- #1
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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
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