Tag: Newton

Would the CIA want a LIGO?

On By rolcottIn Astronomy: Techniques, Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Electromagnetism, Physics: Fundamentals, Physics: Light and Relativity, UncategorizedLeave a comment

So I was telling a friend about the LIGO announcement, going on about how this new “device” will lead to a whole new kind of astronomy.  He suddenly got a far-away look in his eyes and said, “I wonder how many of these the CIA has.”

The CIA has a forest of antennas, but none of them can do what LIGO does.  That’s because of the physics of how it works, and what it can and cannot detect.  (If you’re new to this topic, please read last week’s post so you’ll be up to speed on what follows.  Oh, and then come back here.)

There are remarkable parallels between electromagnetism and gravity.  The ancients knew about electrostatics — amber rubbed by a piece of cat fur will attract shreds of dry grass.  They certainly knew about gravity, too.  But it wasn’t until 100 years after Newton wrote his Principia that Priestly and then Coulomb found that the electrostatic force law, F = ke·q1·q2 / r2, has the same form as Newton’s Law of Gravity, F = G·m1·m2 / r2. (F is the force between two bodies whose centers are distance r apart, the q‘s are their charges and the m‘s are their masses.)

Jim and AlAlmost 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.

But interesting as the parallels may be, there are some fundamental differences between the two forces — fundamental enough that not even Einstein was able to tie the two together.

One difference is in their magnitudes.  Consider, for instance, two protons.  Running the numbers, I found that the gravitational force pulling them together is a factor of 1036 smaller than the electrostatic force pushing them apart.  If a physicist wanted to add up all the forces affecting a particular proton, he’d have to get everything else (nuclear strong force, nuclear weak force, electromagnetic, etc.) nailed down to better than one part in 1036 before he could even detect gravity.

But it’s worse — electromagnetism and gravity don’t even have the same shape.

Electromagneticwave3D
Electric (red) and magnetic (blue) fields in a linearly polarized light wave
(graphic from WikiMedia Commons, posted by Lookang and Fu-Kwun Hwang)

A word first about words.  Electrostatics is about pure straight-line-between-centers (longitudinal) attraction and repulsion — that’s Coulomb’s Law.  Electrodynamics is about the cross-wise (transverse) forces exerted by one moving charged particle on the motion of another one.  Those forces are summarized by combining Maxwell’s Equations with the Lorenz Force Law.  A moving charge gives rise to two distinct forces, electric and magnetic, that operate at right angles to each other.  The combined effect is called electromagnetism.

The effect of the electric force is to vibrate a charge along one direction transverse to the wave.  The magnetic force only affects moving charges; it acts to twist their transverse motion to be perpendicular to the wave.  An EM antenna system works by sensing charge flow as electrons move back and forth under the influence of the electric field.

Gravitostatics uses Newton’s Law to calculate longitudinal gravitational interaction between masses.  That works despite gravity’s relative weakness because all the astronomical bodies we know of appear to be electrically neutral — no electrostatic forces get in the way.  A gravimeter senses the strength of the local gravitostatic field.

Maxwell and EinsteinGravitodynamics 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 this video for a helpful visualization of a gravitational wave.

LIGO is neither a telescope nor an electromagnetic antenna.  It operates by detecting sudden drastic changes in the disposition of matter within a “small” region.  In LIGO’s Sept 14 observation, 1031 kilograms of black hole suddenly ceased to exist, converted to gravitational waves that spread throughout the Universe.  By comparison, the Hiroshima explosion released the energy of 10-6 kilograms.

Seismometers do a fine job of detecting nuclear explosions.  Hey, CIA, they’re a lot cheaper than LIGO.

~~ Rich Olcott

The Force(s) of Geometry

On By rolcottIn Mathematics, Physics: Fundamentals, UncategorizedLeave a comment

There’s a lot more to Geometry than congruent triangles.  Geometry can generate hurricanes and slam you to the floor.

It all starts (of course) with Newton.  His three laws boil down to

Effect is to Cause as Change of Motion is to Force.

They successfully account for the physical movement of pretty much everything bigger than an atom.  But sometimes the forces are a bit weird and it takes Geometry to understand them.

Side forceFor instance, suppose Fred and Ethel collaborate on a narwhale research project.  Fred is based in San Diego CA and Ethel works out of Norfolk VA. They fly to meet their research vessel at the North Pole. Fred’s plane follows the green track, Ethel’s plane follows the yellow one.  At the start of the trip, they’re on parallel paths going straight north (the dotted lines).  After a few hours, though, Ethel notices the two planes pulling closer together.

