I'm just a guy studying a introduction to GR entitled 'Exploring Black Holes' by Edwin Taylor and John Archibald Wheeler. The purpose of this introduction is to use the Schwarzchild and Kerr metrics, with algebra and calculus, to examine spacetimes around nonspinning and spinning bodies of mass. It just so happens I'm working on a project in the text entitled 'Inside the horizon of the nonspinning black hole'. BTWThis is a great book.
I'm going to provide you the steps from the book.
"We want a metric in the coordinates r,f, and train. We make this transition in two jumps from events outside the horizon: from bookkeeper coordinates to shell coordinates, then from shell coordinates to rain coordinates (note: bookkeeper is the far away observer, shell is the local observer outside the event horizon at rest on a spherical shell, and rain is proper time of falling observer). Assume that the resulting metric is valid inside the event horizon as well as outside. The transition from bookkeeper coordinates to shell coordinates is given by equations [C] and [D] (note: these equations are part of the Schwarzchild metric).
drshell=dr/(12M/r)1/2 [D]
dtshell=(12M/r)1/2dt [C]
Now to go from shell coordinates to rain coordinates, use the Lorentz transformation of SR (note:all local observers use SR to do the physics). Choose the 'rocket' coordinates to be those of the rain frame and the 'laboratory' coordinates to be those of the shell frame. The Lorentz transformation for differentials is expressed for motion along the xaxis, which in this case lies along the raidial inward direction:
dtrain= vrelgdrshell+gdtshell [9]
Substitute equations [C] and [D] into the Lorentz transformation equation [9] to obtain:
dtrain=(vrelgdr )/(12M/r)1/2+g(12M/r)1/2dt [10]
Solve for dt:
dt=(dtrain>)/g(12M/r)1/2+(vreldr)/(12M/r) [11]
Substitute vrel from equation [1] into the stretch factor g
vrel=(2M/r)1/2 [12]
g=1/(1v2rel)1/2=1/(12M/r)1/2 [13]
Substitute equations [12] and [13] into [11] to obtain
dt=dtrain(2M/r)1/2dr/(12M/r) [14]
The Schwarzchild metric:
dt2=(12M/r)dt2dr2/(12M/r)r2df2
Substitute expresion [14] into the Schwarzchild metric and collect terms to obtain the global rain metric in r, f, train:
dt2=(12M/r)dt2rain2(2M/r)1/2dtraindrdr2r2df2
This metric can be used anywhere around a nonrotating black hole, not just inside the horizon. Our ability to write the metric in a form without infinities at r=2M is an indication that no jerk or jolt is felt as the plunger passes through the horizon."
