Geodesic
Classification of locally rotationally symmetric Bianchi-I space–times using conformal Ricci collineations
Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 1, pp. 133-145 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We present a complete classification of locally rotationally symmetric (LRS) Bianchi-I space–times in accordance with their conformal Ricci collineations (CRCs). In the case where the Ricci tensor is nondegenerate, we find a general form of the vector field generating CRCs subject to some integrability conditions. Solving the integrability conditions in different cases, we find that the LRS Bianchi-I space–times admit $7$-, $10$-, $11$-, or $15$-dimensional Lie algebras of CRCs in the case where the Ricci tensor is nondegenerate. Moreover, we find that these space–times admit an infinite number of CRCs if the Ricci tensor is degenerate. We give some examples of LRS Bianchi-I space–times that admit nontrivial CRCs and are models of a perfect fluid.
Keywords: conformal Ricci collineation, Ricci collineation, Bianchi-I space–time. 
@article{TMF_2017_193_1_a8,
     author = {T. Hussain and S. S. Akhtar and F. Khan},
     title = {Classification of locally rotationally symmetric {Bianchi-I} space{\textendash}times using conformal {Ricci} collineations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {133--145},
     year = {2017},
     volume = {193},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2017_193_1_a8/}
}
TY  - JOUR
AU  - T. Hussain
AU  - S. S. Akhtar
AU  - F. Khan
TI  - Classification of locally rotationally symmetric Bianchi-I space–times using conformal Ricci collineations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2017
SP  - 133
EP  - 145
VL  - 193
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2017_193_1_a8/
LA  - ru
ID  - TMF_2017_193_1_a8
ER  - 
%0 Journal Article
%A T. Hussain
%A S. S. Akhtar
%A F. Khan
%T Classification of locally rotationally symmetric Bianchi-I space–times using conformal Ricci collineations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2017
%P 133-145
%V 193
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2017_193_1_a8/
%G ru
%F TMF_2017_193_1_a8
T. Hussain; S. S. Akhtar; F. Khan. Classification of locally rotationally symmetric Bianchi-I space–times using conformal Ricci collineations. Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 1, pp. 133-145. http://geodesic.mathdoc.fr/item/TMF_2017_193_1_a8/

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

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

[3] C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, W. H. Freeman and Co., San Francisco, CA, 1973 | MR

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

[5] A. H. Bokhari, A. R. Kashif, “Curvature collineations of some static spherically symmetric space-times”, J. Math. Phys., 37:7 (1996), 3498–3504 | DOI | MR

[6] G. H. Katzin, J. Levine, “Applications of Lie derivatives to symmetries, geodesic mappings, and first integrals in Riemannian spaces”, Colloq. Math., 26 (1972), 21–38 | MR

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

[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

[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

[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

[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

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

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

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

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

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

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

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

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

[22] 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

[23] 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 | MR

[24] 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

[25] 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

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

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

[28] T. Hussain, S. S. Akhtar, S. Khan, “Ricci inheritance collineation in Bianchi type I spacetimes”, Eur. Phys. J. Plus, 130 (2015), 44 | DOI

[29] T. Hussain, S. S. Akhtar, A. H. Bokhari, S. Khan, “Ricci inheritance collineations in Bianchi type II spacetime”, Modern Phys. Lett. A, 31:17 (2016), 1650102, 16 pp. | DOI | MR | Zbl

[30] T. Hussain, A. Musharaf, S. Khan, “Ricci inheritance collineations in Kantowski–Sachs spacetimes”, Internat. J. Geom. Meth. Modern Phys., 13:5 (2016), 1650057, 14 pp. | DOI | MR | Zbl

[31] R. M. Wald, General Relativity, Chicago Univ. Press, Chicago, IL, 1984 | MR

[32] T. Hussain, S. S. Akhtar, S. Khan, “Classification of LRS Bianchi type I spacetimes via conformal Ricci collineations”, arXiv: 1701.06900