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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

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

Rain

[tex]\left(1-\frac{2M}{r}\right)\frac{dt}{d\tau}\ =\ 1[/tex]

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-v^{2}/c^{2})=(1-2M/r) where v=√(2M/r)c for an object in free fall from infinity (see below).

Hail

[tex]\left(1-v_{far}^2\right)^{-1/2}\ >\ 1[/tex]

E_{shell}/m (energy/mass ratio of object relative to shell frame)

Drip

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

Rain

[tex]\left(1-\frac{2M}{r}\right)^{-1/2}[/tex]

Hail

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

v_{shell}(velocity of object relative to shell frame)

Drip

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

Rain

[tex]\left(\frac{2M}{r}\right)^{1/2}[/tex]

Hail

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

multiply by [itex](1-2M/r)[/itex] 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).

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# Properties of in-falling radial plungers

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