Landau-Lifschitz gravitational energy-momentum pseudotensor

Lyudmil Antonov
lantonov.here@gmail.com

This Mathematica notebook is a detailed workout of the main part of [1] (the Book). The derivation of Landau-Lifshitz pseudotensor is rightly considered as one of the hardest and most calculation-intensive parts of the Book. This is especially true for the calculation of the pseudotensor in terms of connections (eq. LL96,8) and metric densities (eq. LL96,9)(LL in front of the equation number means that the equation is from the Book). The Book chapter §96 starts with relatively simple expressions but very quickly they turn into a morass of indices through substitutions and decompositions. The calculation of the final formulae can proceed by several different paths with branching in different steps of the calculation chain. A somewhat different and quite original approach for finding a similar pseudotensor is pursued the J.L.Synge' s masterpiece [2], where a surprising shortcut avoids a large part of the tedium. In a subsequent post, I'll show evidence that Synge's pseudotensor is somewhat different than the Landau-Lifshitz (LL) pseudotensor despite the widespread opinion that they are just different derivations of the same pseudotensor.

Computations in this research field, although relatively uncomplicated, were not, until recently, amenable to computer algebra treatment because most tensor packages in general purpose programs, such as Mathematica and Maple, hadn't the full capability to work directly with indexed tensors (indicial notation) and most functions acted on components. This involved conversion of indexed tensors to matrices of components and operation with the matrices, which is computationally intensive and in many cases, not really required. On the other hand, tensor algebra in textbooks and papers is performed on indexed tensors, whose component structure in many cases is not known or not relevant for the problem at hand. More recent tensor packages, like Tensorial and xTensor for Mathematica, Riegeom for Maple, and itensor (indicial tensor) for Maxima allow direct manipulation of indexed tensors. The following exposition uses xTensor by José M.Martín - García at the xAct site and some of the accompanying packages. The software version used is Mathematica 11.0. The Landau-Lifshitz pseudotensor example is a good test of xTensor as it clearly outlines its strong and weak points.

The simplification procedures in the package xTensor are automated to a very large degree and use the so-called "canonicalization", a process in which individual terms in a tensor sum are converted in some canonical form that makes them more uniform and non-reducible. On the other hand, automatism means less transparency. Therefore, when deriving LL96,8 and LL96,9 the automatic routine will be accompanied with somewhat less automatic and more transparent calculations.

Energy and momentum conservation principle in general relativity

Two of the most popular definitions of the combined Energy and Momentum Conservation Principle (EMCP) are:

  1. the energy and momentum of a closed isolated system are conserved
  2. the stress - energy - momentum tensor of a closed isolated system is a constant quantity

The second definition is preferred in theoretical physics. The stress-energy-momentum (SEM) tensor, $T^k_i$, that combines energy, mass, momentum, their fluxes and stresses (also called "super energy-momentum tensor", etc.), is the content of the Universe. It is both the matter and the electromagnetic fields. SEM is derived by varying the action integral and is defined in eq. LL32,3 as:

$$T_i^k = q_{,i}\frac{\partial \Lambda }{\partial q_{,k}} - \Lambda \delta _i^k$$

where Λ is the Lagrange function and q is a generalised coordinate. EMCP requires that the SEM tensor is constant. Imagine the closed isolated system as a volume (Universe, black hole), enclosed by an impenetrable envelope such that no matter or radiation can pass through. The SEM tensor within the enclosed volume is constant: there is no tensor flow through the envelope. Tensor flow, or tensor divergence, is the sum of partial derivatives of tensor components with respect to coordinates (similar to vector divergence). From the equations of motion it follows that divergence should be zero according to LL32,4:

$$\frac{\partial T_i^k}{\partial x^k} = T_{i,k}^k = 0$$

which is, of course, the EMCP. In the middle of the above equation the ordinary derivatives are given with the simpler comma notation. In a similar mood, covariant derivatives will be marked with semi-colons.

Presence of gravitation spoils this clear picture in several ways. Gravitation curves space. In curved space, tensor (and vector) flows are ill-defined because they change direction in each point. Parallel transfer, which is essential for definition of derivatives, is very complicated on curved paths. Derivatives are only obtained after making corrections for the curvature – one correction for each tensor index. These corrected derivatives are known as covariant derivatives. In curved space, ordinary derivatives are replaced by covariant ones:

$$ T_i^l \Gamma _{kl}^k -\Gamma _{ik}^l T_l^k+\frac{\partial T_i^k}{\partial x^k} \equiv T_{i;k}^k $$

But $T^{ik}$ is a symmetric tensor, and, according to eq. LL86,11 the above becomes (eq. LL96,1):

$$T_{i;k}^k=\frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g} T_i^k}{\partial x^k} - \frac{1}{2}T^{kl}\frac{\partial g_{kl}}{\partial x^i}$$

Does this equation express the conservation of energy and momentum in curved space-time or, in other words, can it be turned to zero? The answer is emphatically NO and that' s why : Let $T^{ik}$ be the true stress - energy tensor and $P^i$ be the 4-momentum (energy + momentum) of the system that must be a conserved quantity. By an analogy with eq. LL32,6

$$P^i=\frac{1}{c} \int \sqrt{-g} T^{ik} dS_k$$

where $\sqrt{-g}dS_k$ is the element of a hypersphere in a curved plane. In order to make the expression for $P^k$ exactly identical to that in eq. LL32, 6, $T^{ik}$ should be equated to the stress-energy of the system (this is only by analogy, and does not reflect the tensor properties of $T^{ik}$, as the Book footnote warns). Then, in order to conserve the 4-vector $P^i$, the integral (LL32,6) must be conserved over the whole hyperplane, that is, over the whole 3-dimensional space. As proved in Book, Ch. 29 with the help of Gauss theorem, the integrand must fulfill the following condition in order to conserve the integral:

$$\frac{\partial T^{ik}}{\partial x^k} = 0.$$

This condition is not, however, met in eq. LL96,1, where the additional term $-\frac{1}{2} T^{kl} \frac{\partial g_{kl}}{\partial x^i}$ and the presence of $\sqrt{-g}$ around $T^{ik}$ do not allow derivation of a global conservation law in a manner, analogous to the Classical Field Theory. This is so, because the gravitational field is also a component of $T^{ik}$ which should be included inside it, together with pure matter and electromagnetic field. The offending terms cannot be eliminated because the metric tensor $g^{ik}$ is not constant – it can be anything, depending on the spacetime topology. So, in the first term in the right hand side $\sqrt{-g}$ cannot be pulled out from inside the differential operator. However, the real difficulty lies in the term $-\frac{1}{2} T^{kl} \frac{\partial g_{kl}}{\partial x^i}.$

Taking up the above analogy, the volume that contains the SEM tensor itself stretches, shrinks, and moves in every imaginable way. It cannot be bound in an envelope with zero volume. The spacetime itself generates tensor flows. Do what you want, you can't just ignore or remove the second term. Failure to do so means nothing less than the startling conclusion: in the curved spacetime of general relativity, the EMCP is no longer valid. Whether this is real or apparent remains subject to interpretation. It is easy to see, however, that this result can be expected from what we know from elementary physics. In the heart of the concept of the energy conservation principle is the assumption that space is isotropic while the momentum conservation principle requires symmetric space. Both isotropy and symmetry are no longer assumed in curved spacetime. The energy and momentum conservation principles fail in principle. The very basis on which they are built sinks into a quicksand. Is there a way to salvage something?

In search of symmetry

Many efforts have been and are being made to sidetrack this important unsolvable problem. Landau-Lifshitz turned their attention to a couple of possible loopholes.

First, as pointed by Einstein, it is always possible to flatten the space (make it pseudo-Euclidic) in the neighbourhood of a point (locally), or, in other words, it is always possible to locally extinguish the field by, e.g., choosing such a frame that all first derivatives of the metric $\frac{\partial g_{ik}}{\partial x^l} \equiv g_{ik,l}$ vanish. Then, at this point, the second term in (LL96,1) disappears and by dint of eq. 86,4 : $\frac{\partial g}{\partial x^l} = gg^{ik} g_{ik,l}$, the first derivatives of g also vanish. Equation LL96,1:

$$T_{i;k}^k=\frac{1}{\sqrt{-g}}\frac{\partial \left(\sqrt{-g} T_i^k\right)}{\partial x^k}-\frac{1}{2}T^{kl} \frac{\partial g_{kl}}{\partial x^i}$$

becomes after eliminating the first derivatives of the metric and metric determinant:

$$T_{i;k}^k=\frac{\partial T_i^k}{\partial x^k}=T_{i,k}^k$$

so that the offending terms disappear; this means that the tiny volume around this point can be enclosed in an envelope into which EMCP is valid. In contravariant form it is:

$$\frac{\partial T^{ik}}{\partial x^k} = 0$$

Second, the above identity does not fix $T^{ik}$ exactly but leaves a quantity undefined. Namely, according to LL32,7, one can always find an antisymmetric tensor $\eta^{ikl} = -\eta^{ilk}$ such that if $T^{ik} = \eta^{ikl}_{,l}$ then $T^{ik}_{,k} = \eta^{ikl}_{,lk} \equiv 0$. It makes use of the fact that the second derivatives of any antisymmetric tensor disappear because of the symmetry of the differentiation operator. We prove this with the automatic simplification routine of xTensor:

DefTensor[η[i,k,l],M4,Antisymmetric[{2,3}]];

** DefTensor: Defining tensor η[i,k,l].

differentiate the above twice by k and l, equate to 0, canonicalize and check if true:

(PD[-l]@PD[-k]@η[i,k,l] == 0) // ToCanonical

true

which proves the identity.

