Please note that throughout we will assume the use of the Einstein summation convention also x⁰ will represent time, while the other coordinates x¹, x² and x³ will be the remaining spacial components For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

The stress-energy tensor is a tensor quantity in physics. It describes the flux of momentum in space. In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalars, vectors, and linear operators. ... Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ... The flux visualized. ... In physics, momentum is a physical quantity related to the velocity and mass of an object. ...

Definition

The Stress-energy tensor is defined as the the tensor T^ab of rank two that gives the flux of the a ^th component of the momentum vector across a surface with constant x^b coordinate. (In the theory of relativity this momentum vector is taken as the four-momentum). It is also important to note that the stress-energy tensor is symmetric, as in In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalars, vectors, and linear operators. ... The flux visualized. ... In physics, momentum is a physical quantity related to the velocity and mass of an object. ... VECTOR is the name of a Human Capital Management tool from the UK company Vector Management Systems. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... Two-dimensional visualization of space-time distortion. ... In special relativity, four-momentum is a four-vector that replaces classical momentum; the four-momentum of a particle is defined as the particles mass times the particles four-velocity. ...

T^ab = T^ba

Examples

Here we will present some specific cases:

T⁰⁰

This represents the energy density. Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ...

T⁰ⁱ

This represents the flux of the i momentum across the xⁱ surface.

As a Noether current

The stress-energy tensor satisfies the continuity equation Note that all the examples given below express the same idea (i. ...

.

The quantity

over a spacelike slice gives the energy-momentum vector. The components T^0b can therefore be interpreted as the local density of (non-gravitational) energy and momentum, and the first component of the continuity equation In the context of special relativity, space-like separated points (or events) in spacetime have a spacetime interval less than 0 (see sign convention). ... In special relativity, four-momentum is a four-vector that replaces classical momentum; the four-momentum of a particle is defined as the particles mass times the particles four-velocity. ...

is simply a statement of energy conservation. The spatial components T^ij (i, j = 1, 2, 3) correspond to components of local non-gravitational stresses, including pressure. This tensor is the conserved Noether current associated with spacetime translations. Energy conservation is the idealistic or economic practice of using energy resources in a sustainable way by considering which processes are wasteful, and addressing those inefficiencies. ... Stress tensor In physics, stress is the internal distribution of forces within a body that balance and react to the loads applied to it. ... Pressure is the application of force to a surface, and the concentration of that force in a given area. ... Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between the symmetries and the conservation laws. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... Translation is an activity comprising the interpretation of the meaning of a text in one language — the source text — and the production of a new, equivalent text in another language — called the target text, or the translation. ...

In general relativity

The relations given above do not uniquely define the tensor. In general relativity, the symmetric form additionally satisfying Two-dimensional visualization of space-time distortion. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

T^ab = T^ba

acts as the source of spacetime curvature, and is the current density associated with gauge transformations (in this case coordinate transformations). If there is torsion, then the tensor is no longer symmetric. In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... // Mathmatics In mathematics, the term torsion has several meanings, mostly unrelated to each other. ...

In general relativity, the partial derivatives given above are actually covariant derivatives. 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. 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 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. ... In category theory, see covariant functor. ... The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inverse-square law. ... Potential energy (U, or Ep) is defined as work of conservative force(s) during change of state of physical system from given static state to another static state (latter is usually called reference state, or reference level). ...

In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energy-momentum vector in a general curved spacetime. In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...

The Einstein Field Equations

In General Relativity, the stress tensor is studied in the context of the Einstein Field Equations which are often written as

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. In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ...

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^αβ = (ρ + p)u^αu^β + pg^αβ,

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_αβ = η_αβ = diag( − 1,1,1,1) and the squared magnitude of the 4-velocity g^αβu^αu^β = diag(1,0,0,0). The stress tensor is then a nice simple diagonal matrix:

External links

• http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec12.html