Stress–energy–momentum pseudotensor: Difference between revisions

clean up, typos fixed: , → , (2) using AWB
(clean up, typos fixed: , → , (2) using AWB)
[[Lev Davidovich Landau|Landau]] & [[Evgeny Mikhailovich Lifshitz|Lifshitz]] were led by four requirements in their search for a gravitational energy momentum pseudotensor, <math>t_{LL}^{\mu \nu}\,</math>:<ref name="LL"/>
# that it be constructed entirely from the [[Metric tensor (general relativity)|metric tensor]], so as to be purely geometrical or gravitational in origin.
# that it be index symmetric , i.e. <math>t_{LL}^{\mu \nu} = t_{LL}^{\nu \mu} \,</math>, (to conserve [[angular momentum]])
# that, when added to the [[stress–energy tensor]] of matter, <math>T^{\mu \nu}\,</math>, its total 4-[[divergence]] vanishes (this is required of any [[conserved current]]) so that we have a conserved expression for the total stress–energy–momentum.
# that it vanish locally in an [[inertial frame of reference]] (which requires that it only contains first and not second or higher [[derivative]]s of the metric). This is because the [[equivalence principle]] requires that the gravitational force field, the [[Christoffel symbols]], vanish locally in some frame. If gravitational energy is a function of its force field, as is usual for other forces, then the associated gravitational pseudotensor should also vanish locally.
* ''G''<sup>&mu;&nu;μν</sup> is the [[Einstein tensor]] (which is constructed from the metric)
* ''g''<sup>&mu;&nu;μν</sup> is the inverse of the [[Metric tensor (general relativity)|metric tensor]]
* ''g''&nbsp;=&nbsp;det(''g''<sub>&mu;&nu;μν</sub>) is the [[determinant]] of the metric tensor and is < 0. Hence its appearance as <math>-g </math>.
* <math>,_{\alpha \beta} = \frac{\partial^2}{\partial x^{\alpha} \partial x^{\beta}}\,</math> are [[partial derivative]]s, not [[covariant derivative]]s.
Examining the 4 requirement conditions we can see that the first 3 are relatively easy to demonstrate:
#Since the Einstein tensor, <math>G^{\mu \nu}\,</math>, is itself constructed from the metric, so therefore is <math>t_{LL}^{\mu \nu} </math>
#Since the Einstein tensor, <math>G^{\mu \nu}\,</math>, is symmetric so is <math>t_{LL}^{\mu \nu} </math> since the additional terms are symmetric by inspection.
#The Landau–Lifshitz pseudotensor is constructed so that when added to the [[stress–energy tensor]] of matter, <math>T^{\mu \nu}\,</math>, its total 4-[[divergence]] vanishes: <math>((-g)(T^{\mu \nu} + t_{LL}^{\mu \nu}))_{,\mu} = 0 </math>. This follows from the cancellation of the Einstein tensor, <math>G^{\mu \nu}\,</math>, with the [[stress–energy tensor]], <math>T^{\mu \nu}\,</math> by the [[Einstein field equations]]; the remaining term vanishes algebraically due the commutativity of partial derivatives applied across antisymmetric indices.
===Cosmological constant===
When the Landau–Lifshitz pseudotensor was formulated it was commonly assumed that the [[cosmological constant]],<math>\Lambda \,</math> , was zero. Nowadays [[accelerating universe|we don't make that assumption]], and the expression needs the addition of a <math>\Lambda \,</math> term, giving:
:<math>t_{LL}^{\mu \nu} = - \frac{c^4}{8\pi G}(G^{\mu \nu}+\Lambda g^{\mu \nu}) + \frac{c^4}{16\pi G (-g)}((-g)(g^{\mu \nu}g^{\alpha \beta} - g^{\mu \alpha}g^{\nu \beta}))_{,\alpha \beta}</math>
This is necessary for consistency with the [[Einstein field equations]].
