and isotropy of space. The transformation sought after thus must possess the property that:


title: Lorentz transformation

text: where are the spacetime coordinates used to define events in one frame, and are the coordinates in another frame. First one observes that () is satisfied if an arbitrary -tuple of numbers are added to events and . Such transformations are called "spacetime translations" and are not dealt with further here. Then one observes that a "linear" solution preserving the origin of the simpler problem solves the general problem too:


title: Lorentz transformation

text: (a solution satisfying the first formula automatically satisfies the second one as well; see polarization identity). Finding the solution to the simpler problem is just a matter of look-up in the theory of classical groups that preserve bilinear forms of various signature. First equation in () can be written more compactly as:


title: Lorentz transformation

text: where refers to the bilinear form of signature on exposed by the right hand side formula in (). The alternative notation defined on the right is referred to as the "relativistic dot product". Spacetime mathematically viewed as endowed with this bilinear form is known as Minkowski space . The Lorentz transformation is thus an element of the group , the Lorentz group or, for those that prefer the other metric signature, (also called the Lorentz group). One has:


title: Lorentz transformation

text: which is precisely preservation of the bilinear form () which implies (by linearity of and bilinearity of the form) that () is satisfied. The elements of the Lorentz group are rotations and "boosts" and mixes thereof. If the spacetime translations are included, then one obtains the "inhomogeneous Lorentz group" or the Poincaré group.


title: Lorentz transformation

text: The relations between the primed and unprimed spacetime coordinates are the Lorentz transformations, each coordinate in one frame is a linear function of all the coordinates in the other frame, and the inverse functions are the inverse transformation. Depending on how the frames move relative to each other, and how they are oriented in space relative to each other, other parameters that describe direction, speed, and orientation enter the transformation equations.


title: Lorentz transformation

text: Transformations describing relative motion with constant (uniform) velocity and without rotation of the space coordinate axes are called "boosts", and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformation is rotation in the spatial coordinates only, these like boosts are inertial transformations since there is no relative motion, the frames are simply tilted (and not continuously rotating), and in this case quantities defining the rotation are the parameters of the transformation (e.g., axis–angle representation, or Euler angles, etc.). A combination of a rotation and boost is a "homogeneous transformation", which transforms the origin back to the origin.


title: Lorentz transformation

text: The full Lorentz group also contains special transformations that are neither rotations nor boosts, but rather reflections in a plane through the origin. Two of these can be singled out; spatial inversion in which the spatial coordinates of all events are reversed in sign and temporal inversion in which the time coordinate for each event gets its sign reversed.


title: Lorentz transformation

text: Boosts should not be conflated with mere displacements in spacetime; in this case, the coordinate systems are simply shifted and there is no relative motion. However, these also count as symmetries forced by special relativity since they leave the spacetime interval invariant. A combination of a rotation with a boost, followed by a shift in spacetime, is an "inhomogeneous Lorentz transformation", an element of the Poincaré group, which is also called the inhomogeneous Lorentz group.


title: Lorentz transformation

text: A "stationary" observer in frame defines events with coordinates . Another frame moves with velocity relative to , and an observer in this "moving" frame defines events using the coordinates .


title: Lorentz transformation

text: The coordinate axes in each frame are parallel (the and axes are parallel, the and axes are parallel, and the and axes are parallel), remain mutually perpendicular, and relative motion is along the coincident axes. At , the origins of both coordinate systems are the same, . In other words, the times and positions are coincident at this event. If all these hold, then the coordinate systems are said to be in standard configuration, or synchronized.


title: Lorentz transformation

text: Here, is the "parameter" of the transformation, for a given boost it is a constant number, but can take a continuous range of values. In the setup used here, positive relative velocity is motion along the positive directions of the axes, zero relative velocity is no relative motion, while negative relative velocity is relative motion along the negative directions of the axes. The magnitude of relative velocity cannot equal or exceed , so only subluminal speeds are allowed. The corresponding range of is .


title: Lorentz transformation

text: The transformations are not defined if is outside these limits. At the speed of light () is infinite, and faster than light () is a complex number, each of which make the transformations unphysical. The space and time coordinates are measurable quantities and numerically must be real numbers.


title: Lorentz transformation

text: As an active transformation, an observer in F′ notices the coordinates of the event to be "boosted" in the negative directions of the axes, because of the in the transformations. This has the equivalent effect of the "coordinate system" F′ boosted in the positive directions of the axes, while the event does not change and is simply represented in another coordinate system, a passive transformation.


title: Lorentz transformation

text: The inverse relations ( in terms of ) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here is the "stationary" frame while is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from to must take exactly the same form as the transformations from to . The only difference is moves with velocity relative to (i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer in notes an event , then an observer in notes the "same" event with coordinates


title: Lorentz transformation

text: and the value of remains unchanged. This "trick" of simply reversing the direction of relative velocity while preserving its magnitude, and exchanging primed and unprimed variables, always applies to finding the inverse transformation of every boost in any direction.


title: Lorentz transformation

text: which shows much more clearly the symmetry in the transformation. From the allowed ranges of and the definition of , it follows . The use of and is standard throughout the literature.


title: Lorentz transformation

text: The Lorentz transformations can also be derived in a way that resembles circular rotations in 3d space using the hyperbolic functions. For the boost in the direction, the results are


title: Lorentz transformation

text: where (lowercase zeta) is a parameter called "rapidity" (many other symbols are used, including ). Given the strong resemblance to rotations of spatial coordinates in 3d space in the Cartesian xy, yz, and zx planes, a Lorentz boost can be thought of as a hyperbolic rotation of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4d Minkowski space. The parameter is the hyperbolic angle of rotation, analogous to the ordinary angle for circular rotations. This transformation can be illustrated with a Minkowski diagram.


title: Lorentz transformation

text: The hyperbolic functions arise from the "difference" between the squares of the time and spatial coordinates in the spacetime interval, rather than a sum. The geometric significance of the hyperbolic functions can be visualized by taking or in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying , which parametrizes the curves according to the identity


title: Lorentz transformation

text: provides the link between a constant value of rapidity, and the slope of the axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor


title: Lorentz transformation

text: Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections between , , and are


title: Lorentz transformation

text: Since , it follows . From the relation between and , positive rapidity is motion along the positive directions of the axes, zero rapidity is no relative motion, while negative rapidity is relative motion along the negative directions of the axes.


