1.
General relativity
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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
2.
Einstein field equations
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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
3.
Introduction to general relativity
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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
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History of general relativity
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General relativity is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915, with contributions by many others after 1915. According to general relativity, the gravitational attraction between masses results from the warping of space and time by those masses. Within a century of Newtons formulation, careful astronomical observation revealed unexplainable variations between the theory and the observations, under Newtons model, gravity was the result of an attractive force between massive objects. Although even Newton was bothered 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, so, while still working at the patent office in 1907, Einstein had what he would call his happiest thought. He realized that the principle of relativity could be extended to gravitational fields, consequently, in 1907 he wrote an article on acceleration under special relativity. In that article, he argued that free fall is really inertial motion, and this argument is called the Equivalence principle. In the same article, Einstein also predicted the phenomenon of time dilation. In 1911, Einstein published another article expanding on the 1907 article and he used special relativity to see that the rate of clocks at the top of a box accelerating upward would be faster than the rate of clocks at the bottom. He concludes that the rates of clocks depend on their position in a field. Also the deflection of light by massive bodies was predicted, although the approximation was crude, it allowed him to calculate that the deflection is nonzero. German astronomer Erwin Finlay-Freundlich publicized Einsteins challenge to scientists around the world and this urged astronomers to detect the deflection of light during a solar eclipse, and gave Einstein confidence that the scalar theory of gravity proposed by Gunnar Nordström was incorrect. But the actual value for the deflection that he calculated was too small by a factor of two, because the approximation he used doesnt work well for things moving at near the speed of light. When Einstein finished the full theory of relativity, he would rectify this error. Another of Einsteins notable thought experiments about the nature of the field is that of the rotating disk
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Mathematics of general relativity
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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
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Tests of general relativity
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At its introduction in 1915, the general theory of relativity did not have a solid empirical foundation. Beginning in 1974, Hulse, Taylor and others have studied the behaviour of binary pulsars experiencing much stronger gravitational fields than those found in the Solar System. Both in the field limit and with the stronger fields present in systems of binary pulsars the predictions of general relativity have been extremely well tested locally. As a consequence of the principle, Lorentz invariance holds locally in non-rotating. Experiments related to Lorentz invariance and thus special relativity are described in Tests of special relativity, in February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from a black hole merger. This discovery along with a second discovery announced in June 2016 tested general relativity in the strong field limit. He also mentioned three classical tests with comments, The chief attraction of the lies in its logical completeness. If a single one of the conclusions drawn from it proves wrong, it must be given up, under Newtonian physics, a two-body system consisting of a lone object orbiting a spherical mass would trace out an ellipse with the spherical mass at a focus. The point of closest approach, called the periapsis, is fixed, a number of effects in the Solar System cause the perihelia of planets to precess around the Sun. The principal cause is the presence of planets which perturb one anothers orbit. Mercury deviates from the precession predicted from these Newtonian effects and this anomalous rate of precession of the perihelion of Mercurys orbit was first recognized in 1859 as a problem in celestial mechanics, by Urbain Le Verrier. A number of ad hoc and ultimately unsuccessful solutions were proposed, in general relativity, this remaining precession, or change of orientation of the orbital ellipse within its orbital plane, is explained by gravitation being mediated by the curvature of spacetime. Einstein showed that general relativity agrees closely with the amount of perihelion shift. This was a factor motivating the adoption of general relativity. Although earlier measurements of planetary orbits were made using conventional telescopes, the total observed precession of Mercury is 574. 10±0.65 arc-seconds per century relative to the inertial ICRF. This precession can be attributed to the causes, The correction by 42.98 is 3/2 multiple of classical prediction with PPN parameters γ = β =0. Thus the effect can be explained by general relativity. More recent calculations based on precise measurements have not materially changed the situation
7.
Theory of relativity
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The theory of relativity usually encompasses two interrelated theories by Albert Einstein, special relativity and general relativity. Special relativity applies to particles and their interactions, describing all their physical phenomena except gravity. General relativity explains the law of gravitation and its relation to other forces of nature and it applies to the cosmological and astrophysical realm, including astronomy. The theory transformed theoretical physics and astronomy during the 20th century and it introduced concepts including spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, with relativity, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves. Max Planck, Hermann Minkowski and others did subsequent work, Einstein developed general relativity between 1907 and 1915, with contributions by many others after 1915. The final form of general relativity was published in 1916, the term theory of relativity was based on the expression relative theory used in 1906 by Planck, who emphasized how the theory uses the principle of relativity. In the discussion section of the paper, Alfred Bucherer used for the first time the expression theory of relativity. By the 1920s, the community understood and accepted special relativity. It rapidly became a significant and necessary tool for theorists and experimentalists in the new fields of physics, nuclear physics. By comparison, general relativity did not appear to be as useful and it seemed to offer little potential for experimental test, as most of its assertions were on an astronomical scale. Its mathematics of general relativity seemed difficult and fully understandable only by a number of people. Around 1960, general relativity became central to physics and astronomy, new mathematical techniques to apply to general relativity streamlined calculations and made its concepts more easily visualized. Special relativity is a theory of the structure of spacetime and it was introduced in Einsteins 1905 paper On the Electrodynamics of Moving Bodies. Special relativity is based on two postulates which are contradictory in classical mechanics, The laws of physics are the same for all observers in motion relative to one another. The speed of light in a vacuum is the same for all observers, the resultant theory copes with experiment better than classical mechanics. For instance, postulate 2 explains the results of the Michelson–Morley experiment, moreover, the theory has many surprising and counterintuitive consequences. Some of these are, Relativity of simultaneity, Two events, simultaneous for one observer, time dilation, Moving clocks are measured to tick more slowly than an observers stationary clock
8.
