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The Landau-Lifshitz Pseudotensor Another Meaningless Concoction of Mathematical Symbols
Stephen J. Crothers 25 January 2020
sjcrothers@plasmaresources.com

Abstract: In an attempt to make A. Einstein’s General Theory of Relativity comply with the usual conservation of energy and momentum for a closed system which a vast array of experiments has ascertained, Mr. L. Landau and Mr. E. Lifshitz constructed, ad hoc, their pseudotensor, as a proposed improvement upon the pseudotensor of Mr. Einstein. Their pseudotensor is symmetric (Mr. Einstein’s is not) and, they say, it permits a conservation law including angular momentum. That it is not a tensor is outside the very mathematical structure of Mr. Einstein’s theory. Beyond that, it violates the rules of pure mathematics. It is therefore a meaningless concoction of mathematical symbols.

Proof of Another Meaningless Concoction of Mathematical Symbols The Riemann-Christoffel symbol of the first kind, denoted by Γαβγ, and of the second kind, denoted Γαβγ, are defined by,

Γαβγ

=

1 2

∂gαβ ∂xγ

−

∂gβγ ∂xα

+

∂gγα ∂xβ

,

Γαβγ

=

1 gαω 2

∂gωγ ∂xβ

−

∂gγβ ∂xω

+

∂gβω ∂xγ

(1)

Note that Γαβγ and Γαβγ are not tensors and that the first is composed solely from the first dreivatives of the metric tensor, the second solely from the components of the metric tensor and their first derivatives.
The Landau-Lifshitz pseudotensor is denoted tik, defined by [1],

tik

=

c4 {
16πk

gilgkm

− gikglm

2ΓnlmΓnpp − ΓnlpΓmp n − ΓnlnΓmp p

+gilgmn ΓklpΓmp n + ΓkmnΓlpp − ΓknpΓlpm − ΓklmΓnpp

(2)

+gklgmn ΓilpΓmp n + ΓimnΓlpp − ΓinpΓlpm − ΓilmΓnpp

+glmgnp ΓilnΓkmp − ΓilmΓknp }

where k in the opening denominator is a constant, c the speed of light in vacuum.

Although tik is not a tensor, Mr. Einstein’s followers claim that it acts ‘like a tensor’ under linear transformations of

coordinates, so it is, they claim, meaningful, both mathematically and physically. Since it acts ‘like a tensor’, a superscript

can be lowered, thus,

tri

=

grk tik

=

c4 {
16πk

grk gil gkm

− grkgikglm

2ΓnlmΓnpp − ΓnlpΓmp n − ΓnlnΓmp p

+gilgmn ΓrlpΓmp n + ΓrmnΓlpp − ΓrnpΓlpm − ΓrlmΓnpp

(3)

+grkgklgmn ΓilpΓmp n + ΓimnΓlpp − ΓinpΓlpm − ΓilmΓnpp

+glmgnp ΓilnΓrmp − ΓilmΓrnp }

and then this can be contracted ‘like a tensor’, to produce an invariant t, thus,

t

=

tii

=

c4 {
16πk

δlk gkm

− glm

2ΓnlmΓnpp − ΓnlpΓmp n − ΓnlnΓmp p

+gilgmn ΓilpΓmp n + ΓimnΓlpp − ΓinpΓlpm − ΓilmΓnpp

(4)

+gklgmn ΓklpΓmp n + ΓkmnΓlpp − ΓknpΓlpm − ΓklmΓnpp

+glmgnp ΓilnΓimp − ΓilmΓinp }

where δlk is the Kronecker delta∗.
∗δlk = 1 when k = l, but = 0 when k l

1

From eqs.(1) it is easily seen that the invariant eq.(4) is a first-order intrinsic differential invariant; that is, it is an invariant composed solely of the components of the metric tensor and their first derivatives. But the pure mathematicians proved [2, 3], in 1900, that first-order intrinsic differential invariants do not exist. Thus, the Landau-Lifshitz pseudotensor is a meaningless concoction of mathematical symbols and therefore cannot be used to represent anything and cannot be used to do calculations; just the same as Mr. Einstein’s meaningless concocted pseudotensor [4, 5]. Nevertheless, Mr. Einstein’s followers sometimes use it, instead of Mr. Einstein’s, to represent the energy-momentum of his gravitational field and hence his gravitational waves, and, miraculously, do calculations with it; a feat beyond the capacity of pure mathematics and rational thought.
General Relativity cannot localise its alleged gravitational energy, so Mr. Einstein’s gravitational waves do not exist [3, 6, 7]. This is the more so since Mr. Einstein’s claim [8] that form-invariance of the Theorem of Pythagoras under Lorentz transformation is form-invariance of his expanding spherical wave of light, is false. Under Lorentz transformation his spherical wave of light is an expanding translated ellipsoidal wave of light with a non-static centre [9]. Consequently the ideological and occult Theory of Relativity is logically inconsistent and therefore false [9–11].
References
[1] Landau, L., Lifshitz, E., The Classical Theory of Fields, Addison-Wesley Press, Inc., Cambridge, Mass., USA, (1951). [2] Ricci-Curbastro, G., Levi-Civita, T., Me´thodes de calcul diffe´rentiel absolu et leurs applications, Matematische Annalen, B. 54, p.162, (1900). [3] Levi-Civita, T., Mechanics. - On the Analytical Expression that Must be Given to the Gravitational Tensor in Einstein’s Theory, Rendiconti della
Reale Accadmeia dei Lincei 26: 381, (1917), eprint: http://arxiv.org/pdf/physics/9906004 [4] Crothers, S.J., Einstein’s Pseudotensor - a Meaningless Concoction of Mathematical Symbols, http://vixra.org/pdf/2001.0499v1.pdf [5] Crothers, S.J., EU 2015 lecture - General Relativity: A Case Study in Numerology, https://www.youtube.com/watch?v=QBorBKDnE3U [6] Crothers, S.J., General Relativity: In Acknowledgement Of Professor Gerardus ’t Hooft, Nobel Laureate, 4 August, (2014),
http://vixra.org/pdf/1409.0072v9.pdf [7] Crothers, S.J., A Critical Analysis of LIGO’s Recent Detection of Gravitational Waves Caused by Merging Black Holes, Hadronic Journal, Vol. 39,
2016, http://vixra.org/pdf/1603.0127v5.pdf [8] Einstein, A., On the electrodynamics of moving bodies, Annalen der Physik, 17, p.891, (1905). [9] Crothers, S.J., Special Relativity and the Lorentz Sphere, Physics Essays, V.33, No.1, p.15, (2020), http://vixra.org/pdf/1911.0013v2.pdf [10] Crothers, S.J., On the Logical Inconsistency of the Special Theory of Relativity, American Journal of Modern Physics, Vol.6, No.3, pp.43-48, (2017),
http://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20170603.12.pdf http://vixra.org/pdf/1703.0047v6.pdf [11] Crothers, S.J., Special Relativity: its Inconsistency with the Standard Wave Equation, Physics Essays, 31, 2, pp. 137, 2018, http://vixra.org/pdf/1708.0055v2.pdf
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