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204-413: 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

408-414: A − 3 {\displaystyle \rho \propto a^{-3}} , where a {\displaystyle a} is the scale factor . For ultrarelativistic particles ("radiation"), the energy density drops more sharply, as ρ ∝ a − 4 {\displaystyle \rho \propto a^{-4}} . This is because in addition to the volume dilution of

612-456: 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

816-477: A cosmological constant in the simplest gravitational models, as a way to explain this late-time acceleration. According to the simplest extrapolation of the currently favored cosmological model, the Lambda-CDM model , this acceleration becomes dominant in the future. In 1912–1914, Vesto Slipher discovered that light from remote galaxies was redshifted , a phenomenon later interpreted as galaxies receding from

1020-487: 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,

1224-539: 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

1428-447: 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

1632-547: A Hubble constant of 73 ± 7 km⋅s ⋅Mpc . In 2003, David Spergel 's analysis of the cosmic microwave background during the first year observations of the Wilkinson Microwave Anisotropy Probe satellite (WMAP) further agreed with the estimated expansion rates for local galaxies, 72 ± 5 km⋅s ⋅Mpc . The universe at the largest scales is observed to be homogeneous (the same everywhere) and isotropic (the same in all directions), consistent with

1836-568: 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

2040-505: 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

2244-529: A completion of its repairs related to the main mirror of the Hubble Space Telescope , allowing for sharper images and, consequently, more accurate analyses of its observations. Shortly after the repairs were made, Wendy Freedman 's 1994 Key Project analyzed the recession velocity of M100 from the core of the Virgo Cluster , offering a Hubble constant measurement of 80 ± 17 km⋅s ⋅Mpc . Later

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2448-558: 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,

2652-595: 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

2856-569: 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

3060-519: 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,

3264-537: 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

3468-464: A distance ct in a time t , as the red worldline illustrates. While it always moves locally at  c , its time in transit (about 13 billion years) is not related to the distance traveled in any simple way, since the universe expands as the light beam traverses space and time. The distance traveled is thus inherently ambiguous because of the changing scale of the universe. Nevertheless, there are two distances that appear to be physically meaningful:

3672-491: 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

3876-404: A finite distance. The comoving distance that such particles can have covered over the age of the universe is known as the particle horizon , and the region of the universe that lies within our particle horizon is known as the observable universe . If the dark energy that is inferred to dominate the universe today is a cosmological constant, then the particle horizon converges to a finite value in

4080-484: 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

4284-540: 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

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4488-518: 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

4692-448: 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

4896-435: A non-zero Riemann curvature tensor in curvature of Riemannian manifolds . Euclidean "geometrically flat" space has a Riemann curvature tensor of zero. "Geometrically flat" space has three dimensions and is consistent with Euclidean space. However, spacetime has four dimensions; it is not flat according to Einstein's general theory of relativity. Einstein's theory postulates that "matter and energy curve spacetime, and there

5100-461: 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,

5304-765: 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

5508-442: 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 ,

5712-433: A priori constraints) on how the space in which we live is connected or whether it wraps around on itself as a compact space . Though certain cosmological models such as Gödel's universe even permit bizarre worldlines that intersect with themselves, ultimately the question as to whether we are in something like a " Pac-Man universe", where if traveling far enough in one direction would allow one to simply end up back in

5916-495: 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

6120-433: 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

6324-413: 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. Metric expansion of space The expansion of the universe is the increase in distance between gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion, so it does not mean that

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6528-506: 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

6732-411: 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

6936-515: 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

7140-489: 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

7344-536: 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

7548-417: 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

7752-525: 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,

7956-503: Is a disagreement between this measurement and the supernova-based measurements, known as the Hubble tension . A third option proposed recently is to use information from gravitational wave events (especially those involving the merger of neutron stars , like GW170817 ), to measure the expansion rate. Such measurements do not yet have the precision to resolve the Hubble tension. In principle,

8160-401: 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.,

8364-475: 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

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8568-423: 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

8772-444: 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

8976-482: Is accelerating in the present epoch. By assuming a cosmological model, e.g. the Lambda-CDM model , another possibility is to infer the present-day expansion rate from the sizes of the largest fluctuations seen in the cosmic microwave background . A higher expansion rate would imply a smaller characteristic size of CMB fluctuations, and vice versa. The Planck collaboration measured the expansion rate this way and determined H 0 = 67.4 ± 0.5 (km/s)/Mpc . There

