Velocity and velocity bounds in static spherically symmetric metrics

• Authors: Ivan Arraut , Davide Batic , Marek Nowakowski
• Languages of publication: EN

Abstracts

EN

We find simple expressions for velocity of massless particles with dependence on the distance, r, in Schwarzschild coordinates. For massive particles these expressions give an upper bound for the velocity. Our results apply to static spherically symmetric metrics. We use these results to calculate the velocity for different cases: Schwarzschild, Schwarzschild-de Sitter and Reissner-Nordström with and without the cosmological constant. We emphasize the differences between the behavior of the velocity in the different metrics and find that in cases with naked singularity there always exists a region where the massless particle moves with a velocity greater than the velocity of light in vacuum. In the case of Reissner-Nordström-de Sitter we completely characterize the velocity and the metric in an algebraic way. We contrast the case of classical naked singularities with naked singularities emerging from metric inspired by noncommutative geometry where the radial velocity never exceeds one. Furthermore, we solve the Einstein equations for a constant and polytropic density profile and calculate the radial velocity of a photon moving in spaces with interior metric. The polytropic case of radial velocity displays an unexpected variation bounded by a local minimum and maximum.

Keywords

EN

relativity and gravitation; classical black holes; singularities and cosmic censorship; post-Newtonian approximation

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2011
• Volume: 9
• Issue: 4
• Pages: 926-938

Dates

published

1 - 8 - 2011

online

30 - 4 - 2011

Contributors

author

Ivan Arraut

• Centro de Investigaciones, Universidad Antonio Nariño, Cra.3 Este No.47A-15, Bogotá, Colombia

author

Davide Batic

• Department of Mathematics, University of the West Indies, Kingston 7, Jamaica

author

Marek Nowakowski

• Departamento de FÍsica, Universidad de los Andes, Cra.1E No.18A-10, Bogotá, Colombia

