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In the theory of [[general relativity]], a '''
Some people object to this derivation on the grounds that [[pseudotensor]]s are inappropriate objects in general relativity, but the conservation law only requires the use of the 4[[divergence]] of a pseudotensor which is, in this case, a tensor (which also vanishes). Also, most pseudotensors are sections of [[jet bundle]]s, which are perfectly valid objects in GR.
==Landau–Lifshitz pseudotensor==
The use of the Landau–Lifshitz combined matter+gravitational
===Requirements===
# 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 [[
# 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.
#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 [[
#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 [[LeviCivita 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 energymomentum.<ref name="LL"/>
