I've been thinking about this issue the last few days. I suspect that the complaint about Stat Mech is correct. My own thinking, and as always, I hope smarter people will correct me, is as follows: Noether's Theorem applies as always - the interesting question is why *time* has an arrow, and position doesn't. Special relativity looks as if it treats space and time similarly, though we all know it doesn't. (Minkowski metric etc. - Though when we go to QFT and perform the Wick rotation, and spacetime becomes Euclidean, then everything changes. But I am not thinking about that now...) Again, I hope a mathematical physicist will kick in at some point, but I think the issue is that spacial translations are accompanied by rotational symmetry, which makes + and - translations identical on the position axes. But since there is no rotational symmetry mixing the spatial axes with the time axis, then the time translation invariance is not required to be symmetric in both directions. Does this make any sense? Or am I full of it? Rowan. Dan Asimov wrote:
Lots of interesting things here.
It does seem Rowan and Harold were discussing *classical* mechanics. (Can there be theoretical newtonian black holes? Actual ones?)
Kind of ironic that using a black hole one could do all kinds of Fields medal-worthy research but be unable to report back about it.
--Dan
Mike wrote:
<< On Sat, Jul 25, 2009 at 10:21 PM, <mcintosh@servidor.unam.mx> wrote:
Quoting Rowan Hamilton <rowanham@gmail.com>:
Classical mechanics is *not* invariant under time reversal, as others have pointed out, since this violates the Second Law of Thermodynamics.
But that is not true, at least for a time independent or symmetric Hamiltonian. The Second Law is a statistical, or even empirical law, and is not part of Classical Mechanics; look at Poincare's recurrence theorem, for example.
From the perspective of an outside observer, the infalling matter never gets into the black hole. Time-reversing that gives a picture in which there's matter just outside the Schwarzschild radius that moves away from the star; that certainly happens all the time, so there's no contradiction with the second law. It's possible, though very difficult, to arrange things to end up in a low entropy state--for instance, Honda's Rube-Goldberg commercial used no computer graphics and required 606 takes.
http://video.google.com/videoplay?docid=-4187430023476942057
For an observer falling into the star, the light outside gets more and more blue-shifted, because from his perspective time outside is passing more quickly. Crossing over the Schwarzschild radius, the light is infinitely blue shifted and infinite time passes. (So if you want to solve the halting problem, leave your computer in orbit around a black hole and jump in! It will do infinitely many calculations in a finite time from your perspective.)
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