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The Lanczos tensor or Lanczos potential is a rank 3 tensor in general relativity that generates the Weyl tensor.〔Hyôitirô Takeno, "On the spintensor of Lanczos", ''Tensor'', 15 (1964) pp. 103–119.〕 It was first introduced by Cornelius Lanczos in 1949.〔Cornelius Lanczos, "Lagrangian Multiplier and Riemannian Spaces", ''Rev. Mod. Phys.'', 21 (1949) pp. 497–502. 〕 The theoretical importance of the Lanczos tensor is that it serves as the gauge field for the gravitational field in the same way that, by analogy, the electromagnetic four-potential generates the electromagnetic field.〔P. O’Donnell and H. Pye, "A Brief Historical Review of the Important Developments in Lanczos Potential Theory", ''EJTP'', 7 (2010) pp. 327–350. 〕〔M. Novello and A. L. Velloso, "The Connection Between General Observers and Lanczos Potential", ''General Relativity and Gravitation'', 19 (1987) pp. 1251-1265. 〕 ==Definition== The Lanczos tensor can be defined in a few different ways. The most common modern definition is through the Weyl–Lanczos equations, which demonstrate the generation of the Weyl tensor from the Lanczos tensor.〔 These equations, presented below, were given by Takeno in 1964.〔 The way that Lanczos introduced the tensor originally was as a Lagrange multiplier〔〔Cornelius Lanczos, "The Splitting of the Riemann Tensor", ''Rev. Mod. Phys.'', 34 (1962) pp. 379–389. 〕 on constraint terms studied in the variational approach to general relativity.〔Cornelius Lanczos, "A Remarkable Property of the Riemann–Christoffel Tensor in Four Dimensions", ''Annals of Mathematics'', 39 (1938) pp. 842-850. 〕 Under any definition, the Lanczos tensor exhibits the following symmetries: : : The Lanczos tensor always exists in four dimensions〔F. Bampi and G. Caviglia, "Third-order tensor potentials for the Riemann and Weyl tensors", ''General Relativity and Gravitation'', 15 (1983) pp. 375-386. 〕 but does not generalize to higher dimensions.〔S. B. Edgar, "Nonexistence of the Lanczos potential for the Riemann tensor in higher dimensions", ''General Relativity and Gravitation'', 26 (1994) pp. 329-332. 〕 This highlights the specialness of four dimensions.〔 Note further that the full Riemann tensor cannot in general be derived from derivatives of the Lanczos potential alone.〔〔E. Massa and E. Pagani, "Is the Riemann tensor derivable from a tensor potential?", ''General Relativity and Gravitation'', 16 (1984) pp. 805-816. 〕 The Einstein field equations must provide the Ricci tensor to complete the components of the Ricci decomposition. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lanczos tensor」の詳細全文を読む スポンサード リンク
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