1.
Spacetime
–
In physics, spacetime is any mathematical model that combines space and time into a single interwoven continuum. Until the turn of the 20th century, the assumption had been that the 3D geometry of the universe was distinct from time, Einsteins theory was framed in terms of kinematics, and showed how measurements of space and time varied for observers in different reference frames. His theory was an advance over Lorentzs 1904 theory of electromagnetic phenomena. A key feature of this interpretation is the definition of an interval that combines distance. Although measurements of distance and time between events differ among observers, the interval is independent of the inertial frame of reference in which they are recorded. The resultant spacetime came to be known as Minkowski space, non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space and which is separate from space. Classical mechanics assumes that time has a constant rate of passage that is independent of the state of motion of an observer, furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field. Mathematically, spacetime is a manifold, which is to say, by analogy, at small enough scales, a globe appears flat. An extremely large scale factor, c relates distances measured in space with distances measured in time, waves implied the existence of a medium which waved, but attempts to measure the properties of the hypothetical luminiferous aether implied by these experiments provided contradictory results. For example, the Fizeau experiment of 1851 demonstrated that the speed of light in flowing water was less than the speed of light in air plus the speed of the flowing water, the partial aether-dragging implied by this result was in conflict with measurements of stellar aberration. By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein were to derive later, but with a fundamentally different interpretation. As a theory of dynamics, his theory assumed actual physical deformations of the constituents of matter. For example, most physicists believed that Lorentz contraction would be detectable by such experiments as the Trouton–Noble experiment or the Experiments of Rayleigh and Brace. However, these negative results, and in his 1904 theory of the electron. Einstein performed his analyses in terms of kinematics rather than dynamics and it would appear that he did not at first think geometrically about spacetime. It was Einsteins former mathematics professor, Hermann Minkowski, who was to provide an interpretation of special relativity. Einstein was initially dismissive of the interpretation of special relativity
2.
General relativity
–
General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of an expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a thought experiment involving an observer in free fall. 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 the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, 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 our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life
3.
Special relativity
–
In physics, special relativity is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time. In Albert Einsteins original pedagogical treatment, it is based on two postulates, The laws of physics are invariant in all inertial systems, the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source. It was originally proposed in 1905 by Albert Einstein in the paper On the Electrodynamics of Moving Bodies, as of today, special relativity is the most accurate model of motion at any speed. Even so, the Newtonian mechanics model is useful as an approximation at small velocities relative to the speed of light. Not until Einstein developed general relativity, to incorporate general frames of reference, a translation that has often been used is restricted relativity, special really means special case. It has replaced the notion of an absolute universal time with the notion of a time that is dependent on reference frame. Rather than an invariant time interval between two events, there is an invariant spacetime interval, a defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other, rather space and time are interwoven into a single continuum known as spacetime. Events that occur at the time for one observer can occur at different times for another. The theory is special in that it applies in the special case where the curvature of spacetime due to gravity is negligible. In order to include gravity, Einstein formulated general relativity in 1915, Special relativity, contrary to some outdated descriptions, is capable of handling accelerations as well as accelerated frames of reference. e. At a sufficiently small scale and in conditions of free fall, a locally Lorentz-invariant frame that abides by special relativity can be defined at sufficiently small scales, even in curved spacetime. Galileo Galilei had already postulated that there is no absolute and well-defined state of rest, Einstein extended this principle so that it accounted for the constant speed of light, a phenomenon that had been recently observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light and the independence of physical laws from the choice of inertial system, the Principle of Invariant Light Speed –. Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. That is, light in vacuum propagates with the c in at least one system of inertial coordinates. Following Einsteins original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations, however, the most common set of postulates remains those employed by Einstein in his original paper
4.
