We propose that the Chern-Simons invariant of the Ashtekar-Sen connection (or its imaginary part in the Lorentzian case) is the natural internal time coordinate for classical and quantum cosmology. The reasons for this are: 1) It is a function on the gauge and di eomorphism invariant con guration space, whose gradient is orthogonal to the two p h ysical degrees of freedom, in the metric de ned by the Ashtekar formulation of general relativity.2) The imaginary part of the Chern-Simons form reduces in the limit of small cosmological constant, , and solutions close to DeSitter spacetime, to the York extrinsic time coordinate. 3) Small matter-eld excitations of the Chern-Simons state satisfy, b y virtue of the quantum constraints, a functional Schroedinger equation in which the matter elds evolve o n a D eSitter background in the Chern-Simons time. We then propose this is the natural vacuum state of the theory for 6 = 0. 4) This time coordinate is periodic on the Euclidean con guration space, due to the large gauge transformations, which means that physical expectation values for all states in non-perturbative quantum gravity will satisfy the KMScondition, and may then be interpreted as thermal states. We also nd evidence that the theory with 6 = 0 is unstable as the cosmological constant term contributes third time derivatives to the Hamiltonian constraint. Finally, forms for the physical hamiltonian and inner product are suggested and a new action principle for general relativity, as a geodesic principle on the connection superspace, is found. smolin@phys.psu.edu y soo@phys.psu.edu 1
An alternative expression for the length operator in loop quantum gravity is presented. The operator is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space. The expression for the regularized length operator can moreover be understood both from a simple geometrical perspective as the average of a formula relating the length to area, volume and flux operators, and also consistently as the result of direct substitution of the densitized triad operator with the functional derivative operator into the regularized expression of the length. Both these derivations are discussed, and the origin of an undetermined overall factor in each case is also elucidated.Comment: 10 pages, 2 figures; version published in PR
An obstruction to the implementation of spatially flat Painlevé-Gullstrand(PG) slicings is demonstrated, and explicitly discussed for Reissner-Nordström and Schwarzschild-anti-deSitter spacetimes. Generalizations of PG slicings which are not spatially flat but which remain regular at the horizons are introduced. These metrics can be obtained from standard spherically symmetric metrics by physical Lorentz boosts. With these generalized PG metrics, problematic contributions to the imaginary part of the action in the Parikh-Wilczek derivation of Hawking radiation due to the obstruction can be avoided.
We investigate the effect on the Hamiltonian structure of general relativity of choosing an intrinsic time to fix the time slicing. 3-covariance with momentum constraint is maintained, but the Hamiltonian constraint is replaced by a dynamical equation for the trace of the momentum. This reveals a very simple structure with a local reduced Hamiltonian. The theory is easily generalized; in particular, the square of the Cotton-York tensor density can be added as an extra part of the potential while at the same time maintaining the classic 2 + 2 degrees of freedom. Initial data construction is simple in the extended theory; we get a generalized Lichnerowicz-York equation with nice existence and uniqueness properties. Adding standard matter fields is quite straightforward.
PACS number(s): 04.62.+v, 11.30.Er In this paper, we examine the coupling of matter fields to gravity within the framework of the Standard Model of particle physics. The coupling is described in terms of Weyl fermions of a definite chirality, and employs only (anti)self-dual or left-handed spin connection fields. It is known from the work of Ashtekar and others that such fields can furnish a complete description of gravity without matter. We show that conditions ensuring the cancellation of perturbative chiral gauge anomalies are not disturbed. We also explore a global anomaly associated with the theory, and argue that its removal requires that the number of fundamental fermions in the theory must be multiples of 16. In addition, we investigate the behavior of the theory under discrete transformations P, C and T; and discuss possible violations of these discrete symmetries, including CPT, in the presence of instantons and the Adler-Bell-Jackiw anomaly.
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