In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is somewhat complex. Dimensional analysis, or more the factor-label method known as the unit-factor method, is a used technique for such conversions using the rules of algebra; the concept of physical dimension was introduced by Joseph Fourier in 1822. Physical quantities that are of the same kind have the same dimension and can be directly compared to each other if they are expressed in differing units of measure. If physical quantities have different dimensions, they cannot be expressed in terms of similar units and cannot be compared in quantity. For example, asking whether a kilogram is larger than an hour is meaningless. Any physically meaningful equation will have the same dimensions on its left and right sides, a property known as dimensional homogeneity.
Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation. Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number – a numerical quantity and a corresponding dimensional unit. A quantity is expressed in terms of several other quantities. Compound relations with "per" are expressed with division, e.g. 60 mi/1 h. Other relations can involve powers, or combinations thereof. A set of base units for a system of measurement is a conventionally chosen set of units, none of which can be expressed as a combination of the others, in terms of which all the remaining units of the system can be expressed. For example, units for length and time are chosen as base units. Units for volume, can be factored into the base units of length, thus they are considered derived or compound units.
Sometimes the names of units obscure the fact. For example, a newton is a unit of force; the newton is defined as 1 N = 1 kg⋅m⋅s−2. Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100. Taking a derivative with respect to a quantity adds the dimension of the variable one is differentiating with respect to, in the denominator. Thus: position has the dimension L. In economics, one distinguishes between stocks and flows: a stock has units of "units", while a flow is a derivative of a stock, has units of "units/time". In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, debt-to-GDP ratios are expressed as percentages: total debt outstanding divided by annual GDP – but one may argue that in comparing a stock to a flow, annual GDP should have dimensions of currency/time, thus Debt-to-GDP should have units of years, which indicates that Debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor. For example, kPa and bar are both units of pressure, 100 kPa = 1 bar; the rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including units. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500, bar/bar cancels out, so 5 bar = 500 kPa; the most basic rule of dimensional analysis is that of dimensional homogeneity. 1. Only commensurable quantities may be compared, added, or subtracted. However, the dimensions form an abelian group under multiplication, so: 2. One may take ratios of incommensurable quantities, multiply or divide them. For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometer, as these have different dimensions, nor to add 1 hour to 1 kilometer.
However, it makes perfect sense to ask whether 1 mile is more, the same, or less than 1 kilometer being the same dimension of physical quantity though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h; the rule implies that in a physically mea
In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams can be used to visualize relativistic effects such as why different observers perceive where and when events occur; until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe was independent of one-dimensional time. However, in 1905, Albert Einstein based his seminal work on special relativity on two postulates: The laws of physics are invariant in all inertial systems; the logical consequence of taking these postulates together is the inseparable joining together of the four dimensions, hitherto assumed as independent, of space and time. Many counterintuitive consequences emerge: in addition to being independent of the motion of the light source, the speed of light has the same speed regardless of the frame of reference in which it is measured. Einstein framed his theory in terms of kinematics.
His theory was a breakthrough advance over Lorentz's 1904 theory of electromagnetic phenomena and Poincaré's electrodynamic theory. Although these theories included equations identical to those that Einstein introduced, they were ad hoc models proposed to explain the results of various experiments—including the famous Michelson–Morley interferometer experiment—that were difficult to fit into existing paradigms. In 1908, Hermann Minkowski—once one of the math professors of a young Einstein in Zürich—presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. A key feature of this interpretation is the formal definition of the spacetime interval. Although measurements of distance and time between events differ for measurements made in different reference frames, the spacetime interval is independent of the inertial frame of reference in which they are recorded. Minkowski's geometric interpretation of relativity was to prove vital to Einstein's development of his 1915 general theory of relativity, wherein he showed how mass and energy curve this flat spacetime to a Pseudo Riemannian manifold.