Equivalence principle
–
Kepler, using Galileos discoveries, showed knowledge of the equivalence principle by accurately describing what would occur if the moon were stopped in its orbit and dropped towards Earth. This can be deduced without knowing if or in what manner gravity decreases with distance, the 1/54 ratio is Keplers estimate of the Moon–Earth mass ratio, based on their diameters. The accuracy of his statement can be deduced by using Newtons inertia law F=ma, setting these accelerations equal for a mass is the equivalence principle. Einstein stated it thus, we assume the physical equivalence of a gravitational field. That is, being on the surface of the Earth is equivalent to being inside a spaceship that is being accelerated by its engines, from this principle, Einstein deduced that free-fall is inertial motion. Objects in free-fall do not experience being accelerated downward but rather weightlessness, in an inertial frame of reference bodies obey Newtons first law, moving at constant velocity in straight lines. Analogously, in a curved spacetime the world line of a particle or pulse of light is as straight as possible. Such a world line is called a geodesic and from the point of view of the frame is a straight line. This is why an accelerometer in free-fall doesnt register any acceleration, as an example, an inertial body moving along a geodesic through space can be trapped into an orbit around a large gravitational mass without ever experiencing acceleration. This is possible because spacetime is curved in close vicinity to a large gravitational mass. In such a situation the geodesic lines bend inward around the center of the mass, an accelerometer on-board would never record any acceleration. By contrast, in Newtonian mechanics, gravity is assumed to be a force and this force draws objects having mass towards the center of any massive body. At the Earths surface, the force of gravity is counteracted by the resistance of the Earths surface. So in Newtonian physics, a person at rest on the surface of an object is in an inertial frame of reference. Einstein also referred to two frames, K and K. This observation was the start of a process that culminated in general relativity, by assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. So the original equivalence principle, as described by Einstein, concluded that free-fall and this form of the equivalence principle can be stated as follows. An observer in a windowless room cannot distinguish between being on the surface of the Earth, and being in a spaceship in deep space accelerating at 1g
5.
Minkowski space
–
Minkowski space is closely associated with Einsteins theory of special relativity, and is the most common mathematical structure on which special relativity is formulated. Because it treats time differently than it treats the three dimensions, Minkowski space differs from four-dimensional Euclidean space. In 3-dimensional Euclidean space, the group is the Euclidean group. It consists of rotations, reflections, and translations, when time is amended as a fourth dimension, the further transformations of translations in time and Galilean boosts are added, and the group of all these transformations is called the Galilean group. All Galilean transformations preserve the 3-dimensional Euclidean distance, Time differences are separately preserved as well. This changes in the spacetime of special relativity, where space, spacetime is equipped with an indefinite non-degenerate bilinear form. Equipped with this product, the mathematical model of spacetime is called Minkowski space. The analogue of the Galilean group for Minkowski space, preserving the interval is the Poincaré group. In summary, Galilean spacetime and Minkowski spacetime are, when viewed as barebones manifolds and they differ in what kind of further structures are defined on them. Here the speed of c is, following Poincare, set to unity. The naming and ordering of coordinates, with the labels for space coordinates. The above expression, while making the expression more familiar. Rotations in planes spanned by two unit vectors appear in coordinate space as well as in physical spacetime appear as Euclidean rotations and are interpreted in the ordinary sense. The analogy with Euclidean rotations is thus only partial and this idea was elaborated by Hermann Minkowski, who used it to restate the Maxwell equations in four dimensions, showing directly their invariance under the Lorentz transformation. He further reformulated in four dimensions the then-recent theory of relativity of Einstein. From this he concluded that time and space should be treated equally, points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point and it is principally this view of spacetime that is current nowadays, although the older view involving imaginary time has also influenced special relativity. An imaginary time coordinate is used also for more subtle reasons in quantum field theory than formal appearance of expressions, in this context, the transformation is called a Wick rotation
6.
Introduction to general relativity
–
General relativity is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915. According to general relativity, the gravitational effect between masses results from their warping of spacetime. By the beginning of the 20th century, Newtons law of gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newtons model, gravity is the result of a force between massive objects. Although even Newton was troubled by the nature of that force. General relativity also predicts novel effects of gravity, such as waves, gravitational lensing. Many of these predictions have been confirmed by experiment or observation, General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes and their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects. General relativity is also part of the framework of the standard Big Bang model of cosmology, although general relativity is not the only relativistic theory of gravity, it is the simplest such theory that is consistent with the experimental data. In September 1905, Albert Einstein published his theory of special relativity, special relativity introduced a new framework for all of physics by proposing new concepts of space and time. Some then-accepted physical theories were inconsistent with that framework, a key example was Newtons theory of gravity, several physicists, including Einstein, searched for a theory that would reconcile Newtons law of gravity and special relativity. Only Einsteins theory proved to be consistent with experiments and observations, a person in a free-falling elevator experiences weightlessness, objects either float motionless or drift at constant speed. Since everything in the elevator is falling together, no effect can be observed. In this way, the experiences of an observer in free fall are indistinguishable from those of an observer in deep space, such observers are the privileged observers Einstein described in his theory of special relativity, observers for whom light travels along straight lines at constant speed. Roughly speaking, the states that a person in a free-falling elevator cannot tell that they are in free fall. Every experiment in such an environment has the same results as it would for an observer at rest or moving uniformly in deep space. Most effects of gravity vanish in free fall, but effects that seem the same as those of gravity can be produced by a frame of reference. Objects are falling to the floor because the room is aboard a rocket in space, the objects are being pulled towards the floor by the same inertial force that presses the driver of an accelerating car into the back of his seat
7.
