DOCSLIB.ORG

The Petrov Classification

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Petrov Classification Appendix A The Petrov Classification The Weyl tensor Cabcd, due to its symmetries, can be represented as a 6 x 6 matrix, denoted by CAB. One can then classify the Weyl tensor by using the following simple approach. The Weyl symmetry c~bc = 0 allows us to write cAB as where A and B are 3 x 3 matrices, such that A is symmetric and both A and B are trace free. Above matrix can also be described by the complex 3 x 3 matrix D = A+ i B for which there are the following three possible Jordan forms: Petrov Type I Q 0 0) ( 0 f3 0 0 0 1 Petrov Type II Q 1 0 ) ( 0 Q 0 0 0 -2a Petrov Type III (H!) where the trace free condition on A and B ( and hence on D) is used. Following are subcases of Petrov Type I and II: Petrov Type D Subcase of type I where a= {3. Petrov Type 0 Subcase of type I where a = f3 = 1 = 0 {:::::::} C = 0. Petrov Type N Subcase of type II where a = 0. 193 194 APPENDIX A. THE PETROV CLASSIFICATION An alternative and very useful version of the Petrov classification is due to the work of L. Bel [10], known as the Bel criteria. In this criteria, null eigen bivectors and their principal null directions play an important role. Define the complex self + dual Weyl tensor Cabcd by + * Cabcd= Cabcd + i Cabcd, where Cis* the dual of C. Let a null direction k satisfy the following k[eCa]bc[dkf] kb kc = 0. (A.1) Then the Bel criteria are as follows: (1) Cis Petrov type I if there are exactly 4 distinct null vectors (called its principal null directions) k satisfying (A.1). (2) C is Petrov type II if there are two coincident null directions k satisfying (A.1). (3) C is Petrov type III if there are three coincident null directions k satisfying . + (A.1). Also CIS of type III {::::::} Cabcd ka kc = 0. (4) Cis Petrov type N if there are all four coincident null directions k satisfying + (A.1). Also Cis of type N {::::::} Cabcd kd = 0. (5) C is Petrov type D if its principal null directions are coincident pairs. (6) The Petrov type 0 is characterized by zero Weyl tensor and it does not single out any null direction. Bibliography [1] Adler, R., Bazin, M., and Schiffer, M. Introduction to General Relativity, McGraw-Hill, New York, 1975. [2] Aichelburg, P. C. Curvature collineations for gravitational pp-waves, J. Math. Phys., 11, 1970, 2458-2462. [3] Alias, L. J., Romero, A., and Sanchez, W. Spacelike hypersurfaces of constant mean curvature in certain spacetimes. Nonlinear Analysis, Theory, Methods & Applications, 30, no. 1, (Proc. 2nd World Congress of Nonlinear Analysis, Elsevier Science Ltd.), 1997, 655-661. [4] Arnold, V.I. Contact geometry: The geometrical method of Gibbs's Thermo­ dynamics, Proc. Gibbs Symposium, Yale University, 1989, 163-179. [5] Barnes, A. Projective collineations in Einstein spaces, Class. Quant. Grav., 10, 1993, 1139-1145. [6] Barrabes, C. and Israel, W. Thin shells in general relativity and cosmology: the lightlike limit, Physical Review D., 43, no. 4, 1991, 1129-1142. [7] Barros, M. and Romero, A. Indefinite Kaehler manifolds, Math. Ann., 261,1982, 55-62. [8] Beem, J. K., Ehrlich, P. E. and Easley, K. L. Global Lorentzian Geometry, Marcel Dekker, Inc. New York, Second Edition, 1996. [9] Bejancu, A. and Duggal, K. L. Real hypersurfaces of indefinite Kaehler man­ ifolds, Internat. J. Math. and Math. Sci., 16, 1993, 545-556. [10] Bel, L. Quelques remarques sur le classification de Petrov, C. R. Acad. Sci. (Paris), 148, 1959, 2561-2564. [11] Benenti, S. and Francaviglia, M. The theory of separability of the Hamilton­ Jacobi equations and its applications to relativity, General relativity and Grav­ itation I (edited by A. Held), 1980, 393-439. [12] Berger, M. Trois remarques sur les varietes riemanniennes a courbure positive, C. R. A cad. Sc. Paris, Series A., 263, 1966, 76-78. [13] Berger, B. Homothetic and conformal motions in spacelike slices of solutions of Einstein equations, J. Math. Phys., 17, 1976, 1268-1273. 