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Bronze
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metal alloy
This article is about the metal alloy. For other uses, see Bronze (disambiguation).
The Artemision Bronze representing either Poseidon or Zeus, c. 460 BC, National Archaeological Museum, Athens. This masterpiece of classical sculpture was found in a sunken ship off the coast of Cape Artemisium in 1928. The figure is more than 2 m in height.
Bronze deer figurine dating from between the 9th and 6th centuries BC, National Archaeological Museum of Sofia, Bulgaria
Chinese ritual bronze, Ding form, Western Zhou (1046–771 BC)
Bronze is an alloy consisting primarily of copper, commonly with about 12% tin and often with the addition of other metals (such as aluminium, manganese, nickel or zinc) and sometimes non-metals or metalloids such as arsenic, phosphorus or silicon. These additions produce a range of alloys that may be harder than copper alone, or have other useful properties, such as stiffness, ductility, or machinability.
The archeological period where bronze was the hardest metal in widespread use is known as the Bronze Age. The beginning of the Bronze Age in western Eurasia and south Asia is conventionally dated to the mid-4th millennium BC, and to the early 2nd millennium BC in China;[1] everywhere it gradually spread across regions. The Bronze Age was followed by the Iron Age starting from about 1300 BC and reaching most of Eurasia by about 500 BC, though bronze continued to be much more widely used than it is in modern times.
Because historical pieces were often made of brasses (copper and zinc) and bronzes with different compositions, modern museum and scholarly descriptions of older objects increasingly use the more inclusive term "copper alloy" instead.[2]
Contents
1 Etymology
2 History
2.1 Transition to iron
3 Technical information
3.1 Composition and alloys
3.2 Properties
3.3 Varieties of fitness
4 Applications
4.1 Industry
4.2 Sculptures
4.3 Mirrors
4.4 Musical instruments
4.5 Coins and medals
5 Image gallery
6 See also
7 References
8 External links
Etymology[edit]
The word bronze (1730–40) is borrowed from French bronze (1511), itself borrowed from Italian bronzo "bell metal, brass" (13th century) (transcribed in Medieval Latin as bronzium) from either,
bróntion, back-formation from Byzantine Greek brontēsíon (11th century), perhaps from Brentḗsion ‘Brindisi’, reputed for its bronze;[3][4] or originally,
in its earliest form from Old Persian birinj, biranj (برنج) "brass" (modern berenj), piring (پرنگ) "copper",[5] from which also came Serbo-Croatian pìrinač "brass",[6] Georgian brinǰao "bronze", Armenian płinj "copper".
History[edit]
A hoard of bronze socketed axes from the Bronze Age found in modern Germany. This was the top tool of the period, and also seems to have been used as a store of value.
The discovery of bronze enabled people to create metal objects which were harder and more durable than previously possible. Bronze tools, weapons, armor, and building materials such as decorative tiles were harder and more durable than their stone and copper ("Chalcolithic") predecessors. Initially, bronze was made out of copper and arsenic, forming arsenic bronze, or from naturally or artificially mixed ores of copper and arsenic,[7] with the earliest artifacts so far known coming from the Iranian plateau in the 5th millennium BCE.[8] It was only later that tin was used, becoming the major non-copper ingredient of bronze in the late 3rd millennium BC.[9]
Tin bronze was superior to arsenic bronze in that the alloying process could be more easily controlled, and the resulting alloy was stronger and easier to cast. Also, unlike arsenic, metallic tin and fumes from tin refining are not toxic. The earliest tin-alloy bronze dates to 4500 BCE in a Vinča culture site in Pločnik (Serbia).[10] Other early examples date to the late 4th millennium BC in Egypt,[11] Susa (Iran) and some ancient sites in China, Luristan (Iran) and Mesopotamia (Iraq).[citation needed]
Ores of copper and the far rarer tin are not often found together (exceptions include one ancient site in Thailand and one in Iran), so serious bronze work has always involved trade. Tin sources and trade in ancient times had a major influence on the development of cultures. In Europe, a major source of tin was the British deposits of ore in Cornwall, which were traded as far as Phoenicia in the eastern Mediterranean.
In many parts of the world, large hoards of bronze artifacts are found, suggesting that bronze also represented a store of value and an indicator of social status. In Europe, large hoards of bronze tools, typically socketed axes (illustrated above), are found, which mostly show no signs of wear. With Chinese ritual bronzes, which are documented in the inscriptions they carry and from other sources, the case is very clear. These were made in enormous quantities for elite burials, and also used by the living for ritual offerings.
Transition to iron[edit]
Though bronze is generally harder than wrought iron, with Vickers hardness of 60–258 vs. 30–80,[12] the Bronze Age gave way to the Iron Age after a serious disruption of the tin trade: the population migrations of around 1200–1100 BC reduced the shipping of tin around the Mediterranean and from Britain, limiting supplies and raising prices.[13] As the art of working in iron improved, iron became cheaper and improved in quality. As cultures advanced from hand-wrought iron to machine-forged iron (typically made with trip hammers powered by water), blacksmiths learned how to make steel. Steel is stronger than bronze and holds a sharper edge longer.[14]
Bronze was still used during the Iron Age, and has continued in use for many purposes to the modern day.
Technical information[edit]
Bronze bell with a visible crystallite structure.
Composition and alloys[edit]
There are many different bronze alloys, but typically modern bronze is 88% copper and 12% tin.[15] Alpha bronze consists of the alpha solid solution of tin in copper. Alpha bronze alloys of 4–5% tin are used to make coins, springs, turbines and blades. Historical "bronzes" are highly variable in composition, as most metalworkers probably used whatever scrap was on hand; the metal of the 12th-century English Gloucester Candlestick is bronze containing a mixture of copper, zinc, tin, lead, nickel, iron, antimony, arsenic with an unusually large amount of silver – between 22.5% in the base and 5.76% in the pan below the candle. The proportions of this mixture suggests that the candlestick was made from a hoard of old coins. The Benin Bronzes are in fact brass, and the Romanesque Baptismal font at St Bartholomew's Church, Liège is described as both bronze and brass.