Einstein field equations
–
First published by Einstein in 1915 as a tensor equation, the EFE equate local spacetime curvature with the local energy and momentum within that spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor, the inertial trajectories of particles and radiation in the resulting geometry are then calculated using the geodesic equation. As well as obeying local energy–momentum conservation, the EFE reduce to Newtons law of gravitation where the field is weak. Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry, special classes of exact solutions are most often studied as they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the actual spacetime as flat spacetime with a small deviation and these equations are used to study phenomena such as gravitational waves. The EFE is an equation relating a set of symmetric 4 ×4 tensors. Each tensor has 10 independent components, although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations, the vacuum field equations define Einstein manifolds. Despite the simple appearance of the equations they are quite complicated. In fact, when written out, the EFE are a system of 10 coupled, nonlinear. The EFE can then be written as G μ ν + Λ g μ ν =8 π G c 4 T μ ν. Using geometrized units where G = c =1, this can be rewritten as G μ ν + Λ g μ ν =8 π T μ ν. The expression on the left represents the curvature of spacetime as determined by the metric, the EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime. These equations, together with the equation, which dictates how freely-falling matter moves through space-time. The above form of the EFE is the established by Misner, Thorne. The sign of the term would change in both these versions, if the metric sign convention is used rather than the MTW metric sign convention adopted here. Taking the trace with respect to the metric of both sides of the EFE one gets R − D2 R + D Λ =8 π G c 4 T where D is the spacetime dimension
8.
Gravity
–
Gravity, or gravitation, is a natural phenomenon by which all things with mass are brought toward one another, including planets, stars and galaxies. Since energy and mass are equivalent, all forms of energy, including light, on Earth, gravity gives weight to physical objects and causes the ocean tides. Gravity has a range, although its effects become increasingly weaker on farther objects. The most extreme example of this curvature of spacetime is a hole, from which nothing can escape once past its event horizon. More gravity results in time dilation, where time lapses more slowly at a lower gravitational potential. Gravity is the weakest of the four fundamental interactions of nature, the gravitational attraction is approximately 1038 times weaker than the strong force,1036 times weaker than the electromagnetic force and 1029 times weaker than the weak force. As a consequence, gravity has an influence on the behavior of subatomic particles. On the other hand, gravity is the dominant interaction at the macroscopic scale, for this reason, in part, pursuit of a theory of everything, the merging of the general theory of relativity and quantum mechanics into quantum gravity, has become an area of research. While the modern European thinkers are credited with development of gravitational theory, some of the earliest descriptions came from early mathematician-astronomers, such as Aryabhata, who had identified the force of gravity to explain why objects do not fall out when the Earth rotates. Later, the works of Brahmagupta referred to the presence of force, described it as an attractive force. Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th and this was a major departure from Aristotles belief that heavier objects have a higher gravitational acceleration. Galileo postulated air resistance as the reason that objects with less mass may fall slower in an atmosphere, galileos work set the stage for the formulation of Newtons theory of gravity. In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. Newtons theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted for by the actions of the other planets. Calculations by both John Couch Adams and Urbain Le Verrier predicted the position of the planet. A discrepancy in Mercurys orbit pointed out flaws in Newtons theory, the issue was resolved in 1915 by Albert Einsteins new theory of general relativity, which accounted for the small discrepancy in Mercurys orbit. The simplest way to test the equivalence principle is to drop two objects of different masses or compositions in a vacuum and see whether they hit the ground at the same time. Such experiments demonstrate that all objects fall at the rate when other forces are negligible
9.