Frame of reference
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In physics, a frame of reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements. In n dimensions, n+1 reference points are sufficient to define a reference frame. Using rectangular coordinates, a frame may be defined with a reference point at the origin. In Einsteinian relativity, reference frames are used to specify the relationship between an observer and the phenomenon or phenomena under observation. In this context, the phrase often becomes observational frame of reference, a relativistic reference frame includes the coordinate time, which does not correspond across different frames moving relatively to each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent, the need to distinguish between the various meanings of frame of reference has led to a variety of terms. For example, sometimes the type of system is attached as a modifier. Sometimes the state of motion is emphasized, as in rotating frame of reference, sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference, in this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a system may be employed for many purposes where the state of motion is not the primary concern. For example, a system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors and it seems useful to divorce the various aspects of a reference frame for the discussion below. A coordinate system is a concept, amounting to a choice of language used to describe observations. Consequently, an observer in a frame of reference can choose to employ any coordinate system to describe observations made from that frame of reference. A change in the choice of coordinate system does not change an observers state of motion. This viewpoint can be found elsewhere as well, which is not to dispute that some coordinate systems may be a better choice for some observations than are others. Choice of what to measure and with what observational apparatus is a separate from the observers state of motion. D. Norton, The discussion is taken beyond simple space-time coordinate systems by Brading, extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory, classical relativistic mechanics, and quantum gravity
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Inertial frame of reference
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In classical physics and special relativity, an inertial frame of reference is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner. The physics of a system in an inertial frame have no causes external to the system, all inertial frames are in a state of constant, rectilinear motion with respect to one another, an accelerometer moving with any of them would detect zero acceleration. Measurements in one frame can be converted to measurements in another by a simple transformation. In general relativity, in any region small enough for the curvature of spacetime and tidal forces to be negligible, systems in non-inertial frames in general relativity dont have external causes because of the principle of geodesic motion. Physical laws take the form in all inertial frames. For example, a ball dropped towards the ground does not go straight down because the Earth is rotating. Someone rotating with the Earth must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion, another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force. The motion of a body can only be described relative to something else—other bodies, observers and these are called frames of reference. If the coordinates are chosen badly, the laws of motion may be more complex than necessary, for example, suppose a free body that has no external forces on it is at rest at some instant. In many coordinate systems, it would begin to move at the next instant, however, a frame of reference can always be chosen in which it remains stationary. Similarly, if space is not described uniformly or time independently, indeed, an intuitive summary of inertial frames can be given as, In an inertial reference frame, the laws of mechanics take their simplest form. In an inertial frame, Newtons first law, the law of inertia, is satisfied, Any free motion has a constant magnitude, the force F is the vector sum of all real forces on the particle, such as electromagnetic, gravitational, nuclear and so forth. The extra terms in the force F′ are the forces for this frame. The first extra term is the Coriolis force, the second the centrifugal force, also, fictitious forces do not drop off with distance. For example, the force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis. All observers agree on the forces, F, only non-inertial observers need fictitious forces. The laws of physics in the frame are simpler because unnecessary forces are not present. In Newtons time the stars were invoked as a reference frame
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Equivalence principle
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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
11.
Special relativity
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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
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World line
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The world line of an object is the path of that object in 4-dimensional spacetime, tracing the history of its location in space at each instant in time. It is an important concept in physics, and particularly theoretical physics. The idea of world lines originates in physics and was pioneered by Hermann Minkowski, the term is now most often used in relativity theories. In physics, a line of an object is the sequence of spacetime events corresponding to the history of the object. A world line is a type of curve in spacetime. Below an equivalent definition will be explained, A world line is a curve in spacetime. Each point of a line is an event that can be labeled with the time. For example, the orbit of the Earth in space is approximately a circle, a curve in space. However, it there at a different time. The world line of the Earth is helical in spacetime and does not return to the same point, spacetime is the collection of points called events, together with a continuous and smooth coordinate system identifying the events. Each event can be labeled by four numbers, a time coordinate, the mathematical term for spacetime is a four-dimensional manifold. The concept may be applied as well to a higher-dimensional space, for easy visualizations of four dimensions, two space coordinates are often suppressed. The event is represented by a point in a Minkowski diagram, which is a plane usually plotted with the time coordinate, say t, upwards. As expressed by F. R. Harvey A curve M in is called a worldline of a particle if its tangent is future timelike at each point, the arclength parameter is called proper time and usually denoted τ. The length of M is called the time of the worldline or particle. If the worldline M is a segment, then the particle is said to be in free fall. A world line traces out the path of a point in spacetime. A world sheet is the analogous two-dimensional surface traced out by a one-dimensional line traveling through spacetime, the world sheet of an open string is a strip, that of a closed string is a volume
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Riemannian geometry
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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
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Gravitoelectromagnetism
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Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge. The most common version of GEM is valid only far from isolated sources, the analogy and equations differing only by some small factors were first published in 1893, before general relativity, by Oliver Heaviside as a separate theory expanding Newtons law. More subtle predictions, such as induced rotation of a falling object, indirect validations of gravitomagnetic effects have been derived from analyses of relativistic jets. Roger Penrose had proposed a frame dragging mechanism for extracting energy, reva Kay Williams, University of Florida, developed a rigorous proof that validated Penroses mechanism. All of those observed properties could be explained in terms of gravitomagnetic effects, Williams application of Penroses mechanism can be applied to black holes of any size. Relativistic jets can serve as the largest and brightest form of validations for gravitomagnetism, a group at Stanford University is currently analyzing data from the first direct test of GEM, the Gravity Probe B satellite experiment, to see if they are consistent with gravitomagnetism. The Apache Point Observatory Lunar Laser-ranging Operation also plans to observe gravitomagnetism effects, for example, to obtain agreement with Mashhoons writings, all instances of Bg in the GEM equations must be multiplied by −1/2c and Eg by −1. These factors variously modify the analogues of the equations for the Lorentz force, no scaling choice allows all the GEM and EM equations to be perfectly analogous. This difference becomes clearer when one compares non-invariance of relativistic mass to electric charge invariance and this can be traced back to the spin-2 character of the gravitational field, in contrast to the electromagnetism being a spin-1 field. From comparison of GEM equations and Maxwells equations it is obvious that −1/ is the analog of vacuum permittivity