title: Psy

text: On November 7, 2012, Psy held a speech in England at the Oxford Union to discuss the inspiration behind "Gangnam Style" and his next album. He told the audience that due to the success of "Gangnam Style" he is now living in both a dream and a nightmare, as it will be difficult for his next song to equal "Gangnam Style"'s success. He also talked about his early life and the moment he realized "Gangnam Style" became famous. According to "The Independent", tickets for his speech were "in such demand they had to be assigned by ballot—a method not required when former presidential candidate John McCain spoke earlier that year, nor when Mother Teresa, the Dalai Lama nor Michael Jackson spoke".


title: Psy

text: On November 12, 2012, Psy became the second South Korean music artist to appear at the MTV Europe Music Awards where he performed "Gangnam Style" and held off competition from Rihanna, Katy Perry, and Lady Gaga to win the "Best Video" award. The event was broadcast worldwide and hosted by the German model and actress Heidi Klum, who introduced Psy to the audience as the "undisputed King of Pop". A few days later, American singer-songwriter Madonna performed a mashup of "Gangnam Style" and "Give It 2 Me" alongside Psy and her backup dancers during a concert in New York City at Madison Square Garden during The MDNA Tour. Psy later told reporters that his gig with Madonna had "topped his list of accomplishments".


title: Psy

text: On November 24, 2012, "Gangnam Style" became the most viewed video in YouTube history, surpassing the previous most-watched video, Justin Bieber's "Baby". The number of views was achieved about eleven times faster than Bieber's. Psy later won four awards at the 2012 Mnet Asian Music Awards in Hong Kong on November 30, 2012. On December 21, 2012, "Gangnam Style" reached 1 billion views on YouTube, becoming the first video to do so in the website's history. He met actor and stunt performer Jackie Chan, who called him a role model that proved that "dreams do come true". Also, he was selected as Singer of the Year in the public survey conducted by Gallup Korea that year.


title: Psy

text: In January 2013, Psy was announced as a winner of the 2013 Korea Image Awards, being awarded the Korea Image Stepping Stone Award for "his contribution to enhancing the national image". Psy made his debut on South American television by giving an interview on the Brazilian news program "Fantástico". It was announced on January 27, 2013, that Psy would perform at South Korea's presidential inauguration ceremony on February 25, 2013.


title: Psy

text: On April 12, 2013, the audio of Psy's follow-up single "Gentleman" was leaked onto the internet, a day before its official international release. On the following day, the music video for 'Gentleman' premiered at Psy's 'Happening' Concert, which was attended by 50,000 people and live streamed on YouTube to an audience of 150,000. Guest performers of the concert included Lee Hi, 2NE1 and G-Dragon. He reportedly invested US$2.7 million into the production of the concert. Psy continued to promote Gentlemen throughout 2013. In May 2013, PSY appeared twice on Live! with Kelly and Michael and taught Kelly Ripa and Michael Strahan how to do the Gentleman Dance. Psy also appeared in the finale of "American Idol" season 12 and sang "Gentleman". Psy also performed his song "Gentleman" on the Finale of Dancing With the Stars, Season 16. Five days later, Psy also performed at the final match of the 2012–13 Coppa Italia. On June 8, 2013, Psy and his troupe performed "Gentleman" on the Finale of Britain's Got Talent, Series 7. Later that month, on June 8, Psy co-hosted the Canadian MuchMusic Video Awards, where he also opened the show with his worldwide hit "Gangnam Style" and ended the show with his "official last performance of 'Gentleman' on TV."


title: Psy

text: Psy was slated to star in a South Korean remake of the Hindi film "", taking over the role of "Vishnu" (Prabhu Deva) in the original film. In April he became the tourism ambassador of South Korea for 2013. On May 9, Psy gave a special lecture at Harvard University. In this lecture, he spoke about his passion and other reasons for his popularity. On May 24, 2013, people began voting daily, at www.psygobibigo.com, for one of the Top Three Chefs, based on more than one hundred ninety video entries, in Psy's "Psy Needs a Chef" video contest. The Top Three Chefs in this contest were Aaron Contreras, Dj Park and Ricardo Caput. Voting ended June 10, 2013, at midnight Greenwich Mean Time. Aaron Contreras received 2,005 votes. Ricardo received 22,384 votes. Dj Park, won the popular vote with 22,533 votes. Winners were to be announced June 14, 2013. However, Psy postponed his decision until June 18, 2013, and chose Ricardo to be his chef, in spite of the popular vote.


title: Psy

text: On June 9, 2014, Psy released a new single, "Hangover", featuring American rapper Snoop Dogg, and announced his intention to release a new song titled "Father".


title: Psy

text: On December 1, 2014, the YouTube video for "Gangnam Style" garnered over 2 − 1 views, overflowing the YouTube counter to a negative number, resulting in a public comment from Google/YouTube saying "We never thought a video would be watched in numbers greater than a 32-bit integer (=2,147,483,647 views), but that was before we met PSY. "Gangnam Style" has been viewed so many times we had to upgrade to a 64-bit integer (9,223,372,036,854,775,808)!" Hovering over the counter of the YouTube video triggered an easter egg. A YouTube representative later revealed that the comment was a joke and that the company had already updated to a 64-bit integer months ago.


title: Psy

text: On March 28, 2015, he released a music video for his song "Father". On December 1, 2015, Psy released his seventh album titled "Chiljip PSY-da", with double title tracks "Daddy" and "Napal Baji". The album included features from label mate CL as well as Will.i.am Promotions for the album included two performances on "Inkigayo", where the single "Daddy" won a triple crown, as well as a performance on "You Hee-yeol's Sketchbook". "Daddy" was also included in the dance video game Just Dance 2017.


title: Psy

text: In May 2017, Psy released his eighth studio album, "4X2=8", with lead singles "I Luv it" and "New Face". The album features collaborations from several artists including label-mates G-Dragon and Taeyang. The music video for "I Luv It" featured actor Lee Byung-hun and comedian Pikotaro while for the "New Face" video Apink's Son Na-eun played the title role.


title: Psy

text: In 2019, he founded his own label P Nation. He signed his first artist Jessi on January 25, 2019. Two days later, on January 27, 2019, he also signed both Hyuna and Dawn, formerly of Cube Entertainment with the latter being a former member of Pentagon under the name E'Dawn. He also signed Crush on July 17, 2019, and Heize on September 16, 2020. On April 26, 2021, it was reported that he and Park Jin-young, the founder of JYP Entertainment, will collaborate together to form a new boy group each in "Loud", which premiered on June 5, 2021 on SBS. In 2022, Psy did his first concert in Dubai as part of the Expo 2020 on Korean national day event.


title: Psy

text: On April 29, 2022, Psy released his ninth studio album, "Psy 9th", with its lead single "That That" featuring Suga of BTS.


title: Psy

text: On June 10, 2022, Psy released the official SNS afternoon tour schedule of 'Psy Soak Show SUMMER SWAG 2022', a domestic tour of 7 cities in the country until August 20.


