Extension Of General Relativity

In this thesis, the extension of General Relativity in Ashtekar's formulation towards degenerate metrics is emphatically studied. The gravitational thermodynamics of space-time foam is also studied in one-loop approximation.The greatest task raised by physics in this century is how to construct a quantum gravity theory. Since the end of 1980s, Ashtekar's new variables have led to a considerable progress in loop quantum gravity. Degenerate metrics play important roles in the quantum descriptions of gravity, while an important feature of the Ashtekar theory is to involve degenerate triads or degenerate metrics.The constructed example in Chapter 2 shows that a degenerate space-time could be evolved from non-degenerate initial data if time evolution is confined to satisfying the Ashtekar evolution equations. But our example could not be a complete Cauchy evolution. The Ashtekar's phase space is studied in Chapter 3, where a new first-class constraint system is found to describe a degenerate sector of the phase space. Using the symplectic geometry of first-class constraint systems, it is shown that only one kind of degeneracy of triads, namely type (a), will never change in the time evolution. The Bengtsson-Jacobson conjecture that "degenerate phase boundaries are always null" is studied by the mapping language in Chapter 4. By distinguishing a "phase boundary" from its image in the "covariant approach", the nulless of the image is proved. Time-like and null hypersurfaces in the degenerate space-times in the Ashtekar theory are denned, and then the phase boundary obtained by the "covariant approach" is shown to be always null.The gravitational thermodynamics of space-time foam is studied in Chapter 5 by one-loop quantum gravity and statistical thermodynamics. It is shown by further discussions of Hawking's results that Hawking-Unruh themal radiation should contain thermal gravitons or the contribution of quantum space-time foam. As a by-product, the one-loop quantum gravity correction to the classical black hole thermodynamics is also given.