标题： 天空不变量：施瓦茨希尔德时空的一个新的共形不变量 显示英文标题

标题： The sky invariant: A new conformal invariant for Schwarzschild spacetime

摘要： 对于给定的时空 $M$，引入了一类新的共形不变量，利用任何光线 $\Gamma$的共形几何。 通过给定点 $p$的每一条光线的集合定义了该点的天区 $S(p)$。 新的共形不变量定义在时空 $M$的天区丛上，因此被称为天区不变量。 在一条光线及其相关协变微积分上定义的自然共形协变导数允许我们证明存在一个自然的共形不变的弧长微分，该微分与共形协变导数的曲率限制一起可以用来构造一个被称为天区曲率的天区不变量。 将讨论一个可以在任何符号操作软件系统上实现的算法来计算天区曲率，并在施瓦茨希尔德时空中说明天区曲率的主要思想和显式计算。 显示英文摘要

摘要： A new class of conformal invariants for a given spacetime $M$ is introduced exploiting the conformal geometry of any light ray $\Gamma$. Each congruence of light rays passing through a given point $p$ defines the sky $S(p)$ of such point. The new conformal invariants are defined on the bundle of skies of the spacetime $M$, being called sky invariants accordingly. The natural conformal covariant derivative defined on a light ray and its associated covariant calculus allows us show the existence of a natural conformal invariant differential of arc that, together with the restriction of the curvature of the conformal covariant derivative, can be used to construct a sky invariant that will be called the sky curvature. An algorithm, that can be implemented on any symbolic manipulation software system, to compute the sky curvature will be discussed and the main ideas and the explicit computation of the sky curvature are illustrated in Schwarzschild spacetime.