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Perfect fluids are used in general relativity to model idealized distributions of matter, such as the interior of a star or an isotropic universe. In the latter case, the equation of state of the perfect fluid may be used in Friedmann–Lemaître–Robertson–Walker equations to describe the evolution of the universe.
Let us assume a static, spherically symmetric perfect fluid. The metric components are similar to those for the Schwarzschild metric: [2] = = By the perfect fluid assumption, the stress-energy tensor is diagonal (in the central spherical coordinate system), with eigenvalues of energy density and pressure:
A scalar field can be viewed as a sort of perfect fluid with equation of state = ˙ ˙ + (), where ˙ is the time-derivative of and () is the potential energy. A free ( V = 0 {\displaystyle V=0} ) scalar field has w = 1 {\displaystyle w=1} , and one with vanishing kinetic energy is equivalent to a cosmological constant: w = − 1 {\displaystyle ...
Fluid solutions: must arise entirely from the stress–energy tensor of a fluid (often taken to be a perfect fluid); the only source for the gravitational field is the energy, momentum, and stress (pressure and shear stress) of the matter comprising the fluid.
Liquid: A mostly non-compressible fluid. Able to conform to the shape of its container but retains a (nearly) constant volume independent of pressure. Gas: A compressible fluid. Not only will a gas take the shape of its container but it will also expand to fill the container. Mesomorphic states: States of matter intermediate between solid and ...
There is a large number of perturbation theory based equations of state available today, [23] [24] e.g. for the classical Lennard-Jones fluid. [ 11 ] [ 25 ] The two most important theories used for these types of equations of state are the Barker-Henderson perturbation theory [ 26 ] and the Weeks–Chandler–Andersen perturbation theory.
The last two are often used as cosmological models for (respectively) matter-dominated and radiation-dominated epochs. Notice that while in general it requires ten functions to specify a fluid, a perfect fluid requires only two, and dusts and radiation fluids each require only one function.
The equations of motion are contained in the continuity equation of the stress–energy tensor: =, where is the covariant derivative. [5] For a perfect fluid, = (+) +. Here is the total mass-energy density (including both rest mass and internal energy density) of the fluid, is the fluid pressure, is the four-velocity of the fluid, and is the metric tensor. [2]