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 corresponding 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. These values will also be used in the later sections while studying some ¯ = + − , Lμνσ Lμνσ gνσ Xμ gμσ Xν (6) possible applications of the tensor. which leaves Eq. (4) invariant for an arbitrary vector field Xα. This arbitrariness in Lμνσ is fixed by the condition (5a) 2.1 Lanczos tensor for Schwarzschild spacetime which gives Xα = 0. Thus the algebraic gauge condition (5a), taken together with (3a) and (3b), reduces the number The simplest example of the Lanczos tensor, which we take of independent components of Lμνσ to 16. from the literature, corresponds to the Schwarzschild line Let us emphasize that the differential gauge condition element (5b), which was adopted by Lanczos just because the diver- κ gence Lμν ;κ does not participate in Eq. (4), does not further 2m dr2 ds2 = 1 − dt2 − reduce the degrees of freedom of Lμνσ , contrary to the wide- r (1 − 2m/r) spread misunderstanding in the literature [3]. Thus ample −r 2dθ 2 − r 2 sin2 θ dφ2, (9) degeneracy remains there in the values of the tensor Lμνσ even after the Lanczos gauge conditions (5a) and (5b)are applied, as has been demonstrated in [3]. In fact, the reason which represents a static spherically symmetric spacetime = for this degeneracy in Lμνσ is due to the presence of an aux- outside an isotropic mass m placed in a Ricci-flat (Rμν iliary potential to the Weyl tensor, discovered by Takeno [4]. 0) manifold. Novello and Velloso [9] have shown that if a α ≡ α/ He noticed that given a Lanczos potential Lμνσ of a space- unit time-like vector field V dx ds tangential to the time, the quantity trajectory of an observer in a given spacetime is irrotational and shear-free, the Lanczos potential of the sapcetime is given ¯ Lμνσ = Lμνσ + Aμνσ (7) by is again a Lanczos potential of that spacetime if the tensor λ λ Lμνσ = Vμ;λV Vν Vσ − Vν;λV VμVσ . (10) Aμνσ (termed as ‘s-tensor’ by Takeno) satisfies By considering V α = √ 1 , 0, 0, 0 , which comes out Aμνσ =−Aνμσ , (8a) 1−2m/r as irrotational and shear-free in the spacetime (9) hence sat- Aμνσ + Aνσμ + Aσμν = 0, (8b) isfying all the requirements of the Novello–Velloso formula A[μν][σ;ρ] + A[σρ][μ;ν] −∗A∗[μν][σ;ρ] α (10): V Vα = 1, shear = rotation = 0, acceleration = 0. −∗A∗[σρ][μ;ν] = 0. (8c) The formula (10) then provides the Lanczos tensor with only one non-vanishing independent component It has been shown that the gauge condition (5b) of divergence- κ = m freeness of the Lanczos tensor, which implies that Aμν ;κ L =− . 100 2 (11) 0, does not fix Aμνσ uniquely and hence causes degen- r eracy in Lμνσ [3]. Thus, the cause of the degeneracy in By chance this satisfies Lanczos’s gauge condition (5b)of Lμνσ is the redundant degrees of freedom of the tensor, and it is not an issue with the gauge conditions. That is, divergence-freeness, though it does not satisfy the condition the conditions (5a) and (5b) alone, taken together with (3a) (5a) of trace-freeness. A trace-free potential can be obtained by using relation (6), which allows to cancel the trace of the and (3b), cannot supply a unique value of Lμνσ inagiven =− λ / spacetime. In order to fix this arbitrariness, one needs addi- tensor by choosing Xμ Lμλ3giving tional conditions/assumptions of plausibility to be imposed ⎫ 2m ⎪ on Lμνσ , which is still missing. In the absence of this, any L¯ =− ⎪ 100 2 ⎪ arbitrary value of Lμνσ , just satisfying the existing con- 3r ⎪ m ⎬ ditions/symmetries, may not constitute a physically viable L¯ =− 122 ( − / ) , (12) quantity. 3 1 2m r ⎪ ⎪ Although there exists no algorithm for finding the Lanc- m sin2 θ ⎪ L¯ =− ⎭ zos potential tensor in general, the tensor has indeed been 133 3(1 − 2m/r) found explicitly in certain special situations, for instance the spacetimes of physical significance like Schwarzschild [9] which satisfy both Lanczos gauge conditions given in (5) and Kerr [10], besides many others (see [11] and the refer- fortunately. We find another solution of Eq. (4), not realized ences therein). In order to elucidate the obscure theory of so far in the literature, for the Schwarzschild line element 123 194 Page 4 of 14 Eur. Phys. J. C (2021) 81 :194

(9): this, let us first consider the Schwarzschild interior solution ⎫ ⎛ ⎞ ˜ 1 m ⎪ 2 L100 =− ⎪ 2 r 2 ⎪ 1 r 2 r 2 dr2 ⎬ ds2 = ⎝3 1 − 0 − 1 − ⎠ dt2 − 1 2 2 − 2/ 2 L˜ = r sin2 θ , (13) 4 R R 1 r R 133 2 ⎪ ⎪ 2 2 2 2 2 1 ⎭⎪ −r dθ − r sin θ dφ , (17) L˜ = r 2 sin 2θ 233 4 which, being a static and isotropic solution of Einstein satisfying the gauge condition (5b) of the divergence freeness equation, represents the spacetime inside a non-rotating, but not satisfying the gauge condition (5a) of trace freeness. incompressible perfect fluid sphere of constant density The trace-free form can similarly be obtained by using (6), √ ρ and radius r .HereR ≡ 3/8πρ .Asthesolu- though at the cost of letting more non-vanishing components 0 0 0 tion is conformally flat, its Weyl tensor vanishes identi- enter it. α = cally. It would not be difficult to check that a V −1 r2 2 3 − 0 − 1 − r , , , 2.2 Lanczos tensor for the Schwarzschild–de Sitter 2 1 R2 2 1 R2 0 0 0 in the spacetime spacetime (17), satisfies all the requirements of the Novello–Velloso α formula (10): V Vα = 1, shear = rotation = 0, acceleration The Lanczos potential of the Schwarzschild spacetime can = 0. The formula then gives easily be generalized by adding the cosmological constant in (9) thereby resulting in the Schwarzschild–de Sitter + = r2+9r2 2 r2 solution of Gμν gμν 0: 5r 1 − 0 − 3 1 − r 1 − 0 10R2 5 R2 R2 2 2 L100 = (18) 2 = − 2m − r 2 − dr ds 1 dt 2 2 r2 r 3 (1 − 2m/r − r 2/3) R2 − r − − r − 0 2 1 R2 3 1 R2 1 R2 −r 2dθ 2 − r 2 sin2 θ dφ2. (14) − / 2 1 2 as the only non-vanishing independent component of Lμνσ , V α = − 2m − r , , , The choice 1 r 3 0 0 0 satisfies which is divergence-free but not trace-free. A trace-free form all the requirements of the Novello–Velloso formula (10)for can similarly be obtained by the use of (6), giving the line element (14). The formula then gives only one non- ⎫ vanishing independent component ⎪ r2+9r2 2 r2 ⎪ 5r 1 − 0 − 3 1 − r 1 − 0 ⎪ 10R2 5 R2 R2 ⎪ r m ¯ ⎪ L100 = ⎪ L100 = − (15) ⎪ 2 2 2 r2 ⎪ 3 r 3R2 1 − r − 3 1 − r 1 − 0 ⎪ R2 R2 R2 ⎪ ⎪ satisfying the gauge condition (5b) but not (5a). A trace-free ⎪ ⎪ r2+9r2 2 r2 ⎪ potential is obtained by the use of (6)giving 10r 3 1 − 0 − 3 1 − r 1 − 0 ⎪ 10R2 5 R2 R2 ⎬⎪ ⎫ ¯ = , L122 3 2 r m ⎪ / / 1/2 ⎪ ¯ ⎪ 2 2 3 2 1 6 r2 ⎪ = − ⎪ 2 − r − − r − 0 ⎪ L100 ⎪ 3R 1 2 3 1 2 1 2 ⎪ 3 3 r 2 ⎪ R R R ⎪ ⎪ ⎪ 3 ⎪ ⎪ r − ⎬ ⎪ m r2+9r2 2 r2 ⎪ ¯ = 3 10r 3 1 − 0 − 3 1 − r 1 − 0 ⎪ L122 , (16) 10R2 5 R2 R2 ⎪ 3(1 − 2m/r − r 2/3)⎪ ⎪ ⎪ L¯ = 2 θ⎪ ⎪ 133 3 sin ⎪ 3 ⎪ / / 1/2 ⎪ r − 2 θ ⎪ 2 2 3 2 1 6 r2 ⎪ m sin ⎪ 2 − r − − r − 0 ⎭⎪ 3 ⎪ 3R 1 2 3 1 2 1 2 L¯ = ⎭ R R R 133 3(1 − 2m/r − r 2/3) (19) which satisfies both gauge conditions (5). which satisfies both gauge conditions documented in (5). 