harrylin said:
If you mean the SR twin paradox: once more, that is very different as the (t, t') sets are finite and agree with each other. It's different however with Einstein's GR twin paradox which is much more interesting and relevant as background for this topic. It would distract too much from this topic to discuss it here, but I encourage you to study it.
I always considered Zeno's paradox as a joke - it may have been serious for philosophers, but not for physicists IMHO.
The funny thing is, you're basically using the same reasoning that Zeno did - without apparently even realizing this fact, not even when it's pointed out!
Let's compare what happens issue by issue.
In Zeno's paradox, we have two times. Let's let the "real" time be represented by tau, and zeno time by Z.
In the infalling black hole case, we have proper time tau, and Schwarzschild time t
The mapping from t to tau that we worked out previously for the Schwarzschild case in great detail is:
t = \tau-4\,\left(-3 \, \tau \right)^{\frac{1}{3}}+4\,\ln \left[ \left(-3 \, \tau \right)^{\frac{1}{3}}+2 \right] -4\,\ln \left[\left(-3 \, \tau \right)^{\frac{1}{3}}-2<br />
\right]
(You can probably find this in a textbook if you want to check my math).
The characteristic features of this mapping is that t increases monotonically with tau, and that infinite range of tau only covers a finite range of t.
This is due specifically to the term
<br />
-4 \ln \left[\left(-3 \, \tau \right)^{\frac{1}{3}}-2<br />
\right]
This is rather complicated, the argument will be clearest if we assume this term, which is the one that approaches infinity, is the dominant term near the event horizon, in which case we can solve for \tau assming that this is the only term that matters.
\tau \approx \frac{1}{3} \, \left[\exp^{-t/4} - 2\right]^3
and we see that \tau approaches -8/3 as t-> infinity (which is when the horizon is reached).
In the Zeno paradox, the mapping is something like
<br />
\tau = a(1-\exp^{z/b})<br />
where a and b are some constants
where \tau approaches some constant a as Z approaches infinity
And we see the issue -in both cases, even though t (in one case), Z (in the other case) cover infnite ranges, \tau does not.
So, essentially Zeno never assigns a label, Z to some events. And he concludes from this that these events don't exist.
And you assume the same thing - because you never assign a coordinate "t" to some events, you assume they don't happen.
And this conclusion is just as unjustified when you do it, as when Zeno does it.
So the not-very-complicated moral of the story is that because you can choose ANY coordinates you want, you need to be careful in your interpretation of the results. Specifically, it's possible to choose coordinates like Zeno did, that exclude important regions from analysis, because the coordinates don't label physically significant events. However, not giving something a label doesn't make it not exist, any more than closing your eyes does. At least not for most defitnitions of "existence".
