We transcribe into the framework of the torsionful version of the epsilonformalism of Infeld and van der Waerden the world definition of the Weyl tensor for a curved spacetime that occurs in the realm of EinsteinCartans theory. The resulting expression shows us that it is not possible to attain any general condition for conformal flatness in such a spacetime even if wave functions for gravitons are eventually taken to vanish identically. A short discussion on the situation concerning the limiting case of general relativity is presented thereafter.
['General Relativity and Quantum Cosmology']
On the Weyl Tensor for a Curved Spacetime Endowed with a Torsionful Affinity
{'J. G. Cardoso ∗': 'Department of Mathematics Centre for Technological Sciences-UDESC Joinville 89223-100 SC Brazil.  \nOctober 29, 2024', 'Abstract': "We transcribe into the framework of the torsionful version of the ε -formalism of Infeld and van der Waerden the world definition of the Weyl tensor for a curved spacetime that occurs in the realm of Einstein-Cartan's theory. The resulting expression shows us that it is not possible to attain any general condition for conformal flatness in such a spacetime even if wave functions for gravitons are eventually taken to vanish identically. A short discussion on the situation concerning the limiting case of general relativity is presented thereafter.", '1 Introduction': "The most important physical significance of the Weyl tensor for a curved spacetime having torsionlessness [1,2] relies upon the fact that its two-component spinor expression, as provided by either of the γε -formalisms of Infeld and van der Waerden [3], carries locally only suitable couplings between metric spinors and wave functions for gravitons [2]. Such wave functions thus show up in both formalisms as totally symmetric parts of certain curvature spinors, being at every spacetime point likewise taken to describe microscopically the degrees of freedom of a metric tensor while vanishing identically in the case of a conformally flat spacetime. In this torsionless context, the vanishing of a Weyl tensor actually constitutes the definitive condition for conformal flatness.  \nIn the present paper, we first transcribe into the framework of the torsionful version of the ε -formalism the world definition of the Weyl tensor for a curved spacetime equipped with a torsionful affinity. The spinor expression which results from our transcription shows us that it is not possible to attain any general  \ncondition for conformal flatness in such a spacetime even if wave functions for gravitons are eventually taken to vanish identically. A short discussion on the situation concerning the limiting case of general relativity is then given. For the sake of consistency, we will have to call for part of the algebraic description of the world and spinor curvatures that take place in the context of EinsteinCartan's theory such as exhibited in Refs. [4,5]. It shall become quite clear that all the spinor expressions arising herein still remain formally valid when the pertinent procedures are occasionally shifted to the framework of the torsional γ -formalism.  \nThe principal motivation for elaborating upon the situation being considered, stems from the belief that our work might provide some original material which can be of some importance as far as the cosmological models based upon Einstein-Cartan's theory are concerned [6-8]. In Section 2, we bring out the curvature structures needed for working out the transcription procedures (Section 3). The limiting case is considered in Section 4 along with some physical remarks. The symbolic configurations displayed in the work of Ref. [4] shall be utilized many times in what follows, but we will refer to them explicitly just a few times. All the conventions adhered to in Ref. [5] will be taken for granted at the outset.", '2 Curvature Structures': "Let us consider a curved spacetime endowed with a symmetric metric tensor g µν of signature (+ ---) together with a torsionful covariant derivative operator ∇ µ that satisfies the metric compatibility condition ∇ µ g λσ = 0 . The Weyl tensor of ∇ µ is defined by [2]  \nC µν λσ = R µν λσ -2 R [ µ [ λ g ν ] σ ] + 1 3 Rg [ µ λ g ν ] σ , (1)  \nwith R µν λσ , R µν and R being, respectively, the Riemann tensor, the Ricci tensor and the Ricci scalar of ∇ µ . The tensor R µνλσ possesses skewness in the indices of the pairs µν and λσ, but the traditional index-pair symmetry ceases holding because of the applicability of the cyclic identity  \n∗ R λ µνλ +2 ∇ λ T ∗ λµν +4 T ∗ µ λτ T λτν = 0 , (2)  \nwhere T µνλ is the torsion tensor of ∇ µ which satisfies T µνλ = T [ µν ] λ by definition, and the starred objects just come from adequate first left-right duals [2,5]. As a consequence, R µν bears asymmetry while the Bianchi identity reads  \n∇ ρ ∗ R ρµλσ +2 T ∗ µ ρτ R ρτλσ = 0 . (3)  \nThus, R µνλσ and R µν possess 36 and 16 independent components, respectively, whilst C µνλσ possesses 20.  \nIn either formalism, the gravitational curvature spinors of ∇ µ are borne by the correspondence  \nR µνλσ ↔ (X ABCD , Ξ A ' B ' CD ) , (4)  \nwhere the XΞ-entries have the symmetries  \nX ABCD = X ( AB )( CD ) , Ξ A ' B ' CD = Ξ ( A ' B ' )( CD ) . (5)  \n/negationslash  \nSince R µνλσ = R λσµν , we also have  \n/negationslash  \n/negationslash  \nX ABCD = X CDAB , Ξ A ' B ' CD = Ξ CDA ' B ' , (6)  \nwhence the XΞ-spinors naively recover the number of degrees of freedom of R µνλσ as 18 + 18. In the ε -formalism, one thus gets the expressions  \nR AA ' BB ' CC ' DD ' =( ε A ' B ' ε C ' D ' X ABCD + ε AB ε C ' D ' Ξ A ' B ' CD ) + c.c. (7)  \nand 1  \n∗ R AA ' BB ' CC ' DD ' = [( -i )( ε A ' B ' ε C ' D ' X ABCD -ε AB ε C ' D ' Ξ A ' B ' CD )] + c.c. , (8) with the ε -objects being the only covariant metric spinors for the formalism allowed for.  \nOn the basis of the valence-reduction devices built up in Ref. [2], we can expand the X-spinor of (4) as  \nX ABCD =Ψ ABCD -ε ( A | ( C ξ D ) | B ) -1 3 κ ε A ( C ε D ) B , (9)  \nwith the individual pieces [4]  \nΨ ABCD = X ( ABCD ) , ξ AB = X M ( AB ) M , κ = X LM LM . (10)  \nThe Ψ-spinor of Eq. (9) is usually taken as a typical wave function for gravitons [1,9], and κ amounts to a complex-valued world-spin invariant. In Ref. [5], ξ AB was supposed to account for a wave function for dark matter. Its occurrence in the expansion (9) is essentially related to the appropriateness of the first of Eqs. (6). The complex valuedness of κ enables us to reinstate promptly the number of complex independent components of X ABCD as 5 + 3 + 1. Hence, it may be said that the objects  \n(Ψ ABCD , ξ AB , κ , Ξ A ' B ' CD ) (11)  \ntogether determine both R µνλσ and ∗ R µνλσ completely. It follows that we can spell out the particular patterns  \nR µν ↔ R AA ' BB ' = ε AB ε A ' B ' Re κ -[( ε A ' B ' ξ AB +Ξ A ' B ' AB ) + c.c.] (12)  \nand  \n∗ R λ µλν ↔ R CC ' AA ' CC ' BB ' = [ i ( ε A ' B ' ξ AB -1 2 ε AB ε A ' B ' κ -Ξ A ' B ' AB )] + c.c. . (13)  \nAccordingly, the parts of κ are subject to [4]  \nR = 4 Re κ , ∗ R µν µν = 4 Im κ , (14)  \nwith Eq. (12) thereby recovering the number of degrees of freedom of R µν as 1 + 6 + 9.", '3 Spinor Transcription': "Equations (7) and (9) give right away the correspondence 2  \nR µν λσ ↔ ε B ' D ' [ ε A ' C ' (Ψ AC BD -ε ( A ( B ξ D ) C ) + 1 3 κ ε ( A B ε C ) D )+ ε AC Ξ A ' C ' BD ]+c.c. , (15)  \nwhich involves making some index substitutions in the expression (7) so as to yield the pattern  \nR µν λσ ↔ R AA ' CC ' BB ' DD ' . (16)  \nBy implementing Eq. (12) and invoking one of the relations (14), we obtain the irreducible relationship between -2 R [ µ [ λ g ν ] σ ] and the configuration  \nε B ' D ' [ ε A ' C ' ( ε ( A ( B ξ D ) C ) -ε ( A B ε C ) D Re κ ) -1 2 ε AC (Ξ A ' C ' BD +Ξ BD A ' C ' )] + c.c. . (17)  \nIt is worth mentioning that the derivation of the arrangement (17) takes up the complex conjugates of the blocks  \nR ( AC ) ( B ' D ' ) = -(Ξ AC B ' D ' +Ξ B ' D ' AC ) (18)  \nand  \nR M M ( A ' ( B ' ε C ' ) D ' ) = 2( ε ( A ' B ' ε C ' ) D ' Re κ -ε ( A ' ( B ' ξ D ' ) C ' ) ) . (19)  \nFor the Rgg -term of Eq. (1), we get  \n1 3 Rg [ µ λ g ν ] σ ↔ 2 3 Re κ ( ε A ' C ' ε B ' D ' ε ( A B ε C ) D +c.c.) . (20)  \nThe combination of Eqs. (15)-(20) with (1) produces the cancellation of the contributions that carry the εξ -pieces and Re κ . We are therefore led to the following expression for C µν λσ  \nε B ' D ' [ ε A ' C ' (Ψ AC BD + 1 3 ε ( A B ε C ) D i Im κ ) + 1 2 ε AC (Ξ A ' C ' BD -Ξ BD A ' C ' )] + c.c. , (21)  \nwhich transparently recovers the number of independent components of C µν λσ as 10 + 1 + 9, and thence C µνλσ gets completely determined out of  \n(Ψ ABCD , Im κ , Ξ A ' B ' CD ) . (22)  \n/negationslash  \nFrom (21), we see that C µν λσ = 0 throughout spacetime since the overall terms  \nε A ' C ' ε B ' D ' (Ψ AC BD + 1 3 ε ( A B ε C ) D i Im κ ) , 1 2 ε AC ε B ' D ' (Ξ A ' C ' BD -Ξ BD A ' C ' ) (23)  \nas well as their conjugates are all nowhere-vanishing spinors in the spacetime background we have been dealing with and, in addition, can not cancel out one  \nanother. Yet, we may possibly have torsional spacetimes wherein Ψ ABCD ≡ 0, in which case the spinor expression for C µν λσ could be reset as  \nε B ' D ' [ 1 3 ( ε A ' C ' ε ( A B ε C ) D i Im κ ) + 1 2 ε AC (Ξ A ' C ' BD -Ξ BD A ' C ' )] + c.c. . (24)  \nThen, we have established that it is not possible to attain any general condition for conformal flatness in a spacetime endowed with a torsionful affinity even if the corresponding gravitational wave functions are taken to vanish identically.", '4 Concluding Remarks': "In general relativity, the property R µνλσ = R [ µν ][ λσ ] and the torsionless cyclic identity ∗ R λ µνλ = 0 entail imparting the index-pair symmetry to R µνλσ . Consequently, in the torsionless limiting case, the curvature spinors of (4) turn out to enjoy also the symmetries X ABCD = X CDAB and Ξ A ' B ' CD = Ξ CDA ' B ' , while Im κ = 0 by the second of Eqs. (14). Under this geometric circumstance, the Ξ-term of (23) vanishes identically and the expression (21) thus supplies us with the correspondence [2,10]  \nC µνλσ ↔ ε A ' B ' ε C ' D ' Ψ ABCD +c.c. ,  \nwhence any generally relativistic Weyl tensor carries 10 = 20 -1 -9 degrees of freedom regardless of whether R = 0 or not. 3  \nA spacetime that admits a Weyl tensor like the one expressed by (24) may, in a more or less conventional way, be designated as a conformally flat spacetime. Nevertheless, its Weyl tensor must be defined locally as a non-vanishing object, in contradistinction to the case of general relativity. The word block ' conformally flat Kopczy'nski solution ', referred to in the concluding Section of Ref. [5], may then look legitimate.  \nWe saw that ξ AB does not occur in the expression (21). Hence, according to the picture proposed in Ref. [5], dark matter does not contribute to any extent to the Weyl tensor of ∇ µ . Neither do it the cosmic microwave and dark energy backgrounds because these can never be thought of as possessing a gravitational character [11].  \nData Availability Statement:  \nthe data on the article at issue are entirely available in the repository arXiv on the location https://doi.org/10.48550/arXiv.2409.07364.", 'References': "- [1] Penrose, R., Ann. Phys. (N.Y.) 10, 171 (1960).  \n- [2] Penrose, R., Rindler, W., Spinors and Space-Time Vol. 1, Cambridge University Press, (1984).\n- [3] Infeld L., Van der Waerden B. L., Sitzber. Akad. Wiss. Physik-math. KL. 9, 380 (1933).\n- [4] Cardoso J. G., Eur. Phys. J. Plus 10, 130 (2015).\n- [5] Cardoso J. G., Eur. Phys. Jour. C 83, 212 (2023).\n- [6] Trautman A., Nature Physical Science 114, 7 (1973).\n- [7] Kopczy'nski W., Phys. Lett. A 39, 219 (1972).\n- [8] Poplawski N. J., Phys. Lett. B 3, 181 (2010).\n- [9] Penrose R., Foundations of Physics 13, 325 (1983).\n- [10] Cardoso J. G., Adv. Appl. Clifford Algebras 22, 985 (2012).\n- [11] Cardoso J. G., De Gruitter Open Phys. 13, 339 (2015)."}