Non-relativistic classical mechanics treats time as a universal quantity of measurement, uniform throughout space and, separate from space. Classical mechanics assumes that time has a constant rate of passage, independent of the state of motion of an observer, or indeed of anything external. Furthermore, it assumes that space is Euclidean, to say, it assumes that space follows the geometry of common sense. In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field. In ordinary space, a position is specified by three numbers, known as dimensions. In the Cartesian coordinate system, these are called x, y, z. A position in spacetime is called an event, requires four numbers to be specified: the three-dimensional location in space, plus the position in time.
Spacetime is thus four dimensional. An event is something that happens instantaneously at a single point in spacetime, represented by a set of coordinates x, y, z and t; the word "event" used in relativity should not be confused with the use of the word "event" in normal conversation, where it might refer to an "event" as something such as a concert, sporting event, or a battle. These are not mathematical "events" in the way the word is used in relativity, because they have finite durations and extents. Unlike the analogies used to explain events, such as firecrackers or lightning bolts, mathematical events have zero duration and represent a single point in spacetime; the path of a particle through spacetime can be considered to be a succession of events. The series of events can be linked together to form a line which represents a particle's progress through spacetime; that line is called the particle's world line. Mathematically, spacetime is a manifold, to say, it appears locally "flat" near each point in the same way that, at small enough scales, a globe appears flat.
An large scale factor, c relates distances measured in space with distances measured in time. The magnitude of this scale factor, along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe, noticeably different from what they might observe if the world were Euclidean, it was only with the advent of sensitive scientific measurements in the mid-1800s, such as the Fizeau experiment and the Michelson–Morley experiment, that puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space. In special relativity, an observer will, in most
The inch is a unit of length in the imperial and United States customary systems of measurement. It is equal to 1⁄12 of a foot. Derived from the Roman uncia, the word inch is sometimes used to translate similar units in other measurement systems understood as deriving from the width of the human thumb. Standards for the exact length of an inch have varied in the past, but since the adoption of the international yard during the 1950s and 1960s it has been based on the metric system and defined as 25.4 mm. The English word "inch" was an early borrowing from Latin uncia not present in other Germanic languages; the vowel change from Latin /u/ to Old English /y/ is known as umlaut. The consonant change from the Latin /k/ to English /tʃ/ is palatalisation. Both were features of Old English phonology. "Inch" is cognate with "ounce", whose separate pronunciation and spelling reflect its reborrowing in Middle English from Anglo-Norman unce and ounce. In many other European languages, the word for "inch" is the same as or derived from the word for "thumb", as a man's thumb is about an inch wide.
Examples include Afrikaans: duim. The inch is a used customary unit of length in the United States and the United Kingdom, it is used in Japan for electronic parts display screens. In most of continental Europe, the inch is used informally as a measure for display screens. For the United Kingdom, guidance on public sector use states that, since 1 October 1995, without time limit, the inch is to be used as a primary unit for road signs and related measurements of distance and may continue to be used as a secondary or supplementary indication following a metric measurement for other purposes; the international standard symbol for inch is in but traditionally the inch is denoted by a double prime, approximated by double quotes, the foot by a prime, approximated by an apostrophe. For example, three feet two inches can be written as 3′ 2″. Subdivisions of an inch are written using dyadic fractions with odd number numerators. 1 international inch is equal to: 10,000 tenths 1,000 thou or mil 100 points or gries 72 PostScript points 10, 12, 16, or 40 lines 6 computer picas 3 barleycorns 25.4 millimetres 0.999998 US Survey inches 1/3 or 0.333 palms 1/4 or 0.25 hands 1/12 or 0.08333 feet 1/36 or 0.02777 yards The earliest known reference to the inch in England is from the Laws of Æthelberht dating to the early 7th century, surviving in a single manuscript, the Textus Roffensis from 1120.