9180-402: 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

9384-497: 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,

9588-405: 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

9792-443: Is enough matter and energy to provide for curvature." In part to accommodate such different geometries, the expansion of the universe is inherently general-relativistic. It cannot be modeled with special relativity alone: Though such models exist, they may be at fundamental odds with the observed interaction between matter and spacetime seen in the universe. The images to the right show two views of spacetime diagrams that show

9996-442: Is essentially pressureless, with | p | ≪ ρ c 2 {\displaystyle |p|\ll \rho c^{2}} , while a gas of ultrarelativistic particles (such as a photon gas ) has positive pressure p = ρ c 2 / 3 {\displaystyle p=\rho c^{2}/3} . Negative-pressure fluids, like dark energy, are not experimentally confirmed, but

10200-424: Is expanding. The words ' space ' and ' universe ', sometimes used interchangeably, have distinct meanings in this context. Here 'space' is a mathematical concept that stands for the three-dimensional manifold into which our respective positions are embedded, while 'universe' refers to everything that exists, including the matter and energy in space, the extra dimensions that may be wrapped up in various strings , and

10404-412: 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

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10608-444: 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

10812-540: 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

11016-534: Is known. The object's distance can then be inferred from the observed apparent brightness . Meanwhile, the recession speed is measured through the redshift. Hubble used this approach for his original measurement of the expansion rate, by measuring the brightness of Cepheid variable stars and the redshifts of their host galaxies. More recently, using Type Ia supernovae , the expansion rate was measured to be H 0   =   73.24 ± 1.74 (km/s)/Mpc . This means that for every million parsecs of distance from

11220-403: 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

11424-450: 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

11628-538: Is often framed as a consequence of general relativity , it is also predicted by Newtonian gravity . According to inflation theory , the universe suddenly expanded during the inflationary epoch (about 10 of a second after the Big Bang), and its volume increased by a factor of at least 10 (an expansion of distance by a factor of at least 10 in each of the three dimensions). This would be equivalent to expanding an object 1  nanometer across ( 10  m , about half

11832-436: 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

12036-400: Is smaller in the past and larger in the future. Extrapolating back in time with certain cosmological models will yield a moment when the scale factor was zero; our current understanding of cosmology sets this time at 13.787 ± 0.020 billion years ago . If the universe continues to expand forever, the scale factor will approach infinity in the future. It is also possible in principle for

12240-423: 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

12444-417: 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 )

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12648-452: 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

12852-584: Is the equation of state parameter . The energy density of such a fluid drops as Nonrelativistic matter has w = 0 {\displaystyle w=0} while radiation has w = 1 / 3 {\displaystyle w=1/3} . For an exotic fluid with negative pressure, like dark energy, the energy density drops more slowly; if w = − 1 {\displaystyle w=-1} it remains constant in time. If w < − 1 {\displaystyle w<-1} , corresponding to phantom energy ,

13056-417: Is the gravitational constant , ρ {\displaystyle \rho } is the energy density within the universe, p {\displaystyle p} is the pressure , c {\displaystyle c} is the speed of light , and Λ {\displaystyle \Lambda } is the cosmological constant. A positive energy density leads to deceleration of

13260-486: 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

13464-408: 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

13668-469: 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

13872-467: 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

14076-477: 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

14280-566: 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

14484-479: 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

14688-541: 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

14892-432: 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

15096-462: 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

15300-599: 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

15504-541: 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 }

15708-452: 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

15912-565: 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

16116-609: 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

16320-587: 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

16524-407: 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

16728-491: The cosmological principle . These constraints demand that any expansion of the universe accord with Hubble's law , in which objects recede from each observer with velocities proportional to their positions with respect to that observer. That is, recession velocities v → {\displaystyle {\vec {v}}} scale with (observer-centered) positions x → {\displaystyle {\vec {x}}} according to where

16932-621: 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

17136-440: The equivalence principle of general relativity, the rules of special relativity are locally valid in small regions of spacetime that are approximately flat. In particular, light always travels locally at the speed  c ; in the diagram, this means, according to the convention of constructing spacetime diagrams, that light beams always make an angle of 45° with the local grid lines. It does not follow, however, that light travels

