Black holes and Schwarzschild and calcs

Factual Questions
1

I was fooling around with the Schwarzschild metric, and I decided to find how long it would take me to fall to the center, if our sun was a BH and the Earth disappeared.

dTau = r[sup]1/2[/sup] dr / (2M) [sup]1/2[/sup] so integrating

Tau = (2/3)*r[sup]3/2[/sup]*1/(2M) [sup]1/2[/sup] from r to r = 0

Convert M to M/M[sub]sun[/sub]*(1477)
Convert meters of time to seconds by dividing by c
convert 93 million miles to 1.5x10[sup]11[/sup] meters

Tau = 4.09x10[sup]-11[/sup]*r[sup]1/2[/sup] / (M/M[sub]sun[/sub])
Tau = 1.6 x 10[sup]-5[/sup] sec

  1. What the hell did I do wrong? I absolutely cannot find it.

  2. r is actually the reduced circumference, how do I find this? I’m sure it’s not 93 million miles as I’d guess that’s the actual distance through curve spacetime.

  3. Why does this equation work right down to r = 0? Without using a Finkelstein or Kruskal coordinate transformation Tau goes to infinity at r = 2M.

  4. Why do GR guys say that Finkelstein showed that stuff falls right in? You still cannot find out how Tau relates to faraway time t.

  5. In flat spacetime if I don’t move through space I’m traveling through time at c, but I’m still not going to reach something that’s 2 feet away. In fact I won’t reach anything that isn’t going to hit me in the head. When the r and t coefficients change sign when passing the EH and you’re now traveling through time- what does this mean?

Please feel to answer as many of these questions as you feel like.

2

Shit the r exponent should be 3/2 not 1/2 so

Tau = 2.38x10[sup]6[/sup] secs

3

At any point in your trajectory you’re traveling through time. Free-fall means you’re in a locally inertial reference frame, which means that you can consider yourself to be “not moving” for the entire fall.