July 2009 - mathfun@mailman.xmission.com

Re: [math-fun] dumb question re GR -- 2-body simulators on the web?
by Eugene Salamin 27 Jul '09

27 Jul '09

This message was sent only to the original poster, so I'm resending it to math-fun. From: Henry Baker <hbaker1(a)pipeline.com> To: Eugene Salamin <gene_salamin(a)yahoo.com> Cc: math-fun <math-fun(a)mailman.xmission.com> Sent: Sunday, July 26, 2009 2:32:34 PM Subject: Re: [math-fun] dumb question re GR -- 2-body simulators on the web? Sounds like the evolution of the banking system. :-) I watched the movie you referenced, but the downloadable version was at too low a resolution & went by too fast to get any better understanding of what was going on. I assume that you mean that _any_ configuration of exactly 2 black holes -- regardless of size ratio, energy & angular momentum -- will eventually merge into one black hole, with the emission of gravitational radiation. [Yes, that is correct. The radiated energy is supplied by increased gravitational binding.] Even dumber question: what eventually happens to the resulting black hole & its gravitational radiation? In the classical GR case, is this an infinite universe? [Assuming the initial conditions to be asymptotically flat space, the same will be true after the merger. The resulting black hole is an ordinary black hole with mass and angular momentum, while the radiation propagates off to infinity. -- Gene] At 02:17 PM 7/26/2009, Eugene Salamin wrote: >The second problem, that of two black holes, is nontrivial, and is a topic of current research. There does not exist a static solution since there will be gravitational radiation, leading to the eventual merger of the black holes. Google search for "black hole merger". The first hit is a movie that may possibly be helpful.

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Re: [math-fun] dumb question re GR -- 2-body simulators on the web?
by Henry Baker 26 Jul '09

26 Jul '09

I'd like to redo Newton's "cannon on a mountain top" thought experiment again, except using GR instead of Newtonian gravity. If you recall, the cannon is on a mountain top on the Earth & it fires a cannonball at various velocities parallel to the Earth's surface. With enough velocity, the cannonball "falls", but not fast enough to actually hit the Earth's surface, and therefore comes all the way around the Earth & its path becomes an orbit. I'd be interested in finding a GR 2-body simulator that handles 1 black hole plus a particle of very small mass, or 2 black holes of various masses -- i.e., it should be capable of handling extremely large gravitational forces. Since there are only 2 bodies, we need only 2 space dimensions plus time. With only 2 bodies, we could also utilize the 3rd dimension to show distortion of spacetime. We can center the picture at the center of mass. I don't know whether it's possible or reasonable to utilize a frame of reference that rotates with angular momentum. We should be able to scale the various space & time dimensions to watch interesting behaviors. I did a Google search, but was unable to find such a simulator. Does anyone know if such a GR 2-body simulator exists on the web?

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[math-fun] dumb question about general relativity
by Henry Baker 26 Jul '09

26 Jul '09

This question is about "classical" "infinitesimal" GR -- i.e., no quantum stuff like Hawking radiation & no Planck length. The laws of GR are symmetric w.r.t. time, so we have the following conundrum: If an object falls towards a black hole, from the perspective of an outside observer, time on the object gets slower & slower & in the limit it stops. I've been told that as a result of this time dilation, everything that "falls into" the black hole doesn't really fall in, but ends up getting painted onto its surface (from the perspective of the outside observer). But if the laws are truly symmetric w.r.t. time, then it should be possible for the black hole to "burp" out objects, as well. I've read expositions of what happens to an observer _on_ the object falling into the black hole, and time supposedly doesn't stop. Also, if the black hole is large enough, then the tidal forces near the surface aren't large enough to destroy even a human observer, so one can ask the question about what such an observer would see. All of this presupposes some sort of invertible transform in going from the perspective of the outside observer to the observer on the moving object. But clearly, such an invertible transform has a singularity at the "time" when the object gets painted onto the black hole (from the outside observer's perspective). Is there an analogy here with Taylor & Laurent series in the complex plane, where you can extend a function beyond its usual convergence by utilizing a different type of series?

