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arXiv 2202.11259gr-qc
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用一种新几何将标量-张量场论转化为标量-标量场

Reformulating Scalar-Tensor Field Theories as Scalar-Scalar Field Theories Using a Novel Geometry

Gregory W. Horndeski

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53 pages
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摘要：

在本文中，我将说明如何芬斯勒几何的概念，可以用来构建一个类似的几何使用标量场，f，余切丛的可微流形M。这将使我能够使用f的二阶垂直导数，以及M上标量场的微分，来构造M上依赖于φ的洛伦兹度规。我把基于具有这种洛伦兹结构的流形的场论称为标量-标量场理论。我们将研究这样一种场论，当选择f使得M上的合成度规具有弗里德曼-勒梅特-罗伯逊-沃克度规的形式，而拉格朗日量具有特别简单的形式。我们将证明，由这个拉格朗日量决定的标量-标量理论可以产生自膨胀宇宙，这些宇宙可以拼凑在一起形成具有非豪斯多夫拓扑的多重宇宙，其中全局时间函数在t=0处多重分叉。这些多重宇宙中的一些宇宙开始爆炸，然后稳定下来，进入一个安静得多的加速膨胀时期，随后可能会坍缩到最初的膨胀前状态。最后，我将讨论如何将概率分配给多元宇宙的各个宇宙。这是通过使用宇宙的作用来实现的，具有接近于零的作用的宇宙比具有大的正值的宇宙更有可能。为了确保类似于我们自己的宇宙的宇宙模型可能存在，我发现有必要在M上引入第二个标量场，并修改原来的拉格朗日量。最后，我的理论有三个标量场，两个在流形M上，一个在M的余切丛上。

英文摘要：

In this paper I shall show how the notions of Finsler geometry can be used to construct a similar geometry using a scalar field, f, on the cotangent bundle of a differentiable manifold M. This will enable me to use the second vertical derivatives of f, along with the differential of a scalar field phi on M, to construct a Lorentzian metric on M that depends upon phi. 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 field theory when f is chosen so that the resultant metric on M has the form of a Friedmann-Lemaitre-Robertson-Walker metric, and the Lagrangian has a particularly simple form. It will be shown that the scalar-scalar theory determined by this 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 t=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. I conclude the paper with a discussion of how probabilities can be assigned to the various universes of a multiverse. This is accomplished by using the action of the universes, with universes having action closer to zero being more likely than universes with large positive values for their action. In order to assure that universe models similar to our own universe are likely to exist I found it necessary to introduce a second scalar field on M, and to modify the original Lagrangian. In the end my theory has three scalar fields, two on the manifold M and one on the cotangent bundle of M.

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