ε0. Adopting Planck units normalizes G, c and 1/ to 1, the two sets of equations then become identical but for the minus sign preceding 4π in the GEM equations and a factor of four in Amperes law. These minus signs stem from a difference between gravity and electromagnetism, electrostatic charges of identical sign repel each other, while masses attract each other. Hence the GEM equations are nearly Maxwells equations with mass substituting for charge, 4π appears in both the GEM and Maxwell equations, because Planck units normalize G and 1/ to 1, and not 4πG and 1/ε0. Some higher-order gravitomagnetic effects can reproduce effects reminiscent of the interactions of more conventional polarized charges. For instance, if two wheels are spun on an axis, the mutual gravitational attraction between the two wheels will be greater if they spin in opposite directions than in the same direction. This can be expressed as an attractive or repulsive gravitomagnetic component, Gravitomagnetic arguments also predict that a flexible or fluid toroidal mass undergoing minor axis rotational acceleration will tend to pull matter through the throat. In theory, this configuration might be used for accelerating objects without such objects experiencing any g-forces, consider a toroidal mass with two degrees of rotation. This represents a case in which gravitomagnetic effects generate a chiral corkscrew-like gravitational field around the object
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Two-body problem in general relativity
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The two-body problem in general relativity is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of light by gravity, solutions are also used to describe the motion of binary stars around each other, and estimate their gradual loss of energy through gravitational radiation. It is customary to assume that both bodies are point-like, so tidal forces and the specifics of their material composition can be neglected. General relativity describes the field by curved space-time, the field equations governing this curvature are nonlinear. No exact solutions of the Kepler problem have been found, but a solution has. This solution pertains when the mass M of one body is greater than the mass m of the other. If so, the mass may be taken as stationary. This is an approximation for a photon passing a star. The motion of the body can then be determined from the Schwarzschild solution. Such geodesic solutions account for the precession of the planet Mercury. They also describe the bending of light in a gravitational field, if both masses are considered to contribute to the gravitational field, as in binary stars, the Kepler problem can be solved only approximately. The earliest approximation method to be developed was the post-Newtonian expansion, more recently, it has become possible to solve Einsteins field equation using a computer instead of mathematical formulae. As the two orbit each other, they will emit gravitational radiation, this causes them to lose energy and angular momentum gradually. For binary black holes numerical solution of the two body problem was achieved after four decades of research, in 2005 when three groups devised the breakthrough techniques, the Kepler problem derives its name from Johannes Kepler, who worked as an assistant to the Danish astronomer Tycho Brahe. Brahe took extraordinarily accurate measurements of the motion of the planets of the Solar System. From these measurements, Kepler was able to formulate Keplers laws, a line joining a planet and the Sun sweeps out equal areas during equal intervals of time. The square of the period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Kepler published the first two laws in 1609 and the law in 1619
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Gravity
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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
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Gravity well
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The Sun is very massive, relative to other bodies in the Solar System, so the corresponding gravity well that surrounds it appears deep and far-reaching. The gravity wells of asteroids and small moons, conversely, are depicted as very shallow. The deeper a gravity well is, the energy any space-bound climber must use to escape it. In astrophysics, a gravity well is specifically the potential field around a massive body. Other types of potential wells include electrical and magnetic potential wells, physical models of gravity wells are sometimes used to illustrate orbital mechanics. Gravity wells are frequently confused with embedding diagrams used in relativity theory. If G is the gravitational constant, the external gravitational potential of a spherically symmetric body of mass M is given by the formula. A plot of this function in two dimensions is shown in the figure, the potential function has a hyperbolic cross section, the sudden dip in the center is the origin of the name gravity well. A black hole would not have this closing dip due to its size being only determined by its event horizon, in a uniform gravitational field, the gravitational potential at a point is proportional to the height. As a result, an object constrained to move on the surface will have roughly the same equation of motion as an object moving in the potential field Φ itself, gravity wells constructed on this principle can be found in many science museums. There are several sources of inaccuracy in this model, The friction between the object and the surface has no analogue in vacuum and this effect can be reduced by using a rolling ball instead of a sliding block. The objects vertical motion contributes kinetic energy which has no analogue and this effect can be reduced by making the gravity well shallower. A rolling balls rotational kinetic energy has no analogue and this effect can be reduced by concentrating the balls mass near its center so that the moment of inertia is small compared to mr². A balls center of mass is not located on the surface but at a distance r. For balls of a size, this effect can be eliminated by constructing the surface so that the center of the ball, rather than the surface itself. Consider an idealized rubber sheet suspended in a gravitational field normal to the sheet. The mass density may be viewed as intrinsic to the sheet or as belonging to objects resting on top of the sheet. This equilibrium condition is identical in form to the gravitational Poisson equation ∇2 Φ = −4 π G ρ where Φ is the gravitational potential, in particular, the embedding diagram most commonly found in textbooks superficially resembles a gravity well
18.
Gravitational lens
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A gravitational lens is a distribution of matter between a distant light source and an observer, that is capable of bending the light from the source as the light travels towards the observer. This effect is known as gravitational lensing, and the amount of bending is one of the predictions of Albert Einsteins general theory of relativity, fritz Zwicky posited in 1937 that the effect could allow galaxy clusters to act as gravitational lenses. It was not until 1979 that this effect was confirmed by observation of the so-called Twin QSO SBS 0957+561. Unlike an optical lens, a lens produces a maximum deflection of light that passes closest to its center. Consequently, a lens has no single focal point, but a focal line. The term lens in the context of light deflection was first used by O. J. Lodge, who remarked that it is not permissible to say that the gravitational field acts like a lens. If the source, the lensing object, and the observer lie in a straight line. If there is any misalignment, the observer will see an arc segment instead and this phenomenon was first mentioned in 1924 by the St. Petersburg physicist Orest Chwolson, and quantified by Albert Einstein in 1936. It is usually referred to in the literature as an Einstein ring, more commonly, where the lensing mass is complex and does not cause a spherical distortion of space–time, the source will resemble partial arcs scattered around the lens. There are three classes of gravitational lensing,1, strong lensing, where there are easily visible distortions such as the formation of Einstein rings, arcs, and multiple images. The lensing shows up statistically as a stretching of the background objects perpendicular to the direction to the center of the lens. By measuring the shapes and orientations of large numbers of distant galaxies and this, in turn, can be used to reconstruct the mass distribution in the area, in particular, the background distribution of dark matter can be reconstructed. Since galaxies are elliptical and the weak gravitational lensing signal is small. They may also provide an important future constraint on dark energy, Microlensing, where no distortion in shape can be seen but the amount of light received from a background object changes in time. The lensing object may be stars in the Milky Way in one case, with the background source being stars in a remote galaxy, or, in another case. The effect is small, such that even a galaxy with a more than 100 billion times that of the Sun will produce multiple images separated by only a few arcseconds. Galaxy clusters can produce separations of several arcminutes, in both cases the galaxies and sources are quite distant, many hundreds of megaparsecs away from our Galaxy
19.
