Burtscher Annegret, García-Heveling Leonardo
Department of Mathematics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP) Radboud University Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands.
Present Address: Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany.
Ann Henri Poincare. 2025;26(5):1533-1572. doi: 10.1007/s00023-024-01461-y. Epub 2024 Jul 1.
In general relativity, time functions are crucial objects whose existence and properties are intimately tied to the causal structure of a spacetime and also to the initial value formulation of the Einstein equations. In this work we establish all fundamental classical existence results on time functions in the setting of Lorentzian (pre-)length spaces (including causally plain continuous spacetimes, closed cone fields and even more singular spaces). More precisely, we characterize the existence of time functions by -causality, show that a modified notion of Geroch's volume functions are time functions if and only if the space is causally continuous, and lastly, characterize global hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our results thus inevitably show that no manifold structure is needed in order to obtain suitable time functions.
在广义相对论中，时间函数是关键对象，其存在性和性质与时空的因果结构以及爱因斯坦方程的初值表述密切相关。在这项工作中，我们在洛伦兹（预）长度空间（包括因果平坦连续时空、闭锥场以及更奇异的空间）的背景下建立了关于时间函数的所有基本经典存在性结果。更确切地说，我们通过因果性来刻画时间函数的存在性，表明如果且仅当空间是因果连续的时，一种修改后的杰罗奇体积函数概念才是时间函数，最后，通过柯西时间函数和柯西集的存在性来刻画全局双曲性。因此，我们的结果不可避免地表明，为了获得合适的时间函数，不需要流形结构。
