# Velocity of object falling into black hole measured in Schwarzschild co-ordinates, depending on co-ordinate time

This is a followup question to the answer given to a similar question where the velocity is given as

$$v(r) = \left(1 - \frac{r_s}{r}\right)\sqrt{\frac{r_s}{r}}c \tag{1}$$

where $$r$$ is the radius from the Schwarzschild co-ordinates and $$r_s$$ is the Schwarzschild radius.

What would be the velocity depending on $$t$$, the co-ordinate time, i.e.

$$\frac{dr(t)}{dt} = \;?$$

Looking at the Wikipedia link given in the referenced answer, section "Local and delayed velocities" I would guess it is

\begin{align} \frac{dr(t)}{dt} &= \frac{dr}{d\tau}\frac{d\tau}{dt}\\ &= v_{\parallel}\gamma\sqrt{1-r_s/r}\cdot\frac{1}{\gamma} \sqrt{1-r_s/r}\\ &= v_{\parallel}(1-r_s/r) \end{align}

but I can't find what $$v_{\parallel}$$ is.

• I wrote that section about the local and delayed velocities, $v_{\parallel}$ is meant to be the radial component, and $v_{\perp}$ the transverse component of the local velocity relative to a Fido. That is also mentioned in that section. Sep 7, 2019 at 3:20
• @Yukterez What is a Fido? Sep 7, 2019 at 5:15
• see universeinproblems.com/index.php/… and perimeterinstitute.ca/images/files/… - that's simply a local observer who is stationary with respect to the central mass Sep 7, 2019 at 9:53
• If you find any of the answers helpful, please consider upvoting them and also accepting the most helpful one. I usually delete my answers that users do not need for a future reference. Thanks! Jan 24, 2020 at 9:17

What would be the velocity depending on $$t$$, the co-ordinate time[?]

A free fall from infinity takes an infinite time. To define the velocity dependence on time, you can drop an object from a finite distance $$R$$ and start your clock $$t$$ at that moment.

A free fall to a Schwarzschild black hole from rest at a distance $$R$$ is given by the radial geodesic in geometrized units ($$c=1$$, $$G=1$$):

$$t(r)=\sqrt{\dfrac{R}{2M}-1}\cdot\left(\left(\dfrac{R}{2}+2M\right)\cdot\arccos\left(\dfrac{2r}{R}-1\right)+\dfrac{R}{2}\sin\left(\arccos\left(\dfrac{2r}{R}-1\right)\right)\right)+$$

$$+\, 2M\ln\left(\left|\dfrac{\sqrt{\dfrac{R}{2M}-1}+\tan\left(\dfrac{1}{2}\arccos\left(\dfrac{2r}{R}-1\right)\right)}{\sqrt{\dfrac{R}{2M}-1}-\tan\left(\dfrac{1}{2}\arccos\left(\dfrac{2r}{R}-1\right)\right)}\right|\right)$$

You can figure the coordinate velocity outside the horizon by differentiating this equation by $$r$$:

$$v=\dfrac{dr}{dt}=\dfrac{1}{\dfrac{dt}{dr}}$$

Then for any $$r$$ you would know both the velocity $$v$$ and coordinate time $$t$$ to define the (non-explicit) dependence of velocity on time.

The Schwarzschild metric in Schwarzschild coordinates $$(t, r, \theta, \phi)$$ shows
$$ds^2 = -(1 - 2M/r) dt^2 + (1 - 2M/r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2)$$
where:
$$c = G = 1$$ natural units
$$M$$ black hole mass
$$r_s = 2M$$ Schwarzschild radius (event horizon)

If we drop an object at rest from infinity, we have a radial free fall $$(d\theta = d\phi = 0)$$ where
$$dt = (1 - 2M/r)^{-1/2} d\tau_{stat}$$ time dilation measured at infinity (far away from the horizon) vs. the proper time of a stationary observer at Schwarzschild radial coordinate $$r = constant$$
$$dt = \gamma (1 - 2M/r)^{-1/2} d\tau_{free fall} = (1 - 2M/r)^{-1} d\tau_{free fall}$$ time dilation measured at infinity (far away from the horizon) vs. the proper time of a radially free falling object
$$\gamma = (1 - v^2)^{-1/2} = (1 - 2M/r)^{-1/2}$$ Lorentz factor of a radially free falling object vs. a stationary observer at Schwarzschild radial coordinate $$r = constant$$
$$v_{stat} = (2M/r)^{1/2}$$ velocity relative to a stationary observer at Schwarzschild radial coordinate $$r = constant$$
$$v_{Schw} = (1 - 2M/r) (2M/r)^{1/2}$$ velocity relative to Schwarzschild coordinates

Coming to your first question, $$dr/dt$$ is $$v_{Schw}$$, i.e. the velocity measured in Schwarzschild coordinates. In the post it is equation (1).
As for your second question, $$v_{\parallel}$$ is $$v_{stat}$$, i.e. the velocity relative to a stationary observer.