Paragraph LXVII sets out the fine for wounds of various depths: one inch, one shilling, two inches, two shillings, etc. An Anglo-Saxon unit of length was the barleycorn. After 1066, 1 inch was equal to 3 barleycorns, which continued to be its legal definition for several centuries, with the barleycorn being the base unit. One of the earliest such definitions is that of 1324, where the legal definition of the inch was set out in a statute of Edward II of England, defining it as "three grains of barley and round, placed end to end, lengthwise". Similar definitions are recorded in both Welsh medieval law tracts. One, dating from the first half of the 10th century, is contained in the Laws of Hywel Dda which superseded those of Dyfnwal, an earlier definition of the inch in Wales. Both definitions, as recorded in Ancient Laws and Institutes of Wales, are that "three lengths of a barleycorn is the inch". King David I of Scotland in his Assize of Weights and Measures is said to have defined the Scottish inch as the width of an average man's thumb at the base of the nail including the requirement to calculate the average of a small, a medium, a large man's measures.
However, the oldest surviving manuscripts date from the early 14th century and appear to have been altered with the inclusion of newer material. In 1814, Charles Butler, a mathematics teacher at Cheam School, recorded the old legal definition of the inch to be "three grains of sound ripe barley being taken out the middle of the ear, well dried, laid end to end in a row", placed the barleycorn, not the inch, as the base unit of the English Long Measure system, from which all other units were derived. John Bouvier recorded in his 1843 law dictionary that the barleycorn was the fundamental measure. Butler observed, that "s the length of the barley-corn cannot be fixed, so the inch according to this method will be uncertain", noting that a standard inch measure was now kept in the Exchequer chamber and, the legal definition of the inch; this was a point made by George Long in his 1842 Penny Cyclopædia, observing that st
The system of imperial units or the imperial system is the system of units first defined in the British Weights and Measures Act of 1824, refined and reduced. The Imperial units replaced the Winchester Standards, which were in effect from 1588 to 1825; the system came into official use across the British Empire. By the late 20th century, most nations of the former empire had adopted the metric system as their main system of measurement, although some imperial units are still used in the United Kingdom and other countries part of the British Empire; the imperial system developed from what were first known as English units, as did the related system of United States customary units. The Weights and Measures Act of 1824 was scheduled to go into effect on 1 May 1825. However, the Weights and Measures Act of 1825 pushed back the date to 1 January 1826; the 1824 Act allowed the continued use of pre-imperial units provided that they were customary known, marked with imperial equivalents. Apothecaries' units are mentioned neither in the act of 1824 nor 1825.
At the time, apothecaries' weights and measures were regulated "in England and Berwick-upon-Tweed" by the London College of Physicians, in Ireland by the Dublin College of Physicians. In Scotland, apothecaries' units were unofficially regulated by the Edinburgh College of Physicians; the three colleges published, at infrequent intervals, the London and Dublin editions having the force of law. Imperial apothecaries' measures, based on the imperial pint of 20 fluid ounces, were introduced by the publication of the London Pharmacopoeia of 1836, the Edinburgh Pharmacopoeia of 1839, the Dublin Pharmacopoeia of 1850; the Medical Act of 1858 transferred to The Crown the right to publish the official pharmacopoeia and to regulate apothecaries' weights and measures. Metric equivalents in this article assume the latest official definition. Before this date, the most precise measurement of the imperial Standard Yard was 0.914398415 metres. In 1824, the various different gallons in use in the British Empire were replaced by the imperial gallon, a unit close in volume to the ale gallon.
It was defined as the volume of 10 pounds of distilled water weighed in air with brass weights with the barometer standing at 30 inches of mercury at a temperature of 62 °F. In 1963, the gallon was redefined as the volume of 10 pounds of distilled water of density 0.998859 g/mL weighed in air of density 0.001217 g/mL against weights of density 8.136 g/mL, which works out to 4.546096 l or 277.4198 cu in. The Weights and Measures Act of 1985 switched to a gallon of 4.54609 L. These measurements were in use from 1826, when the new imperial gallon was defined, but were abolished in the United Kingdom on 1 January 1971. In the US, though no longer recommended, the apothecaries' system is still used in medicine in prescriptions for older medications. In the 19th and 20th centuries, the UK used three different systems for weight. Troy weight, used for precious metals; the distinction between mass and weight is not always drawn. A pound is a unit of mass, although it is referred to as a weight; when a distinction is necessary, the term pound-force may be used to refer to a unit of force rather than mass.
