Exact solutions in general relativityIn general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field.
Fluid solutionIn general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. In astrophysics, fluid solutions are often employed as stellar models. (It might help to think of a perfect gas as a special case of a perfect fluid.) In cosmology, fluid solutions are often used as cosmological models.
IntegralIn mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.
Integral equationIn mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: where is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.
Dust solutionIn general relativity, a dust solution is a fluid solution, a type of exact solution of the Einstein field equation, in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid that has positive mass density but vanishing pressure. Dust solutions are an important special case of fluid solutions in general relativity. A pressureless perfect fluid can be interpreted as a model of a configuration of dust particles that locally move in concert and interact with each other only gravitationally, from which the name is derived.
Electrovacuum solutionIn general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the (curved-spacetime) source-free Maxwell equations appropriate to the given geometry. For this reason, electrovacuums are sometimes called (source-free) Einstein–Maxwell solutions.
Contour integrationIn the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. Contour integration methods include: direct integration of a complex-valued function along a curve in the complex plane; application of the Cauchy integral formula; and application of the residue theorem.
AcousticsAcoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of modern society with the most obvious being the audio and noise control industries.
Kerr metricThe Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find. The Kerr metric is a generalization to a rotating body of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically symmetric, and non-rotating body.
Multiple integralIn mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in (real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