Ethel calls on her Newton knowledge to explain the phenomenon.  “It can’t be Earth’s gravity moving us together, because that force points down to Earth’s center and this is a sideways motion.  Our planes each weigh about 2000 kilograms and we’re still 2,000 kilometers apart.  By Newton’s F = G m1m2/r2 equation, the gravitational force between us should be (6.7×10-11 N m2/kg2) x (2000 kg) x (2000 kg) / (2,000 m)2 = 6.7×10-11 newtons, way too small to account for our speed of approach.  Both planes were electrically grounded when we fueled up, so we’re both carrying a neutral electric charge and it can’t be an electrostatic force.  If it were magnetic my compass would be going nuts and it’s not.  Woo-hoo, I’ve discovered a new kind of force!”

See what I did there?  Fred and Ethel would have stayed a constant distance apart if Earth were a cylinder.  Parallel lines running up a cylinder never meet.  But Earth is a sphere, not a cylinder.  Any pair of lines on a sphere must meet, sooner or later.  Ethel’s “sideways force” is a product of Geometry.

Sandy
Images extracted from NOAA’s SOS Explorer app, available from sos.noaa.gov

Hurricanes, too.  This video shows a day in the life of Hurricane Sandy.  Weather geeks will find several interesting details there, but for now just notice the centers of  counter-clockwise rotation (the one off the Florida coast is Sandy).  Storm centers in the Northern Hemisphere virtually always spin counterclockwise.  Funny thing is, in the Southern Hemisphere those centers go clockwise instead.

The difference has to do with angular momentum.  We could get all formal vector math here, but the easy way is to consider how fast the air is moving in different parts of the world.

We’ve all seen at least one ice show act where skaters form a spinning line. The last skater to join up (usually it’s a short girl) has to push like mad to catch the end of that moving line and everyone applauds her success. Meanwhile the tall girl at the center of the line is barely moving except to fend off dizziness.

YellowknifeThe line rotates as a unit — every skater completes a 360o rotation in the same time. Similarly, everywhere on Earth a day lasts for exactly 24 hours.

Skaters at the end of the line must skate faster than those further in because they have to cover a greater distance in the same amount of time.  The same geometry applies to Earth’s atmosphere.  The Earth is 25,000 miles around at the equator but only 12,500 miles around near the latitude of Whitehorse, Canada.  By and large, a blob of air at the equator must move twice as fast as a blob at 60o north.

chain 2Now suppose our speedy skater hits a slushy patch of ice.  Her end of the line is slowed down, so what happens to the rest of the line?  It deforms — there’s a new center of rotation that forces the entire line to curl around towards the slow spot.  Similarly, that blob near the Equator in the split-Earth diagram curls in the direction of the slower-moving air to its north, which is counter-clockwise.

In the Southern hemisphere, “slower” is southward and clockwise.

If not for Geometry (those differing circle sizes), we wouldn’t have hurricanes.  Or gravity — but that’s another story.

~~ Rich Olcott

Smack-dab in the middle

On By rolcottIn Cosmology, General: Fun Stuff, Mathematics: Dimensions, UncategorizedLeave a comment

BridgeSee that little guy on the bridge, suspended halfway between all the way down and all the way up?  That’s us on the cosmic size scale.

I suspect there’s a lesson there on how to think about electrons and quantum mechanics.

Let’s start at the big end.  The physicists tell us that light travels at 300,000 km/s, and the astronomers tell us that the Universe is about 13.7 billion years old.  Allowing for leap years, the oldest photons must have taken about 4.3×1017 seconds to reach us, during which time they must have covered 1.3×1026 meters.  Double that to get the diameter of the visible Universe, 2.6×1026 meters.  The Universe probably is even bigger than that, but far as I can see that’s as far as we can see.

At the small end there’s the Planck length, which takes a little explaining.  Back in 1899, Max Planck published his epochal paper showing that light happens piecewise (we now call them photons).  In that paper, he combined several “universal constants” to derive a convenient (for him) universal unit of length: 1.6×10-35 meters.  It’s certainly an inconvenient number for day-to-day measurements (“Gracious, Junior, how you’ve grown!  You’re now 8×1034 Planck-lengths tall.”).  However, theoretical physicists have saved barrels of ink and hours of keyboarding by using Planck-lengths and other such “natural units” in their work instead of explicitly writing down all the constants.

