In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz. For example, the following laws and theories respect Lorentz symmetry: The kinematical laws of special relativity Maxwell's field equations in the theory of electromagnetism The Dirac equation in the theory of the electron The Standard model of particle physicsThe Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In general relativity physics, in cases involving small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as that of special relativity physics; the Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are isometries that leave the origin fixed.
Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations. Mathematically, the Lorentz group may be described as the generalized orthogonal group O, the matrix Lie group that preserves the quadratic form ↦ t 2 − x 2 − y 2 − z 2 on R4; this quadratic form is, when put on matrix form, interpreted in physics as the metric tensor of Minkowski spacetime. The Lorentz group is a six-dimensional noncompact non-abelian real Lie group, not connected; the four connected components are not connected, but rather doubly connected. The identity component of the Lorentz group is itself a group, is called the restricted Lorentz group, is denoted SO+; the restricted Lorentz group consists of those Lorentz transformations that preserve the orientation of space and direction of time.
The restricted Lorentz group has been presented through a facility of biquaternion algebra. The restricted Lorentz group arises in other ways in pure mathematics. For example, it arises as the point symmetry group of a certain ordinary differential equation; this fact has physical significance. Because it is a Lie group, the Lorentz group O is both a group and admits a topological description as a smooth manifold; as a manifold, it has four connected components. Intuitively, this means; the four connected components can be categorized by two transformation properties its elements have: some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing timelike vector would be inverted to a past-pointing vector some elements have orientation reversed by improper Lorentz transformations, for example, certain vierbein Lorentz transformations that preserve the direction of time are called orthochronous. The subgroup of orthochronous transformations is denoted O+.
Those that preserve orientation are called proper, as linear transformations they have determinant +1. The subgroup of proper Lorentz transformations is denoted SO; the subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, is denoted by SO+. The set of the four connected components can be given a group structure as the quotient group O/SO+, isomorphic to the Klein four-group; every element in O can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group where P and T are the space inversion and time reversal operators: P = diag T = diag. Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two bits of information, which pick out one of the four connected components; this pattern is typical of finite-dimensional Lie groups. The restricted Lorentz group is the identity component of the Lorentz group, which means that it consists of all Lorentz transformations that can be connected to the identity by a continuous curve lying in the group.
The restricted Lorentz group is a connected normal subgroup of the full Lorentz group with the same dimension, in this case with dimension six. The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts. Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation and a boost, it takes 6 real parameters to specify an arbitrary proper orthochronous Lorentz transformation; this is one way. The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO; the set of all boosts, does not form a sub
In astronomy, perturbation is the complex motion of a massive body subject to forces other than the gravitational attraction of a single other massive body. The other forces can include a third body, resistance, as from an atmosphere, the off-center attraction of an oblate or otherwise misshapen body; the study of perturbations began with the first attempts to predict planetary motions in the sky. In ancient times the causes were a mystery. Newton, at the time he formulated his laws of motion and of gravitation, applied them to the first analysis of perturbations, recognizing the complex difficulties of their calculation. Many of the great mathematicians since have given attention to the various problems involved; the complex motions of gravitational perturbations can be broken down. The hypothetical motion that the body follows under the gravitational effect of one other body only is a conic section, can be described with the methods of geometry; this is called an unperturbed Keplerian orbit.
The differences between that and the actual motion of the body are perturbations due to the additional gravitational effects of the remaining body or bodies. If there is only one other significant body the perturbed motion is a three-body problem. A general analytical solution exists for the two-body problem; the two-body problem becomes insoluble if one of the bodies is irregular in shape. Most systems that involve multiple gravitational attractions present one primary body, dominant in its effects; the gravitational effects of the other bodies can be treated as perturbations of the hypothetical unperturbed motion of the planet or satellite around its primary body. In methods of general perturbations, general differential equations, either of motion or of change in the orbital elements, are solved analytically by series expansions; the result is expressed in terms of algebraic and trigonometric functions of the orbital elements of the body in question and the perturbing bodies. This can be applied to many different sets of conditions, is not specific to any particular set of gravitating objects.
General perturbations were investigated first. The classical methods are known as variation of the elements, variation of parameters or variation of the constants of integration. In these methods, it is considered that the body is always moving in a conic section, however the conic section is changing due to the perturbations. If all perturbations were to cease at any particular instant, the body would continue in this conic section indefinitely. General perturbations takes advantage of the fact that in many problems of celestial mechanics, the two-body orbit changes rather due to the perturbations. General perturbations is applicable only if the perturbing forces are about one order of magnitude smaller, or less, than the gravitational force of the primary body. In the Solar System, this is the case. General perturbation methods are preferred for some types of problems, as the source of certain observed motions are found; this is not so for special perturbations. In methods of special perturbations, numerical datasets, representing values for the positions and accelerative forces on the bodies of interest, are made the basis of numerical integration of the differential equations of motion.
