Max wrote:
Thanks for the information. When I was looking for information on RAIN
frames, the book you wrote of, Exploring Black Holes, was cited. My
campus library doesn't carry it though, so I will see what I can gleam
from the extracts. I'll go and read the document you linked to and
appreciate your help.

Thanks again,

Max
The 2nd project is 'Inside the Black Hole'. The metric for the rain
frame doesn't result in infinities at r = 2M. If you're interested I'll
write out the derivation [not to long or difficult]. Starting with the
Schwarzschild metric you choose Lorentz transform 'rocket coordinates'
for the rain frame and 'laboratory' coordinates for the shell frame.

Bruce

Hi Bruce!

I have a derivation of the RAIN frame from class notes, but I'm not
understanding how to use it because I'm confused about the coordinates.
The rain frame is the frame of an inertial observer that falls from 0
velocity at infinity radially towards the non-rotating central mass of
the SM. However, with the resulting RAIN frame metric, I'm not sure
what is to be meant the "r"coordinates, dt_rain, etc.. are the
increments of the rain observer, or of what is observerd? Also, what is
d_tau measuring?

I'm quite confused as you can see, but I'm still learning. If you could
explain those questions, since I have a derivation, I'd much appreciate
it!

Thanks, I look forward to your response,

Max wrote:
Hi Bruce!

I have a derivation of the RAIN frame from class notes, but I'm not
understanding how to use it because I'm confused about the coordinates.
The rain frame is the frame of an inertial observer that falls from 0
velocity at infinity radially towards the non-rotating central mass of
the SM. However, with the resulting RAIN frame metric, I'm not sure
what is to be meant the "r"coordinates, dt_rain, etc.. are the
increments of the rain observer, or of what is observerd? Also, what is
d_tau measuring?

I'm quite confused as you can see, but I'm still learning. If you could
explain those questions, since I have a derivation, I'd much appreciate
it!

Thanks, I look forward to your response,

Max

For this frame the falling rain observer measures the distance between
shells

dr_rain = dr_shell [1 - (v_rel)^2]^1/2 = dr_shell (1 - 2M/r)^1/2

dr = dr_shell (1 - 2M/r)^1/2

Surprisingly

dr_rain = dr

dT is the proper time [wris****ch time] of the falling rain observer.

To transform from shell coordinates to rain coordinates use the Lorentz
transformation choosing 'rocket coordinates' for the rain frame and
'laboratory coordinates' for the shell frame.

dt_rain = - v_rel gamma dr_shell + gamma dt_shell

or

= [( - v_rel gamma dr )] / (1 - 2M/r)^1/2 + gamma (1 - 2M/r)^1/2 dt

Solve for dt

dt = dt_rain / (gamma (1 - 2M/r)^1/2 + ( v_rel dr ) / (1 - 2M/r)

Make these substitutions

v_rel = - (2M/r)^1/2

and

gamma = (1 - 2M/r )^-1/2

dt = dt_rain - [(2M/r)^1/2 dr] / (1 - 2M/r )

Substitute into the Schwarzschild metric to get the global rain metric
[assume its valid for inside and outside r = 2M].

Here is an equivalent form of the rain metric

dT^2 = - [dr + (1 + {2M/r)^1/2 dt_rain][dr - (1 - {2M/r)^1/2 dt_rain] -
r^2(dphi)^2

Multiplied out it becomes

dT^2 = (1 - 2M/r)dt_rain^2 - 2(2M/r)^1/2 (dt_rain^2 dr) - dr^2 -
r^2(dphi)^2

Hope that answers all the questions.

Send me an email [it's correct]. I've got about 6 copies of EBH [prof
Taylor set me up] and sometimes I like to give a copy away. Most likely
this would be such a time.

Bruce

Max
anep for the dots.

Hi Bruce,

If I wrote out the mathematics for what I don't understand in Latex
format, would you be able to look at it in a reader for comments? Just
curious because writing in ASCII makes the notation for reading very
difficult.