Furthermore, there are theoretical reasons to believe that the smallest possible events in the Universe occur at the scale of Planck lengths.  For instance, some theories suggest that it’s impossible to measure the distance between two points that are closer than a Planck-length apart.  In a sense, then, the resolution limit of the Universe, the ultimate pixel size, is a Planck length.

sizelineSo that’s the size range of the Universe, from 1.6×10-35 up to 2.6×1026 meters. What’s a reasonable way to fix a half-way mark between them?

It makes no sense to just add the two numbers together and divide by two the way we’d do for an arithmetic average. That’d be like adding together the dime I owe my grandson and the US national debt — I could owe him 10¢ or $10, but either number just disappears into the trillions.

The best way is to take the geometrical average — multiply the two numbers and take the square root.  I did that.  It’s the X in the sizeline, at 6.5×10-5 meters, or about the diameter of a fairly large bacterium.  (In the diagram, VSC is the Vega Super Cluster, AG is the Andromeda Galaxy, and the numbers are those exponents of 10.)

That’s worth marveling at.  Sixty orders of magnitude between the size of the Universe and the size of the ultimate pixel.  Yet from blue whales to bacteria, Earth’s life just happens to occupy the half-dozen orders right in the middle of the range.  We think that’s it.

Could this be another case of the geocentric fallacy?  Humans were so certain that Earth was the center of the Universe, before Brahe and Galileo and Newton proved otherwise.  Is there life out there at scales much larger or much smaller than we imagine?

Who knows? But here’s an intriguing physics/quantum angle I’d like to promote.  We know a lot about structures bigger than us — solar systems and binary stars and galaxy clusters on up.  We know a few sizes and structures a bit smaller — viruses and molecules and atoms.  We’re aware of quarks and gluons that reside inside protons and atomic nuclei, but we don’t know their size or structure.

Even a proton is huge on the Planck-length scale.  At 1.8×10-15 meters the proton measures some 1020 Planck-lengths.  There’s as much scale-space between the Planck-length and the proton as there is between the Earth (1.3×107 meters) and the Universe.

It’s hard to believe that Terra infravita’s area has no structure whereas Terra supravita is so … busy.  The Standard Model’s “ultimate particles,” the electrons and photons and neutrinos and quarks and gluons, all operate down there somewhere.   It’s reasonable to suppose that they reflect a deeper architecture somewhere on the way down to the Planck-length foam.

Newton wrote (in Latin), “I do not make hypotheses.”  But golly, it’s tempting.

~~ Rich Olcott

Sir Isaac, The Atom And The Whirlpool

On By rolcottIn Physics: Fundamentals, Physics: Newton and GravityLeave a comment

Newton and atomNewton definitely didn’t see that one coming.  He has an excuse, though.  No-one in in the 17th Century even realized that electricity is a thing, much less that the electrostatic force follows the same inverse-square law that gravity does. So there’s no way poor Isaac would have come up with quantum mechanics.

Lemme ‘splain.  Suppose you have a mathematical model that’s good at predicting some things, like exactly where Jupiter will be next week.  But if the model predicts an infinite value under some circumstances, that tells you it’s time to look for a new model for those particular circumstances.

For example, Newton’s Law of Gravity says that the force between two objects is proportional to 1/r2, where r is the distance between their centers of mass.  The Law does a marvelous job with stars and satellites but does the infinity thing when r approaches zero.  In prior posts I’ve described some physics models that supercede Newton’s gravity law at close distances.

Electrical forces are same song second verse with a coda.  They follow the 1/r2 law, so they also have those infinity singularities.  According to the force law, an electron (the ultimate “particle” of negative charge) that approaches another electron would feel a repulsion that rises to infinity.  The coda is that as an electron approaches a positive atomic nucleus it would feel an attraction that rises to infinity.  Nature abhors infinities, so something else, some new physics, must come into play.

I put that word “particle” in quotes because common as the electron-is-a-particle notion is, it leads us astray.  We tend to think of the electron as this teeny little billiard-ballish thing, but it’s not like that at all.  It’s also not a wave, although it sometimes acts like one.  “Wavicle” is just  a weasel-word.  It’s far better to think of the electron as just a little traveling parcel of energy.  Photons, too, and all those other denizens of the sub-atomic zoo.

An electron can’t crumble or leak mass or deform to merge the way that sizable objects can.  What it does is smear. Quantum mechanics is all about the smear.  Much more about that in later posts.

Newton in whirlpoolIf Newton loved anything (and that question has been discussed at length), he loved an argument.  His battle with Leibniz is legendary.  He even fought with Descartes, who was a decade dead when Newton entered Cambridge.

Descartes had grabbed “Nature abhors a vacuum” from Aristotle and never let it go.  He insisted that the Universe must be filled with some sort of water-like fluid.  He know the planets went round the Sun despite the fluid getting in the way, so he reasoned they moved as they did because of the fluid.