17340-678: 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

17544-501: 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

17748-418: The large-scale structure of the universe . Around 3 billion years ago, at a time of about 11 billion years, dark energy is believed to have begun to dominate the energy density of the universe. This transition came about because dark energy does not dilute as the universe expands, instead maintaining a constant energy density. Similarly to inflation, dark energy drives accelerated expansion, such that

17952-459: 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

18156-531: 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

18360-472: 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

18564-453: 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

18768-429: 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

18972-473: 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

19176-473: 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

19380-470: The Big Bang. During the matter-dominated epoch, cosmic expansion also decelerated, with the scale factor growing as the 2/3 power of the time ( a ∝ t 2 / 3 {\displaystyle a\propto t^{2/3}} ). Also, gravitational structure formation is most efficient when nonrelativistic matter dominates, and this epoch is responsible for the formation of galaxies and

19584-648: 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

19788-463: 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.

19992-554: The Earth. In 1922, Alexander Friedmann used the Einstein field equations to provide theoretical evidence that the universe is expanding. Swedish astronomer Knut Lundmark was the first person to find observational evidence for expansion, in 1924. According to Ian Steer of the NASA/IPAC Extragalactic Database of Galaxy Distances, "Lundmark's extragalactic distance estimates were far more accurate than Hubble's, consistent with an expansion rate (Hubble constant) that

20196-480: 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

20400-401: The Hubble horizon are not dynamical, because gravitational influences do not have time to propagate across them, while perturbations much smaller than the Hubble horizon are straightforwardly governed by Newtonian gravitational dynamics . An object's peculiar velocity is its velocity with respect to the comoving coordinate grid, i.e., with respect to the average expansion-associated motion of

20604-418: The Hubble rate H {\displaystyle H} quantifies the rate of expansion. H {\displaystyle H} is a function of cosmic time . The expansion of the universe can be understood as a consequence of an initial impulse (possibly due to inflation ), which sent the contents of the universe flying apart. The mutual gravitational attraction of the matter and radiation within

20808-495: 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

21012-446: 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

21216-413: 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

21420-403: 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

21624-405: 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

21828-457: The cosmic scale factor grew exponentially in time. In order to solve the horizon and flatness problems, inflation must have lasted long enough that the scale factor grew by at least a factor of e (about 10 ). The history of the universe after inflation but before a time of about 1 second is largely unknown. However, the universe is known to have been dominated by ultrarelativistic Standard Model particles, conventionally called radiation , by

22032-503: The cosmic expansion history can also be measured by studying how redshifts, distances, fluxes, angular positions, and angular sizes of astronomical objects change over the course of the time that they are being observed. These effects are too small to have yet been detected. However, changes in redshift or flux could be observed by the Square Kilometre Array or Extremely Large Telescope in the mid-2030s. At cosmological scales,

22236-456: The decay of particles' peculiar momenta, as discussed above. It can also be understood as adiabatic cooling . The temperature of ultrarelativistic fluids, often called "radiation" and including the cosmic microwave background , scales inversely with the scale factor (i.e. T ∝ a − 1 {\displaystyle T\propto a^{-1}} ). The temperature of nonrelativistic matter drops more sharply, scaling as

22440-444: 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

22644-528: The distance between Earth and the quasar when the light was emitted, and the distance between them in the present era (taking a slice of the cone along the dimension defined as the spatial dimension). The former distance is about 4 billion light-years, much smaller than ct , whereas the latter distance (shown by the orange line) is about 28 billion light-years, much larger than  ct . In other words, if space were not expanding today, it would take 28 billion years for light to travel between Earth and

22848-417: 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

23052-452: 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

23256-416: The energy density grows as the universe expands. Inflation is a period of accelerated expansion hypothesized to have occurred at a time of around 10 seconds. It would have been driven by the inflaton , a field that has a positive-energy false vacuum state. Inflation was originally proposed to explain the absence of exotic relics predicted by grand unified theories , such as magnetic monopoles , because

23460-555: 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

23664-445: 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

23868-405: The evidence that leads to the inflationary model of the early universe also implies that the "total universe" is much larger than the observable universe. Thus any edges or exotic geometries or topologies would not be directly observable, since light has not reached scales on which such aspects of the universe, if they exist, are still allowed. For all intents and purposes, it is safe to assume that

