100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

364 A. AshtekarTo summarize, from the mathematical physics perspective, in the Hamiltonianapproach the crux of dynamics lies in quantum constraints. Thequantum Gauss and diffeomorphism constraints have been solved satisfactorilyand detailed regularization schemes have been proposed for theHamiltonian constraint. This progress is notable; for example, the analogoustasks were spelled out in geometrodynamics 2,3,4 some 35 years ago butstill remain unfulfilled. In spite of this technical success, however, it is notclear if any of the proposed strategies to solve the Hamiltonian constraintincorporates the familiar low energy physics in the full theory, i.e., beyondsymmetry reduced models. Novel ideas are being pursued to address thisissue. I will list them in section 4.3. Applications of Quantum GeometryIn this section, I will summarize two developments that answer several of thequestions raised under first two bullets in section 2.1. The first applicationis to black holes and the second to cosmology. The two are complementary.In the discussion of black holes, one considers full theory but the main issueof interest —analysis of black hole entropy from statistical mechanicalconsiderations— is not sensitive to the details of how the Hamiltonian constraintis solved. In quantum cosmology, on the other hand, one considersonly a symmetry reduced model but the focus is on the Hamiltonian constraintwhich dictates quantum dynamics. Thus, as in all other approachesto quantum gravity, concrete advances can be made because there existphysically interesting problems which can be addressed without having acomplete solution to the full theory.3.1. Black-holesThis discussion is based on work of Ashtekar, Baez, Corichi, Domagala, Engle,Krasnov, Lewandowski, Meissner and Van den Broeck, much of whichwas motivated by earlier work of Krasnov, Rovelli, Smolin and others. 24,31,32As explained in the Introduction, since mid-seventies, a key questionin the subject has been: What is the statistical mechanical origin of theentropy S BH = (a hor /4l 2 Pl ) of large black holes? What are the microscopicdegrees of freedom that account for this entropy? This relation implies thata solar mass black hole must have (exp 10 77 ) quantum states, a numberthat is huge even by the standards of statistical mechanics. Where do allthese states reside? To answer these questions, in the early nineties Wheelerhad suggested the following heuristic picture, which he christened ‘It from