2.3 Lanczos tensor for the Schwarzschild interior spacetime Let us note that if Lμνσ is a Lanczos potential of a conformally flat (or flat) spacetime, then for an arbitrary constant It may appear surprising that the Lanczos tensor, that appears κ, the tensor κ Lμνσ is also its Lanczos potential by virtue as the potential of the Weyl tensor, may be non-vanishing of the vanishing Weyl tensor and the linearity of Eq. (4)in even when the Weyl tensor vanishes. In order to exemplify Lμνσ . 123 Eur. Phys. J. C (2021) 81 :194 Page 5 of 14 194

2.4 Lanczos tensor for the Robertson–Walker spacetimes which gives a unique solution 1 As another example of a spacetime with vanishing Weyl ten- = 1, p =− , 2 sor, let us consider the Robertson–Walker (R–W) line ele- α ment (which too is conformally flat) in the form to be compatible with an arbitrary value of . Thus n 2 rS 2 2 2 dr 2 2 2 2 L100 = κ √ , (23) ds = dt − S (t) + r (dθ + sin θ dφ ) , − α 2 1 − αr 2 1 r (20) constitutes the Lanczos potential for the spacetime (20) with κ, n and α as arbitrary constants. The trace L λ and the where α has been kept as an arbitrary constant (not nor- μλ divergence L λ of the tensor can be calculated, giving the malized, as in the usual standard form) to be used later in μν ;λ non-vanishing components as Sect. 6.2. The Lanczos tensor for some particular cases of rSn this spacetime has been discovered in [3]. Here we dis- λ = κ √ , L1 λ (24) cover its value in a more general case. For this purpose, 1 − αr 2 we follow a heuristic approach and take guidelines from the Schwarzschild case (9). Since both the line elements (9) and rSn−1 S˙ λ =− λ = κ( + )√ . (20) are spherically symmetric, we expect that it should be L10 ;λ L01 ;λ n 2 (25) 1 − αr 2 possible to have Lμνσ for the R–W line element (20) with a single (independent) non-vanishing component, like the one Clearly the solution (23) does not satisfy the condition (5a) given by (11) for the Schwarzschild line element (9). But [nor does it satisfy the condition (5b)]. As we have mentioned apart from the radial coordinate r, now in the case of the line earlier, a trace-free potential can be obtained by using relation element (20) we should also expect contributions from S and (6), which allows to cancel the trace of the tensor by choosing =− λ / possibly from (1 − αr 2) also. Hence, we expect Xμ Lμλ3 giving now the following three independent non-vanishing components: n 2 p L100 = κr S (1 − αr ) , (21) ⎫ κ rSn = 2 √ , ⎪ to form the only independent non-vanishing component of L100 ⎪ 3 1 − αr 2 ⎪ the Lanczos potential for the spacetime (20) for suitable val- + ⎬⎪ κ r 3 Sn 2 ues of the parameters , n and p. These parameters are to L122 = √ , (26) 3 − αr 2 ⎪ be determined by the condition that (21) should satisfy the 1 ⎪ κ 3 n+2 ⎪ Weyl–Lanczos generating Eq. (4) for the line element (20) r S 2 ⎭⎪ L133 = √ sin θ. with Cμνσρ = 0. Let us recall that the spacetime (20) is con- 3 1 − αr 2 formally flat and hence its Weyl tensor vanishes identically. This can be generalized into yet another solution The multiplicative constant κ has been introduced in (21) ⎫ rSn for the same reason, as explained earlier, that if Lμνσ is a = κ √ , ⎪ L100 1 ⎪ solution of (4) for the line element (20), then so is κ Lμνσ . 1 − αr 2 ⎪ + ⎬⎪ If we denote the right hand side of Eq. (4)byWμνσρ, r 3 Sn 2 L = κ √ , then its non-vanishing components generated by the defini- 122 2 ⎪ (27) 1 − αr 2 ⎪ + ⎪ tion (21) for the spacetime (20) yield r 3 Sn 2 ⎪ ⎫ = κ √ 2 θ,⎭⎪ 2κ − − ⎪ L133 2 sin W = r 1(1 − αr 2)p 1 Sn[1 − + αr 2( + 2p)], ⎪ 1 − αr 2 1010 ⎪ 3 ⎪ κ + − + ⎪ W = r 1(1 − αr 2)p 1 Sn 2[1 − + αr 2( + 2p)], ⎪ though at the cost of loosing the gauge symmetry (5a). Here 1212 3 ⎪ ⎪ κ1,κ2 are arbitrary constants. κ + − + ⎪ W = r 1(1 − αr 2)p 1 Sn 2[1 − + αr 2( + 2p)] sin2 θ, ⎬⎪ 1313 3 Our elucidation on the Lanczos tensor is now complete to κ +1 2 p n 2 ⎪ some extent, and it is ready to be involved in applications. W2020 =− r (1 − αr ) S [1 − + αr ( + 2p)], ⎪ 3 ⎪ κ ⎪ Although much attention has been devoted to the Weyl ten- =− +1( − α 2)p n[ − + α 2( + )] 2 θ, ⎪ W3030 r 1 r S 1 r 2p sin ⎪ sor in the research on a gravitational theory, the same did not 3 ⎪ ⎪ 2κ + + ⎪ occur with the Lanczos potential. Since the Lanczos tensor W =− r 3(1 − αr 2)p Sn 2[1 − + αr 2( + 2p)] sin2 θ.⎭ 2323 3 appears as a fundamental property of spacetime itself irre- (22) spective of any particular theory of gravitation, it is expected to be imbued with more interesting physical properties than Obviously, the vanishing of all these components simultane- the Weyl tensor. With the values of this tensor derived in these ously (for all r, S and κ = 0), requires sections, we explore what physical information this potential 1 − + αr 2( + 2p) = 0, tensor may convey in a physical situation. 123 194 Page 6 of 14 Eur. Phys. J. C (2021) 81 :194

˜ μ (s−2) μ 3 Conformal properties of Lanczos tensor W νσρ = W νσρ ( − ) s 3 (s−3) μα Conformal symmetry plays an important role in field theories + g 2 including gravity and leads to interesting insights about the × + + + physical phenomena related to the dynamics of spacetime. 