24072-455: 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.,

24276-464: 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

24480-506: The existence of dark energy is inferred from astronomical observations. In an expanding universe, it is often useful to study the evolution of structure with the expansion of the universe factored out. This motivates the use of comoving coordinates , which are defined to grow proportionally with the scale factor. If an object is moving only with the Hubble flow of the expanding universe, with no other motion, then it remains stationary in comoving coordinates. The comoving coordinates are

24684-469: The expansion, a ¨ < 0 {\displaystyle {\ddot {a}}<0} , and a positive pressure further decelerates expansion. On the other hand, sufficiently negative pressure with p < − ρ c 2 / 3 {\displaystyle p<-\rho c^{2}/3} leads to accelerated expansion, and the cosmological constant also accelerates expansion. Nonrelativistic matter

24888-405: 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

25092-401: The first few billion years of its travel time, also indicating that the expansion of space between Earth and the quasar at the early time was faster than the speed of light. None of this behavior originates from a special property of metric expansion, but rather from local principles of special relativity integrated over a curved surface. Over time, the space that makes up the universe

25296-404: 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

25500-420: The future" over long distances. However, within general relativity , the shape of these comoving synchronous spatial surfaces is affected by gravity. Current observations are consistent with these spatial surfaces being geometrically flat (so that, for example, the angles of a triangle add up to 180 degrees). An expanding universe typically has a finite age. Light, and other particles, can have propagated only

25704-410: 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

25908-474: 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

26112-415: 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

26316-439: 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

26520-418: The infinite extent of the expanse. All that is certain is that the manifold of space in which we live simply has the property that the distances between objects are getting larger as time goes on. This only implies the simple observational consequences associated with the metric expansion explored below. No "outside" or embedding in hyperspace is required for an expansion to occur. The visualizations often seen of

26724-432: The infinite future. This implies that the amount of the universe that we will ever be able to observe is limited. Many systems exist whose light can never reach us, because there is a cosmic event horizon induced by the repulsive gravity of the dark energy. Within the study of the evolution of structure within the universe, a natural scale emerges, known as the Hubble horizon . Cosmological perturbations much larger than

26928-446: 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

27132-512: 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

27336-413: The inverse square of the scale factor (i.e. T ∝ a − 2 {\displaystyle T\propto a^{-2}} ). The contents of the universe dilute as it expands. The number of particles within a comoving volume remains fixed (on average), while the volume expands. For nonrelativistic matter, this implies that the energy density drops as ρ ∝

27540-409: 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

27744-409: The large-scale geometry of the universe according to the ΛCDM cosmological model. Two of the dimensions of space are omitted, leaving one dimension of space (the dimension that grows as the cone gets larger) and one of time (the dimension that proceeds "up" the cone's surface). The narrow circular end of the diagram corresponds to a cosmological time of 700 million years after the Big Bang, while

27948-453: 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

28152-405: 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

28356-472: 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

28560-399: 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

28764-411: 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}

28968-453: 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,

29172-516: 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

29376-449: 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

29580-442: 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

29784-441: 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

29988-423: 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

30192-413: 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

30396-429: 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

30600-545: The observer, recessional velocity of objects at that distance increases by about 73 kilometres per second (160,000 mph). Supernovae are observable at such great distances that the light travel time therefrom can approach the age of the universe. Consequently, they can be used to measure not only the present-day expansion rate but also the expansion history. In work that was awarded the 2011 Nobel Prize in Physics , supernova observations were used to determine that cosmic expansion

30804-459: 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

31008-423: The particle count, the energy of each particle (including the rest mass energy ) also drops significantly due to the decay of peculiar momenta. In general, we can consider a perfect fluid with pressure p = w ρ {\displaystyle p=w\rho } , where ρ {\displaystyle \rho } is the energy density. The parameter w {\displaystyle w}

31212-421: 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

31416-504: 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

31620-435: The present era (less in the past and more in the future). The circular curling of the surface is an artifact of the embedding with no physical significance and is done for illustrative purposes; a flat universe does not curl back onto itself. (A similar effect can be seen in the tubular shape of the pseudosphere .) The brown line on the diagram is the worldline of Earth (or more precisely its location in space, even before it

