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asked Upper bound on the number of convex connected components of the complement of the zero set of a polynomial
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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)