On regular Fréchet-Lie groups. VI: Infinite dimensional Lie groups which appear in general relativity
[For part V see ibid. 6, 39-64 (1983; Zbl 0526.58003).] The Lie group theoretical point of view in general relativistic mechanics permits the authors of this series to show here that this is a more closed system than Newtonian mechanics. If by \((M,\Omega)\) is denoted a symplectic manifold, let \({\mathcal O}\) be the linear space of all observables in a mechanical system \((M,\Omega\),H) and \({\mathcal U}_{{\mathcal O}}\) be the set of all Hamiltonian vector fields \(X_ f,f\in {\mathcal O}\). Natural reasoning leads to the assumption that \({\mathcal O}\) must have the properties: \((A_ 1) {\mathcal O}\) is a Lie algebra under the Poisson bracket; \((A_ 2)\) Each Hamiltonian vector field \(X_ f\), \(f\in {\mathcal O}\), is complete; \((A_ 3)\) (exp t\(X_ f)^*{\mathcal O}={\mathcal O}\) for every \(f\in {\mathcal O}\) and for all \(t\in R\). It is shown that if M is a closed manifold the space \(C^{\infty}(M)\) satisfies \((A_ 1)-(A_ 3)\) and \({\mathcal U}_{{\mathcal O}}\) (\({\mathcal O}=C^{\infty}(M))\) is the Lie algebra of the ILB-Lie group of all canonical transformations on M. However, if M is non-compact, namely if \(M=T^*N\), N being the configuration space with smooth Riemannian metric \((g_{ij})\), the authors give the following result which shows the curious fact that, in this case, there are few potential functions in \({\mathcal O}:\) Let \({\mathcal O}\) be a set of smooth functions on \(T^*N\) satisfying \((A_ 1)\) and \((A_ 2)\), and containing a smooth function \((1/2)g^{ij}p_ ip_ j+V\) for some \(V\in C^{\infty}(N)\). If \(f\in C^{\infty}(N)\cap {\mathcal O}\) satisfies \(j^ 2_ x=0\) at a point \(x\in N\), then \(j^ k_ xf=0\) for all \(k\geq 2\). To resolve the above difficulty, the authors consider on \((T^*N,\Omega)\) a new space \(\Sigma^ 1\) consisting of all smooth functions on \(T^*N\) which have asymptotic expansion of type \(a_ 1r+a_ 0+a_{-1}r^{-1}+..\). for \(r=\sqrt{g^{ij}p_ ip_ j}\geq 0\), where \(a_ j\) are smooth functions on the unit cosphere bundle \(S^*N\) in \(T^*N\) with respect to \(g^{ij}\). This space contains \(C^{\infty}(N)\) and all relativistic Hamiltonians \(H_ t(x,t)\) for every fixed \(t\in R\). The announced result follows from the following main result of this paper: \(\Sigma^ 1\) satisfies \((A_ 1)-(A_ 3)\). Moreover, \({\mathcal U}_{\Sigma^ 1}\), the set of all Hamiltonian vector fields \(X_ f\) of \(f\in \Sigma^ 1\), is a Lie algebra of a regular Fréchet-Lie group.
scientific article; zbMATH DE number 3854041