In effect, the positions and velocities are perturbed directly, no attempt is made to calculate the curves of the orbits or the orbital elements. Special perturbations can be applied to any problem in celestial mechanics, as it is not limited to cases where the perturbing forces are small. Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generated planetary ephemerides of the great astronomical almanacs. Special perturbations are used for modeling an orbit with computers. Cowell's formulation is the simplest of the special perturbation methods. In a system of n mutually interacting bodies, this method mathematically solves for the Newtonian forces on body i by summing the individual interactions from the other j bodies: r ¨ i = ∑ j = 1 j ≠ i n G m j r i j
Physical cosmology is a branch of cosmology concerned with the studies of the largest-scale structures and dynamics of the Universe and with fundamental questions about its origin, structure and ultimate fate. Cosmology as a science originated with the Copernican principle, which implies that celestial bodies obey identical physical laws to those on Earth, Newtonian mechanics, which first allowed us to understand those physical laws. Physical cosmology, as it is now understood, began with the development in 1915 of Albert Einstein's general theory of relativity, followed by major observational discoveries in the 1920s: first, Edwin Hubble discovered that the universe contains a huge number of external galaxies beyond our own Milky Way; these advances made it possible to speculate about the origin of the universe, allowed the establishment of the Big Bang Theory, by Georges Lemaître, as the leading cosmological model. A few researchers still advocate a handful of alternative cosmologies. Dramatic advances in observational cosmology since the 1990s, including the cosmic microwave background, distant supernovae and galaxy redshift surveys, have led to the development of a standard model of cosmology.
This model requires the universe to contain large amounts of dark matter and dark energy whose nature is not well understood, but the model gives detailed predictions that are in excellent agreement with many diverse observations. Cosmology draws on the work of many disparate areas of research in theoretical and applied physics. Areas relevant to cosmology include particle physics experiments and theory and observational astrophysics, general relativity, quantum mechanics, plasma physics. Modern cosmology developed along tandem tracks of observation. In 1916, Albert Einstein published his theory of general relativity, which provided a unified description of gravity as a geometric property of space and time. At the time, Einstein believed in a static universe, but found that his original formulation of the theory did not permit it; this is because masses distributed throughout the universe gravitationally attract, move toward each other over time. However, he realized that his equations permitted the introduction of a constant term which could counteract the attractive force of gravity on the cosmic scale.
Einstein published his first paper on relativistic cosmology in 1917, in which he added this cosmological constant to his field equations in order to force them to model a static universe. The Einstein model describes a static universe. However, this so-called Einstein model is unstable to small perturbations—it will start to expand or contract, it was realized that Einstein's model was just one of a larger set of possibilities, all of which were consistent with general relativity and the cosmological principle. The cosmological solutions of general relativity were found by Alexander Friedmann in the early 1920s, his equations describe the Friedmann–Lemaître–Robertson–Walker universe, which may expand or contract, whose geometry may be open, flat, or closed. In the 1910s, Vesto Slipher interpreted the red shift of spiral nebulae as a Doppler shift that indicated they were receding from Earth. However, it is difficult to determine the distance to astronomical objects. One way is to compare the physical size of an object to its angular size, but a physical size must be assumed to do this.
Another method is to measure the brightness of an object and assume an intrinsic luminosity, from which the distance may be determined using the inverse square law. Due to the difficulty of using these methods, they did not realize that the nebulae were galaxies outside our own Milky Way, nor did they speculate about the cosmological implications. In 1927, the Belgian Roman Catholic priest Georges Lemaître independently derived the Friedmann–Lemaître–Robertson–Walker equations and proposed, on the basis of the recession of spiral nebulae, that the universe began with the "explosion" of a "primeval atom"—which was called the Big Bang. In 1929, Edwin Hubble provided an observational basis for Lemaître's theory. Hubble showed that the spiral nebulae were galaxies by determining their distances using measurements of the brightness of Cepheid variable stars, he discovered a relationship between the redshift of its distance. He interpreted this as evidence that the galaxies are receding from Earth in every direction at speeds proportional to their distance.