title: Lorentz transformation

text: The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity since this is equivalent to negating the relative velocity. Therefore,


title: Lorentz transformation

text: So far the Lorentz transformations have been applied to "one event". If there are two events, there is a spatial separation and time interval between them. It follows from the linearity of the Lorentz transformations that two values of space and time coordinates can be chosen, the Lorentz transformations can be applied to each, then subtracted to get the Lorentz transformations of the differences;


title: Lorentz transformation

text: where (uppercase delta) indicates a difference of quantities; e.g., for two values of coordinates, and so on.


title: Lorentz transformation

text: These transformations on "differences" rather than spatial points or instants of time are useful for a number of reasons:


title: Lorentz transformation

text: A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves. If in the equation for a pulse of light along the direction is , then in the Lorentz transformations give , and vice versa, for any .


title: Lorentz transformation

text: For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation


title: Lorentz transformation

text: in accordance with the correspondence principle. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".


title: Lorentz transformation

text: The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relative velocity vector with a magnitude that cannot equal or exceed , so that .


title: Lorentz transformation

text: Only time and the coordinates parallel to the direction of relative motion change, while those coordinates perpendicular do not. With this in mind, split the spatial position vector as measured in , and as measured in , each into components perpendicular (⊥) and parallel ( ‖ ) to ,


title: Lorentz transformation

text: where is the dot product. The Lorentz factor retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition with magnitude is also used by some authors.


title: Lorentz transformation

text: Introducing a unit vector in the direction of relative motion, the relative velocity is with magnitude and direction , and vector projection and rejection give respectively


title: Lorentz transformation

text: The projection and rejection also applies to . For the inverse transformations, exchange and to switch observed coordinates, and negate the relative velocity (or simply the unit vector since the magnitude is always positive) to obtain


title: Lorentz transformation

text: The unit vector has the advantage of simplifying equations for a single boost, allows either or to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing and . It is not convenient for multiple boosts.


title: Lorentz transformation

text: each of which serves as a useful abbreviation in some contexts. The magnitude of is the absolute value of the rapidity scalar confined to , which agrees with the range .


title: Lorentz transformation

text: taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to


title: Lorentz transformation

text: The velocities and are the velocity of some massive object. They can also be for a third inertial frame (say "F"′′), in which case they must be "constant". Denote either entity by X. Then X moves with velocity relative to F, or equivalently with velocity relative to F′, in turn F′ moves with velocity relative to F. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange and , and change to .


title: Lorentz transformation

text: The transformation of velocity is useful in stellar aberration, the Fizeau experiment, and the relativistic Doppler effect.


title: Lorentz transformation

text: The Lorentz transformations of acceleration can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.


title: Lorentz transformation

text: In general, given four quantities and and their Lorentz-boosted counterparts and , a relation of the form


title: Lorentz transformation

text: implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates;


title: Lorentz transformation

text: The decomposition of (and ) into components perpendicular and parallel to is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange and to switch observed quantities, and reverse the direction of relative motion by the substitution ).


title: Lorentz transformation

text: The quantities collectively make up a "four-vector", where is the "timelike component", and the "spacelike component". Examples of and are the following:


title: Lorentz transformation

text: For a given object (e.g., particle, fluid, field, material), if or correspond to properties specific to the object like its charge density, mass density, spin, etc., its properties can be fixed in the rest frame of that object. Then the Lorentz transformations give the corresponding properties in a frame moving relative to the object with constant velocity. This breaks some notions taken for granted in non-relativistic physics. For example, the energy of an object is a scalar in non-relativistic mechanics, but not in relativistic mechanics because energy changes under Lorentz transformations; its value is different for various inertial frames. In the rest frame of an object, it has a rest energy and zero momentum. In a boosted frame its energy is different and it appears to have a momentum. Similarly, in non-relativistic quantum mechanics the spin of a particle is a constant vector, but in relativistic quantum mechanics spin depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike quantity , however a boosted observer will perceive a nonzero timelike component and an altered spin.


title: Lorentz transformation

text: Not all quantities are invariant in the form as shown above, for example orbital angular momentum does not have a timelike quantity, and neither does the electric field nor the magnetic field . The definition of angular momentum is , and in a boosted frame the altered angular momentum is . Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out transforms with another vector quantity related to boosts, see relativistic angular momentum for details. For the case of the and fields, the transformations cannot be obtained as directly using vector algebra. The Lorentz force is the definition of these fields, and in it is while in it is . A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra, given below.


title: Lorentz transformation

text: Throughout, italic non-bold capital letters are 4×4 matrices, while non-italic bold letters are 3×3 matrices.


title: Lorentz transformation

text: The set of all Lorentz transformations Λ in this article is denoted formula_30. This set together with matrix multiplication forms a group, in this context known as the "Lorentz group". Also, the above expression is a quadratic form of signature (3,1) on spacetime, and the group of transformations which leaves this quadratic form invariant is the indefinite orthogonal group O(3,1), a Lie group. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned are matrix Lie groups. In this context the operation of composition amounts to matrix multiplication.


title: Lorentz transformation

text: and this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. Taking the determinant of the equation using the product rule gives immediately


title: Lorentz transformation

text: Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form,


title: Lorentz transformation

text: carrying out the block matrix multiplications obtains general conditions on to ensure relativistic invariance. Not much information can be directly extracted from all the conditions, however one of the results


title: Lorentz transformation

text: The negative inequality may be unexpected, because multiplies the time coordinate and this has an effect on time symmetry. If the positive equality holds, then is the Lorentz factor.


title: Lorentz transformation

text: The determinant and inequality provide four ways to classify Lorentz Transformations ("herein LTs for brevity"). Any particular LT has only one determinant sign "and" only one inequality. There are four sets which include every possible pair given by the intersections ("n"-shaped symbol meaning "and") of these classifying sets.


title: Lorentz transformation

text: where "+" and "−" indicate the determinant sign, while "↑" for ≥ and "↓" for ≤ denote the inequalities.


title: Lorentz transformation

text: A subgroup of a group must be closed under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformations and from a particular set, the composite Lorentz transformations and must be in the same set as and . This is not always the case: the composition of two antichronous Lorentz transformations is orthochronous, and the composition of two improper Lorentz transformations is proper. In other words, while the sets formula_37, formula_38, formula_39, and formula_40 all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g. formula_41, formula_42, formula_43) do not form subgroups.