The troy pound was made the primary unit of mass by the 1824 Act. The Weights and Measures Act 1855 made the avoirdupois pound the primary unit of mass. In all the systems, the fundamental unit is the pound, all other units are defined as fractions or multiples of it. Although the 1824 act defined the yard and pound by reference to the prototype standards, it defined the values of certain physical constants, to make provision for re-creation of the standards if they were to be damaged. For the yard, the length of a pendulum beating seconds at the latitude of Greenwich at Mean Sea Level in vacuo was defined as 39.01393 inches. For the pound, the mass of a cubic inch of distilled water at an atmospheric pressure of 30 inches of mercury and a temperature of 62° Fahrenheit was defined as 252.458 grains, with there being 7,000 grains per pound. However, following the destruction of the original prototypes in the 1834 Houses of Parliament fire, it proved impossible to recreate the standards from these definitions, a new Weights and Measures Act was passed in 1855 which permitted the recreation of the prototypes from recognized secondary standards.
The imperial system is one of many systems of English units. Although most of the units are defined in more than one system, some subsidiary units were used to a much greater extent, or for different purposes, in one area rather than the other; the distinctions between these systems are not drawn precisely. One such distinction is that between these systems and older British/English units/systems or newer additions; the term imperial should not be applied to English units that were outlawed in the Weights and Measures Act 1824 or earlier, or which had fallen out of use by that time, nor to post-imperial inventions, such as the slug or poundal. The US customary system is derived from the English units that were in use at the time of settlement; because the United States was independent at the time, these units were unaffected b
Loop quantum gravity
Loop quantum gravity is a theory of quantum gravity, merging quantum mechanics and general relativity, making it a possible candidate for a theory of everything. Its goal is to unify gravity in a common theoretical framework with the other three fundamental forces of nature, beginning with relativity and adding quantum features, it competes with string theory that adds gravity. From the point of view of Einstein's theory, all attempts to treat gravity as another quantum force equal in importance to electromagnetism and the nuclear forces have failed. According to Einstein, gravity is not a force – it is a property of spacetime itself. Loop quantum gravity is an attempt to develop a quantum theory of gravity based directly on Einstein's geometric formulation. To do this, in LQG theory space and time are quantized, analogously to the way quantities like energy and momentum are quantized in quantum mechanics; the theory gives a physical picture of spacetime where space and time are granular and discrete directly because of quantization just like photons in the quantum theory of electromagnetism and the discrete energy levels of atoms.
Distance exists with a minimum. Space's structure prefers an fine fabric or network woven of finite loops; these networks of loops are called spin networks. The evolution of a spin network, or spin foam, has a scale on the order of a Planck length 10−35 metres, smaller scales do not exist. Not just matter, but space itself, prefers an atomic structure; the vast areas of research developed in several directions that involve about 30 research groups worldwide. They all share the basic physical assumptions and the mathematical description of quantum space. Research follows two directions: the more traditional canonical loop quantum gravity, the newer covariant loop quantum gravity, called spin foam theory. Physical consequences of the theory proceed in several directions; the most well-developed applies to cosmology, called loop quantum cosmology, the study of the early universe and the physics of the Big Bang. Its greatest consequence sees the evolution of the universe continuing beyond the Big Bang called the Big Bounce.
In 1986, Abhay Ashtekar reformulated Einstein's general relativity in a language closer to that of the rest of fundamental physics. Shortly after, Ted Jacobson and Lee Smolin realized that the formal equation of quantum gravity, called the Wheeler–DeWitt equation, admitted solutions labelled by loops when rewritten in the new Ashtekar variables. Carlo Rovelli and Lee Smolin defined a nonperturbative and background-independent quantum theory of gravity in terms of these loop solutions. Jorge Pullin and Jerzy Lewandowski understood that the intersections of the loops are essential for the consistency of the theory, the theory should be formulated in terms of intersecting loops, or graphs. In 1994, Rovelli and Smolin showed that the quantum operators of the theory associated to area and volume have a discrete spectrum; that is, geometry is quantized. This result defines an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose's spin networks, which are graphs labelled by spins.
