Geodesic
Conformal collineations of the Ricci and energy–momentum tensors in static plane symmetric space–times
Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 1, pp. 117-129 Cet article a éte moissonné depuis la source Math-Net.Ru

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We provide a complete classification of static plane symmetric space–times according to conformal Ricci collineations {(}CRCs{\rm)} and conformal matter collineations (CMCs) in both the degenerate and nondegenerate cases. In the case of a nondegenerate Ricci tensor, we find a general form of the vector field generating CRCs in terms of unknown functions of $t$ and $x$ subject to some integrability conditions. We then solve the integrability conditions in different cases depending upon the nature of the Ricci tensor and conclude that the static plane symmetric space–times have a $7$-, $10$- or $15$-dimensional Lie algebra of CRCs. Moreover, we find that these space–times admit an infinite number of CRCs if the Ricci tensor is degenerate. We use a similar procedure to study CMCs in the case of a degenerate or nondegenerate matter tensor. We obtain the exact form of some static plane symmetric space–time metrics that admit nontrivial CRCs and CMCs. Finally, we present some physical applications of our obtained results by considering a perfect fluid as a source of the energy–momentum tensor.
Keywords: conformal Ricci collineation, conformal matter collineation, Ricci collineation, matter collineation. 
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S. S. Akhtar; T. Hussain; A. H. Bokhari; F. Khan. Conformal collineations of the Ricci and energy–momentum tensors in static plane symmetric space–times. Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 1, pp. 117-129. http://geodesic.mathdoc.fr/item/TMF_2018_195_1_a10/

[1] G. S. Hall, Symmetries and Curvature Structure in General Relativity, Lecture Notes in Physics, 46, World Sci., Singapore, 2004 | MR

[2] M. Tsamparlis, P. S. Apostolopoulos, “Ricci and matter collineations of locally rotationally symmetric space-times”, Gen. Relat. Grav., 36:1 (2004), 47–69 | DOI | MR | Zbl

[3] U. Camci, “Conformal collineations and Ricci inheritance symmetry in string cloud and string fluids”, Internat. J. Modern Phys. D, 11:3 (2002), 353–366 | DOI | MR | Zbl

[4] U. Camci, A. Qadir, K. Saifullah, “Conformal Ricci collineations of static spherically symmetric spacetimes”, Commun. Theor. Phys., 49:6 (2008), 1527–1532 | DOI | MR

[5] G. H. Katzin, J. Levine, W. R. Davis, “Curvature collineations: a fundamental symmetry property of the space-times of general relativity defined by the vanishing Lie derivative of the Riemann curvature tensor”, J. Math. Phys., 10:4 (1969), 617–629 | DOI | MR

[6] A. Z. Petrov, Prostranstva Einshteina, Fizmatgiz, M., 1961 | MR

[7] İ. Yavuz, U. Camci, “Ricci collineations of the Bianchi type II, VIII, and IX space-times”, Gen. Relat. Grav., 28:6 (1996), 691–700 | DOI | MR | Zbl

[8] U. Camci, H. Baysal, .{I}. Tarhan, İ. Yilmaz, İ. Yavuz, “Ricci collineations of the bianchi types I and III, and Kantowski–Sachs spacetimes”, Internat. J. Modern Phys. D, 10:5 (2001), 751–765 | DOI | MR | Zbl

[9] U. Camci, İ. Yavuz, “Classifications of Kantowski–Sachs, Bianchi Types I and III spacetimes according to Ricci collineations”, Internat. J. Modern Phys. D, 12:1 (2003), 89–100 | DOI | MR | Zbl

[10] W. R. Davis, G. H. Katzin, “Mechanical conservation laws and the physical properties of groups of motions in flat and curved space-times”, Amer. J. Phys., 30:10 (1962), 750–764 | DOI | MR | Zbl

[11] W. R. Davis, L. H. Green, L. K. Norris, “Relativistic matter fields admitting Ricci collineations and related conservation laws”, Nuovo Cimento B, 34:2 (1976), 256–280 | DOI | MR

[12] D. R. Oliver Jr., W. R. Davis, “Perfect fluids and symmetry mappings leading to conservation laws”, J. Math. Phys., 17:10 (1976), 1790–1792 | DOI | MR

