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113-472: Patek Philippe is widely considered to be one of the most prestigious watch manufacturers in the world. As of July 2023, among the world's top ten most expensive watches ever sold at auctions , nine were Patek Philippe watches. In particular, Patek Philippe Grandmaster Chime Ref. 6300A-010 currently holds the title of the most expensive watch (and wristwatch) ever sold at auction (US$ 31 million/27 million CHF ), while Patek Philippe Henry Graves Supercomplication ,

226-543: A pair of black holes merging . 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

339-548: A black rubber strap. This signature rubber strap, a first for Patek Philippe, was composed of a new composite material designed for durability and comfort. The Aquanaut collection has grown to encompass a variety of models in stainless steel, rose gold, and white gold, with additional complications like chronograph and travel time functions. The Aquanaut has become a highly successful collection for Patek Philippe, known for its blend of sportiness, luxury, and everyday wearability. Patek Philippe perpetual calendar chronograph

452-570: 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 . Conversely, one might expect that inertial motions, once identified by observing

565-560: 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,

678-508: 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

791-530: 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,

904-539: 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. While general relativity replaces

1017-520: A family business, Patek Philippe then had seven shareholders, five of whom formed the board of directors with J. A. Bénassy-Philippe being the chairman. Joseph E. Philippe's son later joined the company, and he was the last offspring of the founders in the business. In 1915, Albert Einstein ordered a gold pocket watch from Patek Philippe; in that year, he completed his Theory of General Relativity . The Stern family of Switzerland has owned Patek Philippe since 1932, when Charles Stern and Jean Stern acquired

1130-597: 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. The same experimental data shows that time as measured by clocks in

1243-471: 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

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1356-450: A massive central body M is given by A conservative total force can then be obtained as its negative gradient where L is the angular momentum . The first term represents the force of Newtonian 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. There are alternatives to general relativity built upon

1469-660: A new company, Patek & Cie , beginning on 15 May 1845 . On 1 January 1851, the company's name was officially changed to Patek, Philippe & Cie . In the same year, Queen Victoria of the United Kingdom acquired a keyless pendant watch at the Great Exhibition in London . The watch was embellished with rose-cut diamonds set in the pattern of a bouquet of flowers. The Queen had another exclusive Patek Philippe timepiece, to be worn pinned to clothing. This watch

1582-779: 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

1695-442: 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 ,

1808-511: A report assigning environmental ratings for 15 major watch manufacturers and jewelers in Switzerland. Patek Philippe was given the lowest environmental rating as "Latecomers/Non-transparent", suggesting the manufacturer has taken few actions addressing the impact of its manufacturing activities on the environment and climate change. Patek Philippe has invented over 20 basic calibres and has received over 100 patents. The following are some of

1921-474: A successor to Antoni Patek. In March 1877, Antoni Patek died at the age of 65, but his only son, Léon Mecyslas Vincent Patek, did not join the business. As a result, Joseph Antoine Bénassy-Philippe, one of Adrien Philippe's sons-in-law, succeeded to Antoni Patek's position. In 1887, the cross of one of the four Military Orders of Spain, the Order of Calatrava , became the registered company logo of Patek Philippe as

2034-714: A sympathetic allusion to the still-extant order of Catholic knights that fought the Muslims in the Crusades. In 1891, the 76-year-old Adrien Philippe handed over his position in the business to his youngest son, Joseph Emile Philippe, together with François Antoine Conty. Adrien Philippe died in January 1894. 1901 saw the transformation of Patek Philippe into a joint-stock company , Ancienne Manufacture d'horlogerie Patek, Philippe & Cie, Société Anonyme, initiated by J. A. Bénassy-Philippe and Joseph E. Philippe. Still being run as

2147-711: A threshold of 1.5 million US dollars. The following table shows the brand breakdown of the most expensive watches ever sold at auction (for at least US$ 1.5 million). As of December 2022, there are 19 manufacturers that have manufactured watches that qualify: Patek Philippe , Rolex , Breguet , Audemars Piguet , Philippe Dufour , F. P. Journe , George Daniels , Richard Mille , URWERK, Zenith , Vacheron Constantin , Omega , Tag Heuer , Chopard , Bulova , Cartier and Attributed to Piguet & Capt. Among these brands, Philippe Dufour, F.P. Journe, George Daniels, Richard Mille, URWERK and Piguet & Capt are independent watchmakers. The following table contains details of