title: Lorentz transformation

text: If a Lorentz covariant 4-vector is measured in one inertial frame with result formula_44, and the same measurement made in another inertial frame (with the same orientation and origin) gives result formula_45, the two results will be related by


title: Lorentz transformation

text: where the boost matrix formula_47 represents the Lorentz transformation between the unprimed and primed frames and formula_48 is the velocity of the primed frame as seen from the unprimed frame. The matrix is given by


title: Lorentz transformation

text: where formula_50 is the magnitude of the velocity and formula_51 is the Lorentz factor. This formula represents a passive transformation, as it describes how the coordinates of the measured quantity changes from the unprimed frame to the primed frame. The active transformation is given by formula_52.


title: Lorentz transformation

text: If a frame is boosted with velocity relative to frame , and another frame is boosted with velocity relative to , the separate boosts are


title: Lorentz transformation

text: Successive transformations act on the left. If and are collinear (parallel or antiparallel along the same line of relative motion), the boost matrices commute: . This composite transformation happens to be another boost, , where is collinear with and .


title: Lorentz transformation

text: If and are not collinear but in different directions, the situation is considerably more complicated. Lorentz boosts along different directions do not commute: and are not equal. Also, each of these compositions is "not" a single boost, but they are still Lorentz transformations they each preserve the spacetime interval. It turns out the composition of any two Lorentz boosts is equivalent to a boost followed or preceded by a rotation on the spatial coordinates, in the form of or . The and are composite velocities, while and are rotation parameters (e.g. axis-angle variables, Euler angles, etc.). The rotation in block matrix form is simply


title: Lorentz transformation

text: where is a 3d rotation matrix, which rotates any 3d vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It is "not" simple to connect and (or and ) to the original boost parameters and . In a composition of boosts, the matrix is named the Wigner rotation, and gives rise to the Thomas precession. These articles give the explicit formulae for the composite transformation matrices, including expressions for .


title: Lorentz transformation

text: In this article the axis-angle representation is used for . The rotation is about an axis in the direction of a unit vector , through angle (positive anticlockwise, negative clockwise, according to the right-hand rule). The "axis-angle vector"


title: Lorentz transformation

text: Spatial rotations alone are also Lorentz transformations they leave the spacetime interval invariant. Like boosts, successive rotations about different axes do not commute. Unlike boosts, the composition of any two rotations is equivalent to a single rotation. Some other similarities and differences between the boost and rotation matrices include:


title: Lorentz transformation

text: The most general proper Lorentz transformation includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, and . An explicit form of the general Lorentz transformation is cumbersome to write down and will not be given here. Nevertheless, closed form expressions for the transformation matrices will be given below using group theoretical arguments. It will be easier to use the rapidity parametrization for boosts, in which case one writes and .


title: Lorentz transformation

text: with matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is the special indefinite orthogonal group SO(3,1). (The plus sign indicates that it preserves the orientation of the temporal dimension).


title: Lorentz transformation

text: For simplicity, look at the infinitesimal Lorentz boost in the x direction (examining a boost in any other direction, or rotation about any axis, follows an identical procedure). The infinitesimal boost is a small boost away from the identity, obtained by the Taylor expansion of the boost matrix to first order about ,


title: Lorentz transformation

text: where the higher order terms not shown are negligible because is small, and is simply the boost matrix in the "x" direction. The derivative of the matrix is the matrix of derivatives (of the entries, with respect to the same variable), and it is understood the derivatives are found first then evaluated at ,


title: Lorentz transformation

text: For now, is defined by this result (its significance will be explained shortly). In the limit of an infinite number of infinitely small steps, the finite boost transformation in the form of a matrix exponential is obtained


title: Lorentz transformation

text: where the limit definition of the exponential has been used (see also characterizations of the exponential function). More generally


title: Lorentz transformation

text: The axis-angle vector and rapidity vector are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group are and , each vectors of matrices with the explicit forms


title: Lorentz transformation

text: These are all defined in an analogous way to above, although the minus signs in the boost generators are conventional. Physically, the generators of the Lorentz group correspond to important symmetries in spacetime: are the "rotation generators" which correspond to angular momentum, and are the "boost generators" which correspond to the motion of the system in spacetime. The derivative of any smooth curve with in the group depending on some group parameter with respect to that group parameter, evaluated at , serves as a definition of a corresponding group generator , and this reflects an infinitesimal transformation away from the identity. The smooth curve can always be taken as an exponential as the exponential will always map smoothly back into the group via for all ; this curve will yield again when differentiated at .


title: Lorentz transformation

text: It has been stated that the general proper Lorentz transformation is a product of a boost and rotation. At the "infinitesimal" level the product


title: Lorentz transformation

text: is commutative because only linear terms are required (products like and count as higher order terms and are negligible). Taking the limit as before leads to the finite transformation in the form of an exponential


title: Lorentz transformation

text: The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,


title: Lorentz transformation

text: because the generators do not commute. For a description of how to find the factors of a general Lorentz transformation in terms of a boost and a rotation "in principle" (this usually does not yield an intelligible expression in terms of generators and ), see Wigner rotation. If, on the other hand, "the decomposition is given" in terms of the generators, and one wants to find the product in terms of the generators, then the Baker–Campbell–Hausdorff formula applies.


title: Lorentz transformation

text: Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the set of all Lorentz generators


title: Lorentz transformation

text: together with the operations of ordinary matrix addition and multiplication of a matrix by a number, forms a vector space over the real numbers. The generators form a basis set of "V", and the components of the axis-angle and rapidity vectors, , are the coordinates of a Lorentz generator with respect to this basis.


title: Lorentz transformation

text: where the bracket is known as the "commutator", and the other relations can be found by taking cyclic permutations of x, y, z components (i.e. change x to y, y to z, and z to x, repeat).


title: Lorentz transformation

text: These commutation relations, and the vector space of generators, fulfill the definition of the Lie algebra formula_70. In summary, a Lie algebra is defined as a vector space "V" over a field of numbers, and with a binary operation [ , ] (called a Lie bracket in this context) on the elements of the vector space, satisfying the axioms of bilinearity, alternatization, and the Jacobi identity. Here the operation [ , ] is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generators "V" as given previously, and the field is the set of real numbers.


title: Lorentz transformation

text: Linking terminology used in mathematics and physics: A group generator is any element of the Lie algebra. A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis. A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense.


title: Lorentz transformation

text: provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group. In the case of the Lorentz group, the exponential map is just the matrix exponential. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is surjective (onto). Hence any group element in the connected component of the identity can be expressed as an exponential of an element of the Lie algebra.


title: Lorentz transformation

text: which negates the time coordinate only, because these transformations leave the spacetime interval invariant. Here is the 3d identity matrix. These are both symmetric, they are their own inverses (see involution (mathematics)), and each have determinant −1. This latter property makes them improper transformations.


title: Lorentz transformation

text: If is a proper orthochronous Lorentz transformation, then is improper antichronous, is improper orthochronous, and is proper antichronous.


title: Lorentz transformation

text: Two other spacetime symmetries have not been accounted for. For the spacetime interval to be invariant, it can be shown that it is necessary and sufficient for the coordinate transformation to be of the form


title: Lorentz transformation

text: where "C" is a constant column containing translations in time and space. If "C" ≠ 0, this is an inhomogeneous Lorentz transformation or Poincaré transformation. If "C" = 0, this is a homogeneous Lorentz transformation. Poincaré transformations are not dealt further in this article.


title: Lorentz transformation

text: allows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g., tensors or spinors of any order in 4d spacetime, to be defined. In the corresponding tensor index notation, the above matrix expression is


title: Lorentz transformation

text: where lower and upper indices label covariant and contravariant components respectively, and the summation convention is applied. It is a standard convention to use Greek indices that take the value 0 for time components, and 1, 2, 3 for space components, while Latin indices simply take the values 1, 2, 3, for spatial components (the opposite for Landau and Lifshitz). Note that the first index (reading left to right) corresponds in the matrix notation to a "row index". The second index corresponds to the column index.


title: Lorentz transformation

text: The transformation matrix is universal for all four-vectors, not just 4-dimensional spacetime coordinates. If is any four-vector, then in tensor index notation formula_77


title: Lorentz transformation

text: Alternatively, one writes formula_78 in which the primed indices denote the indices of A in the primed frame. For a general -component object one may write formula_79 where is the appropriate representation of the Lorentz group, an matrix for every . In this case, the indices should "not" be thought of as spacetime indices (sometimes called Lorentz indices), and they run from to . E.g., if is a bispinor, then the indices are called "Dirac indices".


title: Lorentz transformation

text: There are also vector quantities with covariant indices. They are generally obtained from their corresponding objects with contravariant indices by the operation of "lowering an index"; e.g.,


title: Lorentz transformation

text: where is the metric tensor. (The linked article also provides more information about what the operation of raising and lowering indices really is mathematically.) The inverse of this transformation is given by


title: Lorentz transformation

text: where, when viewed as matrices, is the inverse of . As it happens, . This is referred to as "raising an index". To transform a covariant vector , first raise its index, then transform it according to the same rule as for contravariant -vectors, then finally lower the index;


title: Lorentz transformation

text: That is, it is the -component of the "inverse" Lorentz transformation. One defines (as a matter of notation),


title: Lorentz transformation

text: is running over "a row index" of the matrix representing . Thus, in terms of matrices, this transformation should be thought of as the "inverse transpose" of acting on the column vector . That is, in pure matrix notation,


title: Lorentz transformation

text: This means exactly that covariant vectors (thought of as column matrices) transform according to the dual representation of the standard representation of the Lorentz group. This notion generalizes to general representations, simply replace with .


title: Lorentz transformation

text: If and are linear operators on vector spaces and , then a linear operator may be defined on the tensor product of and , denoted according to


title: Lorentz transformation

text: The second step uses the bilinearity of the tensor product and the last step defines a 2-tensor on component form, or rather, it just renames the tensor .


title: Lorentz transformation

text: These observations generalize in an obvious way to more factors, and using the fact that a general tensor on a vector space can be written as a sum of a coefficient (component!) times tensor products of basis vectors and basis covectors, one arrives at the transformation law for any tensor quantity . It is given by


title: Lorentz transformation

text: where is defined above. This form can generally be reduced to the form for general -component objects given above with a single matrix () operating on column vectors. This latter form is sometimes preferred; e.g., for the electromagnetic field tensor.


title: Lorentz transformation

text: Lorentz transformations can also be used to illustrate that the magnetic field and electric field are simply different aspects of the same force — the electromagnetic force, as a consequence of relative motion between electric charges and observers. The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.


title: Lorentz transformation

text: The electric and magnetic fields transform differently from space and time, but exactly the same way as relativistic angular momentum and the boost vector.


title: Lorentz transformation

text: in SI units. In relativity, the Gaussian system of units is often preferred over SI units, even in texts whose main choice of units is SI units, because in it the electric field and the magnetic induction have the same units making the appearance of the electromagnetic field tensor more natural. Consider a Lorentz boost in the -direction. It is given by


title: Lorentz transformation

text: where the field tensor is displayed side by side for easiest possible reference in the manipulations below.


title: Lorentz transformation

text: and are independent of the metric signature. For SI units, substitute . refer to this last form as the view as opposed to the "geometric view" represented by the tensor expression


title: Lorentz transformation

text: and make a strong point of the ease with which results that are difficult to achieve using the view can be obtained and understood. Only objects that have well defined Lorentz transformation properties (in fact under "any" smooth coordinate transformation) are geometric objects. In the geometric view, the electromagnetic field is a six-dimensional geometric object in "spacetime" as opposed to two interdependent, but separate, 3-vector fields in "space" and "time". The fields (alone) and (alone) do not have well defined Lorentz transformation properties. The mathematical underpinnings are equations and that immediately yield . One should note that the primed and unprimed tensors refer to the "same event in spacetime". Thus the complete equation with spacetime dependence is


title: Lorentz transformation

text: Length contraction has an effect on charge density and current density , and time dilation has an effect on the rate of flow of charge (current), so charge and current distributions must transform in a related way under a boost. It turns out they transform exactly like the space-time and energy-momentum four-vectors,


title: Lorentz transformation

text: Charge density transforms as the time component of a four-vector. It is a rotational scalar. The current density is a 3-vector.


title: Lorentz transformation

text: Equation hold unmodified for any representation of the Lorentz group, including the bispinor representation. In one simply replaces all occurrences of by the bispinor representation ,


title: Lorentz transformation

text: The above equation could, for instance, be the transformation of a state in Fock space describing two free electrons.


