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 r ...

, a perfect fluid is a

fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...

that can be completely characterized by its rest frame

mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

\rho_m

and ''isotropic''

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

''p''. Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in which these possibilities are neglected. Specifically, perfect fluids have no

shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...

es,

viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...

, or

heat conduction Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted . Heat spontaneously flows along a te ...

Quark–gluon plasma Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a 1 ...

is the closest known substance to a perfect fluid. In space-positive

metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative an ...

tensor notation, the

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 ...

of a perfect fluid can be written in the form :

T^ = \left( \rho_m + \frac \right) \, U^\mu U^\nu + p \, \eta^\,

where ''U'' is the

4-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...

vector field of the fluid and where

\eta_ = \operatorname(-1,1,1,1)

is the metric tensor of

Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...

. In time-positive

tensor notation, the

of a perfect fluid can be written in the form :

T^ = \left( \rho_\text + \frac \right) \, U^\mu U^\nu - p \, \eta^\,

where ''U'' is the 4-velocity of the fluid and where

\eta_ = \operatorname(1,-1,-1,-1)

is the metric tensor of

. This takes on a particularly simple form in the rest frame :

\left \begin \rho_e &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end \right

where

\rho_\text = \rho_\text c^2

is the ''

energy density In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or . Often only the ''useful'' or extract ...

'' and

p

is the ''pressure'' of the fluid. Perfect fluids admit a Lagrangian formulation, which allows the techniques used in field theory, in particular, quantization, to be applied to fluids. Perfect fluids are used in

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

to model idealized distributions of

matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...

, such as the interior of a star or an isotropic universe. In the latter case, the

equation of state In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal ...

of the perfect fluid may be used in Friedmann–Lemaître–Robertson–Walker equations to describe the evolution of the universe. In

, the expression for the

of a perfect fluid is written as :

T^ = \left( \rho_m + \frac \right) \, U^\mu U^\nu + p \, g^\,

where ''U'' is the

vector field of the fluid and where

g^

is the inverse metric, written with a space-positive signature.

References

*The Large Scale Structure of Space-Time, by S.W.Hawking and G.F.R.Ellis, Cambridge University Press, 1973. , (pbk.) *

External links

*Mark D. Roberts, Fluid Generalization of Membranes http://www.arXiv.org/abs/hep-th/0406164 hep-th/0406164 Fluid mechanics Superfluidity {{fluiddynamics-stub

See also

References

External links