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## 物理代写

we write,
$$\bar{h}{k l}(t, \mathbf{x}) \simeq \frac{2 G}{c^{4} r} \ddot{I}{k l}(t-r / c)$$
where the dot represents the time derivative. Note that a factor of $c^{2}$ has cancelled out since $\partial_{0}^{2}=\left(1 / c^{2}\right) \partial / \partial t^{2}$.

The final step is to project the quantities in Eq. (8.5.13) to the TT gauge. This makes the quantities trace-free, in which case, $\bar{h}^{k l}=h^{k l}$. Thus, for plane waves travelling in the spatial direction $n^{i}$ we use the projection operator $P_{j}^{i}$ defined in Eq. (8.4.16). Another quantity which we use later is reduced quadrupole moment tensor,
$$I^{k l}=I^{k l}-\frac{1}{3} \delta^{k l} I$$
where $I=I_{k}^{k}$ is the trace of $I^{k l} .$ We thus arrive at the the famous quadrupole formula,
$$h_{k l}^{T T}(t, \mathbf{x})=\frac{2 G}{c^{4} r} \ddot{I}{k l}^{T T}(t-r / c)$$ where the superscript TT on $I{k l}$ denotes its tranverse-traceless projection. Certain remarks about the formula are in order. First, the gravitational wave amplitude falls with distance as $1 / r$, just like with propagating electromagnetic fields. The energy flux therefore falls off as $1 / r^{2}$. Although, the rate of change of quadrupole moment can be very large for compact binary stars just before merger, the $G / c^{4}$ factor $(\sim$ $8.3 \times 10^{-44}$ in SI units) makes the strain very small, thus making it very difficult to detect the waves. It is also clear that the leading order emission of gravitational waves requires a non-zero second order time derivative of the mass-energy quadrupole moment. Hence, if an object is axially symmetric, any spin about the axis of symmetry will not lead to any GW emission. Thus a spinning sphere with uniform density
$8.5$ Quadrupole Formula
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distribution will not emit GW. Similarly, a spherically symmetric collapse of a star will not produce any GW.

## 物理代考

$$\bar{h}{k l}(t, \mathbf{x}) \simeq \frac{2 G}{c^{4} r} \ddot{I}{k l}(t-r / c)$$

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Matlab代写