"causal future of the world line is also in its causal past, a situation which is akin to time travel. Such a solution was first proposed by Kurt Gödel, a solution known as the Gödel metric, but his (and others') solution requires the universe to have physical characteristics that it does not appear to have, such as rotation and lack of Hubble expansion. Whether general relativity forbids closed time-like curves for all realistic conditions is still being researched. Wormholes are a hypothetical warped spacetime which are permitted by the Einstein field equations of general relativity. A proposed time-travel machine using"
"Gödel metric The Gödel metric is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles (dust solution), and the second associated with a nonzero cosmological constant (see lambdavacuum solution). It is also known as the Gödel solution or Gödel universe. This solution has many unusual properties—in particular, the existence of closed timelike curves that would allow time travel in a universe described by the solution. Its definition is somewhat artificial in that the value of the cosmological constant must"
"Goldberg–Sachs theorem The Goldberg–Sachs theorem is a result in Einstein's theory of general relativity about vacuum solutions of the Einstein field equations relating the existence of a certain type of congruence with algebraic properties of the Weyl tensor. More precisely, the theorem states that ""a vacuum solution of the Einstein field equations will admit a shear-free null geodesic congruence if and only if the Weyl tensor is algebraically special."" The theorem is often used when searching for algebraically special vacuum solutions. It has been shown by Dain and Moreschi (2000) that a corresponding theorem will not hold in linearized gravity,"
"is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang singularity. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed (cf. the section on quantum gravity, below). Kurt Gödel showed that solutions to Einstein's equations exist that contain closed timelike curves (CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR"
"that is, given a solution of the linearised Einstein field equations admitting a shear-free null congruence, then this solution need not be algebraically special. Goldberg–Sachs theorem The Goldberg–Sachs theorem is a result in Einstein's theory of general relativity about vacuum solutions of the Einstein field equations relating the existence of a certain type of congruence with algebraic properties of the Weyl tensor. More precisely, the theorem states that ""a vacuum solution of the Einstein field equations will admit a shear-free null geodesic congruence if and only if the Weyl tensor is algebraically special."" The theorem is often used when searching"
"to partially cancel the matter density of the dust. The Gödel spacetime is a rare example of a ""regular"" (singularity-free) solution of the Einstein field equation. Gödel's original chart (given here) is geodesically complete and singularity free; therefore, it is a global chart, and the spacetime is homeomorphic to R, and therefore, simply connected. In any Lorentzian spacetime, the fourth-rank Riemann tensor is a multilinear operator on the four-dimensional space of tangent vectors (at some event), but a linear operator on the six-dimensional space of bivectors at that event. Accordingly, it has a characteristic polynomial, whose roots are the eigenvalues."
"to solve the geodesic equation, an approach adopted by Einstein and others. The Schwarzschild metric may be written as where the two functions ""w(r)"" = (1 - ""r""/""r"") and its inverse ""v(r)"" = 1/""w(r)"" are defined for brevity. From this metric, the Christoffel symbols Γ may be calculated, and the results substituted into the geodesic equations It may be verified that θ=π/2 is a valid solution by substitution into the first of these four equations. By symmetry, the orbit must be planar, and we are free to arrange the coordinate frame so that the equatorial plane is the plane of"
"Non-exact solutions in general relativity Non-exact solutions in general relativity are solutions of Albert Einstein's field equations of general relativity which hold only approximately. These solutions are typically found by treating the gravitational field, formula_1, as a background space-time, formula_2, (which is usually an exact solution) plus some small perturbation, formula_3. Then one is able to solve the Einstein field equations as a series in formula_3, dropping higher order terms for simplicity. A common example of this method results in the linearised Einstein field equations. In this case we expand the full space-time metric about the flat Minkowski metric, formula_5:"
"cut Gödel off, and moved the hearing on to other questions and a routine conclusion. Gödel became a permanent member of the Institute for Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976. During his many years at the Institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving closed timelike curves, to Einstein's field equations in general relativity. He is said to have given this elaboration to Einstein"
"Al Momin argues that Kurt Gödel's solution to these equations do not describe our universe and are therefore approximations. There are practical as well as theoretical reasons for studying solutions of the Einstein field equations. From a purely mathematical viewpoint, it is interesting to know the set of solutions of the Einstein field equations. Some of these solutions are parametrised by one or more parameters. Solutions of the Einstein field equations Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations (EFE) of general relativity. Solving the field equations actually gives a Lorentz manifold."
