Properties of in-falling radial plungers

[stevebd1]

Due to the blogs being removed, I thought it might be worthwhile posting a few in the forums-


From 'Exploring Black Holes' by John Wheeler and Edwin Taylor; can apply to any object falling radially towards a static spherical mass (where the mass of the infalling object is much smaller than the static spherical mass).


Three types of in-falling radial plunger-

Drip (dropped from rest at r_o)

Rain (dropped from rest at infinity)

Hail (hurled inward at speed v_far from a great distance).


E/m (energy/mass ratio of the object)

Drip

$$\left(1-\frac{2M}{r_o}\right)^{1/2}\ <\ 1$$

Rain

$$\left(1-\frac{2M}{r}\right)\frac{dt}{d\tau}\ =\ 1$$

where dτ is the time dilation for an object in free fall from infinity which in this case is the sum of both the time dilation for gravity and velocity- dτ=√(1-2M/r)·√(1-v²/c²)=(1-2M/r) where v=√(2M/r)c for an object in free fall from infinity (see below).

Hail

$$\left(1-v_{far}^2\right)^{-1/2}\ >\ 1$$


E_shell/m (energy/mass ratio of object relative to shell frame)

Drip

$$\left(1-\frac{2M}{r_o}\right)^{1/2}\left(1-\frac{2M}{r}\right)^{-1/2}$$

Rain

$$\left(1-\frac{2M}{r}\right)^{-1/2}$$

Hail

$$\left(1-v_{far}^2\right)^{-1/2}\left(1-\frac{2M}{r}\right)^{-1/2}$$


v_shell (velocity of object relative to shell frame)

Drip

$$\left(1-\frac{2M}{r_o}\right)^{-1/2}\left(\frac{2M}{r}-\frac{2M}{r_o}\right)^{1/2}$$

Rain

$$\left(\frac{2M}{r}\right)^{1/2}$$

Hail

$$\left[\frac{2M}{r}+v_{far}^2\left(1-\frac{2M}{r}\right)\right]^{1/2}$$


multiply by $(1-2M/r)$ for velocity of in-falling object as observed from infinity- dr/dt.

multiply by c for SI units.


Source-
Sections 3 & B of 'Exploring Black Holes' (Note: The above equations were collected from a draft of chapter 3 for the new edition, since collecting these equations, the draft has been revised and all mention of drip and hail frames has been removed, the authors stated they wanted to focus primarily on the rain frame. They did however send a copy of the draft that includes the drip & hail frame and said that I was welcome to distribute this version (see attachments).

 

[Ambitwistor]

Thank you for sharing this information on the different types of in-falling radial plungers and their corresponding equations. As a scientist studying black holes, I find this to be a very interesting and relevant topic.

I would like to add some thoughts on the significance of these different frames and their corresponding energy/mass ratios and velocities. The key concept here is the idea of time dilation in the presence of a strong gravitational field.

In the case of the drip frame, where the object is dropped from rest at a distance ro from the black hole, the energy/mass ratio must be less than 1 in order for the object to reach the event horizon before being stretched to infinity. This is due to the strong time dilation effect near the event horizon, where time appears to slow down for an outside observer. As the energy/mass ratio approaches 1, the time dilation becomes infinite and the object never reaches the event horizon.

In the rain frame, where the object is dropped from rest at infinity, the energy/mass ratio must be exactly 1 for the object to reach the event horizon in a finite amount of time. This is because the time dilation effect near the event horizon cancels out the velocity of the object, allowing it to reach the event horizon in a finite time.

Finally, in the hail frame, where the object is hurled inward at a high velocity from a great distance, the energy/mass ratio must be greater than 1 in order for the object to reach the event horizon. This is because the object's high velocity counteracts the time dilation effect near the event horizon, allowing it to reach the event horizon in a finite time.

These different frames and their corresponding energy/mass ratios and velocities demonstrate the complex effects of gravity near a black hole. I appreciate you sharing this information with the forum and I look forward to further discussions on this topic.