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195-413: General relativity , also known as the general theory of relativity , and as Einstein's theory of gravity , is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics . General relativity generalizes special relativity and refines Newton's law of universal gravitation , providing a unified description of gravity as

390-456: A Fields medal , made new impacts in mathematics by using topological quantum field theory and string theory to make predictions and provide frameworks for new rigorous mathematics, which has resulted for example in the conjectural mirror symmetry and the Seiberg–Witten invariants . Riemannian geometry studies Riemannian manifolds , smooth manifolds with a Riemannian metric . This is

585-487: A directional derivative of a function from multivariable calculus is extended to the notion of a covariant derivative of a tensor . Many concepts of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry . This notion can also be defined locally , i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However,

780-539: A pair of black holes merging . The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right). Since Einstein's equations are non-linear , arbitrarily strong gravitational waves do not obey linear superposition , making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe

975-447: A vector bundle endomorphism (called an almost complex structure ) It follows from this definition that an almost complex manifold is even-dimensional. An almost complex manifold is called complex if N J = 0 {\displaystyle N_{J}=0} , where N J {\displaystyle N_{J}} is a tensor of type (2, 1) related to J {\displaystyle J} , called

1170-568: A body in accordance with Newton's second law of motion , which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration . The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics , straight world lines in curved spacetime . Conversely, one might expect that inertial motions, once identified by observing

1365-512: A combinatorial and differential-geometric nature. Interest in the subject was also focused by the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation g {\displaystyle g} for a Riemannian metric, and Γ {\displaystyle \Gamma } for

1560-558: A computer, or by considering small perturbations of exact solutions. In the field of numerical relativity , powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes. In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities . Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization,

1755-601: A concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, though they still resemble Euclidean space at each point infinitesimally, i.e. in the first order of approximation . Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all possess natural analogues in Riemannian geometry. The notion of

1950-574: A cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which the universe has evolved from an extremely hot and dense earlier state. Einstein later declared the cosmological constant the biggest blunder of his life. During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity , being consistent with special relativity and accounting for several effects unexplained by

2145-530: A curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection , and this is, in fact,

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2340-537: A curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler , spacetime tells matter how to move; matter tells spacetime how to curve. While general relativity replaces

2535-453: A factor of ten, due to not knowing about the types of Cepheid variables. Given the cosmological principle, Hubble's law suggested that the universe was expanding. Two primary explanations were proposed for the expansion. One was Lemaître's Big Bang theory, advocated and developed by George Gamow. The other explanation was Fred Hoyle 's steady state model in which new matter is created as the galaxies move away from each other. In this model,

2730-497: A field concerned more generally with geometric structures on differentiable manifolds . A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over

2925-489: A geometric property of space and time , or four-dimensional spacetime . In particular, the curvature of spacetime is directly related to the energy and momentum of whatever present matter and radiation . The relation is specified by the Einstein field equations , a system of second-order partial differential equations . Newton's law of universal gravitation , which describes classical gravity, can be seen as

3120-540: A gravitational field (cf. below ). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity. The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle , a crucial guiding principle for generalizing special-relativistic physics to include gravity. The same experimental data shows that time as measured by clocks in

3315-528: A gravitational field— proper time , to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric . As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with

3510-448: A massive central body M is given by A conservative total force can then be obtained as its negative gradient where L is the angular momentum . The first term represents the force of Newtonian gravity , which is described by the inverse-square law. The second term represents the centrifugal force in the circular motion. The third term represents the relativistic effect. There are alternatives to general relativity built upon

3705-404: A modification of gravity on the largest scales. The effect on cosmology of the dark energy that these models describe is given by the dark energy's equation of state , which varies depending upon the theory. The nature of dark energy is one of the most challenging problems in cosmology. A better understanding of dark energy is likely to solve the problem of the ultimate fate of the universe . In

3900-461: A nondegenerate 2- form ω , called the symplectic form . A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: d ω = 0 . A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism . Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2,

4095-776: A number of exact solutions are known, although only a few have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution , the Reissner–Nordström solution and the Kerr metric , each corresponding to a certain type of black hole in an otherwise empty universe, and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes , each describing an expanding cosmos. Exact solutions of great theoretical interest include

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4290-442: A prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics . These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall , and the propagation of light, and include gravitational time dilation , gravitational lensing ,

4485-495: A reputation as a theory of extraordinary beauty. Subrahmanyan Chandrasekhar has noted that at multiple levels, general relativity exhibits what Francis Bacon has termed a "strangeness in the proportion" ( i.e . elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time versus matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were

4680-433: A rudimentary measure of arclength of curves, a concept which did not see a rigorous definition in terms of calculus until the 1600s. Around this time there were only minimal overt applications of the theory of infinitesimals to the study of geometry, a precursor to the modern calculus-based study of the subject. In Euclid 's Elements the notion of tangency of a line to a circle is discussed, and Archimedes applied

4875-580: A sequence of stellar nucleosynthesis reactions, smaller atomic nuclei are then combined into larger atomic nuclei, ultimately forming stable iron group elements such as iron and nickel , which have the highest nuclear binding energies . The net process results in a later energy release , meaning subsequent to the Big Bang. Such reactions of nuclear particles can lead to sudden energy releases from cataclysmic variable stars such as novae . Gravitational collapse of matter into black holes also powers

5070-421: A single bivector-valued one-form called the shape operator . Below are some examples of how differential geometry is applied to other fields of science and mathematics. Physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model , or simply cosmology , provides a description of the largest-scale structures and dynamics of

5265-502: A small, positive cosmological constant. The solution is an expanding universe; due to this expansion, the radiation and matter in the universe cool and become diluted. At first, the expansion is slowed down by gravitation attracting the radiation and matter in the universe. However, as these become diluted, the cosmological constant becomes more dominant and the expansion of the universe starts to accelerate rather than decelerate. In our universe this happened billions of years ago. During

5460-506: A student of Johann Bernoulli, provided many significant contributions not just to the development of geometry, but to mathematics more broadly. In regards to differential geometry, Euler studied the notion of a geodesic on a surface deriving the first analytical geodesic equation , and later introduced the first set of intrinsic coordinate systems on a surface, beginning the theory of intrinsic geometry upon which modern geometric ideas are based. Around this time Euler's study of mechanics in

5655-411: A subject begins at least as far back as classical antiquity . It is intimately linked to the development of geometry more generally, of the notion of space and shape, and of topology , especially the study of manifolds . In this section we focus primarily on the history of the application of infinitesimal methods to geometry, and later to the ideas of tangent spaces , and eventually the development of

5850-523: A symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The phase space of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Joseph Louis Lagrange on analytical mechanics and later in Carl Gustav Jacobi 's and William Rowan Hamilton 's formulations of classical mechanics . By contrast with Riemannian geometry, where

6045-437: A unified description of gravity as a geometric property of space and time. At the time, Einstein believed in a static universe , but found that his original formulation of the theory did not permit it. This is because masses distributed throughout the universe gravitationally attract, and move toward each other over time. However, he realized that his equations permitted the introduction of a constant term which could counteract

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6240-522: A universe with a larger cosmological constant. Many cosmologists find this an unsatisfying explanation: perhaps because while the weak anthropic principle is self-evident (given that living observers exist, there must be at least one universe with a cosmological constant (CC) which allows for life to exist) it does not attempt to explain the context of that universe. For example, the weak anthropic principle alone does not distinguish between: Other possible explanations for dark energy include quintessence or

6435-489: A university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated." General relativity can be understood by examining its similarities with and departures from classical physics. The first step

6630-536: A wave train traveling through empty space or Gowdy universes , varieties of an expanding cosmos filled with gravitational waves. But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models. General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation ( precession ) of planetary orbits, as well as orbital decay caused by

6825-417: A well-known standard definition of metric and parallelism. In Riemannian geometry , the Levi-Civita connection serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be spacetime and the bundles and connections are related to various physical fields. From

7020-525: Is Minkowskian , and the laws of physics exhibit local Lorentz invariance . The core concept of general-relativistic model-building is that of a solution of Einstein's equations . Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi- Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular,

7215-442: Is a fourth "sterile" species of neutrino. The ΛCDM ( Lambda cold dark matter ) or Lambda-CDM model is a parametrization of the Big Bang cosmological model in which the universe contains a cosmological constant, denoted by Lambda ( Greek Λ ), associated with dark energy, and cold dark matter (abbreviated CDM ). It is frequently referred to as the standard model of Big Bang cosmology. The cosmic microwave background

7410-401: Is a function F : T M → [ 0 , ∞ ) {\displaystyle F:\mathrm {T} M\to [0,\infty )} such that: Symplectic geometry is the study of symplectic manifolds . An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e.,

7605-479: Is a price to pay in technical complexity: the intrinsic definitions of curvature and connections become much less visually intuitive. These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem .) In the formalism of geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by

7800-423: Is a scalar parameter of motion (e.g. the proper time ), and Γ μ α β {\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and

7995-444: Is a universality of free fall (also known as the weak equivalence principle , or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties. A simplified version of this is embodied in Einstein's elevator experiment , illustrated in the figure on the right: for an observer in an enclosed room, it

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8190-420: Is a version of MOND that can explain gravitational lensing. If the universe is flat , there must be an additional component making up 73% (in addition to the 23% dark matter and 4% baryons) of the energy density of the universe. This is called dark energy. In order not to interfere with Big Bang nucleosynthesis and the cosmic microwave background, it must not cluster in haloes like baryons and dark matter. There

8385-402: Is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames . But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through

8580-412: Is called baryogenesis . Three required conditions for baryogenesis were derived by Andrei Sakharov in 1967, and requires a violation of the particle physics symmetry , called CP-symmetry , between matter and antimatter. However, particle accelerators measure too small a violation of CP-symmetry to account for the baryon asymmetry. Cosmologists and particle physicists look for additional violations of

8775-402: Is changed only by the increase in volume, but the energy density of radiation is changed both by the increase in volume and by the increase in the wavelength of the photons that make it up. Thus the energy of radiation becomes a smaller part of the universe's total energy than that of matter as it expands. The very early universe is said to have been 'radiation dominated' and radiation controlled

8970-497: Is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system. In this geometric description, tidal effects —the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass. As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics,

9165-405: Is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global inertial frames . Instead there are approximate inertial frames moving alongside freely falling particles. Translated into

9360-412: Is given by all the smooth complex projective varieties . CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds . Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. Differential topology is the study of global geometric invariants without a metric or symplectic form. Differential topology starts from

9555-444: Is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration. Given the universality of free fall, there is no observable distinction between inertial motion and motion under

9750-438: Is known about dark energy. Quantum field theory predicts a cosmological constant (CC) much like dark energy, but 120 orders of magnitude larger than that observed. Steven Weinberg and a number of string theorists (see string landscape ) have invoked the 'weak anthropic principle ': i.e. the reason that physicists observe a universe with such a small cosmological constant is that no physicists (or any life) could exist in

9945-555: Is known as gravitational time dilation. Gravitational redshift has been measured in the laboratory and using astronomical observations. Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks , while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS). Tests in stronger gravitational fields are provided by

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10140-403: Is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor , which includes both energy and momentum densities as well as stress : pressure and shear. Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that

10335-455: Is merely a limiting case of (special) relativistic mechanics. In the language of symmetry : where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group , which includes translations, rotations, boosts and reflections.) The differences between

10530-469: Is radiation left over from decoupling after the epoch of recombination when neutral atoms first formed. At this point, radiation produced in the Big Bang stopped Thomson scattering from charged ions. The radiation, first observed in 1965 by Arno Penzias and Robert Woodrow Wilson , has a perfect thermal black-body spectrum. It has a temperature of 2.7 kelvins today and is isotropic to one part in 10 . Cosmological perturbation theory , which describes

