Abstract:
In this thesis, some aspects of spacetime coordinates are presented. After discussing some
non-singular coordinates for the Schwartzschild, the Reissner-Nordstr¨om and the 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 approach is further ex
tended 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 geom
etry, the possibility of obtaining Carter-like coordinates 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 dimensional 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 or both are 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 ther
modynamics. In Chapter 4, a simpler approach, using the concept of foliation is devel
oped to obtain such results. Using this simpler approach, thermodynamic identities are
established for the Schwarzschild, the Reissner-Nordstr¨om, the Kerr, and 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.
