Keep calm and stay close to home

Again with the fizzing sound.  Her white satin still looked good.  A little travel-worn, but on her that looked even better.  Her voice still sounded like molten silver — “Hello.”White satin and drunkard walk

“Hello, Anne.  Where you been?”

“You wouldn’t believe.  I don’t believe.  I’ve got to get some control over this.”

“What’s the problem?”

“I never know where I’ll be next.  Or when.  Or even how it’ll look when I get there.  We’ve met before, haven’t we?”

“Yes, we have, and you told me your memory works in circles.  We figured out that when you ‘push,’ you relocate to a reality with a different probability.”

“But it could also be a different time.  Future, past, it’s so confusing.  Sometimes I meet myself and I don’t know whether I’m coming or going.  We never know what to say to each other.  It’s horrible way to be.”

“It sounds awful.  Here, have a tissue.  So, how can I help you?”

“You do theory stuff.  Can you physics a way to let me steer through all this?”

<fizzing sound> Another Anne appeared, next to my file cabinet on the far side of the office.  “Don’t mind me, just passing through.”  <more fizzing>  She flickered away.  My ears itched a little.

“See?  And she always knows more than I do, except when I know more than she does.”

“I’m beginning to get the picture.  Mind if I ask you a few questions?”

“Anything, if it’ll help solve this.”

“When you time-hop, do you use the same kind of ‘push’ feeling that sends you to different probabilities?”

“No-o, it’s a little different, but not much.”

“We found that you have to ‘push’ harder to get to a less-probable reality.  Is there the same kind of difference between past and future hopping?”

“Now you mention it, yes!  It’s always easier to jump to the future.  I have to struggle sometimes when I get too far ahead of myself.”

“Can you do time and probability together?”

“Hard to say.  When I hop I mostly just try to work out when I am, much less whether things are odd.”

“Give it a shot.  Try a couple of ‘nearby places’ and come back here/now.  Just use tiny ‘pushes.’ I don’t want you to get lost again.”

“Me neither.  OK, here I go.” <prolonged flickering and fizzing> “Is this the right place?  I tried a couple of hops here in your office, and <charming blush> stole some of your papers.  Here.”

“Perfect, Anne, objective evidence is always best.  Let’s see…  Yep, this report is one I finished a week ago, looks OK, and this one … I recognize the name of a client I’ve not yet hooked, but the spelling!  The letter ‘c’ isn’t there at all — ‘rekognize,’ ‘sirkle,’ ‘siense’ — that’s low probability for sure.”

“Actually, it felt like higher probability.”

“Whatever.  One more question.  I gather that most of your hops are more-or-less good ones but every once in a while you drop into a complete surprise, something you’re totally not used to.”

“Uh-huh.”

“I’ll bet the surprises happen when you’re in a jam and do a get me out of here jump.”

“Huh!  I’d not made that connection, but you’re right.”

“I think I’ve got the picture.  When you ‘push,’ you somehow displace yourself on a surface that has two dimensions — time and probability.  You move around in those two dimensions independently from how you move in 3-D space.  I take it you’re comfortable dong that but you want more control over it, right?”

“Mmm, yeah.  It’s kind of my special superpower, you know?  I don’t want to give it up entirely.”

“Good, because I wouldn’t know how to make that happen for you.  Best I can do is give you some strategy coaching, OK?”

“That’d be a big help.”Drunkard

“Stay calm.”

“That’s it?  Where’s the physics in that?”

“Ever hear of the Drunkard’s Walk?”

“I’ve seen a few.”

“Well, you’re doing one.”

“Beg pardon?”

“It’s math talk for a stepwise process where every step goes in a random direction.  Your problem is that some of the steps are way too big.  Keep the steps small and you’ll stay in familiar territory.”

<molten silver, coming closer> “Like … here?”

“Stay calm.”

~~ Rich Olcott

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Through The Looking Glass, Darkly

The Acme Building is quiet on summer evenings.  I was in my office, using the silence to catch up on paperwork.  Suddenly I heard a fizzing sound.  Naturally I looked around.  She was leaning against the door frame.

White satin looked good on her, and she looked good in it.  A voice like molten silver — “Hello, Mr Moire.”White satin and chessboard 1

“Hello yourself.  What can I do for you?”

