Mathematical Implications of Einstein-Weyl Causality
While working on our common problems, we often used our meetings to d- cuss also the fundamentals of physics and the implications of new results in mathematics for theoretical physics. Out of these discussions grew the work presentedinthismonograph.Itisanattemptatansweringthefollowingqu- tion:WhatmathematicalstructuresdoesEinstein–Weylcausalityimposeona point-set M that has no other structure de?ned on it? In order to address this question, we have, ?rst of all, to de?ne what precisely we mean by Einstein- Weylcausality–thatis,toprovideanaxiomatizationofthisnotion.Itmaybe remarked that we were led to the axiomatization given in Chap. 3 by physical intuition, and not by mathematical analogy. Next, we show that our axiomatization de?nes a topology on the point-set M, and that the topological space M is uniformizable. We then show that, if thetopologyofM is?rst-countableanditsuniformityistotallybounded,then 1 the order completion of the uniformity has a local di?erentiable structure. Examplesshowthattheconditionsof?rstcountabilityandtotalboundedness are su?cient, but not necessary. However, the methods we have developed so far are not applicable to more general situations. Roughly speaking, we can summarize our results as follows: Subject to the above caveat, spaces satis- ing Einstein–Weyl causality are densely embedded in spaces that have locally but not necessarily globally the structure of di?erentiable manifolds.Phy- cal intuition can give no more, since a ?nite number of measurements cannot distinguish between in?nitely-di?erentiable and (continuous but) nowhe- di?erentiable structures.
Offers the first systematic approach to fundamental questions regarding Einstein-Weyl causalityInvestigates the consequences in terms of possible topological spaces Requires graduate-level familiarity with mathematical physics