title: Lorentz transformation

text: A general "noninteracting" multi-particle state (Fock space state) in quantum field theory transforms according to the rule


title: 2014 Scottish independence referendum

text: A referendum on Scottish independence from the United Kingdom was held in Scotland on 18 September 2014. The referendum question was, "Should Scotland be an independent country?", which voters answered with "Yes" or "No". The "No" side won with 2,001,926 (55.3%) voting against independence and 1,617,989 (44.7%) voting in favour. The turnout of 84.6% was the highest recorded for an election or referendum in the United Kingdom since the January 1910 general election, which was held before the introduction of universal suffrage.


title: 2014 Scottish independence referendum

text: The Scottish Independence Referendum Act 2013 set out the arrangements for the referendum and was passed by the Scottish Parliament in November 2013, following an agreement between the devolved Scottish government and the Government of the United Kingdom. The independence proposal required a simple majority to pass. All European Union (EU) or Commonwealth citizens residing in Scotland age 16 or over could vote, with some exceptions, which produced a total electorate of almost 4,300,000 people. This was the first time that the electoral franchise was extended to include 16- and 17-year-olds in Scotland.


title: 2014 Scottish independence referendum

text: Yes Scotland was the main campaign group for independence, while Better Together was the main campaign group in favour of maintaining the union. Many other campaign groups, political parties, businesses, newspapers, and prominent individuals were also involved. Prominent issues raised during the referendum included what currency an independent Scotland would use, public expenditure, EU membership, and North Sea oil. An exit poll revealed that retention of the pound sterling was the deciding factor for those who voted No, while "disaffection with Westminster politics" was the deciding factor for those who voted Yes.


title: 2014 Scottish independence referendum

text: The Kingdom of Scotland and the Kingdom of England were established as independent countries during the Middle Ages. After fighting a series of wars during the 14th century, the two monarchies entered a personal union in 1603 (the Union of the Crowns) when James VI of Scotland also became James I of England. The two nations were temporarily united under one government when Oliver Cromwell was declared Lord Protector of a Commonwealth in 1653, but this was dissolved when the monarchy was restored in 1660. Scotland and England united to form the Kingdom of Great Britain in 1707 (Wales had already been unilaterally incorporated into England in the mid-16th century). The Scots favoured union to solve the economic problems caused by the failure of the Darien scheme and the English favoured it to secure the Hanoverian line of succession. Great Britain in turn united with the Kingdom of Ireland in 1801, forming the United Kingdom of Great Britain and Ireland. Most of Ireland left the Union in 1922 to form the Irish Free State; consequently, the full name of the United Kingdom since 1927 is the "United Kingdom of Great Britain and Northern Ireland".


title: 2014 Scottish independence referendum

text: The Labour Party was committed to home rule for Scotland in the 1920s, but it slipped down its agenda in the following years. The Scottish National Party (SNP) was founded in 1934, but did not achieve significant electoral success until the 1960s. A petition calling for home rule, the Scottish Covenant, was signed by 2,000,000 people (out of a population of 5,000,000) in the late-1940s.


title: 2014 Scottish independence referendum

text: As James Callaghan's Labour Government of the late 1970s was pressured by the SNP, Scottish devolution was, for the first time, seriously proposed. In a 1979 referendum for a devolved Scottish Assembly, a narrow majority of votes were cast in favour of devolution, but this had no effect due to a requirement that the number voting 'Yes' had to exceed 40% of the total electorate.


title: 2014 Scottish independence referendum

text: No further constitutional reform was proposed until Labour returned to power in a landslide electoral victory in May 1997. A second Scottish devolution referendum was held later that year, as promised in the Labour election manifesto. Clear majorities expressed support for both a devolved Scottish Parliament (74.3% in favour) and that Parliament having the power to vary the basic rate of UK income tax (63.5% in favour). The Scotland Act 1998 established the new Scottish Parliament, first elected on 6 May 1999, with power to legislate on unreserved matters within Scotland.


title: 2014 Scottish independence referendum

text: A commitment to hold an independence referendum in 2010 was part of the SNP's election manifesto when it contested the 2007 Scottish Parliament election. The press were largely hostile towards the SNP, with a headline for "The Scottish Sun" in May 2007 stating – along with an image of a hangman's noose – "Vote SNP today and you put Scotland's head in the noose". As a result of that election, the SNP became the largest party in the Scottish Parliament and formed a minority government led by the First Minister, Alex Salmond.


title: 2014 Scottish independence referendum

text: The SNP administration launched a 'National Conversation' as a consultation exercise in August 2007, part of which included a draft referendum bill, the "Referendum (Scotland) Bill". After this, a white paper for the proposed Referendum Bill was published, on 30 November 2009. It detailed 4 possible scenarios, with the text of the Bill and Referendum to be revealed later. The scenarios were: no change; devolution per the Calman Review; further devolution; and full independence. The Scottish government published a draft version of the bill on 25 February 2010 for public consultation; "Scotland's Future: Draft Referendum (Scotland) Bill Consultation Paper" contained a consultation document and a draft version of the bill. The consultation paper set out the proposed ballot papers, the mechanics of the proposed referendum, and how the proposed referendum was to be regulated. Public responses were invited.


title: 2014 Scottish independence referendum

text: The bill outlined three proposals: the first was full devolution or 'devolution max', suggesting that the Scottish Parliament should be responsible for "all laws, taxes and duties in Scotland", except for "defence and foreign affairs; financial regulation, monetary policy and the currency", which would be retained by the UK government. The second proposal outlined Calman-type fiscal reform, gaining the additional powers and responsibilities of setting a Scottish rate of income tax that could vary by up to 10p in the pound compared with the rest of the UK, setting the rate of stamp duty land tax and "other minor taxes", and introducing new taxes in Scotland with the agreement of the UK Parliament, and finally, "limited power to borrow money". The third proposal was for full independence.


title: 2014 Scottish independence referendum

text: In the 3rd Scottish Parliament only 50 of 129 MSPs (47 SNP, 2 Greens, and Margo MacDonald) supported a referendum. The Scottish government withdrew the bill in September 2010, after failing to secure opposition support.


