Radial Infall into a Static Mass: Equations & Guide

From Exploring Black Holes by John Wheeler and Edwin Taylor. The relations below apply to any object falling radially toward a static, spherically symmetric mass (assuming the mass of the infalling object is much smaller than the central mass).

Types of radial infall

• 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 per unit mass)

Drip

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

Rain

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

Here dτ is the proper time for an object in free fall from infinity; in this context dτ = √(1-2M/r)·√(1-v^2/c^2) = (1-2M/r), since 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 per unit mass 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 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 the velocity of the infalling object as observed from infinity (dr/dt).

Multiply by c for SI units.

Source

Sections 3 & B of Exploring Black Holes. (Note: these equations were collected from a draft of chapter 3 for a new edition. Since then the draft has been revised and mentions of the drip and hail frames were removed; the authors chose to focus primarily on the main (rain) frame. The authors did send a copy of the draft that includes the drip & hail frames and said that I was welcome to distribute that version.)

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