“I’m open to suggestions, but first you can help me find myself.”

“Excuse me, but you’re right here.  And besides, who are you?”

“Not where I am but when I am.  Anne.”

“You said it right the first time.”

“No, no, my name is Anne.  At the moment.  I think.  Oh, it’s so confusing when your memory works in circles but not very well.  Do you have the time?”

“Well, I was busy, but you’re here and much more interesting.”

“No, I mean, what time is it?”

I showed her my desk clock — date, time, even the phase of the moon.

“Half past gibbous already?  Oh, bread-and-butter…”

“Wait — circles?  Time’s one-dimensional.  Clock readings increase or decrease, they don’t go sideways.”

“You don’t know Time as well as I do, Mr Moire.  It’s a lot more complicated than that.  Time can be triangular, haven’t you noticed?”

“Can’t say as I have.”

“That paperwork you’re working on, are you near a deadline?”

“Nah.”

“And given that expanse of time, you feel free to permit distractions.  There are so many distractions.”

“You’re very distracting.”

“Thank you, I guess.  But suppose you had an important deadline coming up tomorrow.   That broad flow of possibilities at the beginning of the project has narrowed to just two — finish or don’t finish.  Your Time has closed in until you.”

“So you’re saying we can think of Time as two-dimensional.  The second dimension being…?”

“I don’t know.  I just go there.  That’s the problem.”

“Hmm… When you do, do you feel like you’re turning left or right?”

“No turning or moving forward or backward.  Generally I have to … umm… ‘push’ like I’m going uphill, but that only works if there’s a ‘being pushed’ when I get past that.  Otherwise I’m back where I started, whatever that means.”

“What do you see?  What changes during the episode?”

“Little things. <brief fizzing sound.  She … flickered.>  Like ‘over there’ you’re wearing a bright green T-shirt instead of what you’re wearing here.  And you’re using pen-and-paper instead of that laptop.  Green doesn’t suit you.”

“I know, which is why there’s nothing green in my wardrobe, here.  But that gives me an idea.  Did you always have to ‘push’ to get ‘over there’?”

“Usually.”

“Fine.  OK, I’m going to flip this coin.  While it’s in the air, ‘push’ just lightly and come back to tell me which way the coin fell.”

<fizzing> “Heads.”

“It’s tails here.  OK, we’re going to do that again but this time ‘push’ much harder.”

<louder fizzing> “That was weird.  Your coin rolled off the desk and landed on edge in a crack in the floor so it’s not heads or tails.”

“AaaHAH!”Coins 1

“?”

“Your ‘over theres’ have different levels of probability than ‘over here.’  They’re different realities.  Actually, I’ll bet you travel across ranges of probability.  Or tunnel through them, maybe.  That’d why you have to ‘push’ to get past something that’s less probable in order to get to something that’s more probable.  Like getting past a reality where the coin can just hang in the air or fly apart.”

“I’ve done that.  Once I sneezed while ‘pushing’ and wound up sitting at a tea party where the cream and sugar just refused to stir into the tea.  When I ‘pushed’ from there I practically fell into a coffee shop where the coffee was well-behaved.”

“Case closed.  Now I can answer your question.  Spacewise, you’re in my office on the twelfth floor.  Timewise, I just showed you my clock.  As for which reality, you’re in one with a very high probability because, well, you’re here.”

“So provincial.  Oh, Mr Moire, how little you know.” <fizzing>

On the 12th floor of the Acme Building, high above the city, one man still tries to answer the Universe’s persistent questions — Sy Moire, Physics Eye.

~~ Rich Olcott

Superluminal Superman

Comic book and movie plotlines often make Superman accelerate up to lightspeed and travel backward in time.  Unfortunately, well-known fundamental Physics principles forbid that.  But suppose Green Lantern or Dr Strange could somehow magic him past the Lightspeed Barrier.  Would that let him do his downtimey thing?

Light_s hourglass
Light’s Hourglass

A quick review of Light’s Hourglass.  According to Einstein we live in 4D spacetime.  At any moment you’re at a specific time t relative to some origin time t=0 and a specific 3D location (x,y,z) relative to a spatial origin (0,0,0).  Your spacetime address is (ct,x,y,z) where c is the speed of light.  This diagram shows time running vertically into the future, plus two spatial coordinates x and y.  Sorry, I can’t get z into the diagram so pretend it’s zero.