Gravitational wave
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Gravitational waves are ripples in the curvature of spacetime that propagate as waves at the speed of light, generated in certain gravitational interactions that propagate outward from their source. The possibility of gravitational waves was discussed in 1893 by Oliver Heaviside using the analogy between the law in gravitation and electricity. In 1905 Henri Poincaré first proposed gravitational waves emanating from a body, Gravitational waves cannot exist in the Newtons law of universal gravitation, since that law is predicated on the assumption that physical interactions propagate at infinite speed. On June 15,2016, a detection of gravitational waves from coalescing black holes was announced. Besides LIGO, many other observatories are under construction. In Einsteins theory of relativity, gravity is treated as a phenomenon resulting from the curvature of spacetime. This curvature is caused by the presence of mass, generally, the more mass that is contained within a given volume of space, the greater the curvature of spacetime will be at the boundary of its volume. As objects with mass move around in spacetime, the changes to reflect the changed locations of those objects. In certain circumstances, accelerating objects generate changes in this curvature and these propagating phenomena are known as gravitational waves. As a gravitational wave passes an observer, that observer will find spacetime distorted by the effects of strain, distances between objects increase and decrease rhythmically as the wave passes, at a frequency corresponding to that of the wave. This occurs despite such free objects never being subjected to an unbalanced force, the magnitude of this effect decreases proportional to the inverse distance from the source. Inspiraling binary neutron stars are predicted to be a source of gravitational waves as they coalesce. However, due to the distances to these sources, the effects when measured on Earth are predicted to be very small. Scientists have demonstrated the existence of these waves with ever more sensitive detectors, the most sensitive detector accomplished the task possessing a sensitivity measurement of about one part in 5×1022 provided by the LIGO and VIRGO observatories. A space based observatory, the Laser Interferometer Space Antenna, is currently under development by ESA, Gravitational waves can penetrate regions of space that electromagnetic waves cannot. They are able to allow the observation of the merger of black holes, such systems cannot be observed with more traditional means such as optical telescopes or radio telescopes, and so gravitational-wave astronomy gives new insights into the working of the Universe. In particular, gravitational waves could be of interest to cosmologists as they offer a way of observing the very early Universe. This is not possible with conventional astronomy, since before recombination the Universe was opaque to electromagnetic radiation, precise measurements of gravitational waves will also allow scientists to test more thoroughly the general theory of relativity
20.
Gravitational redshift
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This is a direct result of gravitational time dilation—if one is outside of an isolated gravitational source, the rate at which time passes increases as one moves away from that source. As frequency is inverse of time, frequency of the radiation is reduced in an area of higher gravitational potential. There also exists a corresponding blueshift when electromagnetic radiation propagates from an area of higher gravitational potential to an area of lower gravitational potential. If applied to optical wavelengths, this itself as a change in the colour of visible light as the wavelength of the light is shifted toward the red part of the light spectrum. λ e is the wavelength of the electromagnetic radiation when measured at the source of emission, the redshift is not defined for photons emitted inside the Schwarzschild radius, the distance from the body where the escape velocity is greater than the speed of light. Therefore, this only applies when R e is larger than r s. When the photon is emitted at an equal to the Schwarzschild radius, the redshift will be infinitely large. When the photon is emitted at a large distance, there is no redshift. All of this early work assumed that light could slow down and fall, one way around this conclusion would be if time itself were altered—if clocks at different points had different rates. This was precisely Einsteins conclusion in 1911 and he considered an accelerating box, and noted that according to the special theory of relativity, the clock rate at the bottom of the box was slower than the clock rate at the top. Nowadays, this can be shown in accelerated coordinates. The acceleration at position r is equal to the curvature of the hyperbola at fixed r, the rate is zero at r=0, which is the location of the acceleration horizon. When g is slowly varying, it gives the rate of change of the ticking rate. The constant is chosen to make the rate at infinity equal to 1. Since the gravitational potential is zero at infinity, R =1 − V c 2 where the speed of light has been restored to make the gravitational potential dimensionless, using this approximation, Einstein reproduced the incorrect Newtonian value for the deflection of light in 1909. But since a light beam is a fast moving object, the space-space components contribute too, Einsteins prediction was confirmed by many experiments, starting with Arthur Eddingtons 1919 solar eclipse expedition. The receiving end of the transmission must be located at a higher gravitational potential in order for gravitational redshift to be observed. In other words, the observer must be standing uphill from the source, if the observer is at a lower gravitational potential than the source, a gravitational blueshift can be observed instead
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Redshift
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In physics, redshift happens when light or other electromagnetic radiation from an object is increased in wavelength, or shifted to the red end of the spectrum. Some redshifts are an example of the Doppler effect, familiar in the change of apparent pitches of sirens, a redshift occurs whenever a light source moves away from an observer. Finally, gravitational redshift is an effect observed in electromagnetic radiation moving out of gravitational fields. However, redshift is a common term and sometimes blueshift is referred to as negative redshift. Knowledge of redshifts and blueshifts has been applied to develop several terrestrial technologies such as Doppler radar and radar guns, Redshifts are also seen in the spectroscopic observations of astronomical objects. Its value is represented by the letter z, a special relativistic redshift formula can be used to calculate the redshift of a nearby object when spacetime is flat. However, in contexts, such as black holes and Big Bang cosmology. Special relativistic, gravitational, and cosmological redshifts can be understood under the umbrella of frame transformation laws, the history of the subject began with the development in the 19th century of wave mechanics and the exploration of phenomena associated with the Doppler effect. The effect is named after Christian Doppler, who offered the first known physical explanation for the phenomenon in 1842, the hypothesis was tested and confirmed for sound waves by the Dutch scientist Christophorus Buys Ballot in 1845. Doppler correctly predicted that the phenomenon should apply to all waves, before this was verified, however, it was found that stellar colors were primarily due to a stars temperature, not motion. Only later was Doppler vindicated by verified redshift observations, the first Doppler redshift was described by French physicist Hippolyte Fizeau in 1848, who pointed to the shift in spectral lines seen in stars as being due to the Doppler effect. The effect is called the Doppler–Fizeau effect. In 1868, British astronomer William Huggins was the first to determine the velocity of a moving away from the Earth by this method. In 1871, optical redshift was confirmed when the phenomenon was observed in Fraunhofer lines using solar rotation, about 0.1 Å in the red. In 1887, Vogel and Scheiner discovered the annual Doppler effect, in 1901, Aristarkh Belopolsky verified optical redshift in the laboratory using a system of rotating mirrors. The word does not appear unhyphenated until about 1934 by Willem de Sitter, perhaps indicating that up to point its German equivalent. Beginning with observations in 1912, Vesto Slipher discovered that most spiral galaxies, Slipher first reports on his measurement in the inaugural volume of the Lowell Observatory Bulletin. Three years later, he wrote a review in the journal Popular Astronomy, Slipher reported the velocities for 15 spiral nebulae spread across the entire celestial sphere, all but three having observable positive velocities
