mmcirvin


Matt McIrvin's Steam-Operated World of Yesteryear


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Rubber sheet abuse
mmcirvin
That title ought to get me some Google hits.

The silly automatic hyperlink in this NYT article on general relativity and LIGO seems to be gone, but I was irked by the paragraph it appears in:
Imagine a rubber sheet pulled taut horizontally and then tossing a bowling ball and a tennis ball onto it. The heavier bowling ball sinks deeper, and the tennis ball will move toward the bowling ball not because of a direct attraction between the two, but because the tennis ball rolls into the depression around the bowling ball.
This use of the rubber-sheet metaphor to describe gravity in general relativity shows up a lot in popularizations and especially in news articles. It frequently leads people to conclude that general relativity is a stupid theory, because it's completely circular. Why did the tennis ball roll into the depression? Because gravity's pulling it downward. But isn't gravity what we're trying to explain? Does this mean that general relativity explains gravity as the consequence of some super-gravity pulling in a higher dimension?

Actually there is no super-gravity involved; it's just a bad metaphor. The rubber sheet only shows you what happens to space, but in general relativity, it's space-time that actually gets distorted, and it's the time part that is responsible for gravity in the everyday limit.

What this rubber-sheet picture is good for is to explain certain deviations from Newtonian gravity in general relativity: why, for instance, the deflection of starlight past the Sun is twice what you would calculate from a naive Newtonian treatment in which the photons are just particles traveling at the speed of light and responding to gravity. The other half of the effect comes from distorted spatial geometry. But you don't need to invoke super-gravity pulling downward on the photons to see this.

I must have had the rubber-sheet thing explained to me by a genius or I was just plain unimaginative, because that particular way of looking at the rubber sheet metaphor -- some sort of supergravity pulls things toward the dents in the sheet causing gravity -- had never occurred to me before reading this...

Wow, like wow, I never even thought of thinking like that. Wow!

My, doubtless differently incorrect interpretation, was always that mass warped your spacetime simply by definition -- as represented by the rubber sheet and heavy ball -- therefore all straight lines were no longer straight, and thus when your small mass -- the lighter ball -- proceeding at a constant velocity in a straight line, as per Newton, it went in a curve (which is not really a curve but still a straight line topologically), therefore you get your orbits and falling towards stuff. Thus General Relativity was pretty much Newtons Laws of Motion in a funkier mass-influenced geometry.

...Which always seemed pretty simple stuff, much more comprehensible than Special Relativity which involved bouncing light clocks on trains and platforms maintained by mysterious animated men with bushy beards, possibly because I was less than ten at the time of the explanation, and therefore didn't have to do any of the math or work out what it implied for anything :)

That is the right version! But since it only works if spacetime rather than space is the thing being distorted, it does not lend itself well to visualization.

Though there's a diagram in Misner, Thorne and Wheeler that gets it across pretty well. If you toss a ball into the air at different speeds, it traces different curves through space. But when you take into account that the ball's path slants through the time dimension at 300,000 kilometers per second, suddenly the curvatures in spacetime all look the same, and the idea that the ball is responding to geometry is not so implausible.

Hmmm, I wonder why that never bothered me.

My hazy and hole filled knowledge about this sort of stuff comes from the era when The Open University hijacked BBC2 for 8 hours a day starting at 6am, every Saturday and Sunday, to show the TV programs for the undergraduate courses. All filled with bearded professors in flares with boards full of equations illustrated by animations of the periodic table falling into blackholes shaped like klein bottles, with hairy balls and toruses being brushed by Jackson Pollock to illustrate tologopical concepts. Or something like that. I can still hum the theme tune to 'Graphs Networks and Design MT365' and visualise the small silver model ships firing their cannons, even if I still don't know which classical chamber music piece it was, and what *precisely* naval battles have to do with Graphs, Networks and Design is now very very hazy.

Also: the evolution of Warfarin resistance in rats, I remember that being a good one.

I must have been about 10 at the time, so I mainly skipped the ones that had the dramatised business case studies for the MBA courses.

Alas, they don't hijack BBC2 for huge lumps of time anymore in .uk. At some point I think they suddenly realised that TV programming was valuable and you could actual charge TV companies for it, so no free ride for BBC2 anymore. I think they send the students videos or DVDs or something nowdays. There ought to be a law, I tell you!

Since these were important programs that all the students taking each course had to see, and video records were not ubiquitous, they were all repeated a lot in no particular order, so people could catch the program if they'd missed it before.

So I must have seen various shows involving the rubber sheet lots and lots of times; and I also probably saw the one where one professor fell into a black hole whilst the other professor stood around at a safe distance whilst they observed each others time dilation effects frequently enough before any individual rubber sheet explanation that I'm thinking 'yes, well it has be spacetime doesn't it, otherwise you wouldn't get the time dilation to work properly now would you?'

I suspect that's completely pedalogically the wrong way round. You're supposed to do the rubber sheets and then fall into the black hole :)

Also the rubber sheets -- I think they were even cutting edge *computer animated* rubber sheets in full 8-bit glory, with the Prof. badly chromakeyed over the top -- always, always had straight lines painted on them, so you could see the straight lines bend when the weights got put on.

Also the sprinkling of courses on topology and group theory probably helped, tho' the ones with Jackson Pollock and Monet probably weren't directly relevant :)

[unspellchecked because the livejournal spellchecker seems to be taking forever]

There was a nice pastiche of Seventies Open University programming in the first episode of "Life on Mars".