Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Robert J. McCann, Argam Ohanyan, Felix Rott, Clemens Sämann
{"title":"A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes","authors":"Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Robert J. McCann, Argam Ohanyan, Felix Rott, Clemens Sämann","doi":"arxiv-2408.15968","DOIUrl":null,"url":null,"abstract":"We introduce a variational first-order Sobolev calculus on metric measure\nspacetimes. The key object is the maximal weak subslope of an arbitrary causal\nfunction, which plays the role of the (Lorentzian) modulus of its differential.\nIt is shown to satisfy certain chain and Leibniz rules, certify a locality\nproperty, and be compatible with its smooth analog. In this setup, we propose a\nquadraticity condition termed infinitesimal Minkowskianity, which singles out\ngenuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we\nestablish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian\nin a weak form under the timelike measure contraction property. As a particular\ncase, this extends Eschenburg's classical estimate past the timelike cut locus.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.15968","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce a variational first-order Sobolev calculus on metric measure
spacetimes. The key object is the maximal weak subslope of an arbitrary causal
function, which plays the role of the (Lorentzian) modulus of its differential.
It is shown to satisfy certain chain and Leibniz rules, certify a locality
property, and be compatible with its smooth analog. In this setup, we propose a
quadraticity condition termed infinitesimal Minkowskianity, which singles out
genuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we
establish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian
in a weak form under the timelike measure contraction property. As a particular
case, this extends Eschenburg's classical estimate past the timelike cut locus.