"laws governing whatever matter might be present. Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless, a number of exact solutions are known, although only a few have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner–Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe, and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes, each describing an expanding cosmos. Exact solutions of great theoretical interest include the Gödel universe (which"
"Gauss–Seidel method In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a linear system of equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in"
"may be derived from the Einstein equations. Since the equations of general relativity are non-linear, a lump of energy made out of pure gravitational fields, like a black hole, would move on a trajectory which is determined by the Einstein equations themselves, not by a new law. So Einstein proposed that the path of a singular solution, like a black hole, would be determined to be a geodesic from general relativity itself. This was established by Einstein, Infeld, and Hoffmann for pointlike objects without angular momentum, and by Roy Kerr for spinning objects. In a 1905 paper, Einstein postulated that"
"Solutions of the Einstein field equations Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations (EFE) of general relativity. Solving the field equations actually gives a Lorentz manifold. Solutions are broadly classed as ""exact"" or ""non-exact"". The Einstein field equations are or more generally, if one allows a nonzero cosmological constant, where formula_3 is a constant, and the Einstein tensor on the left side of the equation is equated to the stress–energy tensor representing the energy and momentum present in the spacetime. The Einstein tensor is built up from the metric tensor and"
"and dropping all terms which are of second or higher order in formula_3. Non-exact solutions in general relativity Non-exact solutions in general relativity are solutions of Albert Einstein's field equations of general relativity which hold only approximately. These solutions are typically found by treating the gravitational field, formula_1, as a background space-time, formula_2, (which is usually an exact solution) plus some small perturbation, formula_3. Then one is able to solve the Einstein field equations as a series in formula_3, dropping higher order terms for simplicity. A common example of this method results in the linearised Einstein field equations. In this"
"as a present for his 70th birthday. His ""rotating universes"" would allow time travel to the past and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric (an exact solution of the Einstein field equation). He studied and admired the works of Gottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed. To a lesser extent he studied Immanuel Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of Anselm of Canterbury's ontological proof"
"Theoretical motivation for general relativity A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation ""a priori"". This provides a means to inform and verify the formalism. General relativity addresses two questions: The former question is answered with the geodesic equation. The second question is answered with the Einstein field equation. The geodesic"
"Gross–Pitaevskii equation The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross ) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model. In the Hartree–Fock approximation the total wave-function formula_1 of the system of formula_2 bosons is taken as a product of single-particle functions formula_3, where formula_5 is the coordinate of the formula_6-th boson. The pseudopotential model Hamiltonian of the system is given as where formula_8 is the mass of the boson, formula_9 is the external potential, formula_10 is the boson-boson scattering length, and formula_11 is the Dirac delta-function. If"
"the Euler–Lagrange equation is Multiplying by the inverse metric tensor formula_36 we get that Thus we get the geodesic equation: with the Christoffel symbol defined in terms of the metric tensor as Albert Einstein believed that the geodesic equation of motion can be derived from the field equations for empty space, i.e. from the fact that the Ricci curvature vanishes. He wrote: It has been shown that this law of motion — generalized to the case of arbitrarily large gravitating masses — can be derived from the field equations of empty space alone. According to this derivation the law of"
"by the kinetics and dynamics of a particle orbiting the earth in a circular orbit. They are true in general. Solving the Einstein field equation requires an iterative process. The solution is represented in the metric tensor Typically there is an initial guess for the tensor. The guess is used to calculate Christoffel symbols, which are used to calculate the curvature. If the Einstein field equation is not satisfied, the process is repeated. Solutions occur in two forms, vacuum solutions and non-vacuum solutions. A vacuum solution is one in which the stress–energy tensor is zero. The relevant vacuum solution for"
"mechanics together, was found by all those who discovered what is frequently called the Klein–Gordon equation: by inserting the energy operator and momentum operator into the relativistic energy–momentum relation: The solutions to () are scalar fields. The KG equation is undesirable due to its prediction of ""negative"" energies and probabilities, as a result of the quadratic nature of () – inevitable in a relativistic theory. This equation was initially proposed by Schrödinger, and he discarded it for such reasons, only to realize a few months later that its non-relativistic limit (what is now called the Schrödinger equation) was still of"
"to the non-linear Schrödinger equation. Considering the simplified solutions of the form we obtain the KdV equation as or Integrating and taking the special case in which the integration constant is zero, we have: which is the formula_43 special case of the generalized stationary Gross–Pitaevskii equation (GPE) Therefore, for the certain class of solutions of generalized GPE (formula_45 for the true one-dimensional condensate and formula_46 while using the three dimensional equation in one dimension), two equations are one. Furthermore, taking the formula_47 case with the minus sign and the formula_1 real, one obtains an attractive self-interaction that should yield a"
"easily compatible with models that are supposed to represent real systems, which are described by means of measurements containing experimental errors and are normally expressed as initial-boundary value problems in a Hilbert space. At least three methods have been reported to obtain the boundary functions ""g"", ""g"" that are compatible with any lateral set of conditions {""f"", ""f""} imposed. This makes it possible to find the analytical solution of any PDE boundary problem on a closed rectangle with the required accuracy, so allowing to solve a wide range of problems that the standard Adomian's method was not able to address."