2 Lανσ ,ρ Lναρ ,σ Lσρα ,ν Lρσν ,α Under this symmetry for instance, all the features of grav- κ κ + gαρ (Lνσ+ Lσν) itational wave propagation and the global causal structure κ κ + gνσ (Lαρ+ Lρα) of spacetime remain conserved, as the conformal transfor- κ κ κ κ − gασ (Lνρ+ Lρν) − gνρ (Lασ+ Lσα) ,κ mations leave the null geodesics and hence the light cones κ κ invariant. + gνρ Lσκ− gνσ Lρκ ,α A conformal transformation is a rescaling of the space κ κ + gασ Lρκ− gαρ Lσκ ,ν and time intervals performed without changing the coordi- κ κ + gρν L − gραL ,σ ακ νκ nates used to describe events in a spacetime manifold. This κ κ + gσαLνκ− gσνLακ ,ρ rescaling is brought about through the metric transformation 4 λκ →˜ = 2 , − L κ ,λ gασ gνρ − gνσ gαρ . (31) gμν gμν gμν (28) 3 where the conformal factor (xα) is a dimensionless, posi- tive and smooth but otherwise arbitrary function of the space- Thus in general the potential tensor Lμνσ , defined through ˜ μ = μ time coordinates. This gives rise to a new spacetime and (4), fails to produce W νσρ W νσρ if the tensor is limited to admit the symmetries (3) only. This is due to the presence of its geometry characterized by the metric g˜μν, wherein the inverse metric and the Christoffel symbols lead to the second term on the right hand side of Eq. (31). Although this undesired term vanishes for s = 3, this choice does not μ μ μν − μν ˜ = g˜ = 2g , (29a) fulfill the requirement W νσρ W νσρ. The only way to achieve this goal is to have s = 2 together with imposing a ˜ α α −1 α α αλ βγ = βγ + δβ ,γ + δγ ,β − gβγ g ,λ . condition on Lμνσ given by (29b) 2 Lανσ ,ρ + Lναρ ,σ + Lσρα ,ν + Lρσν ,α κ κ κ κ κ κ + gαρ (Lνσ+ Lσν) + gνσ (Lαρ+ Lρα) − gασ (Lνρ+ Lρν) The tilde over a quantity is going to denote its value in the κ κ κ κ ˜ −gνρ (Lασ + Lσα) ,κ + gνρ Lσκ− gνσ Lρκ ,α conformally rescaled spacetime with metric gμν. As the Weyl κ κ κ κ tensor is fully invariant under the transformation (28), it is + gασ Lρκ− gαρ Lσκ ,ν + gρν Lακ− gραLνκ ,σ κ κ natural to ask how this conformal invariance is manifested in + gσαLνκ− gσνLακ ,ρ the Lanczos tensor which plays the role of the potential for 4 λκ − L κ ,λ gασ gνρ − gνσ gαρ = 0. (32) the Weyl tensor. But, unlike the Weyl tensor, it has not so far 3 been possible to obtain an expression for the Lanczos tensor It is not clear at this point what kind of symmetry this equa- in terms of the metric tensor so that its conformal invariance tion imposes on Lμνσ in addition to the two obligatory sym- can be studied directly. Nevertheless one can still analyze metries (3a) and (3b). In the following we discuss a simple the conformal invariance of the Weyl–Lanczos Eq. (4)by possibility which is suggested by breaking the long Eq. (32) considering a conformal weight for Lμνσ : in four parts according to the symmetries of the terms: ˜ s Lμνσ = Lμνσ , (30) Lανσ ,ρ + Lναρ ,σ + Lσρα ,ν + Lρσν ,α = 0, where s is a real number (see the Appendix D of Reference (33a) μνσρ [12]). If we denote the right hand side of Eq. (4) with W κ κ κ κ κ gαρ (L + L ) + gνσ (L + L ) − gασ (L as we have done earlier, then the application of the rescaling νσ σν αρ ρα νρ μ κ κ κ (30) must render W νσρ unchanged under the transformation + L ) − gνρ (L + L ) ,κ = 0, (33b) ˜ μ μ ρν ασ σα (28), i.e. W νσρ = W νσρ, since so is the left hand side of κ κ κ κ ˜ μ μ gνρ Lσκ− gνσ Lρκ ,α + gασ Lρκ− gαρ Lσκ ,ν Eq. (4): C νσρ = C νσρ. κ κ + gρν L − gραL ,σ In order to look into this matter directly, let us compute ακ νκ κ κ the right hand side of Eq. (4) in the conformally transformed + gσαLνκ− gσνLακ ,ρ = 0, (33c) spacetime. Use of the conformal rescalings (29a, 29b, 30) λκ L κ ,λ = 0. (33d) and a long but straightforward calculation produce The last equation, taken together with the arbitrariness in , implies that the Lanczos potential tensor Lμνσ should be trace-free: 123 Eur. Phys. J. C (2021) 81 :194 Page 7 of 14 194

κ = . GS Lμκ 0 (34) a given spacetime in a weak field, then so is Lμνσ + L μνσ , GS This not only satisfies identically the condition (33c) but also where L μνσ is the Lanczos potential of the Minkowskian appears consistent with the general tenet that the conformally GS spacetime. Thus L μνσ appears as a ‘ground state’ potential invariant tensors must be trace-free, thus contributing further field which makes an essential contribution to the Lanczos to the aesthetic appeal of (34). Let us note that the scaling potential fields of all the curved spacetimes in a weak gravita- law (30), for s = 2, implies that the tensor Lμνσ (with one ˜ μ μ tional field. This provides the Minkowskian Lanczos poten- index raised) is conformally invariant: L νσ = L νσ , akin ˜ μ μ tial a natural and adequate interpretation portraying it as the to C νσρ = C νσρ. (However, this is not so with the tensor ˜ ˜ ‘weight’ or the ‘metrical elasticity’ of spacetime, which being with all covariant indices: Lμνσ = 2 Lμνσ , akin to Cμνσρ = an integral part of all the spacetime geometries, opposes the 2Cμνσρ.) curving of spacetime. Thus under this simple case, the constraints (33a), (33b) Assigning a ‘ground state’ potential field to the and (34) are imposed on the Lanczos tensor by the conformal Minkowskian spacetime in the absence of any curvature, invariance of the Weyl tensor (which must be met) brought may appear puzzling and surprising at the first glance. How- about through the Weyl–Lanczos equation. This may sup- ever, there does exist a parallel to this situation in the quan- ply the missing plausibility conditions to be imposed on the tum realm wherein one can have a vanishing electromagnetic Lanczos tensor as has been mentioned in Sect. 2. Further field in a region but a non-vanishing potential, explaining the study is required to decipher the meaning of the more gen- conspicuous physical effects which do exist in that region eral case (32). (Aharonov–Bohm effect) [3]. A word of caution is in order here. It should be noted that the Lanczos ‘potential’ does not have the dimensions of the 4 Possible interpretations of Lanczos tensor potential energy, in the same way as the Weyl tensor does not have the dimensions of force. Thus the denomination We have derived the Lanczos potential tensor in various cases of the Lanczos tensor as the ‘Lanczos potential’, refers to of spacetime in the above examples. However, unlike the just the electromagnetic analogy. Let us recall that the Lanc- Weyl tensor, it has not so far been possible to obtain a gen- zos tensor is regarded as the gravitational analogue of the eral expression for this tensor in terms of the metric tensor. electromagnetic potential, since it generates the Weyl tensor Though this expression does exist in the case of a weak grav- differentially, in the same way as the electromagnetic poten- itational field, wherein the spacetime metric differs minutely tial generates the Faraday tensor. Hence, the Lanczos tensor from the Minkowskian metric ημν: is generally termed as the potential to the Weyl tensor. gμν = ημν + hμν, where |hμν| << 1. (35) 4.2 Lanczos tensor as a relativistic analogue of the In this case, the Lanczos tensor can be written as [2] Newtonian