2260-470: A unified description of gravity as a geometric property of space and time , or four-dimensional spacetime . In particular, the curvature of spacetime is directly related to the energy and momentum of whatever present matter and radiation . The relation is specified by the Einstein field equations , a system of second-order partial differential equations . Newton's law of universal gravitation , which describes classical gravity, can be seen as

2373-490: 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." General relativity can be understood by examining its similarities with and departures from classical physics. The first step

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2486-539: 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. 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

2599-526: Is Minkowskian , and the laws of physics exhibit local Lorentz invariance . 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,

2712-438: Is "You never actually own a Patek Philippe. You merely look after it for the next generation." The slogan was introduced when the company launched its "Generations" campaign in 1996. Patek Philippe manufactures its own watch components. Like other Swiss manufacturers, the company produces mostly mechanical movements with automatic or manual winding mechanism, but has also been producing quartz watches . In fact, Patek Philippe

2825-423: Is a scalar parameter of motion (e.g. the proper time ), and Γ μ α β {\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} 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

2938-445: 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

3051-422: Is able to add a day to February for leap years while leaving out the extra day for every 100 year interval. Only four pieces of Calibre 89 were ever manufactured by Patek Philippe, with one in white gold, one in yellow gold, one in rose gold, and one in platinum. The yellow-gold and the white-gold Calibre 89 were sold at auctions by Antiquorum in 2009 and 2004, respectively, and both watches currently rank among

3164-402: 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

3277-499: 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. As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics,

3390-405: 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

3503-445: 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. Given the universality of free fall, there is no observable distinction between inertial motion and motion under

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3616-560: Is known as gravitational time dilation. 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

3729-404: 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

3842-456: 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

3955-430: 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

4068-495: Is possibly incomplete. (USD millions) (USD millions) date place Theory of General Relativity General relativity , also known as the general theory of relativity , and as 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

4181-495: Is the Shapiro Time 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

4294-409: Is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories. 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. Assuming that

4407-471: 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. 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

4520-514: The Aquanaut collection , marking a departure from their traditional dress watches. The Aquanaut was designed as a luxury sports watch for a younger generation of watch enthusiasts, featuring a contemporary and sporty aesthetic alongside the refined craftsmanship that Patek Philippe is known for. The first Aquanaut model, the Ref. 5060A, was crafted from stainless steel and equipped with a black dial and

4633-432: 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. In general relativity, the effective gravitational potential energy of an object of mass m revolving around

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4746-634: The Great Depression smoothly. The original Calatrava Ref. 96 was in production for more than 40 years, and its successor models include Ref. 2526, Ref. 3520, Ref. 5196 and so on. Known for its simple and elegant design, the Calatrava wristwatch has been a flagship model of Patek Philippe since its introduction. Notably, the Calatrava Cross has been company's logo since 1887. Patek Philippe World Time ( Heures Universelles ) collection

4859-609: 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 ). Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on

4972-597: 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

5085-682: 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

5198-561: 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

5311-421: The most expensive watches ever sold in auction . In 1989, Patek Philippe created Calibre 89 , then the most complicated mechanical watch ever made, for its 150th anniversary. Calibre 89 holds 33 complications, including the date of Easter, time of sunrise, equation of time , sidereal time , and many other indicators. 1,728 unique parts allow sidereal time , a 2,800 star chart, and more. In addition, Calibre 89

5424-473: 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

5537-454: 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. As it is constructed using tensors, general relativity exhibits general covariance : its laws—and further laws formulated within

5650-429: The summation convention is used for repeated indices α {\displaystyle \alpha } and β {\displaystyle \beta } . 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

5763-399: The 173 timepieces auctioned for at least US$ 1.5 million , ranked according to their original auction prices. The inflation-adjusted price is given by consumer price index inflation-adjusted value of United States dollars in 2023. The following table contains some of the most expensive auctioned watches without public online records from auctioneers (e.g., major auction houses). This list

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5876-464: The 58 world's most expensive watches sold at auction (over 2 million US dollars), with the world record being 4.027 million US dollars (6,603,500 CHF ) made at Antiquorum 's Geneva auction on April 13, 2002. In 1976, Patek Philippe introduced the Nautilus collection after deciding it was time to produce an exclusive sport watch with finishes of the highest quality. The first model was Ref. 3700 and

5989-413: 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

6102-416: The brand image while perpetuating tradition. The target was represented by dynamic business managers of the new generations. The Nautilus wristwatch has become one of the most popular collections from Patek Philippe, and the Ref. 5711 & 5712 models, which the company introduced in 2006 to celebrate the 30th anniversary of the collection, are among the most popular models. In 1997, Patek Philippe introduced