In the Bronze Age, two forms of bronze were commonly used: "classic bronze", about 10% tin, was used in casting; and "mild bronze", about 6% tin, was hammered from ingots to make sheets. Bladed weapons were mostly cast from classic bronze, while helmets and armor were hammered from mild bronze.
Commercial bronze (90% copper and 10% zinc) and architectural bronze (57% copper, 3% lead, 40% zinc) are more properly regarded as brass alloys because they contain zinc as the main alloying ingredient. They are commonly used in architectural applications.[16][17]
Bismuth bronze is a bronze alloy with a composition of 52% copper, 30% nickel, 12% zinc, 5% lead, and 1% bismuth. It is able to hold a good polish and so is sometimes used in light reflectors and mirrors.[18]
Plastic bronze is bronze containing a significant quantity of lead which makes for improved plasticity[19] possibly used by the ancient Greeks in their ship construction.[20]
Silicon bronze has a composition of Si: 2.80–3.80%, Mn: 0.50–1.30%, Fe: 0.80% max., Zn: 1.50% max., Pb: 0.05% max., Cu: balance.[21]
Other bronze alloys include aluminium bronze, phosphor bronze, manganese bronze, bell metal, arsenical bronze, speculum metal and cymbal alloys.
Properties[edit]
Detail of the relief memorial to Cyprian Kamil Norwid, Wawel Cathedral, Kraków, by Czesław Dźwigaj
Benin Bronzes
Bronzes are typically very ductile alloys, considerably less brittle than cast iron. Typically bronze only oxidizes superficially; once a copper oxide (eventually becoming copper carbonate) layer is formed, the underlying metal is protected from further corrosion. This can be seen on statues from the Hellenistic period. However, if copper chlorides are formed, a corrosion-mode called "bronze disease" will eventually completely destroy it.[22] Copper-based alloys have lower melting points than steel or iron and are more readily produced from their constituent metals. They are generally about 10 percent denser than steel, although alloys using aluminium or silicon may be slightly less dense. Bronze is a better conductor of heat and electricity than most steels. The cost of copper-base alloys is generally higher than that of steels but lower than that of nickel-base alloys.
Copper and its alloys have a huge variety of uses that reflect their versatile physical, mechanical, and chemical properties. Some common examples are the high electrical conductivity of pure copper, low-friction properties of bearing bronze (bronze which has a high lead content— 6–8%), resonant qualities of bell bronze (20% tin, 80% copper), and resistance to corrosion by seawater of several bronze alloys.
The melting point of bronze varies depending on the ratio of the alloy components and is about 950 °C (1,742 °F). Bronze is usually nonmagnetic, but certain alloys containing iron or nickel may have magnetic properties.
Varieties of fitness[edit]
Bronze, or bronze-like alloys and mixtures, were used for coins over a longer period. Bronze was especially suitable for use in boat and ship fittings prior to the wide employment of stainless steel owing to its combination of toughness and resistance to salt water corrosion. Bronze is still commonly used in ship propellers and submerged bearings.
In the 20th century, silicon was introduced as the primary alloying element, creating an alloy with wide application in industry and the major form used in contemporary statuary. Sculptors may prefer silicon bronze because of the ready availability of silicon bronze brazing rod, which allows colour-matched repair of defects in castings. Aluminium is also used for the structural metal aluminium bronze.
It is also widely used for casting bronze sculptures. Many common bronze alloys have the unusual and very desirable property of expanding slightly just before they set, thus filling in the finest details of a mould. Bronze parts are tough and typically used for bearings, clips, electrical connectors and springs.
Bronze also has very low friction against dissimilar metals, making it important for cannons prior to modern tolerancing, where iron cannonballs would otherwise stick in the barrel.[23] It is still widely used today for springs, bearings, bushings, automobile transmission pilot bearings, and similar fittings, and is particularly common in the bearings of small electric motors. Phosphor bronze is particularly suited to precision-grade bearings and springs. It is also used in guitar and piano strings.
Unlike steel, bronze struck against a hard surface will not generate sparks, so it (along with beryllium copper) is used to make hammers, mallets, wrenches and other durable tools to be used in explosive atmospheres or in the presence of flammable vapors. Bronze is used to make bronze wool for woodworking applications where steel wool would discolour oak.
Applications[edit]
Industry[edit]
Various kinds of bronze are used in many different industrial applications.
Phosphor bronze is used for ships' propellers, musical instruments, and electrical contacts.[24] Bearings are often made of bronze for its friction properties. It can be filled with oil to make the proprietary Oilite and similar material for bearings. Aluminium bronze is very hard and wear-resistant, and is used for bearings and machine tool ways.[25]
Sculptures[edit]
See also: Bronze sculpture
Fragment of a bronze statue, Vatican Museums, Rome
The Assyrian king Sennacherib (704–681 BC) claims to have been the first to cast monumental bronze statues (of up to 30 tonnes) using two-part moulds instead of the lost-wax method.[26]
Bronze statues were regarded as the highest form of sculpture in Ancient Greek art, though survivals are few, as bronze was a valuable material in short supply in the Late Antique and medieval periods. Many of the most famous Greek bronze sculptures are known through Roman copies in marble, which were more likely to survive.
In India, bronze sculptures from the Kushana (Chausa hoard) and Gupta periods (Brahma from Mirpur-Khas, Akota Hoard, Sultanganj Buddha) and later periods (Hansi Hoard) have been found.[27] Indian Hindu artisans from the period of the Chola empire in Tamil Nadu used bronze to create intricate statues via the lost wax casting method with ornate detailing depicting the deities of Hinduism. The art form survives to this day, with many silpis, craftsmen, working in the areas of Swamimalai and Chennai.
In antiquity other cultures also produced works of high art using bronze. For example: in Africa, the bronze heads of the Kingdom of Benin; in Europe, Grecian bronzes typically of figures from Greek mythology; in east Asia, Chinese ritual bronzes of the Shang and Zhou dynasty—more often ceremonial vessels but including some figurine examples. Bronze sculptures, although known for their longevity, still undergo microbial degradation; such as from certain species of yeasts.[28]
Bronze continues into modern times as one of the materials of choice for monumental statuary.