The possibility to locally fulfill EMCP gives the idea to find a SEM tensor that locally has a zero divergence and then correct it to make it valid globally, in curved spacetime. An inkling to the nature of this correction may be taken from the analogous case with the covariant derivatives. There all corrections contain Christoffel symbols which are composed from first derivatives of the metric tensor. The way to make the required SEM tensor is already obvious: it must be such antisymmetric expression that upon differentiation becomes a second derivative which is identically zero. When constructing it, we must throw out all first metric derivatives that would prevent the expression from turning to zero. To put $T^{ik}$ into the required form ($T^{ik} = \eta^{ikl}_{,l}$), Landau & Lifshitz use the right-hand side of the field (Einstein) equations (LL95,5) in contravariant form

RHSEinstein = EinsteinCD[i, k] // EinsteinToRicci

$$R^{ik}-\frac{1}{2}g^{ik}R$$

Define Ricci tensor and Ricci scalar in terms of Riemann tensor

RicciToRiemann = RicciCD[i_, k_] :> metricg[i, m] metricg[k, p] metricg[l, n] RiemannCD[-l, -m, -n, -p];
RicciScalarToRiemann = RicciScalarCD[] :> metricg[l, n] metricg[m, p] RiemannCD[-l, -m, -n, -p];

Decompose the Einstein tensor as usual to Ricci tensor and Ricci scalar and then turn them to Riemann tensor using the above definitions.
The expression is multiplied by a factor of 2 which should subsequently neutralize the ½ in the decomposition. The Ricci tensor is substituted according to (LL92,9) and the scalar curvature according to (LL92,12) both in terms of the Riemann tensor.

RHSEinstein /. RicciToRiemann;
Energy = 2 % //. RicciScalarToRiemann // Expand

$$2 g^{im} g^{kp} g^{ln} R_{lmnp} - g^{ik} g^{ln} g^{mp} R_{lmnp}$$

Define Riemann tensor as second metric derivatives and Christoffel symbols (first metric derivatives) according to LL92, 1.

RiemannSecondDerivatives = RiemannCD[-i_, -k_, -l_, -m_] :> 1/2 (-PD[-m][PD[-k][metricg[-i, -l]]] + PD[-l][PD[-k][metricg[-i, -m]]] + PD[-m][PD[-i][metricg[-k, -l]]] - PD[-l][PD[-i][metricg[-k, -m]]]) + metricg[-r, -q] (ChristoffelCD[r, -k, -l] ChristoffelCD[ q, -i, -m] - ChristoffelCD[r, -k, -m] ChristoffelCD[q, -i, -l]);

Express the curvature part of the Einstein equations (the Einstein tensor) as second metric derivatives and Christoffels

EnergyTensor = Energy /. RiemannSecondDerivatives // ToCanonical

$$ \begin{multline} \Gamma^{l}{}_{mn} \Gamma^{p}{}_{qr} g^{im} g^{kq} g_{lp} g^{nr} - \tfrac{1}{2} \Gamma^{l}{}_{mn} \Gamma^{p}{}_{qr} g^{ik} g_{lp} g^{mq} g^{nr} - \Gamma^{l}{}_{mn} \Gamma^{p}{}_{qr} g^{im} g^{kn} g_{lp} g^{qr} + \\ \tfrac{1}{2} \Gamma^{l}{}_{mn} \Gamma^{p}{}_{qr} g^{ik} g_{lp} g^{mn} g^{qr} - \tfrac{1}{2} g^{il} g^{km} g^{np} g_{np}{}_{,l}{}_{,m} + \tfrac{1}{2} g^{il} g^{km} g^{np} g_{mn}{}_{,l}{}_{,p} + \tfrac{1}{2} g^{il} g^{km} g^{np} g_{ln}{}_{,m}{}_{,p} - \\ \tfrac{1}{2} g^{ik} g^{lm} g^{np} g_{ln}{}_{,m}{}_{,p} - \tfrac{1}{2} g^{il} g^{km} g^{np} g_{lm}{}_{,n}{}_{,p} + \tfrac{1}{2} g^{ik} g^{lm} g^{np} g_{lm}{}_{,n}{}_{,p} \end{multline} $$

In the locality first-order derivatives disappear (this means that the connection terms also disappear) which leaves only terms with second-order derivatives (step 1 elimination of first-order derivatives)

% /. {ChristoffelCD[l_, -m_, -n_] -> 0}

$$ -g^{il} g^{km} g^{np} g_{np,m,l} + g^{il} g^{km} g^{np} g_{mn,p,l} + g^{il} g^{km} g^{np} g_{ln,p,m} - g^{ik} g^{lm} g^{np} g_{ln,p,m} - g^{il} g^{km} g^{np} g_{lm,p,n} + g^{ik} g^{lm} g^{np} g_{lm,p,n} $$

Now the dummy indices are shuffled in such a way that the covariant tensors all become $g_{np,m,l}$ and the whole expression is factored

$$ \left ( -g^{il} g^{km} g^{np} + g^{il} g^{kp} g^{mn} + g^{ip} g^{kl} g^{mn} - g^{ik} g^{lp} g^{mn} - g^{in} g^{kp} g^{lm} + g^{ik} g^{np} g^{lm} \right ) g_{np,m,l} $$

The expression in parenthesis can be abbreviated as a 6-rank tensor $W^{iklmnp}$

$$W^{iklmnp} = -g^{il} g^{km} g^{np} + g^{il} g^{kp} g^{mn} + g^{ip} g^{kl} g^{mn} - g^{ik} g^{lp} g^{mn} - g^{in} g^{kp} g^{lm} + g^{ik} g^{np} g^{lm} $$

and according to the well-known formula from calculus, the differentiation can be distributed over the product:

$$g_{np,m,l} W^{iklmnp} = \left ( g_{np,m} W^{iklmnp} \right )_{,l} - g_{np,m} W^{iklmnp}_{,l} $$

The second term in the left-hand side disappears, because it contains first-order metric derivatives (step 2 elimination of first-order derivatives), and $W^{iklmnp}$ is substituted back. The expression in parentheses becomes:

$$ -g^{il} g^{km} g^{np} g_{np,m} + g^{il} g^{kp} g^{mn} g_{np,m} + g^{ip} g^{kl} g^{mn} g_{np,m} - g^{ik} g^{lp} g^{mn} g_{np,m} - g^{in} g^{kp} g^{lm} g_{np,m} + g^{ik} g^{np} g^{lm} g_{np,m} $$

With the help of (LL86, 7): $g^{ln} g_{np,m} = -g_{np} g_{,m}^{ln}$ and (LL86, 4): $g_{,m} = gg^{np} g_{np,m}$, so that $g^{np} g_{np,m} = g_{,m} /g$

$$-\frac{g_{,m} g^{il} g^{km}}{g} + \frac{g_{,m} g^{ik} g^{ml}}{g} + g^{kp}g^{ml}g_{np}g_{,m}^{in} -g^{ip}g^{kl}g_{np}g_{,m}^{mn} - g^{il}g^{kp}g_{np}g_{,m}^{mn} + g^{ik}g^{lp}g_{np}g_{,m}^{mn}$$

(LL83, 10): $g^{kp} g_{np} = \delta_n^k$, where $\delta_n^k$ is the Kronecker symbol

$-\frac{g_{,m}g^{il}g^{km}}{g} + \frac{g_{,m}g^{ik}g^{ml}}{g} + g^{ml}g_{,m}^{in}\delta_n^k -g^{kl}g_{,m}^{mn}\delta_n^i-g^{il}g_{,m}^{mn}\delta_n^k + g^{ik}g_{,m}^{mn}\delta_n^l$

A property of the Kronecker symbol is $\delta _n^i g_{,m}^{mn} = g_{,m}^{im}$

$-\frac{g_{,m}g^{il}g^{km}}{g} + \frac{g_{,m}g^{ik}g^{ml}}{g} + g^{ml}g_{,m}^{ik} -g^{kl}g_{,m}^{im}-g^{il}g_{,m}^{km} + g^{ik}g_{,m}^{lm}$

l and m are dummy indices in the whole expression (when it is doubly differentiated) which allows exchanging their places

$-\frac{g_{,m}g^{il}g^{km}}{g} + \frac{g_{,m}g^{ik}g^{ml}}{g} + g^{lm}g_{,m}^{ik} -g^{km}g_{,m}^{il}-g^{il}g_{,m}^{km} + g^{ik}g_{,m}^{lm}$

The calculus formula for differentiation of a product (in reverse):

$-\frac{g_{,m} g^{il} g^{km} }{g} + \frac{g_{,m} g^{ik} g^{ml} }{g} + \left( g^{ik} g^{lm} \right)_{,m} - \left( g^{il} g^{km} \right)_{,m}$

The distributive property of the differentiation operator, collection of $g_{,m}$, and substitution $U^{iklm} = g^{ik}g^{lm} - g^{il} g^{km}$

$\frac{1}{g} g_{,m} \left( g^{ik} g^{lm} - g^{il} g^{km} \right) + \left( g^{ik} g^{lm} -g^{il} g^{km} \right)_{,m} = \frac{1}{g} g_{,m} \left( U^{iklm} \right) + \left ( U^{iklm} \right )_{,m}$

Once more the calculus formula for differentiation of a product in reverse and substitutions of $g_{,m} = -(-g)_{,m}$ and $g = -(-g)$

$\frac{1}{(-g)} \left [ (-g) U^{iklm} \right ]_{,m} = \frac{1}{(-g)} \left [(-g) \left( g^{ik} g^{lm} - g^{il} g^{km} \right )\right ]_{,m}$

This leads to (LL96, 2) for $h^{ikl} = \frac{\partial}{\partial x^m} \lambda^{iklm}$, which is the numerator of the above expression, multiplied by the constant $\frac{c^4}{16 \pi k}$

It is easy to check (LL96, 4) $h^{ikl} = -h^{ilk}$. With a precision to a constant factor, $h^{ikl}$ corresponds to the expression in the parentheses of the term $(g_{np,m} W^{iklmnp})_{,l}$. The factor $1/(-g)$ is also a constant in the local case, so it can be pulled from under the operator ,l. For the whole term, which corresponds to the stress-energy tensor in the neighborhood of a point, thus

$\frac{1}{(-g)} h^{ikl}_{,l} = T^{ik}$

$ h^{ikl}_{,l} = (-g) T^{ik}$

$ (-g) T^{ik} = \frac{c^4}{16 \pi k} \left [ (-g) \left ( g^{ik} g^{lm} - g^{il} g^{km} \right ) \right ]_{,m,l}$

