to a real girl, but still not there
It started with the Babylonians. The Greeks abhorred the notion. The Egyptians and Romans couldn’t have gotten along without it. Only 1600 years later did Newton gave final polishing to … The Method of Successive Approximations.
Stay with me, we’ll get to The Chicken soon.
Suppose for some weird reason you wanted to know the square root of 2701. Any Babylonian could see immediately that 2701 is a bit less than 3600 = 602, so as a first approximation they’d guess ½(60 + (2701/60)) = 52.5. They’d do the multiplication to check: 52.5×52.5 = 2756.25.
Well, 52.5 is closer than 60 but not close enough. So they’d plug that number into the same formula to get the next successive approximation: ½(52.5 + 2701/52.5) = 51.97. Check it: 51.97×51.97 = 2700.88. That was probably good enough for government work in Babylonia, but if the boss wanted an even better estimate they could go around the loop again.
Scientists and engineers tackle a complex problem piecewise. Start by looking for a simple problem you know how to solve. Adjust that solution little by little to account for the ways in which the real system differs from the simple case. Successive Approximation is only one of many adjustment strategies invented over the centuries.
The most widely-used technique is called Perturbation Theory (which has nothing to do with the ways kids find to get on their parents’ nerves). The strategy is to find some single parameter, maybe a ratio of two masses or the relative strength of a particle-particle interaction. For a realistic solution, it’s important that the parameter’s value be small compared to other quantities in the problem.
Simplify the original problem by keeping that parameter in the equations but assume that it’s zero. When you’ve found a solution to that problem, you “perturb” the solution — you see what happens to the model when you allow the parameter to be non-zero.
There’s an old story, famous among physicists and engineers, about an association of farmers who wanted to design an optimum chicken-raising operation. Maybe with an optimal chicken house they could heat the place with the birds’ own body heat, things like that. They called in an engineering consultant. He looked around some running farms, took lots of measurements, and went away to compute. A couple of weeks later he came back, with slides. (I told you it’s an old story.) He started to walk the group though his logic, but he lost them when he opened his pitch with, “Assume a spherical chicken…”
Fat Chicken Bank by Becky Zee
Now, he may actually have been on the right track. It’s a known fact that many biological processes (digestion, metabolism, drug dosage, etc.) depend on an organism’s surface area. A chicken’s surface area could be key to calculating her heat production. But chickens (for example, our charming Henrietta) have a complicated shape with a poorly-defined surface area. The engineer’s approximation strategy must have been to estimate each bird as a sphere with a tweakable perturbation parameter reflecting how spherical they aren’t.
Then, of course, he’d have to apply a second adjustment for feathers, but I digress.
Now here’s the thing. In quantum mechanics there’s only a half-dozen generic systems with exact solutions qualifying them to be “simple” Perturbation Theory starters. Johnny’s beloved Particle In A Box (coming next week) is one of them. The others all depend in similar logic — the particle (there’s always only one of them) is confined to a region which contains places where the particle’s not allowed to be. (There’s one exception: the Free Particle has no boundaries and therefore is evenly smeared across the Universe.)
Virtually all other quantum-based results — multi-electron atoms, molecular structures, Feynman diagrams for sub-atomic physics, string theories, whatever — depend on Perturbation Theory. (The exceptions are topology and group-theory techniques that generally attempt to produce qualitative rather quantitative predictions.) They need those tweakable parameters.
In quantum-chemical calculations the perturbation parameters are generally reasonably small or at least controllable. That’s not true for many of the other areas. This issue is especially problematic for string theory. In many of its proposed problem solutions no-one knows whether a first-, second- or higher-level approximation even exists, much less whether it would produce reasonable predictions.
I find that perturbing.
~~ Rich Olcott
I was only 10 years old but already had Space Fever thanks to Chesley Bonestell’s artwork in Collier’s and Life magazines. I eagerly joined the the movie theater ticket line to see George Pal’s Destination Moon. I loved the Woody Woodpecker cartoon (it’s 12 minutes into the 
Their common experimental strategy sounds simple enough — compare two beams of light that had traveled along different paths
Of all the wave varieties we’re familiar with, gravitational waves are most similar to (NOT identical with!!) sound waves. A sound wave consists of cycles of compression and expansion like you see in this graphic. Those dots could be particles in a gas (classic “sound waves”) or in a liquid (sonar) or neighboring atoms in a solid (a xylophone or marimba).
