# The nearly unmoving heavy star?

$$c^2 {d \tau}^{2} = \frac{1}{4} \left( 3 \sqrt{1-\frac {r_s}{r_g}}-\sqrt{1-\frac{r^2 r_s}{r_g^3}} \right)^2 c^2 dt^2 - \left( 1-\frac{r^2 r_s}{r_g^3} \right)^{-1} dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right)$$

approximates the metric inside a planet or star of some homogeneous material (see Wikipedia). Here, $$r_s$$ is the Schwarzschild radius of its mass and $$r_g>r_s$$ its radius, $$r$$ the radius at which we want to know the metric.

To derive the speed of light at the surface $$(r=r_g$$), I follow the derivation shown in this answer and set $$d\tau=0$$ to get

$$0 = (1-\frac{r_s}{r_g})c^2 dt^2 - (1-\frac{r_s}{r_g})^{-1}dr^2 -r_g^2d\Omega^2$$

where $$d\Omega = \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right)$$. Rearranging gives:

$$\left(\frac{dr}{dt}\right)^2 + (1-\frac{r_s}{r_g})r_g^2 \left(\frac{d\Omega}{dt}\right)^2 = (1-\frac{r_s}{r_g})^2c^2$$

Taking the square root on both sides, on the left side we have the speed of light on the surface of the object according to my watch (read: coordinate time) limited by a constant depending on its mass and radius being smaller than the speed of light $$c$$ far away from any gravitational source.

1) I assume that if the speed of light is like that, any massive particle near $$r_g$$ has to obey the same speed limit. In particular the atoms of this object. This is, with regard to coordinate time, i.e. what I would measure when watching it with my telescope and timing with my watch, so to say.

2) Now start reducing $$r_g-r_s$$ by either increasing the mass or reducing its radius such that $$r_s/r_g \to 1$$ such that the speed of light nicely approaches zero. The speed of the surface particles (and by extension the whole object) has to go to zero too.

This looks as if the object has to come to a halt and does not move anymore.

I am rather confident in the interpretation of $$dt$$ as the time my watch shows, so to say. But clearly black holes ($$r_s\to r_g$$) are not nailed to the sky.

In which coordinate system does the object slow down, to a halt in the limit, and how does this coordinate system relate to me observing it with a telescope (undisturbed by any nearby gravitational sources).