10725-436: Is realised, and the first analytical formula for the radius of an osculating circle, essentially the first analytical formula for the notion of curvature , is written down. In the wake of the development of analytic geometry and plane curves, Alexis Clairaut began the study of space curves at just the age of 16. In his book Clairaut introduced the notion of tangent and subtangent directions to space curves in relation to

10920-456: Is roughly equal to the age of the universe at each point in time. Observations suggest that the universe began around 13.8 billion years ago. Since then, the evolution of the universe has passed through three phases. The very early universe, which is still poorly understood, was the split second in which the universe was so hot that particles had energies higher than those currently accessible in particle accelerators on Earth. Therefore, while

11115-447: Is simulations, which cosmologists use to study the gravitational aggregation of matter in the universe, as it clusters into filaments , superclusters and voids . Most simulations contain only non-baryonic cold dark matter , which should suffice to understand the universe on the largest scales, as there is much more dark matter in the universe than visible, baryonic matter. More advanced simulations are starting to include baryons and study

11310-444: Is smaller than, or comparable to, the time scale of the expansion of the universe. The time scale that describes the expansion of the universe is 1 / H {\displaystyle 1/H} with H {\displaystyle H} being the Hubble parameter , which varies with time. The expansion timescale 1 / H {\displaystyle 1/H}

11505-457: Is strong observational evidence for dark energy, as the total energy density of the universe is known through constraints on the flatness of the universe, but the amount of clustering matter is tightly measured, and is much less than this. The case for dark energy was strengthened in 1999, when measurements demonstrated that the expansion of the universe has begun to gradually accelerate. Apart from its density and its clustering properties, nothing

11700-423: Is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i.e. a notion of parallel transport . An important example is provided by affine connections . For a surface in R , tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has

11895-497: Is that dark energy is just the vacuum energy , a component of empty space that is associated with the virtual particles that exist due to the uncertainty principle . There is no clear way to define the total energy in the universe using the most widely accepted theory of gravity, general relativity. Therefore, it remains controversial whether the total energy is conserved in an expanding universe. For instance, each photon that travels through intergalactic space loses energy due to

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12090-417: Is the Levi-Civita connection of g {\displaystyle g} . In this case, ( J , g ) {\displaystyle (J,g)} is called a Kähler structure , and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold . A large class of Kähler manifolds (the class of Hodge manifolds )

12285-457: Is the Riemannian symmetric spaces , whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry . Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite . A special case of this is a Lorentzian manifold , which is

12480-494: Is the Shapiro Time Delay, the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction. In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on

12675-408: Is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories. General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication. Assuming that

12870-469: Is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity. At the base of classical mechanics is the notion that a body 's motion can be described as a combination of free (or inertial ) motion, and deviations from this free motion. Such deviations are caused by external forces acting on

13065-467: Is the study of connections on vector bundles and principal bundles, and arises out of problems in mathematical physics and physical gauge theories which underpin the standard model of particle physics . Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric moduli spaces of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as

13260-480: Is the tangent space at the unit endowed with the Lie bracket between left-invariant vector fields . Beside the structure theory there is also the wide field of representation theory . Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology. Gauge theory

13455-515: Is to measure the brightness of an object and assume an intrinsic luminosity , from which the distance may be determined using the inverse-square law . Due to the difficulty of using these methods, they did not realize that the nebulae were actually galaxies outside our own Milky Way , nor did they speculate about the cosmological implications. In 1927, the Belgian Roman Catholic priest Georges Lemaître independently derived

13650-626: Is to measure the basic parameters of the Lambda-CDM model with increasing accuracy, as well as to test the predictions of the Big Bang model and look for new physics. The results of measurements made by WMAP, for example, have placed limits on the neutrino masses. Newer experiments, such as QUIET and the Atacama Cosmology Telescope , are trying to measure the polarization of the cosmic microwave background. These measurements are expected to provide further confirmation of

13845-528: Is unstable to small perturbations—it will eventually start to expand or contract. It was later realized that Einstein's model was just one of a larger set of possibilities, all of which were consistent with general relativity and the cosmological principle . The cosmological solutions of general relativity were found by Alexander Friedmann in the early 1920s. His equations describe the Friedmann–Lemaître–Robertson–Walker universe, which may expand or contract, and whose geometry may be open, flat, or closed. In

14040-452: Is what caused the universe to contain far more matter than antimatter . Cosmologists can observationally deduce that the universe is not split into regions of matter and antimatter. If it were, there would be X-rays and gamma rays produced as a result of annihilation , but this is not observed. Therefore, some process in the early universe must have created a small excess of matter over antimatter, and this (currently not understood) process

14235-437: Is zero or negligible compared to their kinetic energy , and so move at the speed of light or very close to it; non-relativistic particles have much higher rest mass than their energy and so move much slower than the speed of light. As the universe expands, both matter and radiation become diluted. However, the energy densities of radiation and matter dilute at different rates. As a particular volume expands, mass-energy density

14430-571: The Mechanica lead to the realization that a mass traveling along a surface not under the effect of any force would traverse a geodesic path, an early precursor to the important foundational ideas of Einstein's general relativity , and also to the Euler–Lagrange equations and the first theory of the calculus of variations , which underpins in modern differential geometry many techniques in symplectic geometry and geometric analysis . This theory

14625-485: The Christoffel symbols which describe the covariant derivative in 1868, and by others including Eugenio Beltrami who studied many analytic questions on manifolds. In 1899 Luigi Bianchi produced his Lectures on differential geometry which studied differential geometry from Riemann's perspective, and a year later Tullio Levi-Civita and Gregorio Ricci-Curbastro produced their textbook systematically developing

14820-519: The Cosmic Background Explorer in the early 1990s, few cosmologists have seriously proposed other theories of the origin and evolution of the cosmos. One consequence of this is that in standard general relativity, the universe began with a singularity , as demonstrated by Roger Penrose and Stephen Hawking in the 1960s. An alternative view to extend the Big Bang model, suggesting the universe had no beginning or singularity and