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Blueshift
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A blueshift is any decrease in wavelength, with a corresponding increase in frequency, of an electromagnetic wave, the opposite effect is referred to as redshift. In visible light, this shifts the color from the red end of the spectrum to the blue end, Doppler blueshift is caused by movement of a source towards the observer. The term applies to any decrease in wavelength and increase in frequency caused by relative motion, blazars are known to propel relativistic jets toward us, emitting synchrotron radiation and bremsstrahlung that appears blueshifted. Nearby stars such as Barnards Star are moving toward us, resulting in a very small blueshift, Doppler blueshift of distant objects with a high z can be subtracted from the much larger cosmological redshift to determine relative motion in the expanding universe. There are faraway active galaxies that show a blueshift in their emission lines, one of the largest blueshifts is found in the narrow-line quasar, PG 1543+489, which has a relative velocity of -1150 km/s. These types of galaxies are called blue outliers, gravitational potential Redshift Relativistic Doppler effect
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Time dilation
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A clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observers clock. This effect arises neither from technical aspects of the clocks nor from the time of signals. Clocks on the Space Shuttle run slightly slower than reference clocks on Earth, while clocks on GPS, such time dilation has been repeatedly demonstrated, for instance by small disparities in atomic clocks on Earth and in space, even though both clocks work perfectly. The nature of spacetime is such that time measured along different trajectories is affected by differences in either gravity or velocity – each of which time in different ways. In theory, and to make an example, time dilation could affect planned meetings for astronauts with advanced technologies. The astronauts would have to set their clocks to count exactly 80 days, the astronauts would return to Earth, after their mission, having aged one day less than the people staying on Earth. What is more, the experience of time passing never actually changes for anyone. In other words, the astronauts on the ship as well as the mission control crew on Earth each feel normal, despite the effects of time dilation.005 seconds. The effects would be if the astronauts were traveling nearer to the speed of light, instead of their actual speed – which is the speed of the orbiting ISS. Time dilation is caused by differences in either gravity or relative velocity, in the case of ISS, time is slower due to the velocity in circular orbit, this effect is slightly reduced by the opposing effect of less gravitational potential. When two observers are in uniform motion and uninfluenced by any gravitational mass, the point of view of each will be that the others clock is ticking at a slower rate than the local clock. The faster the relative velocity, the greater the magnitude of time dilation and this case is sometimes called special relativistic time dilation. For instance, two ships speeding past one another in space would experience time dilation. If they could see each others ships, they would see the other ships clocks as going more slowly. That is, inside the frame of reference of Ship A, everything is moving normally, from a local perspective, time registered by clocks that are at rest with respect to the local frame of reference always appears to pass at the same rate. In other words, if a new ship, Ship C, travels alongside Ship A, from the point of view of Ship A, new Ship Cs time would appear normal too. A question arises, If Ship A and Ship B both think each others time is moving slower, who will have aged more if they decided to meet up. With a more sophisticated understanding of relative velocity time dilation, this seeming twin paradox turns out not to be a paradox at all, similarly, understanding the twin paradox would help explain why astronauts on the ISS age slower even though they are experiencing relative velocity time dilation
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Gravitational time dilation
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Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The weaker the gravitational potential, the time passes. Albert Einstein originally predicted this effect in his theory of relativity and this has been demonstrated by noting that atomic clocks at differing altitudes will eventually show different times. The effects detected in such Earth-bound experiments are small, with differences being measured in nanoseconds. Demonstrating larger effects would require greater distances from the Earth or a larger gravitational source, Gravitational time dilation was first described by Albert Einstein in 1907 as a consequence of special relativity in accelerated frames of reference. In general relativity, it is considered to be a difference in the passage of time at different positions as described by a metric tensor of spacetime. The existence of time dilation was first confirmed directly by the Pound–Rebka experiment in 1959. Clocks that are far from massive bodies run more quickly, for example, considered over the total lifetime of the earth, a clock set at the peak of Mount Everest would be about 39 hours ahead of a clock set at sea level. This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects. According to general relativity, inertial mass and gravitational mass are the same, let us consider a family of observers along a straight vertical line, each of whom experiences a distinct constant g-force directed along this line. Let g be the dependence of g-force on height, a coordinate along the aforementioned line. For simplicity, in a Rindlers family of observers in a flat space-time, the dependence would be g = c 2 / with constant H, which yields T d = e ln − ln H = H + h H. On the other hand, when g is constant and g h is much smaller than c 2. See Ehrenfest paradox for application of the formula to a rotating reference frame in flat space-time. In comparison, a clock on the surface of the sun will accumulate around 66.4 fewer seconds in one year, in the Schwarzschild metric, free-falling objects can be in circular orbits if the orbital radius is larger than 32 r s. T0 = t f 1 −32 ⋅ r s r, according to the general theory of relativity, gravitational time dilation is copresent with the existence of an accelerated reference frame. An exception is the center of a distribution of matter. Additionally, all phenomena in similar circumstances undergo time dilation equally according to the equivalence principle used in the general theory of relativity
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Gravitational potential