To summarize, the final expression for the SEM tensor in the special point as a result of the above transformations is:

$\frac{16 \pi k}{c^4} T^{ik} = \frac{1}{(-g)} \left [(-g) \left( g^{ik} g^{lm} - g^{il} g^{km} \right) \right]_{,m,l}$

For the following discussion we need this expression somewhat rearranged. Namely, the constant $\frac{c^4}{16 \pi k}$ is put under ,l (but not under ,m) and both sides are multiplied by (−g). After these "simple transformations" (according to Landau-Lifshitz), we obtained a SEM tensor in a completely antisymmetric form:

$(-g) T^{ik} = \left\{ \frac{c^4}{16 \pi k} \left[(-g) \left(g^{ik} g^{lm} - g^{il} g^{km} \right) \right]_{,m} \right\}_{,l}$

The quantity in the braces corresponds to $h^{ikl} = \eta^{ikl}$

$\eta^{ikl} = \frac{c^4}{16 \pi k} \left[(-g) \left(g^{ik} g^{lm} - g^{il} g^{km} \right) \right]_{,m}$

and is antisymmetric for indices k, l while the quantity under the brackets corresponds to (LL96,3)

$\lambda^{iklm} = \frac{c^4}{16 \pi k} (-g) \left(g^{ik} g^{lm} -g^{il} g^{km} \right)$

Note that first order derivatives we eliminated at two steps. $(−g)T^{ik}$ vanishes upon differentiation by ,k (obeys EMCP) as checked below.

DefTensor[T[i, k], M4, Symmetric[{1, 2}]]

** DefTensor: Defining tensor T[i,k].

As seen from above

$(-g)T^{ik}_{,k} = \eta^{ikl}_{,l,k}$

and $\eta^{ikl}_{,l,k} \equiv 0$ as proven above. To prove that $\eta^{ikl}$ also disappears in the metric form when doubly differentiated

etaikl = PD[-m][-Detmetricg[]*(metricg[i, k]* metricg[l, m] - metricg[i, l]*metricg[k, m])];
Simplification[PD[-l]@PD[-k]@etaikl] == 0

true

Finally, let's prove that $T^{ik}$ expressed in metric derivatives is symmetric

gTik = PD[-l][ PD[-m][-Detmetricg[] (metricg[i, k] metricg[l, m] - metricg[i, l] metricg[k, m])]];
ToCanonical[Antisymmetrize[gTik]] == 0

true

Landau-Lifshitz pseudotensor in canonical form

The nice and pliable SEM tensor obtained above whose divergence disappears automatically just by dint of symmetry (or, more precisely, antisymmetry) so that it fulfills EMCP, does not work outside the very small flat locality of the point that we chose. Remember that in its derivation above we threw out first derivatives of the metric tensor at a few steps. First derivatives usually combine to form connections (Christoffel symbols) and these last are directly connected to curvature. So we may expect that outside our flat neighborhood first derivatives will reappear to play a prominent role as corrections for curvature in the same way as they act in covariant derivatives. We have every reason to expect, then, that $(-g) T^{ik}$ will no longer be equal to $\eta_{,l}^{ikl}$ but there will be some non-zero difference that is the quantity that corrects for the transition from local to global geometry, from flat to curved spacetime: $\eta_{,l}^{ikl} - (-g) T^{ik} \neq 0$. Let's define this difference as $(−g) t^{ik}$; then by definition $(-g) \left( T^{ik} + t^{ik} \right) = \eta_{,l}^{ikl}$ (LL96, 5) so that $t^{ik} = \frac{1}{-g} \eta_{,l}^{ikl} - T^{ik}$. We can't tell much about the quantities $t^{ik}$ except that they have absorbed all the first derivatives which we threw out before, and also that they are symmetric: $t^{ik} = t^{ki}$. Symmetry follows from the definition of $t^{ik}$, since they are composed of $T^{ik}$ and $\eta^{ikl}_{,l}$ which are symmetric as we proved at the end of the last section. The method to find $t^{ik}$ that Landau-Lifshitz propose, is direct but excessively long and tedious : ... after a rather lengthy calculation we find the following expression .... One seldom finds in their book such a phrase at the place of the gory detail of the intermediate calculations though it is an understatement as we see in the derivation of LL96, 8 that follows. The intensive computations include transitions from Christoffel symbols to metrics and vice versa with the active participation of the metric determinant and its derivatives in multiple terms.

Enter the antisymmetric form which was enclosed in the parenthesis of the pseudotensor in the special point ($ g^{ik} g^{lm} - g^{il} g^{km}$ ) and then double differentiate it to obtain the expression for $\eta_{,l}^{ikl} = \left [(-g) \left(g^{ik} g^{lm} - g^{il} g^{km} \right ) \right]_{,m,l}$ in expanded form which is a mixture of first and second metric derivatives

AntisymTerm = (metricg[i, k] metricg[l, m] - metricg[i, l] metricg[k, m]);
DifAntiSym = -1/2/Detmetricg[] PD[-l][PD[-m][-Detmetricg[] AntisymTerm]];

Now obtain the expression $t^{ik} = \frac{1}{-g} \eta_{,l}^{ikl} - T^{ik}$ in first metric derivatives (canonical form) simultaneously turning the Christoffels in $T^{ik}$ to metric derivatives. Because the canonical form is important for the subsequent derivations, we will number the individual terms and colour them with the help of the module ExpressionManipulation in xAct

LLCanonical = (DifAntiSym - 1/2*EnergyTensor) // ChristoffelToGradMetric // ToCanonical;
% // ColorTerms

Simplification is very efficient : out of the 103 terms in the original expanded and unsimplified expression, only 16 very similar terms are left in the canonical form. Each term consists of 4 contravariant metric tensors and 2 first derivatives of covariant metrics. Note that the second metric derivatives cancel completely; this is to be expected since the second derivatives in $\eta^{ikl}_{,l}$ are the same as those in $T^{ik}$ both coming from the decomposed Riemann tensor. Exactly the same canonical form of the Landau-Lifshitz pseudotensor was computed by Sergei A. Klioner using his package EinS in Mathematica [3].

As said above, the Landau-Lifshitz pseudotensor is symmetric for the free indices i and k but this is not evident from the canonical form. This can be checked by antisymmetrizing $t^{ik}$ (i.e. making the expression $\frac{1}{2}\left(t^{ik}-t^{ki}\right)$). If $t^{ik}$ is symmetric, then antisymmetrizing should give 0.

ToCanonical[Antisymmetrize[LLCanonical, {i, k}]] == 0

true

This proves that the Landau-Lifshitz pseudotensor is symmetric; however, it remains pseudotensor in the sense that it changes with change of coordinates, it changes from place to place, or, in other words, it cannot be localized. This is because the canonical form contains ordinary partial derivatives (commas) instead of covariant derivatives (semi-colons). Ordinary derivatives depend on coordinates unlike covariant derivatives which are invariant. If a combination can be found in which all derivatives are covariant then the pseudotensor will turn to tensor. I am not aware of any attempt to do this, nor if there is any chance to succeed.

Landau-Lifshitz pseudotensor in connections (LL96, 8)

Several generations of theoretical physics students (and many of their professors) have repeatedly given up in their efforts to derive the above formula. Arguably, it's the most difficult in the Book. Even LL, in their succinct and dispassionate style couldn't hide the tedium when deriving it: ...after a rather lengthy calculation, we find the following expression for $t^{ik}$ .... Here, we'll first derive LL96,8 following the LL way with a somewhat less automated use of xTensor to show the intermediate calculations, then derive LL96,8 from the canonical pseudotensor in a fully automated way and checking that formulae derived by the two ways are identical. Last, we'll check that LL96,8 in the form given in the Book is identical to the formulae derived by us.

The shortest path to obtain the LL pseudotensor in terms of Christoffel connections is to start from LL96,2 $$h^{ikl} = \lambda^{iklm}{}_{,m}$$ with $\lambda^{iklm}$ from LL96,3 $$\lambda^{iklm} = \frac{c^4}{16 \pi k} (-g) (g^{ik} g^{lm} - g^{il} g^{km}).$$ For shortness of the following formulae, instead of the constant $\frac{c^4}{16 \pi k}$ we'll use just $\frac{1}{2}$. Jumping below, from LL96,7 we have $$t^{ik} = \frac{1}{(-g)} h^{ikl}{}_{,l} - \left( R^{ik} - \frac{1}{2} g^{ik} R \right)$$ For the first term on the right-hand side we have $$\frac{1}{(-g)} h^{ikl}{}_{,l} = \frac{1}{(-g)} \lambda^{iklm}{}_{,m,l} = \frac{1}{(-g)} \left[ \frac{1}{2}(-g) (g^{ik} g^{lm} - g^{il} g^{km}) \right]_{,m,l}$$

Now it's time to use xTensor but before we do it is necessary to make a couple of adjustments in order to avoid its automatic canonicalisation which could prevent us from seeing the intermediate calculations. First, instead of treating $g^{ik}$ as a metric, we'll define it just as an ordinary symmetric tensor. This is because xTensor automatically canonicalises the upper-index metric derivatives $g^{ik}{}_{,l}$ to lower-index metric derivatives $g^{il}g^{km}g_{lm,n}$ and there is no way to switch off this compulsory canonicalisation. Second, we will treat the scalar $g$ not as the metric determinant but just like an ordinary scalar because the derivative $g_{,l}$ automatically becomes $g g^{ik}g_{ik,l}$ (formula LL86,4), and, guess what, this automatic feature cannot be switched off either. So, yes, this automatic canonicalisation eases the calculation of long expressions but intermediate calculations are harder and longer for humans although easier for the machine. Soon, though, we'll use this automatism to obtain LL96,8 from the canonical pseudotensor and to cross-check its correctness.