15015-552: The Disquisitiones generales circa superficies curvas detailing the general theory of curved surfaces. In this work and his subsequent papers and unpublished notes on the theory of surfaces, Gauss has been dubbed the inventor of non-Euclidean geometry and the inventor of intrinsic differential geometry. In his fundamental paper Gauss introduced the Gauss map , Gaussian curvature , first and second fundamental forms , proved

15210-432: The Einstein notation , meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation. In general relativity, the effective gravitational potential energy of an object of mass m revolving around

15405-462: The Euler–Lagrange equations describing the equations of motion of certain physical systems in quantum field theory , and so their study is of considerable interest in physics. The apparatus of vector bundles , principal bundles , and connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle . Loosely speaking, this structure by itself

15600-608: The Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub–NUT solution (a model universe that is homogeneous , but anisotropic ), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture ). Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on

15795-546: The Nijenhuis tensor (or sometimes the torsion ). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas . An almost Hermitian structure is given by an almost complex structure J , along with a Riemannian metric g , satisfying the compatibility condition An almost Hermitian structure defines naturally a differential two-form The following two conditions are equivalent: where ∇ {\displaystyle \nabla }

15990-452: The Poincaré conjecture . During this same period primarily due to the influence of Michael Atiyah , new links between theoretical physics and differential geometry were formed. Techniques from the study of the Yang–Mills equations and gauge theory were used by mathematicians to develop new invariants of smooth manifolds. Physicists such as Edward Witten , the only physicist to be awarded

16185-583: The Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds

16380-614: The Theorema Egregium showing the intrinsic nature of the Gaussian curvature, and studied geodesics, computing the area of a geodesic triangle in various non-Euclidean geometries on surfaces. At this time Gauss was already of the opinion that the standard paradigm of Euclidean geometry should be discarded, and was in possession of private manuscripts on non-Euclidean geometry which informed his study of geodesic triangles. Around this same time János Bolyai and Lobachevsky independently discovered hyperbolic geometry and thus demonstrated

16575-591: The Weyl tensor providing insight into conformal geometry , and first defined the notion of a gauge leading to the development of gauge theory in physics and mathematics . In the middle and late 20th century differential geometry as a subject expanded in scope and developed links to other areas of mathematics and physics. The development of gauge theory and Yang–Mills theory in physics brought bundles and connections into focus, leading to developments in gauge theory . Many analytical results were investigated including

16770-407: The circumference of the Earth around 200 BC, and around 150 AD Ptolemy in his Geography introduced the stereographic projection for the purposes of mapping the shape of the Earth. Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used in geodesy , although in a much simplified form. Namely, as far back as Euclid 's Elements it

16965-417: The cosmic microwave background , structure formation, and galaxy rotation curves suggests that about 23% of the mass of the universe consists of non-baryonic dark matter, whereas only 4% consists of visible, baryonic matter . The gravitational effects of dark matter are well understood, as it behaves like a cold, non-radiative fluid that forms haloes around galaxies. Dark matter has never been detected in

17160-621: The curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the Poincaré–Birkhoff theorem , conjectured by Henri Poincaré and then proved by G.D. Birkhoff in 1912. It claims that if an area preserving map of an annulus twists each boundary component in opposite directions, then

17355-682: The field equation for gravity relates this tensor and the Ricci tensor , which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy –momentum corresponds to the statement that the energy–momentum tensor is divergence -free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved- manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of

17550-506: The gravitational redshift of light, the Shapiro time delay and singularities / black holes . So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology , thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite

17745-465: The method of exhaustion to compute the areas of smooth shapes such as the circle , and the volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders. There was little development in the theory of differential geometry between antiquity and the beginning of the Renaissance . Before the development of calculus by Newton and Leibniz , the most significant development in

17940-531: The natural sciences . Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity , and subsequently by physicists in the development of quantum field theory and the standard model of particle physics . Outside of physics, differential geometry finds applications in chemistry , economics , engineering , control theory , computer graphics and computer vision , and recently in machine learning . The history and development of differential geometry as

18135-472: The post-Newtonian expansion , both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity. An extension of this expansion

18330-464: The redshift effect. This energy is not transferred to any other system, so seems to be permanently lost. On the other hand, some cosmologists insist that energy is conserved in some sense; this follows the law of conservation of energy . Different forms of energy may dominate the cosmos— relativistic particles which are referred to as radiation , or non-relativistic particles referred to as matter. Relativistic particles are particles whose rest mass

18525-453: The scalar gravitational potential of classical physics by a symmetric rank -two tensor , the latter reduces to the former in certain limiting cases . For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation. As it is constructed using tensors, general relativity exhibits general covariance : its laws—and further laws formulated within

18720-429: The summation convention is used for repeated indices α {\displaystyle \alpha } and β {\displaystyle \beta } . The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to Newton's laws of motion which likewise provide formulae for the acceleration of a particle. This equation of motion employs

18915-488: The universe and allows study of fundamental questions about its origin , structure, evolution , and ultimate fate . Cosmology as a science originated with the Copernican principle , which implies that celestial bodies obey identical physical laws to those on Earth, and Newtonian mechanics , which first allowed those physical laws to be understood. Physical cosmology, as it is now understood, began in 1915 with

19110-473: The universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant —to match that observational presumption. By 1929, however, the work of Hubble and others had shown that the universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require

19305-473: The 1600s when calculus was first developed by Gottfried Leibniz and Isaac Newton . At this time, the recent work of René Descartes introducing analytic coordinates to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this time Pierre de Fermat , Newton, and Leibniz began the study of plane curves and the investigation of concepts such as points of inflection and circles of osculation , which aid in

19500-400: The 1910s, Vesto Slipher (and later Carl Wilhelm Wirtz ) interpreted the red shift of spiral nebulae as a Doppler shift that indicated they were receding from Earth. However, it is difficult to determine the distance to astronomical objects. One way is to compare the physical size of an object to its angular size , but a physical size must be assumed in order to do this. Another method

19695-549: The Big Bang cosmology, which is presented in Timeline of the Big Bang . The early, hot universe appears to be well explained by the Big Bang from roughly 10 seconds onwards, but there are several problems . One is that there is no compelling reason, using current particle physics, for the universe to be flat , homogeneous, and isotropic (see the cosmological principle ) . Moreover, grand unified theories of particle physics suggest that there should be magnetic monopoles in