title: Psy

text: Following Psy's departure from YG Entertainment in 2018, Psy founded his own entertainment company, P Nation. Psy announced the first artist signed to his label in January 2019, the rapper Jessi. P Nation then signed Hyuna and Dawn, both formerly of Cube Entertainment.


title: Psy

text: As of November 2022, the label had 10 artists signed to the label, including the six-member boy band, TNX.


title: Psy

text: Park Jae-sang's stage name "Psy" derives from the word "psycho". Explaining his stage name, he said in a BBC interview, "what I thought was, you know, crazy about music, dancing, performance, so that kind of psycho".


title: Psy

text: In a 2012 interview with "The New York Times", Psy called Freddie Mercury of the British rock band Queen his biggest musical influence: "My lifetime role model and hero is Freddie Mercury of Queen. His songwriting skills, I cannot even approach, but his showmanship, I learned it from videos. I'm No. 1 in the U.K. right now, so if I have any chance to go there, I want to meet Queen and to tell them how much I got inspired by their music." Psy also revealed that the one celebrity he wants to meet most is the American actor Tom Cruise, who helped popularize "Gangnam style" on Twitter and tweeted whether Psy would "make a good future co-star Gangnam Style?".


title: Psy

text: Psy is known for his sense of humor in his concerts, where he imitates female singers such as Park Ji-yoon, Lee Hyo-ri, Lady Gaga, and Beyoncé.


title: Psy

text: Although his music is part of the K-pop (Korean popular music) genre, Beth Hong from "The Vancouver Observer" noted that Psy doesn't fit the standard K-pop idol image of being "incredibly young, good-looking, and able to carry a melodramatic note".


title: Psy

text: Lucy Williamson from the BBC recognized Psy as South Korea's "newest and biggest music star", but also described him as "unpolished, unpredictable and he doesn't look like your typical Korean idol". Sarah Charlton from Reuters called him a "chubby South Korean pop singer" that has found fame and popularity in a "sea of pretty K-pop stars". In South Korea, some have called him the "Bizarre Singer" while others consider him to be "the antithesis of what is popular in Korean pop music".


title: Psy

text: Chelsea Handler from "Chelsea Lately" jokingly described Psy as "Korea's Ricky Martin, as well as a sex symbol" during his introduction on the show, while Gil Kaufman from MTV described the singer as one of the "biggest pop sensations in the world".


title: Psy

text: As a result of his achievements, Psy is considered to be the first K-pop artist to make a breakthrough in the Western music industry. In an interview with Agence France-Presse, Psy affirmed that "[i]t will be only a matter of time before K-Pop will produce many others like Psy".


title: Psy

text: According to Hugo Swire, the British Minister of State for the Foreign Office, Psy's music has given the world a glimpse of the dynamism and vibrancy of modern Korea. South Korea's Ministry of Culture, Sports and Tourism recognized Psy's cultural achievements by bestowing upon him the Okgwan Order, a 4th grade Order of Cultural Merit.


title: Psy

text: On May 25, 2017, Psy was presented with a Diamond Play Button from YouTube, in recognition of his surpassing 10 million subscribers to his YouTube channel, making him the first Asian solo artist to ever do so.


title: Psy

text: On October 14, 2006, Psy married Yoo Hye-yeon, a cello major at Yonsei University and his girlfriend of three and a half years. According to the Korean Broadcasting System, the couple were introduced by a mutual acquaintance. They have twin daughters. Psy considers "Gangnam Style" to be the greatest achievement of his life. Psy finalized the purchase of a condominium unit in Westwood, Los Angeles, California, in November 2012. Psy made the purchase to relocate to Los Angeles following his signing with US-based music manager Scooter Braun to further his career.


title: Psy

text: In a July 2013 interview with "The Sunday Times", Psy stated that he has a drinking problem, noting that the only time he is not drinking alcohol is when he is hung over.


title: Psy

text: In 2022, he moved back to South Korea to start a record label, located in Gangnam. He lives in the north part of Gangnam area, north of the river.


title: Psy

text: In late 2001, Psy was arrested for smoking and possessing marijuana and was sentenced to 25 days in jail. As a result, he was unable to be at his grandfather's deathbed as he died of cancer, or attend his funeral. During an interview in 2012 Psy stated "I was very close to him. I was not there at the funeral: I will regret this for the rest of my life, because my grandpa loved me so much, and I couldn't be there for him on his deathbed." Psy was also fined in January 2002 for the marijuana incident.


title: Psy

text: In the early 2010s, South Korea's Sunshine Policy towards North Korea was downscaled as tensions between the two countries continued to increase. As a result, Psy was questioned by the media on several occasions regarding his views on North Korea.


title: Psy

text: On April 13, 2013, Psy attended a press conference where he expressed regret about his country's conflict with North Korea and described the situation as a "tragedy". He also expressed hope that North Koreans would one day be able to enjoy his music before elaborating that his job is to make everyone, including North Koreans, laugh. Right before the start of a concert in Seoul, Psy added: "Tonight me and 50,000 Korean people... we are going to sing out loud. We are going to shout out loud and we are really close to them, so they [the North Koreans] can hear."


title: Psy

text: In an interview with "The Daily Beast" in April 2013, Psy was asked to give his take on North Korean leader Kim Jong-un's threats against South Korea and the United States, to which he replied: "Well, as an entertainer, I don't want to talk about politics. As a Korean citizen, I want peace. That's all I can say. I want permanent peace."


title: Psy

text: On August 11, 2022, Psy donated to help those affected by the 2022 South Korean floods through the Hope Bridge Korea Disaster Relief Association.


title: Psy

text: In 2002, Psy participated in an anti-American concert after a U.S. military convoy accidentally struck and killed two 14-year-old South Korean schoolgirls in the Yangju highway incident. The soldiers involved in the incident were acquitted by U.S. military courts, which fueled a significant amount of anti-American sentiment in South Korea. Inspired by that incident, Psy lifted up a miniature model of an 'American tank' and smashed it against the stage.


title: Psy

text: In 2004, the South Korean translator and Christian missionary Kim Sun-il was kidnapped and beheaded in Iraq after the South Korean government refused to reconsider sending its armed forces to support the Iraq War. Although initial protests were only directed towards the South Korean government and towards extremists in Iraq, anti-U.S. military protesters decided to seize the moment to trigger a much larger wave of anti-Americanism. During a concert, Psy admonished the Iraqi kidnappers, condemned South Korea's former president Roh Moo-hyun, and also sang along to lyrics of the song "Dear American" by South Korean rock band N.EX.T, which criticizes the United States military for its actions in the Iraq war. An initial translation of the lyrics was posted by an iReporter onto CNN's iReport site. Some of the lyrics, referring to the guards who tortured Iraqi prisoners, were translated by CNN as follows: "Kill those fucking Yankees who have been torturing Iraqi captives and those who ordered them to torture" and "Kill [the Yankees'] daughters, mothers, daughters-in-law and fathers / Kill them all slowly and painfully."