Einstein noticed that implication of his Theory of General Relativity and in 1916 predicted that the path of starlight would be bent when it passed close to a heavy object like the Sun. The graphic shows a wave front passing through a static gravitational structure. Two points on the front each progress at one graph-paper increment per step. But the increments don’t match so the front as a whole changes direction. Sure enough, three years after Einstein’s prediction, Eddington observed just that effect while watching a total solar eclipse in the South Atlantic.
We’re being dynamic here, so the simulation has to include the fact that changes in the mass configuration aren’t felt everywhere instantaneously. Einstein showed that space transmits gravitational waves at the speed of light, so I used a scaled “speed of light” in the calculation. You can see how each of the new features expands outward at a steady rate.
The second question is harder. The best the aLIGO team could do was point to a “banana-shaped region” (their words, not mine) that covers about 1% of the sky. The team marshaled a world-wide collaboration of observatories to scan that area (a huge search field by astronomical standards), looking for electromagnetic activities concurrent with the event they’d seen. Nobody saw any. That was part of the evidence that this collision involved two black holes. (If one or both of the objects had been something other than a black hole, the collision would have given off all kinds of photons.)
In contrast, a LIGO facility is (roughly speaking) omni-directional. When a LIGO installation senses a gravitational pulse, it could be coming down from the visible sky or up through the Earth from the other hemisphere — one signal doesn’t carry the “which way?” information. The diagram above shows that situation. (The “chevron” is an image of the LIGO in Hanford WA.) Models based on the signal from that pair of 4-km arms can narrow the source field to a “banana-shaped region,” but there’s still that 180o ambiguity.
The great “if only” is that the VIRGO installation in Italy was not recording data when the Hanford WA and Livingston LA saw that September signal. With three recordings to reconcile, the aLIGO+VIRGO combination would have had enough information to slice that banana and localize the event precisely.
Nonetheless, mathematicians and cryptographers have forged ahead, calculating π to more than a trillion digits. Here for your enjoyment are the 99 digits that come after digit million….
Back to π. The Greeks knew that the circumference of a circle (c) divided by its diameter (d) is π. Furthermore they knew that a circle’s area divided by the square of its radius (r) is also π. Euclid was too smart to try calculating the area of the visible sky in his astronomical work. He had two reasons — he didn’t know the radius of the horizon, and he didn’t know the height of the sky. Later geometers worked out the area of such a spherical cap. I was pleased to learn that π is the ratio of the cap’s area to the square of its chord, s2=r2+h2.
Astrophysicists and cosmologists look at much bigger figures, ones so large that curvature has to be figured in. There are three possibilities
We can investigate things that take longer than an instrument’s characteristic time by making repeated measurements, but we can’t use the instrument to resolve successive events that happen more quickly than that. We also can’t resolve events that take place much closer together than the instrument’s characteristic length.
A wave happens in a system when a driving force and a restoring force take turns overshooting an equilibrium point AND the away-from-equilibrium-ness gets communicated around the system. The system could be a bunch of springs tied together in a squeaky old bedframe, or labor and capital in an economic system, or the network of water molecules forming the ocean surface, or the fibers in the fabric of space (whatever those turn out to be).
An isolated black hole is surrounded by an intense gravitational field and a corresponding compression of spacetime. A pair of black holes orbiting each other sends out an alternating series of tensions, first high, then extremely high, then high…
Almost a century later, James Clerk Maxwell (the bearded fellow at left) wrote down his electromagnetism equations that explain how light works. Half a century later, Einstein did the same for gravity.
Gravitodynamics is completely unlike electrodynamics. Gravity’s transverse “force” doesn’t act to move a whole mass up and down like Maxwell’s picture at left. Instead, as shown by Einstein’s picture, gravitational waves stretch and compress while leaving the center of mass in place. I put “force” in quotes because what’s being stretched and compressed is space itself. See 
Hard Science Ain't Hard 
