Stress–energy–momentum pseudotensor: Difference between revisions

m (link to pseudotensor)
#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.
#The Landau-Lifshitz pseudotensor appears to include second derivative terms in the metric, but in fact the explicit second derivative terms in the pseudotensor cancel with the implicit second derivative terms contained within the [[Einstein tensor]], <math>G^{\mu \nu}\,</math>. This is more evident when the pseudotensor is directly expressed in terms of the metric tensor or the [[Levi-Civita connection]]; only the first derivative terms in the metric survive and these vanish where the frame is locally inertial around any chosen point. As a result the entire pseudotensor vanishes locally (again, around any chosen point) <math>t_{LL}^{\mu \nu} = 0</math>, which demonstrates the delocalisation of gravitational energy-momentum.<ref name="LL"/>
==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 Einstein tensor, <math>G^{\mu \nu}\,</math>, is modified by the addition of <math>\Lambda \,</math>, giving:
:<math>16 \pi t_{LL}^{\mu \nu} = - 2(G^{\mu \nu} + \Lambda g^{\mu \nu}) + (-g)^{-1}((-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 and affine connection versions===
Landau & Lifshitz also provide two equivalent but longer expressions for the Landau-Lifshitz pseudotensor: