I have a problem in calculating the module of the velocity of a particle measured by a static observer in a specific metric. This metric is


Because both $\partial_t$ and $\partial_\varphi$ are Killing vectors, we have two conserved quantities

$$E=-g_{\mu\nu}U^\mu\delta^{\nu t}=(r^2-R^2)\frac{\partial t}{\partial \tau}$$ $$L=g_{\mu\nu}U^\mu\delta^{\nu \varphi}=r^2\frac{\partial\varphi}{\partial\tau}$$

where $U^\mu$ is the four velocity. The equations of motion for a massive particle come from $U^2=1$ and are

$$E^2=\left(\frac{\partial r}{\partial\tau}\right)^2+(r^2-R^2)(1+\frac{L^2}{r^2}).$$

My problem is that i dont understand how to calculate the velocity of that a static observer in $r_1>r_0$ (where $r_0$ is the initial point of the particle).

Any help is appreciatted.

  • $\begingroup$ From the two conserved quantities and the fact that $d\tau^2=(r^2-R^2)dt^2-\frac{dr^2}{r^2-R^2}-r^2d\varphi^2$, you can calculate $dr/d\tau$ as a function of $r$, $E$, and $L$. $\endgroup$ Dec 3, 2023 at 14:36
  • $\begingroup$ Do you mean the local 3-velocity v relative to a static observer at the same coordinate as the test particle so that a photon has v=c? Then the covariant momentum pᵤ=vᵤ√|gᵤᵤ|/√(1-μ²|v|²) with |μ|=1 for particles and 0 for photons. The coordinate velocity dx/dt may be smaller and the proper velocity dx/dτ higher than the local v. $\endgroup$
    – Yukterez
    Dec 3, 2023 at 18:28


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