some homework problems for Farsight, part 4
It's time to offer more homework problems. As before (except for exercise 8), these exercises assume nothing beyond basic skill in differential calculus and high school algebra.
This sequence of exercises began in a different thread, so you might want to review
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References.[/size]
In
epilogue 2, I cited and linked to an English translation of Lemaître's 1933 paper. Today's exercises are derived from
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Why more exercises are needed.[/size]
Farsight made the case for more exercises:
Painlevé-Gullstrand coordinates suffer from the same problem to Kruskal-Szekeres coordinates which we've discussed previously.
Farsight's problem is that neither Painlevé-Gullstrand nor Kruskal-Szekeres coordinates exhibit the coordinate singularity that
Farsight believes to be a real singularity of spacetime. He rejects those coordinate systems because they prove him wrong.
Rejecting mathematics because you're afraid it will prove you wrong is not a good way to understand physics. It's better to learn what the math has to teach you.
That is not
Farsight's way:
There you go hiding behind mathematics again. It cuts no ice, sol.
Change the record, Clinger. Taking refuge behind you don't understand the mathematics just isn't good enough.
The two situations are not the same. You're misinterpreting the mathematics used to describe a black hole because you're ignoring the undefined result at R=2M. Then you're claiming that the mathematics supports your description when it doesn't.
Note well that
Farsight has made mathematical claims about the event horizon (where r=2M). Note also that
Farsight has made an implicit claim that he understands the math well enough to know others are misinterpreting the math.
Because
Farsight has made both explicit and implicit claims about the math, I don't see any reason to take him seriously when he dismisses math's relevance. Before we get to the exercises, however, let's take care of these red herrings:
I've provided actual evidence. Optical clocks do go slower when they're lower. The
Shapiro delay is a delay, the radar signal goes slower when it goes past the sun. We have hard scientific evidence that the speed of light varies...
Et cetera.
Farsight has mentioned plenty of empirical facts, but they are not evidence for his idiosyncratic views because the mainstream interpretation of general relativity predicts those observations as well or better than his.
In particular, I have already proved (in exercises 20 and 21) that Lemaître charts predict every physical observation that can be predicted using
Farsight's beloved Schwarzschild charts.
Changing to Lemaître coordinates removes the coordinate singularity at the Schwarzschild radius, which is present in Schwarzschild coordinates.
Again you've evaded the undefined result at R=2M. I will reiterate, to understand why it's incorrect to do this you have to look at the empirical evidence of moving light, and what Einstein said, and what clocks do.
No. Lemaître coordinates explain everything that Schwarzschild coordinates explain, and more. That's a mathematical fact.
Had
Farsight done his homework (
exercises 20 and 21), he'd know that. He hasn't, doesn't, and probably won't.
The "undefined result at R=2M" occurs only in Schwarzschild coordinates. That's a purely mathematical defect of Schwarzschild coordinates, with no more physical significance than the coordinate singularity at the north pole of spherical coordinates. People are often willing to tolerate that defect because Schwarzschild coordinates are convenient when you want to ignore a star or black hole's negligible far-field gravity.
That purely mathematical, non-physical defect can be avoided by using Lemaître, Painlevé-Gullstrand, Eddington-Finkelstein, or Kruskal-Szekeres coordinates. All of those coordinate systems are in perfect agreement with Schwarzschild coordinates on the submanifold that Schwarzschild coordinates can describe, but (unlike Schwarzschild) are well-behaved and smooth all the way down to the black hole's central singularity.
Don't take my word for it. Prove it to yourself.
Exercises 20 and 21 established that Lemaître charts agree with Schwarzschild charts on the submanifold covered by Schwarzschild charts. The following exercises prove the analogous result for Painlevé-Gullstrand coordinates.
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Painlevé-Gullstrand coordinates.[/size]
With the notational conventions of the previous 23 exercises, the Painlevé-Gullstrand metric around a black hole of mass M is
[latex]
\[
ds^2 = - dt_f^2 + (dr + \beta dt_f)^2 + r^2 d\Omega^2
\]
[/latex]
where the spatial coordinates are the same as in Schwarzschild coordinates, t
f is the proper time of a free-falling object, and
[latex]
\[
\beta = \sqrt{\frac{2M}{r}}
\]
[/latex]
Those are equations (1) and (2) of the Hamilton and Lisle paper; Visser's equation (49) is basically the same as Hamilton&Lisle's equation (1). Their equation (5) relates the free-fall time t
f to Schwarzschild time t and radius r:
[latex]
\[
t_f = t - \int_r^\infty \frac{\beta}{1 - \beta^2} dr
\]
[/latex]
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Exercises.[/size]
Exercise 24 is Visser's equation (52) for an infalling object:
Exercise 24. Prove:
[latex]
\[
dt_f = dt + \frac{\beta}{1 - \beta^2} dr
\]
[/latex]
Hint: Differentiate both sides of equation (5) with respect to r.
Exercise 25. Prove: Given any Schwarzschild chart, there exists a Painlevé-Gullstrand chart whose restriction to r > 2M is isometric to the Schwarzschild chart.
Hint: trivial algebra, as in exercise 20.
Exercise 26. Prove: The manifold covered by Schwarzschild charts is (isometric to) a proper submanifold of the manifold covered by Painlevé-Gullstrand charts.
Hint: see exercise 21.
Exercise 27. Find a minor and inconsequential miscalculation in the
arXiv version of Visser's paper.
Hint: see section 7.
! 
Mathematics is the language of science. If you don't understand the math, you don't understand the idea. Period. Full stop. Even in paleontology we acknowledge that, and we use less math than any other field of science I've yet encountered. 
