From: Rowan Hamilton <rowanham@gmail.com> To: Dan Asimov <dasimov@earthlink.net>; math-fun <math-fun@mailman.xmission.com> Sent: Sunday, July 26, 2009 9:31:26 PM Subject: Re: [math-fun] dumb question about general relativity 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:

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Quoting Rowan Hamilton <rowanham@gmail.com>:

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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.

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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.)

The full Lorentz group, i.e. those linear transformations that preserve the metric M = x^2+y^2+z^2-t^2, consists of 4 connected components. The connected component containing the identity, called the proper Lorentz group, preserves the direction of time and the orientation of 3-space. Acting on vectors in spacetime, the proper Lorentz group has 6 orbits: the future light cone (M=0, t>0), the future timelike vectors (M<0, t>0, which lie inside the future light cone), the past light cone (M=0, t<0), the past timelike vectors (M<0, t<0), the spacelike vectors (M>0), and the origin. The usual Lorentz boost is [t'] = [cosh b sinh b] [t] [x'] [sinh b cosh b] [x] where b is related to the conventional velocity v by tanh b = v/c. The full Lorentz group augments the proper group with space inversion and time reversal. The quotient group FLG/PLG is the four-group {1, P, T, PT}. In the curved spacetime of general relativity, the Lorentz group acts on the tangent spaces at each point. Each small neighborhood has a time orientation, the future light cone direction. Physical influences can propagate only along paths that lie on or within the future light cone; that is the principle of causality. Time orientation can be transported along paths throughout the spacetime manifold. One can imagine that the time orientations transported from A to B along two different paths are opposite. This would be troublesome only if the paths are causal. Another way to get into trouble is to have a causal closed loop. I will not attempt to speculate about what one might observe if a visitor from a faraway part of the universe arrived here with time reversed orientation. One can hope that these scenarios are forbidden by some consequence of quantum theory, either of the matter present in the spacetime (without which there could be no observers), or of spacetime itself (i.e. quantum gravity). The geometry of spacetime is dynamical. The viewpoint 30 years ago, at the time of publication of Misner, Thorn, and Wheeler, "Gravitation", was a Hamiltonian formulation in which the position variables are the metric of a 3-dimensional spacelike slice, and the momentum variables describe the evolution of the slice. These slices stack together to form the 4-dimensional spacetime obeying Einstein's equations in tensor form. These dynamics are time reversible. While it is true that the time reversal of a black hole satisfies the equations of general relativity, such an object would gravitationally repel, and the m parameter, which normally is the mass of the black hole, would be negative. The actual mass of the white hole would be -m > 0. Whether white holes could actually exist, and what sort of stuff they are compelled to emit, are questions of quantum theory. An observer falling past the event horizon does not get to view the entire infinite future of any part of the external universe. Having set up his computer simulation, he will only see a finite number of cycles before reaching the end of time at the singularity. -- Gene