物理代写| Geodesics 相对论代考代写 - 代考代写：100%准时可靠您的作业代写专家

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By admin Physics代写数学代写物理物理代写相对论代写 2022年3月1日

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1 物理代写| Geodesics 相对论代考

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:

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.
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
在弯曲空间中，“直线”称为测地线。当然，当测地线嵌入到平坦的高维空间中时，它们可能看起来并不直。例如，大圆是两个球体上的测地线。对于在球体上移动的小蚂蚁来说，测地线确实就像一条直线。但是当两个球体嵌入到 3 维欧几里得空间中时，它们会画出圆，这就是为什么我们写“大圆是两个球体上的测地线”。

我们现在问一个问题，一般曲面上的蚂蚁如何绘制测地线？扩展我们对欧几里得几何的理解，我们可以想到两种方法：
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。

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。 ${ }^{1}$

当度量是正定的，因为它对于球体来说是正定的，两点之间的测地线与其邻域中的曲线相比给出了最短的距离。但是当度量不是我们在广义相对论中的正定时，测地线给出了极值距离。例如，如果两个点是时间间隔的，则测地线给出最大距离。然而，即使对于不确定的度量，第一个标准也保持不变。

让测地线由变量 $\lambda$ 参数化。可以将 $\lambda$ 视为里程碑，这不是唯一的，因为 $\lambda=0,1,2, \cdots$ 标记的位置取决于单位（公里，英里等.)，但是在给定的单位中，$\lambda$ 唯一地指定了曲线上的一个点。我们将根据参数 $\lambda$ 推导出测地线曲线的方程。
4.3.1 平行传输切线
曲线 $x^{i}(\lambda)$ 的切向量 $u^{i}$ 的分量是，
$$
u^{i}=\mathrm{d} x^{i} / \mathrm{d} \lambda
$$
（因为每个 $x^{i}$ 只是 $\lambda$ 的函数！）。平行传输切线给出，
通过缓慢改变方向，在第二个点向另一只蚂蚁移动。一旦第二只蚂蚁收到它，它就会通过缓慢旋转镜子将其反射回第一只蚂蚁。激光的路径可以通过放置中间观察者来追踪，称为测地线。 $4.3$ 测地线 $$ u_{; k}^{i} u^{k}=0 。 $$
由于这个微分是沿着一条由 $\lambda$ 参数化的曲线，所以上面的方程可以写成
$$
\frac{\partial}{\partial x^{k}}\left(\frac{\mathrm{d} x^{i}}{\mathrm{~d} \lambda}\right) \frac{\mathrm {d} x^{k}}{\mathrm{~d} \lambda}+\Gamma_{jk}^{i} u^{j} u^{k}=\frac{\mathrm{d}^{ 2} x^{i}}{\mathrm{~d} \lambda^{2}}+\Gamma_{jk}^{i} \frac{\mathrm{d} x^{j}}{\mathrm{ ~d} \lambda} \frac{\mathrm{d} x^{k}}{\mathrm{~d} \lambda}=0
$$
这个二阶微分方程称为测地线方程。该微分方程的解给出测地线或测地线曲线。