On the Schwarzschild singularity

Abstract

The Schwarzschild ABSTRACT metric d<2�� (1- 2; }'dt' (I-�� t'-r' (d02+sin2 Od��' ) appears to behave badly near r = 2m, where gtt becomes zero, and grr tends to infinity1• There is a pathology in the line element that is due to a pathology in the space-time geometry itself. The womsome region of Schwarzschild metric, r = 2m , IS called the "event horizon". It is also called the "Schwarzschild singularity"1• There are many coordinate systems that have been found to overcome the Schwarzschild singularity1,2. By using the Schwarzschild metric in Schwarzschild coordinates, m Eddington-Filkelstein coordinates and m Kruslal-Szekeres coordinates, we have obtained some expressions for geodesics to check the behavior of a test particle at r = 2m , and in the two regions, the region outside r = 2m and the region inside r = 2m . W h h h · 11 h d . · · · ak dr e ave s own t at m. a t e coor mate systems It IS consistent to t e - < 0 ds when r > 2m and dr > 0 when r < 2m. The coefficient of dr2 becomes negative ds when r < 2m , making r a time like coordinate in that region. Thus r has to increase in this region. Further I: I becomes greater than c , the speed of light when r = 2mk , where k is a constant that depends on the initial condition, in the case of Schwarzschild coordinates and Eddington-Finkelstein coordinates, and there is jump at 2 . dr r = m, In - ds from -cl to cl , where l is a constant. These results suggest that once the particle crosses the event horizon at r = 2m it tends to remain there as dr > 0 , when r < 2m in all the three coordinate systems. ds A transformation of coordinate does not change this fact and we may suggest that the particle does not cross the event horizon, making it more than a mere coordinate singularity. The fact that I: I becomes greater than c m the neighborhood of r = 2m at least m two coordinate systems also suggest that the particle 1s changed physically around r = 2m . Hence we may say that the singularity at r = 2m is a physical singularity and not merely a coordinate singularity.