195 196 BIBLIOGRAPHY [14] Bishop, R. L. and Goldberg, S. I. A characterization of the Euclidean sphere, Bull. Am. Math. Soc., 72, 1966, 122-124. [15] Bishop, R. L. and Goldberg, S. I. Tensor Analysis on Manifolds, The MacMil­ lan Company, New York, 1986. [16] Bishop, R. L. and O'Neill, B. Manifolds of negative curvature, 'frans. Amer. Math. Soc., 145, 1969, 1-49. [17] Blair, D. E. Contact manifolds in Riemannian geometry, Lecture notes in Math., 509, 1976, Springer-Verlag, Berlin. [18] Blair, D. E. Geometry of manifolds with structure group U(n) x O(s), J. Diff. Geom., 4, 1970, 155-167. [19] Blair, D. E., Koufogiorgos, T. and Sharma, R. A classification of 3-dimensional contact metric manifolds with Q¢ = ,PQ, Kodai Math. J., 13, 1990, 391-401. [20] Blair, D. E. and Sharma, R. Three dimensional locally symmetric contact metric manifolds, Boll. U.M.I.,(7)4-A, 1990, 385-390. [21] Blair, D. E. and Sierra, J. M. 5-dimensionallocally symmetric contact metric manifolds, Boll. U.M.I.(7)7-A, 1993, 299-311. [22] Bochner, S. Curvature and Betti numbers, Ann. Math., 49, 1948, 379-390. [23] Bonnor, W. B. Null hypersurfaces in Minkowski spacetime, Tensor N.S., 24, 1972, 329-345. [24] Brocker, Th. and Jiinich, K. Introduction to Differential Topology, Cambridge University Press, Cambridge, 1982. [25] Cagnac, Fr. Geometrie de hypersurfaces isotropes, C. R. Acad. Sci. Paris, 201, 1965, 3045-3048. [26] Candelas, P., Horowitz, G., Strominger, A. and Witten, E. Vacuum configu­ rations for superstrings, Nucl. Phys., B 258, 1985, 46-74. [27] Capocci, M.S. and Hall, G. S. Conformal vector fields on decomposable space­ times, Gravitation & Cosmology, 3, no. 1(9), 1997, 001-010. [28] Carot, J., da Costa, J. and Vaz, E.G.L. Matter Collineations: The inverse "symmetry inheritance" problem, J. Math. Phys., 35, 1994, 4832-4838. [29] Carter, B. Killing horizons and orthogonally transitive groups in spacetimes, J. Math. Phys., 10, 1969, 70-81. [30] Castejon-Amenedo, J. and Coley, A. A. Exact solutions with conformal Killing vector fields, Class. Quant. Grav., 9, 1992, 2203-2215. [31] Coley, A. A. and Tupper, B. 0. J. Special conformal Killing vector spacetimes and symmetry inheritance, J. Math. Phys., 30, 1989, 2616-2625. [32] Coley, A. A. and Tupper, B.O.J. Spacetimes admitting special affine conformal vectors, J. Math. Phys., 31, 1990, 649-652. BIBLIOGRAPHY 197 [33] Coley, A. A. and Tupper, B.O.J. Spacetimes admitting inheriting conformal Killing vector fields, Class. Quant. Grav., 7, 1990, 1961-1981. [34] Coley, A. A. and Tupper, B.O.J. Spherical symmetric spacetimes admitting inheriting conformal Killing vector fields, Class. Quant. Grav., 7, 1990, 2195- 2214. [35] Collinson, C. D. Curvature collineations in empty spacetimes, J. Math. Phys., 11, 1970, 818-819. [36] Collinson, C.D. Proper affine collineation in Robertson-Walker spacetimes, J. Math. Phys., 29, 1988, 1972-73. [37] Collinson, C. D. and French, D. C. Null tetrad approach to motions in empty spacetime, J. Math. Phys., 8, 1967, 701-708. [38] Dallmer, F. Zur Geometrie der Metrish Singularen Hyperflachenim Vierdi­ mensionalen Pseudoeuklidischen Raum, diplomarbeit, Universitat Karlsruhe, 1987. [39] Dautcourt, G. Characteristic hypersurfaces in general relativity, J. Math. Phys., 8, 1967, 1492-1501. [40] Debney, G. C. and Zund, J.D. A note on the classification of electromagnetic field, Tensor N.S., 22, 1971, 333-340. [41] Davis, W. R. and Moss, M. K. Conservation laws of the general theory of