論文の公開元へA Lie algebra-based approach to asymptotic symmetries in general relativity
概要
In this Ph.D thesis, we propose a useful approach to construct integrable and non-gauge charges in general spacetime shown in Chapter 4. Our approach may significantly reduce the effort involved in finding proper asymptotic conditions by trials and errors in the conventional approach.
Our approach has two key ingredients. One of them is to use Eq. (4.1.5) to find an algebra of symmetries with a non-vanishing Poisson bracket at the background metric ¯gµν. The metrics connected to the background metric through a diffeomorphism generated by the Lie algebra A satisfying Eq. (4.1.5) can be physically distinguished from each other since the Poisson brackets do not vanish. In our analysis, we have investigated a set of metrics which are connected to a fixed background metric by diffeomorphisms generated by a Lie algebra of vector fields. Since all the metrics are diffeomorphic to the background metric, it is possible to investigate the properties of the asymptotic symmetries of the background spacetime. The set in our approach is different from that in the conventional approach, where the set of metrics are defined by their asymptotic behaviors. The other key ingredient of our approach is making use of Eqs. (4.2.23) to check the integrability. As we have seen in Sec. 4.2.1, Eq. (4.2.23) provides a sufficient condition for the charges to be integrable, which can be checked at the background metric. This saves the efforts of calculating all the diffeomorphisms generated by A. In addition, the Poisson brackets of the charges can be calculated at the background metric and hence the algebra of the charges can be fully identified without calculating the diffeomorphisms generated by A explicitly. Our approach contains two cases depending on the purpose. One of them is used to obtain the charges themselves, and is shown in Fig. 4.4. The other one is used to get only the algebra of charges, and is shown in Fig. 4.5. In particular, in the latter case, we can carry out all the steps only for a background metric.
As a demonstration of our approach, we have investigated asymptotic symmetries on a Rindler horizon and that of spacetimes with the Killing horizon with metrics given in Eq. (4.3.33). In both cases, we found a new asymptotic symmetry, which we term superdilatation. In the former case, we got the expression of charges explicitly. In the latter case, we obtained the algebra of charges and found that it is a central extension of vector fields algebra A. We expect that our approach will be helpful to investigate other important spacetimes with non-Killing horizon. Although our approach can be applied to any spacetime as long as we consider diffeomorphisms which do not shift the boundary on which charges are defined, there may be asymptotic symmetries which cannot be found in our approach since Eqs. (4.1.5) and (4.2.23) are sufficient conditions for the charges to be integrable and form a non-trivial algebra. Therefore we will explore the possibility of extending our approach to identify all the asymptotic symmetries as the future work. One possibility is to derive a condition under which Eq. (4.1.3) holds at a particular metric gµν but not at the background metric ¯gµν. In our approach, we have started with two vector fields satisfying Eq. (4.1.5) and constructed a minimal Lie algebra A spanned by the vector fields and their commutators. To proceed the classification of the symmetry in general relativity, it will be quite interesting to investigate how the charge algebra changes by adding other elements to A.
As we have introduced in Chapter 1, in order to get the BH entropy, we must identify the number of microstates generated by asymptotic symmetries on a horizon. The representation of the algebra of charges is required to construct the Hilbert space of the corresponding quantum system. As we have seen in Eqs. (4.3.66) and (4.3.67), in general, the algebra of charges is an infinite dimensional Lie algebra. Thus we need the representation theory of an infinite dimensional Lie algebra, which is more difficult to treat than finite dimensional one. To the author’s knowledge, this task has been done for only a few cases, e.g. Virasoro like asymptotic symmetries. We hope that mathematicians and physicists will tackle this problem together.