force 1 1 1 Lμνσ = hμσ,ν − hνσ,μ + h,μηνσ − h,νημσ , (36) 4 6 6 Since the Lanczos tensor generates the Weyl tensor and since the latter is linked with the (free) gravitational field, one may ≡ to linear order in the metric perturbations hμν.Hereh naturally expect that the Lanczos tensor too has something ημν hμν . to do with the gravitational field. In this view, the quantity Lμνσ given by expression (36) in terms of the derivatives 4.1 Minkowskian Lanczos tensor as a ground state of the metric tensor, gives a clue that the Lanczos tensor ‘Potential’ may represent a relativistic analogue of the Newtonian force. However if this is so, the tensor should reduce to the New- Let us note that the covariant and ordinary differentiations tonian gravitational force in a physical situation in the weak coincide in the first order of approximation in the case of field and low velocity limit. As the Newtonian theory of grav- a weak gravitational field and hence the differential and itation provides excellent approximations under a wide range algebraic tensor operations are performed by the use of the of astrophysical cases, the first crucial test of any theory of Minkowskian metric ημν, as is the case with Lμνσ defined by gravitation is that it reduces to the Newtonian gravitation. (36). This is exactly the case with the Lanczos potential for In order to check this, let us consider a static spherical the Minkowski spacetime, which has been elaborated on in mass m placed at the origin of a centrally symmetric coordi- 3 [3,4] . This implies that if Lμνσ is the Lanczos potential of nate system r,θ,φ. In the Newtonian theory of gravity, the

3 The Lanczos tensor of any conformally flat spacetime including the standing of Aμνσ denoting the Lanczos tensor of the corresponding flat spacetime itself can be defined by Eqs. (8a–8c) (with the under- spacetime). 123 194 Page 8 of 14 Eur. Phys. J. C (2021) 81 :194 gravitational field produced by the mass at a point r is repre- This analogy however also poses a question: How is the sented in terms of the gravitational potential (r) =−m/r gravitational information encoded in the Lanczos tensor? It at that point. In a relativistic theory of gravitation, for exam- is clear that in the above-described example, the Newtonian ple GR, the gravitational field of the mass is well-described force on the test mass is estimated in terms of the mass of by the Schwarzschild line element (9). In a weak field, the the gravitating ball, which enters the scene through the Ein- line element (9) reduces to stein field equation. In other words, the information on this force (and hence that on matter and gravity) gets encoded in 2 μ ν 2m 2 2 ds = ημνdx dx − (dt + dr ), (37) Lanczos tensor through Einstein’s tensor, i.e. through Ricci’s r tensor and scalar. Let us also recall that the Riemann tensor is at large r so that 2m/r << 1. Its comparison with the condi- decomposed in terms of its two mutually independent parts tion (35) gives the only non-vanishing components of hμν as - its trace-free part (the Weyl tensor) and its trace (the Ricci h = h =−2m/r. For this, the definition (36) provides 00 11 tensor and Ricci scalar) as m L100 =− , (38) 1 2r 2 Rμνσρ = Cμνσρ − gμ[ρ Rσ]ν − gν[σ Rρ]μ − Rgμ[σ gρ]ν. 3 as the only non-vanishing independent component of Lμνσ , (41) which amounts to 1 ∂ As the energy–momentum tensor of matter is correlated with L =− , (39) 100 2 ∂r the trace of Riemann tensor (through the field equation), Eq. (41) indicates that the Weyl tensor is algebraically inde- by virtue of the well-known relation g = (1 + 2) valid in 00 pendent of the matter tensor. Hence so is the Lanczos tensor the weak field and low velocity limit. This can be written in (which is the potential of Weyl tensor), since the Ricci’s ten- terms of F , the corresponding Newtonian force experienced N sor and scalar cannot admit a Lanczos tensor by virtue of the by a unit test mass placed at a distancer from the source mass: absence of a Lanczos potential tensor for the Riemann tensor 1 [13,14]. Then how is the above-mentioned encoding of the L100 = FN. (40) 2 gravitational information into the Lanczos tensor realized? Thus the Lanczos tensor indeed represents the relativistic Perhaps some another unknown feature of the tensor is at analogue of the Newtonian gravitational force in a curved work, further studies may unearth it. spacetime in the weak field and low velocity limit (the factor It may however be noted that the force analogy of the Lanc- 1 zos tensor has limitations. The analogy does not imply that 2 can be considered as a relativistic effect). It may be mentioned that conventionally it is the Christoffel symbol which a geometric theory of gravitation, for instance GR, would is considered as the analogue of the Newtonian force. How- mimic the Newtonian action-at-a-distance concept (which ever, as the Christoffel symbols are not tensors, the Lanczos requires an infinite speed of propagation of gravity) for tensor suits better to a covariant theory in representing the explaining observations. The domain of applicability of the analogue of the Newtonian force when the analogue of the force analogy of the Lanczos tensor is the same as that of the Newtonian potential is ascribed to a tensor (the metric ten- potential analogy of the metric tensor, i.e. the slow motion sor). and the weak field limit of gravity. This new understanding of the Lanczos tensor also makes a prediction: the tensor should match at the boundary of the two spacetime regions on the surface of a gravitating body. 5 Gravitational wave and its energy–momentum In order to exemplify this, let us apply the force analogy on a perfect fluid ball of constant density wherein the spacetime Existence of gravitation waves is one of the most important inside the ball is given by the Schwarzschild interior line features of GR, which was first predicted by Einstein. In the element (17), and that outside the ball by the Schwarzschild light of the direct detection of the gravitational waves from (exterior) line element (9). According to the force analogy, binary mergers, it is now possible to extract information about the Lanczos tensor would approximate the force experienced matter and spacetime under extreme conditions. by a unit test mass placed on the surface of the ball. But the On the theoretical side however, there is no general agree- Lanczos tensor