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Abstract
摘要 |
It has long been suggested that one should break the arbitrariness of the time function in standard Hamiltonian G.R. to ease the transition to canonical quantum gravity. Misner, in 1969, introduced $T = \ln g^{1/3}$, where $g$ is the determinant of the 3-metric, as a special time. This has many interesting and attractive features. While $T$ itself is not covariant, $\Delta T = \Delta \ln g^{1/3}$, the time interval, is. The trace of the momentum, $\pi$, is canonically conjugate to $T$, so defines a local energy. We split the metric into a unimodular $\bar{g}_{ij}$, and trace part and the momentum into a tracefree and trace part. and the symplectic form neatly factors. We then `solve' the Hamiltonian constraint via $-\sqrt{6}\pi = \sqrt {\bar{g}_{ab}\bar{g}_{cd}\bar{\pi}^{ac}\bar{\pi}^{bd} - gR} to find a well-defined reduced Hamiltonian. Finally, this reduced Hamiltonian generates evolution which is equivalent to standard GR under `inverse mean curvature flow'. This is extremely well behaved in the direction of contracting volume, towards the big bang. If the initial singularity is `crushing' and the 3-volume goes to zero, we get a foliation which strikes the singularity all at once and along which $-\pi$ smoothly approaches + infinity. Adding a positive cosmological constant and regular matter is quite straightforward.
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