辐射区

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定义 · 5.3 制造波

这里我们用到了 ${\partial }_{j}r=x_j/r={\hat{x}}_j$ 这一事实。(5.34) 中这两项哪一个更大？随着我们离源越来越远，可以预期第二项，即 $1/r$ 项，会压过第一项，即 $1/r^2$ 项。但第二项多了一个时间导数，这意味着多出一个源的特征频率 ω 的因子。因此，只要 $r\gg 1/\omega$，或者说，用所发出的引力波的波长 λ 来表示，只要 $r\gg \lambda$，第二项就占主导。这被称为远场区(far-field zone)，有时也称为辐射区(radiation zone)。在这一区域内，我们有

$${\bar{h}}_{0i}\approx -\frac{2G{\hat{x}}_j}{r}{\ddot{I}}_{ij}(t-r)$$

英文原文

where we’ve used the fact that ${\partial }_{j}r=x_j/r={\hat{x}}_j$ . Which of these two terms in (5.34) is bigger? As we get further from the source, we would expect the second, $1/r$ , term to dominate over the first, $1/r^2$ term. But the second term has an extra time derivative, which means an extra factor of the characteristic frequency of the source, ω. This means that the second term dominates provided that $r\gg 1/\omega$ or, in terms of the wavelength λ of the emitted gravitational wave, $r\gg \lambda$ . This is known as the far-field zone or, sometimes, the radiation zone. In this regime, we have

$${\bar{h}}_{0i}\approx -\frac{2G{\hat{x}}_j}{r}{\ddot{I}}_{ij}(t-r)$$

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