Abstract
The gravitational deflection of light ray is an important prediction of general theory of relativity. In this paper we have developed an analytical expression of the deflection of light ray without any weak field approximation due to a charged gravitating body represented by Reissner-Nordström (RN) and Janis-Newman-Winicour (JNW) space-time geometry, using material medium approach. It is concluded that although both the geometries represent the charged, non-rotating, spherically symmetric gravitating body, the effect of charge on the gravitational deflection is just opposite to each other. The gravitational deflection decreases with charge in the RN geometry and increases with charge in the JNW geometry. The calculations obtained here are compared with other methods done by different authors. The formalism is applied to an arbitrarily selected gravitating body, as a test case and compared with the standard Schwarzschild geometry for comparison purposes.
Acknowledgments
SR acknowledges Trishna Bordaloi and Samujwal Das, MSc. Student of Assam University, Silchar who had done MSc. project under this topic. SR deeply acknowledges S Chakraborty, Assam University, Silchar also for useful discussions. Finally we acknowledge grants from the UGC-SAP under which the work was done.
Appendix A
Bending Angle Due to RN Geometry
We rewrite (22) as
where
with D=n(v, q)v.
To evaluate the above integral, we follow a procedure similar to what was done by Sen [38] and Roy and Sen [39].
Thus with the value of refractive index from (12) and D=Dr [where Dr=n(v, q)v, n(v, q) is the refractive index due to RN geometry at the limit of impact parameter] we have
where D0=n0(v)·v (corresponding to Schwarzschild deflection). And we have also denoted
Here, we can show that K(x)≪1. To evaluate the value of K(x) of (A4), the value of ((1+Cx)−2−1) is as follows:
Substituting the value of ((1+Cx)−2−1) from (A5) and n0(x)=x/(x−1) we can write the value of K(x) as
At this stage we can show that K(x)≪1. As K(x) is discontinuous at x=v, we can remove its discontinuity and evaluate its value by applying L’Hospital’s rule.
Therefore, from (A1) and (A3) one can write:
where, we have introduced some other notations:
and so on.
Now I0 can be evaluate by following the same procedure as Sen [38] and Roy and Sen [39]. According to Roy and Sen [39] I0 can be split into two integrals as I01 and I02. Here, Dk is replaced by Dr. Thus the value of I01 and I02 will be :
and
where we change the variable as
Now substituting the value of K(x) from (A6),
Applying the same procedure, the other integrals I2, I3 etc. are as follows:
Therefore, substituting all the values of I0, I1, I2, I3 etc. (A7) becomes
The above expression represents the light deflection angle due to charged gravitating mass in RN space-time.
Appendix B
Bending Angle Due to JNW Geometry
Here also we will follow the same procedure as Appendix A. Now with the value of refractive index from (18) and D=Dj [where Dj=n(v, q)v, n(v, q) is the refractive index due to JNW geometry at the limit of impact parameter], we have
Now, let
so that
Thus the limit changes to y=0 and
Now by applying the change of variable as
Thus from expression (A1) and (B1) the light deflection angle due to JNW space-time can be written as
References
[1] H. Reissner, Ann. Phys. 50, 106 (1916).10.1002/andp.19163550905Search in Google Scholar
[2] G. Nordström, Proc. Kon. Ned. Akad. Wet. 20, 1238 (1918).Search in Google Scholar
[3] A. I. Janis, E. T. Newman, and J. Winicour, Phys. Rev. Lett. 20, 878 (1968).10.1103/PhysRevLett.20.878Search in Google Scholar
[4] O. Bergmann, Am. J. Phys. 24, 38 (1956).10.1119/1.1934129Search in Google Scholar
[5] O. Bergmann and R. Leipnik, Phys. Rev 107, 1157 (1957).10.1103/PhysRev.107.1157Search in Google Scholar
[6] H. A. Buchdahl, Phys. Rev. 115, 1325 (1959).10.1103/PhysRev.115.1325Search in Google Scholar
[7] R. Penney, Phys. Rev. 182, 1383 (1969).10.1103/PhysRev.182.1383Search in Google Scholar