Newton's law of universal gravitation
–
This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. It is a part of classical mechanics and was formulated in Newtons work Philosophiæ Naturalis Principia Mathematica, in modern language, the law states, Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them, the first test of Newtons theory of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798. It took place 111 years after the publication of Newtons Principia, Newtons law of gravitation resembles Coulombs law of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both are inverse-square laws, where force is proportional to the square of the distance between the bodies. Coulombs law has the product of two charges in place of the product of the masses, and the constant in place of the gravitational constant. Newtons law has since been superseded by Albert Einsteins theory of general relativity, at the same time Hooke agreed that the Demonstration of the Curves generated thereby was wholly Newtons. In this way, the question arose as to what, if anything and this is a subject extensively discussed since that time and on which some points, outlined below, continue to excite controversy. And that these powers are so much the more powerful in operating. Thus Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, Hookes statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hookes gravitation was also not yet universal, though it approached universality more closely than previous hypotheses and he also did not provide accompanying evidence or mathematical demonstration. It was later on, in writing on 6 January 1679|80 to Newton, Newton, faced in May 1686 with Hookes claim on the inverse square law, denied that Hooke was to be credited as author of the idea. Among the reasons, Newton recalled that the idea had been discussed with Sir Christopher Wren previous to Hookes 1679 letter, Newton also pointed out and acknowledged prior work of others, including Bullialdus, and Borelli. D T Whiteside has described the contribution to Newtons thinking that came from Borellis book, a copy of which was in Newtons library at his death. Newton further defended his work by saying that had he first heard of the inverse square proportion from Hooke, Hooke, without evidence in favor of the supposition, could only guess that the inverse square law was approximately valid at great distances from the center. Thus Newton gave a justification, otherwise lacking, for applying the inverse square law to large spherical planetary masses as if they were tiny particles, after his 1679-1680 correspondence with Hooke, Newton adopted the language of inward or centripetal force. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s, the lesson offered by Hooke to Newton here, although significant, was one of perspective and did not change the analysis. This background shows there was basis for Newton to deny deriving the inverse square law from Hooke, on the other hand, Newton did accept and acknowledge, in all editions of the Principia, that Hooke had separately appreciated the inverse square law in the solar system
10.
Minkowski diagram
–
It allows a qualitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical equations. The term Minkowski diagram is used in both a generic and particular sense, in general, a Minkowski diagram is a graphical depiction of a portion of Minkowski space, usually where space has been curtailed to a single dimension. These two-dimensional diagrams portray worldlines as curves in a plane that correspond to motion along the spatial axis. The vertical axis is usually temporal, and the units of measurement are taken such that the cone at an event consists of the lines of slope plus or minus one through that event. The horizontal lines correspond to the notion of simultaneous events for a stationary observer at the origin. A particular Minkowski diagram illustrates the result of a Lorentz transformation, the Lorentz transformation relates two inertial frames of reference, where an observer stationary at the event makes a change of velocity along the x-axis. The new time axis of the forms a angle α with the previous time axis. In the new frame of reference the simultaneous events lie parallel to a line inclined by α to the lines of simultaneity. Both the original set of axes and the set of axes have the property that they are orthogonal with respect to the Minkowski inner product or relativistic dot product. Whatever the magnitude of α, the line t = x forms the universal bisector, for simplification in Minkowski diagrams, usually only events in a universe of one space dimension and one time dimension are considered. Unlike common distance-time diagrams, the distance will be displayed on the horizontal axis, in this manner the events happening in the one dimension of space can be transferred easily to a horizontal line in the diagram. Objects plotted on the diagram can be thought of as moving from bottom to top as time passes, in this way each object, like an observer or a vehicle, follows in the diagram a certain curve which is called its world line. Each point in the diagram represents a position in space. Such a position is called an event whether or not anything happens at that position and this way light paths are represented by lines bisecting the axes. The black axes labelled x and ct on the diagram are the coordinate system of an observer which we will refer to as at rest. His world line is identical with the time axis, each parallel line to this axis would correspond also to an object at rest but at another position. The blue line, however, describes an object moving with constant speed v to the right and this blue line labelled ct′ may be interpreted as the time axis for the second observer. Together with the path axis it represents his coordinate system, both observers agree on the location of the origin of their coordinate systems
11.
Differential geometry
–
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century, since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas, Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. These unanswered questions indicated greater, hidden relationships, initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric and this is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Various concepts based on length, such as the arc length of curves, area of plane regions, the notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. Many concepts and techniques 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. 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 is the Riemannian symmetric spaces, whose curvature is not necessarily constant. These are the closest analogues to the 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 the mathematical basis of Einsteins general relativity theory of gravity. Finsler geometry has the Finsler manifold as the object of study. This is a manifold with a Finsler metric, i. e. 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 is a function F, TM → [0, ∞) such that, F = |m|F for all x, y in TM, F is infinitely differentiable in TM −, symplectic geometry is the study of symplectic manifolds. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed, 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, in dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism
12.