can be calculated from either of the two line ment on the description of the energy–momentum carried elements, which of course match at the boundary. It is then by the gravitational waves. The reason is that there is no expected that the Lanczos tensor too should match smoothly existence of a localizable (i.e., tensor-represented) point- at the boundary, if the force analogy is correct. This is indeed wise gravitational energy–momentum density in GR. The the case, as we shall witness in Sect. 6.1. This scenario is also use of the non-covariant energy–momentum pseudo-tensors corroborated by the case of the collapsing ball discussed in totally obscures the analysis of the subject. Besides their Sect. 6.2. non-uniqueness, they can be annihilated at will at any given 123 Eur. Phys. J. C (2021) 81 :194 Page 9 of 14 194 spacetime point by the right choice of coordinates. Thus, in In this case, the plane wave solution of Eq. (42) traveling the framework of pseudo tensor-formulation of the energy– along the z = x3-direction can be written as momentum, one concludes that the energy carried by a grav- hμν = Aμν cos ω(t − z) , (44) itational wave is coordinate dependent. Thus, the pseudo tensor-formulation of the energy–momentum density of the where Aμν is the constant amplitude of the wave and ω its gravitational waves is though practically useful, it is not angular frequency. satisfactory from the theoretical point of view. We need a As has been mentioned earlier, the description of the fully covariant tensorial formulation of this quantity with the energy–momentum carried by the gravitational waves is still requirement that it reduces to the pseudo tensor-formulation a disputed topic. Generally, the energy–momentum tensor in a weak field. for gravitational waves is obtained from Isaacson’s stress- In our search for a covariant tensorial formulation of the energy pseudo-tensor τμν [15,16] by averaging the squared energy–momentum of the gravitational waves, we get a clue gradient of the wave field over a length much larger than the from the results obtained in the preceding section that the typical gravitational wavelength: Lanczos tensor is a relativistic analogue of the Newtonian 1 αβ force. Hence its square has the same dimensions as that of τμν = ∇μhαβ ∇ν h . π (45) the energy density (in geometric units). This means that, fol- 32 lowing the gravitation-electrodynamics correspondence, an In the above-defined TT gauge, the non-vanishing compo- energy–momentum tensor can be formulated in terms of the nents of τμν are Lanczos tensor, along the lines of the energy–momentum 1 ˙ 2 ˙ 2 tensor of the electromagnetic field, in which the Lanczos τ00 =−τ03 =−τ30 = τ33 = (h+) + (h×) , (46) 16π tensor appears quadratically. It is shown in the following that the so framed energy–momentum tensor gives the same where an overhead dot denotes derivative with respect to t. results in the linearized theory as do the conventional pseudo tensor-based formulations for the energy–momentum of the 5.2 Lanczos wave and its energy–momentum gravitational waves. As has been mentioned earlier, there exists a perfect correspondence between electrodynamics and gravitation and 5.1 Linearized gravity and Isaacson’s energy–momentum various analogues of electromagnetic phenomena have been pseudo tensor discovered in gravitation [17–19]. Electrodynamics admits a wave equation for the electromagnetic 4-potential. Note that A simple wave equation in GR, is a linearized approximation this is so not only in flat spacetime, but also in the curved of Einstein’s equation where the velocities are small and the one. It is already known (see, for example, [12,20]) that a gravitational fields are weak, so that the spacetime metric is Killing vector field Aμ in a Ricci-flat spacetime plays the given by (35). Einstein’s equation, in a Ricci-flat spacetime, role of the electromagnetic 4-potential and the source-free then yields κ Maxwell equations, in Lorenz gauge (A ;κ = 0), reduce to κ ¯ ¯ the homogeneous wave equation ∂ ∂κ hμν = 0 with hμν ≡ hμν − ημν h/2, (42) κ ∇ ∇κ Aμ = 0. (47) which is a spin-2 field equation. The linearized Einstein equation takes the simple form (42) only if the Lorenz gauge con- As the Lanczos potential constitute a gravitational analogue ¯μν ditions ∂μh = 0 are satisfied. This requirement makes the of electromagnetic 4-potential, we expect a similar equation metric perturbation in Eq. (42) look like a transverse wave. for Lμνσ if the correspondence between electrodynamics By imposing the Lorenz gauge conditions, we have reduced and gravity has any real meaning. Remarkably, the Lanczos ¯ the 10 independent components of the symmetric tensor hμν potential indeed satisfies a homogeneous wave equation κ to six. Since there are really only two independent compo- ∇ ∇κ Lμνσ = 0, (48) nents of the Riemann tensor in the present case, the remaining freedom is used to choose a gauge in which the perturbation in Lanczos gauge, in any Ricci-flat spacetime [11,21](not ¯ ¯ μν μν becomes traceless (h ≡ hμνη = 0 = h ≡ hμνη ). This necessarily in a weak field), strengthening our belief in the defines the transverse and traceless (TT) gauge leaving the correspondence. Akin to the wave Eq. (42), this simple and ¯ only independent components of hαβ (= hαβ )as beautiful exact analytical solution is admitted in a Ricci-flat spacetime in any metric theory of gravity formulated in a 4-dimensional pseudo-Riemannian manifold. h =−h ≡ h+, (43a) 11 22 This cannot be just a formal accident and must have = ≡ . h12 h21 h× (43b) some deeper meaning. We know that a plane wave describes, 123 194 Page 10 of 14 Eur. Phys. J. C (2021) 81 :194 from the quantum point of view, a collection of massless Thus the tensor Lμνσ does admit the Lanczos gauge condi- quanta – photons – with helicity ±1 for electrodynam- tions and hence the wave Eq. (48) in the TT gauge in a weak ics. Hence, the electrodynamics-gravity correspondence, in field. We can now come back to Eq. (51). In order to calculate it in a weak field, we use the definition of Lμνσ given by view of Eqs. (47) and (48), insinuates that the plane wave (36). In the TT gauge, this evaluates the following two terms (48) transports the excitations of a massless field. That is, appearing in (51)as it describes at the quantum level, a collection of massless σρ = 1 + + + quanta - gravitons – with helicity ±2 mediating gravity. Lα Lβσρ 2h11,α h11,β 2h12,α h12,β hα1,3 hβ1,3 hα2,3 hβ2,3 16 Hence, its energy–momentum must necessarily be repre- − hα1,0 hβ1,0 − hα2,0 hβ2,0 , (53a) sented by a trace-free symmetric tensor, which is expected to κσρ 1 2 2 2 2 L Lκσρ = (h11,0) + (h12,0) − (h11,3) − (h12,3) . (53b) reduce to the pseudo tensor-formulation (45) in a weak field. 