Stress-energy tensor

The components of the stress-energy tensor.

The stress-energy tensor (sometimes stress-energy-momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different In the various subfields of Physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Stress is a measure of the average amount of Force exerted per unit Area. It is an attribute of matter, radiation, and non-gravitational force fields. Matter is commonly defined as being anything that has mass and that takes up space. Radiation, as in Physics, is Energy in the form of waves or moving Subatomic particles emitted by an atom or other body as it changes from a higher energy Originally a term coined by Michael Faraday to provide an intuitive paradigm but theoretical construct (in the Kuhnian sense for the behavior of electromagnetic fields The stress-energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass is the source of such a field in Newtonian gravity. A gravitational field is a model used within Physics to explain how gravity exists in the universe The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass

1 Definition
2 Examples
3 As a Noether current
4 In general relativity
- 4.1 The Einstein field equations
5 Relativistic stress tensor for an idealized fluid
6 The various stress-energy tensors
7 See also
8 External links

Definition

Please note that throughout we will assume the use of the Einstein summation notation. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational When using coordinates, x⁰ will represent time, while the other coordinates x¹, x² and x³ will be the remaining spatial components.

The Stress-energy tensor is defined as the tensor $T a b$ of rank two that gives the flux of the a ^th component of the momentum vector across a surface with constant x^b coordinate. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In the various subfields of Physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point (In the theory of relativity this momentum vector is taken as the four-momentum). General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In Special relativity, four-momentum is the generalization of the classical three-dimensional Momentum to four-dimensional Spacetime. The stress-energy tensor is symmetric (when the spin tensor is zero), as in

T a b = T b a

If the spin tensor S is nonzero, then

$\partial_{\alpha}S^{\mu\nu\alpha}=T^{\mu\nu}-T^{\nu\mu}$

Examples

Here we will present some specific cases:

T 00

This represents the energy density. In Mathematics and Mathematical physics, the Euclidean group SE(d of Direct isometries is generated by Translations In Mathematics and Mathematical physics, the Euclidean group SE(d of Direct isometries is generated by Translations Energy density is the amount of Energy stored in a given system or region of space per unit Volume, or per unit Mass, depending on the context although

T 0 i

This represents the flux of energy across the xⁱ surface, which is equivalent to

T i 0,

the density of the i^th momentum.

The components

T i j

represent flux of i momentum across the x^j surface. In particular,

T i i

represents a pressure-like quantity, normal stress, whereas

$T^{ij}, \quad i \ne j$

represents shear stress (compare with the stress tensor). Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface Stress is a measure of the average amount of Force exerted per unit Area. A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material Stress is a measure of the average amount of Force exerted per unit Area.

Warning: In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress-energy tensor in the comoving frame of reference. Solid-state physics, the largest branch of Condensed matter physics, is the study of rigid Matter, or Solids The bulk of solid-state physics theory and Fluid mechanics is the study of how Fluids move and the Forces on them A proper frame, or comoving frame, is a Frame of reference that is attached to an object In other words, the stress energy tensor in engineering differs from the stress energy tensor here by a momentum convective term. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and

As a Noether current

In general

$\partial_b T^{ba}\ne0$

but the stress-energy tensor satisfies the continuity equation

$\nabla_b T^{ab}=T^{ab}{}_{;b}=0$ . A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity

The quantity

$\int d^3x T^{a0}$

over a spacelike slice gives the energy-momentum vector. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Special relativity, four-momentum is the generalization of the classical three-dimensional Momentum to four-dimensional Spacetime. The components $T a 0$ can therefore be interpreted as the local density of (non-gravitational) energy and momentum, and the first component of the continuity equation

$\nabla_b T^{0b} = \nabla \cdot \mathbf{p} + \frac{\partial E}{\partial t} = 0$

is simply a statement of energy conservation. Energy conservation is the practice of decreasing the quantity of energy used The spatial components $T i j$ (i, j = 1, 2, 3) correspond to components of local non-gravitational stresses, including pressure. Stress is a measure of the average amount of Force exerted per unit Area. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface This tensor is the conserved Noether current associated with spacetime translations. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Translation is the interpreting of the meaning of a text and the subsequent production of an equivalent text likewise called a " translation

In general relativity

The relations given above do not uniquely define the tensor. In general relativity, the symmetric form additionally satisfying

T a b = T b a

acts as the source of spacetime curvature, and is the current density associated with gauge transformations (in this case coordinate transformations). General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations If there is torsion, then the tensor is no longer symmetric. The term torsion may refer the following In geometry Torsion of curves Torsion tensor in differential geometry This corresponds to the case with a nonzero spin tensor. In Mathematics and Mathematical physics, the Euclidean group SE(d of Direct isometries is generated by Translations See Einstein-Cartan gravity. Einstein–Cartan theory in Theoretical physics extends General relativity to correctly handle Spin angular momentum.

In general relativity, the partial derivatives given above are actually covariant derivatives. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant What this means is that the continuity equation no longer implies that the energy and momentum expressed by the tensor are absolutely conserved. In the classical limit of Newtonian gravity, this has a simple interpretation: energy is being exchanged with gravitational potential energy, which is not included in the tensor, and momentum is being transferred through the field to other bodies. Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass Potential energy can be thought of as Energy stored within a physical system However, in general relativity there is no way to define physical quantities corresponding to densities of gravitational field energy and field momentum; any "pseudo-tensor" purporting to define them can be made to vanish locally by a coordinate transformation. In the general case, we must remain satisfied with a partial "covariant conservation" of the stress-energy tensor.

