Stress-energy Tensor

The stress-energy tensor (sometimes stress-energy-momentum tensor) is a Tensor quantity in Physics . It describes the Density and Flux of Energy and Momentum in space, generalizing the Stress Tensor of Newtonian physics. It is the source of the Gravitational Field of General Relativity --just as mass is the source of such field in the Newtonian theory. One very important use of the stress-energy tensor is in the Einstein Field Equations .

DEFINITION

''Please note that throughout we will assume the use of the Einstein Summation Notation . 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^{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 (in the absence of a nonzero Spin Tensor ), as in
:

T^{ab} = T^{ba}

If the Spin Tensor S is nonzero, then

:

\partial_{\alpha}S^{\mu
u\alpha}=T^{\mu
u}-T^{
u\mu}

EXAMPLES

Here we will present some specific cases:
:

T^{00}

This represents the Energy Density .

:

T^{0i}

This represents the flux of energy across the ''x''^''i'' surface, which is equivalent to
:

T^{i0},

the density of the ''i''^th momentum.

The components
:

T^{ij}

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

T^{ii}

represents a Pressure -like quantity, Normal Stress , whereas
:

T^{ij}, \quad i 
e j

represents Shear Stress (compare with the Stress Tensor ).

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. In other words, the stress energy tensor in Engineering differs from the stress energy tensor here by a momentum convective term.

AS A NOETHER CURRENT

The stress-energy tensor satisfies the Continuity Equation
:

abla_b T^{ab}=T^{ab}{}_{;b}=0

.

The quantity
:

\int d^3x T^{a0}

over a Spacelike slice gives the Energy-momentum Vector . The components

T^{a0}

can therefore be interpreted as the local density of (non-gravitational) energy and momentum, and the first component of the continuity equation

:

abla_b T^{0b} = 
abla \cdot \mathbf{p} - rac{\partial E}{\partial t} = 0

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 Translation s.

IN GENERAL RELATIVITY

The relations given above do not uniquely define the tensor. In General Relativity , the Symmetric form additionally satisfying
:

T^{ab} = T^{ba}

acts as the source of spacetime Curvature , and is the current density associated with Gauge Transformation s (in this case Coordinate Transformation s). If there is Torsion , then the tensor is no longer symmetric. This corresponds to the case with a nonzero Spin Tensor . See Einstein-Cartan Gravity .

In general relativity, the , 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 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.

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

:

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

where

R_{\alpha \beta}

is the Ricci Tensor ,

R

is the Ricci scalar (the Tensor Contraction of the Ricci tensor), and

G

is the Universal Gravitational Constant .

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} \, = (
ho + p)u^{\alpha}u^{\beta} + pg^{\alpha \beta}

,

where

ho

is the mass-energy density (mass per unit 3-volume),

p

is the hydrostatic pressure,

u^{\alpha}

is the fluid's 4-velocity, and

g^{\alpha \beta}

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}(1,0,0,0)

.

The stress tensor is then a diagonal matrix:

:

T^{\alpha \beta} = \left( \begin{matrix}
                   
ho & 0 & 0 & 0 \
                   0 & p & 0 & 0 \
                   0 & 0 & p & 0 \
                   0 & 0 & 0 & p    
      \end{matrix} 
ight)

THE VARIOUS STRESS-ENERGY TENSORS

There are a number of inequivalent stress-energy tensors out there.

Canonical stress-energy tensor

This is the Noether Current associated with spacetime translations. 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 Transformation s obviously don't commute with spatial translations. In General Relativity , the translations are with respect to the coordinate system and as such, don't transform covariantly. This is called a Pseudostress-energy Tensor .

Hilbert stress-energy tensor

This stress-energy tensor can only be defined in General Relativity with a dynamical metric. It is defined as a Functional Derivative

:

T^{\mu

	CATEGORIES ABOUT STRESS-ENERGY TENSOR

	fundamental physics concepts

	tensors in general relativity

	variational formalism of general relativity

	tensors

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME

	MORE SHOPPING...

Information About

Stress-energy Tensor