Change of Variable: Physics Math in GR Problem

[topsquark]

What I am about to do is write an example of what I call "Physics Math."

I am working on a GR problem backward: I have a motion I want to exhibit and am trying to work out the connection coefficients. The problem is that they are in terms of the wrong variable...The equations are expressed in terms of a parameter \(\displaystyle \tau\) and I want them to be in terms of r and t. (The motion is independent of \(\displaystyle \theta\) and \(\displaystyle \phi\) so I am dropping time derivatives of them.)

So. I saw this in one of my Physics texts and if it works it's a good short-cut for me.

Assume a metric
\(\displaystyle d \tau ^2 = -a(r) dt^2 + b(r) dr^2 + r^2 d \theta ^2 + r^2 ~sin^2(\theta) d \phi ^2\).

Thus:
\(\displaystyle d \tau = \sqrt{ -a(r) dt^2 + b(r) dr^2 + r^2 d \theta ^2 + r^2 ~sin^2(\theta) d \phi ^2}\)

\(\displaystyle \frac{d \tau}{dt} = \sqrt{ -a(r) \left ( \frac{dt}{dt} \right ) ^2 + b(r) \left ( \frac{dr}{dt} \right ) ^2 } = \sqrt{ -a(r) + b(r) \left ( \frac{dr}{dt} \right ) ^2 }\)

and
\(\displaystyle \frac{d \tau}{dr} = \sqrt{ -a(r) \left ( \frac{dt}{dr} \right ) ^2 + b(r) \left ( \frac{dr}{dr} \right ) ^2 } = \sqrt{ -a(r) \left ( \frac{dt}{dr} \right ) ^2 + b(r) } \)

How many rules and I breaking and how badly am I breaking them?

-Dan

 

[Euge]

Hmm, it looks like you're dealing with a Schwarzschild-type metric. In any case, I don't see any "rule breaking" but you may want to think carefully about the meaning of those differentials.

 

[topsquark]

Hmm, it looks like you're dealing with a Schwarzschild-type metric. In any case, I don't see any "rule breaking" but you may want to think carefully about the meaning of those differentials.

Yes, eventually I will putting the whole mess into a Schwarzschild-like problem. The premise is that I'm going to visit a black hole and turn on a rocket to add a constant acceleration downward. It seems that the metric in this case is going to be time-dependent but so far I'm awash in a sea of indices written in terms of the parameter \(\displaystyle \tau\). There is a much more complex method to change the variables but it's kind of frustrating. So I decided to look at a possible shortcut.

And also yes, GR is rather unforgiving about talking about the wrong variables. I have already determined that I need to set \(\displaystyle \frac{d^2 x}{dt^2} = a\) rather than \(\displaystyle \frac{d^2 x}{d \tau ^2} = a\).

I am a little surprised to find that this method works. This is one of those "trick" methods that strongly reminds me of black magic. Green's functions are another! (Wasntme)

Thanks for the help!

-Dan