The canonical version of the dynamics was put on firm ground by Thomas Thiemann, who defined an anomaly-free Hamiltonian operator, showing the existence of a mathematically consistent background-independent theory. The covariant or spin foam version of the dynamics developed during several decades, crystallized in 2008, from the joint work of research groups in France, Canada, UK, Germany, leading to the definition of a family of transition amplitudes, which in the classical limit can be shown to be related to a family of truncations of general relativity; the finiteness of these amplitudes was proven in 2011. It requires the existence of a positive cosmological constant, this is consistent with observed acceleration in the expansion of the Universe. In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations; the essential idea is that coordinates are only artifices used in describing nature, hence should play no role in the formulation of fundamental physical laws.
A more significant requirement is the principle of general relativity that states that the laws of physics take the same form in all reference systems. This is a generalization of the principle of special relativity which states that the laws of physics take the same form in all inertial frames. In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds, it is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. These are the defining symmetry transformations of General Relativity since the theory is formulated only in terms of a differentiable manifold. In general relativity, general covariance is intimately related to "diffeomorphism invariance"; this symmetry is one of the defining features of the theory. However, it is a common misunderstanding that "diffeomorphism invariance" refers to the invariance of the physical predictions of a theory under arbitrary coordinate transformations. Diffeomorphisms, as mathematicians define them, correspond to something much more radical.
Diffeomorphisms are the true symmetry transformations of general relativity, come about from the assertion that the formulation of the theory is based on a bare different
A bracket is a tall punctuation mark used in matched pairs within text, to set apart or interject other text. The matched pair is best described as closing. Less formally, in a left-to-right context, it may be described as left and right, in a right-to-left context, as right and left. Forms include round, square and angle brackets, as well as various other pairs of symbols. In addition to referring to the class of all types of brackets, the unqualified word bracket is most used to refer to a specific type of bracket. Chevrons, ⟨ ⟩, were the earliest type of bracket to appear in written English. Desiderius Erasmus coined the term lunula to refer to the rounded parentheses, recalling the shape of the crescent moon; some of the following names are contextual. – parentheses, parens, round brackets, first brackets, or circle brackets – braces are "two connecting marks used in printing". – square brackets, closed brackets, hard brackets, third brackets, crotchets, or brackets ⟨ ⟩ – pointy brackets, angle brackets, triangular brackets, diamond brackets, tuples, or chevrons < > – guillemets, inequality signs, pointy brackets, or brackets.
Sometimes referred to as angle brackets, in such cases as HTML markup. Known as broken brackets or "brokets". ⸤ ⸥. The characters ‹ › and « », known as guillemets or angular quote brackets, are quotation mark glyphs used in several European languages. Which one of each pair is the opening quote mark and, the closing quote varies between languages; the corner-brackets ｢ ｣ are quotation marks used in East Asian languages. In English, typographers prefer not to set brackets in italics when the enclosed text is italic. However, in other languages like German, if brackets enclose text in italics, they are also set in italics. Parentheses contain material, aside from the main point. A milder effect may be obtained by using a pair of commas as the delimiter, though if the sentence contains commas for other purposes, visual confusion may result. In American usage, parentheses are considered separate from other brackets, calling them "brackets" is unusual. Parentheses may be used in formal writing to add supplementary information, such as "Sen. John McCain spoke at length".