References

• [1] S.M. Carroll, Spacetime and Geometry (Addison Wesley, San Francisco, 2004)
• [2] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley, New York, 1972)
• [3] A. Balaguera-Antolinez, C.G. Böhmer, M. Nowakowski, Classical Quant. Grav. 23, 485 (2006) http://dx.doi.org/10.1088/0264-9381/23/2/013[Crossref]
• [4] L. Lehner, F. Pretorius, Phys. Rev. Lett. 105, 101102 (2010) http://dx.doi.org/10.1103/PhysRevLett.105.101102[Crossref]
• [5] C.V. Johnson, R.C. Myers, A.W. Peet, S.F. Ross, Phys. Rev. D 64, 106001 (2001) http://dx.doi.org/10.1103/PhysRevD.64.106001[Crossref]
• [6] C.V. Johnson, R.C. Myers, Phys. Rev. D 64, 106002 (2001) http://dx.doi.org/10.1103/PhysRevD.64.106002[Crossref]
• [7] D. Astefanesei, R.C. Myers, J. High Energy Phys. 0202, 043 (2002) http://dx.doi.org/10.1088/1126-6708/2002/02/043[Crossref]
• [8] E. Boyda, S. Ganguli, P. Horava, U. Varadarajan, Phys. Rev. D 67, 106003 (2003) http://dx.doi.org/10.1103/PhysRevD.67.106003[Crossref]
• [9] D. Brecher, P.A. DeBoer, D.C. Page, M. Rozali, J. High Energy Phys. 0310, 031 (2003) http://dx.doi.org/10.1088/1126-6708/2003/10/031[Crossref]
• [10] D. Astefanesei, R.B. Mann, E. Radu, J. High Energy Phys. 0501, 049 (2005) http://dx.doi.org/10.1088/1126-6708/2005/01/049[Crossref]
• [11] F. Kottler, Ann. Phys.-Leipzig 56, 410 (1918)
• [12] I. Arraut, D. Batic, M. Nowakowski, Classical Quant. Grav. 26, 125006 (2009) http://dx.doi.org/10.1088/0264-9381/26/12/125006[Crossref]
• [13] K.M. Evenson et al., Phys. Rev. Lett. 29, 1346 (1972) http://dx.doi.org/10.1103/PhysRevLett.29.1346[Crossref]
• [14] P. Nicolini, A. Smailagic, E. Spallucci, Phys. Lett. B 632, 547 (2006) http://dx.doi.org/10.1016/j.physletb.2005.11.004[Crossref]
• [15] S. Ansoldi, P. Nicolini, A. Smailagic, E. Spallucci, Phys. Lett. B 645, 261 (2007) http://dx.doi.org/10.1016/j.physletb.2006.12.020[Crossref]
• [16] P. Nicolini, Int. J. Mod. Phys. A 24, 1229 (2009) http://dx.doi.org/10.1142/S0217751X09043353[Crossref]
• [17] I. Arraut, D. Batic, M. Nowakowski, Classical Quant. Grav. 26, 245006 (2009) http://dx.doi.org/10.1088/0264-9381/26/24/245006[Crossref]
• [18] I. Arraut, D. Batic, M. Nowakowski, J. Math. Phys. 51, 022503 (2010) http://dx.doi.org/10.1063/1.3317913[Crossref]
• [19] H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics 138 (Springer Verlag, Berlin Heidelberg New York, 1986)
• [20] J.E. Cremona, J. Comput. Math. 2, 62 (1999)
• [21] H. Reissner, Ann. Phys.-Leipzig 50, 106 (1916) http://dx.doi.org/10.1002/andp.19163550905[Crossref]
• [22] G. Nordström, K. Ned. Akad. Van Wet. 20, 1238 (1918)
• [23] S.W. Hawking, Phys. Rev. D 14, 2460 (1976) http://dx.doi.org/10.1103/PhysRevD.14.2460[Crossref]
• [24] R.M. Wald, Quantum Field Theory in Curved Space-time and Black Hole Thermodynamics (University of Chicago Press, Chicago, 1994)
• [25] J. Earman, World Enough and Spacetime: Absolute vs. Relational Theories of Space and Time (MIT Press, Cambridge, MA, 1992)
• [26] J. Earman, Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Space-times (MIT Press, Cambridge, MA, 1992)
• [27] G. Belot, J. Earman, L. Ruetsche, Brit. J. Philos. Sci. 50, 189 (1999) http://dx.doi.org/10.1093/bjps/50.2.189[Crossref]
• [28] M. Bojowald, Living Rev. Relativ. 8, 11 (2005)
• [29] R. Goswami, P. Joshi, Phys. Rev. D 76, 084026 (2007) http://dx.doi.org/10.1103/PhysRevD.76.084026[Crossref]
• [30] R. Goswami, P. Joshi, P. Singh, Phys. Rev. Lett. 96, 031302 (2006) http://dx.doi.org/10.1103/PhysRevLett.96.031302[Crossref]
• [31] D. Eardley, L. Smarr, Phys. Rev. D 19, 2239 (1979) http://dx.doi.org/10.1103/PhysRevD.19.2239[Crossref]
• [32] A. Krolak, Prog. Theor. Phys. Supp. 136, 45 (1999) http://dx.doi.org/10.1143/PTPS.136.45[Crossref]
• [33] A. Ori, T. Piran, Phys. Rev. D 42, 1068 (1989) http://dx.doi.org/10.1103/PhysRevD.42.1068[Crossref]
• [34] C.J.S. Clarke, F. de Felice, Gen. Relat. Gravit. 16, 139 (1984) http://dx.doi.org/10.1007/BF00762443[Crossref]
• [35] P.S. Joshi, Global Aspects in Gravitation and Cosmology (Oxford University Press, Oxford, 1993)
• [36] G.M. Shore, Nucl. Phys. B 460, 379 (1996) http://dx.doi.org/10.1016/0550-3213(95)00646-X[Crossref]
• [37] A.S. Sefiedgar, K. Atazadeh, H.R. Sepangi, Phys. Rev. D 80, 064010 (2009) http://dx.doi.org/10.1103/PhysRevD.80.064010[Crossref]