relativity II, Nuovo Cimento, 38, 1965, 1558-1569. [42] Duggal, K. L. On almost contingent manifolds of second class with applica­ tions in relativity, Can. Math. Bull., 21, 1978, 289-295. [43] Duggal, K. L. Existence of two Killing vector fields on the spacetime of general relativity, Tensor N.S., 32, 1978, 318-322. [44] Duggal, K. L. Einstein Maxwell equations compatible with certain Killing vectors with light velocity, Annali di Matematica Pure ed Applicata, (lV), 120, 1979, 263-268. [45] Duggal, K. L. On the four current source of the electromagnetic fields, Annali di Matematica pureed Applicata, (IV), 120, 1979, 305-313. [46] Duggal, K. L. Relativistic fluids with shear and timelike conformal collineations, J. Math. Phys., 28, 1987, 2700-2704. [47] Duggal, K. L. Some applications of globally framed structures to relativity, Annali di Matematica pureed Applicata, (lV), 1988, 77-90. [48] Duggal, K. L. Lorentzian geometry of CR-submanifolds, Acta. Appl. Math., 17, 1989, 171-193. [49] Duggal, K. L. Relativistic fluids and metric symmetries, J. Math. Phys., 30, 1989, 1316-1322. 198 BIBLIOGRAPHY [50] Duggal, K. L. Lorentzian geometry of globally framed manifolds, Acta Appl. Math., 19, 1990, 131-148. [51] Duggal, K. L. Spacetime manifolds and contact structures, Internat. J. Math. & Math. Sci., 13, 1990, 545-554. [52] Duggal, K. L. Affine conformal vector fields in semi-Riemannian manifolds, Acta. Appl. Math., 23, 1991, 275-294. [53] Duggal, K. L. Curvature inheritance symmetry in Riemannian spaces with applications to fluid spacetimes, J. Math. Phys., 3:3, 1992, 2989-2997. [54] Duggal, K. L. Symmetry inheritance in Riemannian manifolds with applica­ tions, Acta. Appl. Math., 31, 1993, 225-247. [55] Duggal, K. L., Asgekar, G. G. and Aherkar, S. M. A class of shear-free, non­ static models for anisotropic magnetofl.uid system, J. Math. Phys., 36, 1995, 2929-2940. [56] Duggal, K. L. and Bejancu, A. Spacetime geometry of CR-structures, in Dif­ ferential Geometry and Mathematical Physics (J. K. Beem and K. L. Duggal, eds.), Contemporary Math. Series Vol.
Load more

Recommended publications
  • Riemannian Curvature and the Petrov Classification G
    Riemannian Curvature and the Petrov Classification G. S. Hall Department of Mathematics, The Edward Wright Building, Dunbar Street, University of Aber- deen, Aberdeen AB9 2TY, Scotland Z. Naturforsch. 33a, 559-562 (1978); received November 11, 1977 A connection is shown between the Riemannian Curvatures of the wave surfaces of a null vector I at a point p in space-time and the Petrov type of the space-time at p. Some other results on Riemannian Curvature are discussed. 1. Introduction at the point p under consideration. (A full discussion of the Petrov classification and the geometry of Let M be a Lorentzian space-time manifold. If null congruences and wave surfaces can be found p G 31, let TV(M) denote the tangent space to M in the works of Ehlers and Kundt [5], Sachs [6, 7], at p and let I e Tp (M) be a null vector. Because of Pirani [8, 9] and Petrov [10]. The importance of the the identification of radiation in General Relativity Petrov classification in General Relativity in the with null geodesic congruences of curves in M, study of exact solutions of Einstein's equations is one is led to study the geometrical properties of the dealt with thoroughly in Ref. [5] and in a more wave surfaces of I. These are the two dimensional recent paper by Kinnersley [11].) subspaces of TP(M), each member of which is spacelike and orthogonal to I. There is in fact a two parameter family of such wave surfaces for a 2. Auxiliary Results given null vector leTv(M) which, from the physical viewpoint, might be thought of as the The notation will be the usual one.