[13] M. Tsamparlis, D. P. Mason, “Ricci collineation vectors in fluid space-times”, J. Math. Phys., 31:7 (1990), 1707–1722 | DOI | MR | Zbl

[14] A. Qadir, K. Saifullah, M. Ziad, “Classification of cylindrically symmetric static spacetimes according to their Ricci collineations”, Gen. Relat. Grav., 35:11 (2003), 1927–1975 | DOI | MR | Zbl

[15] A. H. Bokhari, A. Qadir, “Collineations of the Ricci tensor”, J. Math. Phys., 34:8 (1993), 3543–3552 | DOI | MR | Zbl

[16] U. Camci, İ. Türkyilmaz, “Ricci collineations in perfect fluid Bianchi V spacetime”, Gen. Relat. Grav., 36:9 (2004), 2005–2019 | DOI | MR | Zbl

[17] M. J. Amir, A. H. Bokhari, A. Qadir, “Ricci collineations of static spherically symmetric spacetimes”, J. Math. Phys., 35:6 (1994), 3005–3012 ; Erratum 37:2 (1996), 1089 | DOI | MR | Zbl | DOI

[18] G. Contreras, L. A. Nùñez, U. Percoco, “Ricci collineations for non-degenerate, diagonal and spherically symmetric Ricci tensors”, Gen. Relat. Grav., 32:2 (2000), 285–294 | DOI | MR | Zbl

[19] U. Camci, A. Barnes, “Ricci collineations in Friedmann–Robertson–Walker spacetimes”, Class. Quantum Grav., 19:2 (2002), 393–404 | DOI | MR | Zbl

[20] T. B. Farid, A. Qadir, M. Ziad, “The classification of static plane symmetric space-times according to their Ricci collineations”, J. Math. Phys., 36:10 (1995), 5812–5828 | DOI | MR | Zbl

[21] A. H. Bokhari, “Ricci tensor with six collineations”, Internat. J. Theor. Phys., 31:12 (1992), 2091–2094 | DOI | MR | Zbl

[22] J. Llosa, “Collineations of a symmetric 2-covariant tensor: Ricci collineations”, J. Math. Phys., 54:7 (2013), 072501, 13 pp. | DOI | MR | Zbl

[23] M. Tsamparlis, A. Paliathanasis, L. Karpathopoulos, “Exact solutions of Bianchi I spacetimes which admit conformal Killing vectors”, Gen. Relat. Grav., 47:2 (2015), 15, 21 pp. | DOI | MR | Zbl

[24] S. Moopanar, S. D. Maharaj, “Conformal symmetries of spherical spacetimes”, Internat. J. Theor. Phys., 49:8 (2010), 1878–1885 | DOI | MR | Zbl

[25] S. Moopanar, S. D. Maharaj, “Relativistic shear-free fluids with symmetry”, J. Eng. Math., 82 (2013), 125–131 | DOI | MR | Zbl

[26] K. L. Duggal, R. Sharma, “Conformal Killing vector fields on spacetime solutions of Einstein's equations and initial data”, Nonlinear Anal., 63:5–7 (2005), e447–e454 | DOI | Zbl

[27] R. Maartens, S. D. Maharaj, B. O. J Tupper, “General solution and classification of conformal motions in static spherical spacetimes.”, Class. Quantum Grav., 12:10 (1995), 2577–2586 | DOI | MR | Zbl

[28] S. Khan, T. Hussain, A. H. Bokhari, G. A. Khan, “Conformal Killing vectors in LRS Bianchi type V spacetimes”, Commun. Theor. Phys., 65:3 (2016), 315–320 | DOI | MR | Zbl

[29] S. Khan, T. Hussain, A. H. Bokhari, G. A. Khan, “Conformal Killing vectors of plane symmetric four dimensional lorentzian manifolds”, Eur. Phys. J. C, 75 (2015), 523, 9 pp. | DOI

[30] M. Sharif, N. Tehseen, “Conformal Ricci and matter collineations for an anisotropic fluid”, Chinese J. Phys., 45:6-I (2007), 592–605, arXiv: 0707.2989

[31] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of Einstein's Field Equations, Cambridge Univ. Press, Cambridge, 2003 | DOI | MR

[32] S. S. Akhtar, T. Hussain, A. H. Bokhari, F. Khan, Conformal collineations of the Ricci and energy-momentum tensors in static plane symmetric spacetimes, arXiv: 1702.04637