Surely you once played with toy boats in the bathtub.  You may have noticed that when you pulled your arm quickly through the water little whirlpools followed your arm.  If a whirlpool encountered a very small boat, the boat might get caught in it and move in the same direction.  Descartes held that the Solar System worked like that, with the Sun as your arm and the planets caught in Sun-stirred vortices within that watery fluid.

Newton knew that couldn’t be right.  The planets don’t run behind the Sun, they share the same plane.  Furthermore, comets orbit in from all directions.  Crucially, Descartes’ theory conflicted with his own and that settled the matter for Newton.  Much of Principia‘s “Book II” is about motions of and through fluid media.  He laid out there what a trajectory would look like under a variety of conditions.  As you’d expect, none of the paths do what planets, moons and comets do.

From Newton’s point of view, the only use for Book II was to demolish Descartes.  For us in later generations, though, he’d invented the science of hydrodynamics.

Which was a good thing so long as you don’t go too far upstream towards the center of the whirlpool.  As you might expect (or I wouldn’t even be writing this section), Book II is littered with 1/rn formulas that go BLOOIE when the distances get short.  What happens near the center?  That’s where the new physics of turbulence kicks in.

~~ Rich Olcott

Squeezing past Newton’s infinity

On By rolcottIn Astronomy: Planetary Science, Physics: Fundamentals, Physics: Newton and GravityLeave a comment

One of the most powerful moments in musical theater — Philip Quast Quastin his Les Miz role of Inspector Javert, praising the stars for the steadfastness and reverence for law that they signify for him.  The performance is well worth a listen.

Javert’s certitude came from Newton’s sublimely reliable mechanics — the notion that every star’s and planet’s motion is controlled by a single law, F~(1/r2).  The law says that the attractive force between any pair of bodies is inversely proportional to the square of the distance between their centers.  But as Javert’s steel-clad resolve hid a fatal spark of mercy towards Jean Valjean, so Newton’s clockworks hold catastrophe at their axles.

Newton’s gravity law has a problem.  As the distance approaches zero, the predicted force approaches infinity.  The law demands that nearby objects accelerate relentlessly at each other to collide with infinite force, after which their combined mass attracts other objects.  In time, everything must collapse in a reverse of The Big Bang.

Victor Hugo wrote Les Misérables about 180 years after Newton published his Principia.  A decade before Hugo’s book, Professeur Édouard Roche (pronounced rōsh) solved at least part of Newton’s problem.

Roche realized that Newton had made an important but crucial simplification.  Early in the Principia, he’d proven that for many purposes you can treat an entire object as though all of its mass were concentrated at a single point (the “center of mass”).  But in real gravity problems every particle of one object exerts an attraction for every particle of the other.

That distinction makes no difference when the two objects are far apart.  However, when they’re close together there are actually two opposing forces in play:

  • gravity, which preferentially affects the closest particles, and
  • tension, which maintains the integrity of each structure.
contact_binary_1
Binary star pair demonstrating Roche lobes, image courtesy of Cronodon.com

Roche noted that the gravity fields of any pair of objects must overlap.  There will always be a point on the line between them where a particle will be tugged equally in either direction.  If two bodies are close and one or both are fluid (gases and plasmas are fluid in this sense), the tension force is a weak competitor.  The partner with the less intense gravity field will lose material across that bridge to the other partner. Binary star systems often evolve by draining rather than collision.

Now suppose both bodies are solid.  Tension’s game is much stronger.  Nonetheless, as they approach each other gravity will eventually start ripping chunks off of one or both objects.  The only question is the size of the chunks — friable materials like ices will probably yield small flakes, as opposed to larger lumps made from silicates and other rocky materials.  Roche described the final stage of the process, where the less-massive body shatters completely.  The famous rings of Saturn and the less famous rings of Neptune, Uranus and Jupiter all appear to have been formed by this mechanism.

Roche was even able to calculate how close the bodies need to be for that final stage to occur. The threshold, now called the Roche Limit, depends on the size and mass of each body. You can get more detail here.

Klingon3And then there’s spaghettification.  That’s a non-relativistic tidal phenomenon that occurs near an extremely dense body like a neutron star or a black hole.  Because these objects pack an enormous amount of mass into a very small volume, the force of gravity at a close-in point is significantly greater than the force just a little bit further out. Any object, say a Klingon Warbird that ignored peril markings on a space map (Klingons view warnings as personal challenges), would find itself stretched like a noodle between high gravity on the side near the black hole and lower gravity on the opposite side.  (In this cartoon, notice how the stretching doesn’t care which way the pin-wheeling ship is pointed.)