Mirrors[edit]
See also: Bronze mirror
Before it became possible to produce glass with acceptably flat surfaces, bronze was a standard material for mirrors. The reflecting surface was typically made slightly convex so that the whole face could be seen in a small mirror. Bronze was used for this purpose in many parts of the world, probably based on independent discoveries.
Bronze mirrors survive from the Egyptian Middle Kingdom (2040–1750 BCE). In Europe, the Etruscans were making bronze mirrors in the sixth century BCE, and Greek and Roman mirrors followed the same pattern. Although other materials such as speculum metal had come into use, bronze mirrors were still being made in Japan in the eighteenth century AD.
Musical instruments[edit]
Chinese bells:Bianzhong of Marquis Yi of Zeng, Spring and Autumn period (476–221 BC)
Singing bowls from the 16th to 18th centuries. Annealed bronze continues to be made in the Himalayas
Bronze is the preferred metal for bells in the form of a high tin bronze alloy known colloquially as bell metal, which is about 23% tin.
Nearly all professional cymbals are made from bronze, which gives a desirable balance of durability and timbre. Several types of bronze are used, commonly B20 bronze, which is roughly 20% tin, 80% copper, with traces of silver, or the tougher B8 bronze which is made from 8% tin and 92% copper. As the tin content in a bell or cymbal rises, the timbre drops.[29]
Bronze is also used for the windings of steel and nylon strings of various stringed instruments such as the double bass, piano, harpsichord, and guitar. Bronze strings are commonly reserved on pianoforte for the lower pitch tones, as they possess a superior sustain quality to that of high-tensile steel.[30]
Bronzes of various metallurgical properties are widely used in struck idiophones around the world, notably bells, singing bowls, gongs, cymbals, and other idiophones from Asia. Examples include Tibetan singing bowls, temple bells of many sizes and shapes, gongs, Javanese gamelan, and other bronze musical instruments. The earliest bronze archeological finds in Indonesia date from 1–2 BCE, including flat plates probably suspended and struck by a wooden or bone mallet.[30][31] Ancient bronze drums from Thailand and Vietnam date back 2,000 years. Bronze bells from Thailand and Cambodia date back to 3,600 BCE.
Some companies are now making saxophones from phosphor bronze (3.5 to 10% tin and up to 1% phosphorus content).[32] Bell bronze is used to make the tone rings of many professional model banjos.[citation needed] The tone ring is a heavy (usually 3 lbs.) folded or arched metal ring attached to a thick wood rim, over which a skin, or most often, a plastic membrane (or head) is stretched – it is the bell bronze that gives the banjo a crisp powerful lower register and clear bell-like treble register, especially in bluegrass music.[citation needed]
Coins and medals[edit]
See also: Bronze medal
Bronze has also been used in coins; most “copper” coins are actually bronze, with about 4 percent tin and 1 percent zinc.[33]
As with coins, bronze has been used in the manufacture of various types of medals for centuries, and are known in contemporary times for being awarded for third place in sporting competitions and other events. The later usage was in part attributed to the choices of gold, silver and bronze to represent the first three Ages of Man in Greek mythology: the Golden Age, when men lived among the gods; the Silver age, where youth lasted a hundred years; and the Bronze Age, the era of heroes, and was first adopted at the 1904 Summer Olympics. At the 1896 event, silver was awarded to winners and bronze to runners-up, while at 1900 other prizes were given, not medals.
Image gallery[edit]
Bronze flag found in Shahdad, Kerman, (now Iran), 3rd millennium BC
Ewer from 7th-century Iran. Cast, chased, and inlaid bronze. Metropolitan Museum of Art, New York
Corinthian helmet from the tomb of Denda, Staatliche Antikensammlungen museum in Munich
Woman resting on Bill Woodrow's 'Sitting on History' utilitarian sculpture at the British Library
Croatian Apoxyomenos (detail), Mimara Museum, Zagreb
Bronze mirror with support, Metropolitan Museum of Art, New York
Tirthankara from 7th-century Akota Hoard
The inscription, which was written in ancient bronze script, documented the land system in western Zhou dynasty in detail. 9th century BC. National Palace Museum, Taipei.
See also[edit]
Art object
Bell founding
Bronze and brass ornamental work
Bronzing
Chinese bronze inscriptions
Dezincification Resistant Brass
French Empire mantel clock
List of copper alloys
Ormolu
Seagram Building
UNS C69100
Yoruba art
References[edit]
Jump up ^ Robert L. Thorp, China in the Early Bronze Age: Shang Civilization, University of Pennsylvania Press (2013).
Jump up ^ "British Museum, "Scope Note" for "copper alloy"". British Museum. Archived from the original on 18 August 2014. Retrieved 14 September 2014.
Jump up ^ Henry and Renée Kahane, "Byzantium's Impact on the West: The Linguistic Evidence", Illinois Classical Studies 06 (2) 1981, p. 395.
Jump up ^ Originally M.P.E. Berthelot, "Sur le nom du bronze chez les alchimistes grecs", in Revue archéologique, 1888, pp. 294–8.
Jump up ^ Originally Karl Lokotsch, Etymologisches Wörterbuch der europäischen Wörter orientalischen Ursprungs. (Heidelberg: Carl Winter’s Universitätsbuchhandlung, 1927), p. 1657.
Jump up ^ Wolfgang Pfeifer, ed., Etymologisches Wörterbuch des Deutschen, s.v. "Bronze" (Munich: Deutscher Taschenbucher Vertrag, 2005).
Jump up ^ Tylecote, R. F. (1992). A History of Metallurgy, Second Edition. London: Maney Publishing, for the Institute of Materials. ISBN 1-902653-79-3. Archived from the original on 2015-04-02.
Jump up ^ Thornton, C.; Lamberg-Karlovsky, C.C.; Liezers, M.; Young, S.M.M. (2002). "On pins and needles: tracing the evolution of copper-based alloying at Tepe Yahya, Iran, via ICP-MS analysis of Common-place items". Journal of Archaeological Science. 29 (12): 1451–1460. doi:10.1006/jasc.2002.0809.
Jump up ^ Kaufman, Brett. "Metallurgy and Archaeological Change in the Ancient Near East". Backdirt: Annual Review. 2011: 86.
Jump up ^ Radivojević, Miljana; Rehren, Thilo (December 2013). "Tainted ores and the rise of tin bronzes in Eurasia, c. 6500 years ago". Antiquity Publications Ltd. Archived from the original on 2014-02-05.
Jump up ^ History of Africa#Metallurgy
Jump up ^ Smithells Metals Reference Book, 8th Edition, ch. 22
Jump up ^ Clayton E. Cramer. What Caused The Iron Age? Archived 2010-12-28 at the Wayback Machine. claytoncramer.com. December 10, 1995
Jump up ^ Oleg D. Sherby and Jeffrey Wadsworth. Ancient Blacksmiths, the Iron Age, Damascus Steels, and Modern Metallurgy Archived 2007-06-26 at the Wayback Machine.. Tbermec 2000, Las Vegas, Nevada December 4–8, 2000. Retrieved on 2012-06-09.
Jump up ^ Knapp, Brian. (1996) Copper, Silver and Gold. Reed Library, Australia.
Jump up ^ "Copper alloys". Archived from the original on 11 September 2013. Retrieved 14 September 2014.
Jump up ^ "CDA UNS Standard Designations for Wrought and Cast Copper and Copper Alloys: Introduction". Archived from the original on 24 September 2013. Retrieved 14 September 2014.
Jump up ^ "Bismuth Bronze". Archived from the original on 16 March 2015. Retrieved 14 September 2014.
Jump up ^ plastic bronze definition of plastic bronze in the Free Online Encyclopedia
Jump up ^ Adams, Jonathan R. (2012). "The Belgammel Ram, a Hellenistic-Roman BronzeProembolionFound off the Coast of Libya: test analysis of function, date and metallurgy, with a digital reference archive" (PDF). International Journal of Nautical Archaeology. 42 (1): 60–75. doi:10.1111/1095-9270.12001. Archived (PDF) from the original on 2016-08-28.
Jump up ^ ASTM B124 / B124M – 15. ASTM International. 2015.
Jump up ^ "Bronze Disease, Archaeologies of the Greek Past". Retrieved 14 September 2014. [permanent dead link]
Jump up ^ A. Alavudeen; N. Venkateshwaran; J. T. Winowlin Jappes (1 January 2006). A Textbook of Engineering Materials and Metallurgy. Firewall Media. pp. 136–. ISBN 978-81-7008-957-5. Archived from the original on 10 June 2016. Retrieved 25 June 2013.
Jump up ^ Resources: Standards & Properties – Copper & Copper Alloy Microstructures: Phosphor Bronze Archived 2015-12-08 at the Wayback Machine.
Jump up ^ Resources: Standards & Properties – Copper & Copper Alloy Microstructures: Aluminum Bronzes Archived 2013-12-05 at the Wayback Machine.
Jump up ^ for a translation of his inscription see the appendix in Stephanie Dalley, (2013) The Mystery of the Hanging Garden of Babylon: an elusive World Wonder traced, OUP. ISBN 978-0-19-966226-5
Jump up ^ Indian bronze masterpieces: the great tradition: specially published for the Festival of India, Asharani Mathur, Sonya Singh, Festival of India, Brijbasi Printers, Dec 1, 1988
Jump up ^ Francesca Cappitelli; Claudia Sorlini (2008). "Microorganisms Attack Synthetic Polymers in Items Representing Our Cultural Heritage". Applied and Environmental Microbiology. 74 (3): 564–9. doi:10.1128/AEM.01768-07. PMC 2227722  . PMID 18065627.
Jump up ^ Von Falkenhausen, Lothar (1993). Suspended Music: Chime-Bells in the Culture of Bronze Age China. Berkeley and Los Angeles: University of California Press. p. 106. ISBN 978-0-520-07378-4. Archived from the original on 2016-05-26.
^ Jump up to: a b McCreight, Tim. Metals technic: a collection of techniques for metalsmiths. Brynmorgen Press, 1992. ISBN 0-9615984-3-3
Jump up ^ LaPlantz, David. Jewelry – Metalwork 1991 Survey: Visions – Concepts – Communication: S. LaPlantz: 1991. ISBN 0-942002-05-9
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Bronze at Curlie (based on DMOZ)
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"Lost Wax, Found Bronze": lost-wax casting explained
"Flash animation of the lost-wax casting process". James Peniston Sculpture. Retrieved 2008-11-03.
Viking Bronze – Ancient and Early Medieval bronze casting
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text:
Frame of reference
From Wikipedia, the free encyclopedia
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Not to be confused with Inertial frame of reference.
For other uses, see Framing.
Classical mechanics
F → = m a → {\displaystyle {\vec {F}}=m{\vec {a}}}
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v
t
e
In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements.
In n dimensions, n+1 reference points are sufficient to fully define a reference frame. Using rectangular (Cartesian) coordinates, a reference frame may be defined with a reference point at the origin and a reference point at one unit distance along each of the n coordinate axes.
In Einsteinian relativity, reference frames are used to specify the relationship between a moving observer and the phenomenon or phenomena under observation. In this context, the phrase often becomes "observational frame of reference" (or "observational reference frame"), which implies that the observer is at rest in the frame, although not necessarily located at its origin. A relativistic reference frame includes (or implies) the coordinate time, which does not correspond across different frames moving relatively to each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent.
Contents
[hide]
1 Different aspects of "frame of reference"
1.1 Coordinate systems
1.2 Observational frames of reference
1.3 Measurement apparatus
2 Types
3 Examples of inertial frames of reference
3.1 Simple example
3.2 Additional example
3.3 Remarks
4 Non-inertial frames
5 Particular frames of reference in common use
5.1 Other frames
6 See also
7 Notes
Different aspects of "frame of reference"[edit]
The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in Cartesian frame of reference. Sometimes the state of motion is emphasized, as in rotating frame of reference. Sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference.[1]
In this article, the term observational frame of reference is used when emphasis is upon the state of motion rather than upon the coordinate choice or the character of the observations or observational apparatus. In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a coordinate system may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors, which are only indirectly related to space and time. It seems useful to divorce the various aspects of a reference frame for the discussion below. We therefore take observational frames of reference, coordinate systems, and observational equipment as independent concepts, separated as below:
An observational frame (such as an inertial frame or non-inertial frame of reference) is a physical concept related to state of motion.
