- Stress-energy tensor
The stress-energy tensor (sometimes stress-energy-momentum tensor) is a
tensor quantity inphysics that describes thedensity andflux ofenergy andmomentum inspacetime , generalizing the stress tensor of Newtonian physics. It is an attribute ofmatter ,radiation , and non-gravitational force fields. The stress-energy tensor is the source of thegravitational field in theEinstein field equations ofgeneral relativity , just as mass is the source of such a field inNewtonian gravity .Definition
In the following, the Einstein summation notation is used. The components of the position
4-vector are given by: "x"0 = "t" (time in seconds), "x"1 = "x" (in meters), "x"2 = "y" (in meters), and "x"3 = "z" (in meters).The Stress-energy tensor is defined as the
tensor T^{alpha eta} of rank two that gives theflux of the αth component of themomentum vector across a surface with constant "x"βcoordinate . In the theory of relativity this momentum vector is taken as thefour-momentum . The stress-energy tensor is symmetric,:T^{alpha eta} = T^{eta alpha} !.Some people have speculated that it could be non-symmetric. In those hypotheses, when the
spin tensor S is nonzero,:partial_{alpha}S^{mu ualpha} = T^{mu u} - T^{ umu} !.Identifying the components of the contravariant tensor
The time-time component is the density of relativistic mass, i.e. the
energy density divided by the speed of light squared,:T^{00} = ho. !The flux of relativistic mass across the "x""i" surface is equivalent to the density of the "i"th component of linear momentum, :T^{0i} = T^{i0}. !
The components:T^{ik} !represent flux of "i" momentum across the "x""k" surface. In particular,:T^{ii} !(not summed) represents normal stress which is called
pressure when it is independent of direction. Whereas:T^{ik}, quad i e k representsshear stress (compare with the stress tensor).Warning: In
solid state physics andfluid mechanics , the stress tensor is defined to be the spatial components ofthe stress-energy tensor in thecomoving frame of reference. In other words, the stress energy tensor inengineering differs from the stress energy tensor here by a momentum convective term.Covariant and mixed forms
In most of this article we work with the contravariant form, T^{mu u}! of the stress-energy tensor. However, it is often necessary to work with the covariant form:T_{mu u} = g_{mu alpha} g_{ u eta} T^{alpha eta}!
or the mixed form:T_{mu}^{ u} = g_{mu alpha} T^{alpha u}.
Indeed, one could argue that the most correct form is the mixed density:mathfrak{T}_{mu}^{ u} = T_{mu}^{ u} sqrt{-g}.
Conservation law
In special relativity
The stress-energy tensor is the conserved Noether current associated with
spacetime translations.When gravity is negligible and using a
Cartesian coordinate system for spacetime, the divergence of the non-gravitational stress-energy will be zero. In other words, non-gravitational energy and momentum are conserved,:0 = T^{mu u}{}_{, u} = partial_{ u} T^{mu u}. !The integral form of this is:0 = int_{partial N} T^{mu u} mathrm{d}^3 s_{ u} !
where "N" is any compact four-dimensional region of spacetime; partial N is its boundary, a three dimensional hypersurface; and mathrm{d}^3 s_{ u} is an element of the boundary regarded as the outward pointing normal.
If one combines this with the symmetry of the stress-energy tensor, one can show that
angular momentum is also conserved,:0 = (x^{alpha} T^{mu u} - x^{mu} T^{alpha u})_{, u} . !In general relativity
However, when gravity is non-negligible or when using arbitrary coordinate systems, the divergence of the non-gravitational stress-energy may fail to be zero. In this case, we have to use a more general
continuity equation which incorporates thecovariant derivative :0 = T^{mu u}{}_{; u} = abla_{ u} T^{mu u} = T^{mu u}{}_{, u} + T^{sigma u} Gamma^{mu}{}_{sigma u} + T^{mu sigma} Gamma^{ u}{}_{sigma u}where Gamma^{mu}{}_{sigma u} is the
Christoffel symbol which is the gravitational force field.Consequently, if xi^{mu} is any
Killing vector field , then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as:0 = (xi^{mu} mathfrak{T}_{mu}^{ u})_{, u} .The integral form of this is:0 = int_{partial N} xi^{mu} mathfrak{T}_{mu}^{ u} mathrm{d}^3 s_{ u} .