Mathematics of general relativity
–
The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying and formulating Albert Einsteins theory of general relativity. The main tools used in this theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a description of the mathematics of general relativity. Note, General relativity articles using tensors will use the index notation. The term general covariance was used in the formulation of general relativity. This will be discussed further below, most modern approaches to mathematical general relativity begin with the concept of a manifold. More precisely, the basic physical construct representing gravitation - a curved spacetime - is modelled by a four-dimensional, smooth, connected, other physical descriptors are represented by various tensors, discussed below. The rationale for choosing a manifold as the mathematical structure is to reflect desirable physical properties. For example, in the theory of manifolds, each point is contained in a chart. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, for cosmological problems, a coordinate chart may be quite large. An important distinction in physics is the difference between local and global structures, an important problem in general relativity is to tell when two spacetimes are the same, at least locally. This problem has its roots in manifold theory where determining if two Riemannian manifolds of the dimension are locally isometric. This latter problem has been solved and its adaptation for general relativity is called the Cartan–Karlhede algorithm, one of the profound consequences of relativity theory was the abolition of privileged reference frames. The description of phenomena should not depend upon who does the measuring - one reference frame should be as good as any other. Special relativity demonstrated that no reference frame was preferential to any other inertial reference frame. General relativity eliminated preference for inertial reference frames by showing that there is no preferred reference frame for describing nature, any observer can make measurements and the precise numerical quantities obtained only depend on the coordinate system used. This suggested a way of formulating relativity using invariant structures, those that are independent of the system used. The most suitable mathematical structure seemed to be a tensor, mathematically, tensors are generalised linear operators - multilinear maps
13.
Euclidean geometry
–
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
14.
Riemannian geometry
–
This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture Ueber die Hypothesen and it is a very broad and abstract generalization of the differential geometry of surfaces in R3. Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century and it deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of Non-Euclidean geometry. Any smooth manifold admits a Riemannian metric, which helps to solve problems of differential topology. It also serves as a level for the more complicated structure of pseudo-Riemannian manifolds. Other generalizations of Riemannian geometry include Finsler geometry, there exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and Disclinations produce torsions and curvature, the choice is made depending on its importance and elegance of formulation. Most of the results can be found in the monograph by Jeff Cheeger. The formulations given are far from being very exact or the most general and this list is oriented to those who already know the basic definitions and want to know what these definitions are about. Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ where χ denotes the Euler characteristic of M and this theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem. Nash embedding theorems also called fundamental theorems of Riemannian geometry and they state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn. If M is a connected compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then M is diffeomorphic to a sphere. Given constants C, D and V, there are finitely many compact n-dimensional Riemannian manifolds with sectional curvature |K| ≤ C, diameter ≤ D. There is an εn >0 such that if an n-dimensional Riemannian manifold has a metric with sectional curvature |K| ≤ εn, G. Perelman in 1994 gave an astonishingly elegant/short proof of the Soul Conjecture, M is diffeomorphic to Rn if it has positive curvature at only one point. There is a constant C = C such that if M is a compact connected n-dimensional Riemannian manifold with sectional curvature then the sum of its Betti numbers is at most C. Given constants C, D and V, there are finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D. It implies that any two points of a connected complete Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic
15.