title: Psy

text: A few days later "The Washington Post" raised questions about the accuracy of the translation of the lyrics into English, which originated from CNN's citizen journalism initiative iReport. Max Fisher of "The Washington Post" foreign staff reported that he solicited native Korean and English speakers, academics accustomed to the sensitivity of word-for-word translations, young Koreans familiar with the cultural connotations of the lyrics, and a professional interpreter to offer their translation of the lyrics, and found out that the lyrics may have actually slurred the American servicemen rather than calling for their deaths, although he did also go on to opine that, "using a racial slur to accuse Americans of killing Iraqis' family members is still pretty serious". Fisher also states that the word translated 'Yankee' in the CNN iReport was underplayed, with one Korean American describing the slur as a "nearly untranslatable" racist "epithet," perhaps best approximated as "foreign barbarian." Fisher notes that one translator indicates that the slur is meant as "a derogatory term for American", and that others have translated the slur alternately as "Yankee", "big nose", and "despicable Western women and men".


title: Psy

text: Although Psy's actions did not receive any significant international media coverage at that time, this changed after the media reported about it in early December 2012. On December 7, 2012, Psy issued an apology directed towards members of the U.S. military and to the American people for his "inflammatory and inappropriate" language, and expressed hope that the American public would accept his apology.


title: Psy

text: Despite initial public outrage, White House spokesman Josh Earnest told the media that U.S. President Barack Obama and First Lady Michelle Obama will attend Psy's performance at the "Christmas in Washington" charity concert as planned. As a result of the controversy, a petition circulated on the website of the White House demanding that he be dropped from the concert, although the petition was deleted later in the day because the White House website claimed that the petition violated the website's terms of participation. "Time" magazine's Nick Carbone asserted that it is "unlikely that these newly dug-up anecdotes will depose Psy from his king-like level of stardom" although Carbone did go on to write that the atmosphere at the White House concert would be "somewhat subdued".


title: Psy

text: Psy was invited to perform on February 11, 2013, in Penang, Malaysia for the ruling Barisan Nasional party in an attempt to reach out to young voters in the area. Media reports estimated a cost of at least US$300,000 for the performance. Opposition backers called for supporters of the Democratic Action Party to attend wearing opposition colors. Despite the controversy and calls by Malaysian citizens to cancel the concert, Psy completed a Chinese New Year open house performance at Han Chiang School for a crowd of 100,000.


title: Psy

text: Following the concert, a media report revealed that Psy received death threats prior to entering the country and this was the reason for the delay in Psy's on-stage appearance. Psy eventually performed for ten minutes to an excited crowd. The media report also reported on a press conference at the Jelutong Gerakan Chinese New Year open house that revealed that questions were raised over the financing of the event. State Barisan Nasional (BN) party chief Teng Chang Yeow explained that the event "was sponsored by private sponsors who preferred to remain anonymous as they are afraid of being 'victimised' if their identities were revealed". Teng also stated that he was given the choice of a Justin Bieber visit, but chose Psy as he didn't know who Bieber was at the time.


title: Psy

text: In March 2013, South Korean media reported that the title of Psy's upcoming single would contain the word "Assarabia", a slang used by South Koreans to express thrills, or simply to describe something satisfying. There were numerous objections to that upcoming title, and worries have risen that people of Arabic descent might misinterpret the title and find it derogatory.


title: Psy

text: After being questioned by a correspondent of the Voice of America about the upcoming track's potential to offend, Psy said that there has been some misunderstanding and his upcoming song will undergo a major revamp. On March 19, 2013, he revised his song's title and lyrics over "worries it could offend Arabs". Many people consider the change of his song title deliberate, aimed solely at building publicity for a strong follow-up to "Gangnam Style". He subsequently renamed the song "Gentleman" and released it on April 12, 2013.


title: Psy

text: According to "Korea Times" in 2013, Psy earned more than US$40 million in 2012 via sales through concerts, TV commercials, and social media revenue through his hit song "Gangnam Style", while his estimated personal income in 2012 was US$28 million under his 7:3 profit-sharing contract with YG Entertainment.


title: Sapphire

text: Sapphire is a precious gemstone, a variety of the mineral corundum, consisting of aluminium oxide () with trace amounts of elements such as iron, titanium, chromium, vanadium, or magnesium. The name sapphire is derived via the Latin "sapphirus" from the Greek "sappheiros", which referred to lapis lazuli. It is typically blue, but natural "fancy" sapphires also occur in yellow, purple, orange, and green colors; "parti sapphires" show two or more colors. Red corundum stones also occur, but are called rubies rather than sapphires. Pink-colored corundum may be classified either as ruby or sapphire depending on locale. Commonly, natural sapphires are cut and polished into gemstones and worn in jewelry. They also may be created synthetically in laboratories for industrial or decorative purposes in large crystal boules. Because of the remarkable hardness of sapphires 9 on the Mohs scale (the third hardest mineral, after diamond at 10 and moissanite at 9.5) sapphires are also used in some non-ornamental applications, such as infrared optical components, high-durability windows, wristwatch crystals and movement bearings, and very thin electronic wafers, which are used as the insulating substrates of special-purpose solid-state electronics such as integrated circuits and GaN-based blue LEDs. Sapphire is the birthstone for September and the gem of the 45th anniversary. A sapphire jubilee occurs after 65 years.


title: Sapphire

text: Sapphire is one of the two gem-varieties of corundum, the other being ruby (defined as corundum in a shade of red). Although blue is the best-known sapphire color, they occur in other colors, including gray and black, and also can be colorless. A pinkish orange variety of sapphire is called padparadscha.


title: Sapphire

text: Significant sapphire deposits are found in Australia, Afghanistan, Cambodia, Cameroon, China (Shandong), Colombia, Ethiopia, India (Kashmir), Kenya, Laos, Madagascar, Malawi, Mozambique, Myanmar (Burma), Nigeria, Rwanda, Sri Lanka, Tanzania, Thailand, United States (Montana) and Vietnam. Sapphire and rubies are often found in the same geographical settings, but they generally have different geological formations. For example, both ruby and sapphire are found in Myanmar's Mogok Stone Tract, but the rubies form in marble, while the sapphire forms in granitic pegmatites or corundum syenites.


title: Sapphire

text: Every sapphire mine produces a wide range of quality, and origin is not a guarantee of quality. For sapphire, Kashmir receives the highest premium, although Burma, Sri Lanka, and Madagascar also produce large quantities of fine quality gems.


title: Sapphire

text: The cost of natural sapphires varies depending on their color, clarity, size, cut, and overall quality. Sapphires that are completely untreated are worth far more than those that have been treated. Geographical origin also has a major impact on price. For most gems of one carat or more, an independent report from a respected laboratory such as GIA, Lotus Gemology, or SSEF, is often required by buyers before they will make a purchase.


title: Sapphire

text: Gemstone color can be described in terms of hue, saturation, and tone. Hue is commonly understood as the "color" of the gemstone. Saturation refers to the vividness or brightness of the hue, and tone is the lightness to darkness of the hue. Blue sapphire exists in various mixtures of its primary (blue) and secondary hues, various tonal levels (shades) and at various levels of saturation (vividness).