*[[Metric tensor]] version:
:<math>(-g)(t_{LL}^{\mu \nu} + \frac{c^4\Lambda g^{\mu \nu}}{8\pi G}) = \frac{c^4}{16\pi G}((\sqrt{-g}g^{\mu \nu}),_{\alpha }(\sqrt{-g}g^{\alpha \beta}),_{\beta}- </math>
::<math>- (\sqrt{-g}g^{\mu \alpha }),_{\alpha }(\sqrt{-g}g^{\nu \beta}),_{\beta} +\frac{1}{2}g^{\mu \nu}g_{\alpha \beta}(\sqrt{-g}g^{\alpha \sigma }),_{\rho }(\sqrt{-g}g^{\rho \beta }),_{ \sigma }-</math>
::<math>-(g^{\mu \alpha }g_{\beta \sigma }(\sqrt{-g}g^{\nu \sigma }),_{\rho }(\sqrt{-g}g^{\beta \rho }),_{\alpha }+g^{\nu \alpha }g_{\beta \sigma}(\sqrt{-g}g^{\mu \sigma }),_{\rho }(\sqrt{-g}g^{\beta \rho }),_{\alpha })+</math>
::<math>+g_{\alpha \beta }g^{ \sigma \rho }(\sqrt{-g}g^{\mu \alpha }),_{ \sigma }(\sqrt{-g}g^{\nu \beta }),_{\rho }+\,</math>
::<math>+\frac{1}{8}(2g^{\mu \alpha }g^{\nu \beta }-g^{\mu \nu}g^{\alpha \beta })(2g_{ \sigma \rho }g_{\lambda \omega}-g_{\rho \lambda }g_{ \sigma \omega})(\sqrt{-g}g^{ \sigma \omega}),_{\alpha }(\sqrt{-g}g^{\rho \lambda }),_{\beta })</math><ref>Landau–Lifshitz equation 96.9 </ref>
*[[Christoffel_symbolsChristoffel symbols|Affine connection]] version:
:<math>t_{LL}^{\mu \nu} + \frac{c^4\Lambda g^{\mu \nu}}{8\pi G}= \frac{c^4}{16\pi G}((2\Gamma^{ \sigma }_{\alpha \beta }\Gamma^{\rho }_{ \sigma \rho }-\Gamma^{ \sigma }_{\alpha \rho }\Gamma^{\rho }_{\beta \sigma }-\Gamma^{ \sigma }_{\alpha \sigma }\Gamma^{\rho }_{\beta \rho})(g^{\mu \alpha }g^{\nu \beta }-g^{\mu \nu}g^{\alpha \beta })+</math>
::<math>+g^{\mu \alpha }g^{\beta \sigma }(\Gamma^{\nu}_{\alpha \rho }\Gamma^{\rho }_{\beta \sigma }+\Gamma^{\nu}_{\beta \sigma } \Gamma^{\rho }_{\alpha \rho } - \Gamma^{\nu}_{ \sigma \rho } \Gamma^{\rho }_{\alpha \beta } - \Gamma^{\nu}_{\alpha \beta } \Gamma^{\rho }_{ \sigma \rho })+</math>
::<math>+g^{\nu \alpha }g^{\beta \sigma }(\Gamma^{\mu}_{\alpha \rho }\Gamma^{\rho }_{\beta \sigma }+\Gamma^{\mu}_{\beta \sigma } \Gamma^{\rho }_{\alpha \rho } - \Gamma^{\mu}_{ \sigma \rho } \Gamma^{\rho }_{\alpha \beta } - \Gamma^{\mu}_{\alpha \beta } \Gamma^{\rho }_{ \sigma \rho })+</math>
::<math>+g^{\alpha \beta }g^{ \sigma \rho}(\Gamma^{\mu}_{\alpha \sigma } \Gamma^{\nu}_{\beta \rho } - \Gamma^{\mu}_{\alpha \beta } \Gamma^{\nu}_{ \sigma \rho }))</math><ref>Landau–Lifshitz equation 96.8 </ref>
This definition of energy-momentum is covariantly applicable not just under Lorentz transformations, but also under general coordinate transformations.