title: 2014 Scottish independence referendum

text: The SNP repeated its commitment to hold an independence referendum when it published its manifesto for the 2011 Scottish Parliament election. Days before the election, Salmond stated that legislation for a referendum would be proposed in the "second half of the parliament", as he wanted to secure more powers for the Scottish Parliament via the Scotland Bill first. In the election, the SNP won the largest percentage of votes. UK Prime Minister David Cameron congratulated Salmond on his "emphatic win", but pledged that he would campaign for the Union if the SNP carried out their pledge to hold a referendum.


title: 2014 Scottish independence referendum

text: In January 2012, the UK government offered to legislate to provide the Scottish Parliament with the powers to hold a referendum, providing it was "fair, legal and decisive". This would set "terms of reference for the referendum", such as its question(s), elector eligibility and which body would organise the vote. As the UK government worked on legal details, including the timing of the vote, Salmond announced an intention to hold the referendum in the autumn of 2014. Negotiations continued between the two governments until October 2012, when the Edinburgh Agreement was reached.


title: 2014 Scottish independence referendum

text: The Scottish Independence Referendum (Franchise) Act 2013 was passed by the Scottish Parliament on 27 June 2013 and received Royal Assent on 7 August 2013. On 26 November 2013, the Scottish government published "Scotland's Future", a 670-page white paper laying out the case for independence and the means through which Scotland might become an independent country.


title: 2014 Scottish independence referendum

text: The Scottish Government announced on 21 March 2013 that the referendum would be held on Thursday 18 September 2014. Some media reports mentioned that 2014 would be the 700th anniversary of the Battle of Bannockburn and that Scotland would also host the 2014 Commonwealth Games and the 2014 Ryder Cup. Salmond agreed that the presence of these events made 2014 a "good year to hold a referendum".


title: 2014 Scottish independence referendum

text: Under the terms of the 2010 Draft Bill, the following people were entitled to vote in the referendum:


title: 2014 Scottish independence referendum

text: Convicted prisoners were not able to vote in the referendum. The European Court of Human Rights (ECHR) had ruled that this restriction was unlawful, but Scottish judge Lord Glennie said that he believed the ECHR judgment would apply only to parliamentary elections. Appeals against his ruling were rejected by the Court of Session in Edinburgh and the UK Supreme Court.


title: 2014 Scottish independence referendum

text: The normal voting age was reduced from 18 to 16 for the referendum, as it was a Scottish National Party policy to reduce the voting age for all elections in Scotland. The move was supported by Labour, the Liberal Democrats and the Scottish Greens.


title: 2014 Scottish independence referendum

text: In January 2012, Labour MSP Elaine Murray led a debate arguing that the franchise should be extended to Scots living outside Scotland, including the approximately 800,000 living in the other parts of the UK. This was opposed by the Scottish government, which argued that it would greatly increase the complexity of the referendum and stated that there was evidence from the United Nations Human Rights Committee that other nations "might question the legitimacy of a referendum if the franchise is not territorial".


title: 2014 Scottish independence referendum

text: In the House of Lords, Baroness Symons argued that the rest of the UK should be allowed to vote on Scottish independence because it would affect the whole country. This argument was rejected by the UK government, as the Advocate General for Scotland Lord Wallace said that "whether or not Scotland should leave the United Kingdom is a matter for Scotland". Wallace also pointed to the fact that only two of 11 referendums since 1973 had been across all of the United Kingdom. Professor John Curtice cited the precedent of the 1973 Northern Ireland sovereignty referendum (the "border poll"), which allowed only those resident in a part of the UK to vote on its sovereignty.


title: 2014 Scottish independence referendum

text: There was debate as to whether the Scottish Parliament had the power to legislate for a referendum relating to the issue of Scottish independence, as the constitution is a matter reserved to the UK Parliament. The Scottish government insisted in 2010 that they could legislate for a referendum, as it would be an "advisory referendum on extending the powers of the Scottish Parliament", whose result would "have no legal effect on the Union". Lord Wallace, Advocate General for Scotland, said in January 2012 that holding a referendum concerning the constitution would be outside the legislative power of the Scottish Parliament and that private individuals could challenge a Scottish Parliament referendum bill.


title: 2014 Scottish independence referendum

text: The two governments signed the Edinburgh Agreement, which allowed for the temporary transfer of legal authority. Per the Edinburgh Agreement, the UK government drafted an Order in Council granting the Scottish Parliament the necessary powers to hold, on or before 31 December 2014, an independence referendum. The draft Order was approved by resolutions of both Houses of Parliament, and the Order ("The Scotland Act 1998 (Modification of Schedule 5) Order 2013"), was approved by Queen Elizabeth II at a meeting of the Privy Council on 12 February 2013. Under the powers temporarily transferred from Westminster under the section 30 Order, the Scottish Parliament adopted the Scottish Independence Referendum Act 2013. The Act was passed by the Scottish Parliament on 14 November 2013 and received Royal Assent on 17 December 2013. Under section 36 of the Act, it came into force the day after Royal Assent.


title: 2014 Scottish independence referendum

text: The Electoral Commission was responsible for overseeing the referendum, "with the exception of the conduct of the poll and announcement of the result, and the giving of grants. In its role of regulating the campaign and campaign spending, the Electoral Commission will report to the Scottish Parliament. (...) The poll and count will be managed in the same way as [local] elections, by local returning officers (...) and directed by a Chief Counting Officer".


title: 2014 Scottish independence referendum

text: The Edinburgh Agreement stated that the wording of the question would be decided by the Scottish Parliament and reviewed for intelligibility by the Electoral Commission. The Scottish government stated that its preferred question was "Do you agree that Scotland should be an independent country?" The Electoral Commission tested the proposed question along with three other possible versions. Their research found that the "Do you agree" preface made it a leading question, which would be more likely to garner a positive response. The question was amended to "Should Scotland be an independent country?", which the Electoral Commission found was the most neutral and concise of the versions tested.


title: 2014 Scottish independence referendum

text: In the 2010 Draft Bill, the Scottish government proposed that there would be one designated campaign organisation for each outcome, both of which would be permitted to spend up to £750,000 on their campaign and to send one free mailshot to every household or voter in the referendum franchise. There was to be no public funding for campaigns. Registered political parties were each to be allowed to spend £100,000. This proposed limit on party spending was increased to £250,000 in 2012.