Nature abhors singularities.  Where a mathematical model like Newton’s gravity law predicts an infinity, Nature generally says, “You forgot something.”  Newton assumed that objects collide as coherent units.  Real bodies drain, crumble, or deform to slide together.  Look to the apparent singularities to find new physics.

~~ Rich Olcott

The direction Newton avoided facing

On By rolcottIn Physics: Fundamentals, Physics: Newton and Gravity, UncategorizedLeave a comment

Reading Newton’s Philosophiæ Naturalis Principia Mathematica is less challenging than listening to Vogon poetry.  You just have to get your head working like a 17th Century genius who had just invented Calculus and who would have deep-fried his right arm in rancid skunk oil before he’d admit to using any of his rival Leibniz’ math notations or techniques.

Newton II-II ellipseNewton was essentially a geometer. These illustrations (from Book 1 of the Principia) will give you an idea of his style.  He’d set himself a problem then solve it by constructing sometimes elaborate diagrams by which he could prove that certain components were equal or in strict proportion.

Newton XII-VII hyperbolaFor instance, in the first diagram (Proposition II, Theorem II), we see an initial glimpse of his technique of successive approximation.  He defines a sequence of triangles which as they proliferate get closer and closer to the curve he wants to characterize.

The lines and trig functions escalate in the second diagram (Prop XII, Problem VII), where he calculates the force  on a body traveling along a hyperbola.

Newton XLIV-XIV precessionThe third diagram is particularly relevant to the point I’ll finally get to when I get around to it.  In Prop XLIV, Theorem XIV he demonstrates something weird.  Suppose two objects A and B are orbiting around attractive center C, but B is moving twice as fast as A.  If C exerts an additional force on B that is inversely dependent on the cube of the B-C distance, then A‘s orbit will be a perfect circle (yawn) but B‘s will be an ellipse that rotates around C, even though no external force pushes it laterally.

In modern-day math we’d write the additional force as F∼(1/rBC3), but Newton verbalized it as “in a triplicate ratio of their common altitudes inversely.”  See what I mean about Vogon poetry?

Now, about that point I was going to get to.  It’s C, in the center of that circle.  If the force is proportional to 1/r3, what happens when r approaches zero?  BLOOIE, the force becomes infinite.

In the previous post we used geometry to understand the optical singularity at the center of the Christmas ball.  I said there that my modeling project showed me a deeper reason for a BLOOIE.  That reason showed up partway through the calculation for the angle between the axis and the ring of reflected  light.  A certain ratio came out to be (1-x)/2x, where x is proportional to the distance between the LED and the ball’s center.  Same problem: as the LED approaches the center, x approaches zero and BLOOIE.  (No problem when x is one, because the ratio is 0/2 which is zero which is OK.)

Singularities happen when the formula for something goes to infinity.

Now, Newton recognized that his central-force (1/rn)-type equations covered gravity and magnetism and even the inward force on the rim of a rotating wheel.  It’s surprising that he didn’t seem too worried about BLOOIE.

I think he had two excuses.  First, he was limited by his graphical methodology.  In most of his constructions, when a certain distance goes to zero there’s a general catastrophe — rectangles and triangles collapse to lines or even points, radii whirl aimlessly without a vertex to aim at…  His lovely derivations devolve into meaninglessness.  Further advances would depend on the  algebraic approach to Calculus taken by the detested Leibniz.

Second (here’s the hook for this post’s title), Newton was looking outward, not inward.  He was considering the orbits of planets and other sizable objects.  r is always the distance between object centers.  For sizable objects you don’t have to worry about r=0 because “center-to-center equals zero” never occurs.  If the Moon (radius 1080 miles) were to drop down to touch the Earth (radius 3960 miles), their centers would still be 5000 miles apart.  No BLOOIE.

Actually, there would be CRUMBLE instead of BLOOIE because a different physical model would apply — but that’s a tale for another post.

The moral of the story is this.  Mathematical models don’t care about infinities, but Nature does.  Any conditions where the math predicts an infinite value (for instance, where a denominator can become zero) are prime territory for new models that make better predictions.

~~ Rich Olcott

Circular Logic

On By rolcottIn General: Fun Stuff, Mathematics, Uncategorized2 Comments

We often read “singularity” and “black hole” in the same pop-science article.  But singularities are a lot more common and closer to us than you might think. That shiny ball hanging on the Christmas tree over there, for instance.  I wondered what it might look like from the inside.  I got a surprise when I built a mathematical model of it.