A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations.[2] Consequently, an observer in an observational frame of reference can choose to employ any coordinate system (Cartesian, polar, curvilinear, generalized, …) to describe observations made from that frame of reference. A change in the choice of this coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's observational frame of reference. This viewpoint can be found elsewhere as well.[3] Which is not to dispute that some coordinate systems may be a better choice for some observations than are others.
Choice of what to measure and with what observational apparatus is a matter separate from the observer's state of motion and choice of coordinate system.
Here is a quotation applicable to moving observational frames R {\displaystyle {\mathfrak {R}}} and various associated Euclidean three-space coordinate systems [R, R′, etc.]:[4]
“ We first introduce the notion of reference frame, itself related to the idea of observer: the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted R {\displaystyle {\mathfrak {R}}} , is said to move with the observer.… The spatial positions of particles are labelled relative to a frame R {\displaystyle {\mathfrak {R}}} by establishing a coordinate system R with origin O. The corresponding set of axes, sharing the rigid body motion of the frame R {\displaystyle {\mathfrak {R}}} , can be considered to give a physical realization of R {\displaystyle {\mathfrak {R}}} . In a frame R {\displaystyle {\mathfrak {R}}} , coordinates are changed from R to R′ by carrying out, at each instant of time, the same coordinate transformation on the components of intrinsic objects (vectors and tensors) introduced to represent physical quantities in this frame. ”
and this on the utility of separating the notions of R {\displaystyle {\mathfrak {R}}} and [R, R′, etc.]:[5]
“ As noted by Brillouin, a distinction between mathematical sets of coordinates and physical frames of reference must be made. The ignorance of such distinction is the source of much confusion… the dependent functions such as velocity for example, are measured with respect to a physical reference frame, but one is free to choose any mathematical coordinate system in which the equations are specified. ”
and this, also on the distinction between R {\displaystyle {\mathfrak {R}}} and [R, R′, etc.]:[6]
“ The idea of a reference frame is really quite different from that of a coordinate system. Frames differ just when they define different spaces (sets of rest points) or times (sets of simultaneous events). So the ideas of a space, a time, of rest and simultaneity, go inextricably together with that of frame. However, a mere shift of origin, or a purely spatial rotation of space coordinates results in a new coordinate system. So frames correspond at best to classes of coordinate systems. ”
and from J. D. Norton:[7]
“ In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. … Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime.…Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion, then little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.…More recently, to negotiate the obvious ambiguities of Einstein’s treatment, the notion of frame of reference has reappeared as a structure distinct from a coordinate system. ”
The discussion is taken beyond simple space-time coordinate systems by Brading and Castellani.[8] Extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations[9] of quantum field theory, classical relativistic mechanics, and quantum gravity.[10][11][12][13][14]
Coordinate systems[edit]
Main article: Coordinate systems
See also: Generalized coordinates and Axes conventions
An observer O, situated at the origin of a local set of coordinates – a frame of reference F. The observer in this frame uses the coordinates (x, y, z, t) to describe a spacetime event, shown as a star.
Although the term "coordinate system" is often used (particularly by physicists) in a nontechnical sense, the term "coordinate system" does have a precise meaning in mathematics, and sometimes that is what the physicist means as well.
A coordinate system in mathematics is a facet of geometry or of algebra,[15][16] in particular, a property of manifolds (for example, in physics, configuration spaces or phase spaces).[17][18] The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:[19][20]
r = [ x 1 , x 2 , … , x n ] . {\displaystyle \mathbf {r} =[x^{1},\ x^{2},\ \dots \ ,x^{n}]\ .}
In a general Banach space, these numbers could be (for example) coefficients in a functional expansion like a Fourier series. In a physical problem, they could be spacetime coordinates or normal mode amplitudes. In a robot design, they could be angles of relative rotations, linear displacements, or deformations of joints.[21] Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:
x j = x j ( x , y , z , … ) , {\displaystyle x^{j}=x^{j}(x,\ y,\ z,\ \dots )\ ,} j = 1 , … , n {\displaystyle j=1,\ \dots \ ,\ n\ }
where x, y, z, etc. are the n Cartesian coordinates of the point. Given these functions, coordinate surfaces are defined by the relations:
x j ( x , y , z , … ) = c o n s t a n t , {\displaystyle x^{j}(x,y,z,\dots )=\mathrm {constant} \ ,} j = 1 , … , n . {\displaystyle j=1,\ \dots \ ,\ n\ .}
The intersection of these surfaces define coordinate lines. At any selected point, tangents to the intersecting coordinate lines at that point define a set of basis vectors {e1, e2, …, en} at that point. That is:[22]
e i ( r ) = lim ϵ → 0 r ( x 1 , … , x i + ϵ , … , x n ) − r ( x 1 , … , x i , … , x n ) ϵ , {\displaystyle \mathbf {e} _{i}(\mathbf {r} )=\lim _{\epsilon \rightarrow 0}{\frac {\mathbf {r} \left(x^{1},\ \dots ,\ x^{i}+\epsilon ,\ \dots ,\ x^{n}\right)-\mathbf {r} \left(x^{1},\ \dots ,\ x^{i},\ \dots ,\ x^{n}\right)}{\epsilon }}\ ,} i = 1 , … , n {\displaystyle i=1,\ \dots \ ,\ n\ }
which can be normalized to be of unit length. For more detail see curvilinear coordinates.
Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.[23] If the basis vectors are orthogonal at every point, the coordinate system is an orthogonal coordinate system.
An important aspect of a coordinate system is its metric tensor gik, which determines the arc length ds in the coordinate system in terms of its coordinates:[24]
( d s ) 2 = g i k d x i d x k , {\displaystyle (ds)^{2}=g_{ik}\ dx^{i}\ dx^{k}\ ,}
where repeated indices are summed over.
As is apparent from these remarks, a coordinate system is a mathematical construct, part of an axiomatic system. There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can include time as a coordinate, and can be used to describe motion. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations.