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It is analogous to the electric potential with mass playing the role of charge. The reference location, where the potential is zero, is by convention infinitely far away from any mass, in mathematics, the gravitational potential is also known as the Newtonian potential and is fundamental in the study of potential theory. The gravitational potential is the potential energy per unit mass, U = m V. Potential energy is equal to the work done by the field moving a body to its given position in space from infinity. If the body has a mass of 1 unit, then the energy to be assigned to that body is equal to the gravitational potential. So the potential can be interpreted as the negative of the work done by the field moving a unit mass in from infinity. In some situations, the equations can be simplified by assuming a field that is independent of position. For instance, in a close to the surface of the Earth. In that case, the difference in energy from one height to another is, to a good approximation, linearly related to the difference in height. The potential has units of energy per mass, e. g. J/kg in the MKS system. By convention, it is always negative where it is defined, the gravitational field, and thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential. Thus the negative of a gradient yields positive acceleration toward a massive object. The magnitude of the acceleration therefore follows a square law. The potential associated with a distribution is the superposition of the potentials of point masses. If the mass distribution is a collection of point masses. Mn, then the potential of the distribution at the point x is V = ∑ i =1 n − G m i | x − x i |. If the mass distribution is given as a mass measure dm on three-dimensional Euclidean space R3, then the potential is the convolution of −G/|r| with dm. In good cases this equals the integral V = − ∫ R3 G | x − r | d m, where |x − r| is the distance between the points x and r
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Gravitational collapse
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Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of mass. Gravitational collapse is a mechanism for structure formation in the universe. A star is born through the gravitational collapse of a cloud of interstellar matter. The star then exists in a state of dynamic equilibrium, once all its energy sources are exhausted, a star will again collapse until it reaches a new equilibrium state. An interstellar cloud of gas will remain in equilibrium as long as the kinetic energy of the gas pressure is in balance with the potential energy of the internal gravitational force. Mathematically this is expressed using the theorem, which states that, to maintain equilibrium. If a pocket of gas is enough that the gas pressure is insufficient to support it. The mass above which a cloud will undergo such collapse is called the Jeans mass and this mass depends on the temperature and density of the cloud, but is typically thousands to tens of thousands of solar masses. At what is called the death of the star, it will undergo a contraction that can be halted if it reaches a new state of equilibrium. If it has a star, a white dwarf-sized object can accrete matter from the companion star until it reaches the Chandrasekhar limit at which point gravitational collapse takes over again. While it might seem that the white dwarf might collapse to the stage, they instead undergo runaway carbon fusion. Neutron stars are formed by collapse of the cores of larger stars. They are so compact that a Newtonian description is inadequate for an accurate treatment, hence, the collapse continues with nothing to stop it. Once a body collapses to within its Schwarzschild radius it forms what is called a black hole and it follows from a theorem of Roger Penrose that the subsequent formation of some kind of singularity is inevitable. On the other hand, the nature of the kind of singularity to be expected inside a hole remains rather controversial. According to some theories, at a stage, the collapsing object will reach the maximum possible energy density for a certain volume of space or the Planck density. This is when the laws of gravity cease to be valid. There are competing theories as to what occurs at this point, the radii of larger mass neutron stars are estimated to be about 12-km, or approximately 2.0 times their equivalent Schwarzschild radius
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Frame-dragging
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Frame-dragging is an effect on spacetime, predicted by Einsteins general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary field is one that is in a steady state, the first frame-dragging effect was derived in 1918, in the framework of general relativity, by the Austrian physicists Josef Lense and Hans Thirring, and is also known as the Lense–Thirring effect. They predicted that the rotation of an object would distort the spacetime metric. This does not happen in Newtonian mechanics for which the field of a body depends only on its mass. The Lense–Thirring effect is very small—about one part in a few trillion, to detect it, it is necessary to examine a very massive object, or build an instrument that is very sensitive. More generally, the subject of effects caused by mass–energy currents is known as gravitomagnetism, in 2015, new general-relativistic extensions of Newtonian rotation laws were formulated to describe geometric dragging of frames which incorporates a newly discovered antidragging effect. Rotational frame-dragging appears in the principle of relativity and similar theories in the vicinity of rotating massive objects. Under the Lense–Thirring effect, the frame of reference in which a clock ticks the fastest is one which is revolving around the object as viewed by a distant observer. This also means that light traveling in the direction of rotation of the object will move past the object faster than light moving against the rotation. It is now the best known frame-dragging effect, partly thanks to the Gravity Probe B experiment, qualitatively, frame-dragging can be viewed as the gravitational analog of electromagnetic induction. Also, a region is dragged more than an outer region. This produces interesting locally rotating frames, for example, imagine that a north-south–oriented ice skater, in orbit over the equator of a black hole and rotationally at rest with respect to the stars, extends her arms. The arm extended toward the hole will be torqued spinward due to gravitomagnetic induction. Likewise the arm extended away from the hole will be torqued anti-spinward. She will therefore be rotationally sped up, in a sense to the black hole. This is the opposite of what happens in everyday experience and this frame is rotating with respect to the fixed stars and counter-rotating with respect to the black hole. This effect is analogous to the structure in atomic spectra due to nuclear spin. A useful metaphor is a gear system with the black hole being the sun gear, the ice skater being a planetary gear
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Geodetic effect
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The geodetic effect represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. For example, the vector could be the momentum of a gyroscope orbiting the Earth. The geodetic effect was first predicted by Willem de Sitter in 1916, de Sitters work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker. It can also be applied to a particular secular precession of astronomical orbits, the term geodetic effect has two slightly different meanings as the moving body may be spinning or non-spinning. Non-spinning bodies move in geodesics, whereas spinning bodies move in different orbits. The total precession is calculated by combining the de Sitter precession with the Lense–Thirring precession. The geodetic effect was verified to a precision of better than 0. 5% percent by Gravity Probe B, the first results were announced on April 14,2007 at the meeting of the American Physical Society. To derive the precession, assume the system is in a rotating Schwarzschild metric, the nonrotating metric is d s 2 = d t 2 − d r 2 −1 − r 2, where c = G =1. We introduce a rotating system, with an angular velocity ω. This gives us d ϕ = d ϕ ′ − ω d t, in this coordinate system, an observer at radial position r sees a vector positioned at r as rotating with angular frequency ω. This observer, however, sees a vector positioned at some value of r as rotating at a different rate. For a body orbiting in the θ = π/2 plane, we will have β =1, now, the metric is in the canonical form d s 2 = e 2 Φ2 − k i j d x i d x j. This leads to Φ, i =2 m / r 2 −2 r β ω22 =0, solving this equation for ω yields ω2 = m r 3 β. This is essentially Keplers law of periods, which happens to be exact when expressed in terms of the time coordinate t of this particular rotating coordinate system. In the rotating frame, the remains at rest. This observer also sees the distant stars as rotating, but they rotate at a different rate due to time dilation. Let τ be the proper time. Then Δ τ =1 /2 d t =1 /2 d t, the −2m/r term is interpreted as the gravitational time dilation, while the additional −m/r is due to the rotation of this frame of reference
29.