In general relativity
In
general relativity , thesymmetric stress-energy tensor acts as the source of spacetime curvature, and is the current density associated withgauge transformation s of gravity which are general curvilinearcoordinate transformation s. (If there istorsion , then the tensor is no longer symmetric. This corresponds to the case with a nonzerospin tensor . SeeEinstein-Cartan gravity .)In general relativity, the
partial derivatives used in special relativity are replaced bycovariant derivatives. What this means is that the continuity equation no longer implies that the non-gravitational energy and momentum expressed by the tensor are absolutely conserved, i.e. the gravitational field can do work on matter and vice versa. In the classical limit ofNewtonian gravity , this has a simple interpretation: energy is being exchanged with gravitationalpotential 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 not a unique way to define 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 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 eta} - {1 over 2}R,g_{alpha eta} = {8 pi G over c^4} T_{alpha eta},
where R_{alpha eta} is the
Ricci tensor , R is the Ricci scalar (thetensor contraction of the Ricci tensor), and G is theuniversal gravitational constant .tress-energy in special situations
Isolated particle
In special relativity, the stress-energy of a non-interacting particle with mass "m" is:T^{alpha eta} [t,x,y,z] = frac{m , v^{alpha} [t] v^{eta} [t] }{sqrt{1 - (v/c)^2 delta(x - x [t] ) delta(y - y [t] ) delta(z - z [t] )
where δ is the
Dirac delta function and v^{alpha} ! is the velocity vector:egin{pmatrix}v^0 [t] \ v^1 [t] \ v^2 [t] \ v^3 [t] end{pmatrix} = egin{pmatrix}1 \ {d x [t] over d t} \ {d y [t] over d t} \ {d z [t] over d t}end{pmatrix}.tress-energy of a fluid in equilibrium
For a fluid in
thermodynamic equilibrium , the stress-energy tensor takes on a particularly simple form:T^{alpha eta} , = ( ho + {p over c^2})u^{alpha}u^{eta} + p g^{alpha eta}where ho is the mass-energy density (kilograms per cubic meter), p is the hydrostatic pressure (Newtons per square meter), u^{alpha} is the fluid's
four velocity , and g^{alpha eta} is the reciprocal of the metric tensor.The four velocity satisfies:u^{alpha} u^{eta} g_{alpha eta} = - c^2 ,.
In an
inertial frame of reference comoving with the fluid, the four velocity is :u^{alpha} = (1, 0, 0, 0) ,,the reciprocal of the metric tensor is simply:g^{alpha eta} , = left( egin{matrix} - c^{-2} & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{matrix} ight),,
and the stress-energy tensor is a diagonal matrix:
T^{alpha eta} = left( egin{matrix} ho & 0 & 0 & 0 \ 0 & p & 0 & 0 \ 0 & 0 & p & 0 \ 0 & 0 & 0 & p end{matrix} ight).
Electromagnetic stress-energy tensor
The stress-energy tensor of a source-free electromagnetic field is:T^{mu u} (x) = frac{1}{mu_0} left( F^{mu alpha} g_{alpha eta} F^{ u eta} - frac{1}{4} g^{mu u} F_{delta gamma} F^{delta gamma} ight)
where F_{mu u} is the
electromagnetic field tensor .calar Field
The stress-energy tensor for a scalar field phi which satisfies the Klein–Gordon equation is:T^{mu u} = frac{hbar^2}{m} (g^{mu alpha} g^{ u eta} + g^{mu eta} g^{ u alpha} - g^{mu u} g^{alpha eta}) partial_{alpha}arphi partial_{eta}phi - g^{mu u} m c^2 arphi phi .
Variant definitions of stress-energy
There are a number of inequivalent definitions of non-gravitational stress-energy.
Hilbert stress-energy tensor
This stress-energy tensor can only be defined in
general relativity with a dynamical metric. It is defined as afunctional derivative :T^{mu u} = frac{2}{sqrt{-gfrac{delta (mathcal{L}_{mathrm{matter sqrt{-g}) }{delta g_{mu u = 2 frac{delta mathcal{L}_mathrm{matter{delta g_{mu u + g^{mu u} mathcal{L}_mathrm{matter}.where "L"matter is the nongravitational part of the
Lagrangian density of the action. This is symmetric and gauge-invariant. SeeEinstein–Hilbert action for more information.Canonical stress-energy tensor
Noether's theorem implies that there is a conserved current associated with translations through space and time. This is called the canonical stress-energy tensor. Generally, this is not symmetric and if we have some gauge theory, it may not begauge invariant because space-dependentgauge transformation s do not commute with spatial translations.In
general relativity , the translations are with respect to the coordinate system and as such, do not transform covariantly. See the section below on the gravitational stress-energy pseudo-tensor.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.Gravitational stress-energy
By the
equivalence principle gravitational stress-energy will always vanish locally at any chosen point in some chosen frame, therefore gravitational stress-energy cannot be expressed as a non-zero tensor; instead we have to use apseudotensor .In general relativity, there are many possible distinct definitions of the gravitational stress-energy-momentum
pseudotensor . These include the Einstein pseudotensor and the Landau-Lifschitz pseudotensor. The Landau-Lifschitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.ee also
*
Energy condition
*Maxwell stress tensor
*Poynting vector
*Energy density of electric and magnetic fields
*Electromagnetic stress-energy tensor
*Segre classification External links
* [http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec12.html Lecture, Stephan Waner]
* [http://www.black-holes.org/numrel1.html Caltech Tutorial on Relativity] — A simple discussion of the relation between the Stress-Energy tensor of General Relativity and the metric
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