    [Show full text]
  • Arxiv:1311.7110V2 [Gr-Qc] 16 May 2014
    On the Pursuit of Generalizations for the Petrov Classification and the Goldberg-Sachs Theorem Carlos Batista arXiv:1311.7110v2 [gr-qc] 16 May 2014 Doctoral Thesis Universidade Federal de Pernambuco, Departamento de Física Supervisor: Bruno Geraldo Carneiro da Cunha Brazil - November - 2013 Thesis presented to the graduation program of the Physics Department of Universidade Federal de Per- nambuco as part of the duties to obtain the degree of Doctor of Philosophy in Physics. Examining Board: Prof. Amilcar Rabelo de Queiroz (IF-UNB, Brazil) Prof. Antônio Murilo Santos Macêdo (DF-UFPE, Brazil) Prof. Bruno Geraldo Carneiro da Cunha (DF-UFPE, Brazil) Prof. Fernando Roberto de Luna Parisio Filho (DF-UFPE, Brazil) Prof. Jorge Antonio Zanelli Iglesias (CECs, Chile) Abstract The Petrov classification is an important algebraic classification for the Weyl tensor valid in 4-dimensional space-times. In this thesis such classification is gen- eralized to manifolds of arbitrary dimension and signature. This is accomplished by interpreting the Weyl tensor as a linear operator on the bundle of p-forms, for any p, and computing the Jordan canonical form of this operator. Throughout this work the spaces are assumed to be complexified, so that different signatures correspond to different reality conditions, providing a unified treatment. A higher- dimensional generalization of the so-called self-dual manifolds is also investigated. The most important result related to the Petrov classification is the Goldberg- Sachs theorem. Here are presented two partial generalizations of such theorem valid in even-dimensional manifolds. One of these generalizations states that cer- tain algebraic constraints on the Weyl “operator” imply the existence of an in- tegrable maximally isotropic distribution.
    [Show full text]
  • Axisymmetric Spacetimes in Relativity
    9, tr' Axisymmetric Spacetimes in Relativity S. P. Drake Department of Physics and Mathematical Physics îåäi,î31 tÏfåi" Australia 22 July 1998 ACKNO\MTEDGEMENTS It is an unfortunate fact that only my name Soes on this thesis' This is not the work of one person, it is an accumulation of the labor of many' AII those who have helped me in ways that none of us could image should be acknowledged, but how. None of this would have been possible were it not for the wisdom and guidance of my supervisor Peter Szekeres. I have often pondered over the difficulty of supervising students. It must be heart-bleaking to watch as students make necessary mistakes. Patience I'm sure must be the key' So for his wisdom, patience and kindness I am deeply indebted to Peter Szekeres. Without the love and support of my family this thesis would never have begun, much less been completed, to them I owe my life, and all that comes with it. It would take too long to write the names of all those I wish to thank. Those who are special to me, have helped me through so much over the years will receive the thanks in Person. I would like to the department of physics here at Adelaide where most of my thesis work was done. I would like to thank the department of physics at Melbourne university, where I did my undergraduate degree and began my PhD. I would like to thank the university of Padova for there hospitality dur- ing my stay there.
    [Show full text]
  • Conformal Einstein Spaces and Bach Tensor Generalizations in N Dimensions
    LinkÄoping Studies in Science and Technology. Theses No. 1113 Conformal Einstein spaces and Bach tensor generalizations in n dimensions Jonas Bergman Matematiska institutionen LinkÄopings universitet, SE-581 83 LinkÄoping, Sweden LinkÄoping 2004 August 20, 2004 (15:33) ii Conformal Einstein spaces and Bach tensor generalizations in n dimensions °c 2004 Jonas Bergman Matematiska institutionen LinkÄopings universitet SE-581 83 LinkÄoping, Sweden [email protected] LiU-TEK-LIC-2004:42 ISBN 91-85295-28-0 ISSN 0280-7971 Printed by UniTryck, LinkÄoping 2004 August 20, 2004 (15:33) iii Abstract In this thesis we investigate necessary and su±cient conditions for an n- dimensional space, n ¸ 4, to be locally conformal to an Einstein space. After reviewing the classical results derived in tensors we consider the four-dimensional spinor result of Kozameh, Newman and Tod. The in- volvement of the four-dimensional Bach tensor (which is divergence-free and conformally well-behaved) in their result motivates a search for an n-dimensional generalization of the Bach tensor Bab with the same prop- erties. We strengthen a theorem due to Belfag¶onand Ja¶en and give a basis (U ab, V ab and W ab) for all n-dimensional symmetric, divergence-free 2- index tensors quadratic in the Riemann curvature tensor. We discover the 1 1 simple relationship Bab = 2 U ab + 6 V ab and show that the Bach tensor is the unique tensor with these properties in four dimensions. Unfortunately we have to conclude, in general that there is no direct analogue in higher dimension with all these properties.
