We investigate stress–energy tensors constructed from the delta function on a worldline. We concentrate on quadrupoles as they make an excellent model for the dominant source of gravitational waves and have significant novel features. Unlike the dipole, we show that the quadrupole has 20 free components which are not determined by the properties of the stress–energy tensor. These need to be derived from an underlying model and we give an example motivated from a divergent-free dust. We show that the components corresponding to the partial derivatives representation of the quadrupole, have a gauge like freedom. We give the change of coordinate formula which involves second derivatives and two integrals. We also show how to define the quadrupole without reference to a coordinate systems or a metric. For the representation using covariant derivatives, we show how to split a quadrupole into a pure monopole, pure dipole and pure quadrupole in a coordinate free way.

我们研究了由世界线上的 delta 函数构造的应力-能量张量。我们专注于四极杆，因为它们为引力波的主要来源提供了出色的模型，并且具有重要的新颖特征。与偶极子不同，我们表明四极子有 20 个不受应力-能量张量性质决定的自由分量。这些需要从一个基础模型中推导出来，我们给出了一个来自无发散尘埃的例子。我们表明，对应于四极杆的偏导数表示的分量具有类似规范的自由度。我们给出了涉及二阶导数和二次积分的坐标变换公式。我们还展示了如何在不参考坐标系或度量的情况下定义四极。对于使用协变导数的表示，