4 This might be helpful to develop an effective description of By virtue of ∂hμν/∂z =−∂hμν/∂t, which readily fol- the physics of graviton which captures some quantum effects κσρ lows from solutions (44), L Lκσρ vanishes identically and but is otherwise based on classical concepts. This would also hence the second term on the r.h.s. of (51) does not con- fit nicely with the earlier findings that the Lanczos potential tribute anything to Tαβ . Similarly, by the use of this identity, is impregnated with various signatures of quantum physics Eq. (53a) reduces to [2,3,5]. Supplemented with the novel insight that the energy– σρ 1 Lα Lβσρ = ∂αh+ ∂β h+ + ∂αh× ∂β h× , momentum tensor for the wave (48) must be symmetric and 8 trace-free, and the earlier-noted cue that this tensor should giving have Lμνσ appearing quadratically, we define the following κ tensor along the lines of the energy–momentum tensor of the Tαβ = ∂αh+ ∂β h+ + ∂αh× ∂β h× . (54) EM σ σ 4 electromagnetic field ( T αβ = Fα Fβσ +∗Fα ∗Fβσ): Thus the only non-vanishing components of Tαβ are obtained σ σ Tαβγ δ = κ(LαγLβσδ +∗Lαγ∗Lβσδ), (49) as κ ˙ 2 ˙ 2 which is though not symmetric in all pair of indices. [A T =−T =−T = T = (h+) + (h×) . (55) 00 03 30 33 4 dimensional constant κ has been inserted in Eq. (49), which is given in terms of G and c in ordinary units.] A symmetric Remarkably, these values do match with their pseudo-tensor tensor can be obtained from (49) by contracting over the last counterparts given by (46)forκ = 1/4π. But, here they pair of indices of Tαβγ δ ,giving emerge from a fully covariant tensorial definition, which is intuitively appealing. One can guess, how this energy– = κ σρ +∗ σρ ∗ , Tαβ Lα Lβσρ Lα Lβσρ (50) momentum tensor interacts with that of the dynamical matter fields when the gravitational wave travels through matter. which is indeed symmetric and trace-free (even if Lμνσ There will necessarily be an exchange of energy and momen- does not satisfy the trace-free condition) like the energy– tum between matter and the Lanczos field, and what should momentum tensor of the electromagnetic field. By evaluating be conserved is not the energy and momentum of the matter the duals, Eq. (50) can be recast in the form alone, but that of the sum of all the fields contributing to the σρ 1 κσρ structure of spacetime. Tαβ = κ 2Lα Lβσρ − gαβ L Lκσρ . (51) 2 In order to compare this tensor-based definition of the energy–momentum of the gravitational waves with the tradi- 6 Matching of the Lanczos tensor at the surface of a tional pseudo tensor one, let us calculate it in the weak grav- gravitating body ity case (35). But before that, let us first check if the wave Eq. (48) is admitted in a weak field. That is, if the Lanczos In the general-relativistic stellar modelling, the exterior and gauge conditions are admitted in a weak gravity field. For the interior spacetime geometries are glued together at the this purpose, we calculate the trace and the divergence of the hypersurface by using the Israel–Darmois junction condi- Lanczos tensor field given by (36) in a weak field. They lead tions [22]. This requires that the extrinsic curvature and the to induced metric on the hypersurface must be the same on both sides of the hypersurface. Well-known examples are: λ 1 ¯ λ Lμλ= hμ,λ, (52a) 4 1. The Schwarzschild interior metric (17), representing the λ 1 ¯ λ ¯ λ spacetime inside an idealized star of constant density, is Lμν ,λ = hμ,λ,ν− hν,λ,μ . (52b) 4 joined with the Schwarzschild exterior metric (9). 123 Eur. Phys. J. C (2021) 81 :194 Page 11 of 14 194

2. The dynamical cosmological interior (20), modeling an tively for the interior and the exterior spacetimes: idealized collapsing star of spatially constant density, is r2+9r2 2 r2 joined with the Schwarzschild static exterior metric (9). − 0 − 3 − r − 0 5r 1 2 5 1 2 1 2 int 10R R R L 100 = Why are the junction conditions limited to matching only the 2 2 r2 R2 − r − − r − 0 extrinsic curvature and the induced metric at the boundary? 2 1 R2 3 1 R2 1 R2 Because the union of the two metrics form a valid solution to Einstein field equation in this case. However in a geomet- ext m ↔ L 100 =− , ric theory of gravitation wherein gravity appears through the r 2 curvature, we should be able to calculate the gravitational which become equal at the boundary r = r0 by virtue of 2 effect on a test mass placed at the boundary, from either of R = 3/8πρ0 and realizing that the total mass of the interior the two metrics. Should we then expect the curvature of the = π 3ρ / 4 r0 0 3 constitutes the source mass m for the exterior spacetimes too to match on the both sides of the boundary? spacetime. The same is true for the Lanczos tensor satisfying This expectation is however not fulfilled in the above exam- both Lanczos gauge conditions (5a) and (5b). In this case, ples. In both these examples, the interiors are given by the the respective values of the tensor are given by Eqs. (19) and conformally flat spacetimes giving vanishing Weyl tensor. (12) respectively for the interior and the exterior spacetimes. Whereas the exteriors are Ricci-flat spacetimes. Thus, the At the boundary r = r0, these values reduce to Riemann tensor cannot be matched at the boundary, as is int r ext 2m ascertained by Eq. (41). L =− 0 ↔ L =− , 100 2 100 2 How else can we measure the gravitational effect experi- 3R 3r0 enced by the test mass? In terms of a quantity which is an int r 3 ext 0 m analogue of Newtonian gravitational force – the Lanczos ten- L 122 =− ↔ L 122 =− , r2 3(1 − 2m/r ) R2 − 0 0 sor! Interestingly, the Lanczos tensor indeed matches at the 6 1 R2 boundary, as we shall see in the following. int 3 2 θ ext 2 θ r0 sin m sin L 133 =− ↔ L 133 =− , r2 3(1 − 2m/r ) 6.1 Matching the Lanczos tensor on an idealized static star R2 − 0 0 6 1 R2