The two cones depict all the addresses which can communicate with the origin using a flash of light.  Any point on either cone is at just the right distance d=√(++) to match the distance that light can travel in time t.  The bottom cone is in the past, which is why we can see the light from old stars.  The top cone is in the future, which is why we can’t see light from stars that aren’t born yet.

If he obeys the Laws of Physics as we know them, Superman can travel anywhere he wants to inside the top cone.  He goes upward into the future at the rate of one second per second, just like anybody.  On the way, he can travel in space as far from (ct,0,0,0) as he likes so long as it’s not farther than the distance that light can travel the same route at his current t.

From our perspective, the Hourglass is a stack of circles (spheres in 3D space) centered on (ct,0,0,0).  From Supey’s perspective at time t he’s surrounded by a figure with radius ct that Physics won’t let him break through.  That’s his Lightspeed Barrier, like the Sonic Barrier but 900,000 times faster.

Suppose Green Lantern has magicked Supey up to twice lightspeed along the x-axis.  At moment t, he’s at (ct,2ct,0,0), twice as far as light can get.  In the diagram he’s outside the top cone but above the central disk.

Now GL pours on the power to accelerate Superman.  Each increment gets the Man of Steel closer to that disk.  He’s always “above” it, though, because he’s still moving into the future.  Only if he were to get to infinite speed could he reach the disk.

However, at infinite speed he’d go anywhere/everywhere instantaneously which would be confusing to even his Kryptonian intellect.  On the way he might run into things (stars, black holes,…) with literally zero time to react.

But the plotlines have Tall-Dark-and-Muscular flying into the past, breaching that disk and traveling downwards into the bottom cone.  Can GL make that happen?

Enter the Lorentz correction.  If you have rest mass m0 and you’re traveling at speed v, your effective mass is m=m0/√[1-(v/c)²]. That raises a couple of issues when you exceed lightspeed.

Suppose GL decelerates Superluminal Supey down towards lightspeed.  The closer he approaches c from higher speeds, the smaller that square root gets and the greater the effective mass.  It’s the same problem Superman faced when accelerating up to lightspeed.  That last mile per second down to c requires an infinite amount of braking energy — the Lightspeed Barrier is impermeable in both directions.

The other problem is that if v>c there’s a negative number inside that square root.    Above lightspeed, your effective mass becomes Bombelli-imaginary.  Remember Newton’s famous F=m·a?  Re-arrange it to a=F/m.  A real force applied to an object with imaginary mass produces an imaginary acceleration.  “Imaginary” in Physics generally means “perpendicular in some sense” and remember we’re in 4D here with time perpendicular to space.

GL might be able to shove Superman downtime, but he’d have to

  1. squeeze inward at hiper-lightspeed with exactly the same force along all three spatial dimensions, to make sure that “perpendicular” is only along the time axis
  2. start Operation Squish at some time in his own future to push towards the past.

Nice trick.  Would Superman buy in?

~~ Rich Olcott

Does a photon experience time?

My brother Ken asked me, “Is it true that a photon doesn’t experience time?”  Good question.  As I was thinking about it I wondered if the answer could have implications for Einstein’s bubble.

When Einstein was a grad student in Göttingen, he skipped out on most of the classes given by his math professor Hermann Minkowski.  Then in 1905 Einstein’s Special Relativity paper scooped some work that Minkowski was doing.  In response, Minkowski wrote his own paper that supported and expanded on Einstein’s.  In fact, Minkowski’s contribution changed Einstein’s whole approach to the subject, from algebraic to geometrical.

But not just any geometry, four-dimensional geometry — 3D space AND time.  But not just any space-AND-time geometry — space-MINUS-time geometry.  Wait, what?Pythagoras1

Early geometer Pythagoras showed us how to calculate the hypotenuse of a right triangle from the lengths of the other two sides. His a2+b2 = c2 formula works for the diagonal of the enclosing rectangle, too.

