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# 物理代写|广义相对论代写General Relativity代考|PHY475 Perturbation Calculation

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## 物理代写|广义相对论代写General Relativity代考|Perturbation Calculation

Perturbation Calculation. We cannot solve equation 11.6 analytically, but we can solve it using an approach known as a perturbation calculation. This is a powerful technique for solving difficult equations that is often used in physics and is therefore a useful addition to your repertoire as a working physicist.

Consider first a circular orbit with a constant radius $r_c$. The value of $u_c=1 / r_c$ must satisfy equation 11.6 , so we have
$$\frac{d^2 u_c}{d \phi^2}+u_c=\frac{G M}{\ell^2}+3 G M u_c^2$$
Now consider a perturbation of this orbit that is slightly non-circular. Define
$$u(\phi)=u_c+u_c w(\phi)$$
where the unitless “wobble” function $w(\phi) \ll 1$. If you plug equation 11.9 into equation 11.6 and use equation 11.8 to eliminate some terms involving $u_c$, you should find (see box 11.3) that the perturbation (“wobble”) function must satisfy
$$\frac{d^2 w}{d \phi^2}+w=6 G M u_c w+3 G M u_c w^2$$

## 物理代写|广义相对论代写General Relativity代考|The Effect of Spatial Curvature

The Effect of Spatial Curvature. Physically, this precession arises both because $g_{t t}=1-2 G M / r$ (clock rates depend on $r$ ) and because $g_{r r}=(1-2 G M / r)^{-1}$ (the spatial part of the Schwarzschild geometry is curved). The goal of this section is to show that the latter effect contributes $1 / 3$ of the total precession.

Since our orbit remains in the equatorial plane, we will focus on the set of events on the equatorial plane $\theta=\pi / 2$ at some arbitrary instant of time $t$. The geometry for this set of events is found by setting $d t=0, \theta=\pi / 2$, and $d \theta=0$ in the Schwarzschild metric: the metric for this particular spatial plane is thus
$$d s^2=\frac{d r^2}{(1-2 G M / r)}+r^2 d \phi^2$$
As discussed in chapter 9 , we consider the geometry described by this metric to be “curved” because the radial distance $s=\int(1-2 G M / r)^{-1} d r$ between two concentric circles is not equal to the difference $\Delta r$ in their circumferential radii, indicating that the two-dimensional space described by this metric is not Euclidean.

## 物理代写|广义相对论代写General Relativity代考|Perturbation Calculation

$$\frac{d^2 u_c}{d \phi^2}+u_c=\frac{G M}{\ell^2}+3 G M u_c^2$$

$$u(\phi)=u_c+u_c w(\phi)$$

$$\frac{d^2 w}{d \phi^2}+w=6 G M u_c w+3 G M u_c w^2$$

## 物理代写|广义相对论代写General Relativity代考|The Effect of Spatial Curvature

$$d s^2=\frac{d r^2}{(1-2 G M / r)}+r^2 d \phi^2$$

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