As has been mentioned earlier, the line element given by (17), / 2 = / 3 which coincide by virtue of 1 R 2m r0 as before. Thus i.e. the Lanczos tensor does match smoothly at the common ⎛ ⎞ 2 boundary despite the Weyl tensor differing in the two regions. 1 r 2 r 2 ds2 = ⎝3 1 − 0 − 1 − ⎠ dt2 This strengthens our observation that the Lanczos tensor is a 4 R2 R2 relativistic analogue of the Newtonian gravitational force. It however appears that the interior (17) considered in dr2 − − r 2dθ 2 − r 2 sin2 θ dφ2, these examples, which was derived by Schwarzschild to have − 2/ 2 1 r R a mathematically simple solution, is not physically viable. As represents the static and isotropic spacetime inside a non- this solution assumes a static sphere of matter consisting of a perfect fluid of constant density ρ but a variable pressure p(r) rotating star of radius r0, composed of an incompressible √ that vanishes at the boundary, the speed of sound = dp/dρ perfect fluid of constant density ρ0 and variable pressure p. This spacetime is glued with the Schwarzschild static exterior becomes infinite in the fluid. In order to avoid this situation, spacetime given by the line element (9), i.e. let us consider, in the following section, an interior composed of a dust cloud with vanishing pressure, which gives rise to a collapsing interior. 2m dr2 ds2 = 1 − dt2 − − r2dθ2 − r2 sin2 θ dφ2. r (1 − 2m/r) 6.2 Matching the Lanczos tensor on a collapsing star It is well-known that these interior and exterior spacetimes have a perfect matching at the boundary r = r0 (see for The solution of Einstein field equation for a homogeneous example, [23]). dust cloud of spherical symmetry, leads to the well-known We have already calculated the Lanczos tensor for these problem of spherical gravitational collapse. The spacetime interior and exterior spacetimes in Sects. 2.1 and 2.3 respec- inside the collapsing sphere is represented by the dynami- tively. Let us first consider the values of the tensor in these cal R–W metric (20) with the spatially closed case, which spacetimes satisfying the gauge condition of divergence- is matched with the Schwarzschild static exterior (9). This freeness (5b). These are given by Eqs. (18) and (11) respec- problem of the imploding dust ball was considered by Datt in 123 194 Page 12 of 14 Eur. Phys. J. C (2021) 81 :194

1938 [24] and by Oppenheimer and Snyder in 1939 [25]. For Let us now consider the matching of the Lanczos tensor the sake of completeness, we have summarized the formal having more than one non-vanishing components in the two aspects of the theory in the “Appendix”. regions of spacetime, for instance (27)versus(12): Let us note that unlike the case studied in the preced- int κ1r ext 2m 2m ing section, here the matching of the two spacetime regions L 100 = √ ↔ L 100 =− =− , 1 − αr 2 S2 3r¯2 3r 2 S2 take place at a moving boundary between the dynamical cos- int κ r 3 ext m m = √ 2 ↔ =− =− , mological interior (20) and the static Schwarzschild exterior L 122 L 122 1 − αr 2 3(1 − 2m/r¯) 3 1 − 2m (A-3). In order to study the matching of the accompanied rS int κ r 3 2 θ ext m 2 θ m 2 θ = √2 sin ↔ =− sin =− sin . Lanczos potentials in the two regions, we first consider the L 133 L 133 1 − αr 2 3(1 − 2m/r¯) 3 1 − 2m potentials with single non-vanishing components. These are rS (59) given by Eqs. (23) and (11) for the interior and the exterior spacetimes respectively: An inspection reveals that only the first component (i.e., L100) of the interior and exterior potentials diverge at S = 0, int κrSn ext m but the rest two components do not. While the last two inte- L 100 = √ ↔ L 100 =− . (56) 1 − αr 2 r¯2 rior ones remain fixed on the boundary, the corresponding exterior ones vanish as S → 0. Hence they do not seem to match in this form. Let us recall that in the absence of suf- As the coordinates used in the two spacetime regions are dif- ficient conditions of plausibility that should be imposed to ferent, the accompanied potentials shown in (56)alsohave give a unique value of the tensor Lμνσ , there remain redun- different coordinates. In order to make the comparison pos- dant degrees of freedom in it and hence some of its values sible, we use the transformation (A-5) to bring the potentials (in some arbitrary gauge) may not be physically meaningful. in a single (comoving) coordinate system. This gives However, it would be interesting to see under what conditions the values given in (59) can match. In this context, one int κrSn ext m can check that if the collapsing S can halt at a non-vanishing L 100 = √ ↔ L 100 =− . (57) minimum of S,sayS , then all the three components of 1 − αr 2 r 2 S2 min the tensor match at the boundary after the ball settles to the 2 minimum radius. This determines κ1 =−α 1 − αr /3 and Let us recall that the coordinates r,θ,φ are the comoving b 2 α 1−αr coordinates of the spacetime (20) and hence are indepen- κ =− b . 2 6 (1−αr2/S ) dent of time. Hence, a continuous contraction in the surface b min The above result does not mean that the Lanczos tensor (boundary) of the ball can be brought about only through acts as a repulsive agent that can halt the collapse. Rather the function S = S(t), which is a decreasing function of the it provides an indication that there must exist an era when proper time t, as is indicated by (A-1). This implies that both the quantum gravity-effects take over at some S , arresting the potentials in (57) must have the same S-dependence to min the collapse and avoiding the singularity. There have been ensure a proper matching and continuity across the bound- various claims that the singularities of GR can be avoided by ary throughout the collapse. This uniquely determines the using quantum effects. Although a self-consistent theory of parameter n appearing in (23)asn =−2. Interestingly, this quantum gravity remains elusive, there is a general agreement value of n brings the potential (23) in the same (divergence- that removal of classical gravitational singularities is not only free) gauge, as is (11), which is clear from Eq. (25). Thus the a crucial conceptual test of any theory of quantum gravity final form of the potentials to be matched is but also a prerequisite for any reasonable theory of quantum gravity. Let us note that the existence of an Smin is also not int κr ext m inconsistent with the matching discussed in Eq. (58). L 100 = √ ↔ L 100 =− , (58) 1 − αr 2 S2 r 2 S2