Metric expansion of space
–
The metric expansion of space is the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself changes, metric expansion is a key feature of Big Bang cosmology, is modeled mathematically with the Friedmann-Lemaître-Robertson-Walker metric and is a generic property of the universe we inhabit. However, the model is only on large scales. At smaller scales matter has become bound together under the influence of gravitational attraction, the source of this acceleration is currently unknown. Physicists have postulated the existence of energy, appearing as a cosmological constant in the simplest gravitational models as a way to explain the acceleration. According to the simplest extrapolation of the cosmological model, this acceleration becomes more dominant into the future. The definition of distance used here is the summation or integration of local comoving distances, all done at constant local proper time. For example, galaxies that are more than the Hubble radius, approximately 4.5 gigaparsecs or 14.7 billion light-years, visibility of these objects depends on the exact expansion history of the universe. Because of the rate of expansion, it is also possible for a distance between two objects to be greater than the value calculated by multiplying the speed of light by the age of the universe. These details are a frequent source of confusion among amateurs and even professional physicists, in June 2016, NASA and ESA scientists reported that the universe was found to be expanding 5% to 9% faster than thought earlier, based on studies using the Hubble Space Telescope. To understand the expansion of the universe, it is helpful to discuss briefly what a metric is. A metric defines how a distance can be measured between two points in space, in terms of the coordinate system. Coordinate systems locate points in a space by assigning unique positions on a grid, known as coordinates, the metric is then a formula which describes how displacement through the space of interest can be translated into distances. For example, consider the measurement of distance between two places on the surface of the Earth and this is a simple, familiar example of spherical geometry. Because the surface of the Earth is two-dimensional, points on the surface of the Earth can be specified by two coordinates — for example, the latitude and longitude, specification of a metric requires that one first specify the coordinates used. In our simple example of the surface of the Earth, we could choose any kind of coordinate system we wish, for example latitude and longitude, in general, such shortest-distance paths are called geodesics. In Euclidean geometry, the geodesic is a line, while in non-Euclidean geometry such as on the Earths surface. Indeed, even the great circle path is always longer than the Euclidean straight line path which passes through the interior of the Earth
16.
Shape of the universe
–
The shape of the universe is the local and global geometry of the Universe, in terms of both curvature and topology. The shape of the universe is related to general relativity which describes how spacetime is curved and bent by mass, cosmologists distinguish between the observable universe and the global universe. The observable universe consists of the part of the universe that can, in principle, simply connected space or multiply connected. There are certain logical connections among these properties, for example, a universe with positive curvature is necessarily finite. Although it is assumed in the literature that a flat or negatively curved universe is infinite. Theorists have been trying to construct a mathematical model of the shape of the universe. In formal terms, this is a 3-manifold model corresponding to the section of the 4-dimensional space-time of the universe. The model most theorists currently use is the Friedmann–Lemaître–Robertson–Walker model, ideally, one can continue to look back all the way to the Big Bang, in practice, however, the farthest away one can look is the cosmic microwave background, as anything past that was opaque. Experimental investigations show that the universe is very close to isotropic. If the observable universe encompasses the entire universe, we may be able to determine the structure of the entire universe by observation. The universe may be small in dimensions and not in others. For example, if the universe is a closed loop, one would expect to see multiple images of an object in the sky. The curvature of space is a description of length relationships in spatial coordinates. In mathematics, any geometry has three possible curvatures, so the geometry of the universe has the three possible curvatures. Flat Positively curved Negatively curved An example of a flat curvature would be any Euclidean geometry, curved geometries are in the domain of Non-Euclidean geometry. An example of a curved surface would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will result in at least two angles being 90°, making the sum of the 3 angles greater than 180°, an example of a negative curved surface would be the shape of a saddle or mountain pass. A triangle drawn on a saddle shape will result in the sum of the angles adding up to less than 180° due to the curving away as the triangle moves away from the center
17.
Pythagorean theorem
–
In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
18.
Degrees of freedom (physics and chemistry)
–
In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all dimensions of a system is known as a phase space, a degree of freedom of a physical system is an independent parameter that is necessary to characterize the state of a physical system. In general, a degree of freedom may be any property that is not dependent on other variables. The location of a particle in space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. On the other hand, a system with an object that can rotate or vibrate can have more than six degrees of freedom. In statistical mechanics, a degree of freedom is a scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the phase space. In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer and it is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a function to the energy of the system. In three-dimensional space, three degrees of freedom are associated with the movement of a particle, a diatomic gas molecule has 7 degrees of freedom. This set may be decomposed in terms of translations, rotations, the center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two degrees of motion and two vibrational modes. The rotations occur around the two axes perpendicular to the line between the two atoms, the rotation around the atom–atom bond is not a physical rotation. This yields, for a molecule, a decomposition of. In special cases, such as adsorbed large molecules, the degrees of freedom can be limited to only one. As defined above one can also count degrees of freedom using the number of coordinates required to specify a position. This is done as follows, For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space, thus its degree of freedom in a 3-D space is 3
19.
Euclidean space
–
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
20.