This fact is now known as Hubble's law, though the numerical factor Hubble found relating recessional velocity and distance was off by a factor of ten, due to not knowing about the types of Cepheid variables. Given the cosmological principle, Hubble's law suggested. Two primary explanations were proposed for the expansion. One was Lemaître's Big Bang theory and developed by George Gamow; the other explanation was Fred Hoyle's steady state model in which new matter is created as the galaxies move away from each other. In this model, the universe is the same at any point in time. For a number of years, support for these theories was evenly divided. However, the observational evidence began to support the idea that the universe evolved from a hot dense state; the discovery of the cosmic microwave background in 1965 lent strong support to the Big Bang model, since the precise measurements of the cosmic microwave background by the Cosmic Background Explorer in the early 1990s, few cosmologists ha
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 gravity. Gravitational collapse is a fundamental mechanism for structure formation in the universe. Over time an initial smooth distribution of matter will collapse to form pockets of higher density creating a hierarchy of condensed structures such as clusters of galaxies, stellar groups and planets. A star is born through the gradual gravitational collapse of a cloud of interstellar matter; the compression caused by the collapse raises the temperature until thermonuclear fusion occurs at the center of the star, at which point the collapse comes to a halt as the outward thermal pressure balances the gravitational forces. The star 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 hydrostatic 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 virial theorem, which states that, to maintain equilibrium, the gravitational potential energy must equal twice the internal thermal energy. If a pocket of gas is massive enough that the gas pressure is insufficient to support it, the cloud will undergo gravitational collapse; the mass above which a cloud will undergo such collapse is called the Jeans mass. This mass depends on the temperature and density of the cloud, but is 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 only if it reaches a new state of equilibrium. Depending on the mass during its lifetime, these stellar remnants can take one of three forms: White dwarfs, in which gravity is opposed by electron degeneracy pressure Neutron stars, in which gravity is opposed by neutron degeneracy pressure and short-range repulsive neutron–neutron interactions mediated by the strong force Black hole, in which there is no force strong enough to resist gravitational collapse The collapse of the stellar core to a white dwarf takes place over tens of thousands of years, while the star blows off its outer envelope to form a planetary nebula.
If it has a companion star, a white dwarf-sized object can accrete matter from the companion star. Before it reaches the Chandrasekhar limit, the increasing density and temperature within a carbon-oxygen white dwarf initiates a new round of nuclear fusion, not regulated because the star's weight is supported by degeneracy rather than thermal pressure, allowing temperature to rise exponentially; the resulting runaway carbon detonation blows the star apart in a Type Ia supernova. Neutron stars are formed by gravitational collapse of the cores of larger stars, are the remnant of other types of supernova, they are so compact that a Newtonian description is inadequate for an accurate treatment, which requires the use of Einstein's general relativity. According to Einstein's theory, for larger stars, above the Landau-Oppenheimer-Volkoff limit known as the Tolman–Oppenheimer–Volkoff limit no known form of cold matter can provide the force needed to oppose gravity in a new dynamical equilibrium. 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, meaning a space-time region from which not light can escape. It follows from a theorem of Roger Penrose that the subsequent formation of some kind of singularity is inevitable. According to Penrose's cosmic censorship hypothesis, the singularity will be confined within the event horizon bounding the black hole, so the space-time region outside will still have a well behaved geometry, with strong but finite curvature, expected to evolve towards a rather simple form describable by the historic Schwarzschild metric in the spherical limit and by the more discovered Kerr metric if angular momentum is present. On the other hand, the nature of the kind of singularity to be expected inside a black 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. There are competing theories as to what occurs at this point, but it can no longer be considered gravitational collapse at that stage.
The radii of larger mass neutron stars are estimated to be about 12-km, or 2.0 times their equivalent Schwarzschild radius. It might be thought that a sufficiently massive neutron star could exist within its Schwarzschild radius and appear like a black hole without having all the mass compressed to a singularity at the center. Within the event horizon, matter would have to move outward faster than the speed of light in order to remain stable and avoid collapsing to the center. No physical force therefore can prevent a star smaller than 1.0 SR from collapsing to a singularity. A model for nonspherical collapse in general relativity with emission of matter and gravitational waves has been presented. Big Crunch Gravitational compression Stellar evolution Thermal runawa
Astrophysics is the branch of astronomy that employs the principles of physics and chemistry "to ascertain the nature of the astronomical objects, rather than their positions or motions in space". Among the objects studied are the Sun, other stars, extrasolar planets, the interstellar medium and the cosmic microwave background. Emissions from these objects are examined across all parts of the electromagnetic spectrum, the properties examined include luminosity, density and chemical composition; because astrophysics is a broad subject, astrophysicists apply concepts and methods from many disciplines of physics, including mechanics, statistical mechanics, quantum mechanics, relativity and particle physics, atomic and molecular physics. In practice, modern astronomical research involves a substantial amount of work in the realms of theoretical and observational physics; some areas of study for astrophysicists include their attempts to determine the properties of dark matter, dark energy, black holes.
