Existence of geodesics for the Lorentz metric of a stationary gravitational field
Let g be a Lorentzian metric (of type \((+,-,-,-))\) on \({\mathbb{R}}^ 4={\mathbb{R}}\times {\mathbb{R}}^ 3\) which is stationary and not too far from the Minkowski metric. More precisely, \(g=(g_{\alpha \beta})\) is \(C^ 1\) and does not depend on the time coordinate \(x_ 0\) (``stationary metric''), and the timelike part \(g_{00}\) is bounded from below, the spacelike part \((g_{ij})\) (1\(\leq i,j\leq 3)\) is bounded from above, and the cross terms \(g_{0i}\) are bounded from both sides. The authors prove that any two points \(a,b\in {\mathbb{R}}^ 4\) can be joined by a geodesic of such a metric. The typical Lorentzian structure (Causality etc.) is not important in the paper; there is no distinction between spacelike, timelike and null geodesics. Therefore we expect that a similar method works for any indefinite metric on \({\mathbb{R}}^ n\). The idea of the proof is as follows. Consider the set of [0,1]-parametrized \(H^ 1\)-curves \(M=\bar z+H^ 1_ 0\) where \(\bar z\) is the line segment \(\bar z(s)=a+s\cdot (b-a)\) and \(H^ 1_ 0\) are the \(H^ 1\)-loops starting and ending at the origin. One looks for a critical point of the energy functional \[ f(z)=\int^{1}_{0}g(z'(s),z'(s))ds \] on M. Take a certain finite dimensional approximation \(M_ n\subset M\). The bounds on the metric imply the Palais-Smale condition for \(f| M_ n\). Let \(S_ n\) (resp. \(Q_ n)\) be the subset of curves \(\bar z+\zeta\) in \(M_ n\) with \(\zeta '(s)\in {\mathbb{R}}\times 0\) (resp. \(\in 0\times {\mathbb{R}}^ 3)\) for all s. On \(Q_ n\), the functional f is bounded from above and f(z) goes to -\(\infty\) as \(\| z\|_ 1\to \infty\) while f is bounded from below on \(S_ n\). Thus we get a critical point \(z_ n\) of \(f| M_ n\) by using a saddle point theorem. The sequence \((z_ n)_{n\in N}\) is bounded in the \(H^ 1\)-norm, and one shows that it converges to a critical point z of f.
scientific article; zbMATH DE number 4143074