19890-487: The CP-symmetry in the early universe that might account for the baryon asymmetry . Both the problems of baryogenesis and cosmic inflation are very closely related to particle physics, and their resolution might come from high energy theory and experiment , rather than through observations of the universe. Big Bang nucleosynthesis is the theory of the formation of the elements in the early universe. It finished when

20085-660: The Christoffel symbols, both coming from G in Gravitation . Élie Cartan helped reformulate the foundations of the differential geometry of smooth manifolds in terms of exterior calculus and the theory of moving frames , leading in the world of physics to Einstein–Cartan theory . Following this early development, many mathematicians contributed to the development of the modern theory, including Jean-Louis Koszul who introduced connections on vector bundles , Shiing-Shen Chern who introduced characteristic classes to

20280-468: The Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system. Complex differential geometry is the study of complex manifolds . An almost complex manifold is a real manifold M {\displaystyle M} , endowed with a tensor of type (1, 1), i.e.

20475-485: The Einstein field equations, which form the core of Einstein's general theory of relativity. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present. A version of non-Euclidean geometry , called Riemannian geometry , enabled Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity. This idea

20670-533: The Friedmann–Lemaître–Robertson–Walker equations and proposed, on the basis of the recession of spiral nebulae, that the universe began with the "explosion" of a "primeval atom " —which was later called the Big Bang. In 1929, Edwin Hubble provided an observational basis for Lemaître's theory. Hubble showed that the spiral nebulae were galaxies by determining their distances using measurements of

20865-501: The Newtonian limit and treating the orbiting body as a test particle . For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations. Differential geometry Since the late 19th century, differential geometry has grown into

21060-449: The Newtonian theory. Einstein showed in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters (" fudge factors "), and in 1919 an expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of 29 May 1919 , instantly making Einstein famous. Yet

21255-413: The actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction ), can be used to define the geometry of space, as well as a time coordinate . However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment ), there

21450-473: The age of the universe is infinite, has been presented. In September 2023, astrophysicists questioned the overall current view of the universe , in the form of the Standard Model of Cosmology , based on the latest James Webb Space Telescope studies. The lightest chemical elements , primarily hydrogen and helium , were created during the Big Bang through the process of nucleosynthesis . In

21645-441: The attractive force of gravity on the cosmic scale. Einstein published his first paper on relativistic cosmology in 1917, in which he added this cosmological constant to his field equations in order to force them to model a static universe. The Einstein model describes a static universe; space is finite and unbounded (analogous to the surface of a sphere, which has a finite area but no edges). However, this so-called Einstein model

21840-418: The basic features of this epoch have been worked out in the Big Bang theory, the details are largely based on educated guesses. Following this, in the early universe, the evolution of the universe proceeded according to known high energy physics . This is when the first protons, electrons and neutrons formed, then nuclei and finally atoms. With the formation of neutral hydrogen, the cosmic microwave background

22035-403: The beginning and through the middle of the 19th century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with

22230-407: The brightness of Cepheid variable stars. He discovered a relationship between the redshift of a galaxy and its distance. He interpreted this as evidence that the galaxies are receding from Earth in every direction at speeds proportional to their distance from Earth. This fact is now known as Hubble's law , though the numerical factor Hubble found relating recessional velocity and distance was off by

22425-405: The connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates , the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish). Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source

22620-449: The current cosmological epoch, the accelerated expansion due to dark energy is preventing structures larger than superclusters from forming. It is not known whether the acceleration will continue indefinitely, perhaps even increasing until a big rip , or whether it will eventually reverse, lead to a Big Freeze , or follow some other scenario. Gravitational waves are ripples in the curvature of spacetime that propagate as waves at

22815-405: The deceleration of expansion. Later, as the average energy per photon becomes roughly 10 eV and lower, matter dictates the rate of deceleration and the universe is said to be 'matter dominated'. The intermediate case is not treated well analytically . As the expansion of the universe continues, matter dilutes even further and the cosmological constant becomes dominant, leading to an acceleration in

23010-478: The development of Albert Einstein 's general theory of relativity , followed by major observational discoveries in the 1920s: first, Edwin Hubble discovered that the universe contains a huge number of external galaxies beyond the Milky Way ; then, work by Vesto Slipher and others showed that the universe is expanding . These advances made it possible to speculate about the origin of the universe, and allowed

23205-763: The development of a standard model of cosmology . This model requires the universe to contain large amounts of dark matter and dark energy whose nature is currently not well understood, but the model gives detailed predictions that are in excellent agreement with many diverse observations. Cosmology draws heavily on the work of many disparate areas of research in theoretical and applied physics . Areas relevant to cosmology include particle physics experiments and theory , theoretical and observational astrophysics , general relativity, quantum mechanics , and plasma physics . Modern cosmology developed along tandem tracks of theory and observation. In 1916, Albert Einstein published his theory of general relativity , which provided

23400-444: The directions which lie along a surface on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of the tangent space of a surface and studied this idea using calculus for the first time. Importantly Clairaut introduced the terminology of curvature and double curvature , essentially the notion of principal curvatures later studied by Gauss and others. Around this same time, Leonhard Euler , originally

23595-417: The earlier observation of Euler that masses under the effect of no forces would travel along geodesics on surfaces, and predicting Einstein's fundamental observation of the equivalence principle a full 60 years before it appeared in the scientific literature. In the wake of Riemann's new description, the focus of techniques used to study differential geometry shifted from the ad hoc and extrinsic methods of

23790-436: The earliest moments of the universe, the average energy density was very high, making knowledge of particle physics critical to understanding this environment. Hence, scattering processes and decay of unstable elementary particles are important for cosmological models of this period. As a rule of thumb, a scattering or a decay process is cosmologically important in a certain epoch if the time scale describing that process