Extending the idea, the body diagonal of an x×y×z cube is √(x2+y2+z2) and the hyperdiagonal  of a an ct×x×y×z tesseract is √(c2t2+x2+y2+z2) where t is time.  Why the “c“?  All terms in a sum have to be in the same units.  x, y, and z are lengths so we need to turn t into a length.  With c as the speed of light, ct is the distance (length) that light travels in time t.

But Minkowski and the other physicists weren’t happy with Pythagorean hyperdiagonals.  Here’s the problem they wanted to solve.  Suppose you’re watching your spacecraft’s first flight.  You built it, you know its tip-to-tail length, but your telescope says it’s shorter than that.  George FitzGerald and Hendrik Lorentz explained that in 1892 with their length contraction analysis.

What if there are two observers, Fred and Ethel, each of whom is also moving?  They’d better be able to come up with the same at-rest (intrinsic) size for the object.

Minkowski’s solution was to treat the ct term differently from the others.  Think of each 4D address (ct,x,y,z) as a distinct event.  Whether or not something happens then/there, this event’s distinct from all other spatial locations at moment t, and all other moments at location (x,y,z).

To simplify things, let’s compare events to the origin (0,0,0,0).  Pythagoras would say that the “distance” between the origin event and an event I’ll call Lucy at (ct,x,y,z) is √(c2t2+x2+y2+z2).

Minkowski proposed a different kind of “distance,” which he called the interval.  It’s the difference between the time term and the space terms: √[c2t2 + (-1)*(x2+y2+z2)].

If Lucy’s time is t=0 [her event address (0,x,y,z)], then the origin-to-Lucy interval is  √[02+(-1)*(x2+y2+z2)]=i(x2+y2+z2).  Except for the i=√(-1) factor, that matches the familiar origin-to-Lucy spatial distance.

Now for the moment let’s convert the sum from lengths to times by dividing by c2.  The expression becomes √[t2-(x/c)2-(y/c)2-(z/c)2].  If Lucy is at (ct,0,0,0) then the origin-to-Lucy interval is simply √(t2)=t, exactly the time difference we’d expect.

Finally, suppose that Lucy departed the origin at time zero and traveled along x at the speed of light.   At any time t, her address is (ct,ct,0,0) and the interval for her trip is √[(ct)2-(ct)2-02-02] = √0 = 0.  Both Fred’s and Ethel’s clocks show time passing as Lucy speeds along, but the interval is always zero no matter where they stand and when they make their measurements.

Feynman diagramOne more step and we can answer Ken’s question.  A moving object’s proper time is defined to be the time measured by a clock affixed to that object.  The proper time interval between two events encountered by an object is exactly Minkowski’s spacetime interval.  Lucy’s clock never moves from zero.

So yeah, Ken, a photon moving at the speed of light experiences no change in proper time although externally we see it traveling.

Now on to Einstein’s bubble, a lightwave’s spherical shell that vanishes instantly when its photon is absorbed by an electron somewhere.  We see that the photon experiences zero proper time while traversing the yellow line in this Feynman diagram.  But viewed from any other frame of reference the journey takes longer.  Einstein’s objection to instantaneous wave collapse still stands.

~~ Rich Olcott

Is cyber warfare imaginary?

Rule One in hooking the reader with a query headline is: Don’t answer the question immediately.  Let’s break that one.  Yes, cyber warfare is imaginary, but only for a certain kind of “imaginary.”  What kind is that, you ask.  AaaHAH!

spy1
Antonio Prohías’ Mad Magazine spies
didn’t normally use cyber weaponry

It all has to do with number lines.  If the early Greek theoreticians had been in charge, the only numbers in the Universe would have been the integers: 1, 2, 3,….  Life is simple when your only calculating tool is an abacus without a decimal point.  Zero hadn’t been invented in their day, nor had negative numbers.

Then Pythagoras did his experiments with harmony and harp strings, and the Greeks had to admit that ratios of integers are rational.

More trouble from Pythagoras: his a2+b2=c2 equation naturally led to c=√(a2+b2).  Unfortunately, for most integer values of a and b, c can’t be expressed as either an integer or a ratio of integers.  The Greeks labeled such numbers (including π) as irrational and tried to ignore them.