7 Concluding remarks which indeed match at the boundary r = rb throughout the collapse as the matching becomes independent of S(t), and Einstein’s revolutionary insight – the local equivalence of hence independent of time. This determines the constant κ gravitation and inertia – paved the way to the lofty goal of the κ =−α − α 2/ α = πρ / = uniquely as 1 rb 2 by virtue of 8 0 3 geometrization of gravitation by introducing (pseudo) Rie- / 3 2m rb , thus resulting in a smooth fit which appropriately mannian geometry wherein the Riemann–Christoffel curva- accounts for continuity in the Lanczos potentials across the ture tensor plays the central role for the unfolding of gravity sphere boundary. in a four-dimensional spacetime manifold. However, it is not 123 Eur. Phys. J. C (2021) 81 :194 Page 13 of 14 194 the full curvature tensor which takes part in the field equations Data Availability Statement This manuscript has no associated data of gravitation. Rather, only its trace – the Ricci tensor and or the data will not be deposited. [Authors’ comment: This article does Ricci scalar – appear in Einstein’s equation. It is the Lanc- not do any data analysis. Hence it has not used any data.] zos tensor which fulfills this gap by generating the trace-free Open Access This article is licensed under a Creative Commons Attri- part of the Riemann tensor – exactly those components of the bution 4.0 International License, which permits use, sharing, adaptation, tensor which are missing in Einstein’s equation. distribution and reproduction in any medium or format, as long as you Despite its importance and novelty, the Lanczos tensor give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes has not been paid proper attention as it deserves. One of the were made. The images or other third party material in this article reasons of its obscurity is that it is not clear what it represents are included in the article’s Creative Commons licence, unless indi- physically. By deriving expressions for the tensor in some cated otherwise in a credit line to the material. If material is not particularly chosen spacetimes, we have attempted to find included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit- its physical meaning and an adequate interpretation. It has ted use, you will need to obtain permission directly from the copy- been shown that the especial property of the Weyl tensor – right holder. To view a copy of this licence, visit http://creativecomm its conformal invariance – imposes additional constraints on ons.org/licenses/by/4.0/. 3 the Lanczos tensor, not realized before. Funded by SCOAP . The traditional viewpoint on the Lanczos tensor comes from the well-studied correspondence between electrodynamics and gravitation. Thence appears the perspective that Appendix: Dutt–Oppenheimer–Snyder model of collaps- the Lanczos tensor is a gravitational analogue of the elec- ing dust cloud: a brief review tromagnetic 4-potential since it generates Weyl tensor differentially. A complementary perspective on the tensor has This is a simple model of spherical gravitational collapse of been discovered which emerges from the weak field and low a dust ball, which has considerable methodological impor- velocity limit of GR: one may interpret the Lanczos ten- tance. Let us consider a non-rotating ball of electrically neu- sor in a curved spacetime as a relativistic, tensorial ana- tral matter having a spherical symmetry in its physical param- logue of the Newtonian gravitational force. Additionally, the eters such as density and pressure. Let us ignore the effect Minkowskian Lanczos tensor appears as part and parcel of the of pressure as an opposing agency to gravity. This is a rea- Lanczos tensor field of any curved spacetime in a weak field sonable assumption for the stars in which the fusion process and hence can be interpreted as the ‘weight’ of the space- has nearly ceased and hence little radiation pressure remains. time, providing a substantive status to it. This emphasizes Moreover, the pressure generated during collapse is not ade- the importance of the Lanczos tensor in terms of a multi- quate to halt the collapse. featured quantity, which is also corroborated by its applica- By solving Einstein’s field equation for a dynamic, spher- tions devised in the preceding sections. ically symmetric line element with the energy–momentum The relation of this tensor to the energy–momentum of tensor for a pressure-less perfect fluid, it has been shown the gravitational waves, and its matching on the surface of that such a ball of pressure-less dust of uniform but time- ρ a gravitating body give added proof of its fundamental sig- dependent density , must contract without limit under self nificance. Interestingly, the matching of the potential at the gravitation [26]. The spacetime interior to the dust ball turns boundary of a collapsing mass provides an indication of the out to be given by the spatially closed R–W line element (20): existence of quantum gravity-effects avoiding the singular- dr2 ds2 = dt2 − S2(t) + r 2(dθ 2 + sin2 θ dφ2) , ity. This finding attributes a quantum signature to the Lanczos 1 − αr 2 potential, which appears consistent with some other studies α ≡ πρ / ρ revealing a quantum character of the potential tensor. where 8 0 3 with 0 being the initial density of the It appears that the theory of the Lanczos tensor field con- dust particles measured by the observer riding the surface of tains surprises and insights which have only just started to the collapsing ball. Clearly such an observer uses comoving reveal. The performed study has introduced a new domain coordinates r,θ,φand thus measures proper time t. He thus = πρ 3/ of applicability of this tensor field. It reveals that the ten- measures the total mass of the ball as m 4 0rb 3, with = sor field, which requires for its construction merely the basic r rb being the radial coordinate of the boundary of the concepts of spacetime, is impregnated with interesting phys- dust ball. The dynamics of the collapse is regulated by the ical properties and opens up a new research avenue with rich dynamics of the function S(t) through prospects. / √ 1 1 2 S˙(t) =− α − , ( ) 1 (A-1) Acknowledgements The author would like to thank Sanjeev V. Dhu- S t randhar, Jayant V.Narlikar and José M. M. Senovilla for many insightful obtained by normalizing the radial coordinate so that S(0) = discussions. 1, and by assuming that the dust particles are at rest initially 123 194 Page 14 of 14 Eur. Phys. J. C (2021) 81 :194 in the chosen coordinates, so that S˙(0) = 0. 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