23985-452: The emission of gravitational waves and effects related to the relativity of direction. In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass ) will precess ; the orbit is not an ellipse , but akin to an ellipse that rotates on its focus, resulting in a rose curve -like shape (see image). Einstein first derived this result by using an approximate metric representing

24180-555: The energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations: G μ ν ≡ R μ ν − 1 2 R g μ ν = κ T μ ν {\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,} On

24375-445: The equivalence principle holds, gravity influences the passage of time. Light sent down into a gravity well is blueshifted , whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted ; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect

24570-438: The establishment of the Big Bang theory, by Georges Lemaître , as the leading cosmological model. A few researchers still advocate a handful of alternative cosmologies ; however, most cosmologists agree that the Big Bang theory best explains the observations. Dramatic advances in observational cosmology since the 1990s, including the cosmic microwave background , distant supernovae and galaxy redshift surveys , have led to

24765-414: The evolution of slight inhomogeneities in the early universe, has allowed cosmologists to precisely calculate the angular power spectrum of the radiation, and it has been measured by the recent satellite experiments ( COBE and WMAP ) and many ground and balloon-based experiments (such as Degree Angular Scale Interferometer , Cosmic Background Imager , and Boomerang ). One of the goals of these efforts

24960-455: The exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by 10 − 21 {\displaystyle 10^{-21}} or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed . Some exact solutions describe gravitational waves without any approximation, e.g.,

25155-469: The existence of consistent geometries outside Euclid's paradigm. Concrete models of hyperbolic geometry were produced by Eugenio Beltrami later in the 1860s, and Felix Klein coined the term non-Euclidean geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing. Implicitly, the spherical geometry of the Earth that had been studied since antiquity

25350-405: The exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion), several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity. Closely related to light deflection

25545-436: The formation of individual galaxies. Cosmologists study these simulations to see if they agree with the galaxy surveys, and to understand any discrepancy. Other, complementary observations to measure the distribution of matter in the distant universe and to probe reionization include: These will help cosmologists settle the question of when and how structure formed in the universe. Evidence from Big Bang nucleosynthesis ,

25740-546: The formation of the early universe shortly after the Big Bang. In 2016, the LIGO Scientific Collaboration and Virgo Collaboration teams announced that they had made the first observation of gravitational waves , originating from a pair of merging black holes using the Advanced LIGO detectors. On 15 June 2016, a second detection of gravitational waves from coalescing black holes

25935-404: The foundations of topology . At the start of the 1900s there was a major movement within mathematics to formalise the foundational aspects of the subject to avoid crises of rigour and accuracy, known as Hilbert's program . As part of this broader movement, the notion of a topological space was distilled in by Felix Hausdorff in 1914, and by 1942 there were many different notions of manifold of

26130-410: The general relativistic framework—take on the same form in all coordinate systems . Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent . It thus satisfies a more stringent general principle of relativity , namely that the laws of physics are the same for all observers. Locally , as expressed in the equivalence principle, spacetime

26325-418: The geometry of space. Predicted in 1916 by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves . On 11 February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from

26520-415: The groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, eventually resulting in the Reissner–Nordström solution , which is now associated with electrically charged black holes . In 1917, Einstein applied his theory to

26715-439: The image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer -independent. In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure or conformal geometry. Special relativity

26910-446: The influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential . Space, in this construction, still has

27105-512: The introduction of a number of alternative theories , general relativity continues to be the simplest theory consistent with experimental data . Reconciliation of general relativity with the laws of quantum physics remains a problem, however, as there is a lack of a self-consistent theory of quantum gravity . It is not yet known how gravity can be unified with the three non-gravitational forces: strong , weak and electromagnetic . Einstein's theory has astrophysical implications, including

27300-492: The laboratory, and the particle physics nature of dark matter remains completely unknown. Without observational constraints, there are a number of candidates, such as a stable supersymmetric particle, a weakly interacting massive particle , a gravitationally-interacting massive particle, an axion , and a massive compact halo object . Alternatives to the dark matter hypothesis include a modification of gravity at small accelerations ( MOND ) or an effect from brane cosmology. TeVeS

27495-409: The language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry. A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory

27690-492: The largest objects, such as superclusters, are still assembling. One way to study structure in the universe is to survey the visible galaxies, in order to construct a three-dimensional picture of the galaxies in the universe and measure the matter power spectrum . This is the approach of the Sloan Digital Sky Survey and the 2dF Galaxy Redshift Survey . Another tool for understanding structure formation

27885-457: The left-hand side is the Einstein tensor , G μ ν {\displaystyle G_{\mu \nu }} , which is symmetric and a specific divergence-free combination of the Ricci tensor R μ ν {\displaystyle R_{\mu \nu }} and the metric. In particular, is the curvature scalar. The Ricci tensor itself is related to

28080-405: The level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point p , a hyperplane distribution is determined by a nowhere vanishing 1-form α {\displaystyle \alpha } , which is unique up to multiplication by a nowhere vanishing function: A local 1-form on M is a contact form if

28275-475: The light of stars or distant quasars being deflected as it passes the Sun . This and related predictions follow from the fact that light follows what is called a light-like or null geodesic —a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity. As one examines suitable model spacetimes (either

28470-399: The map has at least two fixed points. Contact geometry deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A contact structure on a (2 n + 1) -dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle that is as far as possible from being associated with

28665-411: The mathematical basis of Einstein's general relativity theory of gravity . Finsler geometry has Finsler manifolds as the main object of study. This is a differential manifold with a Finsler metric , that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M {\displaystyle M}

28860-453: The matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present. Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless,

29055-521: The measurement of curvature . Indeed, already in his first paper on the foundations of calculus, Leibniz notes that the infinitesimal condition d 2 y = 0 {\displaystyle d^{2}y=0} indicates the existence of an inflection point. Shortly after this time the Bernoulli brothers , Jacob and Johann made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at

29250-453: The modern formalism of the subject in terms of tensors and tensor fields . The study of differential geometry, or at least the study of the geometry of smooth shapes, can be traced back at least to classical antiquity . In particular, much was known about the geometry of the Earth , a spherical geometry , in the time of the ancient Greek mathematicians. Famously, Eratosthenes calculated

