Energy-momentum tensor

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In general relativity the curvature of space-time is given by energy-impulse tensor.

The stress-energy tensor, also called the momentum-energy tensor (or momentum-energy tensor ) is a tensor quantity in Einstein's theory of relativity that is used to describe the linear flow of energy and momentum in the context of relativity theory, as well as being of paramount importance in Einstein's equations for the gravitational field.

Introduction

Fixed a set of coordinates or a base ${displaystyle scriptstyle {{mathbf {e} }^{0},{mathbf {e} }^{1},{mathbf {e} }^{2},{mathbf {e} }^{3}}}$ at each point of space-time (the elements of this base would be mathematically 1-forms), the energy-impulse tensor is a range 2 tensor that can be described as a type matrix:

${displaystyle mathbf {T} (mathbf {x})=T_{alpha beta }(mathbf {x}){mathbf {e} }^{alpha }otimes {mathbf {e} }^{beta },qquad T_{alpha beta }={begin{bmatrix}T_{00}&T_{01}&T_{02}&T_{03}\T_{10}&T_{11}&T_{12}&T_{13}\T_{20}&T_{21}&T_{22}&T_{23}\T_{30}&T_{31}&T_{32}&T_{33}end{bmatrix}}}$

Usual interpretation of the countervariant components of the energy-impulsive tensor

Where the previous expression has used Einstein's summation agreement. If we now consider an observer who moves quadrantly ${displaystyle scriptstyle mathbf {u} =u^{alpha }mathbf {e} _{alpha }}$ we have the measured energy density at a point ${displaystyle scriptstyle mathbf {x} }$ by that observer is given by:

${displaystyle e=T_{alpha beta }(mathbf {x}){frac {u^{alpha }u^{beta }}{c^{2}}}}$

And the flow of energy through a surface (of a space type and at rest with respect to the observer) whose normal vector is given by ${displaystyle scriptstyle mathbf {n} }$ is given by:

${displaystyle -T_{alpha beta }(mathbf {x})u^{alpha }n^{beta }}$

Law of Conservation

In the context of the theory of relativity, the law of conservation of energy and the law of conservation of momentum can be expressed very simply in terms of the energy-momentum tensor. Specifically, both laws can be written together as a continuity equation of the type:

${displaystyle nabla _{mu }T^{mu nu }=0}$

The quantity

${displaystyle P^{mu }={frac {1}{c}}int _{V}T^{0mu } d^{3}mathbf {x} }$

over a space-like slice gives the energy-momentum fourvector or quadmomentum. This tensor is the Noether current associated with translations in space-time. In general relativity, this quantity acts as the source of the curvature of space-time, and is the current density associated with gauge transformations (in this case coordinate transformations) by Noether's theorem. Now, in curved spacetime, the space-like integral depends on the space-like slice, in general. There is in fact no way to define a global energy-momentum vector in a generally curved spacetime.

Related Tensors

The three-dimensional part of the energy-momentum tensor coincides with the stress tensor of continuum mechanics.

Examples

In the theory of relativity the energy-impulse tensor of a perfect fluid is expressable in terms of its quadrantility, masic density and pressure:

(1) ${displaystyle T_{mu nu }=left(rho +{frac {p}{c^{2}}}right)u_{mu }u_{nu }+pg_{mu nu }}$

Different types of energy-momentum tensor

There are several non-equivalent ways to define the stress tensor for ordinary matter. Among the most common are:

Hilbert's energy-impulse tensor.
The canonical energy-impulse tensor.
Belifante-Rosenfelder's energy-impulse tensor.

Hilbert momentum-energy tensor

This type of energy-momentum tensor can only be defined for a system that is described by a relativistic Lagrangian in the form of a functional derivative:

${displaystyle T^{mu nu }={frac {2}{sqrt {-g}}}{frac {delta ({mathcal {L}}_{mathrm {matter} }{sqrt {-g}})}{delta g_{mu nu }}}=2{frac {delta {mathcal {L}}_{mathrm {matter} }}{delta g_{mu nu }}}+g^{mu nu }{mathcal {L}}_{mathrm {matter} }.}$

where ${displaystyle {mathcal {L}}_{mathrm {matter} }}$ is the lagrangiana density of matter, which appears in the integral of action, for the gravitational part it is not possible to define a similar tensor. This tensor in a wide set of circumstances is symmetrical and invariant gauge.

Canonical energy-momentum tensor

This tensor results from the application of Noether's theorem. If the local spatiotemporal translations are a local symmetry of the Lagrangian, the conserved current associated with said symmetry is the canonical energy-momentum tensor. This tensor does not turn out to be symmetric for some gauge theories, and therefore may not be gauge invariant under local gauge transformations that do not commute with space-time translations.

In general relativity, translations can only be written in terms of coordinates, so they generally do not exhibit covariance.

Belinfante–Rosenfeld energy-momentum tensor

In the presence of spin or other intrinsic angular momentum, the canonical Noether energy-force tensor is not symmetric as anticipated in the previous section. The Belifante-Rosenfeld tensor is a construction from the canonical tensor and the conserved spin current in such a way that a new symmetric and conserved tensor is obtained. In general relativity, this modified tensor coincides with the Hilbert energy-momentum tensor.

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