31824-416: The present universe conforms to Euclidean space , what cosmologists describe as geometrically flat , to within experimental error. Consequently, the rules of Euclidean geometry associated with Euclid's fifth postulate hold in the present universe in 3D space. It is, however, possible that the geometry of past 3D space could have been highly curved. The curvature of space is often modeled using

32028-428: 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

32232-555: 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

32436-418: The quasar, while if the expansion had stopped at the earlier time, it would have taken only 4 billion years. The light took much longer than 4 billion years to reach us though it was emitted from only 4 billion light-years away. In fact, the light emitted towards Earth was actually moving away from Earth when it was first emitted; the metric distance to Earth increased with cosmological time for

32640-416: The rapid expansion would have diluted such relics. It was subsequently realized that the accelerated expansion would also solve the horizon problem and the flatness problem . Additionally, quantum fluctuations during inflation would have created initial variations in the density of the universe, which gravity later amplified to yield the observed spectrum of matter density variations . During inflation,

32844-400: The recession rates of cosmologically distant objects. Cosmic expansion is a key feature of Big Bang cosmology. It can be modeled mathematically with the Friedmann–Lemaître–Robertson–Walker metric (FLRW), where it corresponds to an increase in the scale of the spatial part of the universe's spacetime metric tensor (which governs the size and geometry of spacetime). Within this framework,

33048-417: 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

33252-419: 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

33456-563: The same place like going all the way around the surface of a balloon (or a planet like the Earth), is an observational question that is constrained as measurable or non-measurable by the universe's global geometry . At present, observations are consistent with the universe having infinite extent and being a simply connected space , though cosmological horizons limit our ability to distinguish between simple and more complicated proposals. The universe could be infinite in extent or it could be finite; but

33660-445: 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

33864-522: The same velocity as its own. More generally, the peculiar momenta of both relativistic and nonrelativistic particles decay in inverse proportion with the scale factor. For photons, this leads to the cosmological redshift . While the cosmological redshift is often explained as the stretching of photon wavelengths due to "expansion of space", it is more naturally viewed as a consequence of the Doppler effect . The universe cools as it expands. This follows from

34068-485: The same year, Adam Riess et al. used an empirical method of visual-band light-curve shapes to more finely estimate the luminosity of Type Ia supernovae . This further minimized the systematic measurement errors of the Hubble constant, to 67 ± 7 km⋅s ⋅Mpc . Reiss's measurements on the recession velocity of the nearby Virgo Cluster more closely agree with subsequent and independent analyses of Cepheid variable calibrations of Type Ia supernova , which estimates

34272-424: The scale factor grows exponentially in time. The most direct way to measure the expansion rate is to independently measure the recession velocities and the distances of distant objects, such as galaxies. The ratio between these quantities gives the Hubble rate, in accordance with Hubble's law. Typically, the distance is measured using a standard candle , which is an object or event for which the intrinsic brightness

34476-403: The separation of objects over time is associated with the expansion of space itself. However, this is not a generally covariant description but rather only a choice of coordinates . Contrary to common misconception, it is equally valid to adopt a description in which space does not expand and objects simply move apart while under the influence of their mutual gravity. Although cosmic expansion

34680-541: The size of the known universe in the 1940s, doubling the previous calculation made by Hubble in 1929. He announced this finding to considerable astonishment at the 1952 meeting of the International Astronomical Union in Rome. For most of the second half of the 20th century, the value of the Hubble constant was estimated to be between 50 and 90 km⋅s ⋅ Mpc . On 13 January 1994, NASA formally announced

34884-537: 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

35088-616: The spatial coordinates in the FLRW metric . The universe is a four-dimensional spacetime, but within a universe that obeys the cosmological principle, there is a natural choice of three-dimensional spatial surface. These are the surfaces on which observers who are stationary in comoving coordinates agree on the age of the universe . In a universe governed by special relativity , such surfaces would be hyperboloids , because relativistic time dilation means that rapidly receding distant observers' clocks are slowed, so that spatial surfaces must bend "into