To get something I could model, I chose a simple case.  (Physicists love to do that.  Einstein said, “You should make things as simple as possible, but no simpler.”)

I imagined that somehow I was inside the ball and that I had suspended a tiny LED somewhere along the axis opposite me.  Here’s a sketch of a vertical slice through the ball, and let’s begin on the left half of the diagram…Mirror ball sketch

I’m up there near the top, taking a picture with my phone.

To start with, we’ll put the LED (that yellow disk) at position A on the line running from top to bottom through the ball.  The blue lines trace the light path from the LED to me within this slice.

The inside of the ball is a mirror.  Whether flat or curved, the rule for every mirror is “The angle of reflection equals the angle of incidence.”  That’s how fun-house mirrors work.  You can see that the two solid blue lines form equal angles with the line tangent to the ball.  There’s no other point on this half-circle where the A-to-me route meets that equal-angle condition.  That’s why the blue line is the only path the light can take.  I’d see only one point of yellow light in that slice.

But the ball has a circular cross-section, like the Earth.  There’s a slice and a blue path for every longitude, all 360o of them and lots more in between.  Every slice shows me one point of yellow light, all at the same height.  The points all join together as a complete ring of light partway down the ball.  I’ve labeled it the “A-ring.”

Now imagine the ball moving upward to position B.  The equal-angles rule still holds, which puts the image of B in the mirror further down in the ball.  That’s shown by the red-lined light path and the labeled B-ring.

So far, so good — as the LED moves upward, I see a ring of decreasing size.  The surprise comes when the LED reaches C, the center of the ball.  On the basis of past behavior, I’d expect just a point of light at the very bottom of the ball (where it’d be on the other side of the LED and therefore hidden from me).

Nup, doesn’t happen.  Here’s the simulation.  The small yellow disk is the LED, the ring is the LED’s reflected image, the inset green circle shows the position of the LED (yellow) and the camera (black), and that’s me in the background, taking the picture…g6z

The entire surface suddenly fills with light — BLOOIE! — when the LED is exactly at the ball’s center.  Why does that happen?  Scroll back up and look at the right-hand half of the diagram.  When the ball is exactly at C, every outgoing ray of light in any direction bounces directly back where it came from.  And keeps on going, and going and going.  That weird display can only happen exactly at the center, the ball’s optical singularity, that special point where behavior is drastically different from what you’d expect as you approach it.

So that’s using geometry to identify a singularity.  When I built the model* that generated the video I had to do some fun algebra and trig.  In the process I encountered a deeper and more general way to identify singularities.

<Hint> Which direction did Newton avoid facing?

* – By the way, here’s a shout-out to Mathematica®, the Wolfram Research company’s software package that I used to build the model and create the video.  The product is huge and loaded with mysterious special-purpose tools, pretty much like one of those monster pocket knives you can’t really fit into a pocket.  But like that contraption, this software lets you do amazing things once you figure out how.

~~ Rich Olcott

And now for some completely different dimensions

On By rolcottIn Mathematics: Dimensions, Physics: Fundamentals, UncategorizedLeave a comment

Terry Pratchett wrote that Knowledge = Power = Energy = Matter = Mass.  Physicists don’t agree because the units don’t match up.

Physicists check equations with a powerful technique called “Dimensional Analysis,” but it’s only theoretically related to the “travel in space and time” kinds of dimension we discussed earlier.

Place setting LMTIt all started with Newton’s mechanics, his study of how objects affect the motion of other objects.  His vocabulary list included words like force, momentum, velocity, acceleration, mass, …, all concepts that seem familiar to us but which Newton either originated or fundamentally re-defined. As time went on, other thinkers added more terms like power, energy and action.

They’re all linked mathematically by various equations, but also by three fundamental dimensions: length (L), time (T) and mass (M). (There are a few others, like electric charge and temperature, that apply to problems outside of mechanics proper.)

Velocity, for example.  (Strictly speaking, velocity is speed in a particular direction but here we’re just concerned with its magnitude.)   You can measure it in miles per hour or millimeters per second or parsecs per millennium — in each case it’s length per time.  Velocity’s dimension expression is L/T no matter what units you use.

Momentum is the product of mass and velocity.  A 6,000-lb Escalade SUV doing 60 miles an hour has twice the momentum of a 3,000-lb compact car traveling at the same speed.  (Insurance companies are well aware of that fact and charge accordingly.)  In terms of dimensions, momentum is M*(L/T) = ML/T.