"opens up the intriguing possibility of time travel in curved spacetimes), the Taub-NUT solution (a model universe that is homogeneous, but anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture). Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes."
"Galois group over the rationals in other ways). The inverse Galois problem remains unsolved. Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity, a geometrical theory of gravitation developed mainly by Albert Einstein. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, because gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics, with Noether's first theorem, which she proved in 1915, but did not publish"
"Schrödinger equation, it has been found that a linear combination of Gaussons can very accurately approximate excited states as well. Gausson (physics) Gausson is a soliton which is the solution of the Schrödinger equation with the logarithmic nonlinearity which describes a quantum particle in the possible nonlinear quantum mechanics but still preserving the dimensional homogeneity of the equation i.e. the property that the product of the independent solutions in one dimension remains the solution in two dimensions etc. despite of the nonlinearity or that the nonlinearity alone cannot cause the quantum entanglement between dimensions without the interaction and therefore it"
"The idea was further developed by Tetrode and Fokker in the 1920s and Wheeler and Feynman in the 1940s and constitutes an alternative/equivalent formulation of electrodynamics. Einstein himself was pleasantly surprised to learn that the field equations admitted exact solutions, because of their ""prima facie"" complexity, and because he himself had only produced an approximate solution. Einstein's approximate solution was given in his famous 1915 article on the advance of the perihelion of Mercury. There, Einstein used rectangular coordinates to approximate the gravitational field around a spherically symmetric, non-rotating, non-charged mass. Schwarzschild, in contrast, chose a more elegant ""polar-like"" coordinate"
"is ""d"" − 2. In 2 + 1 dimensions, where the exponent is zero, the analogous solution starts with 2 + 1 de Sitter space, cuts out a wedge, and pastes the two sides of the wedge together to make a conical space. The geodesic equation gives for the radial, and for the time component. de Sitter space is the simplest solution of Einstein's equation with a positive cosmological constant. It is spherically symmetric and it has a cosmological horizon surrounding any observer, and describes an inflating universe. The Schwarzschild solution is the simplest spherically symmetric solution of the Einstein"
"those equations between 1861 and 1862. The Einstein field equations (EFE; also known as ""Einstein's equations"") are a set of ten partial differential equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. First published by Einstein in 1915 as a tensor equation, the EFE equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor). In quantum mechanics, the analogue of Newton's law is Schrödinger's equation (a partial differential equation) for"
"of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional"
"is, up to local isometry, the ""only"" perfect fluid solution of the Einstein field equation admitting a five-dimensional Lie algebra of Killing vectors. The Weyl tensor of the Gödel solution has Petrov type D. This means that for an appropriately chosen observer, the tidal forces have ""Coulomb form"". To study the tidal forces in more detail, we compute the Bel decomposition of the Riemann tensor into three pieces, the tidal or electrogravitic tensor (which represents tidal forces), the magnetogravitic tensor (which represents spin-spin forces on spinning test particles and other gravitational effects analogous to magnetism), and the topogravitic tensor (which"
"Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the principle of least action. With the metric signature, the gravitational part of the action is given as where formula_2 is the determinant of the metric tensor matrix, formula_3 is the Ricci scalar, and formula_4 is Einstein's constant (formula_5 is the gravitational constant and formula_6 is the speed of light in vacuum). If it converges, the integral is taken over the whole spacetime. If it does not converge, formula_7 is no longer well-defined, but a modified"
"exist classical solutions to the Einstein equations that allow values of the entropy larger than those allowed by an area law, hence in principle larger than those of a black hole. These are the so-called ""Wheeler's bags of gold"". The existence of such solutions conflicts with the holographic interpretation, and their effects in a quantum theory of gravity including the holographic principle are not yet fully understood. The anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) which are"