6215-414: The buyer and has to be approved by Patek Philippe President Thierry Stern. List of most expensive watches sold at auction This list of most expensive watches sold at auction documents the watches sold at auction worldwide for at least 1.5 million US dollars . The final price listed is the total price paid by the buyer converted to US dollars, according to the currency exchange rate at

6328-521: The buyer's premium, and so the rankings which follow are based on the aggregated price paid by the buyer: for the watch itself (the hammer or sale price) and for the auction house's services and administrative costs (the buyer's premium). Inflation-adjusted prices are also listed for reference. If a watch has been sold at auction for several times, only the highest final price is included. Finally, any auctioned watch without public online records from auctioneers (e.g. major auction houses) will not be included in

6441-564: The company announced that all of its future mechanical timepieces would be imprinted with the Patek Philippe Seal which requires a precision of -3/+2 seconds per day for diameters no less than 20 mm and -5/+4 seconds per day for diameters less than 20 mm, surpassing the highest industry standard of watch manufacturing and thus abandoning the Geneva Seal. In December 2018, World Wide Fund for Nature (WWF) released

6554-413: The company during the Great Depression . The Stern brothers' company, Fabrique de Cadrans Sterns Frères , had been a business partner of Patek Philippe as its supplier of watch dials. In 1935, Patek Philippe was brought to American markets by New York -based Henri Stern Watch Agency , where it was sold as a sister brand alongside Universal Genève . In 1958, Henri Stern, the son of Charles Stern, became

6667-412: The company was liquidated on 18 April 1845. At that point, Czapek founded Czapek & Cie on 1 May 1845 with a new partner, Juliusz Gruzewski. Subsequently, Patek was joined by French watchmaker Adrien Philippe , the inventor of the keyless winding mechanism (although this had been discovered previously by Abraham Louis Breguet but not patented by him), and continued the watchmaking business with

6780-472: The company. He initiated the publication of the twice-a-year Patek Philippe Magazine in 1996, which is reserved for the watch owners and has received contributions from various prominent writers including Nobel Laureates Gao Xingjian and José Saramago . In 2009, Philippe Stern's son, Thierry Stern, took over the reins from his father. In 2010, the company produced 40,000 timepieces and, according to Thierry Stern, it produced 58,000 pieces in 2017. In 2018,

6893-406: 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). Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source

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7006-653: The course of 35 years, four series of Ref. 2499 were introduced, exerting a strong influence on the watch designs of many other renowned watchmakers. As of June 2019, Ref. 2499 holds eleven spots among the 59 world's most expensive watches sold at auction (over 2 million US dollars) and at least 18 pieces of Ref. 2499 have been auctioned for more than 1.5 million US dollars, with the world record being 3.880 million US dollars (3,915,000 CHF ) made at Sotheby's Geneva auction on November 13, 2018. The successors to Ref. 2499 include, in chronological order, Ref. 3970, Ref. 5970 and Ref. 5270. Patek Philippe Sky Moon Tourbillon wristwatch

7119-560: 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

7232-452: The emission of gravitational waves and effects related to the relativity of direction. 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

7345-555: The energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations: G μ ν ≡ R μ ν − 1 2 R g μ ν = κ T μ ν {\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,} On

7458-446: 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

7571-456: 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 10 − 21 {\displaystyle 10^{-21}} or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed . Some exact solutions describe gravitational waves without any approximation, e.g.,

7684-408: 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. Closely related to light deflection

7797-433: 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

7910-412: 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

8023-484: The geometry of space. 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

8136-441: The image), and a set of events for which such an influence is impossible (such as event C 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. Special relativity

8249-561: The important contributions from Patek Philippe to the watchmaking industry. In 1932, Patek Philippe launched the first Calatrava model Ref. 96, which was designed by English horologist David Penney who was influenced by the Bauhaus art movement in Germany. Originally, the Stern brothers, soon after they acquired Patek Philippe in 1932, introduced the Calatrava to help the company pass through

8362-446: 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

8475-512: The introduction of a number of alternative theories , general relativity continues to be the simplest theory consistent with experimental data . 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 . Einstein's theory has astrophysical implications, including

8588-417: 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. 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

8701-410: 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. 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

8814-457: The left-hand side is the Einstein tensor , G μ ν {\displaystyle G_{\mu \nu }} , which is symmetric and a specific divergence-free combination of the Ricci tensor R μ ν {\displaystyle R_{\mu \nu }} and the metric. In particular, is the curvature scalar. The Ricci tensor itself is related to

