Wow ... I found myself engrossed in that vitriolic exchange of 2004. Sounds exactly like the mess we got into with Joy Christian's result, and for the same reasons. I mean, it appears that when someone -- anyone, it seems -- proposes a purely classical framework (which both Mitra's and Christian's are), it's like waving a red cape in the face of a bull.
I respect John Baez's knowledge and skill in mathematical physics; however, I think he may have been a bit unfair to say things like, "Starting from the solution which describes a black hole of mass m, he attempts by a calculation to show that m = 0. It's a bit like taking an arbitrary prime number and proving that it must equal 37."
However, Mitra didn't assume a black hole of mass M (it should be M in this context). So Baez's criticism amounts to saying that Mitra made an erroneous assumption, not that the calculation is wrong. Nevertheless, Baez then uses what he regards as the erroneous assumption to show that Mitra makes the elementary mathematical error of dividing by zero.
Baez quotes Mitra: "For the benefit of the serious readers, I give below the essence of my proof: In Lemaitre coordinates, the radial geodesic (angular part=0), the metric of a test particle around a BH is ds^2 = dt^2 - g_rr dr^2 (1)"
To which Baez replies, " ... the phrase 'the metric of a test particle' makes no sense. The metric is something on spacetime, and it applies to all particles moving in spacetime, so one never speaks of the metric 'of a test particle'."
While this is exactly true, I think it's ungenerous and a bit condescending. I get the impression that Mitra is using the word metric to mean "trajectory." This would have to be so, in order to fix time coordinates for the endpoints of a geodesic. That is, a particle trajectory on the event horizon traces a metric; it doesn't define a metric. My suspicions is confirmed later on when John says:
"Somehow M = 0 has popped out. It's popped out because in equation (10) he gets ds^2 = 0 at R = 2M, 'following the radial geodesic'. He's not very clear about (what) that means ...
(Well, it's quite clear to me that Mitra means the metric trace of a massless particle, for which t = 0 both at the event horizon *and* for a distant observer.)
" ... but interpreting him generously I'd say he's concluding the change in proper time vanishes for a test particle freely falling into a black hole as it crosses the horizon."
(Mitra's test particle never crosses the horizon.)
"This would indeed be a contradition since general relativity (quoting Mitra here) 'demands that the geodesic must remain timelike there and we should have had ds^2 > 0.'
"So, his mistake may lie in his derivation of ds^2 = 0. Where does this come from? He says it comes from ds^2 = dt^2 - dr^2 at R=R_g (6) and (dr/dt)^2 = 1 at R = R_g (9) I've already said I see no flaw in (6) so probably the flaw is in (9). And indeed, (9) is false for a test particle freely falling into the black hole: in LeMaitre coordinates, r is constant for such a particle, so dr = 0 contradicting (9).
"The rest is a mopping-up operation ..."
Only with John's assumptions. And those assumptions make his criticism look suspiciously like a straw man argument to me, another common characteristic of the Joy Christian controversy. Ah, well.
Professor Corda, if you find this diatribe irrelevant to your forum discussion, please just pull the plug on it.
All best,
Tom