35292-471: 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}

35496-583: 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

35700-401: 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

35904-446: 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

36108-402: 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

36312-432: 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

36516-474: The surrounding material. It is a measure of how a particle's motion deviates from the Hubble flow of the expanding universe. The peculiar velocities of nonrelativistic particles decay as the universe expands, in inverse proportion with the cosmic scale factor . This can be understood as a self-sorting effect. A particle that is moving in some direction gradually overtakes the Hubble flow of cosmic expansion in that direction, asymptotically approaching material with

36720-466: 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

36924-517: 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

37128-419: 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

37332-456: 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

37536-525: 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

37740-413: The time of neutrino decoupling at about 1 second. During radiation domination, cosmic expansion decelerated, with the scale factor growing proportionally with the square root of the time. Since radiation redshifts as the universe expands, eventually nonrelativistic matter came to dominate the energy density of the universe. This transition happened at a time of about 50 thousand years after

37944-413: The time through which various events take place. The expansion of space is in reference to this 3D manifold only; that is, the description involves no structures such as extra dimensions or an exterior universe. The ultimate topology of space is a posteriori – something that in principle must be observed – as there are no constraints that can simply be reasoned out (in other words there cannot be any

38148-400: 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

38352-486: 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

38556-457: 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 ,

38760-425: The universe expands "into" anything or that space exists "outside" it. To any observer in the universe, it appears that all but the nearest galaxies (which are bound to each other by gravity) move away at speeds that are proportional to their distance from the observer , on average. While objects cannot move faster than light , this limitation applies only with respect to local reference frames and does not limit

38964-415: The universe gradually slows this expansion over time, but expansion nevertheless continues due to momentum left over from the initial impulse. Also, certain exotic relativistic fluids , such as dark energy and inflation, exert gravitational repulsion in the cosmological context, which accelerates the expansion of the universe. A cosmological constant also has this effect. Mathematically, the expansion of

39168-462: The universe is infinite in spatial extent, without edge or strange connectedness. Regardless of the overall shape of the universe, the question of what the universe is expanding into is one that does not require an answer, according to the theories that describe the expansion; the way we define space in our universe in no way requires additional exterior space into which it can expand, since an expansion of an infinite expanse can happen without changing

39372-402: The universe is quantified by the scale factor , a {\displaystyle a} , which is proportional to the average separation between objects, such as galaxies. The scale factor is a function of time and is conventionally set to be a = 1 {\displaystyle a=1} at the present time. Because the universe is expanding, a {\displaystyle a}

39576-475: The universe to stop expanding and begin to contract, which corresponds to the scale factor decreasing in time. The scale factor a {\displaystyle a} is a parameter of the FLRW metric , and its time evolution is governed by the Friedmann equations . The second Friedmann equation, shows how the contents of the universe influence its expansion rate. Here, G {\displaystyle G}

39780-489: 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}

39984-412: The wide end is a cosmological time of 18 billion years, where one can see the beginning of the accelerating expansion as a splaying outward of the spacetime, a feature that eventually dominates in this model. The purple grid lines mark cosmological time at intervals of one billion years from the Big Bang. The cyan grid lines mark comoving distance at intervals of one billion light-years in

40188-422: The width of a molecule of DNA ) to one approximately 10.6  light-years across (about 10  m , or 62 trillion miles). Cosmic expansion subsequently decelerated to much slower rates, until around 9.8 billion years after the Big Bang (4 billion years ago) it began to gradually expand more quickly , and is still doing so. Physicists have postulated the existence of dark energy , appearing as

40392-482: 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

40596-410: 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

40800-443: Was formed). The yellow line is the worldline of the most distant known quasar . The red line is the path of a light beam emitted by the quasar about 13 billion years ago and reaching Earth at the present day. The orange line shows the present-day distance between the quasar and Earth, about 28 billion light-years, which is a larger distance than the age of the universe multiplied by the speed of light,  ct . According to

41004-444: 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

41208-477: 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

41412-551: 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

41616-533: Was within 1% of the best measurements today." In 1927, Georges Lemaître independently reached a similar conclusion to Friedmann on a theoretical basis, and also presented observational evidence for a linear relationship between distance to galaxies and their recessional velocity . Edwin Hubble observationally confirmed Lundmark's and Lemaître's findings in 1929. Assuming the cosmological principle , these findings would imply that all galaxies are moving away from each other. Astronomer Walter Baade recalculated

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