Some Aspects of Spacetime Coordinates

Abstract

In this thesis, some aspects of spacetime coordinates are presented. After discussing some non-singular coordinates for the Schwartzschild, the Reissner-Nordstr ̈m and the o Kerr black hole spacetimes, non-singular Kruskal-like coordinates for different cases of general circularly symmetric black holes in (2 + 1) dimensions are constructed. The ap- proach is further extended to construct non-singular coordinates for the rotating BTZ black hole. As Kruskal-like coordinates do not remove the coordinate singularity for the extreme BTZ spacetime geometry, the possibility of obtaining Carter-like coordi- nates is discussed. It is found that these coordinates also do not remove the coordinate singularity for this geometry. The Double-null form has great importance in general relativity (GR), especially in solar-terrestrial relationships, investigation of black hole spacetimes, formulating the Newman-Penrose formalism and Numerical Relativity etc. In Chapter 3, three di- mensional spacetimes are classified according to the possibility of converting them to double-null form. It is found that a class of (2 + 1)−dimensional spacetimes in which coefficient g02 or g12 is non-zero, cannot be transformed to the double-null form. In black hole thermodynamics, it has been shown earlier for different spacetimes that the Einstein field equations at the horizon can be expressed as the first law of black hole thermodynamics. In Chapter 4, a simpler approach, using the concept of folia- tion is developed to obtain such results. Using this simpler approach, thermodynamic identities are established for the Schwarzschild, the Reissner-Nordstr ̈m, the Kerr, and o the Kerr-Newmann black holes. An important aspect of this approach is that one has to essentially deal with an (n − 1)−dimensional induced metric for an n−dimensional spacetime, which significantly simplifies the calculations to obtain such results.