DefTensor[g[i, k], {M4}, Symmetric[{1, 2}], PrintAs -> "g"];

** DefTensor: Defining tensor g[i,k].

Above we defined an ordinary symmetric $g^{ik}$ tensor on the 4-manifold

lambdaiklm = -1/2 g (g[i, k] g[l, m] - g[i, l] g[k, m]);

and expressed $\lambda^{iklm}$ in terms of the just defined $g^{ik}$ and the ordinary scalar $g$. Differentiating $\lambda^{iklm}$ by index $m$ and canonicalising

FirstPD = PD[-m][lambdaiklm] // ToCanonical

$$\tfrac{1}{2}g^{il}g^{km}g_{,m} - \tfrac{1}{2}g^{ik}g^{lm}g_{,m} - \tfrac{1}{2}gg^{lm}g^{ik}{}_{,m} + \tfrac{1}{2}gg^{km}g^{il}{}_{,m} + \tfrac{1}{2}gg^{il}g^{km}{}_{,m} - \tfrac{1}{2}gg^{ik}g^{lm}{}_{,m}$$

we obtained 6 terms with 3 multipliers each (excluding the coefficients). Thus, if nothing is cancelled, we expect 18 terms after the second differentiation by $l$. Let's see

SecondPD = PD[-l][FirstPD] // ToCanonical

$$\begin{multline} -g^{lm}g_{,l}g^{ik}{}_{,m} + \tfrac{1}{2}g^{km}g_{,l}g^{il}{}_{,m} + \tfrac{1}{2}gg^{km}{}_{,l}g^{il}{}_{,m} + \tfrac{1}{2}g^{kl}g_{,l}g^{im}{}_{,m} + \tfrac{1}{2}g^{im}g_{,l}g^{kl}{}_{,m} + \tfrac{1}{2}g^{il}g_{,l}g^{km}{}_{,m} + \tfrac{1}{2}gg^{il}{}_{,l}g^{km}{}_{,m} - \\ g^{ik}g_{,l}g^{lm}{}_{,m} - gg^{ik}{}_{,l}g^{lm}{}_{,m} + \tfrac{1}{2}g^{il}g^{km}g_{,l,m} - \tfrac{1}{2}g^{ik}g^{lm}g_{,l,m} - \tfrac{1}{2}gg^{lm}g^{ik}{}_{,l,m} + \tfrac{1}{2}gg^{kl}g^{im}{}_{,l,m} + \tfrac{1}{2}gg^{il}g^{km}{}_{,l,m} - \tfrac{1}{2}gg^{ik}g^{lm}{}_{,l,m} \end{multline}$$

These are 15 terms, so canonicalisation has effectively cancelled 3 terms (those without the coefficient $\frac{1}{2}$). Next, we'll transform all derivatives to Christoffel connections using LL86,8 $$g^{ik}{}_{,l} = -\Gamma^{i}_{ml}g^{mk} - \Gamma^{k}_{ml}g^{im}$$ and LL86,5 in the form $$g_{,k} = 2 g \Gamma^{i}_{ki}$$ and then divide the whole to $(-g)$. This is the major part of the calculation because of the many substitutions. xTensor, however, remains unimpressed.

SecondPD //. MakeRule[{PD[-l][g[i, k]], - g[k, m] ChristoffelCD[i, -m, -l] - g[i, m] ChristoffelCD[k, -m, -l]}];
% //. MakeRule[{PD[-l][g], 2 g ChristoffelCD[p, -l, -p]}]

$\begin{multline} -2 g \Gamma^{p}{}_{lp} (- \Gamma^{k}{}_{nm} g^{in} - \Gamma^{i}{}_{nm} g^{kn}) g^{lm} + g \Gamma^{p}{}_{lp} g^{km} (- \Gamma^{l}{}_{nm} g^{in} - \Gamma^{i}{}_{nm} g^{ln}) + g \Gamma^{p}{}_{lp} g^{im} (- \Gamma^{l}{}_{nm} g^{kn} - \Gamma^{k}{}_{nm} g^{ln}) +\\ g \Gamma^{p}{}_{lp} g^{kl} (- \Gamma^{m}{}_{nm} g^{in} - \Gamma^{i}{}_{nm} g^{mn}) + \tfrac{1}{2} g (- \Gamma^{l}{}_{pm} g^{ip} - \Gamma^{i}{}_{pm} g^{lp}) (- \Gamma^{m}{}_{nl} g^{kn} - \Gamma^{k}{}_{nl} g^{mn}) +\\ g \Gamma^{p}{}_{lp} g^{il} (- \Gamma^{m}{}_{nm} g^{kn} - \Gamma^{k}{}_{nm} g^{mn}) - 2 g \Gamma^{p}{}_{lp} g^{ik} (- \Gamma^{m}{}_{nm} g^{ln} - \Gamma^{l}{}_{nm} g^{mn}) + \tfrac{1}{2} g (- \Gamma^{l}{}_{nl} g^{in} -\\ \Gamma^{i}{}_{nl} g^{ln}) (- \Gamma^{m}{}_{pm} g^{kp} - \Gamma^{k}{}_{pm} g^{mp}) - g (- \Gamma^{k}{}_{nl} g^{in} - \Gamma^{i}{}_{nl} g^{kn}) (- \Gamma^{m}{}_{pm} g^{lp} - \Gamma^{l}{}_{pm} g^{mp}) -\\ \tfrac{1}{2} g g^{lm} [- \Gamma^{k}{}_{nl} (-\Gamma^{n}{}_{pm} g^{ip} - \Gamma^{i}{}_{pm} g^{np}) - \Gamma^{i}{}_{nl} (- \Gamma^{n}{}_{qm} g^{kq} - \Gamma^{k}{}_{qm} g^{nq}) - g^{kn} \Gamma^{i}{}_{nl}{}_{,m} - g^{in} \Gamma^{k}{}_{nl}{}_{,m} ] +\\ \tfrac{1}{2} g g^{kl} [- \Gamma^{m}{}_{nl} (- \Gamma^{n}{}_{pm} g^{ip} - \Gamma^{i}{}_{pm} g^{np}) - \Gamma^{i}{}_{nl} (- \Gamma^{n}{}_{qm} g^{mq} - \Gamma^{m}{}_{qm} g^{nq}) - g^{mn} \Gamma^{i}{}_{nl}{}_{,m} - g^{in} \Gamma^{m}{}_{nl}{}_{,m} ] + \\ \tfrac{1}{2} g g^{il} [- \Gamma^{m}{}_{nl} (- \Gamma^{n}{}_{pm} g^{kp} - \Gamma^{k}{}_{pm} g^{np}) - \Gamma^{k}{}_{nl} (- \Gamma^{n}{}_{qm} g^{mq} - \Gamma^{m}{}_{qm} g^{nq}) - g^{mn} \Gamma^{k}{}_{nl}{}_{,m} - g^{kn} \Gamma^{m}{}_{nl}{}_{,m}] - \\ \tfrac{1}{2} g g^{ik} [- \Gamma^{m}{}_{nl} (- \Gamma^{n}{}_{pm} g^{lp} - \Gamma^{l}{}_{pm} g^{np}) - \Gamma^{l}{}_{nl} (- \Gamma^{n}{}_{qm} g^{mq} - \Gamma^{m}{}_{qm} g^{nq}) - g^{mn} \Gamma^{l}{}_{nl}{}_{,m} - g^{ln} \Gamma^{m}{}_{nl}{}_{,m} ] +\\ g^{il} g^{km} (2 g \Gamma^{n}{}_{ln} \Gamma^{p}{}_{mp} + g \Gamma^{n}{}_{ln}{}_{,m}) - g^{ik} g^{lm} (2 g \Gamma^{n}{}_{ln} \Gamma^{p}{}_{mp} + g \Gamma^{n}{}_{ln}{}_{,m}) \end{multline}$

Now divide to $(-g)$ and canonicalise to obtain a more tractable expression

hikll = %/(-g) // ToCanonical

$$\begin{multline} - \tfrac{1}{2} \Gamma^{l}{}_{lm} \Gamma^{n}{}_{np} g^{im} g^{kp} - \tfrac{3}{2} \Gamma^{l}{}_{mn} \Gamma^{m}{}_{lp} g^{in} g^{kp} + 2 \Gamma^{l}{}_{lm} \Gamma^{m}{}_{np} g^{in} g^{kp} + \tfrac{1}{2} \Gamma^{k}{}_{lm} \Gamma^{n}{}_{np} g^{ip} g^{lm} + \\ \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{n}{}_{np} g^{kp} g^{lm} + \tfrac{1}{2} \Gamma^{l}{}_{lm} \Gamma^{n}{}_{np} g^{ik} g^{mp} - \tfrac{1}{2} \Gamma^{k}{}_{lm} \Gamma^{n}{}_{np} g^{il} g^{mp} - \tfrac{1}{2} \Gamma^{k}{}_{lm} \Gamma^{l}{}_{np} g^{in} g^{mp} - \\ \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{n}{}_{np} g^{kl} g^{mp} - \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{l}{}_{np} g^{kn} g^{mp} + \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{k}{}_{np} g^{ln} g^{mp} + \Gamma^{l}{}_{mn} \Gamma^{m}{}_{lp} g^{ik} g^{np} - \\ \tfrac{3}{2} \Gamma^{l}{}_{lm} \Gamma^{m}{}_{np} g^{ik} g^{np} + \tfrac{1}{2} \Gamma^{k}{}_{lm} \Gamma^{l}{}_{np} g^{im} g^{np} + \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{l}{}_{np} g^{km} g^{np} - \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{k}{}_{np} g^{lm} g^{np} - \\ g^{il} g^{km} \Gamma^{n}{}_{ln}{}_{,m} + \tfrac{1}{2} g^{ik} g^{lm} \Gamma^{n}{}_{ln}{}_{,m} + g^{il} g^{km} \Gamma^{n}{}_{lm}{}_{,n} - \tfrac{1}{2} g^{ik} g^{lm} \Gamma^{n}{}_{lm}{}_{,n} \end{multline}$$

This is 15 terms with single derivatives and 4 terms with double derivatives (the bottom row).