title: Sapphire

text: Blue sapphires are evaluated based upon the purity of their blue hue. Violet, and green are the most common secondary hues found in blue sapphires. The highest prices are paid for gems that are pure blue and of vivid saturation. Gems that are of lower saturation, or are too dark or too light in tone are of less value. However, color preferences are a personal taste, like a flavor of ice cream.


title: Sapphire

text: The Logan sapphire in the National Museum of Natural History, in Washington, D.C., is one of the largest faceted gem-quality blue sapphires in existence. The 422.66-ct Siren of Serendip in the Houston Museum of Natural Science is another stunning example of a Sri Lankan sapphire on public display.


title: Sapphire

text: Particolored sapphires (or bi-color sapphires) are those stones that exhibit two or more colors within a single stone. The desirability of particolored or bi-color sapphires is usually judged based on the zoning or location of their colors, the colors’ saturation, and the contrast of their colors. Australia is the largest source of particolored sapphires; they are not commonly used in mainstream jewelry and remain relatively unknown. Particolored sapphires cannot be created synthetically and only occur naturally. The vast majority of particolored sapphires occur naturally, but it is possible to replicate the appearance of a particolored sapphire in a synthetic sapphire.


title: Sapphire

text: Pink sapphires occur in shades from light to dark pink, and deepen in color as the quantity of chromium increases. The deeper the pink color, the higher their monetary value. In the United States, a minimum color saturation must be met to be called a ruby, otherwise the stone is referred to as a "pink sapphire".


title: Sapphire

text: "Padparadscha" is a delicate, light to medium toned, pink-orange to orange-pink hued corundum, originally found in Sri Lanka, but also found in deposits in Vietnam and parts of East Africa. Padparadscha sapphires are rare; the rarest of all is the totally natural variety, with no sign of artificial treatment.


title: Sapphire

text: The name is derived from the Sanskrit "padma ranga" (padma = lotus; ranga = color), a color akin to the lotus flower ("Nelumbo nucifera").


title: Sapphire

text: Among the fancy (non-blue) sapphires, natural padparadscha fetch the highest prices. Since 2001, more sapphires of this color have appeared on the market as a result of artificial lattice diffusion of beryllium.


title: Sapphire

text: A "star sapphire" is a type of sapphire that exhibits a star-like phenomenon known as asterism; red stones are known as "star rubies". Star sapphires contain intersecting needle-like inclusions following the underlying crystal structure that causes the appearance of a six-rayed "star"-shaped pattern when viewed with a single overhead light source. The inclusion is often the mineral rutile, a mineral composed primarily of titanium dioxide. The stones are cut "en cabochon", typically with the center of the star near the top of the dome. Occasionally, twelve-rayed stars are found, typically because two different sets of inclusions are found within the same stone, such as a combination of fine needles of rutile with small platelets of hematite; the first results in a whitish star and the second results in a golden-colored star. During crystallization, the two types of inclusions become preferentially oriented in different directions within the crystal, thereby forming two six-rayed stars that are superimposed upon each other to form a twelve-rayed star. Misshapen stars or 12-rayed stars may also form as a result of twinning.


title: Sapphire

text: The inclusions can alternatively produce a cat's eye effect if the girdle plane of the cabochon is oriented parallel to the crystal's c-axis rather than perpendicular to it. To get a cat's eye, the planes of exsolved inclusions must be extremely uniform and tightly packed. If the dome is oriented in between these two directions, an off-center star will be visible, offset away from the high point of the dome.


title: Sapphire

text: At 1404.49 carats, The Star of Adam is the largest known blue star sapphire. The gem was mined in the city of Ratnapura, southern Sri Lanka. The Black Star of Queensland, the second largest star sapphire in the world, weighs 733 carats. The Star of India mined in Sri Lanka and weighing 563.4 carats is thought to be the third-largest star sapphire, and is currently on display at the American Museum of Natural History in New York City. The 182-carat Star of Bombay, mined in Sri Lanka and located in the National Museum of Natural History in Washington, D.C., is another example of a large blue star sapphire. The value of a star sapphire depends not only on the weight of the stone, but also the body color, visibility, and intensity of the asterism. The color of the stone has more impact on the value than the visibility of the star. Since more transparent stones tend to have better colors, the most expensive star stones are semi-transparent "glass body" stones with vivid colors.


title: Sapphire

text: On 28 July 2021, the world's largest cluster of star sapphires, weighing 510 kg, was unearthed from Ratnapura, Sri Lanka. This star sapphire cluster was named "Serendipity Sapphire".


title: Sapphire

text: A rare variety of natural sapphire, known as color-change sapphire, exhibits different colors in different light. Color change sapphires are blue in outdoor light and purple under incandescent indoor light, or green to gray-green in daylight and pink to reddish-violet in incandescent light. Color change sapphires come from a variety of locations, including Madagascar, Myanmar, Sri Lanka and Tanzania. Two types exist. The first features the chromium chromophore that creates the red color of ruby, combined with the iron + titanium chromophore that produces the blue color in sapphire. A rarer type, which comes from the Mogok area of Myanmar, features a vanadium chromophore, the same as is used in Verneuil synthetic color-change sapphire.


title: Sapphire

text: Virtually all gemstones that show the "alexandrite effect" (color change; a.k.a. 'metamerism') show similar absorption/transmission features in the visible spectrum. This is an absorption band in the yellow (~590 nm), along with valleys of transmission in the blue-green and red. Thus the color one sees depends on the spectral composition of the light source. Daylight is relatively balanced in its spectral power distribution (SPD) and since the human eye is most sensitive to green light, the balance is tipped to the green side. However incandescent light (including candle light) is heavily tilted to the red end of the spectrum, thus tipping the balance to red.


title: Sapphire

text: Color-change sapphires colored by the Cr + Fe/Ti chromophores generally change from blue or violetish blue to violet or purple. Those colored by the V chromophore can show a more pronounced change, moving from blue-green to purple.


title: Sapphire

text: Certain synthetic color-change sapphires have a similar color change to the natural gemstone alexandrite and they are sometimes marketed as "alexandrium" or "synthetic alexandrite". However, the latter term is a misnomer: synthetic color-change sapphires are, technically, not synthetic alexandrites but rather alexandrite "simulants". This is because genuine alexandrite is a variety of chrysoberyl: not sapphire, but an entirely different mineral.


title: Sapphire

text: Large rubies and sapphires of poor transparency are frequently used with suspect appraisals that vastly overstate their value. This was the case of the "Life and Pride of America Star Sapphire". Circa 1985, Roy Whetstine claimed to have bought the 1905-ct stone for $10 at the Tucson gem show, but a reporter discovered that L.A. Ward of Fallbrook, CA, who appraised it at the price of $1200/ct, had appraised another stone of the exact same weight several years before Whetstine claimed to have found it.


title: Sapphire

text: Bangkok-based Lotus Gemology maintains an updated listing of world auction records of ruby, sapphire, and spinel. As of November 2019, no sapphire has ever sold at auction for more than $17,295,796.


title: Sapphire

text: Rubies are corundum with a dominant red body color. This is generally caused by traces of chromium (Cr) substituting for the (Al) ion in the corundum structure. The color can be modified by both iron and trapped hole color centers.


title: Sapphire

text: Unlike localized ("intra-atomic") absorption of light, which causes color for chromium and vanadium impurities, blue color in sapphires comes from intervalence charge transfer, which is the transfer of an electron from one transition-metal ion to another via the conduction or valence band. The iron can take the form Fe or Fe, while titanium generally takes the form Ti. If Fe and Ti ions are substituted for Al, localized areas of charge imbalance are created. An electron transfer from Fe and Ti can cause a change in the valence state of both. Because of the valence change, there is a specific change in energy for the electron, and electromagnetic energy is absorbed. The wavelength of the energy absorbed corresponds to yellow light. When this light is subtracted from incident white light, the complementary color blue results. Sometimes when atomic spacing is different in different directions, there is resulting blue-green dichroism.


title: Sapphire

text: Purple sapphires contain trace amounts of chromium and iron plus titanium and come in a variety of shades. Corundum that contains extremely low levels of chromophores is near colorless. Completely colorless corundum generally does not exist in nature. If trace amounts of iron are present, a very pale yellow to green color may be seen. However, if both titanium and iron impurities are present together, and in the correct valence states, the result is a blue color.


title: Sapphire

text: Intervalence charge transfer is a process that produces a strong colored appearance at a low percentage of impurity. While at least 1% chromium must be present in corundum before the deep red ruby color is seen, sapphire blue is apparent with the presence of only 0.01% of titanium and iron.


title: Sapphire

text: The most complete description of the causes of color in corundum extant can be found in Chapter 4 of "Ruby & Sapphire: A Gemologist's Guide" (chapter authored by John Emmett, Emily Dubinsky and Richard Hughes).


title: Sapphire

text: Sapphires can be treated by several methods to enhance and improve their clarity and color. It is common practice to heat natural sapphires to improve or enhance their appearance. This is done by heating the sapphires in furnaces to temperatures between for several hours, or even weeks at a time. Different atmospheres may be used. Upon heating, the stone becomes bluer in color, but loses some of the rutile inclusions (silk). When high temperatures (1400 °C+) are used, exsolved rutile silk is dissolved and it becomes clear under magnification. The titanium from the rutile enters solid solution and thus creates with iron the blue color The inclusions in natural stones are easily seen with a jeweler's loupe. Evidence of sapphire and other gemstones being subjected to heating goes back at least to Roman times. Un-heated natural stones are somewhat rare and will often be sold accompanied by a certificate from an independent gemological laboratory attesting to "no evidence of heat treatment".


title: Sapphire

text: Yogo sapphires do not need heat treating because their cornflower blue color is attractive out of the ground; they are generally free of inclusions, and have high uniform clarity. When Intergem Limited began marketing the Yogo in the 1980s as the world's only guaranteed untreated sapphire, heat treatment was not commonly disclosed; by the late 1980s, heat treatment became a major issue. At that time, much of all the world's sapphires were being heated to enhance their natural color. Intergem's marketing of guaranteed untreated Yogos set them against many in the gem industry. This issue appeared as a front-page story in the "Wall Street Journal" on 29 August 1984 in an article by Bill Richards, "Carats and Schticks: Sapphire Marketer Upsets The Gem Industry". However, the biggest problem the Yogo mine faced was not competition from heated sapphires, but the fact that the Yogo stones could never produce quantities of sapphire above one carat after faceting. As a result, it has remained a niche product, with a market that largely exists in the US.


title: Sapphire

text: Lattice ('bulk') diffusion treatments are used to add impurities to the sapphire to enhance color. This process was originally developed and patented by Linde Air division of Union Carbide and involved diffusing titanium into synthetic sapphire to even out the blue color. It was later applied to natural sapphire. Today, titanium diffusion often uses a synthetic colorless sapphire base. The color layer created by titanium diffusion is extremely thin (less than 0.5 mm). Thus repolishing can and does produce slight to significant loss of color. Chromium diffusion has been attempted, but was abandoned due to the slow diffusion rates of chromium in corundum.


title: Sapphire

text: In the year 2000, beryllium diffused "padparadscha" colored sapphires entered the market. Typically beryllium is diffused into a sapphire under very high heat, just below the melting point of the sapphire. Initially ("c." 2000) orange sapphires were created, although now the process has been advanced and many colors of sapphire are often treated with beryllium. Due to the small size of the beryllium ion, the color penetration is far greater than with titanium diffusion. In some cases, it may penetrate the entire stone. Beryllium-diffused orange sapphires may be difficult to detect, requiring advanced chemical analysis by gemological labs ("e.g.", Gübelin, SSEF, GIA, American Gemological Laboratories (AGL), Lotus Gemology.


title: Sapphire

text: According to United States Federal Trade Commission guidelines, disclosure is required of any mode of enhancement that has a significant effect on the gem's value.


title: Sapphire

text: There are several ways of treating sapphire. Heat-treatment in a reducing or oxidizing atmosphere (but without the use of any other added impurities) is commonly used to improve the color of sapphires, and this process is sometimes known as "heating only" in the gem trade. In contrast, however, heat treatment combined with the deliberate addition of certain specific impurities (e.g. beryllium, titanium, iron, chromium or nickel, which are absorbed into the crystal structure of the sapphire) is also commonly performed, and this process can be known as "diffusion" in the gem trade. However, despite what the terms "heating only" and "diffusion" might suggest, both of these categories of treatment actually involve diffusion processes.


title: Sapphire

text: The most complete description of corundum treatments extant can be found in Chapter 6 of "Ruby & Sapphire: A Gemologist's Guide" (chapter authored by John Emmett, Richard Hughes and Troy R. Douthit).


title: Sapphire

text: Sapphires are mined from alluvial deposits or from primary underground workings. Commercial mining locations for sapphire and ruby include (but are not limited to) the following countries: Afghanistan, Australia, Myanmar/Burma, Cambodia, China, Colombia, India, Kenya, Laos, Madagascar, Malawi, Nepal, Nigeria, Pakistan, Sri Lanka, Tajikistan, Tanzania, Thailand, United States, and Vietnam. Sapphires from different geographic locations may have different appearances or chemical-impurity concentrations, and tend to contain different types of microscopic inclusions. Because of this, sapphires can be divided into three broad categories: classic metamorphic, non-classic metamorphic or magmatic, and classic magmatic.


title: Sapphire

text: Sapphires from certain locations, or of certain categories, may be more commercially appealing than others, particularly classic metamorphic sapphires from Kashmir, Burma, or Sri Lanka that have not been subjected to heat-treatment.