    [Show full text]
  • Riemannian Curvature and the Petrov Classification G
    Riemannian Curvature and the Petrov Classification G. S. Hall Department of Mathematics, The Edward Wright Building, Dunbar Street, University of Aber- deen, Aberdeen AB9 2TY, Scotland Z. Naturforsch. 33a, 559-562 (1978); received November 11, 1977 A connection is shown between the Riemannian Curvatures of the wave surfaces of a null vector I at a point p in space-time and the Petrov type of the space-time at p. Some other results on Riemannian Curvature are discussed. 1. Introduction at the point p under consideration. (A full discussion of the Petrov classification and the geometry of Let M be a Lorentzian space-time manifold. If null congruences and wave surfaces can be found p G 31, let TV(M) denote the tangent space to M in the works of Ehlers and Kundt [5], Sachs [6, 7], at p and let I e Tp (M) be a null vector. Because of Pirani [8, 9] and Petrov [10]. The importance of the the identification of radiation in General Relativity Petrov classification in General Relativity in the with null geodesic congruences of curves in M, study of exact solutions of Einstein's equations is one is led to study the geometrical properties of the dealt with thoroughly in Ref. [5] and in a more wave surfaces of I. These are the two dimensional recent paper by Kinnersley [11].) subspaces of TP(M), each member of which is spacelike and orthogonal to I. There is in fact a two parameter family of such wave surfaces for a 2. Auxiliary Results given null vector leTv(M) which, from the physical viewpoint, might be thought of as the The notation will be the usual one.
    [Show full text]
  • Spinor-Helicity and the Algebraic Classification of Higher-Dimensional Spacetimes Arxiv:1809.03906V2 [Gr-Qc] 1 Apr 2019
    Prepared for submission to JHEP QMUL-PH-18-22 Spinor-helicity and the algebraic classification of higher-dimensional spacetimes Ricardo Monteiro,a Isobel Nicholson,b and Donal O'Connellb aCentre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK bHiggs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK E-mail: [email protected],[email protected], [email protected] Abstract: The spinor-helicity formalism is an essential technique of the ampli- tudes community. We draw on this method to construct a scheme for classifying higher-dimensional spacetimes in the style of the four-dimensional Petrov classifi- cation and the Newman-Penrose formalism. We focus on the five-dimensional case for concreteness. Our spinorial scheme naturally reproduces the full structure previ- ously seen in both the CMPP and de Smet classifications, and resolves longstanding questions concerning the relationship between the two classifications. arXiv:1809.03906v2 [gr-qc] 1 Apr 2019 Contents 1 Introduction1 2 Review of the four-dimensional story3 2.1 Spinors in four dimensions3 2.2 The four-dimensional Newman-Penrose tetrad4 2.3 The Petrov classification for 2-forms and the Weyl spinor5 3 A Newman-Penrose basis in five dimensions7 3.1 Spinors in five dimensions7 3.2 Polarisation vectors 10 3.3 Reality conditions 11 3.4 Lorentz transformations and the little group 12 3.4.1 Boosts and spins 12 3.4.2 The
    [Show full text]
  • Complete Dissertation
    VU Research Portal Kundt Spacetimes in General Relativity and Supergravity Fuster Perez, A. 2007 document version Publisher's PDF, also known as Version of record Link to publication in VU Research Portal citation for published version (APA) Fuster Perez, A. (2007). Kundt Spacetimes in General Relativity and Supergravity. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. E-mail address: [email protected] Download date: 27. Sep. 2021 VRIJE UNIVERSITEIT Kundt Spacetimes in General Relativity and Supergravity ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. L.M. Bouter, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der Exacte Wetenschappen op donderdag 6 december 2007 om 15.45 uur in de aula van de universiteit, De Boelelaan 1105 door Andrea Fuster P´erez geboren te Barakaldo, Spanje promotor: prof.dr.