Further, for the decomposition of the Einstein tensor $G^{ik} = R^{ik} - \frac{1}{2}g^{ik}R$ we'll go not as low as Riemann tensor but will stay at the level of Ricci tensor. We'll need the latter with lower indices achieved with $R^{ik} = g^{il}g^{km}R_{lm}$. For the Ricci scalar, we'll use LL92,9 $R = g^{ik}R_{ik}$. The decomposition of the Ricci tensor is to the level of Christoffels and not lower according to LL92,7 $$R_{ik} = \Gamma^{l}{}_{ik,l} - \Gamma^{l}{}_{il,k} + \Gamma^{l}{}_{ik} \Gamma^{m}{}_{lm} - \Gamma^{m}{}_{il} \Gamma^{l}{}_{km}$$ The whole Einstein tensor becomes $$G^{ik} = \left(g^{il}g^{km} - \tfrac{1}{2}g^{ik}g^{lm}\right) \left(\Gamma^{n}{}_{lm,n} - \Gamma^{n}{}_{ln,m} + \Gamma^{n}{}_{lm} \Gamma^{p}{}_{np} - \Gamma^{p}{}_{ln} \Gamma^{n}{}_{mp}\right)$$ In the language of xTensor this becomes

Gik = RicciCD[i, k] - 1/2 g[i, k] RicciScalarCD[];
RicciDown = RicciCD[i_, k_] :> g[i, l] g[k, m] RicciCD[-l, -m];
RicciScalarDown = RicciScalarCD[] :> g[l, m] RicciCD[-l, -m];
RicciToChristoffel = MakeRule[{RicciCD[-i, -k], PD[-l][ChristoffelCD[l, -i, -k]] - PD[-k][ChristoffelCD[l, -i, -l]] + ChristoffelCD[l, -i, -k] ChristoffelCD[m, -l, -m] - ChristoffelCD[m, -i, -l] ChristoffelCD[l, -k, -m]}];
Gik /. {RicciDown, RicciScalarDown};
Gik = % /. RicciToChristoffel // ToCanonical

$$\begin{multline} - \Gamma^{l}{}_{mn} \Gamma^{m}{}_{lp} g^{in} g^{kp} + \Gamma^{l}{}_{lm} \Gamma^{m}{}_{np} g^{in} g^{kp} + \tfrac{1}{2} \Gamma^{l}{}_{mn} \Gamma^{m}{}_{lp} g^{ik} g^{np} - \tfrac{1}{2} \Gamma^{l}{}_{lm} \Gamma^{m}{}_{np} g^{ik} g^{np} - \\ g^{il} g^{km} \Gamma^{n}{}_{ln}{}_{,m} + \tfrac{1}{2} g^{ik} g^{lm} \Gamma^{n}{}_{ln}{}_{,m} + g^{il} g^{km} \Gamma^{n}{}_{lm}{}_{,n} - \tfrac{1}{2} g^{ik} g^{lm} \Gamma^{n}{}_{lm}{}_{,n} \end{multline}$$

Note that the bottom rows of $\frac{1}{(-g)} h^{ikl}{}_{,l}$ and $R^{ik} - \frac{1}{2} g^{ik}R$, the terms that contain the second derivatives, are identical. Therefore, when we subtract $G^{ik}$ from $\frac{1}{(-g)} h^{ikl}{}_{,l}$ in the process of obtaining $t^{ik}$, the second derivatives cancel and we are left with $t^{ik}$ containing only first derivatives.

tik = (hikll - Gik) // ToCanonical

$$\begin{multline} - \tfrac{1}{2} \Gamma^{l}{}_{lm} \Gamma^{n}{}_{np} g^{im} g^{kp} - \tfrac{1}{2} \Gamma^{l}{}_{mn} \Gamma^{m}{}_{lp} g^{in} g^{kp} + \Gamma^{l}{}_{lm} \Gamma^{m}{}_{np} g^{in} g^{kp} + \tfrac{1}{2} \Gamma^{k}{}_{lm} \Gamma^{n}{}_{np} g^{ip} g^{lm} + \\ \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{n}{}_{np} g^{kp} g^{lm} + \tfrac{1}{2} \Gamma^{l}{}_{lm} \Gamma^{n}{}_{np} g^{ik} g^{mp} - \tfrac{1}{2} \Gamma^{k}{}_{lm} \Gamma^{n}{}_{np} g^{il} g^{mp} - \tfrac{1}{2} \Gamma^{k}{}_{lm} \Gamma^{l}{}_{np} g^{in} g^{mp} - \\ \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{n}{}_{np} g^{kl} g^{mp} - \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{l}{}_{np} g^{kn} g^{mp} + \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{k}{}_{np} g^{ln} g^{mp} + \tfrac{1}{2} \Gamma^{l}{}_{mn} \Gamma^{m}{}_{lp} g^{ik} g^{np} - \\ \Gamma^{l}{}_{lm} \Gamma^{m}{}_{np} g^{ik} g^{np} + \tfrac{1}{2} \Gamma^{k}{}_{lm} \Gamma^{l}{}_{np} g^{im} g^{np} + \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{l}{}_{np} g^{km} g^{np} - \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{k}{}_{np} g^{lm} g^{np} \end{multline}$$

This is the expression for $t^{ik}$ corresponding to LL96,8.

Obtaining LL96,8 from the canonical pseudotensor (the one with lower metric derivatives) is rather straightforward and automatic with the function GradMetricToChristoffel. This function is nothing more that an inbuilt formula $g_{ik,l} = g_{mk} \Gamma^{m}{}_{il} + g_{im} \Gamma^{m}_{kl}$. The intermediate result without canonicalisation is expected to have $16 \times 4 = 64$ terms.

PseudoInChristoffels = LLCanonical // GradMetricToChristoffel

$$\begin{multline} \tfrac{1}{2} g^{il} g^{km} (\Gamma^{s}{}_{qn} g_{ls} + \Gamma^{u}{}_{ql} g_{nu}) g^{np} (\Gamma^{v}{}_{rp} g_{mv} + \Gamma^{w}{}_{rm} g_{pw}) g^{qr} - \tfrac{1}{8} g^{ik} g^{lm} (\Gamma^{s}{}_{qn} g_{ls} + \Gamma^{u}{}_{ql} g_{nu}) g^{np} (\Gamma^{v}{}_{rp} g_{mv} + \Gamma^{w}{}_{rm} g_{pw}) g^{qr} - \\ \tfrac{1}{2} g^{il} g^{km} (\Gamma^{s}{}_{pn} g_{ls} + \Gamma^{u}{}_{pl} g_{nu}) g^{np} (\Gamma^{v}{}_{rq} g_{mv} + \Gamma^{w}{}_{rm} g_{qw}) g^{qr} + \tfrac{1}{2} g^{il} g^{km} (\Gamma^{s}{}_{nm} g_{ls} + \Gamma^{u}{}_{nl} g_{mu}) g^{np} (\Gamma^{v}{}_{rq} g_{pv} + \Gamma^{w}{}_{rp} g_{qw}) g^{qr} - \\ \tfrac{1}{2} g^{ik} g^{lm} (\Gamma^{s}{}_{nm} g_{ls} + \Gamma^{u}{}_{nl} g_{mu}) g^{np} (\Gamma^{v}{}_{rq} g_{pv} + \Gamma^{w}{}_{rp} g_{qw}) g^{qr} - \tfrac{1}{2} g^{il} g^{km} (\Gamma^{v}{}_{qn} g_{mv} + \Gamma^{w}{}_{qm} g_{nw}) g^{np} g^{qr} (\Gamma^{s}{}_{lr} g_{ps} + \Gamma^{u}{}_{lp} g_{ru}) + \\ \tfrac{1}{2} g^{il} g^{km} (\Gamma^{v}{}_{pn} g_{mv} + \Gamma^{w}{}_{pm} g_{nw}) g^{np} g^{qr} (\Gamma^{s}{}_{lr} g_{qs} + \Gamma^{u}{}_{lq} g_{ru}) - \tfrac{1}{2} g^{il} g^{km} (\Gamma^{v}{}_{qn} g_{lv} + \Gamma^{w}{}_{ql} g_{nw}) g^{np} g^{qr} (\Gamma^{s}{}_{mr} g_{ps} + \Gamma^{u}{}_{mp} g_{ru}) + \\ \tfrac{1}{2} g^{il} g^{km} (\Gamma^{v}{}_{pn} g_{lv} + \Gamma^{w}{}_{pl} g_{nw}) g^{np} g^{qr} (\Gamma^{s}{}_{mr} g_{qs} + \Gamma^{u}{}_{mq} g_{ru}) + \tfrac{1}{4} g^{ik} g^{lm} (\Gamma^{v}{}_{qn} g_{lv} + \Gamma^{w}{}_{ql} g_{nw}) g^{np} g^{qr} (\Gamma^{s}{}_{pr} g_{ms} + \Gamma^{u}{}_{pm} g_{ru}) + \\ \tfrac{1}{4} g^{il} g^{km} g^{np} (\Gamma^{s}{}_{lq} g_{ns} + \Gamma^{u}{}_{ln} g_{qu}) g^{qr} (\Gamma^{v}{}_{mr} g_{pv} + \Gamma^{w}{}_{mp} g_{rw}) - \tfrac{1}{2} g^{il} g^{km} g^{np} (\Gamma^{s}{}_{lp} g_{ns} + \Gamma^{u}{}_{ln} g_{pu}) g^{qr} (\Gamma^{v}{}_{mr} g_{qv} + \Gamma^{w}{}_{mq} g_{rw}) - \\ \tfrac{3}{4} g^{il} g^{km} (\Gamma^{s}{}_{nm} g_{ls} + \Gamma^{u}{}_{nl} g_{mu}) g^{np} g^{qr} (\Gamma^{v}{}_{pr} g_{qv} + \Gamma^{w}{}_{pq} g_{rw}) + \tfrac{3}{8} g^{ik} g^{lm} (\Gamma^{s}{}_{nm} g_{ls} + \Gamma^{u}{}_{nl} g_{mu}) g^{np} g^{qr} (\Gamma^{v}{}_{pr} g_{qv} + \Gamma^{w}{}_{pq} g_{rw}) + \\ \tfrac{1}{4} g^{il} g^{km} (\Gamma^{s}{}_{ln} g_{ms} + \Gamma^{u}{}_{lm} g_{nu}) g^{np} g^{qr} (\Gamma^{v}{}_{pr} g_{qv} + \Gamma^{w}{}_{pq} g_{rw}) + \tfrac{1}{4} g^{il} g^{km} (\Gamma^{s}{}_{mn} g_{ls} + \Gamma^{u}{}_{ml} g_{nu}) g^{np} g^{qr} (\Gamma^{v}{}_{pr} g_{qv} + \Gamma^{w}{}_{pq} g_{rw}) \end{multline}$$

Next, we'll canonicalise (simplify) the above expression with the function ToCanonical and then use function ColorTerms from subpackage ExpressionManipulation in xAct to colorise and number terms.