title: Sapphire

text: The Logan sapphire, the Star of India, The Star of Adam and the Star of Bombay originate from Sri Lankan mines. Madagascar is the world leader in sapphire production (as of 2007) specifically its deposits in and around the town of Ilakaka. Prior to the opening of the Ilakaka mines, Australia was the largest producer of sapphires (such as in 1987). In 1991 a new source of sapphires was discovered in Andranondambo, southern Madagascar. That area has been exploited for its sapphires started in 1993, but it was practically abandoned just a few years later—because of the difficulties in recovering sapphires in their bedrock.


title: Sapphire

text: In North America, sapphires have been mined mostly from deposits in Montana: fancies along the Missouri River near Helena, Montana, Dry Cottonwood Creek near Deer Lodge, Montana, and Rock Creek near Philipsburg, Montana. Fine blue Yogo sapphires are found at Yogo Gulch west of Lewistown, Montana. A few gem-grade sapphires and rubies have also been found in the area of Franklin, North Carolina.


title: Sapphire

text: The sapphire deposits of Kashmir are well known in the gem industry, although their peak production took place in a relatively short period at the end of the nineteenth and early twentieth centuries. They have a superior vivid blue hue, coupled with a mysterious and almost sleepy quality, described by some gem enthusiasts as ‘blue velvet”. Kashmir-origin contributes meaningfully to the value of a sapphire, and most corundum of Kashmir origin can be readily identified by its characteristic silky appearance and exceptional hue. The unique blue appears lustrous under any kind of light, unlike non-Kashmir sapphires which may appear purplish or grayish in comparison. Sotheby's has been in the forefront overseeing record-breaking sales of Kashmir sapphires worldwide. In October 2014, Sotheby's Hong Kong achieved consecutive per-carat price records for Kashmir sapphires – first with the 12.00 carat Cartier sapphire ring at US$193,975 per carat, then with a 17.16 carat sapphire at US$236,404, and again in June 2015 when the per-carat auction record was set at US$240,205. At present, the world record price-per-carat for sapphire at auction is held by a sapphire from Kashmir in a ring, which sold in October 2015 for approximately US$242,000 per carat (HK$52,280,000 in total, including buyer's premium, or more than US$6.74 million).


title: Sapphire

text: In 1902, the French chemist Auguste Verneuil announced a process for producing synthetic ruby crystals. In the flame-fusion (Verneuil process), fine alumina powder is added to an oxyhydrogen flame, and this is directed downward against a ceramic pedestal. Following the successful synthesis of ruby, Verneuil focussed his efforts on sapphire. Synthesis of blue sapphire came in 1909, after chemical analyses of sapphire suggested to Verneuil that iron and titanium were the cause of the blue color. Verneuil patented the process of producing synthetic blue sapphire in 1911.


title: Sapphire

text: The key to the process is that the alumina powder does not melt as it falls through the flame. Instead it forms a sinter cone on the pedestal. When the tip of that cone reaches the hottest part of the flame, the tip melts. Thus the crystal growth is started from a tiny point, ensuring minimal strain.


title: Sapphire

text: Next, more oxygen is added to the flame, causing it to burn slightly hotter. This expands the growing crystal laterally. At the same time, the pedestal is lowered at the same rate that the crystal grows vertically. The alumina in the flame is slowly deposited, creating a teardrop shaped "boule" of sapphire material. This step is continued until the desired size is reached, the flame is shut off and the crystal cools. The now elongated crystal contains a lot of strain due to the high thermal gradient between the flame and surrounding air. To release this strain, the now finger-shaped crystal will be tapped with a chisel to split it into two halves.


title: Sapphire

text: Due to the vertical layered growth of the crystal and the curved upper growth surface (which starts from a drop), the crystals will display curved growth lines following the top surface of the boule. This is in contrast to natural corundum crystals, which feature angular growth lines expanding from a single point and following the planar crystal faces.


title: Sapphire

text: Chemical dopants can be added to create artificial versions of the ruby, and all the other natural colors of sapphire, and in addition, other colors never seen in geological samples. Artificial sapphire material is identical to natural sapphire, except it can be made without the flaws that are found in natural stones. The disadvantage of the Verneuil process is that the grown crystals have high internal strains. Many methods of manufacturing sapphire today are variations of the Czochralski process, which was invented in 1916 by Polish chemist Jan Czochralski. In this process, a tiny sapphire seed crystal is dipped into a crucible made of the precious metal iridium or molybdenum, containing molten alumina, and then slowly withdrawn upward at a rate of 1 to 100 mm per hour. The alumina crystallizes on the end, creating long carrot-shaped boules of large size up to 200 kg in mass.


title: Sapphire

text: Synthetic sapphire is also produced industrially from agglomerated aluminum oxide, sintered and fused (such as by hot isostatic pressing) in an inert atmosphere, yielding a transparent but slightly porous polycrystalline product.


title: Sapphire

text: In 2003, the world's production of synthetic sapphire was 250 tons (1.25 × 10 carats), mostly by the United States and Russia. The availability of cheap synthetic sapphire unlocked many industrial uses for this unique material.


title: Sapphire

text: Synthetic sapphire sometimes referred to as "sapphire glass" is commonly used as a window material, because it is both highly transparent to wavelengths of light between 150 nm (UV) and 5500 nm (IR) (the visible spectrum extends about 380 nm to 750 nm), and extraordinarily scratch-resistant.


title: Sapphire

text: Some sapphire-glass windows are made from pure sapphire boules that have been grown in a specific crystal orientation, typically along the optical axis, the "c" axis, for minimum birefringence for the application.


title: Sapphire

text: The boules are sliced up into the desired window thickness and finally polished to the desired surface finish. Sapphire optical windows can be polished to a wide range of surface finishes due to its crystal structure and its hardness. The surface finishes of optical windows are normally called out by the scratch-dig specifications in accordance with the globally adopted MIL-O-13830 specification.


title: Sapphire

text: The sapphire windows are used in both high-pressure and vacuum chambers for spectroscopy, crystals in various watches, and windows in grocery-store barcode scanners, since the material's exceptional hardness and toughness makes it very resistant to scratching.


title: Sapphire

text: In 2014 Apple consumed "one-fourth of the world’s supply of sapphire to cover the iPhone’s camera lens and fingerprint reader".


title: Sapphire

text: Several attempts have been made to make sapphire screens for smartphones viable. Apple contracted GT Advanced Technologies, Inc. to manufacture sapphire screens for iPhones, the venture failed resulting in the bankruptcy of GTAT. The Kyocera Brigadier was the first production smartphone to feature a sapphire screen.


title: Sapphire

text: It is used for end windows on some high-powered laser tubes, as its wide-band transparency and thermal conductivity allow it to handle very high power densities in the infrared and UV spectrum without degrading due to heating.


title: Sapphire

text: Along with zirconia and aluminum oxynitride, synthetic sapphire is used for shatter-resistant windows in armored vehicles and various military body armor suits, in association with composites.


title: Sapphire

text: One type of xenon arc lamp originally called the "Cermax" and now known generically as the "ceramic-body xenon lamp" uses sapphire crystal output windows. This product tolerates higher thermal loads and thus higher output powers when compared with conventional Xe lamps with pure silica window.