    [Show full text]
  • Algebraic Classification of Robinson–Trautman Spacetimes
    Algebraic classification of Robinson–Trautman spacetimes J. Podolsk´y∗ and R. Svarcˇ † Institute of Theoretical Physics, Charles University in Prague, Faculty of Mathematics and Physics, V Holeˇsoviˇck´ach 2, 18000 Prague 8, Czech Republic. June 13, 2016 Abstract We consider a general class of four-dimensional geometries admitting a null vector field that has no twist and no shear but has an arbitrary expansion. We explicitly present the Petrov classification of such Robinson–Trautman (and Kundt) gravitational fields, based on the algebraic properties of the Weyl tensor. In particular, we determine all algebraically special subcases when the optically privileged null vector field is a multiple principal null direction (PND), as well as all the cases when it remains a single PND. No field equations are a priori applied, so that our classification scheme can be used in any metric theory of gravity in four dimensions. In the classic Einstein theory this reproduces previous results for vacuum spacetimes, possibly with a cosmological constant, pure radiation and electromagnetic field, but can be applied to an arbitrary matter content. As non-trivial explicit examples we investigate specific algebraic properties of the Robinson–Trautman spacetimes with a free scalar field, and also black hole spacetimes in the pure Einstein–Weyl gravity. PACS class: 04.20.Jb, 04.50.–h, 04.30.–w, 04.40.Nr Keywords: Robinson–Trautman class, Kundt class, algebraic classification, general relativity and other theories of gravity, black holes, gravitational waves ∗[email protected] †[email protected] 1 1 Introduction Robinson–Trautman family of spacetimes was discovered more than half a century ago [1, 2], soon after the advent of new powerful techniques and concepts in general relativity such as geomet- rical optics of null congruences, null tetrad formalism, and related algebraic classification of the Weyl tensor.
    [Show full text]
  • The Type III Stress-Energy Tensor: Ugly Duckling of the Hawking–Ellis
    The type III stress-energy tensor: Ugly Duckling of the Hawking–Ellis classification Prado Mart´ın–Moruno1 and Matt Visser2 1Departamento de F´ısica Te´orica and IPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain 2School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand E-mail: [email protected], [email protected] 14 November 2019 Abstract. We present some advances in the understanding of type III stress-energy tensors as per the Hawking–Ellis classification. Type I and type II naturally appear in classical situations, and can also describe semiclassical effects. Type IV often shows up in semiclassical gravity. Type III is much more subtle. We focus our attention on type III0 stress-energy tensors, which capture the essence (“essential core”) of type III. Reflecting on known purely phenomenological examples, (“gyratons”), we are able to generalize the geometry generated by those type III0 stress-energy tensors. Moreover, we also succeed in extending work by Griffiths based on massless Weyl spinors by finding a fundamental classical bosonic Lagrangian description of these type III0 stress- energy tensors. To the best of our knowledge this is the first time in the literature that a consistent classical bosonic Lagrangian formulation for type III0 stress-energy has been found. arXiv:1907.01269v3 [gr-qc] 13 Nov 2019 Keywords: type III stress-energy; type III0 stress-energy; Hawking–Ellis classification; gyratons; Kundt spacetimes. The type III stress-energy tensor 2 1. Introduction The understanding of gravity as a property of spacetime shook up the very basis of our understanding of fundamental physics.
    [Show full text]
  • Spinor Classification of the Weyl Tensor in Five Dimensions
    Spinor classification of the Weyl tensor infive dimensions Mahdi Godazgar To cite this version: Mahdi Godazgar. Spinor classification of the Weyl tensor in five dimensions. Classical andQuan- tum Gravity, IOP Publishing, 2010, 27 (24), pp.245013. 10.1088/0264-9381/27/24/245013. hal- 00654152 HAL Id: hal-00654152 https://hal.archives-ouvertes.fr/hal-00654152 Submitted on 21 Dec 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spinor classification of the Weyl tensor in five dimensions Mahdi Godazgar Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences Wilberforce Road, Cambridge CB3 0WA, UK [email protected] August 19, 2010 Abstract We investigate the spinor classification of the Weyl tensor in five dimensions due to De Smet. We show that a previously overlooked reality condition reduces the number of possible types in the classification. We classify all vacuum solutions belonging to the most special algebraic type. The connection between this spinor and the tensor classification due to Coley, Milson, Pravda and Pravdov´ais investigated and the relation between most of the types in each of the classifications is given. We show that the black ring is algebraically general in the spinor classification.
