Global geodesic properties of Gödel-type space times. (English) Zbl 1282.53058
Sánchez, Miguel (ed.) et al., Recent trends in Lorentzian geometry. Based on the presentations at the 6th international meeting on Lorentzian geometry, Granada, Spain, September 6–9, 2011. New York, NY: Springer (ISBN 978-1-4614-4896-9/hbk; 978-1-4614-4897-6/ebook). Springer Proceedings in Mathematics & Statistics 26, 179-193 (2013).
A Lorentzian manifold \((M,\langle\cdot , \cdot\rangle_L)\) is a Gödel-type space-time, briefly GTS, if a smooth, finite-dimensional Riemannian manifold \((M_0,\langle\cdot , \cdot\rangle_R)\) exists such that \(M=M_0\times \mathbb R^2\) and the metric \(\langle\cdot ,\cdot\rangle_L\) is described as
\[
\langle\cdot ,\cdot\rangle_L=\langle\cdot ,\cdot\rangle_R+A(x)dy^2+2B(x)dydt-C(x)dt^2,
\]
where \(x\in M_0\), \((y,t)\) are the natural coordinates of \(\mathbb R^2\) and \(A\), \(B\), \(C\) are \(C^1\) scalar fields on \(M_0\) satisfying
\[
B^2(x)+A(x)C(x)>0,\quad\forall x\in M_0.
\]
In the paper under review the authors prove two new theorems concerning geodesic connectedness and geodesic completeness for GTS.
For the entire collection see [Zbl 1253.53003].
For the entire collection see [Zbl 1253.53003].
Reviewer: Gabriel Eduard Vilcu (Ploieşti)
MSC:
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
| 53C22 | Geodesics in global differential geometry |
| 58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |
