Stress–energy–momentum pseudotensor

Landau and Lifshitz were led by four requirements in their search for a gravitational energy momentum pseudotensor, ${\displaystyle t_{LL}^{\mu \nu }\,}$:[1]

1. that it be constructed entirely from the metric tensor, so as to be purely geometrical or gravitational in origin.
2. that it be index symmetric, i.e. ${\displaystyle t_{LL}^{\mu \nu }=t_{LL}^{\nu \mu }\,}$, (to conserve angular momentum)
3. that, when added to the stress–energy tensor of matter, ${\displaystyle T^{\mu \nu }\,}$, 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.
4. that it vanish locally in an inertial frame of reference (which requires that it only contains first order and not second or higher order derivatives of the metric). This is because the equivalence principle requires that the gravitational force field, the Christoffel symbols, vanish locally in some frames. 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.

Landau & Lifshitz showed that there is a unique construction that satisfies these requirements, namely

$${\displaystyle t_{LL}^{\mu \nu }=-{\frac {c^{4}}{8\pi G}}G^{\mu \nu }+{\frac {c^{4}}{16\pi G(-g)}}\left((-g)\left(g^{\mu \nu }g^{\alpha \beta }-g^{\mu \alpha }g^{\nu \beta }\right)\right)_{,\alpha \beta }}$$

where:

• G^μν is the Einstein tensor (which is constructed from the metric)
• g^μν is the inverse of the metric tensor, g_μν
• g = det(g_μν) is the determinant of the metric tensor. g < 0, hence its appearance as ${\displaystyle -g}$.
• ${\textstyle {}_{,\alpha \beta }={\frac {\partial ^{2}}{\partial x^{\alpha }\partial x^{\beta }}}\,}$ are partial derivatives, not covariant derivatives.
• G is Newton's gravitational constant.

Examining the 4 requirement conditions we can see that the first 3 are relatively easy to demonstrate:

1. Since the Einstein tensor, ${\displaystyle G^{\mu \nu }\,}$, is itself constructed from the metric, so therefore is ${\displaystyle t_{LL}^{\mu \nu }}$
2. Since the Einstein tensor, ${\displaystyle G^{\mu \nu }\,}$, is symmetric so is ${\displaystyle t_{LL}^{\mu \nu }}$ since the additional terms are symmetric by inspection.
3. The Landau–Lifshitz pseudotensor is constructed so that when added to the stress–energy tensor of matter, ${\displaystyle T^{\mu \nu }\,}$, its total 4-divergence vanishes: ${\displaystyle \left(\left(-g\right)\left(T^{\mu \nu }+t_{LL}^{\mu \nu }\right)\right)_{,\mu }=0}$. This follows from the cancellation of the Einstein tensor, ${\displaystyle G^{\mu \nu }\,}$, with the stress–energy tensor, ${\displaystyle T^{\mu \nu }\,}$ by the Einstein field equations; the remaining term vanishes algebraically due to the commutativity of partial derivatives applied across antisymmetric indices.
4. 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, ${\displaystyle G^{\mu \nu }\,}$. 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) ${\displaystyle t_{LL}^{\mu \nu }=0}$, which demonstrates the delocalisation of gravitational energy–momentum.[1]

When the Landau–Lifshitz pseudotensor was formulated it was commonly assumed that the cosmological constant, ${\displaystyle \Lambda \,}$, was zero. Nowadays we don't make that assumption, and the expression needs the addition of a ${\displaystyle \Lambda }$ term, giving:

$${\displaystyle t_{LL}^{\mu \nu }=-{\frac {c^{4}}{8\pi G}}\left(G^{\mu \nu }+\Lambda g^{\mu \nu }\right)+{\frac {c^{4}}{16\pi G(-g)}}\left(\left(-g\right)\left(g^{\mu \nu }g^{\alpha \beta }-g^{\mu \alpha }g^{\nu \beta }\right)\right)_{,\alpha \beta }}$$

This is necessary for consistency with the Einstein field equations.

Landau & Lifshitz also provide two equivalent but longer expressions for the Landau–Lifshitz pseudotensor:

• Metric tensor version:[2]
  $${\displaystyle {\begin{aligned}(-g)\left(t_{LL}^{\mu \nu }+{\frac {c^{4}\Lambda g^{\mu \nu }}{8\pi G}}\right)={\frac {c^{4}}{16\pi G}}{\bigg [}&\left({\sqrt {-g}}g^{\mu \nu }\right)_{,\alpha }\left({\sqrt {-g}}g^{\alpha \beta }\right)_{,\beta }-\left({\sqrt {-g}}g^{\mu \alpha }\right)_{,\alpha }\left({\sqrt {-g}}g^{\nu \beta }\right)_{,\beta }+{}\\&{\frac {1}{8}}\left(2g^{\mu \alpha }g^{\nu \beta }-g^{\mu \nu }g^{\alpha \beta }\right)\left(2g_{\sigma \rho }g_{\lambda \omega }-g_{\rho \lambda }g_{\sigma \omega }\right)\left({\sqrt {-g}}g^{\sigma \omega }\right)_{,\alpha }\left({\sqrt {-g}}g^{\rho \lambda }\right)_{,\beta }-{}\\&\left(g^{\mu \alpha }g_{\beta \sigma }\left({\sqrt {-g}}g^{\nu \sigma }\right)_{,\rho }\left({\sqrt {-g}}g^{\beta \rho }\right)_{,\alpha }+g^{\nu \alpha }g_{\beta \sigma }\left({\sqrt {-g}}g^{\mu \sigma }\right)_{,\rho }\left({\sqrt {-g}}g^{\beta \rho }\right)_{,\alpha }\right)+{}\\&\left.{\frac {1}{2}}g^{\mu \nu }g_{\alpha \beta }\left({\sqrt {-g}}g^{\alpha \sigma }\right)_{,\rho }\left({\sqrt {-g}}g^{\rho \beta }\right)_{,\sigma }+g_{\alpha \beta }g^{\sigma \rho }\left({\sqrt {-g}}g^{\mu \alpha }\right)_{,\sigma }\left({\sqrt {-g}}g^{\nu \beta }\right)_{,\rho }\right]\end{aligned}}}$$

  
• Affine connection version:[3]
  $${\displaystyle {\begin{aligned}t_{LL}^{\mu \nu }+{\frac {c^{4}\Lambda g^{\mu \nu }}{8\pi G}}={\frac {c^{4}}{16\pi G}}{\Big [}&\left(2\Gamma _{\alpha \beta }^{\sigma }\Gamma _{\sigma \rho }^{\rho }-\Gamma _{\alpha \rho }^{\sigma }\Gamma _{\beta \sigma }^{\rho }-\Gamma _{\alpha \sigma }^{\sigma }\Gamma _{\beta \rho }^{\rho }\right)\left(g^{\mu \alpha }g^{\nu \beta }-g^{\mu \nu }g^{\alpha \beta }\right)+{}\\&\left(\Gamma _{\alpha \rho }^{\nu }\Gamma _{\beta \sigma }^{\rho }+\Gamma _{\beta \sigma }^{\nu }\Gamma _{\alpha \rho }^{\rho }-\Gamma _{\sigma \rho }^{\nu }\Gamma _{\alpha \beta }^{\rho }-\Gamma _{\alpha \beta }^{\nu }\Gamma _{\sigma \rho }^{\rho }\right)g^{\mu \alpha }g^{\beta \sigma }+\\&\left(\Gamma _{\alpha \rho }^{\mu }\Gamma _{\beta \sigma }^{\rho }+\Gamma _{\beta \sigma }^{\mu }\Gamma _{\alpha \rho }^{\rho }-\Gamma _{\sigma \rho }^{\mu }\Gamma _{\alpha \beta }^{\rho }-\Gamma _{\alpha \beta }^{\mu }\Gamma _{\sigma \rho }^{\rho }\right)g^{\nu \alpha }g^{\beta \sigma }+\\&\left.\left(\Gamma _{\alpha \sigma }^{\mu }\Gamma _{\beta \rho }^{\nu }-\Gamma _{\alpha \beta }^{\mu }\Gamma _{\sigma \rho }^{\nu }\right)g^{\alpha \beta }g^{\sigma \rho }\right]\end{aligned}}}$$

  

This definition of energy–momentum is covariantly applicable not just under Lorentz transformations, but also under general coordinate transformations.