论文标题
嵌入一级的空间的周期性签名更改
Periodic signature change in spacetimes of embedding class one
论文作者
论文摘要
即使在发现黑暗能源之后，振荡宇宙的想法仍然是一个有趣的话题。本文通过另一种公认的一般相对论证实了这一想法，弯曲的空间嵌入更高维的平坦空间中：据说$ n $维的利马尼亚空间是嵌入$ m $的$ m $ n $，如果$ m+n $是最低的尺寸$ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d。这里$ d = \ frac {1} {2} n（n-1）$。因此，四维的riemannian空间是二级的，因为它可以嵌入六维平面空间中。第二类的线元素可以通过合适的坐标转换将其简化为一类的线元素。额外的尺寸可以是空位般的或及时的，分别导致加速和减速的扩展。因此，在本文中提出，转换中发生的自由参数是时间的周期性函数。结果是一个数学模型，可以将其解释为嵌入空间签名的周期性变化。这种签名变化可能是振荡宇宙的最佳模型，并补充了文献中提出的各种模型。
The idea of an oscillating Universe has remained a topic of interest even after the discovery of dark energy. This paper confirms this idea by means of another well-established theory in general relativity, the embedding of curved spacetimes in higher-dimensional flat spacetimes: an $n$-dimensional Riemannian space is said to be of embedding class $m$ if $m+n$ is the lowest dimension $d$ of the flat space in which the given space can be embedded; here $d=\frac{1}{2}n(n-1)$. So a four-dimensional Riemannian space is of class two since it can be embedded in a six-dimensional flat space. A line element of class two can be reduced to a line element of class one by a suitable coordinate transformation. The extra dimension can be either spacelike or timelike, leading to accelerating and decelerating expansions, respectively. Accordingly, it is proposed in this paper that the free parameter occurring in the transformation be a periodic function of time. The result is a mathematical model that can be interpreted as a periodic change in the signature of the embedding space. This signature change may be the best model for an oscillating Universe and complements various models proposed in the literature.