Move ahead to the Middle Ages, after Europe had imported zero and the decimal point from Brahmagupta’s work in India, and after the post-Medieval rise of trade spawned bookkeepers who had to cope with debt.  At that point we had a continuous number line running from “minus a whole lot” to “plus you couldn’t believe” (infinity wasn’t seriously considered in Western math until the 17th century).

By then European mathematicians had started playing around with algebraic equations and had stumbled into a problem.  They had Brahmagupta’s quadratic formula (you know, that [-b±√(b2-4a·c)]/2a thing we all sang-memorized in high school).  What do you do when b2 is less than 4a·c and you’re looking at the square root of a negative number?

Back in high school they told us, “Well, that means there’s no solution,” but that wasn’t good enough for Renaissance Italy.  Rafael Bombelli realized there’s simply no room for weird quadratic solutions on the conventional number line.  He made room by building a new number line perpendicular to it.  The new line is just like the old one, except everything on it is multiplied by i=√(-1).

(Bombelli used words rather than symbols, calling his creation “plus of minus.”  Eighty years later, René Descartes derisively called Bombelli’s numbers “imaginary,” as opposed to “real” numbers, and pasted them with that letter i.  Those labels have stuck for 380 years.  Except for electricity theoreticians who use j instead because i is for current.)

AxesSuppose you had a graph with one axis for counting animal things and another for counting vegetable things.  Animals added to animals makes more animals; vegetables added to vegetables makes more vegetables.  If you’ve got a chicken, two potatoes and an onion, and you share with your buddy who has a couple of carrots, some green beans and another onion, you’re on your way to a nice chicken stew.

Needs salt, but that’s on yet another axis.

Bombelli’s rules for doing arithmetic on two perpendicular number lines work pretty much the same.  Real numbers added to reals make reals, imaginaries added to imaginaries make more imaginaries.  If you’ve got numbers like x+i·y that are part real and part imaginary, the separate parts each follow their own rule.  Multiplication and division work, too, but I’ll let you figure those out.

The important point is that what happens on each number line can be specified independently of what happens on the other, just like the x and y axes in Descartes’ charts.  Together, Bombelli’s and Descartes’ concepts constitute a nutritious dish for physicists and mathematicians.

Scientists love to plot different experimental results against each other to see if there’s an interesting relationship in play.  For certain problems, for example, it’s useful to plot real-number energy of motion (kinetic energy) against some other variable on the i-axis.

Two-time Defense Secretary Donald Rumsfeld used to speak of “kinetic warfare,” where people get killed, as opposed to the “non-kinetic” kind.  Apparently, he would have visualized cyber somewhere up near the i-axis.  In that scheme, cyber warriors with their ones and zeros are Bombelli-imaginary even if they’re real.

 ~~ Rich Olcott

 

Throwing a Summertime curve

All cats are gray in the dark, and all lines are straight in one-dimensional space.  Sure, you can look at a garden hose and see curves (and kinks, dammit), but a short-sighted snail crawling along on it knows only forward and backward.  Without some 2D notion of sideways, the poor thing has no way to sense or cope with curvature.

Up here in 3D-land we can readily see the hose’s curved path through all three dimensions.  We can also see that the snail’s shell has two distinct curvatures in 3D-space — the tube has an oval cross-section and also spirals perpendicular to that.

But Einstein said that our 3D-space itself can have curvature.  Does mass somehow bend space through some extra dimension?  Can a gravity well be a funnel to … somewhere else?

No and no.  Mathematicians have come up with a dozen technically different kinds of curvature to fit different situations.  Most have to do with extrinsic non-straightness, apparent only from a higher dimension.  That’s us looking at the hose in 3D.

Einstein’s work centered on intrinsic curvature, dependent only upon properties that can be measured within an object’s “natural” set of dimensions.Torus curvature

On a surface, for instance, you could draw a triangle using three straight lines.  If the figure’s interior angles sum up to exactly 180°, you’ve got a flat plane, zero intrinsic curvature.  On a sphere (“straight line” = “arc from a great circle”) or the outside rim of a doughnut, the sum is greater than 180° and the curvature is positive.
Circle curvatures
If there’s zero curvature and positive curvature, there’s gotta be negative curvature, right?  Right — you’ll get less-than-180° triangles on a Pringles chip or on the inside rim of a doughnut.

