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物理代写|广义相对论代写General Relativity代考|PHYS503 The Metric for Global Rain Coordinates

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物理代写|广义相对论代写General Relativity代考|The Metric for Global Rain Coordinates

The Metric for Global Rain Coordinates. The new time coordinate $t$ must be a function of the original Schwarzschild coordinates. The transformation cannot depend on $\theta$ or $\phi$, because spherical symmetry means that all angular directions are equivalent, so $\overparen{t}$ must be a function $\grave{t}(t, r)$ of the coordinates $t$ and $r$ alone.

Now, the metric involves differentials of coordinate quantities. We can express the
$$d t=\left(\frac{\partial i}{\partial t}\right) d t+\left(\frac{\partial \grave{t}}{\partial r}\right) d r$$
We can actually determine the partial derivatives directly from the description of the coordinate system.given in the previous section. Consider the first partial derivative. Physically, this derivative poses the following question: For two events separated by an infinitesimal Schwarzschild coordinate time $d t$ but at fixed radial coordinate $r$, what is the rain coordinate time difference $d i$ between those events? To answer this question, note that the rain time between these events is measured by the internal clocks of two falling robots, one passing radius $r$ at the time of the first event, and the other falling past the same radius at the time of the second event $d t$ later. Now, each robot requires the same Schwarzschild time to fall from infinity to a given $r$, so if the second robot passed $r$ a time $d t$ after the first, it must have been dropped a time $d t$ after the first. Since each robot clock was set to Schwarzschild time when it was dropped, the second robot’s clock will initially be set a time $d t$ later than the first robot’s clock. The additional proper times that each robot registers during the drop to the same $r$ must be the same. Therefore, the difference $d \dot{t}$ in the times they register between the arrival events at $r$ must be the same as the Schwarzschild time difference $d t$ between the $d r o p$ events, so $d t=d t$ for these events, implying that
$$\frac{\partial \dot{t}}{\partial t}=1$$

物理代写|广义相对论代写General Relativity代考|Kruskal-Szekeres Coordinates

Kruskal-Szekeres Coordinates. While we can work with global rain coordinates, its non-diagonal metric can be awkward and misleading. Kruskal-Szekeres (henceforth KS) coordinates have a nice, diagonal metric, and are probably the most useful coordinate system for understanding and displaying the behavior of photons. Start by defining KS coordinates $u$ and $v$ such that
$$\left(\frac{r}{2 G M}-1\right) e^{r / 2 G M}=u^2-v^2 \quad \text { and } \quad t=2 G M \ln \left|\frac{u+v}{u-v}\right|$$
where the $r$ coordinate is now considered to be a function $r(u, v)$ that is implicitly defined by the first of equations 15.8. If you take differentials of both sides of these expressions and substitute the results for $d r$ and $d t$ into the Schwarzschild metric, you will obtain the KS metric for the empty spacetime around a spherically symmetric star or black hole (see box 15.4).
$$d s^2=-\frac{32(G M)^3}{r} e^{-r / 2 G M}\left(d v^2-d u^2\right)+r^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)$$

物理代写|广义相对论代写General Relativity代考|The Metric for Global Rain Coordinates

$$d t=\left(\frac{\partial i}{\partial t}\right) d t+\left(\frac{\partial t}{\partial r}\right) d r$$

$$\frac{\partial \dot{t}}{\partial t}=1$$

物理代写|广义相对论代写General Relativity代考|KruskalSzekeres Coordinates

Kruskal-Szekeres 坐标。虽然我们可以使用全球降雨坐标，但它的非对角线度量可能会很烻尬和误 导。Kruskal-Szekeres (以下简称 KS) 坐标有一个很好的对角线度量，可能是理解和显示光子行 为最有用的坐标系。从定义 $\mathrm{KS}$ 坐标开始 $u$ 和 $v$ 这样
$$\left(\frac{r}{2 G M}-1\right) e^{r / 2 G M}=u^2-v^2 \quad \text { and } \quad t=2 G M \ln \left|\frac{u+v}{u-v}\right|$$

$$d s^2=-\frac{32(G M)^3}{r} e^{-r / 2 G M}\left(d v^2-d u^2\right)+r^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)$$

MATLAB代写

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