FromCanonical = PseudoInChristoffels // ToCanonical // ColorTerms

Visual comparison between the LL pseudotensor obtained by the way given in the Book and the one obtained from the canonical pseudotensor shows that they are identical even down to the term ordering. But we can use the machine to check it.

(tik - FromCanonical) == 0

true

We proved that the pseudotensors obtained by two different ways are identical but we still have not proven that these are identical to LL96,8 as printed in the Book. For this purpose, we'll manually enter LL96,8 exactly as in the Book:

ManualEnter = (-(ChristoffelCD[n, -l, -p]*ChristoffelCD[p, -m, -n])/ 2 - (ChristoffelCD[n, -l, -n]*ChristoffelCD[p, -m, -p])/2 + ChristoffelCD[n, -l, -m]*ChristoffelCD[p, -n, -p])*metricg[i, l]* metricg[k, m] + ((ChristoffelCD[n, -l, -p]*ChristoffelCD[p, -m, -n] + ChristoffelCD[n, -l, -n]*ChristoffelCD[p, -m, -p] - 2*ChristoffelCD[n, -l, -m]*ChristoffelCD[p, -n, -p])* metricg[i, k]*metricg[l, m])/ 2 + ((-(ChristoffelCD[k, -n, -p]*ChristoffelCD[p, -l, -m]) + ChristoffelCD[k, -m, -n]*ChristoffelCD[p, -l, -p] + ChristoffelCD[k, -l, -p]*ChristoffelCD[p, -m, -n] - ChristoffelCD[k, -l, -m]*ChristoffelCD[p, -n, -p])* metricg[i, l]*metricg[m, n])/ 2 + ((-(ChristoffelCD[i, -n, -p]*ChristoffelCD[p, -l, -m]) + ChristoffelCD[i, -m, -n]*ChristoffelCD[p, -l, -p] + ChristoffelCD[i, -l, -p]*ChristoffelCD[p, -m, -n] - ChristoffelCD[i, -l, -m]*ChristoffelCD[p, -n, -p])* metricg[k, l]*metricg[m, n])/ 2 + ((ChristoffelCD[i, -l, -n]*ChristoffelCD[k, -m, -p] - ChristoffelCD[i, -l, -m]*ChristoffelCD[k, -n, -p])* metricg[l, m]*metricg[n, p])/2

If the automatically found pseudotensor is identical to the manually entered one, the difference between the two should be zero.

ToCanonical[PseudoInChristoffels - ManualEnter] == 0

true

Yes, they are identical.

Another way to prove the manually entered LL96,8 is to transform it to the canonical form with the xTensor routine ChristoffelToGradMetric, simplify it with the tensorial simplification (contractions, correct re-indexing), expand it and then subtract the automatically obtained canonical pseudotensor. The result should be again zero.

ManualEnter // ChristoffelToGradMetric // Simplification // Expand;
Simplification[% - LLCanonical] == 0

true

Correct again for both manual and automatic.
All this proves that the pseudotensor in Christoffels that we obtained automatically with GradMetricToChristoffel in principle can be made the same as LL96,8 by rearranging the dummy indexes individually in the 16 terms so that the metric combinations with equal indexes could be factored in the same terms as in LL96,8. However, xTensor automatically re-indexes with the default index combinations putting the Christoffel indices in front of the metric tensors and I could not find a way to switch this off. Correspondingly, the index ordering begins at the Christoffel symbols and metric tensors take the later dummies. So the custom indexing and then collecting has to be done manually as I have done in previous chapters. This is done below and it is easy to see that if we factor by metric tensors, we shall come exactly to LL96,8.

$$\begin{multline} - \tfrac{1}{2} \Gamma^{n}{}_{ln} \Gamma^{p}{}_{mp} g^{il} g^{km} - \tfrac{1}{2} \Gamma^{n}{}_{lp} \Gamma^{p}{}_{mn} g^{il} g^{km} + \Gamma^{n}{}_{lm} \Gamma^{p}{}_{np} g^{il} g^{km} + \tfrac{1}{2} \Gamma^{k}{}_{mn} \Gamma^{p}{}_{lp} g^{il} g^{mn} + \tfrac{1}{2} \Gamma^{i}{}_{mn} \Gamma^{p}{}_{lp} g^{kl} g^{mn} + \\ \tfrac{1}{2} \Gamma^{n}{}_{ln} \Gamma^{p}{}_{mp} g^{ik} g^{lm} - \tfrac{1}{2} \Gamma^{k}{}_{lm} \Gamma^{p}{}_{np} g^{il} g^{mn} - \tfrac{1}{2} \Gamma^{k}{}_{np} \Gamma^{p}{}_{lm} g^{il} g^{mn} - \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{p}{}_{np} g^{kl} g^{mn} - \tfrac{1}{2} \Gamma^{i}{}_{np} \Gamma^{p}{}_{lm} g^{kl} g^{mn} + \\ \tfrac{1}{2} \Gamma^{i}{}_{ln} \Gamma^{k}{}_{mp} g^{lm} g^{np} + \tfrac{1}{2} \Gamma^{n}{}_{lp} \Gamma^{p}{}_{mn} g^{ik} g^{lm} - \Gamma^{n}{}_{lm} \Gamma^{p}{}_{np} g^{ik} g^{lm} + \tfrac{1}{2} \Gamma^{k}{}_{lp} \Gamma^{p}{}_{mn} g^{il} g^{mn} + \tfrac{1}{2} \Gamma^{i}{}_{lp} \Gamma^{p}{}_{mn} g^{kl} g^{mn} - \tfrac{1}{2} \Gamma^{i}{}_{lm} \Gamma^{k}{}_{np} g^{lm} g^{np} \end{multline}$$

Landau-Lifshitz pseudotensor in metric densities (LL96,9)

The most used form of the LL pseudotensor is the one derived in terms of metric densities, or, what some authors call "gothic g metric" because this metric density is defined as $\mathfrak{g}^{ik} = \sqrt{-g} g^{ik}$. As we see above in the formula LL96,9 lifted directly from the Book, all the ordinary derivatives are on the gothic g and there is no derivative on the metric. 

First we need to define the tensor $\mathfrak{g}$ and expand the expression in terms because it's too much parenthesised.

DefTensor[\[GothicG][i, k], {M4}, Symmetric[{1, 2}]];

** DefTensor: Defining tensor \[GothicG][i,k].

In order to show more intermediate calculations, we'll again use a general symmetric tensor and a scalar instead of metric tensor and its determinant.

g[] LLCanonical //. MakeRule[{metricg[-i, -k], g[-i, -k]}];

It is most important to transform all $g_{ik,l}$ into $\mathfrak{g}^{ik}{}_{,l}$. Unlike the $\Gamma^{l}{}_{ik}$ form, for the $\mathfrak{g}^{ik}{}_{,l}$ form there are no ready transformation formulae, and we must invent our own. First, we'll transform expressions of the kind $g^{ik}g_{ik,l}$ through a reverse transformation.

$$\begin{multline} g_{ik}\mathfrak{g}^{ik}{}_{,l} = g_{ik}\left( \sqrt{-g} g^{ik} \right)_{,l} = g_{ik} \left[ g^{ik} \left(\sqrt{-g} \right)_{,l} + \sqrt{-g} g^{ik}{}_{,l} \right] = -\frac{g_{ik} g^{ik} g_{,l}}{2 \sqrt{-g}} - \sqrt{-g} g^{ik} g_{ik,l} = \\ - \frac{4 g g^{ik} g_{ik,l}}{2 \sqrt{-g}} - \sqrt{-g} g^{ik} g_{ik,l} = 2 \sqrt{-g} g^{ik} g_{ik,l} - \sqrt{-g} g^{ik} g_{ik,l} = \sqrt{-g} g^{ik} g_{ik,l} \end{multline}$$

Reversing this chain, we have the needed transform

$$g^{ik} g_{ik,l} = \frac{g_{ik}\mathfrak{g}^{ik}{}_{,l}}{\sqrt{-g}}$$

Substituting this transform in the canonical pseudotensor multiplied by $g$ is straightforward

