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Let's say we are working with the Schwarzschild metric and we have an emitter of light falling into a Schwarzschild black hole.

Suppose we define the quantity $$u=t- v$$ where $$dv/dr= 1/(1-r_{s}/r)$$ where $r_s$ is the Schwarzschild radius. What is the $u$ as observed by the emitter? I just need a definition of $u_e$. I have problems identifying the quantities as measured by an observer at large $r$ and that of the emitter. Would I be right at least to say that $$t_{e}=\tau$$ the proper time? Many thanks.

In fact, I've been told that

$$du_o/d\tau=du_e/d\tau$$

Why is it?

My apologies for leaving out a subscript $e$ in the question. It has been added.

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What are $u$ and $v$? Whose proper time is $\tau$? I think we need more background to the setup of your question. – Nathan Reed Mar 9 at 21:58
@NathanReed: $u,v$ are defined in the question...! I don't know whose proper time that is, unfortunately. I am only given the metric! I have already included all the info I know... – Sad confused person Mar 9 at 22:26
No, $u, v$ are not defined in the question. :) You've given a couple of relationships between $u, v$ and the coordinates $r, t$, but I still have no idea what you intend $u, v$ to actually mean. And how can you not know whose proper time you're talking about? It's your question, after all. Sorry, but you're going to have to do a better job of asking what you want to know. Where is all of this coming from - a textbook, a homework problem? Give us more context. – Nathan Reed Mar 9 at 22:29
@NathanReed: :) It is taken from some old notes. Let's assume the proper time is that of the emitter and the observer located at a constant large r. – Sad confused person Mar 9 at 23:07
If you define $u$ by your first two equations, then $u,r$ form a system of Outgoing Eddington Finkelstein coordinates. $u$=const describes an outgoing null ray, which I guess in your picture is the path which represents a light pulse emitted by the emitter and received by the observer who sits fixed at large $r$. Like Nathan though, I'm not sure whose proper time we're talking about. – twistor59 Mar 10 at 8:57
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