title: 2014 Scottish independence referendum

text: In 2013, new proposals by the Electoral Commission for the 16-week regulated period preceding the poll were accepted. They allowed the two designated campaign organisations to spend up to £1.5 million each and for the parties in Scotland to spend the following amounts: £1,344,000 (SNP); £834,000 (Labour); £396,000 (Conservatives); £201,000 (Liberal Democrats); £150,000 (Greens). An unlimited number of other organisations could register with the Electoral Commission, but their spending was limited to £150,000. Groups spending more than £250,000 were required to submit audited returns to the Commission by 18 March 2015.


title: 2014 Scottish independence referendum

text: According to the Scottish government's consultation paper published on 25 February 2010, the cost of the referendum was "likely to be around £9.5 million", mostly spent on running the poll and the count. Costs would also include the posting of one neutral information leaflet about the referendum to every Scottish household, and one free mailshot to every household or voter in the poll for the designated campaign organisations. In April 2013, the projected cost of the referendum was £13.3 million; the final administrative cost was £15.85 million.


title: 2014 Scottish independence referendum

text: Its chief executive was Blair Jenkins, formerly the Director of Broadcasting at STV and Head of News and Current Affairs at both STV and BBC Scotland. The campaign was supported by the SNP, the Scottish Green Party (which also created "its own pro-independence campaign to run alongside Yes Scotland") and the Scottish Socialist Party.


title: 2014 Scottish independence referendum

text: At its launch, Salmond stated that he hoped one million people in Scotland would sign a declaration of support for independence. On 22 August 2014, Yes Scotland announced that the one million target had been surpassed.


title: 2014 Scottish independence referendum

text: The campaign in favour of Scotland remaining in the UK, Better Together, was launched on 25 June 2012. It was led by Alistair Darling, former Chancellor of the Exchequer, and had support from the Conservative Party, Labour Party and Liberal Democrats.


title: 2014 Scottish independence referendum

text: Political advertising on television and radio in the UK was prohibited by the Communications Act 2003, with the exception of permitted party political broadcasts. Three major cinema chains stopped showing adverts by referendum campaign groups after receiving negative feedback from their customers.


title: 2014 Scottish independence referendum

text: In December 2013, the Better Together campaign declared that it had received donations of £2,800,000. Six-figure contributions were made by businessmen Ian Taylor and Donald Houston, and by author C. J. Sansom. Almost 27,000 donations of under £7,500 had been received by the same date. A later donation came from writer J. K. Rowling, who announced in June 2014 that she had given £1,000,000 to the Better Together campaign. In the following month, whisky distiller William Grant & Sons announced a donation of approximately £100,000. On 12 August 2014, Better Together announced that it had raised enough money to cover the maximum spending permitted and was no longer accepting donations. This was attributed in part to a large number of small donations being received after the first televised debate between First Minister Alex Salmond and Labour MP Alistair Darling.


title: 2014 Scottish independence referendum

text: As of May 2014, the Yes Scotland campaign had declared £4,500,000 in cash donations. EuroMillions lottery-winners Chris and Colin Weir gave £3,500,000. A six-figure donation was given by investment fund manager Angus Tulloch; approximately 18,000 donations of less than £7,500 had been made by the same date.


title: 2014 Scottish independence referendum

text: Voting for the referendum commenced on 27 August 2014, with the receipt of ballots by postal voters. As of 15 August, 680,235 eligible voters had registered for postal voting, a 20% increase compared with March 2014. During the postal vote phase, Police Scotland arrested a man from Glasgow on suspicion of selling his vote on eBay.


title: 2014 Scottish independence referendum

text: The registration deadline for referendum voters was 2 September 2014. Several councils reported the processing of "unprecedented" numbers of new registrations, while others received "tens of thousands" of applications in the final week.


title: 2014 Scottish independence referendum

text: The UK Government stated that, if a simple majority of the votes cast were in favour of independence, then "Scotland would become an independent country after a process of negotiations". If the majority was against independence, Scotland would continue to be a part of the United Kingdom. Further powers would be devolved to the Scottish Parliament as a result of the Scotland Act 2012. The Electoral Commission prepared an information leaflet which confirmed that the UK and Scottish governments had reached agreement on these points. David Cameron said in May 2014 that he believed that the referendum would be "'irreversible and binding". In the week before the referendum, both sides said that they would abide by the result. Alex Salmond said that it was a "once in a generation" event, citing the precedent of the two devolution referendums in 1979 and 1997.


title: 2014 Scottish independence referendum

text: In the event of a majority for Yes, the Scottish Government had proposed an independence date of 24 March 2016. It was suggested that, following the conclusion of negotiations, the UK Parliament would legislate for Scottish independence to take place on the date that had been negotiated. A report by a UK House of Lords committee, published in May 2014, said that the UK could opt to delay the independence date.


title: 2014 Scottish independence referendum

text: In 2013, as part of a European Union (EU) member state, Scottish farmers received £583 million in subsidy payments from the EU under the Common Agricultural Policy (CAP). Annual CAP payments are made to the UK, which then determines how much to allocate to each of the devolved administrations, including Scotland. In the last CAP agreement, farmers in the UK qualified for additional convergence payments because Scottish farmers received a lower average single farm payment per hectare, mainly due to the mountainous terrain in Scotland. Supporters of independence therefore believed that an independent Scotland would receive greater agricultural subsidies as an independent state. Opponents of independence believed that Scottish farmers benefited from the Union because the UK was one of the larger EU member states, and therefore had a greater say in CAP negotiations. They also questioned whether an independent Scotland would immediately receive full subsidy payments from the EU, as recent new member states had their subsidies phased in.


title: 2014 Scottish independence referendum

text: At the time of the independence referendum, the UK had some opt-outs from EU policies. One was the opt-out from the Schengen Area, meaning there were full passport checks for travellers from other EU countries except the Republic of Ireland, which was part of a Common Travel Area (CTA) with the UK. The Scottish government proposed that an independent Scotland would remain outside the Schengen Area and remain inside the CTA, ensuring that no passport controls would be needed at the Anglo-Scottish border. Nicola Sturgeon commented that an independent Scotland would negotiate with the EU to have the same visa arrangements as the UK has.


title: 2014 Scottish independence referendum

text: Alistair Carmichael, the Secretary of State for Scotland, said in January 2014 that it would make sense for Scotland to be in the CTA, but it would have to operate similar immigration policies to the rest of the UK. This position was supported by Home Secretary Theresa May, who said in March 2014 that passport checks should be introduced if Scotland adopted a looser immigration policy. The Conservative MP Richard Bacon said there would be "no reason" for border controls to be implemented.


