标题： 零标量曲率时空的超能量张量 显示英文标题

标题： Super-energy tensor for space-times with vanishing scalar curvature

摘要： 一个四指标张量被构造出来，其项既二次于黎曼张量，又一次于其二阶导数，对于标量曲率为零的时空，该张量的散度为零。 该张量在真空中退化为Bel-Robinson张量。 此外，相对于任何观察者的完全类时分量为正，当且仅当时空是平直的（排除一些不物理的时空）。 我们还证明了该张量是唯一能具有这些性质的张量。 对于一般的引力场，这样的张量不存在。 最后，我们在几个例子中研究了该张量：充满辐射的弗里德曼-勒梅特-罗伯逊-沃尔克时空、平面前掠引力波以及瓦伊达辐射度规。 显示英文摘要

摘要： A four-index tensor is constructed with terms both quadratic in the Riemann tensor and linear in its second derivatives, which has zero divergence for space-times with vanishing scalar curvature. This tensor reduces in vacuum to the Bel-Robinson tensor. Furthermore, the completely timelike component referred to any observer is positive, and zero if and only if the space-time is flat (excluding some unphysical space-times). We also show that this tensor is the unique that can be constructed with these properties. Such a tensor does not exist for general gravitational fields. Finally, we study this tensor in several examples: the Friedmann-Lema\^{\i}tre-Robertson-Walker space-times filled with radiation, the plane-fronted gravitational waves, and the Vaidya radiating metric.