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# 物理代写| Geodesics 相对论代考

## 物理代写

4.3 Geodesics
In curved spaces, the “straight lines” are called geodesics. Of course, geodesics may not look straight when they are embedded in a flat higher dimensional space. For example, great circles are geodesics on a two-sphere. For a tiny ant moving on the sphere, a geodesic is indeed like a straight line. But when the two sphere is embedded in 3 dimensional Euclidean space, they trace out circles, and that is why we write “great circles are geodesics on a two-sphere”.

We now ask the question how would an ant on a general curved surface draw geodesics? Extending our perception of Euclidean geometry, we can think of two ways:

1. Parallelly transporting the tangent: The ant takes the first step in a given direction and keeps on moving in the same direction in such a way that the tangent keeps the same direction, that is, the tangent is parallelly transported.
2. Shortest Distance: The ant makes large number (tending to infinity) of trips to another point via all possible routes and, calls the route with shortest distance, requiring minimum number of steps, as the geodesic. ${ }^{1}$

When the metric is positive definite as it is for a sphere, the geodesic between two points gives the shortest distance as compared with curves in its neighbourhood. But when the metric is not positive definite as we have in general relativity, the geodesic gives the extremal distance. For example if the two points are timelike separated, the geodesic gives the maximum distance. However, the first of the criterion remains the same even for an indefinite metric.

Let a geodesic be parameterised by a the variable $\lambda$. One can consider the $\lambda$ ‘s to be milestones, which are not unique, in the sense that the locations of the $\lambda=0,1,2, \cdots$ marks depend on the units (km, miles etc.), nevertheless in given units $\lambda$ uniquely specifies a point on the curve. We shall derive the equations for the geodesic curves in terms of the parameter $\lambda$.
4.3.1 Parallelly Transporting the Tangent
The components of the tangent vector $u^{i}$ of a curve $x^{i}(\lambda)$ is,
$$u^{i}=\mathrm{d} x^{i} / \mathrm{d} \lambda$$
(since each $x^{i}$ is a function of $\lambda$ only!). Parallelly transporting the tangent gives,
towards another ant at a second point by slowly changing the direction. As soon as the second ant receives it, it reflects it back to the first ant by slowly rotating a mirror. The path of the laser can be traced by putting intermediate observers and called the geodesic. $4.3$ Geodesics $$u_{; k}^{i} u^{k}=0 .$$
Since this differential is along a curve parameterised by $\lambda$, the above equation can be written as
$$\frac{\partial}{\partial x^{k}}\left(\frac{\mathrm{d} x^{i}}{\mathrm{~d} \lambda}\right) \frac{\mathrm{d} x^{k}}{\mathrm{~d} \lambda}+\Gamma_{j k}^{i} u^{j} u^{k}=\frac{\mathrm{d}^{2} x^{i}}{\mathrm{~d} \lambda^{2}}+\Gamma_{j k}^{i} \frac{\mathrm{d} x^{j}}{\mathrm{~d} \lambda} \frac{\mathrm{d} x^{k}}{\mathrm{~d} \lambda}=0$$
This second order differential equation is called the geodesic equation. The solutions to this differential equation give the geodesics or geodesic curves.

## 物理代考

4.3 测地线

1.平行输送切线：蚂蚁在给定方向上迈出第一步，并继续沿同一方向移动，使切线保持同一方向，即平行输送切线。

1. 最短距离：蚂蚁通过所有可能的路线多次（趋于无穷大）到达另一点，并将最短距离、需要最少步数的路线称为测地线。 ${ }^{1}$

4.3.1 平行传输切线

$$u^{i}=\mathrm{d} x^{i} / \mathrm{d} \lambda$$
（因为每个 $x^{i}$ 只是 $\lambda$ 的函数！）。平行传输切线给出，

$$\frac{\partial}{\partial x^{k}}\left(\frac{\mathrm{d} x^{i}}{\mathrm{~d} \lambda}\right) \frac{\mathrm {d} x^{k}}{\mathrm{~d} \lambda}+\Gamma_{jk}^{i} u^{j} u^{k}=\frac{\mathrm{d}^{ 2} x^{i}}{\mathrm{~d} \lambda^{2}}+\Gamma_{jk}^{i} \frac{\mathrm{d} x^{j}}{\mathrm{ ~d} \lambda} \frac{\mathrm{d} x^{k}}{\mathrm{~d} \lambda}=0$$

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