Schwarzschild’s metric for the black hole:
<img src=../Ss.png>
<br>For constant <i>t, &theta; &phi;</i> we compute the arc length of a radial curve to the horizon by integrating dr to the surface, r → r<sub>s</sub>.
<p><a href=http://integrals.wolfram.com/index.jsp>Wolfram says</a> that the integral of 1/(√(1-1/x)) is
((√x)(x-1) + √(x-1) log(2(√(x-1) + √x)))/√(x-1).
<p>
Query expression is 1/(Sqrt[1-1/x])
<p>
Scheme:
(define (f x) (/ (+ (* (sqrt x)(- x 1)) (* (sqrt (- x 1)) (log (* 2 (+ (sqrt (- x 1)) (sqrt x)))))) (sqrt (- x 1))))
<br>Metric: (define (g x) (/ (sqrt (- 1 (/ x)))))
<table><tr><td>x<td>f(x)
<tr><td>1000<td>1004.34
<tr><td>100<td>103.185
<tr><td>10<td>11.998
<tr><td>5<td>6.609
<tr><td>2<td>2.989
<tr><td>1.2<td>1.617
<tr><td>1.15<td>1.487
<tr><td>1.1<td>1.334
<tr><td>1.01<td>.8934
<tr><td>1.001<td>.7564
<tr><td>1.0001<td>.7131
<tr><td>1.00001<td>.6995
<tr><td>1.000001<td>.669515
</table>
You could find an exact formula for the limit by l’Hôpital’s rule but I am too lazy.
It seems that the distance in finite.
<p>
To verify Wolfram and my transcription we differentiate the integral numerically and get
<br>
(define (df x) (/ (- (f (+ x dx)) (f x)) dx))
<br>
(define dx 0.0000001)

<table><tr><td>x<td>(df x)<td>(g x)
<tr><td>1000<td>1.0004987416323274<td>1.0005003753127737
<tr><td>100<td>1.00503783073691<td>1.005037815259212
<tr><td>10<td>1.0540925288182734<td>1.0540925533894598
<tr><td>7<td>1.080123457342097<td>1.0801234497346435
<tr><td>4<td>1.154700530747732<td>1.1547005383792517
<tr><td>2<td>1.4142135418282464<td>1.414213562373095
<tr><td>1.5<td>1.732050751712677<td>1.732050807568877
<tr><td>1.1<td>3.3166240398507796<td>3.3166247903553994</table>
This calculation is via the Schwarzschield metric and other metrics would provide a different answer or none.