Dirac’s observables for the rest-frame instant form of tetrad gravity in a completely fixed 3-orthogonal gauge. (English) Zbl 1005.83002
In a previous paper by one of the authors (L. L.), the definition of the rest-frame instant form of metric gravity was given. In particular, therein it was shown that this form can only be reached when the requirement of the existence of a well-defined and unique asymptotic Poincaré group is imposed. By selecting the class of Christodoulou-Klainermann space-times, where there exist three first-class constraints defining the rest frame, a consistent rest-frame instant form of metric gravity was given. As a consequence, the author could partly transfer methods from parametrized quantum field theory in flat space-time to curved space-time. In particular, this led to the existence of an ultraviolet cut-off in Einstein’s general relativity theory (which is not power counting renormalizable) in such space-times.
In the present review paper the authors show that restricting oneself to Christodoulou-Klainermann space-times, in a similar way one can establish the rest-frame instant form of tetrad gravity. Then, the gauge transformations generated by the 14 first-class constraints of the theory are studied and, afterwards, the rotation and space diffeomorphism constraints are analyzed. As a result, it is seen how the cotriads and their momenta depend on the corresponding gauge variables. This allows the authors to find a point canonical transformation to the class of 3-orthogonal gauges and to construct the superspace Dirac observables in these gauges. It is shown that the explicit construction of this transformation and of the solution of the rotation and supermomentum constraints is reduced to the task to solve a system of elliptic linear and quasi-linear partial differential equations, whose solution would give the expressions of the cotriad momenta in terms of the gauge variables and the Dirac observables in these gauges. The authors then demonstrate that, after a canonical transformation of the previous canonical basis of the Dirac observables, the superhamiltonian constraint takes the form of Lichnerowicz equation for the conformal factor of the 3-metric. The momentum canonically conjugate to this factor is determined. The paper is completed by two appendices (on 3-tensors in the special 3-orthogonal gauge and on the strong and weak ADM Poincaré charges in this gauge) and 130 references.
In the present review paper the authors show that restricting oneself to Christodoulou-Klainermann space-times, in a similar way one can establish the rest-frame instant form of tetrad gravity. Then, the gauge transformations generated by the 14 first-class constraints of the theory are studied and, afterwards, the rotation and space diffeomorphism constraints are analyzed. As a result, it is seen how the cotriads and their momenta depend on the corresponding gauge variables. This allows the authors to find a point canonical transformation to the class of 3-orthogonal gauges and to construct the superspace Dirac observables in these gauges. It is shown that the explicit construction of this transformation and of the solution of the rotation and supermomentum constraints is reduced to the task to solve a system of elliptic linear and quasi-linear partial differential equations, whose solution would give the expressions of the cotriad momenta in terms of the gauge variables and the Dirac observables in these gauges. The authors then demonstrate that, after a canonical transformation of the previous canonical basis of the Dirac observables, the superhamiltonian constraint takes the form of Lichnerowicz equation for the conformal factor of the 3-metric. The momentum canonically conjugate to this factor is determined. The paper is completed by two appendices (on 3-tensors in the special 3-orthogonal gauge and on the strong and weak ADM Poincaré charges in this gauge) and 130 references.
Reviewer: Horst-Heino von Borzeszkowski (Berlin)
MSC:
| 83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |
| 83C45 | Quantization of the gravitational field |
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
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