In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space There is in fact no way to define a global energy-momentum vector in a general curved spacetime.

The Einstein field equations

Main article: Einstein field equations

In general relativity, the stress tensor is studied in the context of the Einstein field equations which are often written as

$R_{\alpha \beta} - {1 \over 2}R\,g_{\alpha \beta} = {8 \pi G \over c^4} T_{\alpha \beta},$

where $R αβ$ is the Ricci tensor, $R$ is the Ricci scalar (the tensor contraction of the Ricci tensor), and $G$ is the universal gravitational constant. The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined In Multilinear algebra, a tensor contraction is an operation on one or more Tensors that arises from the natural pairing of a finite- Dimensional The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass

Relativistic stress tensor for an idealized fluid

For an idealized fluid, with no viscosity and no heat conduction, the stress tensor takes on a particularly simple form:

$T^{\alpha \beta} \, = (\rho + {p\over c^2})u^{\alpha}u^{\beta} + pg^{\alpha \beta}$ ,

where $ρ$ is the mass-energy density (mass per unit 3-volume), $p$ is the hydrostatic pressure, $u α$ is the fluid's 4-velocity, and $g αβ$ is the inverse metric of the manifold.

Furthermore, if the tensor components are being measured in a local inertial frame comoving with the fluid, then the metric tensor is simply Minkowski's metric

$g^{\alpha \beta} \, = \eta^{\alpha \beta} = \mathrm{diag}(-1,1,1,1)$

and the squared magnitude of the 4-velocity

$u^{\alpha}u^{\beta} \, = \mathrm{diag}(c^2,0,0,0)$ .

The stress tensor is then a diagonal matrix:

$T^{\alpha \beta} = \left( \begin{matrix} \rho c^2 & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right).$

The various stress-energy tensors

There are a number of inequivalent stress-energy tensors.

Canonical stress-energy tensor

This is the Noether current associated with spacetime translations. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In flat spacetime, this isn't symmetric in general and if we have some gauge theory, it won't be gauge invariant because space-dependent gauge transformations obviously don't commute with spatial translations. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In general relativity, the translations are with respect to the coordinate system and as such, don't transform covariantly. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 This is called a pseudostress-energy tensor. In the theory of General relativity, a stress-energy-momentum pseudotensor, such as the Landau-Lifshitz pseudotensor, is an extension of the non-gravitational

Hilbert stress-energy tensor

This stress-energy tensor can only be defined in general relativity with a dynamical metric. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 It is defined as a functional derivative

$T^{\mu\nu}(x)=\frac{2}{\sqrt{-g}}\frac{\delta \mathcal{S}_{\mathrm{matter}}}{\delta g_{\mu\nu}(x)}$

where S_matter is the nongravitational part of the action. In Mathematics and theoretical Physics, the functional derivative is a generalization of the Directional derivative. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation This is symmetric and gauge-invariant. See Einstein–Hilbert action for more information. The Einstein-Hilbert action in General relativity is the action that yields the Einstein's field equations when varied to obtain Equations

Belinfante-Rosenfeld stress-energy tensor

This is a symmetric and gauge-invariant stress energy tensor defined over flat spacetimes. There is a construction to get the Belinfante-Rosenfeld tensor from the canonical stress-energy tensor. In GR, this tensor agrees with the Hilbert stress-energy tensor. See the article Belinfante-Rosenfeld stress-energy tensor for more details.

Pseudotensors

All useful pseudotensors in general relativity are stress-energy-momentum pseudotensors, of which the Einstein pseudotensor and the Landau-Lifschitz pseudotensor are examples. In Physics and Mathematics, a pseudotensor is usually a quantity that transforms like a Tensor under a Proper rotation, but gains an additional In the theory of General relativity, a stress-energy-momentum pseudotensor, such as the Landau-Lifshitz pseudotensor, is an extension of the non-gravitational In the theory of General relativity, a stress-energy-momentum pseudotensor, such as the Landau-Lifshitz pseudotensor, is an extension of the non-gravitational In the theory of General relativity, a stress-energy-momentum pseudotensor, such as the Landau-Lifshitz pseudotensor, is an extension of the non-gravitational

External links

Lecture, Stephan Waner
Caltech Tutorial on Relativity — A simple discussion of the relation between the Stress-Energy tensor of General Relativity and the metric

In relativistic classical field theories of Gravitation, particularly General relativity, an energy condition is one of various alternative conditions The Maxwell Stress Tensor (also known as Maxwell's Stress Tensor is used to calculate the stresses on objects in magnetic or electrical fields In Physics, the Poynting vector can be thought of as representing the Energy Flux (in W/m2 of an Electromagnetic field. Energy density is the amount of Energy stored in a given system or region of space per unit Volume, or per unit Mass, depending on the context although In Physics, the electromagnetic stress-energy tensor is the portion of the Stress-energy tensor due to the Electromagnetic field. The Segre classification is an algebraic classification of rank two Symmetric tensors The resulting types are then known as Segre types.

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