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Lanczos Potential of Weyl Field: Interpretations and Applications Eur. Phys. J. C (2021) 81:194 https://doi.org/10.1140/epjc/s10052-021-08981-5 Regular Article - Theoretical Physics Lanczos potential of Weyl field: interpretations and applications Ram Gopal Vishwakarmaa Unidad Académica de Matemáticas, Universidad Autónoma de Zacatecas, Zacatecas, ZAC, Mexico Received: 27 August 2020 / Accepted: 16 February 2021 / Published online: 27 February 2021 © The Author(s) 2021 Abstract An attempt is made to uncover the physical mean- tion towards an unnoticed ingredient of the spacetime, which ing and significance of the obscure Lanczos tensor field is impregnated with extraordinary scientific and philosophi- which is regarded as a potential of the Weyl field. Despite cal values and is expected to contribute to our understanding being a fundamental building block of any metric theory of of the fundamental nature of gravitation itself. This missing gravity, the Lanczos tensor has not been paid proper atten- link is a rank three tensor – the Lanczos tensor Lμνσ – appear- tion as it deserves. By providing an elucidation on this ten- ing as a potential for the Weyl tensor (which represents the sor field through its derivation in some particularly chosen true gravitational degrees of freedom). As we know, the Rie- spacetimes, we try to find its adequate interpretation. Though mann curvature tensor plays the nodal point for the unfold- the Lanczos field is traditionally introduced as a gravita- ing of gravity in the spacetime manifold. However, strangely tional analogue of the electromagnetic 4-potential field, the enough, the full curvature tensor has been pushed into back- performed study unearths its another feature – a relativistic ground. What comes forth, through the field equations of analogue of the Newtonian gravitational force field. A new gravitation, is only the trace of the Riemann tensor (the Ricci domain of applicability of the Lanczos tensor is introduced tensor and Ricci scalar), due to its relation to the energy– which corroborates this new feature of the tensor. momentum tensor of matter. This has led to a certain eclipse of the full curvature tensor. It is the Lanczos tensor which fulfills this gap by providing the trace-free part of the Rie- 1 Introduction mann tensor (the Weyl tensor) – exactly those components of the tensor that are not embraced by the field equations. Thus, General relativity (GR), which constitutes the current descrip- the understanding of the spacetime structure, and hence that tion of gravitation in modern physics, appears to be afflicted of gravity, cannot be complete without the Lanczos tensor with at least two major difficulties. Firstly, it shows an field. intrinsic difficulty in its unification with the rest of physics, Interestingly the Lanczos tensor, discovered by Cornelius which is perhaps the most profound foundational problem in Lanczos in 1962 [2], happens to exist, by coincidence or physics. Secondly, its cosmological application – the CDM providence, in any Riemannian manifold with Lorentzian model – faces challenges on both theoretical and observa- signature in four dimensions. It generates the Weyl tensor tional grounds, despite its overall success and simplicity. It Cμνσρ of the manifold through the Weyl–Lanczos equation1 is now claimed that the current observations show evidence of physics beyond the CDM model [1]. This is an alarm- ing signal to scrutinize the very foundations of the theory. It seems that out of the four fundamental interactions, the most 1 Notations adopted: the starred symbol denotes the dual operation ∗ ∗ = 1 ρστδ familiar to us is the most mysterious one. Perhaps the present defined by N αβμν 4 eαβρσ eμντδ N ,witheμνσρ representing description of this interaction is incomplete just because we the Levi–Civita tensor. The semicolon (comma) followed by an index are not taking into account the whole system of elements denotes covariant (ordinary) derivative with respect to the correspond- ing variable. (Sometimes we shall also use ∇α to represent covariant responsible for the structure of spacetime. derivative with respect to xα.) The square brackets [] enclosing indices ≡ 1 ( − ) Science thrives on crisis. Hard times call for new ideas and denote skew-symmetrization, for instance, X[μν] 2! Xμν Xνμ . insights. In this view, it would be worthwhile to draw atten- Similarly, the round brackets () enclosing indices denote symmetriza- ≡ 1 ( + ) tion, i.e., X(μν) 2! Xμν Xνμ . The Greek indices range over the values 0, 1, 2, 3 where 0 is temporal and 1, 2, 3 are spatial. For simplic- a e-mail: [email protected] (corresponding author) ity, we have considered the geometric units with G = c = 1. 