Gravitational singularity
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The quantities used to measure gravitational field strength are the scalar invariant curvatures of space-time, which includes a measure of the density of matter. Since such quantities become infinite within the singularity, the laws of normal space-time could not exist, the Penrose–Hawking singularity theorems define a singularity to have geodesics that cannot be extended in a smooth manner. The termination of such a geodesic is considered to be the singularity, according to modern general relativity, the initial state of the universe, at the beginning of the Big Bang, was a singularity. Many theories in physics have mathematical singularities of one kind or another, equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a piece in the theory, as in the Ultraviolet Catastrophe, re-normalization. Some theories, such as the theory of quantum gravity suggest that singularities may not exist. A conical singularity occurs when there is a point where the limit of every diffeomorphism invariant quantity is finite, thus, space-time looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere if a suitable system is used. An example of such a singularity is a cosmic string. Solutions to the equations of general relativity or another theory of gravity often result in encountering points where the metric blows up to infinity, however, many of these points are completely regular, and the infinities are merely a result of using an inappropriate coordinate system at this point. In order to test whether there is a singularity at a certain point, such quantities are the same in every coordinate system, so these infinities will not go away by a change of coordinates. An example is the Schwarzschild solution that describes a non-rotating, uncharged black hole, in coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the event horizon. However, space-time at the event horizon is regular, the regularity becomes evident when changing to another coordinate system, where the metric is perfectly smooth. On the other hand, in the center of the hole, where the metric becomes infinite as well. The existence of the singularity can be verified by noting that the Kretschmann scalar, being the square of the Riemann tensor i. e. R μ ν ρ σ R μ ν ρ σ, such a singularity may also theoretically become a wormhole. For example, any observer inside the event horizon of a black hole would fall into its center within a finite period of time. The classical version of the Big Bang cosmological model of the universe contains a causal singularity at the start of time, extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite space-time curvature. Until the early 1990s, it was believed that general relativity hides every singularity behind an event horizon
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Event horizon
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In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In laymans terms, it is defined as the shell of points of no return, i. e. the points at which the gravitational pull becomes so great as to make escape impossible, an event horizon is most commonly associated with black holes. Light emitted from inside the event horizon can never reach the outside observer, likewise, any object approaching the horizon from the observers side appears to slow down and never quite pass through the horizon, with its image becoming more and more redshifted as time elapses. This means that the wavelength is getting longer as the object away from the observer. The traveling object, however, experiences no strange effects and does, in fact, more specific types of horizon include the related but distinct absolute and apparent horizons found around a black hole. Often, this is described as the boundary within which the black holes escape velocity is greater than the speed of light, the surface at the Schwarzschild radius acts as an event horizon in a non-rotating body that fits inside this radius. The Schwarzschild radius of an object is proportional to its mass, theoretically, any amount of matter will become a black hole if compressed into a space that fits within its corresponding Schwarzschild radius. For the mass of the Sun this radius is approximately 3 kilometers, in practice, however, neither the Earth nor the Sun has the necessary mass and therefore the necessary gravitational force, to overcome electron and neutron degeneracy pressure. The minimal mass required for a star to be able to collapse beyond these pressures is the Tolman-Oppenheimer-Volkoff limit, black hole event horizons are widely misunderstood. As with any mass in the Universe, matter must come within its scope for the possibility to exist of capture or consolidation with any other mass. Equally common is the idea that matter can be observed “falling into” a black hole, astronomers can only detect accretion disks around black holes, where material moves with such speed that friction creates high-energy radiation which can be detected. Furthermore, a distant observer will never see something cross the horizon. Instead, while approaching the hole, the object will seem to go ever more slowly, in cosmology, the event horizon of the observable universe is the largest comoving distance from which light emitted now can ever reach the observer in the future. This differs from the concept of particle horizon, which represents the largest comoving distance from which light emitted in the past could have reached the observer at a given time. For events beyond that distance, light has not had time to reach our location, how the particle horizon changes with time depends on the nature of the expansion of the Universe. If the expansion has certain characteristics, there are parts of the Universe that will never be observable, the boundary past which events cannot ever be observed is an event horizon, and it represents the maximum extent of the particle horizon. The criterion for determining whether a particle horizon for the Universe exists is as follows, define a comoving distance d p by d p = ∫0 t 0 c a d t. In this equation, a is the factor, c is the speed of light
31.
Naked singularity
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In general relativity, a naked singularity is a gravitational singularity without an event horizon. In a black hole, the singularity is completely enclosed by a known as the event horizon. Hence, objects inside the event horizon—including the singularity itself—cannot be directly observed, a naked singularity, by contrast, is observable from the outside. The theoretical existence of naked singularities is important because their existence would mean that it would be possible to observe the collapse of an object to infinite density. It would also cause problems for general relativity, because general relativity cannot make predictions about the future evolution of space-time near a singularity. In generic black holes, this is not a problem, as a viewer cannot observe the space-time within the event horizon. Some research has suggested that if loop quantum gravity is correct, then naked singularities could exist in nature, numerical calculations and some other arguments have also hinted at this possibility. At LIGO, first observation of gravitational waves were detected after the two black holes, known as event GW150914. This event did not produce a naked singularity based on observation, from concepts drawn from rotating black holes, it is shown that a singularity, spinning rapidly, can become a ring-shaped object. This results in two event horizons, as well as an ergosphere, which draw closer together as the spin of the singularity increases, when the outer and inner event horizons merge, they shrink toward the rotating singularity and eventually expose it to the rest of the universe. A singularity rotating fast enough might be created by the collapse of dust or by a supernova of a fast-spinning star, studies of pulsars and some computer simulations have been performed. This is an example of a difficulty which reveals a more profound problem in our understanding of the relevant physics involved in the process. A workable theory of gravity should be able to solve problems such as these. Shaw Prize winning mathematician Demetrios Christodoulou has shown that contrary to what had been expected, however, he then showed that such naked singularities are unstable. Disappearing event horizons exist in the Kerr metric, which is a black hole in a vacuum. Specifically, if the momentum is high enough, the event horizons could disappear. In this case, event horizons disappear means when the solutions are complex for r ±, disappearing event horizons can also be seen with the Reissner–Nordström geometry of a charged black hole. In this metric, it can be shown that the singularities occur at r ± = μ ±1 /2, where μ = G M / c 2, and q 2 = G Q2 /
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Black hole
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A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it. The theory of relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of the region from which no escape is possible is called the event horizon, although the event horizon has an enormous effect on the fate and circumstances of an object crossing it, no locally detectable features appear to be observed. In many ways a black hole acts like a black body. Moreover, quantum theory in curved spacetime predicts that event horizons emit Hawking radiation. This temperature is on the order of billionths of a kelvin for black holes of stellar mass, objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace. Black holes were considered a mathematical curiosity, it was during the 1960s that theoretical work showed they were a generic prediction of general relativity. The discovery of neutron stars sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality, black holes of stellar mass are expected to form when very massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings, by absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses may form. There is general consensus that supermassive black holes exist in the centers of most galaxies, despite its invisible interior, the presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light. Matter that falls onto a black hole can form an accretion disk heated by friction. If there are other stars orbiting a black hole, their orbits can be used to determine the black holes mass, such observations can be used to exclude possible alternatives such as neutron stars.3 million solar masses. On 15 June 2016, a detection of a gravitational wave event from colliding black holes was announced. The idea of a body so massive that light could not escape was briefly proposed by astronomical pioneer John Michell in a letter published in 1783-4. Michell correctly noted that such supermassive but non-radiating bodies might be detectable through their effects on nearby visible bodies. In 1915, Albert Einstein developed his theory of general relativity, only a few months later, Karl Schwarzschild found a solution to the Einstein field equations, which describes the gravitational field of a point mass and a spherical mass. A few months after Schwarzschild, Johannes Droste, a student of Hendrik Lorentz, independently gave the solution for the point mass. This solution had a peculiar behaviour at what is now called the Schwarzschild radius, the nature of this surface was not quite understood at the time
