Stress–energy–momentum pseudotensor: Difference between revisions

→‎Verification: Typo fixing, replaced: due the → due to the
m (→‎Verification: Typo fixing, replaced: due the → due to the)
#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 to 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 at any chosen point. As a result, the entire pseudotensor vanishes locally (again, at any chosen point) <math>t_{LL}^{\mu \nu} = 0</math>, which demonstrates the delocalisation of gravitational energy–momentum.<ref name="LL"/>