% //. MakeRule[{g[i, k] PD[-l][g[-i, -k]], g[-i, -k] PD[-l][\[GothicG][i, k]]/Sqrt[-g[]]}] // ToCanonical

$$\begin{multline} \tfrac{1}{4} g g^{il} g^{km} g^{np} g^{qr} g_{nq}{}_{,l} g_{pr}{}_{,m} + \tfrac{1}{2} g^{il} g^{km} g_{np} g_{qr} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{qr}{}_{,m} - \tfrac{3}{8} g^{ik} g^{lm} g_{np} g_{qr} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{qr}{}_{,m} - \\ \tfrac{1}{2} \sqrt{-g} g^{il} g^{km} g^{np} g_{qr} \mathfrak{g}^{qr}{}_{,m} g_{ln}{}_{,p} - \tfrac{1}{2} \sqrt{-g} g^{il} g^{km} g^{np} g_{qr} \mathfrak{g}^{qr}{}_{,l} g_{mn}{}_{,p} - \tfrac{1}{4}\sqrt{-g} g^{il} g^{km} g^{np} g_{qr} g_{mn}{}_{,l} \mathfrak{g}^{qr}{}_{,p} - \\ \tfrac{1}{4}\sqrt{-g} g^{il} g^{km} g^{np} g_{qr} g_{ln}{}_{,m} \mathfrak{g}^{qr}{}_{,p} + \tfrac{1}{2} \sqrt{-g} g^{ik} g^{lm} g^{np} g_{qr} g_{ln}{}_{,m} \mathfrak{g}^{qr}{}_{,p} + \tfrac{3}{4} \sqrt{-g} g^{il} g^{km} g^{np} g_{qr} g_{lm}{}_{,n} \mathfrak{g}^{qr}{}_{,p} - \\ \tfrac{1}{2} g g^{il} g^{km} g^{np} g^{qr} g_{pr}{}_{,m} g_{ln}{}_{,q} + \tfrac{1}{4} g g^{ik} g^{lm} g^{np} g^{qr} g_{mr}{}_{,p} g_{ln}{}_{,q} - \tfrac{1}{2} g g^{il} g^{km} g^{np} g^{qr} g_{pr}{}_{,l} g_{mn}{}_{,q} + \\ \tfrac{1}{2} g g^{il} g^{km} g^{np} g^{qr} g_{ln}{}_{,q} g_{mp}{}_{,r} - \tfrac{1}{8} g g^{ik} g^{lm} g^{np} g^{qr} g_{ln}{}_{,q} g_{mp}{}_{,r} - \tfrac{1}{2} g g^{il} g^{km} g^{np} g^{qr} g_{ln}{}_{,p} g_{mq}{}_{,r} + \tfrac{1}{2} g g^{il} g^{km} g^{np} g^{qr} g_{lm}{}_{,n} g_{pq}{}_{,r} \end{multline}$$

The 16 term of the canonical pseudotensor are preserved but the substitution is acting in about half of them. Now we'll make the more frequent transform $g^{im}g_{ik,l}$

$$\begin{multline} g_{im}\mathfrak{g}^{ik}{}_{,l} = g_{im}\left( \sqrt{-g} g^{ik} \right)_{,l} = g_{im} \left[ g^{ik} \left(\sqrt{-g} \right)_{,l} + \sqrt{-g} g^{ik}{}_{,l} \right] = - \frac{g_{im} g^{ik} g_{,l}}{2 \sqrt{-g}} - \sqrt{-g} g^{im} g_{ik,l} = \\ - \frac{g g_{im} g^{ik} g^{np}g_{np,l}}{2 \sqrt{-g}} - \sqrt{-g} g_{im} g^{iq} g^{kr}g_{qr,l} = \tfrac{1}{2}\sqrt{-g} \delta^{k}{}_{m} g^{np} g_{np,l} - \sqrt{-g} \delta^{q}{}_{m} g^{kr} g_{qr,l} = \tfrac{1}{2}\sqrt{-g} \delta^{k}{}_{m} g^{in} g_{in,l} - \sqrt{-g} g^{ik} g_{im,l} \end{multline}$$

From the above we have in reverse

$$g^{ik} g_{im,l} = \tfrac{1}{2}\delta^{k}{}_{m} g^{in} g_{in,l} - \frac{g_{im}\mathfrak{g}^{ik}{}_{,l}}{\sqrt{-g}}$$

However, there is a problem in this transform. The derivative $g_{in,l}$ in the right hand side would cause the apearance of 3 types of terms with 0, 1, and 2 $\mathfrak{g}$ derivatives, respectively. These would not cancel during the canonicalisation. This can be remedied by observing the previous substitution $g^{in} g_{in,l} = g_{in}\mathfrak{g}^{in}{}_{,l} / \sqrt{-g}$. So we have finally

$$g^{ik} g_{im,l} = \tfrac{1}{2}\delta^{k}{}_{m} \frac{g_{in}\mathfrak{g}^{in}{}_{,l}}{\sqrt{-g}} - \frac{g_{im}\mathfrak{g}^{ik}{}_{,l}}{\sqrt{-g}}$$

Raw substitution of this transform is

% //. MakeRule[{g[i, k] PD[-l][g[-i, -m]], 1/2 delta[k, -m] g[-i, -n] PD[-l][\[GothicG][i, n]]/Sqrt[-g[]] - g[-i, -m] PD[-l][\[GothicG][i, k]]/Sqrt[-g[]]}]

$$\begin{multline} \tfrac{1}{2} g^{il} g^{km} g_{np} g_{qr} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{qr}{}_{,m} - \tfrac{3}{8} g^{ik} g^{lm} g_{np} g_{qr} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{qr}{}_{,m} + \\ \tfrac{1}{4} g g^{il} g^{km} \left(\frac{\sqrt{-g} g_{nq} \mathfrak{g}^{np}{}_{,l}}{g} - \frac{\delta_{q}{}^{p} \sqrt{-g} g_{ns} \mathfrak{g}^{ns}{}_{,l}}{2 g}\right) \left(\frac{\sqrt{-g} g_{rp} \mathfrak{g}^{rq}{}_{,m}}{g} - \frac{\delta_{p}{}^{q} \sqrt{-g} g_{ru} \mathfrak{g}^{ru}{}_{,m}}{2 g}\right) - \\ \tfrac{1}{2} \sqrt{-g} g^{km} g^{np} g_{qr} \mathfrak{g}^{qr}{}_{,m} \left(\frac{\sqrt{-g} g_{ln} \mathfrak{g}^{li}{}_{,p}}{g} - \frac{\delta_{n}{}^{i} \sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,p}}{2 g}\right) - \tfrac{1}{2} \sqrt{-g} g^{il} g^{np} g_{qr} \mathfrak{g}^{qr}{}_{,l} \left(\frac{\sqrt{g} g_{mn} \mathfrak{g}^{mk}{}_{,p}}{g} - \frac{\delta_{n}{}^{k} \sqrt{-g} g_{ms} \mathfrak{g}^{ms}{}_{,p}}{2 g}\right) - \\ \tfrac{1}{4} \sqrt{-g} g^{il} g^{np} g_{qr} \left(\frac{\sqrt{-g} g_{mn} \mathfrak{g}^{mk}{}_{,l}}{g} - \frac{\delta_{n}{}^{k} \sqrt{-g} g_{ms} \mathfrak{g}^{ms}{}_{,l}}{2 g}\right) \mathfrak{g}^{qr}{}_{,p} - \tfrac{1}{4} \sqrt{-g} g^{km} g^{np} g_{qr} \left(\frac{\sqrt{-g} g_{ln} \mathfrak{g}^{li}{}_{,m}}{g} - \frac{\delta_{n}{}^{i} \sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,m}}{2 g}\right) \mathfrak{g}^{qr}{}_{,p} + \\ \tfrac{1}{2} \sqrt{-g} g^{ik} g^{np} g_{qr} \left(\frac{\sqrt{-g} g_{ln} \mathfrak{g}^{lm}{}_{,m}}{g} - \frac{\sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,n}}{2 g}\right) \mathfrak{g}^{qr}{}_{,p} + \tfrac{3}{4} \sqrt{-g} g^{km} g^{np} g_{qr} \left(\frac{\sqrt{-g} g_{lm} \mathfrak{g}^{li}{}_{,n}}{g} - \\ \frac{\delta_{m}{}^{i} \sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,n}}{2 g}\right) \mathfrak{g}^{qr}{}_{,p} - \\ \tfrac{1}{2} g g^{km} g^{qr} \left(\frac{\sqrt{-g} g_{pr} \mathfrak{g}^{pn}{}_{,m}}{g} - \frac{\delta_{r}{}^{n} \sqrt{-g} g_{pu} \mathfrak{g}^{pu}{}_{,m}}{2 g}\right) \left(\frac{\sqrt{-g} g_{ln} \mathfrak{g}^{li}{}_{,q}}{g} - \frac{\delta_{n}{}^{i} \sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,q}}{2 g}\right) + \\ \tfrac{1}{4} g g^{ik} g^{np} \left(\frac{\sqrt{-g} g_{rm} \mathfrak{g}^{rq}{}_{,p}}{g} - \frac{\delta_{m}{}^{q} \sqrt{-g} g_{ru} \mathfrak{g}^{ru}{}_{,p}}{2 g}\right) \left(\frac{\sqrt{-g} g_{ln} \mathfrak{g}^{lm}{}_{,q}}{g} - \frac{\delta_{n}{}^{m} \sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,q}}{2 g}\right) - \\ \tfrac{1}{2} g g^{il} g^{qr} \left(\frac{\sqrt{-g} g_{pr} \mathfrak{g}^{pn}{}_{,l}}{g} - \frac{\delta_{r}{}^{n} \sqrt{-g} g_{pu} \mathfrak{g}^{pu}{}_{,l}}{2 g}\right) \left(\frac{\sqrt{-g} g_{mn} \mathfrak{g}^{mk}{}_{,q}}{g} - \frac{\delta_{n}{}^{k} \sqrt{-g} g_{ms} \mathfrak{g}^{ms}{}_{,q}}{2 g}\right) + \\ \tfrac{1}{2} g g^{np} g^{qr} \left(\frac{\sqrt{-g} g_{ln} \mathfrak{g}^{li}{}_{,q}}{g} - \frac{\delta_{n}{}^{i} \sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,q}}{2 g}\right) \left(\frac{\sqrt{-g} g_{mp} \mathfrak{g}^{mk}{}_{,r}}{g} - \frac{\delta_{p}{}^{k} \sqrt{-g} g_{mu} \mathfrak{g}^{mu}{}_{,r}}{2 g}\right) - \\ \tfrac{1}{2} g g^{np} g^{qr} \left(\frac{\sqrt{-g} g_{ln} \mathfrak{g}^{li}{}_{,p}}{g} - \frac{\delta_{n}{}^{i} \sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,p}}{2 g}\right) \left(\frac{\sqrt{-g} g_{mq} \mathfrak{g}^{mk}{}_{,r}}{g} - \frac{\delta_{q}{}^{k} \sqrt{-g} g_{mu} \mathfrak{g}^{mu}{}_{,r}}{2 g}\right) - \\ \tfrac{1}{8} g g^{ik} g^{qr} \left(\frac{\sqrt{-g} g_{ln} \mathfrak{g}^{lm}{}_{,q}}{g} - \frac{\delta_{n}{}^{m} \sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,q}}{2 g}\right) \left(\frac{\sqrt{-g} g_{pm} \mathfrak{g}^{pn}{}_{,r}}{g} - \frac{\delta_{m}{}^{n} \sqrt{-g} g_{pu} \mathfrak{g}^{pu}{}_{,r}}{2 g}\right) + \\ \tfrac{1}{2} g g^{km} g^{qr} \left(\frac{\sqrt{-g} g_{lm} \mathfrak{g}^{li}{}_{,n}}{g} - \frac{\delta_{m}{}^{i} \sqrt{-g} g_{ls} \mathfrak{g}^{ls}{}_{,n}}{2 g}\right) \left(\frac{\sqrt{-g} g_{pq} \mathfrak{g}^{pn}{}_{,r}}{g} - \frac{\delta_{q}{}^{n} \sqrt{-g} g_{pu} \mathfrak{g}^{pu}{}_{,r}}{2 g}\right) \end{multline}$$