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White hole
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In general relativity, a white hole is a hypothetical region of spacetime which cannot be entered from the outside, although matter and light can escape from it. In this sense, it is the reverse of a black hole, white holes appear in the theory of eternal black holes. In addition to a black hole region in the future, such a solution of the Einstein field equations has a white hole region in its past. However, this region does not exist for black holes that have formed through gravitational collapse, no white hole has ever been observed. Like black holes, white holes have properties like mass, charge and they attract matter like any other mass, but objects falling towards a white hole would never actually reach the white holes event horizon. Imagine a gravitational field, without a surface, acceleration due to gravity is the greatest on the surface of any body. But since black holes lack a surface, acceleration due to gravity increases exponentially, in quantum mechanics, the black hole emits Hawking radiation and so can come to thermal equilibrium with a gas of radiation. Because a thermal-equilibrium state is time-reversal-invariant, Stephen Hawking argued that the reverse of a black hole in thermal equilibrium is again a black hole in thermal equilibrium. This may imply that black holes and white holes are the same object, the Hawking radiation from an ordinary black hole is then identified with the white-hole emission. The possibility of the existence of white holes was put forward by Russian cosmologist Igor Novikov in 1964, there is little evidence of white holes, though. All four regions can be seen in a diagram which uses Kruskal–Szekeres coordinates. When the infalling matter is added to a diagram of a black holes history. But because the equations of general relativity are time-reversible, general relativity must also allow the time-reverse of this type of black hole that forms from collapsing matter. The time-reversed case would be a hole that has existed since the beginning of the universe. A view of black holes first proposed in the late 1980s might be interpreted as shedding light on the nature of classical white holes. Some researchers have proposed that when a black hole forms, a big bang may occur at the core, torsion naturally accounts for the quantum-mechanical, intrinsic angular momentum of matter. According to general relativity, the collapse of a sufficiently compact mass forms a singular black hole. Such an interaction prevents the formation of a gravitational singularity, instead, the collapsing matter on the other side of the event horizon reaches an enormous but finite density and rebounds, forming a regular Einstein–Rosen bridge
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Spacetime
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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
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Space
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, in Isaac Newtons view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the visibility of spatial depth in his Essay Towards a New Theory of Vision. Kant referred to the experience of space in his Critique of Pure Reason as being a pure a priori form of intuition. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in space is conceived as curved. According to Albert Einsteins theory of relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a model for the shape of space. In the seventeenth century, the philosophy of space and time emerged as an issue in epistemology. At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac Newton, unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people, but since there would be no observational way of telling these universes apart then, according to the identity of indiscernibles, there would be no real difference between them. According to the principle of sufficient reason, any theory of space that implied that there could be two possible universes must therefore be wrong. Newton took space to be more than relations between objects and based his position on observation and experimentation
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Time
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Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future. Time is often referred to as the dimension, along with the three spatial dimensions. Time has long been an important subject of study in religion, philosophy, and science, nevertheless, diverse fields such as business, industry, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems. Two contrasting viewpoints on time divide prominent philosophers, one view is that time is part of the fundamental structure of the universe—a dimension independent of events, in which events occur in sequence. Isaac Newton subscribed to this realist view, and hence it is referred to as Newtonian time. This second view, in the tradition of Gottfried Leibniz and Immanuel Kant, holds that time is neither an event nor a thing, Time in physics is unambiguously operationally defined as what a clock reads. Time is one of the seven fundamental physical quantities in both the International System of Units and International System of Quantities, Time is used to define other quantities—such as velocity—so defining time in terms of such quantities would result in circularity of definition. The operational definition leaves aside the question there is something called time, apart from the counting activity just mentioned, that flows. Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy. Furthermore, it may be there is a subjective component to time. Temporal measurement has occupied scientists and technologists, and was a motivation in navigation. Periodic events and periodic motion have long served as standards for units of time, examples include the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, and the beat of a heart. Currently, the unit of time, the second, is defined by measuring the electronic transition frequency of caesium atoms. Time is also of significant social importance, having economic value as well as value, due to an awareness of the limited time in each day. In day-to-day life, the clock is consulted for periods less than a day whereas the calendar is consulted for periods longer than a day, increasingly, personal electronic devices display both calendars and clocks simultaneously. The number that marks the occurrence of an event as to hour or date is obtained by counting from a fiducial epoch—a central reference point. Artifacts from the Paleolithic suggest that the moon was used to time as early as 6,000 years ago. Lunar calendars were among the first to appear, either 12 or 13 lunar months, without intercalation to add days or months to some years, seasons quickly drift in a calendar based solely on twelve lunar months
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Minkowski diagram
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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
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Minkowski space
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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
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Closed timelike curve
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In mathematical physics, a closed timelike curve is a world line in a Lorentzian manifold, of a material particle in spacetime that is closed, returning to its starting point. When discussing the evolution of a system in general relativity, or more specifically Minkowski space, a light cone represents any possible future evolution of an object given its current state, or every possible location given its current location. An objects possible future locations are limited by the speed that the object can move, for instance, an object located at position p at time t0 can only move to locations within p + c by time t1. This is commonly represented on a graph with physical locations along the axis and time running vertically, with units of t for time. Light cones in this representation appear as lines at 45 degrees centered on the object, on such a diagram, every possible future location of the object lies within the cone. Additionally, every location has a future time, implying that an object may stay at any location in space indefinitely. Any single point on such a diagram is known as an event, separate events are considered to be timelike if they are separated across the time axis, or spacelike if they differ along the space axis. If the object were in free fall, it would travel up the t-axis, if it accelerates, the actual path an object takes through spacetime, as opposed to the ones it could take, is known as the worldline. Another definition is that the light cone represents all possible worldlines, in simple examples of spacetime metrics the light cone is directed forward in time. This corresponds to the case that an object cannot be in two places at once, or alternately that it cannot move instantly to another location. In these spacetimes, the worldlines of physical objects are, by definition, however this orientation is only true of locally flat spacetimes. In curved spacetimes the light cone will be tilted along the spacetimes geodesic, for instance, while moving in the vicinity of a star, the stars gravity will pull on the object, affecting its worldline, so its possible future positions lie closer to the star. This appears as a slightly tilted lightcone on the spacetime diagram. In extreme examples, in spacetimes with suitably high-curvature metrics, the cone can be tilted beyond 45 degrees. That means there are potential future positions, from the frame of reference. From this outside viewpoint, the object can move instantaneously through space, in these situations the object would have to move, since its present spatial location would not be in its own future light cone. Additionally, with enough of a tilt, there are event locations that lie in the past as seen from the outside, with a suitable movement of what appears to it its own space axis, the object appears to travel though time as seen externally. An object in such an orbit would repeatedly return to the point in spacetime if it stays in free fall