[8] J. R. Rao, A. R. Roy, and R. N. Tiwari, Ann. Phys. 69, 473 (1972).10.1016/0003-4916(72)90187-XSearch in Google Scholar
[9] D. R. K. Reddy and V. U. M. Rao, Austr. Math. Soc. Ser. B 24, 461 (1983).10.1017/S0334270000003817Search in Google Scholar
[10] M. Wyman, Phys. Rev. D 24, 839 (1981).10.1103/PhysRevD.24.839Search in Google Scholar
[11] K. S. Virbhadra, Int. J. Modern Phys. A 12, 4831 (1997).10.1142/S0217751X97002577Search in Google Scholar
[12] K. S. Virbhadra, D. Narasimha, and S. M. Chitre, Astron. Astrophys. 337, 1 (1998).Search in Google Scholar
[13] K. S. Virbhadra, and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000).10.1103/PhysRevD.62.084003Search in Google Scholar
[14] C. M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Math. Phys. 42, 818 (2001).10.1063/1.1308507Search in Google Scholar
[15] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 65, 103004 (2002).10.1103/PhysRevD.65.103004Search in Google Scholar
[16] V. Bozza, Phys. Rev. D 66, 103001 (2002).10.1103/PhysRevD.66.103001Search in Google Scholar
[17] P. Amore and S. Arceo, Phys. Rev. D 73, 083004 (2006).10.1103/PhysRevD.73.083004Search in Google Scholar
[18] A. N. Chowdhury, M. Palit, D. Malafarina, and P. S. Joshi, Phys. Rev. D 85, 104031 (2012).10.1103/PhysRevD.85.104031Search in Google Scholar
[19] E. F. Eiroa, G. E. Romero, and D. F. Torres, Phys. Rev. D 66, 024010 (2002).10.1103/PhysRevD.66.024010Search in Google Scholar
[20] A. Bhadra, Phys. Rev. D 67, 103009 (2003).10.1103/PhysRevD.67.103009Search in Google Scholar
[21] S. Chakraborty and S. S. Gupta, Phys. Rev. D 89, 026003 (2014).10.1103/PhysRevD.89.026003Search in Google Scholar
[22] M. Sereno, Phys. Rev. D 69, 023002 (2004).10.1103/PhysRevD.69.023002Search in Google Scholar
[23] J. E. Tamm, J. Russ. Phys. Chem. Soc. 56, 284 (1924).Search in Google Scholar
[24] N. L. Balazs, Phys. Rev. 110, 236 (1958).10.1103/PhysRev.110.236Search in Google Scholar
[25] J. Plebanski, Phys. Rev. 118, 1396 (1960).10.1103/PhysRev.118.1396Search in Google Scholar
[26] F. de Felice, Gen. Relat. Gravit. 2, 347 (1971).10.1007/BF00758153Search in Google Scholar
[27] B. Mashhoon, Phys. Rev. D 7, 2807 (1973).10.1103/PhysRevD.7.2807Search in Google Scholar
[28] B. Mashhoon, Phys. Rev. D 11, 2679 (1975).10.1103/PhysRevD.11.2679Search in Google Scholar
[29] E. Fischbach and B. S. Freeman, Phys. Rev. D 22, 2950 (1980).10.1103/PhysRevD.22.2950Search in Google Scholar
[30] J. Evans and M. Rosenquist, Am. J. Phys. 54, 876 (1986).10.1119/1.14861Search in Google Scholar
[31] K. K. Nandi and A. Islam, Am. J. Phys. 63, 251 (1995).10.1119/1.17934Search in Google Scholar
[32] J. Evans, K. K. Nandi, and A. Islam, Am. J. Phys. 64, 1404 (1996).10.1119/1.18366Search in Google Scholar
[33] J. Evans, K. K. Nandi, and A. Islam, Gen. Relat. Gravit. 28, 413 (1996).10.1007/BF02105085Search in Google Scholar
[34] P. M. Alsing, Am. J. Phys. 66, 779 (1998).10.1119/1.18957Search in Google Scholar
[35] M. Sereno, Phys. Rev. D 67, 064007 (2003).10.1103/PhysRevD.67.064007Search in Google Scholar
[36] M. Sereno, Phys. Rev. D 69, 087501 (2004).10.1103/PhysRevD.69.087501Search in Google Scholar
[37] X-H. Ye and Q. Lin, J. Mod. Opt. 55, 1119 (2008).10.1080/09500340701618395Search in Google Scholar
[38] A. K. Sen, Astrophysics 53, 560 (2010).10.1007/s10511-010-9148-3Search in Google Scholar
[39] S. Roy and A. K. Sen, Astrophys. Space Sci. 360, 23 (2015).10.1007/s10509-015-2538-6Search in Google Scholar
[40] d’Inverno Ray, in: Introducing Einstein’s Relativity, Oxford University Press, New York 1998 (Reprint).Search in Google Scholar
[41] P. D. Nunez and M. Nowakowski, J. Astrophys. Astron. 31, 105 (2010).10.1007/s12036-010-0006-9Search in Google Scholar
[42] M. Born and E. Wolf, in: Principles of Optics, 7th Edn., Cambridge University Press, Cambridge 1999, p. 131.Search in Google Scholar
[43] S. Chakraborty and A. K. Sen, Class. Quantum Grav. 32, 115011 (2015).10.1088/0264-9381/32/11/115011Search in Google Scholar
©2017 Walter de Gruyter GmbH, Berlin/Boston