Some surfaces don’t have intersecting straight lines, but you can still classify their curvature by using a different criterion.  Visualize our snail gliding along the biggest “circle” he/she/it (with snails it’s complicated) can get to while tethered by a thread pinned to a point on the surface. Divide the circle’s circumference by the length of the thread.  If the ratio’s equal to 2π then the snail’s on flat ground.  If the ratio is bigger than ,  the critter’s on a saddle surface (negative curvature). If it’s smaller, then he/she/it has found positive curvature.

In a sense, we’re comparing the length of a periphery and a measure of what’s inside it.  That’s the sense in which Einsteinian space is curved — there are regions in which the area inside a circle (or the volume inside a sphere) is greater than or less than what would be expected from the size of its boundary.

Here’s an example.  The upper panel’s dotted grid represents a simple flat space being traversed by a “disk.”  See how the disk’s location has no effect on its size or shape.  As a result, dividing its circumference by its radius always gives you 2π.Curvature 3

In the bottom panel I’ve transformed* the picture to represent space in the neighborhood of a black hole (the gray circle is its Event Horizon) as seen from a distance.  Close-up, every row of dots would appear straight.  However, from afar the disk’s apparent size and shape depend on where it is relative to the BH.

By the way, the disk is NOT “falling” into the BH.  This is about the shape of space itself — there’s no gravitational attraction or distortion by tidal spaghettification.

Visually, the disk appears to ooze down one of those famous 3D parabolic funnels.  But it doesn’t — all of this activity takes place within the BH’s equatorial plane, a completely 2D place.  The equations generate that visual effect by distorting space and changing the local distance scale near our massive object.  This particular distortion generates positive curvature — at 90% through the video, the disk’s C/r ratio is about 2% less than 2π.

As I tell Museum visitors, “miles are shorter near a black hole.”

~~ Rich Olcott

* – If you’re interested, here are the technical details.  A Schwarzchild BH, distances as multiples of the EH radius.  The disk (diameter 2.0) is depicted at successive time-free points in the BH equatorial plane.  The calculation uses Flamm’s paraboloid to convert each grid point’s local (r,φ) coordinates to (w,φ) to represent the spatial configuration as seen from r>>w.

And now for some completely different dimensions

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 Liebniz 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

Dimensional Venturing, Part 6 – Tiny Dimensions

“The Universe is much larger than is generally supposed.”  

What a great opening line, eh?  Decades later I still recall reading that in a technical paper about then-recent adjustments in the way astronomical distances were measured.

The authors didn’t know the half of it.  They were thinking in only three dimensions.  That’s so last-century.

If you read science articles in the popular press you’ve probably run into statements like this one from Brian Green’s article “Hanging by a String” in the January 2015 Smithsonian:

String theory’s equations require that the universe has extra dimensions beyond the three of everyday experience – left/right, back/forth and up/down…. [T]heorists realized that there might be two kinds of spatial dimensions: those that are large and extended, which we directly experience, and others that are tiny and tightly wound, too small for even our most refined equipment to reveal.

Tightly wound dimensions?  What’s that about?  And what’s it got to do with strings?

The “large extended” dimensions are the kind we discussed in Part 1 of this series.  The essential point is that (in principle) once you or a light ray start moving in a particular direction you can keep going in that direction forever.

Seems obvious, how else could it be?

tiny dimension 1Well, suppose that we bend one of those three familiar “large” dimensions around in a circle, as in the drawing to the right. Our little guy could walk straight out of the page “forever” in the X direction. He could walk straight up the page “forever” in the Z direction. However, if he tries to walk along the Y track perpendicular to both of those two, in a while he’ll wind up right back where he started.

That’s an example of a “tightly wound” dimension.

Because it makes the math easier, physicists usually don’t calculate the absolute distance traveled around the circle.  Instead they write equations that depend on the angle from zero as the starting point. Notice that 360 degrees is exactly the same as zero — that’ll be important in a later post here.  Anyhow, there’s reason to believe that the effective circumference of a “tightly wound” dimension is really, really small.

