Horndeski Gregory W
Department of Applied Math, University of Waterloo, Waterloo, Ontario, Canada.
Philos Trans A Math Phys Eng Sci. 2022 May 2;380(2222):20210183. doi: 10.1098/rsta.2021.0183. Epub 2022 Mar 14.
In this paper, I shall show how the notions of Finsler geometry can be used to construct a similar geometry using a scalar field, , on the cotangent bundle of a differentiable manifold . This will enable me to use the second vertical derivatives of , along with the differential of a scalar field on , to construct a Lorentzian metric on that depends upon . I refer to a field theory based upon a manifold with such a Lorentzian structure as a scalar-scalar field theory. We shall study such a theory when is chosen so that the resultant metric on has the form of a Friedmann-Lemaître-Robertson-Walker metric, and the Lagrangian has a particularly simple form. It will be shown that the scalar-scalar theory determined by the Lagrangian can generate self-inflating universes, which can be pieced together to form multiverses with non-Hausdorff topologies, in which the global time function multifurcates at = 0. Some of the universes in these multiverses begin explosively, and then settle down to a period of much quieter accelerated expansion, which can be followed by a collapse to its original, pre-expansion state. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.
在本文中，我将展示如何利用芬斯勒几何的概念，通过一个标量场，在一个可微流形的余切丛上构建一种类似的几何。这将使我能够利用该标量场的二阶垂直导数，以及该流形上一个标量场的微分，在该流形上构建一个依赖于该标量场的洛伦兹度量。我将基于具有这种洛伦兹结构的流形的场论称为标量 - 标量场论。当选择该标量场使得在该流形上得到的度量具有弗里德曼 - 勒梅特 - 罗伯逊 - 沃克度量的形式，并且拉格朗日量具有特别简单的形式时，我们将研究这样一种理论。结果表明，由该拉格朗日量确定的标量 - 标量理论可以产生自我膨胀的宇宙，这些宇宙可以拼接在一起形成具有非豪斯多夫拓扑的多元宇宙，其中全局时间函数在(t = 0)处多分叉。这些多元宇宙中的一些宇宙开始时爆发性地膨胀，然后进入一段更为平静的加速膨胀期，随后可能会坍缩到其原始的、膨胀前的状态。本文是主题为“数学宇宙学的未来，第一卷”的一部分。