8927-477: The light of stars or distant quasars being deflected as it passes the Sun . 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

9040-455: 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. Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless,

9153-442: The more general Riemann curvature tensor as On the right-hand side, κ {\displaystyle \kappa } is a constant and T μ ν {\displaystyle T_{\mu \nu }} 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

9266-424: The most beautiful of all existing physical theories. Henri Poincaré 's 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that gravitational waves propagate at

9379-562: The number went up to 62,000, and in order to maintain quality and exclusivity, Patek Philippe would only slowly increase the number of timepieces produced each year (by 1-3 percent per year) but with a ceiling. Currently, the company is an active member of the Federation of the Swiss Watch Industry FH . Since the 2000s, Patek Philippe timepieces have repeatedly fetched high prices in auctions worldwide. As of 2023, among

9492-432: 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. 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

9605-459: 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

9718-500: The other six pieces was 2.5 million CHF (2.6 million US dollars). In 2016, Patek Philippe introduced the Grandmaster Chime Ref. 6300, succeeding the Grandmaster Chime Ref. 5175. Ref. 6300 also has 20 complications, including grande sonnerie , minute repeater , and alarm with time strike (but without tourbillon), with the sale price over 2.2 million US dollars. The purchase of each piece requires an application from

9831-421: 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

9944-511: The preface to Relativity: The Special and the General Theory , 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

10057-399: The president of Patek Philippe. Alan Banbery, who previously designed Universal's "Compax" movements and worked as a horologist for London's Garrard & Co , would take on the position of Director of Sales in 1965 and later authored official reference books on vintage Patek Philippe pocket watches and chronographs . In 1993, Philippe Stern, the son of Henri Stern, became the president of

10170-430: 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. In

10283-906: The ranking. As of December 2022, the most expensive watch (and wristwatch ) ever sold at auction is the Patek Philippe Grandmaster Chime Ref. 6300A-010, fetching US$ 31.19 million (31,000,000 CHF ) in Geneva on 9 November 2019. The most expensive pocket watch ever sold at auction is the Patek Philippe Henry Graves Supercomplication , fetching US$ 23.98 million (23,237,000 CHF) in Geneva on 11 November 2014. As of December 2022, at least 115 timepieces have been sold at auction for over US$ 2 million, and at least 173 timepieces have been sold at auction for over $ 1.5 million. Among

10396-419: 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 . Widely acknowledged as a theory of extraordinary beauty , general relativity has often been described as

10509-446: 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 . 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

10622-472: 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, 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. The geodesic equation is: where s {\displaystyle s}

10735-399: The speed of light. Soon afterwards, Einstein started thinking about how to incorporate gravity into his relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall (FFO), 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

10848-518: The theory can be used for model-building. 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

10961-644: The theory's predictive power, and relativistic cosmology also became amenable to direct observational tests. 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

11074-477: The time of auction. This price is the aggregate of the hammer price (i.e., the winning bid or sale price at the auction) plus any buyer's premium paid to the auction houses (where levied, and in accordance with the rates charged by the relevant auction house). While the rates of buyer's vary between auction houses (which rates can also vary within each auction house based on the nature of the lot and its value), most auction houses publish their results inclusive of

11187-465: The top 10 of these watches, nine are Patek Philippe timepieces and one is Rolex 's. The following table shows the breakdown of auction houses that have sold the most expensive watches at auctions (for at least US$ 2 million). As of December 2022, there are at least six auction houses that have sold watches for no less than $ 2 million: Christie's , Phillips , Antiquorum , Sotheby's , Poly Auction , and Monaco Legend Auctions. The following table has

11300-533: The top ten most expensive watches ever sold at auction , with final prices over 5 million US dollars. In 2014, Patek Philippe introduced the Grandmaster Chime Ref. 5175, the most complicated wristwatch ever built by the manufacturer (with 20 complications, but no tourbillon ), to celebrate its 175th anniversary. Only seven pieces of Ref. 5175 were created, with one permanently residing in the Patek Philippe Museum in Geneva. The sale price for each of

11413-487: The two become significant when dealing with speeds approaching the speed of light , and with high-energy phenomena. 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 A , there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in

11526-489: The 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 the universe has evolved from an extremely hot and dense earlier state. Einstein later declared the cosmological constant the biggest blunder of his life. During that period, general relativity remained something of

11639-494: The weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant κ {\displaystyle \kappa } is found to be κ = 8 π G c 4 {\textstyle \kappa ={\frac {8\pi G}{c^{4}}}} , where G {\displaystyle G} is the Newtonian constant of gravitation and c {\displaystyle c}