Canonicalising this longish and awkward-looking expression with possible 44 terms gives a surprisingly regular expression with only 14 terms (2 terms less than the canonical LL pseudotensor)

% // ToCanonical

$$\begin{multline} - \tfrac{1}{4} g^{il} g^{km} g_{np} g_{qr} \mathfrak{g}^{nq}{}_{,l} \mathfrak{g}^{pr}{}_{,m} + \tfrac{1}{8} g^{ik} g^{lm} g_{np} g_{qr} \mathfrak{g}^{nq}{}_{,l} \mathfrak{g}^{pr}{}_{,m} + \tfrac{1}{8} g^{il} g^{km} g_{np} g_{qr} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{qr}{}_{,m} - \tfrac{1}{16} g^{ik} g^{lm} g_{np} g_{qr} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{qr}{}_{,m} + \\ \tfrac{1}{2} g^{kl} g_{mp} g^{mn} g_{qr} \mathfrak{g}^{pr}{}_{,l} \mathfrak{g}^{iq}{}_{,n} + \tfrac{1}{2} g^{il} g_{mp} g^{mn} g_{qr} \mathfrak{g}^{pr}{}_{,l} \mathfrak{g}^{kq}{}_{,n} - \tfrac{1}{4} g^{kl} g_{lm} g^{np} g_{qr} \mathfrak{g}^{im}{}_{,n} \mathfrak{g}^{qr}{}_{,p} + \tfrac{1}{4} g^{il} g_{lm} g^{np} g_{qr} \mathfrak{g}^{km}{}_{,n} \mathfrak{g}^{qr}{}_{,p} + \\ \tfrac{1}{2} g_{ln} g^{lm} g_{pr} g^{pq} \mathfrak{g}^{in}{}_{,m} \mathfrak{g}^{kr}{}_{,q} - \tfrac{1}{2} g^{kl} g_{lm} g_{nq} g^{np} \mathfrak{g}^{qr}{}_{,p} \mathfrak{g}^{im}{}_{,r} - \tfrac{1}{2} g_{ln} g^{lm} g_{mp} g^{qr} \mathfrak{g}^{in}{}_{,q} \mathfrak{g}^{kp}{}_{,r} - \\ \tfrac{1}{4} g^{ik} g_{ln} g^{lm} g_{pq} \mathfrak{g}^{qr}{}_{,m} \mathfrak{g}^{np}{}_{,r} - \tfrac{3}{8} g^{ik} g_{ln} g^{lm} g_{pq} \mathfrak{g}^{pq}{}_{,m} \mathfrak{g}^{nr}{}_{,r} + \tfrac{3}{8} g^{ik} g_{ln} g^{lm} g_{pq} \mathfrak{g}^{nr}{}_{,m} \mathfrak{g}^{pq}{}_{,r} \end{multline}$$

Because instead of metric we put a regular symmetric tensor, expressions of the type $g^{ik} g_{il}$ are not automatically transformed into $\delta^{k}{}_{l}$. We do this substitution manually to finally obtain

GothicMetric = % //. MakeRule[{g[-m, -p] g[m, n], delta[n, -p]}] // ToCanonical

$$\begin{multline} \tfrac{1}{2} \mathfrak{g}^{il}{}_{,l} \mathfrak{g}^{km}{}_{,m} - \tfrac{1}{2} g^{lm} g_{np} \mathfrak{g}^{in}{}_{,l} \mathfrak{g}^{kp}{}_{,m} - \tfrac{1}{2} \mathfrak{g}^{ik}{}_{,l} \mathfrak{g}^{lm}{}_{,m} - \tfrac{1}{4} g^{il} g^{km} g_{np} g_{qr} \mathfrak{g}^{nq}{}_{,l} \mathfrak{g}^{pr}{}_{,m} + \tfrac{1}{8} g^{ik} g^{lm} g_{np} g_{qr} \mathfrak{g}^{nq}{}_{,l} \mathfrak{g}^{pr}{}_{,m} + \\ \tfrac{1}{8} g^{il} g^{km} g_{np} g_{qr} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{qr}{}_{,m} - \tfrac{1}{16} g^{ik} g^{lm} g_{np} g_{qr} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{qr}{}_{,m} + \tfrac{1}{2} g^{kl} g_{mn} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{im}{}_{,p} + \tfrac{1}{2} g^{il} g_{mn} \mathfrak{g}^{np}{}_{,l} \mathfrak{g}^{km}{}_{,p} - \tfrac{1}{4} g^{ik} g_{lm} \mathfrak{g}^{mp}{}_{,n} \mathfrak{g}^{ln}{}_{,p} \end{multline}$$

Looks like this is it. These are 10 terms like in LL96,9 and their structure at first look is the same. To check this, we'll manually enter LL96,9 exactly in the form given in the Book.

ManualLLDensity = -1/2 (PD[-l][\[GothicG][i, k]] PD[-m][\[GothicG][l, m]] - PD[-l][\[GothicG][i, l]] PD[-m][\[GothicG][k, m]] + 1/2 metricg[i, k] metricg[-l, -m] PD[-p][\[GothicG][l, n]] PD[-n][\[GothicG][p, m]] - metricg[i, l] metricg[-m, -n] PD[-p][\[GothicG][k, n]] PD[-l][\[GothicG][m, p]] - metricg[k, l] metricg[-m, -n] PD[-p][\[GothicG][i, n]] PD[-l][\[GothicG][m, p]] + metricg[-l, -m] metricg[n, p] PD[-n][\[GothicG][i, l]] PD[-p][\[GothicG][k, m]] + 1/8 (2 metricg[i, l] metricg[k, m] -metricg[i, k] metricg[l, m]) (2 metricg[-n, -p] metricg[-q, -r] - metricg[-p, -q] metricg[-n, -r]) PD[-l][\[GothicG][n, r]] PD[-m][\[GothicG][p, q]])

$$\begin{multline} \tfrac{1}{2} \bigl[\mathfrak{g}^{il}{}_{,l} \mathfrak{g}^{km}{}_{,m} - \mathfrak{g}^{ik}{}_{,l} \mathfrak{g}^{lm}{}_{,m} - \tfrac{1}{8} (2 g^{il} g^{km} - g^{ik} g^{lm}) (- g_{nr} g_{pq} + 2 g_{np} g_{qr}) \mathfrak{g}^{nr}{}_{,l} \mathfrak{g}^{pq}{}_{,m} + \\ g^{kl} g_{mn} \mathfrak{g}^{mp}{}_{,l} \mathfrak{g}^{in}{}_{,p} - g_{lm} g^{np} \mathfrak{g}^{il}{}_{,n} \mathfrak{g}^{km}{}_{,p} + g^{il} g_{mn} \mathfrak{g}^{mp}{}_{,l} \mathfrak{g}^{kn}{}_{,p} - \tfrac{1}{2} g^{ik} g_{lm} \mathfrak{g}^{pm}{}_{,n} \mathfrak{g}^{ln}{}_{,p}\bigr] \end{multline}$$

To emphasize the structure, we'll number and colorize terms with ColorTerms

ManualLLDensity // Expand;
% // ColorTerms

If the manually entered pseudotensor is equal to the one obtained here, then their difference should be zero.

ToCanonical[ManualLLDensity - GothicMetric] == 0

true

This proves that the $\mathfrak{g}$ pseudotensor obtained from the canonical pseudotensor is the same as LL96,9. For me, LL96,9 is more difficult than LL96,8 because there is no ready substitution.

Let's check if the substitution $\mathfrak{g}^{ik} = \sqrt{-g} g^{ik}$ can change the gothic g form back in the canonical form, thus checking the legitimacy of the gothic g pseudotensor. Note also that in the comparison we have to multiply the canonical pseudotensor by $g$.

ToCanonical[ManualLLDensity //. {\[GothicG][i_, k_] -> Sqrt[Detmetricg[]] metricg[i, k]}] == ToCanonical[Detmetricg[] LLCanonical]

true

This is a proof that the $\mathfrak{g}$ pseudotensor can be brought back to the canonical pseudotensor.

In conclusion, this post gave the complete derivation of the Landau-Lifshitz pseudotensor in its various forms: in metrics, in Christoffel symbols, and in metric densities. The exposition was directed exclusively towards the computational technicalities and nothing was said about the physical meaning of these forms, e.g., why the gothic metric form is preferred. Such physical considerations, and also a comparison of the LL pseudotensor with the closely related Synge's and Fock pseudotensors will be included in the next post.


1.Chapter 96 "Stress-energy pseudotensor" in The classical theory of fields by Lev Davidovich Landau and Evgeniy Mikhaylovich Lifshitz.Translated by Morton Hamermesh.Edition : 4, illustrated, revised.Published by Butterworth - Heinemann, 1975. ISBN 0750627689, 9780750627689. 402 pages
2.J. L. Synge. Relativity : The General Theory, 1960, Ch.VI, Sect.7