    [Show full text]
  • Gravitational Waves in Non-Local Gravity
    Gravitational waves in non-local gravity Salvatore Capozziello ∗1,2,3 and Maurizio Capriolo † 4 1Dipartimento di Fisica "E. Pancini", Università di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy, 2INFN Sezione di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy, 3 Lab. Theor. Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia. 4Dipartimento di Matematica Università di Salerno, via Giovanni Paolo II, 132, Fisciano, SA I-84084, Italy. July 16, 2021 Abstract We derive gravitational waves in a theory with non-local curvature corrections to the Hilbert-Einstein Lagrangian. In addition to the standard two massless tensor modes, with plus and cross polarizations, helicity 2 and angular frequency ω1, we obtain a further scalar massive mode with helicity 0 and angular frequency ω2, whose polarization is transverse. It is a breathing mode, which, at the lowest order of an effective parameter γ, presents a speed difference between nearly null and null plane waves. Finally, the quasi-Lorentz E(2)-invariant class for the non-local gravity is type N3, according to the Petrov classification. This means that the presence (or absence) of gravitational wave modes is observer-independent. PACS numbers: 04.50.Kd, 04.30.-w, 98.80.-k Keywords: Non-local gravity; modified gravity; gravitational waves. 1 Introduction arXiv:2107.06972v1 [gr-qc] 14 Jul 2021 Theories of physics describing elementary interactions are local, that is, fields are evaluated at the same point, and are governed by point-like Lagrangians from which one derives equations of motion.
    [Show full text]
  • The Weyl Tensor Introduction, Properties and Applications
    The Weyl Tensor Introduction, properties and applications David Wichmann Dec 06, 2016 1 / 24 Overview 1. Motivation 2. Derivation and properties of the Weyl Tensor 3. Where the Weyl Tensor appears 4. Application: Classification of Spacetimes using the Weyl Tensor (Petrov Classification) 2 / 24 Motivation 1 G = R − Rg = κT µν µν 2 µν µν I Algebraic equations for the traces of the Riemann Tensor I Determine 10 components of the Riemann Tensor I No direct visibility of curvature propagation Traceless part of Rαβγδ is the Weyl tensor, Cαβγδ . 3 / 24 Decomposition of Curvature Tensor Definition For Aαβ symmetric, define: ∗ Aαβγδ = Aαγgβδ + Aβδgαγ − Aαδgβγ − Aβγgαδ ∗ Aαβγδ fulfills: ∗ ∗ 1. Aαβγδ = −Aαβδγ ∗ ∗ 2. Aαβγδ = Aγδαβ ∗ ∗ ∗ 3. Aαβγδ + Aαγδβ + Aαδβγ = 0 Possible Ansatz ∗ ∗ Rαβγδ = Cαβγδ + ARαβγδ + BRgαβγδ; A; B 2 R. 4 / 24 The Weyl Tensor 1 C = R − (R g + R g − R g − R g ) αβγδ αβγδ n − 2 αγ βδ βδ αγ αδ βγ βγ αδ 1 + R(g g − g g ) (n − 1)(n − 2) αγ βδ βγ αδ See for example: H.-J. Schmidt, Gravitoelectromagnetism and other decompositions of the Riemann tensor (2004, arXiv:gr-qc/04070531v1) Get a propagation equation: n − 3 1 rαC = κ r T − r T − (r Tg − r Tg ) αβγδ n − 2 γ βδ δ βγ n − 1 γ βδ δ βγ See for example: Matthias Blau, Lecture Notes on General Relativity (ttp://www.blau.itp.unibe.ch/Lecturenotes.html) 5 / 24 Weyl Tensor Properties 1. Same algebraic symmetries as Riemann Tensor αγ 2. Traceless: g Cαβγδ = 0 3. Conformally invariant: I That means: 2 ~α α g~αβ = Ω (x)gαβ ) C βγδ = C βγδ 6( 2 I C = 0 is sufficient condition for gαβ = Ω ηαβ in n ≥ 4 4.
    [Show full text]