title: 2014 Scottish independence referendum

text: In the white paper "Scotland's Future", the Scottish government pledged to expand childcare provision in an independent Scotland. The paper stated that this policy would cost £700 million, but that this would be financed by increased tax revenue from an additional 100,000 women returning to work.


title: 2014 Scottish independence referendum

text: Scottish Labour leader Johann Lamont said that the policy should be implemented immediately if the Scottish government believed it would have a beneficial effect, but Alex Salmond responded that under devolution the costs of the policy would have to be financed by cuts elsewhere in public expenditure. In March 2014, the National Day Nurseries Association said that the plan could not be implemented unless greater funding was provided by local authorities to private nurseries. A report by the Scottish Parliament Information Centre questioned the economic benefit of the policy, pointing out that there were only 64,000 mothers of children aged between 1 and 5 who were economically inactive. A spokesman for Salmond said that the estimated total 104,000 women would enter the workforce over a longer period, as future generations of mothers would also be able to work, stating: "The key point about the policy is that it doesn't happen on one day or one year and then cease."


title: 2014 Scottish independence referendum

text: The Scottish government proposed that all British citizens who had been born in Scotland would automatically become Scottish citizens on the date of independence, regardless of whether or not they were then living in Scotland. British citizens "habitually resident" in Scotland would also be considered Scottish citizens, even if they already held the citizenship of another country. Every person who would automatically be considered a Scottish citizen would be able to opt-out of Scottish citizenship provided they already held the citizenship of another country. The Scottish government also proposed that anyone with a Scottish parent or grandparent would be able to apply for registration as a Scottish citizen, and any foreign national living in Scotland legally, or who had lived in Scotland for at least 10 years at any time and had an ongoing connection to Scotland, should be able to apply for naturalisation as a Scottish citizen. UK Home Secretary Theresa May said the future policies of an independent Scottish government would affect whether Scottish citizens would be allowed to retain British citizenship. An analysis paper published by the UK government in January 2014 stated that it was likely that Scots would be able to hold dual citizenship; however, the duality was considered for all other countries, not specifically to the rest of the UK. The possibility of holding dual UK–Scotland citizenships could be subject to the "proof of affinity".


title: 2014 Scottish independence referendum

text: Conspiracy theories were proliferated by supporters of Scottish independence included the claim that MI5, the British government, and other intelligence agencies were pretending to be Scottish nationalists online, pretending to be "cybernats" (a derogatory term for supporters of Scottish nationalism online, especially trolls) or engaging in ballot tampering or other forms of vote fraud. A YouGov poll in early September showed 25 percent of the electorate believed MI5 was working with the British government to block independence, with many voters so fearful the marks made by the pencils provided in polling stations would be tampered with, they brought their own pens (see: Voting pencil conspiracy theory).


title: 2014 Scottish independence referendum

text: The SNP said that there was a defence underspend of "at least £7.4 billion" between 2002 and 2012 in Scotland and that independence would allow the Scottish government to correct this imbalance. In its white paper, the Scottish government planned that an independent Scotland would have a total of 15,000 regular and 5,000 reserve personnel across land, air and maritime forces by 2026. In July 2013, the SNP proposed that there would be a £2.5 billion annual military budget in an independent Scotland. The House of Commons Defence Select Committee said that the £2.5bn budget was too low. Andrew Murrison, UK Minister for International Security Strategy agreed and said it was "risible" for the SNP to suggest it could create an independent force by "salami-slicing" from current British armed forces units.


title: 2014 Scottish independence referendum

text: The House of Commons defence committee also stated that Scottish independence would have a negative effect on its industry, while the UK government said it would not be willing to build warships in a foreign country. Geoff Searle, the director of BAE Systems' Type 26 Global Combat Ship programme, said in June 2014 that the company had no alternative plan for shipbuilding, but this position was later revised by the Chairman of BAE, who stated that they could resume shipbuilding in the English city of Portsmouth if an independent Scotland was established. The chief executive of Thales, one of Britain's largest defence suppliers, said that if Scotland became independent that this might raise questions about continued investment from his firm.


title: 2014 Scottish independence referendum

text: The Royal United Services Institute said in 2012 that an independent Scotland could set up a Scottish Defence Force, comparable in size and strength to those of other small European states like Denmark, Norway and Ireland, at an annual cost of £1.8 billion. The authors acknowledged that an independent Scotland would "need to come to some arrangement with the rest of the UK" on intelligence-gathering, cyber-warfare and cyber-defence, that the future cost of purchasing and maintaining equipment of its forces might be higher due to smaller orders, and that recruitment and training "may prove problematic" in the early years.


title: 2014 Scottish independence referendum

text: Dorcha Lee, a former colonel in the Irish Army, said that Scotland could eschew forming an army based on inherited resources from the British Army and instead follow an Irish model of a limited self-defence force.


title: 2014 Scottish independence referendum

text: At the time of the referendum, the Trident nuclear missile system was based at Coulport weapons depot and naval base of Faslane in the Firth of Clyde area. The SNP objected to having nuclear weapons on Scottish territory, but British military leaders said that there was no alternative site for the missiles. In April 2014, several British military leaders co-signed a letter stating that forcing Trident to leave Scottish waters would place the UK nuclear deterrent in jeopardy. "Nowhere to Go", a report by Scottish CND, concluded that the removal of Trident from Scotland would force unilateral nuclear disarmament by the United Kingdom, as the weapons would have no viable alternative base. A report by the Royal United Services Institute said that relocating Trident would be "very difficult, but not impossible" and estimated that it would take about 10 years and create an additional cost of around £3 billion.


title: 2014 Scottish independence referendum

text: A seminar hosted by the Carnegie Endowment for International Peace stated that the Royal Navy would have to consider a range of alternatives, including disarmament. A report in 2013 from the Scotland Institute think tank suggested a future Scottish government could be convinced to lease the Faslane nuclear base to the rest of the UK, in order to maintain good diplomatic relations and expedite NATO entry negotiations.


title: 2014 Scottish independence referendum

text: SNP policy had historically stated that an independent Scotland should not seek to obtain membership of NATO, until this was dropped after a vote by party members in 2012. MSPs John Finnie and Jean Urquhart resigned from the SNP over this policy change. The Scottish Green Party and Scottish Socialist Party continued their opposition to NATO membership.