11752-496: The world's most complicated mechanical watch until 1989, currently holds the title of the most expensive pocket watch ever sold at auction (US$ 24 million/21 million CHF ). The company traces its origins to the mid-19th century, when Polish watchmaker Antoni Patek and his Czech -born Polish business partner Franciszek Czapek formed Patek, Czapek & Cie in Geneva on 1 May 1839 and started manufacturing pocket watches . The two eventually separated due to disagreements, and

11865-455: The world's top ten most expensive watches ever sold at auctions , nine were Patek Philippe watches. Among the top 58 most expensive watches sold at auction (over 2 million US dollars), 46 are Patek Philippe watches. A small part of the demand for auction pieces is driven by Patek Philippe themselves, as they are purchasing in the auction market to add to the collection of the Patek Philippe Museum in Geneva. One of Patek Philippe's company slogans

11978-403: Was auctioned in 2023 by the descendants of Adrien Philippe. The watch was purchased most probably by the Patek Philippe Museum though this is not confirmed and may be owned by a private collector. The wedding between Louise Philippe and Joseph Antoine Bénassy where the watch was presented is noted as the wedding that helped Patek Philippe survive the founding generation by providing the company with

12091-617: Was introduced in 1939, with Ref.1415 being the first model. The "World Time" complication is able to show the time for all 24 time zones on the same watch and was invented by Swiss watchmaker Louis Cottier in 1931. Cottier's invention attracted several watchmaking companies, while Patek Philippe was the first company to introduce a series of World Time wristwatches after forging a partnership with Mr. Cottier. The successors to Ref. 1415 include Ref. 2523, Ref. 5230, Ref. 5531 (with minute repeater ), and so on. As of 2018, Patek Philippe World Time collection (Ref. 1415, Ref. 2523) holds 6 spots among

12204-503: Was introduced in 1941, with Ref. 1518 being the first model. On November 12, 2016, a Ref. 1518 in steel became the most expensive wristwatch ever sold at auction (until October 2017), fetching a record-breaking US$ 11.14 million (11,002,000 CHF ) in Geneva (through Phillips ' auction house). As the successor, Ref. 2499 is widely considered to be one of the greatest watch models in the world. Historically, only 349 pieces of Ref. 2499 were made from 1950 to 1985 - around 9 pieces each year. Over

12317-458: Was introduced in 2001, with Ref. 5002 being the first model. The wristwatch has two dials and contains 12 complications including tourbillon , minute repeater , sky chart, and moon phase & orbit. In 2013, the company introduced the second model of Sky Moon Tourbillon, Ref. 6002, also with 12 complications. The new model took 7 years of research and development, while the engraving on each piece alone took more than 100 hours. Sky Moon Tourbillon

12430-536: Was made of steel. It was designed by Swiss watchmaker Gérald Genta , who previously designed the Royal Oak collection for Audemars Piguet , and was released by Patek Philippe during the quartz crisis in the hope that it would help re-attract people's attention to high-end Swiss mechanical watches. The Nautilus collection played a key role in Patek Philippe's overall marketing strategy as it had to refresh

12543-645: Was one of the twenty Swiss watch companies that founded the Centre Electronique Horloger and collaboratively developed the first Swiss quartz movements, such as the Beta 21 movement (1969) which was used by several manufacturers in their watches. In 1950s, the company even produced a prototype for a mechanical digital wristwatch, Ref. 3414. Patek Philippe popularized complications such as perpetual calendar , split-seconds hand , chronograph , and minute repeater in mechanical watches. In 2009,

12656-418: Was suspended from a diamond and enamel brooch . In 1868, Patek Philippe created the first Swiss wristwatch for Countess Koscowicz of Hungary . In 1875, Adrien Philippe commissioned a watch that he later gave to his daughter Louise as a wedding present, which is known as the only wristwatch dating back to be owned by either Patek or Philippe. This historical watch is known as “The Watchmaker’s Daughter” and

12769-536: Was the most complicated wristwatch from Patek Philippe until 2014. It is estimated that only 3 - 5 pieces are made each year, and the price for each piece is over US$ 1.2 million. However, every purchase of the piece requires an application from the buyer and has to be approved by Patek Philippe President Thierry Stern. On October 2, 2018, a Sky Moon Tourbillon 6002G was auctioned by Poly Auction in Hong Kong , fetching US$ 2.707 million ( HKD 21,240,000), making it one of

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