General and specific topics of coordinate systems can be pursued following the See also links below.
Observational frames of reference[edit]
Main article: Inertial frame of reference
Three frames of reference in special relativity. The black frame is at rest. The primed frame moves at 40% of light speed, and the double primed frame at 80%. Note the scissors-like change as speed increases.
An observational frame of reference, often referred to as a physical frame of reference, a frame of reference, or simply a frame, is a physical concept related to an observer and the observer's state of motion. Here we adopt the view expressed by Kumar and Barve: an observational frame of reference is characterized only by its state of motion.[25] However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an observer and a frame. According to this view, a frame is an observer plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran.[26] This restricted view is not used here, and is not universally adopted even in discussions of relativity.[27][28] In general relativity the use of general coordinate systems is common (see, for example, the Schwarzschild solution for the gravitational field outside an isolated sphere[29]).
There are two types of observational reference frame: inertial and non-inertial. An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. In special relativity these frames are related by Lorentz transformations, which are parametrized by rapidity. In Newtonian mechanics, a more restricted definition requires only that Newton's first law holds true; that is, a Newtonian inertial frame is one in which a free particle travels in a straight line at constant speed, or is at rest. These frames are related by Galilean transformations. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of representations of the Poincaré group and of the Galilean group.
In contrast to the inertial frame, a non-inertial frame of reference is one in which fictitious forces must be invoked to explain observations. An example is an observational frame of reference centered at a point on the Earth's surface. This frame of reference orbits around the center of the Earth, which introduces the fictitious forces known as the Coriolis force, centrifugal force, and gravitational force. (All of these forces including gravity disappear in a truly inertial reference frame, which is one of free-fall.)
Measurement apparatus[edit]
A further aspect of a frame of reference is the role of the measurement apparatus (for example, clocks and rods) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in quantum mechanics, where the relation between observer and measurement is still under discussion (see measurement problem).
In physics experiments, the frame of reference in which the laboratory measurement devices are at rest is usually referred to as the laboratory frame or simply "lab frame." An example would be the frame in which the detectors for a particle accelerator are at rest. The lab frame in some experiments is an inertial frame, but it is not required to be (for example the laboratory on the surface of the Earth in many physics experiments is not inertial). In particle physics experiments, it is often useful to transform energies and momenta of particles from the lab frame where they are measured, to the center of momentum frame "COM frame" in which calculations are sometimes simplified, since potentially all kinetic energy still present in the COM frame may be used for making new particles.
In this connection it may be noted that the clocks and rods often used to describe observers' measurement equipment in thought, in practice are replaced by a much more complicated and indirect metrology that is connected to the nature of the vacuum, and uses atomic clocks that operate according to the standard model and that must be corrected for gravitational time dilation.[30] (See second, meter and kilogram).
In fact, Einstein felt that clocks and rods were merely expedient measuring devices and they should be replaced by more fundamental entities based upon, for example, atoms and molecules.[31]
Types[edit]
Body-fixed frames of reference
Space-fixed frames of reference
Inertial frames of reference
Non-Inertial frames of reference
Examples of inertial frames of reference[edit]
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Simple example[edit]
Figure 1: Two cars moving at different but constant velocities observed from stationary inertial frame S attached to the road and moving inertial frame S′ attached to the first car.
Consider a situation common in everyday life. Two cars travel along a road, both moving at constant velocities. See Figure 1. At some particular moment, they are separated by 200 metres. The car in front is travelling at 22 metres per second and the car behind is travelling at 30 metres per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose.
First, we could observe the two cars from the side of the road. We define our "frame of reference" S as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance d = 200 m apart. Since neither of the cars is accelerating, we can determine their positions by the following formulas, where x 1 ( t ) {\displaystyle x_{1}(t)} is the position in meters of car one after time t in seconds and x 2 ( t ) {\displaystyle x_{2}(t)} is the position of car two after time t.
x 1 ( t ) = d + v 1 t = 200 + 22 t ; x 2 ( t ) = v 2 t = 30 t {\displaystyle x_{1}(t)=d+v_{1}t=200\ +\ 22t\ ;\quad x_{2}(t)=v_{2}t=30t}
Notice that these formulas predict at t = 0 s the first car is 200 m down the road and the second car is right beside us, as expected. We want to find the time at which x 1 = x 2 {\displaystyle x_{1}=x_{2}} . Therefore, we set x 1 = x 2 {\displaystyle x_{1}=x_{2}} and solve for t {\displaystyle t} , that is:
200 + 22 t = 30 t {\displaystyle 200+22t=30t\quad }
8 t = 200 {\displaystyle 8t=200\quad }
t = 25 s e c o n d s {\displaystyle t=25\quad \mathrm {seconds} }
Alternatively, we could choose a frame of reference S′ situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of v2 − v1 = 8 m / s. In order to catch up to the first car, it will take a time of d/v2 − v1 = 200/8 s, that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable frame of reference. The third possible frame of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at 8 m / s.
It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one is able to convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you are able to deduct five minutes from the time displayed on your watch in order to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).
Additional example[edit]
Figure 2: Simple-minded frame-of-reference example
For a simple example involving only the orientation of two observers, consider two people standing, facing each other on either side of a north-south street. See Figure 2. A car drives past them heading south. For the person facing east, the car was moving toward the right. However, for the person facing west, the car was moving toward the left. This discrepancy is because the two people used two different frames of reference from which to investigate this system.
For a more complex example involving observers in relative motion, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's frame of reference is considered an inertial frame of reference because he is not accelerating (ignoring effects such as Earth's rotation and gravity).
Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving – for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.
Finally, as an example of non-inertial observers, assume Candace is accelerating her car. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same as Alfred in her frame of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a′ = a − A in the negative y-direction - a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a′ = a + A in the negative y-direction – a larger value than Alfred's measurement.
Frames of reference are especially important in special relativity, because when a frame of reference is moving at some significant fraction of the speed of light, then the flow of time in that frame does not necessarily apply in another frame. The speed of light is considered to be the only true constant between moving frames of reference.
