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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 tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields.
The local conservation of non-gravitational linear momentum and energy in a free-falling reference frame is expressed by the vanishing of the covariant divergence of the stress–energy tensor. Another important conserved quantity, discovered in studies of the celestial mechanics of astronomical bodies, is the Laplace–Runge–Lenz vector .
The electromagnetic stress–energy tensor allows a compact way of writing the conservation laws of linear momentum and energy in electromagnetism. The divergence of the stress–energy tensor is: ∂ ν T μ ν + η μ ρ f ρ = 0 {\displaystyle \partial _{\nu }T^{\mu \nu }+\eta ^{\mu \rho }\,f_{\rho }=0\,} where f ρ {\displaystyle f_{\rho ...
If the energy–momentum tensor T μν is that of an electromagnetic field in free space, i.e. if the electromagnetic stress–energy tensor = (+) is used, then the Einstein field equations are called the Einstein–Maxwell equations (with cosmological constant Λ, taken to be zero in conventional relativity theory): + = (+).
Since T 00 is the energy density, T j0 for j = 1, 2, 3 is the jth component of the object's 3d momentum per unit volume, and T ij form components of the stress tensor including shear and normal stresses, the orbital angular momentum density about the position 4-vector X β is given by a 3rd order tensor = (¯) (¯)
Using the Maxwell equations, one can see that the electromagnetic stress–energy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current four-vector , + = or , + =, which expresses the conservation of linear momentum and energy by electromagnetic interactions.
This is an accepted version of this page This is the latest accepted revision, reviewed on 14 January 2025. Law of physics and chemistry This article is about the law of conservation of energy in physics. For sustainable energy resources, see Energy conservation. Part of a series on Continuum mechanics J = − D d φ d x {\displaystyle J=-D{\frac {d\varphi }{dx}}} Fick's laws of diffusion Laws ...
Calculating the Minkowski norm squared of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass: = = = + | | = where = is the metric tensor of special relativity with metric signature for definiteness chosen to be (–1, 1, 1, 1).