They can indicate shorthand for "either singular or plural" for nouns, e.g. "the claim". It can be used for gender neutral language in languages with grammatical gender, e.g. "he agreed with his physician". Parenthetical phrases have been used extensively in informal writing and stream of consciousness literature. Examples include the southern American author William Faulkner as well as poet E. E. Cummings. Parentheses have been used where the dash is used in alternatives, such as "parenthesis) educational testing, b) technical writing and diagrams, c) market research, d) elections. Parentheses are used in mathematical notation to indicate grouping inducing a different order of operations. For example: in the usual order of algebraic operations, 4 x 3 + 2 equals 14, since the multiplication is done before the addi
A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not particles and electromagnetic radiation such as light—can escape from inside it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole; the boundary of the region from which no escape is possible is called the event horizon. Although the event horizon has an enormous effect on the fate and circumstances of an object crossing it, no locally detectable features appear to be observed. In many ways, a black hole acts like an ideal black body. Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass; this temperature is on the order of billionths of a kelvin for black holes of stellar mass, making it impossible to observe. Objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace.
The first modern solution of general relativity that would characterize a black hole was found by Karl Schwarzschild in 1916, although its interpretation as a region of space from which nothing can escape was first published by David Finkelstein in 1958. Black holes were long considered a mathematical curiosity; the discovery of neutron stars by Jocelyn Bell Burnell in 1967 sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality. Black holes of stellar mass are expected to form when massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings. By absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses may form. There is general consensus. Despite its invisible interior, the presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light.
Matter that falls onto a black hole can form an external accretion disk heated by friction, forming some of the brightest objects in the universe. If there are other stars orbiting a black hole, their orbits can be used to determine the black hole's mass and location; such observations can be used to exclude possible alternatives such as neutron stars. In this way, astronomers have identified numerous stellar black hole candidates in binary systems, established that the radio source known as Sagittarius A*, at the core of the Milky Way galaxy, contains a supermassive black hole of about 4.3 million solar masses. On 11 February 2016, the LIGO collaboration announced the first direct detection of gravitational waves, which represented the first observation of a black hole merger; as of December 2018, eleven gravitational wave events have been observed that originated from ten merging black holes. On 10 April 2019, the first direct image of a black hole and its vicinity was published, following observations made by the Event Horizon Telescope in 2017 of the supermassive black hole in Messier 87's galactic centre.
Larry Kimura, a Hawaiian language professor at the University of Hawaii at Hilo, named the hole Pōwehi—a Hawaiian phrase referring to an "embellished dark source of unending creation." The idea of a body so massive that light could not escape was proposed by astronomical pioneer and English clergyman John Michell in a letter published in November 1784. Michell's simplistic calculations assumed that such a body might have the same density as the Sun, concluded that such a body would form when a star's diameter exceeds the Sun's by a factor of 500, the surface escape velocity exceeds the usual speed of light. Michell noted that such supermassive but non-radiating bodies might be detectable through their gravitational effects on nearby visible bodies. Scholars of the time were excited by the proposal that giant but invisible stars might be hiding in plain view, but enthusiasm dampened when the wavelike nature of light became apparent in the early nineteenth century. If light were a wave rather than a "corpuscle", it became unclear what, if any, influence gravity would have on escaping light waves.
Modern relativity discredits Michell's notion of a light ray shooting directly from the surface of a supermassive star, being slowed down by the star's gravity and free-falling back to the star's surface. In 1915, Albert Einstein developed his theory of general relativity, having earlier shown that gravity does influence light's motion. Only a few months Karl Schwarzschild found a solution to the Einstein field equations, which describes the gravitational field of a point mass and a spherical mass. A few months after Schwarzschild, Johannes Droste, a student of Hendrik Lorentz, independently gave the same solution for the point mass and wrote more extensively about its properties; this solution had a peculiar behaviour at what is now called the Schwarzschild radius, where it became singular, meaning that some of the terms in the Einstein equations became infinite. The nature of this surface was not quite understood at the time. In 1924, Arthur Eddington showed that the singularity disappeared after a change of coordinates, although it took until 1933 for Georges Lemaître to realize that this meant the singularity at the Schwarzschild radius was a non-physical coordinate singularity.
Arthur Eddington did however comment on the possibility of a star with mass c