Acceleration is how rapidly velocity changes — a car clocked at “zero to 60 in 6 seconds” accelerated an average of 10 miles per hour per second.  Time’s in the denominator twice (who cares what the units are?), so the dimensional expression for acceleration is L/T2.

Physicists and chemists and engineers pay attention to these dimensional expressions because they have to match up across an equal sign.  Everyone knows Einstein’s equation, E = mc2. The c is the velocity of light.  As a velocity its dimension expression is L/T.  Therefore, the expression for energy must be M*(L/T)2 = ML2/T2.  See how easy?

Now things get more interesting.  Newton’s original Second Law calculated force on an object by how rapidly its momentum changed: (ML/T)/T.  Later on (possibly influenced by his feud with Leibniz about who invented calculus), he changed that to mass times acceleration M*(L/T2).  Conceptually they’re different but dimensionally they’re identical — both expressions for force work out to ML/T2.

Something seductively similar seems to apply to Heisenberg’s Area.  As we’ve seen, it’s the product of uncertainties in position (L) and momentum (ML/T) so the Area’s dimension expression works out to L*(ML/T) = ML2/T.

SeductiveThere is another way to get the same dimension expression but things aren’t not as nice there as they look at first glance.  Action is given by the amount of energy expended in a given time interval, times the length of that interval.  If you take the product of energy and time the dimensions work out as (ML2/T2)*T = ML2/T, just like Heisenberg’s Area.

It’s so tempting to think that energy and time negotiate precision like position and momentum do.  But they don’t.  In quantum mechanics, time is a driver, not a result.  If you tell me when an event happens (the t-coordinate), I can maybe calculate its energy and such.  But if you tell me the energy, I can’t give you a time when it’ll happen.  The situation reminds me of geologists trying to predict an earthquake.  They’ve got lots of statistics on tremor size distribution and can even give you average time between tremors of a certain size, but when will the next one hit?  Lord only knows.

File the detailed reasoning under “Arcane” — in technicalese, there are operators for position, momentum and energy but there’s no operator for time.  If you’re curious, John Baez’s paper has all the details.  Be warned, it contains equations!

Trust me — if you’ve spent a couple of days going through a long derivation, totting up the dimensions on either side of equations along the way is a great technique for reassuring yourself that you probably didn’t do something stupid back at hour 14.  Or maybe to detect that you did.

~~ Rich Olcott

The Universe and Werner H.

On By rolcottIn Physics: Quantum Mechanics1 Comment

Heisenberg’s Area ( about 10-34 Joule-second) is small, one ten-millionth of the explosive action in a single molecule of TNT.  OK, that’s maybe important for sub-atomic physics, but it’s way too small to have any implications for anything bigger, right?  Well, it could be responsible for shaping our Universe.

Quick recap: The Heisenberg Uncertainty Principle (HUP) says that certain quantities (for instance, position and momentum) are linked in a remarkable way.  We can’t measure either of them perfectly accurately, but we can make repeated more-or-less sloppy measurements that give us average values.  The linkage is in that sloppiness.  Each repeated measurement lands somewhere in a range of values around the average.  HUP says that even with very careful measurement the product of those two spans must be greater than Heisenberg’s Area.

So now let’s head out to empty space, shall we?  I mean, really empty space, out there between the galaxies, where there’s only about one hydrogen atom per cubic meter.

Here’s a good cubic meter … sure enough, it’s got exactly one hydrogen atom in it.

g25For practice using Heisenberg’s Area, what can we say about the atom? (If you’re checking my math it’ll help to know that the Area, h/4π, can also be expressed as 0.5×10-34 kg m2/s; the mass of one hydrogen atom is 1.7×10-27 kg; and the speed of light is 3×108 m/s.)  On average the atom’s position is at the cube’s center.  Its position range is one meter wide.  Whatever the atom’s average momentum might be, our measurements would be somewhere within a momentum range of (h/4π kg m2/s) / (1 m) = 0.5×10-34 kg m/s. A moving particle’s momentum is its mass times its velocity, so the velocity range is (0.5×10-34 kg m/s) / (1.7×10-27 kg) = 0.3×10-7 m/s.

With really good tools we could determine the atom’s velocity within plus or minus 0.000 000 03 m/s.  Pretty good.

Now zoom in.  Dial that one-meter cube down a billion-fold to a nanometer (10-9 meters, which is still about ten times the atom’s width).  Yeah, the atom’s still in the box, but now its velocity range is 300 m/s.  The atom could be just hanging out at the center, or it could zoom out of the cube a microsecond after we looked — we just can’t tell which.