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Re: [math-fun] dumb question about general relativity
by Henry Baker 26 Jul '09

26 Jul '09

My original question regarded "classical" GR, where interchanging "positive" and "negative" time doesn't change any laws. I.e., if you take a GR movie and simply run it backwards, it still agrees with all the GR laws. So, you don't need any notion of "white holes" -- I'm still talking about regular old "black" holes. So, nothing in these laws keeps such a black hole from burping an object out into space. We can ask what initial conditions would enable such a burp & may complain that they are highly unlikely, but they aren't ruled out by the laws themselves. Now I'm curious about what this implies for the "classical" energy, momentum, etc., of the object being burped. At 03:40 PM 7/24/2009, Dan Asimov wrote: >Re time reversal: > >The fact that in our neighborhood in spacetime we see stars radiating >only in positive time -- and all the entropic consequences of this -- is >usually attributed to initial conditions, not to any actual "law" of physics. > >(The Ehrenfest urn model is usually considered a good microcosm >of thermodynamics, and shows time reversal symmetry.) > >It's just a guess, but it seems likely to me that we are aware of only a small >or infinitesimal fraction of the universe, and that some other time and place >have stars radiating in negative time. > >(Or is it possible that entropically opposite parts of the universe somehow >always end up pinching off from each other by black holes?) > >--Dan

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Re: [math-fun] binary operation seeks symbol, name
by James Propp 26 Jul '09

26 Jul '09

I agree with Erich that "smin" and "smax" are fun to say, but I think I will follow Joerg's suggestion and write the operations as min^- and max^+ (and not give them names at all). As for Henry's request >Tell us more about the interesting properties of these operators. the operations come up in the following context: Suppose we know that m and n are distinct integers, where m lies in the interval [a,b] and n lies in the interval [c,d]. Then max(m,n) lies in the interval [max^+(a,c), max(b,d)] while min(m,n) lies in the interval [min(a,c), min^-(b,d)]. Jim

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[math-fun] Short story related to math
by Dan Asimov 25 Jul '09

25 Jul '09

The current (August, 2009) issue of Harper's magazine has a short story related to math, which I haven't read yet. It's "Too Much Happiness" by Alice Munro, 20 pages long. At the very beginning is this quote: "Many persons who have not studied mathematics confuse it with arithmetic and consider it a dry and arid science. Actually, however, this science requires great fantasy." --Sophia Kovalevsky --Dan P.S. When you finish that, you might want to tackle the (cryptic) crossword, which I've edited since 1996. For how to solve cryptics, see for example < http://www3.sympatico.ca/tagies/cryptics/solvingguide/solvingguide.html >. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] dumb question about general relativity
by Dan Asimov 25 Jul '09

25 Jul '09

Re time reversal: The fact that in our neighborhood in spacetime we see stars radiating only in positive time -- and all the entropic consequences of this -- is usually attributed to initial conditions, not to any actual "law" of physics. (The Ehrenfest urn model is usually considered a good microcosm of thermodynamics, and shows time reversal symmetry.) It's just a guess, but it seems likely to me that we are aware of only a small or infinitesimal fraction of the universe, and that some other time and place have stars radiating in negative time. (Or is it possible that entropically opposite parts of the universe somehow always end up pinching off from each other by black holes?) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] dumb question about general relativity
by Dan Asimov 25 Jul '09

25 Jul '09

No big mystery. I'm just using "positive" time to mean what it usually means: the direction of time that our clocks go forward. --Dan << Re time reversal: The fact that in our neighborhood in spacetime we see stars radiating only in positive time -- and all the entropic consequences of this -- is usually attributed to initial conditions, not to any actual "law" of physics. (The Ehrenfest urn model is usually considered a good microcosm of thermodynamics, and shows time reversal symmetry.) It's just a guess, but it seems likely to me that we are aware of only a small or infinitesimal fraction of the universe, and that some other time and place have stars radiating in negative time. [What exactly does this mean? How do you experimentally determine whether your stars are radiating in positive or negative time? >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] CandyFab 6000: Yum!
by Robert Baillie 25 Jul '09

25 Jul '09

http://threesixty360.wordpress.com/2009/06/18/calculus-demonstration-3d-pri… I thought you all might enjoy this!

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[math-fun] binary operation seeks symbol, name
by James Propp 24 Jul '09

24 Jul '09

Has anyone encountered the operation { min(x,y) if x neq y, f(x,y) = { { x+1 if x = y before, or the three related operations in which min is replaced by max and/or +1 is replaced by -1? These operations have come up in my work, and I need symbols (and possibly names) for a couple of them; I'm guessing I'm not the first person to stumble upon them. E.g., note that if x, y, and z are the highest powers of 2 dividing integers a, b, and c that sum to zero, then z = f(x,y). But the two I need names and symbols for are actually { min(x,y) if x neq y, x-1 if x = y } and { max(x,y) if x neq y, x+1 if x = y }. (In my writeup-in-progress I've been calling them "smin" and "smax", but I no longer recall why I chose "s"; for "special", maybe?) Jim Propp

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