Remarks[edit]
See also: Special theory of relativity and General theory of relativity
It is important to note some assumptions made above about the various inertial frames of reference. Newton, for instance, employed universal time, as explained by the following example. Suppose that you own two clocks, which both tick at exactly the same rate. You synchronize them so that they both display exactly the same time. The two clocks are now separated and one clock is on a fast moving train, traveling at constant velocity towards the other. According to Newton, these two clocks will still tick at the same rate and will both show the same time. Newton says that the rate of time as measured in one frame of reference should be the same as the rate of time in another. That is, there exists a "universal" time and all other times in all other frames of reference will run at the same rate as this universal time irrespective of their position and velocity. This concept of time and simultaneity was later generalized by Einstein in his special theory of relativity (1905) where he developed transformations between inertial frames of reference based upon the universal nature of physical laws and their economy of expression (Lorentz transformations).
It is also important to note that the definition of inertial reference frame can be extended beyond three-dimensional Euclidean space. Newton's assumed a Euclidean space, but general relativity uses a more general geometry. As an example of why this is important, let us consider the geometry of an ellipsoid. In this geometry, a "free" particle is defined as one at rest or traveling at constant speed on a geodesic path. Two free particles may begin at the same point on the surface, traveling with the same constant speed in different directions. After a length of time, the two particles collide at the opposite side of the ellipsoid. Both "free" particles traveled with a constant speed, satisfying the definition that no forces were acting. No acceleration occurred and so Newton's first law held true. This means that the particles were in inertial frames of reference. Since no forces were acting, it was the geometry of the situation which caused the two particles to meet each other again. In a similar way, it is now common to describe[32] that we exist in a four-dimensional geometry known as spacetime. In this picture, the curvature of this 4D space is responsible for the way in which two bodies with mass are drawn together even if no forces are acting. This curvature of spacetime replaces the force known as gravity in Newtonian mechanics and special relativity.
Non-inertial frames[edit]
Main articles: Fictitious force, Non-inertial frame, and Rotating frame of reference
Here the relation between inertial and non-inertial observational frames of reference is considered. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below.
An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x′, y′, a′.
The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r′. From the geometry of the situation, we get
r = R + r ′ {\displaystyle \mathbf {r} =\mathbf {R} +\mathbf {r} '}
Taking the first and second derivatives of this with respect to time, we obtain
v = V + v ′ {\displaystyle \mathbf {v} =\mathbf {V} +\mathbf {v} '}
a = A + a ′ {\displaystyle \mathbf {a} =\mathbf {A} +\mathbf {a} '}
where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.
These equations allow transformations between the two coordinate systems; for example, we can now write Newton's second law as
F = m a = m A + m a ′ {\displaystyle \mathbf {F} =m\mathbf {a} =m\mathbf {A} +m\mathbf {a} '}
When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in a direction orthogonal to an object's motion, the Coriolis effect).
A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation (see Fictitious force for a derivation):
a = a ′ + ω ˙ × r ′ + 2 ω × v ′ + ω × ( ω × r ′ ) + A 0 {\displaystyle \mathbf {a} =\mathbf {a} '+{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '+2{\boldsymbol {\omega }}\times \mathbf {v} '+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')+\mathbf {A} _{0}}
or, to solve for the acceleration in the accelerated frame,
a ′ = a − ω ˙ × r ′ − 2 ω × v ′ − ω × ( ω × r ′ ) − A 0 {\displaystyle \mathbf {a} '=\mathbf {a} -{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '-2{\boldsymbol {\omega }}\times \mathbf {v} '-{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')-\mathbf {A} _{0}}
Multiplying through by the mass m gives
F ′ = F p h y s i c a l + F E u l e r ′ + F C o r i o l i s ′ + F c e n t r i p e t a l ′ − m A 0 {\displaystyle \mathbf {F} '=\mathbf {F} _{\mathrm {physical} }+\mathbf {F} '_{\mathrm {Euler} }+\mathbf {F} '_{\mathrm {Coriolis} }+\mathbf {F} '_{\mathrm {centripetal} }-m\mathbf {A} _{0}}
where
F E u l e r ′ = − m ω ˙ × r ′ {\displaystyle \mathbf {F} '_{\mathrm {Euler} }=-m{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '} (Euler force)
F C o r i o l i s ′ = − 2 m ω × v ′ {\displaystyle \mathbf {F} '_{\mathrm {Coriolis} }=-2m{\boldsymbol {\omega }}\times \mathbf {v} '} (Coriolis force)
F c e n t r i f u g a l ′ = − m ω × ( ω × r ′ ) = m ( ω 2 r ′ − ( ω ⋅ r ′ ) ω ) {\displaystyle \mathbf {F} '_{\mathrm {centrifugal} }=-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')=m(\omega ^{2}\mathbf {r} '-({\boldsymbol {\omega }}\cdot \mathbf {r} '){\boldsymbol {\omega }})} (centrifugal force)
Particular frames of reference in common use[edit]
International Terrestrial Reference Frame
International Celestial Reference Frame
In fluid mechanics, Lagrangian and Eulerian specification of the flow field
Other frames[edit]
Frame fields in general relativity
Linguistic frame of reference
Moving frame in Mathematics
See also[edit]
Analytical mechanics
Applied mechanics
Cartesian coordinate system
Center-of-momentum frame
Centrifugal force
Centripetal force
Classical mechanics
Coriolis force
Curvilinear coordinates
Dynamics (physics)
Fictitious force
Frenet–Serret formulas
Galilean invariance
General relativity
Generalized coordinates
Generalized forces
Inertial frame of reference
Rod and frame test
Kinematics
Laboratory frame of reference
Lorentz transformation
Mach's principle
Orthogonal coordinates
Principle of relativity
Quantum reference frame
Special relativity
Theory of relativity
Notes[edit]
Jump up ^ The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where constitutive relations of various time and length scales are used to determine the current and charge densities entering Maxwell's equations. See, for example, Kurt Edmund Oughstun (2006). Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media. Springer. p. 165. ISBN 0-387-34599-X. . These distinctions also appear in thermodynamics. See Paul McEvoy (2002). Classical Theory. MicroAnalytix. p. 205. ISBN 1-930832-02-8. .
