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Register a new userThe global properties of the finiteness and continuity of the Lorentzian distance
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It is well-known that global hyperbolicity implies that the Lorentzian distance is finite and continuous. By carefully analysing the causes of discontinuity of the Lorentzian distance, we show that in most other respects the finiteness and continuity of the Lorentzian distance is independent of the causal structure. The proof of these results relies on the properties of a class of generalised time functions introduced by the authors in cite{RennieWhale2016}.
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We show that finiteness of the Lorentzian distance is equivalent to the existence of generalised time functions with gradient uniformly bounded away from light cones. To derive this result we introduce new techniques to construct and manipulate achronal sets. As a consequence of these techniques we obtain a functional description of the Lorentzian distance extending the work of Franco and Moretti.
The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian causality theory beyond the manifold level. We then study Gromov-Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.
We analyze the global behaviour of the growth index of cosmic inhomogeneities in an isotropic homogeneous universe filled by cold non-relativistic matter and dark energy (DE) with an arbitrary equation of state. Using a dynamical system approach, we find the critical points of the system. That unique trajectory for which the growth index $gamma$ is finite from the asymptotic past to the asymptotic future is identified as the so-called heteroclinic orbit connecting the critical points $(Omega_m=0,~gamma_{infty})$ in the future and $(Omega_m=1,~gamma_{-infty})$ in the past. The first is an attractor while the second is a saddle point, confirming our earlier results. Further, in the case when a fraction of matter (or DE tracking matter) $varepsilon Omega^{rm tot}_m$ remains unclustered, we find that the limit of the growth index in the past $gamma_{-infty}^{varepsilon}$ does not depend on the equation of state of DE, in sharp contrast with the case $varepsilon=0$ (for which $gamma_{-infty}$ is obtained). We show indeed that there is a mathematical discontinuity: one cannot obtain $gamma_{-infty}$ by taking $lim_{varepsilon to 0} gamma^{varepsilon}_{-infty}$ (i.e. the limits $varepsilonto 0$ and $Omega^{rm tot}_mto 1$ do not commute). We recover in our analysis that the value $gamma_{-infty}^{varepsilon}$ corresponds to tracking DE in the asymptotic past with constant $gamma=gamma_{-infty}^{varepsilon}$ found earlier.
In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.
We show that the first law for the rotating Taub-NUT is straightforwardly established with the surface charge method. The entropy is explicitly found as a charge, and its value is not proportional to the horizon area. We conclude that there are unavoidable contributions from the Misner strings to the charges, still, the mass and angular momentum gets standard values. However, there are no independent charges associated with the Misner strings.
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