m4r35n357 said:
OK, I'm going to rewind to my original attempt as it might be clearer what I am trying to do. Starting from a Kerr (Boyer-Lindquist) metric:
##
d\tau^2 = g_{tt}dt^2 + 2 g_{t\phi}dtd\phi + g_{rr}dr^2 + g_{\theta\theta}d\theta^2 + g_{\phi\phi}d\phi^2
##
4-velocity norm
##
1 = g_{tt}\frac {dt^2} {d\tau^2} + 2 g_{t\phi}\frac {dt} {d\tau} \frac {d\phi} {d\tau} + g_{rr}\frac {dr^2} {d\tau^2} + g_{\theta\theta}\frac {d\theta^2} {d\tau^2} + g_{\phi\phi}\frac {d\phi^2} {d\tau^2}
##
let
##
\gamma = \frac {dt} {d\tau}
##
OK so far
##
1 = \gamma^2 g_{tt} + 2 \gamma^2 g_{t\phi} \frac {d\phi} {dt} + \gamma^2 g_{rr}\frac {dr^2} {dt^2} + \gamma^2 g_{\theta\theta}\frac {d\theta^2} {dt^2} + \gamma^2 g_{\phi\phi}\frac {d\phi^2} {dt^2}
##
I don't follows this at all, I get:
##1 = \gamma^2 g_{tt} + 2 \gamma g_{t\phi} \frac {d\phi} {dt} + g_{rr}\frac {dr^2} {dt^2} + g_{\theta\theta}\frac {d\theta^2} {dt^2} + g_{\phi\phi}\frac {d\phi^2} {dt^2}##
...
OK, so now I've got two formulae for ##v## (is it ##v/c##?). The one using ##g_{tt}## is simpler. Trouble is, this expression is not bounded between 0 and 1 when I use it. Hence my arm-waving justification (cop-out) in post #15 for using the simpler (modulo known metric signature errors!) Minkowski-type expression:
##
v^2 = \frac {\gamma^2 - 1} {\gamma^2} = 1 - \gamma^2
##
I'm really not following this, I see a "v" has appeared at the right of your expresssion, somehow, but I'm not sure how it got there.
The logical consequence of your cop-out the way I see it is that an observer with constant r, theta, phi does not have a "velocity" of zero. I would agree you've calculated a number, but I don't understand why you think this number represents a velocity - for instance, it doesn't seem to have any physical interpretation as the velocity between a pair of worldlines. And the "velocity" of an observer with constant spatial coordinates isn't zero.
However, if we keep the factor of ##g_{tt}## in your equation, I see something that happens to work out to be equivalent for the expression for "little v":
$$\Gamma = \frac{ \frac{dt}{d\tau} }{\sqrt{g_{tt} } }$$
$$v = \sqrt{1-\frac{1}{\Gamma^2}}$$
"It happens to work out" is a bit vague, so I'll give a more formal but brief sketch of why it works out. If we have two four velocities ##p## and ##q##, ##p \cdot q = g_{ij} \, p^i \, q^j## is a measure of the angle between p and q. In flat Minkowskii space-time, ##p \cdot q = \gamma##, modulo sign issues. So ##p \cdot q## it's a function of relative velocity, expressed in a coordinate independent manner.
Since the 4-velocity v of a shell observer has the components ##\left[1/\sqrt{|g_{tt}|},0,0,0\right]## the dot product of ##\left[1/\sqrt{|g_{tt}|},0,0,0\right]## with u is ## u^0 / \sqrt{g_{tt}} = \left(dt / d\tau \right) / \sqrt{g_{tt}} ##
"Little v" does has a physical interpretation as the velocity between shell observer with constant r, theta, phi coordinates relative to the velocity of an observer with a 4-velocity u as measured by either the shell observer , or the observer with 4-velocity u.
You point out that "little v", in your application (which isn't Schwarzschild) sometimes has a magnitude that's greater than 1. I believe this happens because in your application, an observer with constant r, theta, phi coordinates does not have a timelike worldline.