123 194 Page 2 of 14 Eur. Phys. J. C (2021) 81 :194 Cμνσρ = L[μν][σ;ρ] + L[σρ][μ;ν] −∗L∗[μν][σ;ρ] the connections, then how is the Lanczos tensor related with −∗L∗[σρ][μ;ν]. (1) the connections? After exemplifying this tensor field by deriving its values The Lanczos tensor Lμνσ receives a natural and adequate in various spacetimes in the following sections, we discover interpretation in terms of the deep analogy between grav- some possible applications thereof in some gravitational phe- itation and electrodynamics. In electrodynamics, a crucial nomena. This may help to assign an adequate physical inter- ingredient of the electromagnetic field is its 4-potential Aμ, pretation to the tensor. whence emanates the Faraday tensor Fμν: Fμν = Aμ;ν − Aν;μ = Aμ,ν − Aν,μ, (2) 2 Lanczos potential tensor and its non-uniqueness which measures the strength of the electromagnetic field. As Lanczos derived his tensor from a variational principle Let us recall that the Weyl tensor Cμνσρ is the gravitational and since the latter need not be physically consistent always, analogue of the electromagnetic Fμν. Hence, the relativis- a more robust existence theorem was needed. This theorem tic potential Lμνσ generating the Weyl tensor differentially, was provided by Bampi and Caviglia, which guarantees the should be regarded as the gravitational analogue of the elec- existence of such a potential in any 4-dimensional2 Rieman- tromagnetic potential Aμ generating the field strength tensor nian manifold [8]. It should however be noted that this tensor Fμν. This constitutes the Lanczos potential as a more funda- is not unique in a given spacetime. Let us first emphasize that mental geometrical object than the Weyl tensor. the tensor Lμνσ satisfies the generating Eq. (1) only if it pos- The theory of this potential tensor contains many sur- sesses the following two symmetries: prises and insights. For example, it has been shown that the Minkowski spacetime also admits non-trivial values of Lμνσ =−Lνμσ , (3a) κ the Lanczos potential [3,4], proclaiming that the Lanczos L[μνσ] = 0 ⇔∗Lμκ= 0. (3b) potential is a property of spacetime itself rather than an out- come of a particular theory of gravity. This simple but far- Let us simplify the Weyl–Lanczos Eq. (1) by evaluating the reaching insight of mathematical and philosophical value, duals appearing in it, giving revolutionizes our views on the foundational nature of space- = + − − time itself indicating towards an important substantive nature Cμνσρ Lμνσ;ρ Lσρμ;ν Lμνρ;σ Lσρν;μ of spacetime and its geometry. This also seems consistent +gνσ L(μρ) + gμρ L(νσ) with quantum theory wherein the vacuum state of a quantum −gνρ L(μσ) − gμσ L(νρ) theory is not nothing. It would be up to the further studies to 2 λκ + L (gμσ gνρ − gνσ gμρ ), (4) decide whether this revolutionary insight is just a mathemat- 3 λ;κ ical curiosity or leads to a path to something deeper. Inter- κ κ where Lμν ≡ L − L . One may check that this estingly, the Lanczos potential has already been shown to μν;κ μκ;ν equation indeed admits, by virtue of the two symmetry con- be impregnated with various signatures of quantum physics ditions of Lμνσ givenby(3), all the symmetries of the Weyl [2,3,5]. tensor: Cμνσρ = C[μν][σρ], Cμνσρ = Cσρμν, Cμ[νσρ] = 0, As mentioned above, the Lanczos tensor emerges as a Cλ = 0. While the symmetry condition (3a) reduces the fundamental geometric ingredient in any metric theory of μνλ number of independent components of Lμνσ to 24, the condi- gravity (formulated in a 4-dimensional pseudo Riemannian tion (3b) reduces it down to 20. As the Weyl tensor has only spacetime), irrespective of the field equations of the theory. 10 degrees of freedom, Lanczos considered the following This provides the tensor a status of an inherent structural additional symmetries element in the geometric embodiment of gravity. Albeit its novelty and importance, this remarkable discovery is com- κ Lμκ= 0, (5a) paratively unfamiliar even now – some 60 years after Lanczos κ = , first introduced it, and his ingenious discovery has remained Lμν ;κ 0 (5b) more or less a mathematical curiosity. The main reason for this virtual obscurity is the absence of the physical properties as two gauge conditions in order to reduce the number of of the tensor. It has not been possible so far to ascertain what degrees of freedom present in Lμνσ . The condition (5a) abol- the tensor represents physically, although some attempts have been made recently in this direction [3]. There is also another 2 While the Lanczos potential tensor does not exist in general for dimen- basic question related to the Lanczos potential that needs to sions higher than four [6], a potential tensor of rank 5 indeed exists in all dimensions ≥ 4. Interestingly, the trace of this potential (which amounts be answered. As the Lanczos tensor appears to be the poten- to its double dual) corresponds, in the case of 4-dimensions, to the usual tial of the Weyl field and the latter can be written in terms of Lanczos potential [7]. 123 Eur. Phys. J. C (2021) 81 :194 Page 3 of 14 194 ishes the degeneracy in Lμνσ appearing through the gauge Lanczos potential tensor, we derive its values in some par- transformation ticularly chosen spacetimes in the following.
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