Jump up ^ In very general terms, a coordinate system is a set of arcs xi = xi (t) in a complex Lie group; see Lev Semenovich Pontri͡agin. L.S. Pontryagin: Selected Works Vol. 2: Topological Groups (3rd ed.). Gordon and Breach. p. 429. ISBN 2-88124-133-6. . Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {e1, e2,… en}; see Edoardo Sernesi; J. Montaldi (1993). Linear Algebra: A Geometric Approach. CRC Press. p. 95. ISBN 0-412-40680-2. As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.
Jump up ^ J X Zheng-Johansson; Per-Ivar Johansson (2006). Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces. Nova Publishers. p. 13. ISBN 1-59454-260-0.
Jump up ^ Jean Salençon; Stephen Lyle (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity. Springer. p. 9. ISBN 3-540-41443-6.
Jump up ^ Patrick Cornille (Akhlesh Lakhtakia, editor) (1993). Essays on the Formal Aspects of Electromagnetic Theory. World Scientific. p. 149. ISBN 981-02-0854-5.
Jump up ^ Graham Nerlich (1994). What Spacetime Explains: Metaphysical essays on space and time. Cambridge University Press. p. 64. ISBN 0-521-45261-9.
Jump up ^ John D. Norton (1993). General covariance and the foundations of general relativity: eight decades of dispute, Rep. Prog. Phys., 56, pp. 835-7.
Jump up ^ Katherine Brading; Elena Castellani (2003). Symmetries in Physics: Philosophical Reflections. Cambridge University Press. p. 417. ISBN 0-521-82137-1.
Jump up ^ Oliver Davis Johns (2005). Analytical Mechanics for Relativity and Quantum Mechanics. Oxford University Press. Chapter 16. ISBN 0-19-856726-X.
Jump up ^ Donald T Greenwood (1997). Classical dynamics (Reprint of 1977 edition by Prentice-Hall ed.). Courier Dover Publications. p. 313. ISBN 0-486-69690-1.
Jump up ^ Matthew A. Trump; W. C. Schieve (1999). Classical Relativistic Many-Body Dynamics. Springer. p. 99. ISBN 0-7923-5737-X.
Jump up ^ A S Kompaneyets (2003). Theoretical Physics (Reprint of the 1962 2nd ed.). Courier Dover Publications. p. 118. ISBN 0-486-49532-9.
Jump up ^ M Srednicki (2007). Quantum Field Theory. Cambridge University Press. Chapter 4. ISBN 978-0-521-86449-7.
Jump up ^ Carlo Rovelli (2004). Quantum Gravity. Cambridge University Press. p. 98 ff. ISBN 0-521-83733-2.
Jump up ^ William Barker; Roger Howe (2008). Continuous symmetry: from Euclid to Klein. American Mathematical Society. p. 18 ff. ISBN 0-8218-3900-4.
Jump up ^ Arlan Ramsay; Robert D. Richtmyer (1995). Introduction to Hyperbolic Geometry. Springer. p. 11. ISBN 0-387-94339-0.
Jump up ^ According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." Stephen W. Hawking; George Francis Rayner Ellis (1973). The Large Scale Structure of Space-Time. Cambridge University Press. p. 11. ISBN 0-521-09906-4. A mathematical definition is: A connected Hausdorff space M is called an n-dimensional manifold if each point of M is contained in an open set that is homeomorphic to an open set in Euclidean n-dimensional space.
Jump up ^ Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore. p. 12. ISBN 0-8218-1045-6.
Jump up ^ Granino Arthur Korn; Theresa M. Korn (2000). Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review. Courier Dover Publications. p. 169. ISBN 0-486-41147-8.
Jump up ^ See Encarta definition. Archived 2009-10-31.
Jump up ^ Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer. pp. 12–13. ISBN 3-540-20317-6.
Jump up ^ Achilleus Papapetrou (1974). Lectures on General Relativity. Springer. p. 5. ISBN 90-277-0540-2.
Jump up ^ Wilford Zdunkowski; Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. p. 84. ISBN 0-521-00666-X.
Jump up ^ A. I. Borisenko; I. E. Tarapov; Richard A. Silverman (1979). Vector and Tensor Analysis with Applications. Courier Dover Publications. p. 86. ISBN 0-486-63833-2.
Jump up ^ See Arvind Kumar; Shrish Barve (2003). How and Why in Basic Mechanics. Orient Longman. p. 115. ISBN 81-7371-420-7.
Jump up ^ Chris Doran; Anthony Lasenby (2003). Geometric Algebra for Physicists. Cambridge University Press. p. §5.2.2, p. 133. ISBN 978-0-521-71595-9. .
Jump up ^ For example, Møller states: "Instead of Cartesian coordinates we can obviously just as well employ general curvilinear coordinates for the fixation of points in physical space.…we shall now introduce general "curvilinear" coordinates xi in four-space…." C. Møller (1952). The Theory of Relativity. Oxford University Press. p. 222 and p. 233.
Jump up ^ A. P. Lightman; W. H. Press; R. H. Price; S. A. Teukolsky (1975). Problem Book in Relativity and Gravitation. Princeton University Press. p. 15. ISBN 0-691-08162-X.
Jump up ^ Richard L Faber (1983). Differential Geometry and Relativity Theory: an introduction. CRC Press. p. 211. ISBN 0-8247-1749-X.
Jump up ^ Richard Wolfson (2003). Simply Einstein. W W Norton & Co. p. 216. ISBN 0-393-05154-4.
Jump up ^ See Guido Rizzi; Matteo Luca Ruggiero (2003). Relativity in rotating frames. Springer. p. 33. ISBN 1-4020-1805-3. .
Jump up ^ That is, both descriptions are equivalent and can be used as needed. This equivalence does not hold outside of general relativity, e.g., in entropic gravity.
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Einstein field equations: G μ ν + Λ g μ ν = 8 π G c 4 T μ ν {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }} and their analytical solution by Ernst equation: ℜ ( u ) ( u r r + u r / r + u z z ) = ( u r ) 2 + ( u z ) 2 . {\displaystyle \displaystyle \Re (u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}.}
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