Topics studied by theoretical astrophysicists include Solar System formation and evolution. Astronomy is an ancient science, long separated from the study of terrestrial physics. In the Aristotelian worldview, bodies in the sky appeared to be unchanging spheres whose only motion was uniform motion in a circle, while the earthly world was the realm which underwent growth and decay and in which natural motion was in a straight line and ended when the moving object reached its goal, it was held that the celestial region was made of a fundamentally different kind of matter from that found in the terrestrial sphere. During the 17th century, natural philosophers such as Galileo and Newton began to maintain that the celestial and terrestrial regions were made of similar kinds of material and were subject to the same natural laws, their challenge was. For much of the nineteenth century, astronomical research was focused on the routine work of measuring the positions and computing the motions of astronomical objects.
A new astronomy, soon to be called astrophysics, began to emerge when William Hyde Wollaston and Joseph von Fraunhofer independently discovered that, when decomposing the light from the Sun, a multitude of dark lines were observed in the spectrum. By 1860 the physicist, Gustav Kirchhoff, the chemist, Robert Bunsen, had demonstrated that the dark lines in the solar spectrum corresponded to bright lines in the spectra of known gases, specific lines corresponding to unique chemical elements. Kirchhoff deduced that the dark lines in the solar spectrum are caused by absorption by chemical elements in the Solar atmosphere. In this way it was proved that the chemical elements found in the Sun and stars were found on Earth. Among those who extended the study of solar and stellar spectra was Norman Lockyer, who in 1868 detected bright, as well as dark, lines in solar spectra. Working with the chemist, Edward Frankland, to investigate the spectra of elements at various temperatures and pressures, he could not associate a yellow line in the solar spectrum with any known elements.
He thus claimed the line represented a new element, called helium, after the Greek Helios, the Sun personified. In 1885, Edward C. Pickering undertook an ambitious program of stellar spectral classification at Harvard College Observatory, in which a team of woman computers, notably Williamina Fleming, Antonia Maury, Annie Jump Cannon, classified the spectra recorded on photographic plates. By 1890, a catalog of over 10,000 stars had been prepared that grouped them into thirteen spectral types. Following Pickering's vision, by 1924 Cannon expanded the catalog to nine volumes and over a quarter of a million stars, developing the Harvard Classification Scheme, accepted for worldwide use in 1922. In 1895, George Ellery Hale and James E. Keeler, along with a group of ten associate editors from Europe and the United States, established The Astrophysical Journal: An International Review of Spectroscopy and Astronomical Physics, it was intended that the journal would fill the gap between journals in astronomy and physics, providing a venue for publication of articles on astronomical applications of the spectroscope.
Around 1920, following the discovery of the Hertsprung-Russell diagram still used as the basis for classifying stars and their evolution, Arthur Eddington anticipated the discovery and mechanism of nuclear fusion processes in stars, in his paper The Internal Constitution of the Stars. At that time, the source of stellar energy was a complete mystery; this was a remarkable development since at that time fusion and thermonuclear energy, that stars are composed of hydrogen, had not yet been discovered. In 1
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 Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present; the relation is specified by the Einstein field equations, a system of partial differential equations. Some predictions of general relativity differ from those of classical physics concerning the passage of time, the geometry of space, the motion of bodies in free fall, the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the gravitational time delay; the predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date.
Although general relativity is not the only relativistic theory of gravity, it is the simplest theory, consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity. Einstein's 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 light, can escape—as an end-state for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes; the bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity predicts the existence of gravitational waves, which have since been observed directly by the physics collaboration LIGO.
In addition, general relativity is the basis of current cosmological models of a expanding universe. Acknowledged as a theory of extraordinary beauty, general relativity has been described as the most beautiful of all existing physical theories. 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 simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations; these equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, form the core of Einstein's general theory of relativity. The Einstein field equations are nonlinear and 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 final stages of gravitational collapse, the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, which resulted in the Reissner–Nordström solution, now associated with electrically charged black holes. In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, the work of Hubble and others had shown that our universe is expanding; this is 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 hot and dense earlier state. Einstein declared the cosmological constant the biggest blunder of his life. During that period, general relativity remained something of a curiosity among physical theories, it was superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein himself had shown in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters. A 1919 expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of May 29, 1919, making Einstein famous, yet the theory entered the mainstream of theoretical physics and astrophysics only with the developments between 1960 and 1975, now known as the golden age of general relativity. Physicists began to understand the concept of a black hole, to identify quasars as one of these objects' astrophysical manifestations.
More precise solar system tests confirmed the theory's predictive power, relativistic cosmology, became amenable to direct observational tests. Over the years, general relativity has acqui