title: Sapphire

text: Thin sapphire wafers were the first successful use of an insulating substrate upon which to deposit silicon to make the integrated circuits known as silicon on sapphire or "SOS"; now other substrates can also be used for the class of circuits known more generally as silicon on insulator. Besides its excellent electrical insulating properties, sapphire has high thermal conductivity. CMOS chips on sapphire are especially useful for high-power radio-frequency (RF) applications such as those found in cellular telephones, public-safety band radios, and satellite communication systems. "SOS" also allows for the monolithic integration of both digital and analog circuitry all on one IC chip, and the construction of extremely low power circuits.


title: Sapphire

text: In one process, after single crystal sapphire boules are grown, they are core-drilled into cylindrical rods, and wafers are then sliced from these cores.


title: Sapphire

text: Wafers of single-crystal sapphire are also used in the semiconductor industry as substrates for the growth of devices based on gallium nitride (GaN). The use of sapphire significantly reduces the cost, because it has about one-seventh the cost of germanium. Gallium nitride on sapphire is commonly used in blue light-emitting diodes (LEDs).


title: Sapphire

text: The first laser was made in 1960 by Theodore Maiman with a rod of synthetic ruby. Titanium-sapphire lasers are popular due to their relatively rare capacity to be tuned to various wavelengths in the red and near-infrared region of the electromagnetic spectrum. They can also be easily mode-locked. In these lasers a synthetically produced sapphire crystal with chromium or titanium impurities is irradiated with intense light from a special lamp, or another laser, to create stimulated emission.


title: Sapphire

text: Monocrystalline sapphire is fairly biocompatible and the exceptionally low wear of sapphire–metal pairs has led to the introduction (in Ukraine) of sapphire monocrystals for hip


title: Sapphire

text: Extensive tables listing over a hundred important and famous rubies and sapphires can be found in Chapter 10 of "Ruby & Sapphire: A Gemologist's Guide".


title: General relativity

text: General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the "" is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of second order partial differential equations.


title: General relativity

text: Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics. These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light, and include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the Shapiro time delay and singularities/black holes. So far, all tests of general relativity have been shown to be in agreement with the theory. The time dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology, thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories, general relativity continues to be the simplest theory consistent with experimental data.


title: General relativity

text: Reconciliation of general relativity with the laws of quantum physics remains a problem, however, as there is a lack of a self-consistent theory of quantum gravity. It is not yet known how gravity can be unified with the three non-gravitational forces: strong, weak and electromagnetic.


title: General relativity

text: Einstein's theory has astrophysical implications, including the prediction of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape from them. Black holes are the end-state for massive stars. Microquasars and active galactic nuclei are believed to be stellar black holes and supermassive black holes. It also predicts gravitational lensing, where the bending of light results in multiple images of the same distant astronomical phenomenon. Other predictions include the existence of gravitational waves, which have been observed directly by the physics collaboration LIGO and other observatories. In addition, general relativity has provided the base of cosmological models of an expanding universe.


title: General relativity

text: In the preface to , Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated."


title: General relativity

text: Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.


title: General relativity

text: Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present. A version of non-Euclidean geometry, called Riemannian Geometry, enabled Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity. This idea was pointed out by mathematician Marcel Grossmann and published by Grossmann and Einstein in 1913.


title: General relativity

text: The Einstein field equations are nonlinear and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, eventually resulting in the Reissner–Nordström solution, which is now associated with electrically charged black holes. In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot and dense earlier state. Einstein later declared the cosmological constant the biggest blunder of his life.


title: General relativity

text: During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters ("fudge factors"), and in 1919 an expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of 29 May 1919, instantly making Einstein famous. Yet the theory remained outside the mainstream of theoretical physics and astrophysics until developments between approximately 1960 and 1975, now known as the golden age of general relativity. Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations. Ever more precise solar system tests confirmed the theory's predictive power, and relativistic cosmology also became amenable to direct observational tests.


title: General relativity

text: General relativity has acquired a reputation as a theory of extraordinary beauty. Subrahmanyan Chandrasekhar has noted that at multiple levels, general relativity exhibits what Francis Bacon has termed a "strangeness in the proportion" ("i.e". elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time "versus" matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were the principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory. Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency.


title: General relativity

text: General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.


title: General relativity

text: At the base of classical mechanics is the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second law of motion, which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration. The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics, straight world lines in curved spacetime.


title: General relativity

text: Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space, as well as a time coordinate. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties. A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in an enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.


title: General relativity

text: Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, space"time" as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can deduce that spacetime is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system. In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.


title: General relativity

text: As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics. In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group, which includes translations, rotations, boosts and reflections.) The differences between the two become significant when dealing with speeds approaching the speed of light, and with high-energy phenomena.


title: General relativity

text: With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event , there is a set of events that can, in principle, either influence or be influenced by via signals or interactions that do not need to travel faster than light (such as event in the image), and a set of events for which such an influence is impossible (such as event in the image). These sets are observer-independent. In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure or conformal geometry.


title: General relativity

text: Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.


title: General relativity

text: A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. below). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity. The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.


title: General relativity

text: The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).


title: General relativity

text: Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor, which includes both energy and momentum densities as well as stress: pressure and shear. Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy–momentum corresponds to the statement that the energy–momentum tensor is divergence-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest set of equations are what are called Einstein's (field) equations:


title: General relativity

text: On the left-hand side is the Einstein tensor, formula_1, which is symmetric and a specific divergence-free combination of the Ricci tensor formula_2 and the metric. In particular,


title: General relativity

text: is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as


title: General relativity

text: On the right-hand side, "formula_5" is the energy–momentum tensor. All tensors are written in abstract index notation. Matching the theory's prediction to observational results for planetary orbits or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant is found to be formula_6, where formula_7 is the gravitational constant and formula_8 the speed of light in vacuum. When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,


title: General relativity

text: In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic.


title: General relativity

text: where formula_11 is a scalar parameter of motion (e.g. the proper time), and formula_12 are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and the summation convention is used for repeated indices formula_13 and formula_14. The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to Newton's laws of motion which likewise provide formulae for the acceleration of a particle. This equation of motion employs the Einstein notation, meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation.


title: General relativity

text: In general relativity, the effective gravitational potential energy of an object of mass "m" rotating around a massive central body "M" is given by


title: General relativity

text: where "L" is the angular momentum. The first term represents the Newton's force of gravity, which is described by the inverse-square law. The second term represents the centrifugal force in the circular motion. The third term represents the relativistic effect.


title: General relativity

text: There are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Whitehead's theory, Brans–Dicke theory, teleparallelism, "f"("R") gravity and Einstein–Cartan theory.


title: General relativity

text: The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building.


title: General relativity

text: General relativity is a metric theory of gravitation. At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve.


title: General relativity

text: While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases. For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.


title: General relativity

text: As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems. Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers. Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance.


title: General relativity

text: The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.


