Pedro Lauridsen Ribeiro
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 Sep 29 comment Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary Hence, one has to see if the available results on causality for continuous and / or $C^1$ metrics suffice to yield a characterization of maximal causal curves on the double, and use reflection across the boundary to complete the job. Sep 29 comment Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary I believe the loss of causal convexity in the double is not such a big deal, though. If you have a (say, locally Lipschitz) causal curve in the double linking two points of the original manifold, you can reflect back the pieces in the other half. The result, it seems to me, is another locally Lipschitz curve (probably with more breaks) with the same Lorentzian arc length since reflection across the boundary in the double is an isometry. The big problem is really the lack of regularity of the double's metric across the boundary. Sep 29 comment Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary Think, for instance, of the causal diamond $U=\{(t,x)\in M\ |\ |t|<1-|x|\}$ in the 2-dim. half-Minkowski space-time $M=\{(t,x)\ |\ x\geq 0\}$. It is obviously geodesically convex and causally convex, but its image in the full 2-dim. Minkowski space-time (which is the double of $M$) is not causally convex since one can slightly deform any timelike curve with a boundary segment into a timelike curve whose corresponding segment has its interior inside the other half of the double. Sep 29 comment Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary Well, you can use the double of the space-time manifold, but in this case one has two problems: (1) the metric in the double extending the original metric is at best continuous (it is at least $C^1$ if the boundary is totally geodesic), and (2) even if it is smooth, the image of $U$ won't be causally convex, no matter how small, precisely because the boundary is timelike. It is possible to generalize results from causality theory to space-times with rough metrics, but I'm not sure how much of the theory survives with metrics this rough (specially if the boundary is not totally geodesic). Sep 29 comment Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary I believe the usual characterization still holds if the space-time is strongly causal and the boundary is totally geodesic. In a geodesically and causally convex neighborhood, two causally related points are connected by a unique causal geodesic segment which must be the unique maximal causal segment connecting both points, whose interior either belongs to the interior (if one of the endpoints also is) or to the boundary (if both endpoints also do). See, for instance, Propositions 4.32, 4.33 and Theorem 4.34, pp. 165-166 of Beem-Ehrlich-Easley's "Global Lorentzian Geometry" (2.ed., CRC, 1996). Sep 29 comment Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary Ah yes, actually I meant $\gamma(0)=p$ and $\gamma(1)=q$ there, so it fits your context better. Unfortunately, I can no longer edit that comment... Sep 29 comment Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary I'm not aware of any characterization of maximal causal curves in space-times with timelike boundary. At least, once again due to the fact that segments of maximal causal curves are also maximal, one has that any segment with nonvoid interior belonging either to the interior or to the boundary should be a causal (pre)geodesic in the submanifold to which it belongs. However, in principle there is nothing preventing the intersection of the curve with (say) the boundary from happening in a pretty nasty subset of parameters in $[0,1]$, so it's hard to say more without a more detailed analysis. Sep 29 comment Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary Your question essentially reduces to the following: in a time-oriented Lorentzian manifold $(M,g)$ with timelike boundary, given two chronologically related points $q\ll p$, is it possible to have a maximal, past-directed causal curve $\gamma:[0,1]\rightarrow M$ such that $\gamma(0)=q$, $\gamma(1)=p$ and $\gamma|_{[\epsilon,1]}$ is a(n achronal) null geodesic for some $\epsilon\in(0,1)$? It is clear that strict monotonicity of $(*)$ holds iff the answer is negative, since any segment of $\gamma$ must be maximal (non-strict monotonicity always holds by the reverse triangle inequality). Sep 12 comment Wavelet-like Schauder basis for standard spaces of test functions? Interesting question! For $\mathscr{D}$ the answer, if positive, should not be trivial - for instance, it cannot be an orthonormal basis, for there is no orthonormal set of infinitely differentiable wavelets with exponential decay, let alone of compact support (Corollary 5.5.3 of Ingrid Daubechies's "Ten Lectures on Wavelets", apparently independently due to Battle and Meyer). As for $\mathscr{S}$, the Littlewood-Paley wavelets as defined in Meyer's book "Wavelets and Operators" should do the job, thanks to the $L^\infty$ Bernstein inequalities. Sep 3 comment Positive Elements of a $\ast$-Algebra Oops, I hadn't paid atention to the second part... Sep 3 revised Positive Elements of a $\ast$-Algebra Added remark that the answer applies to the first part of the question. Sep 3 answered Positive Elements of a $\ast$-Algebra Jul 12 revised Wave front set from the FBI or Segal-Bergman transform (and a motivation) small typos corrected Jul 11 revised Wave front set from the FBI or Segal-Bergman transform (and a motivation) Added explanation Jul 11 revised Wave front set from the FBI or Segal-Bergman transform (and a motivation) Added explanation Jul 11 answered Wave front set from the FBI or Segal-Bergman transform (and a motivation) Jun 17 revised Do locally convex topological vector spaces embed into diffeological spaces? Added reference to complementary answer Jun 17 comment Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology @JochenWengenroth or perhaps a "convenient-calculus" tag... Jun 17 comment Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology @KathrinL. Igor's comment is still relevant. The graph topology on the space of continuous maps is the Whitney $C^0$ topology, and Peter Michor's argument works in the continuous case as well. Jun 16 comment Do locally convex topological vector spaces embed into diffeological spaces? I have edited my answer to better suit your original question, based on your comments. I've also added a remark at the end concerning your remaining question, to which unfortunately I don't know an answer. Papers who cite Glöckner's work quoted above, according to MathSciNet, apparently don't even raise that question, let alone answer it.