29445-442: The more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of Carl Friedrich Gauss and Bernhard Riemann , and also in the important contributions of Nikolai Lobachevsky on hyperbolic geometry and non-Euclidean geometry and throughout the same period the development of projective geometry . Dubbed the single most important work in the history of differential geometry, in 1827 Gauss produced

29640-441: The more general Riemann curvature tensor as On the right-hand side, κ {\displaystyle \kappa } is a constant and T μ ν {\displaystyle T_{\mu \nu }} is the energy–momentum tensor. All tensors are written in abstract index notation . Matching the theory's prediction to observational results for planetary orbits or, equivalently, assuring that

29835-423: The most beautiful of all existing physical theories. Henri Poincaré 's 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that gravitational waves propagate at

30030-408: The most energetic processes, generally seen in the nuclear regions of galaxies, forming quasars and active galaxies . Cosmologists cannot explain all cosmic phenomena exactly, such as those related to the accelerating expansion of the universe , using conventional forms of energy . Instead, cosmologists propose a new form of energy called dark energy that permeates all space. One hypothesis

30225-413: The natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms . Beside Lie algebroids , also Courant algebroids start playing a more important role. A Lie group is a group in the category of smooth manifolds. Beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which

30420-429: The observation of binary pulsars . All results are in agreement with general relativity. However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid. General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing

30615-459: The ordinary Euclidean geometry . However, space time as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable . From this, one can deduce that spacetime

30810-421: The prediction of black holes —regions of space in which space and time are distorted in such a way that nothing, not even light , can escape from them. Black holes are the end-state for massive stars . Microquasars and active galactic nuclei are believed to be stellar black holes and supermassive black holes . It also predicts gravitational lensing , where the bending of light results in multiple images of

31005-511: The preface to Relativity: The Special and the General Theory , Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of

31200-428: The principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory. Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency. In

31395-563: The proof of the Atiyah–Singer index theorem . The development of complex geometry was spurred on by parallel results in algebraic geometry , and results in the geometry and global analysis of complex manifolds were proven by Shing-Tung Yau and others. In the latter half of the 20th century new analytic techniques were developed in regards to curvature flows such as the Ricci flow , which culminated in Grigori Perelman 's proof of

31590-417: The restriction of its exterior derivative to H is a non-degenerate two-form and thus induces a symplectic structure on H p at each point. If the distribution H can be defined by a global one-form α {\displaystyle \alpha } then this form is contact if and only if the top-dimensional form is a volume form on M , i.e. does not vanish anywhere. A contact analogue of

31785-419: The same distant astronomical phenomenon. Other predictions include the existence of gravitational waves , which have been observed directly by the physics collaboration LIGO and other observatories. In addition, general relativity has provided the base of cosmological models of an expanding universe . Widely acknowledged as a theory of extraordinary beauty , general relativity has often been described as

31980-445: The same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Whitehead's theory , Brans–Dicke theory , teleparallelism , f ( R ) gravity and Einstein–Cartan theory . The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how

32175-471: The signal can be entirely attributed to interstellar dust in the Milky Way. Understanding the formation and evolution of the largest and earliest structures (i.e., quasars, galaxies, clusters and superclusters ) is one of the largest efforts in cosmology. Cosmologists study a model of hierarchical structure formation in which structures form from the bottom up, with smaller objects forming first, while

32370-537: The space. Differential geometry is closely related to, and is sometimes taken to include, differential topology , which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equations , otherwise known as geometric analysis . Differential geometry finds applications throughout mathematics and

32565-471: The speed of light in vacuum. When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations, In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic. The geodesic equation is: where s {\displaystyle s}

32760-438: The speed of light, generated in certain gravitational interactions that propagate outward from their source. Gravitational-wave astronomy is an emerging branch of observational astronomy which aims to use gravitational waves to collect observational data about sources of detectable gravitational waves such as binary star systems composed of white dwarfs , neutron stars , and black holes ; and events such as supernovae , and

32955-596: The speed of light. Soon afterwards, Einstein started thinking about how to incorporate gravity into his relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall (FFO), he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as

33150-401: The straight line paths on his map. Mercator noted that the praga were oblique curvatur in this projection. This fact reflects the lack of a metric-preserving map of the Earth's surface onto a flat plane, a consequence of the later Theorema Egregium of Gauss . The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from calculus began around

33345-446: The study of curves and surfaces to a more systematic approach in terms of tensor calculus and Klein's Erlangen program, and progress increased in the field. The notion of groups of transformations was developed by Sophus Lie and Jean Gaston Darboux , leading to important results in the theory of Lie groups and symplectic geometry . The notion of differential calculus on curved spaces was studied by Elwin Christoffel , who introduced

33540-402: The subject and began the study of complex manifolds , Sir William Vallance Douglas Hodge and Georges de Rham who expanded understanding of differential forms , Charles Ehresmann who introduced the theory of fibre bundles and Ehresmann connections , and others. Of particular importance was Hermann Weyl who made important contributions to the foundations of general relativity, introduced

33735-432: The subject, making great use of the theory of quadratic forms in his investigation of metrics and curvature. At this time Riemann did not yet develop the modern notion of a manifold, as even the notion of a topological space had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric of spacetime through the analysis of masses within spacetime, linking with

33930-466: The systematic study of differential geometry in higher dimensions. This intrinsic point of view in terms of the Riemannian metric, denoted by d s 2 {\displaystyle ds^{2}} by Riemann, was the development of an idea of Gauss's about the linear element d s {\displaystyle ds} of a surface. At this time Riemann began to introduce the systematic use of linear algebra and multilinear algebra into

34125-669: The theory as well as information about cosmic inflation, and the so-called secondary anisotropies, such as the Sunyaev-Zel'dovich effect and Sachs-Wolfe effect , which are caused by interaction between galaxies and clusters with the cosmic microwave background. On 17 March 2014, astronomers of the BICEP2 Collaboration announced the apparent detection of B -mode polarization of the CMB, considered to be evidence of primordial gravitational waves that are predicted by

34320-517: The theory can be used for model-building. General relativity is a metric theory of gravitation. At its core are Einstein's equations , which describe the relation between the geometry of a four-dimensional pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall , orbital motion, and spacecraft trajectories ), correspond to inertial motion within

34515-419: The theory of absolute differential calculus and tensor calculus . It was in this language that differential geometry was used by Einstein in the development of general relativity and pseudo-Riemannian geometry . The subject of modern differential geometry emerged from the early 1900s in response to the foundational contributions of many mathematicians, including importantly the work of Henri Poincaré on

34710-588: The theory of inflation to occur during the earliest phase of the Big Bang. However, later that year the Planck collaboration provided a more accurate measurement of cosmic dust , concluding that the B-mode signal from dust is the same strength as that reported from BICEP2. On 30 January 2015, a joint analysis of BICEP2 and Planck data was published and the European Space Agency announced that

34905-456: The theory of plane curves, surfaces, and studied surfaces of revolution and envelopes of plane curves and space curves. Several students of Monge made contributions to this same theory, and for example Charles Dupin provided a new interpretation of Euler's theorem in terms of the principle curvatures, which is the modern form of the equation. The field of differential geometry became an area of study considered in its own right, distinct from

35100-525: The theory remained outside the mainstream of theoretical physics and astrophysics until developments between approximately 1960 and 1975, now known as the golden age of general relativity . Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations. Ever more precise solar system tests confirmed the theory's predictive power, and relativistic cosmology also became amenable to direct observational tests. General relativity has acquired

35295-400: The time, later collated by L'Hopital into the first textbook on differential calculus , the tangents to plane curves of various types are computed using the condition d y = 0 {\displaystyle dy=0} , and similarly points of inflection are calculated. At this same time the orthogonality between the osculating circles of a plane curve and the tangent directions

35490-486: The two become significant when dealing with speeds approaching the speed of light , and with high-energy phenomena. With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A , there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in

35685-462: The understanding of differential geometry came from Gerardus Mercator 's development of the Mercator projection as a way of mapping the Earth. Mercator had an understanding of the advantages and pitfalls of his map design, and in particular was aware of the conformal nature of his projection, as well as the difference between praga , the lines of shortest distance on the Earth, and the directio ,

35880-414: The universe is roughly the same at any point in time. For a number of years, support for these theories was evenly divided. However, the observational evidence began to support the idea that the universe evolved from a hot dense state. The discovery of the cosmic microwave background in 1965 lent strong support to the Big Bang model, and since the precise measurements of the cosmic microwave background by

36075-443: The universe was about three minutes old and its temperature dropped below that at which nuclear fusion could occur. Big Bang nucleosynthesis had a brief period during which it could operate, so only the very lightest elements were produced. Starting from hydrogen ions ( protons ), it principally produced deuterium , helium-4 , and lithium . Other elements were produced in only trace abundances. The basic theory of nucleosynthesis

36270-499: The universe's expansion. The history of the universe is a central issue in cosmology. The history of the universe is divided into different periods called epochs, according to the dominant forces and processes in each period. The standard cosmological model is known as the Lambda-CDM model . Within the standard cosmological model , the equations of motion governing the universe as a whole are derived from general relativity with

36465-612: The universe, which have not been found. These problems are resolved by a brief period of cosmic inflation , which drives the universe to flatness , smooths out anisotropies and inhomogeneities to the observed level, and exponentially dilutes the monopoles. The physical model behind cosmic inflation is extremely simple, but it has not yet been confirmed by particle physics, and there are difficult problems reconciling inflation and quantum field theory . Some cosmologists think that string theory and brane cosmology will provide an alternative to inflation. Another major problem in cosmology

36660-494: The weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant κ {\displaystyle \kappa } is found to be κ = 8 π G c 4 {\textstyle \kappa ={\frac {8\pi G}{c^{4}}}} , where G {\displaystyle G} is the Newtonian constant of gravitation and c {\displaystyle c}

36855-482: The work of Riemann , the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium , to the effect that Gaussian curvature is an intrinsic invariant. The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic. However, there

37050-410: Was a non-Euclidean geometry, an elliptic geometry . The development of intrinsic differential geometry in the language of Gauss was spurred on by his student, Bernhard Riemann in his Habilitationsschrift , On the hypotheses which lie at the foundation of geometry . In this work Riemann introduced the notion of a Riemannian metric and the Riemannian curvature tensor for the first time, and began

37245-495: Was developed in 1948 by George Gamow, Ralph Asher Alpher , and Robert Herman . It was used for many years as a probe of physics at the time of the Big Bang, as the theory of Big Bang nucleosynthesis connects the abundances of primordial light elements with the features of the early universe. Specifically, it can be used to test the equivalence principle , to probe dark matter , and test neutrino physics. Some cosmologists have proposed that Big Bang nucleosynthesis suggests there

37440-469: Was emitted. Finally, the epoch of structure formation began, when matter started to aggregate into the first stars and quasars , and ultimately galaxies, clusters of galaxies and superclusters formed. The future of the universe is not yet firmly known, but according to the ΛCDM model it will continue expanding forever. Below, some of the most active areas of inquiry in cosmology are described, in roughly chronological order. This does not include all of

37635-444: Was pointed out by mathematician Marcel Grossmann and published by Grossmann and Einstein in 1913. The Einstein field equations are nonlinear and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric . This solution laid

37830-477: Was understood that a straight line could be defined by its property of providing the shortest distance between two points, and applying this same principle to the surface of the Earth leads to the conclusion that great circles , which are only locally similar to straight lines in a flat plane, provide the shortest path between two points on the Earth's surface. Indeed, the measurements of distance along such geodesic paths by Eratosthenes and others can be considered

38025-551: Was used by Lagrange , a co-developer of the calculus of variations, to derive the first differential equation describing a minimal surface in terms of the Euler–Lagrange equation. In 1760 Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem . Later in the 1700s, the new French school led by Gaspard Monge began to make contributions to differential geometry. Monge made important contributions to

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