All of which illuminates the contrast between physics Newton-style and the physics that has bloomed since Einstein’s 1905 “miracle year.”  If Newton were in charge of the Universe, Heisenberg’s Area would be zero.  We could determine that atom’s position and momentum with complete accuracy.  In fact in principle we could accurately determine everything’s position and momentum and then calculate where everything would be at any time in the future.  But he isn’t and it’s not and we can’t.

Theorists and experimenters use the word “measurement” in different ways. A measurement done by a theoretician is generally based on fundamental constants and an Valueselaborate mathematical structure. If the measurement is a quantum mechanical result, part of that structure is our familiar bell-shaped curve.  It’s an explicit recognition that way down in the world of the very small, we can’t know what’s really going on.  Most calculations have to be statistical, predicting an average and an expected range about that average. That prediction may or may not pan out, depending on what the experimentalists find.

By contrast, when experimenters measure something, even as an average of multiple tests, it’s an estimate of the real distribution.  The research group (usually it’s a group these days) reports a distribution that they claim overlaps well with a real one out there in the Universe.  Then another group dives in to prove they or the theoreticians or both are wrong.  That’s how Science works.

You are hereSo there could be a collection of bell-curves gathered about the experimental result. Remember those extra dimensions we discussed earlier?  One theory that’s been floated is that along those extra dimensions the fundamental constants like h might take on different values.  Maybe further along “Dimension W” the value of h is bigger than it is in our Universe, and quantum effects are even more important than they are here.

Now how can we test that?

BTW, Heisenberg will be 114 on Dec 5.  Alles Gute zum Geburtstag, Werner!

~~ Rich Olcott

Dimensional Venturing Part 4 – To infini-D and beyond!

On By rolcottIn Cosmology, Mathematics: Dimensions, UncategorizedLeave a comment

apple plumNow that you’ve read my previous posts and have the 4-D thing working well, you’re ready to go for a few more dimensions.  Consider the apple that struck Isaac Newton’s head.  The event occurred in 1665, in England at 52°55´N by 0°38´E, roughly three feet above ground level.  The apple, variety “Flower of Kent,” weighed about 8 ounces and was probably somewhat past fully ripened.  Got that picture in your head?  You’re doing great.

Now visualize the apple taking thirty seconds to move twenty feet diagonally upward, northward and eastward as it morphs to an underripe 4-ounce Damson plum.

The change you just imagined followed an eight-dimensional path: three dimensions of space, one of time, one of weight, one for degree of ripeness, and two category dimensions, species and variety.

Length in a given direction is only one kind of dimension, as Sir Isaac’s example demonstrates.  A mathematician would say that a dimension is a set of values that can be traversed independently of any other set of values. A dimension can be confined to a limited range (360 degrees in a circle) or be infinite like … well, “infinitely far away.”  A dimension might be continuous (think how loudness can vary smoothly from sleeping-baby hush to stadium ROAR and beyond) or be in discrete steps like the click-stops on a digital controller.  The physicists are arguing now whether, at the smallest of scales, space itself is continuous or discrete.

colors_post
Photo by Becky Ziemer

Color vision’s a good example of dimensions in action.  For most of us, our eyes have three types of cone cells, respectively optimized for red, green and blue light.  We see a specific color as some mixture of the three and that’s how the screen you’re looking at now can fake 16 million colors using just three kinds of color-emitting elements (phosphor dots in old-style TVs, LEDs in most devices these days).

Where did that 16 million number come from?  The signal-processing math is seriously techie, but at the bottom the technology uses 256 intensity levels of red, 256 levels of green and 256 levels of blue — each is a discrete dimension with a limited range.  Together they define a 256x256x256-point cube.  Any point in that cube represents a unique mix of primary colors.  One of the colors in the little girl’s hat, for instance, is at the intersection of 249/256 red, 71/256 green, and 48/256 blue.  The arithmetic tells us there are 16,777,216 points (possible mixed colors) in that cube.

Well, actually, there’s one more dimension to color vision because our eyes also have rod cells that simply sense light or darkness.  Neither brown nor grey are in the spectrum that cones care about.  A good printer uses four separate inks to produce browns and greys as mixtures of three dimensions of red-green-blue plus one of black.

So color is 3-dimensional, mostly.  But that’s just the start of color vision because most of us have millions of cone cells in each eye.  A mathematician would say that any scene you look at has that number of dimensions, because the intensity registered by one cone can vary in its range independently of all the other cones.

Ain’t it wonderful that you’re perfectly OK with living in a multi-million-dimensional world?

Next week – a word from the other side

~~ Rich Olcott