Elliptic geometry
–
Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a line must intersect. In fact, the perpendiculars on one side all intersect at the pole of the given line. There are no points in elliptic geometry. Every point corresponds to a polar line of which it is the absolute pole. Any point on this line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant, the distance between a pair of points is proportional to the angle between their absolute polars. As explained by H. S. M. Coxeter The name elliptic is possibly misleading and it does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes, analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity. A simple way to picture elliptic geometry is to look at a globe, neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles. More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, with this identification of antipodal points, the model satisfies Euclids first postulate, which states that two points uniquely determine a line. Metaphorically, we can imagine geometers who are like living on the surface of a sphere. Even if the ants are unable to move off the surface, they can still construct lines, the existence of a third dimension is irrelevant to the ants ability to do geometry, and its existence is neither verifiable nor necessary from their point of view. Another way of putting this is that the language of the axioms is incapable of expressing the distinction between one model and another. In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the figures are similar, i. e. they have the same angles. In elliptic geometry this is not the case, for example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere. A line segment therefore cannot be scaled up indefinitely, a geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space
21.
Hyperbolic geometry
–
In mathematics, hyperbolic geometry is a non-Euclidean geometry. Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature, a modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space. In Russia it is commonly called Lobachevskian geometry, named one of its discoverers. This page is mainly about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry, Hyperbolic geometry can be extended to three and more dimensions, see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is closely related to Euclidean geometry than it seems. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry, there are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of geometry, including the first 28 propositions of book one of Euclids Elements, are valid in Euclidean. Propositions 27 and 28 of Book One of Euclids Elements prove the existence of parallel/non-intersecting lines and this difference also has many consequences, concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry, new concepts need to be introduced. Further, because of the angle of parallelism hyperbolic geometry has an absolute scale, single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points define a line, and lines can be infinitely extended. Two intersecting lines have the properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, when we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are many lines that do not intersect either of the given lines. While in some models lines look different they do have these properties, non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry, For any line R and any point P which does not lie on R. In the plane containing line R and point P there are at least two lines through P that do not intersect R. This implies that there are through P an infinite number of lines that do not intersect R. All other non-intersecting lines have a point of distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting. Some geometers simply use parallel lines instead of limiting parallel lines and these limiting parallels make an angle θ with PB, this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism
22.
CAT(k) space
–
In mathematics, a CAT space, where k is a real number, is a specific type of metric space. Intuitively, triangles in a CAT space are slimmer than corresponding model triangles in a space of constant curvature k. In a CAT space, the curvature is bounded from above by k, a notable special case is k =0 complete CAT spaces are known as Hadamard spaces after the French mathematician Jacques Hadamard. Originally, Alexandrov called these spaces “ R k domain”, the terminology CAT was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov. For a real number k, let M k denote the unique simply connected surface with constant curvature k, denote by D k the diameter of M k, which is + ∞ if k ≤0 and π k for k >0. Let Δ be a triangle in X with geodesic segments as its sides, the geodesic metric space is said to be a CAT space if every geodesic triangle Δ in X with perimeter less than 2 D k satisfies the CAT inequality. A metric space is said to be a space with curvature ≤ k if every point of X has a geodesically convex CAT neighbourhood, a space with curvature ≤0 may be said to have non-positive curvature. Any CAT space is also a CAT space for all ℓ > k, in fact, the converse holds, if is a CAT space for all ℓ > k, then it is a CAT space. N -dimensional Euclidean space E n with its usual metric is a CAT space. More generally, any inner product space is a CAT space, conversely, if a real normed vector space is a CAT space for some real k. N -dimensional hyperbolic space H n with its usual metric is a CAT space, the n -dimensional unit sphere S n is a CAT space. More generally, the standard space M k is a CAT space, so, for example, regardless of dimension, the sphere of radius r is a CAT space. Note that the diameter of the sphere is π r not 2 r, the closed subspace X of E3 given by X = E3 ∖ equipped with the induced length metric is not a CAT space for any k. Any product of CAT spaces is CAT , as a special case, a complete CAT space is also known as a Hadamard space, this is by analogy with the situation for Hadamard manifolds. A Hadamard space is contractible and, between any two points of a Hadamard space, there is a geodesic segment connecting them. Most importantly, distance functions in Hadamard spaces are convex, if σ1, CAT spaces Let be a CAT space. Then the following hold, Given any two points x, y ∈ X, there is a unique geodesic segment that joins x to y, moreover. Every local geodesic in X with length at most D k is a geodesic, the d -balls in X of radius less than 12 D k are convex