OK, having a closed-off dimension is a little strange but it’s just not real-world, is it?

tiny dimension 2Actually, our real world is like that but moreso. Look at this drawing where we’ve got a pair of perpendicular wound-up dimensions. The little guy on the Y track can go from Denver down to Mazatlan in Mexico and proceed all the way around the world back up to Denver. On the X track he’s going from Denver westward to Chico CA and could continue across the Pacific and onward until he gets back to Denver The only way he can travel in one direction “forever” is to go along the Z track, straight upward, and that’s why NASA builds rocket ships.

Back to the strings. Depending on which variety of string theory you choose, the strings wriggle in a space of three Z-style “extended” dimensions, plus time, plus half-a-dozen or more wound-up or “compactified” (look it up) dimensions.  If string-theory strings can wriggle in all those directions, then how much room does each one have to move around in?  We’ve all learned the formulas for area of a rectangle and volume of a cube — [length times height] and [length times height times depth].  To extend the notion of “volume” to more dimensions you just keep multiplying.

Back to the size of the Universe. You may think that just with straight-line space it’s pretty good-sized.  With those stringy dimensions in play, for every single cube-shaped region you pick in straight-line space you need to multiply that volume by [half-a-dozen or more dimensions] times [many possible angles] to account for all the “space” in all the enhanced regions you could choose from when you include those wound-up dimensions. The total multi-dimensional volume is very, very huge.

The universe is indeed much larger than is generally supposed.

Next week — buttered cats.

~~ Rich Olcott

Dimensional Venturing Part 5 – You Ain’t From Around Here, Are You?

OK, I’ll admit it, back in the day I read a lot of comics.  Even then, though, I was skeptical — “Wait, how could Superman just pick up that building?  It’d fall apart!”

But I was intrigued by one recurring character, Mr Mxyzptlk, a pixie-like “visitor from the 5th dimension.”   His primary purpose in life (other than getting us to buy more comics) seemed to be to play tricks on or otherwise torment Our Hero.

Mxycus 2Mxy wasn’t the only comics character coming in “from another dimension.” It seemed like the entire Marvel team (both sides) was continually flickering out of and into our universe that way. How often did Jane Grey die and then somehow get cloned or refreshed?  (BTW, if the accompanying cartoon is a little obscure, show it to the friendly clerks at your local comics store — it may give them a chuckle.)

But my question was, where was that dimension Mxy came from?  I got an answer, sort of, when our geometry teacher explained that a dimension is just a direction you could travel.  Different dimensions are directions at right angles to each other.  She was right (see my first post in this series), at least in the context of then-HS math, but that explanation opened an editorial issue that’s never been properly settled.

A dimension is a direction, not a location.  You can’t be “from” a fifth or sixth or nth dimension any more than you can be from up.  If there is a spatial fifth dimension, we’re already “in” it in the same sense that we’re already somewhere along east-to-west and somewhen along past-to-future.

What’s going on is that for the purpose of the story, the authors want the character to come from somewhere very else.  We often associate a place with the direction to it — the sun rises in the east, Frodo departs to the west,  Heaven is up, Hell is down — but those are all directions relative to our current location.  We even associate future times as being in front of us and past times behind us (there’s that 4th dimension again).

Mash_sign_post
The M*A*S*H signpost, now at the Smithsonian. Photo by Steven Williamson., in commons.wikimedia.org/wiki/File:Mash_sign.jpg

But a place is more specific than a direction — to navigate to a certain there you need to know the direction and the distance (or another quantity that stands in for a distance).  That matters.  Jimmie Rodgers sang, “Twelve more miles to Tucumcari” as he kept track of the distance left to go along the road he was traveling.  Or away from the town, as it turned out.

Physicists have lots of uses for the combination of a direction and a magnitude, so many that they gave the combination a name — a vector.  The vector may represent a direction and a distance, a direction and the strength of a magnetic field, or a direction and any quantity that happens to be useful in the application at hand.  A wind map uses vectors of direction and wind speed to show air flow.  Here’s a very nice wind map of the US, and I love NOAA’s wind map of the world.  Vectors will be real useful when we start talking about black holes.

OK, so Mr Mxyzpltk (the spelling seemed to vary from issue to issue of the comic) comes from somewhere along a fifth dimension, but they never tell us from how far away.

Next week –As Steve Martin said, “Let’s get small, really small.”

~~ Rich Olcott

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

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