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Distorted Schwarzschild Metric
In physics, the distorted Schwarzschild metric is the metric of a standard/isolated Schwarzschild spacetime exposed in external fields. In numerical simulation, the Schwarzschild metric can be distorted by almost arbitrary kinds of external energy–momentum distribution. However, in exact analysis, the mature method to distort the standard Schwarzschild metric is restricted to the framework of Weyl metrics. Standard Schwarzschild as a vacuum Weyl metric All static axisymmetric solutions of the Einstein–Maxwell equations can be written in the form of Weyl's metric,Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 10. (1)\quad ds^2=-e^dt^2+e^(d\rho^2+dz^2)+e^\rho^2 d\phi^2\,, From the Weyl perspective, the metric potentials generating the standard Schwarzschild solution are given byR Gautreau, R B Hoffman, A Armenti. ''Static multiparticle systems in general relativity''. IL NU ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarzschild Metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916, and around the same time independently by Johannes Droste, who published his more complete and modern-looking discussion four months after Schwarzschild. According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum. A Sch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress–energy Tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity. Definition The stress–energy tensor involves the use of superscripted variables (''not'' exponents; see tensor index notation and Einstein summation notation). If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: , , , and , where ''t'' is time in seconds, and ''x'', ''y'', and ''z'' are distances in meters. The stress–energy tensor is defined as the tensor ''T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Metrics
In general relativity, the Weyl metrics (named after the German-American mathematician Hermann Weyl) are a class of ''static'' and ''axisymmetric'' solutions to Einstein's field equation. Three members in the renowned Kerr–Newman family solutions, namely the Schwarzschild, nonextremal Reissner–Nordström and extremal Reissner–Nordström metrics, can be identified as Weyl-type metrics. Standard Weyl metrics The Weyl class of solutions has the generic formJeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 10.Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt. ''Exact Solutions of Einstein's Field Equations''. Cambridge: Cambridge University Press, 2003. Chapter 20. where \psi(\rho,z) and \gamma(\rho,z) are two metric potentials dependent on ''Weyl's canonical coordinates'' \. The coordinate system \ serves best for symmetries of Weyl's spacet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein–Maxwell Equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form of a tensor equation which related the local ' (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h. This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measured from a fixed zenith direction, and the '' azimuthal angle'' of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is also called the ''radius'' or ''radial coordinate''. The polar angle may be called '' colatitude'', '' zenith angle'', '' normal angle'', or '' inclination angle''. When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called spherical polar coordinates. The use of symbols and the order of the coordinates differs among sources and disciplines. This article wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series Solution Of Differential Equations
In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients. Method Consider the second-order linear differential equation a_2(z)f''(z)+a_1(z)f'(z)+a_0(z)f(z)=0. Suppose is nonzero for all . Then we can divide throughout to obtain f''+f'+f=0. Suppose further that and are analytic functions. The power series method calls for the construction of a power series solution f=\sum_^\infty A_kz^k. If is zero for some , then the Frobenius method, a variation on this method, is suited to deal with so called " singular points". The method works analogously for higher order equations as well as for systems. Example usage Let us look at the Hermite differential equation, f''-2zf'+\lambda f=0; \; \lambda=1 We can try to construct a serie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multipole Expansion
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on \R^3, or less often on \R^n for some other Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space. The multipole expansion is expressed as a sum of terms with progressively finer angula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions. Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval 1,1/math>. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional standardization condi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multipole Moment
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on \R^3, or less often on \R^n for some other Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space. The multipole expansion is expressed as a sum of terms with progressively finer angular features ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