title: General relativity

text: Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless, a number of exact solutions are known, although only a few have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner–Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe, and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes, each describing an expanding cosmos. Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub–NUT solution (a model universe that is homogeneous, but anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).


title: General relativity

text: Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes. In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity. An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.


title: General relativity

text: General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication.


title: General relativity

text: Assuming that the equivalence principle holds, gravity influences the passage of time. Light sent down into a gravity well is blueshifted, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.


title: General relativity

text: Gravitational redshift has been measured in the laboratory and using astronomical observations. Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks, while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS). Tests in stronger gravitational fields are provided by the observation of binary pulsars. All results are in agreement with general relativity. However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.


title: General relativity

text: General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing the light of stars or distant quasars being deflected as it passes the Sun.


title: General relativity

text: This and related predictions follow from the fact that light follows what is called a light-like or null geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity. As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion), several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity.


title: General relativity

text: Closely related to light deflection is the gravitational time delay (or Shapiro delay), the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction. In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.


title: General relativity

text: Predicted in 1916 by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves. On 11 February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from a pair of black holes merging.


title: General relativity

text: The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right). Since Einstein's equations are non-linear, arbitrarily strong gravitational waves do not obey linear superposition, making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by formula_17 or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed.


title: General relativity

text: Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space or Gowdy universes, varieties of an expanding cosmos filled with gravitational waves. But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.


title: General relativity

text: General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (precession) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction.


title: General relativity

text: In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass) will precess; the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rose curve-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a test particle. For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.


title: General relativity

text: The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass) or the much more general post-Newtonian formalism. It is due to the influence of gravity on the geometry of space and to the contribution of self-energy to a body's gravity (encoded in the nonlinearity of Einstein's equations). Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth), as well as in binary pulsar systems, where it is larger by five orders of magnitude.


title: General relativity

text: In general relativity the perihelion shift formula_18, expressed in radians per revolution, is approximately given by


title: General relativity

text: According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the Solar System or for ordinary double stars, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation.


title: General relativity

text: The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor, using the binary pulsar PSR1913+16 they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 Nobel Prize in physics. Since then, several other binary pulsars have been found, in particular the double pulsar PSR J0737−3039, where both stars are pulsars and which was last reported to also be in agreement with general relativity in 2021 after 16 years of observations.


title: General relativity

text: Several relativistic effects are directly related to the relativity of direction. One is geodetic precession: the axis direction of a gyroscope in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("parallel transport"). For the Moon–Earth system, this effect has been measured with the help of lunar laser ranging. More recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 0.3%.


title: General relativity

text: Near a rotating mass, there are gravitomagnetic or frame-dragging effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for rotating black holes where, for any object entering a zone known as the ergosphere, rotation is inevitable. Such effects can again be tested through their influence on the orientation of gyroscopes in free fall. Somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction. Also the Mars Global Surveyor probe around Mars has been used.


title: General relativity

text: Examples of prominent physicists who support neo-Lorentzian explanations of general relativity are Franco Selleri and Antony Valentini.


title: General relativity

text: The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing. Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an Einstein ring, or partial rings called arcs.


title: General relativity

text: The earliest example was discovered in 1979; since then, more than a hundred gravitational lenses have been observed. Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such "microlensing events" have been observed.


title: General relativity

text: Gravitational lensing has developed into a tool of observational astronomy. It is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data provide valuable insight into the structural evolution of galaxies.


title: General relativity

text: Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see Orbital decay, above). Detection of these waves is a major goal of current relativity-related research. Several land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (two detectors), TAMA 300 and VIRGO. Various pulsar timing arrays are using millisecond pulsars to detect gravitational waves in the 10 to 10 hertz frequency range, which originate from binary supermassive blackholes. A European space-based detector, eLISA / NGO, is currently under development, with a precursor mission (LISA Pathfinder) having launched in December 2015.


title: General relativity

text: Observations of gravitational waves promise to complement observations in the electromagnetic spectrum. They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of supernova implosions, and about processes in the very early universe, including the signature of certain types of hypothetical cosmic string. In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger.


title: General relativity

text: Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution, neutron stars of around 1.4 solar masses, and stellar black holes with a few to a few dozen solar masses, are thought to be the final state for the evolution of massive stars. Usually a galaxy has one supermassive black hole with a few million to a few billion solar masses in its center, and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.


title: General relativity

text: Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation. Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as microquasars. In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed.


title: General relativity

text: General relativity plays a central role in modelling all these phenomena, and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.


title: General relativity

text: Black holes are also sought-after targets in the search for gravitational waves (cf. Gravitational waves, above). Merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances. The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive black hole's geometry.


title: General relativity

text: The current models of cosmology are based on Einstein's field equations, which include the cosmological constant formula_24 since it has important influence on the large-scale dynamics of the cosmos,


title: General relativity

text: where "formula_26" is the spacetime metric. Isotropic and homogeneous solutions of these enhanced equations, the Friedmann–Lemaître–Robertson–Walker solutions, allow physicists to model a universe that has evolved over the past 14 billion years from a hot, early Big Bang phase. Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation, further observational data can be used to put the models to the test. Predictions, all successful, include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis, the large-scale structure of the universe, and the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation.


title: General relativity

text: Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90% of all matter appears to be dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly. There is no generally accepted description of this new kind of matter, within the framework of known particle physics or otherwise. Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, known as dark energy, the nature of which remains unclear.


title: General relativity

text: An inflationary phase, an additional phase of strongly accelerated expansion at cosmic times of around 10 seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation. Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario. However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations. An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang singularity. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed (cf. the section on quantum gravity, below).


title: General relativity

text: Kurt Gödel showed that solutions to Einstein's equations exist that contain closed timelike curves (CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes. Stephen Hawking introduced chronology protection conjecture, which is an assumption beyond those of standard general relativity to prevent time travel.


title: General relativity

text: Some exact solutions in general relativity such as Alcubierre drive present examples of warp drive but these solutions requires exotic matter distribution, and generally suffers from semiclassical instability.


title: General relativity

text: The spacetime symmetry group for special relativity is the Poincaré group, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, "viz.", the Poincaré group.


title: General relativity

text: In 1962 Hermann Bondi, M. G. van der Burg, A. W. Metzner and Rainer K. Sachs addressed this asymptotic symmetry problem in order to investigate the flow of energy at infinity due to propagating gravitational waves. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making no "a priori" assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as "supertranslations". This implies the conclusion that General Relativity (GR) does "not" reduce to special relativity in the case of weak fields at long distances. It turns out that the BMS symmetry, suitably modified, could be seen as a restatement of the universal soft graviton theorem in quantum field theory (QFT), which relates universal infrared (soft) QFT with GR asymptotic spacetime symmetries.


title: General relativity

text: In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event "A" can reach any other location "X" before light sent out at "A" to "X". In consequence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using Penrose–Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.


title: General relativity

text: Aware of the importance of causal structure, Roger Penrose and others developed what is known as global geometry. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, and additional non-specific assumptions about the nature of matter (usually in the form of energy conditions) are used to derive general results.