• [38] P.C. Stichel, W.J. Zakrzewski, Phys. Rev. D 80, 083513 (2009) http://dx.doi.org/10.1103/PhysRevD.80.083513[Crossref]
• [39] D.F. Mota, J. Cosmol. Astropart. P. 0809, 006 (2008) http://dx.doi.org/10.1088/1475-7516/2008/09/006[Crossref]
• [40] R.A. Sussman, arXiv:0809.3314 [gr-qc]
• [41] C.G. Boehmer, T. Harko, F.S.N. Lobo, J. Cosmol. As-tropart. P. 0803, 024 (2008) http://dx.doi.org/10.1088/1475-7516/2008/03/024[Crossref]
• [42] C.G. Boehmer, G. Fodor, Phys. Rev. D 77, 064008, (2008) http://dx.doi.org/10.1103/PhysRevD.77.064008[Crossref]
• [43] V. Faraoni, A. Jacques, Phys. Rev. D 76, 063510 (2007) http://dx.doi.org/10.1103/PhysRevD.76.063510[Crossref]
• [44] D.F. Mota, D.J. Shaw, J. Silk, Astrophys. J. 675, 29 (2008) http://dx.doi.org/10.1086/524401[Crossref]
• [45] V. Cardoso, L. Gualtieri, Classical Quant. Grav. 23, 7151 (2006) http://dx.doi.org/10.1088/0264-9381/23/24/001[Crossref]
• [46] R. Caimmi, Serbian Astronomical Journal 174, 13 (2007) http://dx.doi.org/10.2298/SAJ0774013C[Crossref]
• [47] M. Sereno, Phys. Rev. D 77, 043004 (2008) http://dx.doi.org/10.1103/PhysRevD.77.043004[Crossref]
• [48] C.G. Boehmer, G. Fodor, Phys. Rev. D 77, 064008 (2008) http://dx.doi.org/10.1103/PhysRevD.77.064008[Crossref]
• [49] C.G. Boehmer, T. Harko, Mon. Not. R. Astron. Soc. 379, 393 (2007) http://dx.doi.org/10.1111/j.1365-2966.2007.11977.x[Crossref]
• [50] M. Sereno, P. Jetzer, Phys. Rev. D 73, 063004 (2006) http://dx.doi.org/10.1103/PhysRevD.73.063004[Crossref]
• [51] E. Hackmann, C. Lammerzahl, Phys. Rev. D 78, 024035 (2008) http://dx.doi.org/10.1103/PhysRevD.78.024035[Crossref]
• [52] M. Sereno, Phys. Rev. Lett. 102, 021301 (2009) http://dx.doi.org/10.1103/PhysRevLett.102.021301[Crossref]
• [53] M. Sereno, Phys. Rev. D 77, 043004 (2008) http://dx.doi.org/10.1103/PhysRevD.77.043004[Crossref]
• [54] T. Harko, K.S. Cheng, Phys. Rev. D 76, 044013 (2007) http://dx.doi.org/10.1103/PhysRevD.76.044013[Crossref]
• [55] U. Debnath, S. Nath, S. Chakraborty, Mon. Not. R. Astron. Soc. 369, 1961 (2006) http://dx.doi.org/10.1111/j.1365-2966.2006.10457.x[Crossref]
• [56] L. Iorio, Int. J. Mod. Phys. D 15, 473 (2006) http://dx.doi.org/10.1142/S021827180600819X[Crossref]
• [57] H. Andreasson, C.G. Boehmer, Classical Quant. Grav. 26, 195007 (2009) http://dx.doi.org/10.1088/0264-9381/26/19/195007[Crossref]
• [58] C.G. Boehmer, T. Harko, Mon. Not. R. Astron. Soc. 379, 393 (2007) http://dx.doi.org/10.1111/j.1365-2966.2007.11977.x[Crossref]
• [59] C.G. Boehmer, T. Harko, Gen. Relat. Gravit. 39, 757 (2007) http://dx.doi.org/10.1007/s10714-007-0417-3[Crossref]
• [60] C. Massa, Astrophys. Space Sci. 306, 175 (2006) http://dx.doi.org/10.1007/s10509-006-9219-4[Crossref]
• [61] C.G. Boehmer, T. Harko, Classical Quant. Grav. 23, 6479 (2006) http://dx.doi.org/10.1088/0264-9381/23/22/023[Crossref]
• [62] C.G. Boehmer, T. Harko, Phys. Lett. B 630, 73 (2005)
• [63] M.K. Mak, T. Harko, Pramana 65, 185 (2005) http://dx.doi.org/10.1007/BF02898610[Crossref]
• [64] C.G. Boehmer, T. Harko, Phys. Rev. D 71, 084026 (2005) http://dx.doi.org/10.1103/PhysRevD.71.084026[Crossref]
• [65] C.G. Boehmer, Ukrainian Journal of Physics 50, 1219 (2005)
• [66] R.A. Sussman, X. Hernandez, Mon. Not. R. Astron. Soc. 345, 871, 2003 http://dx.doi.org/10.1046/j.1365-8711.2003.07010.x[Crossref]
• [67] M. Lopez-Corredoira, arXiv:1002.0525
• [68] M. Nowakowski, J.-C. Sanabria, A. Gar D 66, 023003 (2002)
• [69] A. Balaguera-Antolinez, C.G. Böhmer M. Nowakowski, Int. J. Mod. Phys. D 14, 1507 (2005) http://dx.doi.org/10.1142/S0218271805007383[Crossref]
• [70] A. Balaguera-Antolinez, M. Nowakowski, Astron. As-trophys. 441, 23 (2005) http://dx.doi.org/10.1051/0004-6361:20053544[Crossref]
• [71] A. Balaguera-Antolinez, D.F. Mota, M. Nowakowski, Classical Quant. Grav. 23, 4497 (2006) http://dx.doi.org/10.1088/0264-9381/23/13/013[Crossref]
• [72] A. Balaguera-Antolinez, D.F. Mota, M. Nowakowski, Mon. Not. R. Astron. Soc. 382, 621 (2007) http://dx.doi.org/10.1111/j.1365-2966.2007.12383.x[Crossref]
• [73] A. Balaguera-Antolinez, M. Nowakowski, AIP Conf. Proc. 861, 1001 (2006) http://dx.doi.org/10.1063/1.2399690[Crossref]
• [74] M. Nowakowski, I. Arraut, Acta Phys. Pol. B 41, 911 (2010)
• [75] T. Fließbach, Allgemeine Relativitätstheorie (Spectrum Akademischer Verlag, Heidelberg, 2003)

Identifiers

DOI

10.2478/s11534-010-0147-0