The Titanic Winds of Titan (And Venus)

On October 3, 2016November 28, 2025 By rolcottIn Astronomy: Planetary Science, Uncategorized2 Comments

Last week we saw that the atmosphere of Saturn’s moon Titan wasn’t quite as weird as we thought. But there another way it’s really weird, completely unlike Earth but yet very much like Venus. Titan’s a superrotater, a world whose atmosphere circles the planet much faster than its surface does.

Let’s start with a relatively simple Earthside phenomenon, a hurricane. Warm air rises, right? When the warmth comes from bathtub-temperature sea-water, it’s wet warm air. As the air rises it cools and releases the moisture as rain. But the air can’t just keep rising forever or we’d squirt out all our atmosphere. So where does it go?

From a physicist’s perspective, that’s the key question. If we can track/predict the path of a small parcel of air molecules through a weather system, then we’ve got at least a rough understanding of how that system works.

For the past half-century, atmosphere physicists have been engaged on a project grandly entitled the General Circulation Model (GCM), a software mash-up of the Ideal Gas Law, Newton’s Laws of motion, thermodynamic data for solid/liquid/gas transformations, the notoriously difficult Navier-Stokes equations for viscous fluids, and careful data management for input streams from thousands of disparate sources. Oh, and it’s important that the Earth is a rotating spheroid rather than a flat plane.

hurricane-en-svg — How a tropical cyclone works
Illustration by Kevin Song, from Wikimedia Commons

Kevin Song’s diagram summarizes much of what we know about hurricanes. An air packet rises until it hits the tropopause (the top edge of the troposphere), then expands horizontally. While the packet’s spreading out, the planet’s rotation generates Coriolis “forces” that bend straight-line radial paths into the spirals we’ve seen so often in satellite photos.

A hurricane may look big on your weathercaster’s screen, but it’s less than 0.1% of Earth’s surface area. Nonetheless, many of the same principles that drive a hurricane underlie global weather patterns.

Air warmed by the equatorial Sun rises, only to sink as it heads poleward. Our packet loops between the Equator and about 30ºN (see the diagram).

Actually that loop is a slice through a big doughnut that stretches all the way around the Earth. Another doughnut lies southward just below the Equator. Two more pairs of doughnuts reside polewards of those as indicated by the other arrows in the diagram. The doughnuts act like a set of interlocking gears, each reinforcing and moderating the motion of its neighbors.

Thanks to the same geometric phenomenon that spins a hurricane, air packets in these doughnuts don’t loop back to the points they started from. The Earth turns under the packets as they journey, so each packet takes a spiraling tour around the planet.

Because of all those doughnuts, on average Earth wears a set of cloud-top necklaces. Regions within 15º of the Equator are rain-forested, as are the Canadian and Siberian forested belts near 60ºN. The world’s most prominent deserts cluster beneath the dry downdrafts near the 30º latitudes. Jupiter, “the Easter egg planet,” gets its pink and blue bands from similar doughnuts except that Jupiter has room for many more of them.

Those green circles in the diagram are important, too. They also represent Earth-circling doughnuts, but ones whose winds flow parallel to Earth’s surface rather than perpendicular to it. The ones close to the surface give rise to the trade winds. The high-altitude ones are the jet streams that steer storm systems and give the weathercasters something to talk about, especially in the wintertime.

Jet streams flow briskly — 60 to 200 mph, on a par with a middling hurricane. Here’s a benchmark: Earth’s equatorial circumference is 25,000 miles, so Ecuadorian palm trees circle the planet at (25000 miles/24 hours)=1041 mph. Our jet streams go about 15% of that. Theory and GCM agree that the jets are powered by the Coriolis effect — spiraling air packets in the primary donuts cooperate to push jet stream air packets like oars on a galley ship. That adds up.

Titan and Venus can’t possibly work that way. Both of them rotate much more slowly than Earth (Titan about 30 mph, Venus only 4), so Coriolis forces are negligible. But Titan’s jet streams do 75 mph and Venus’ race at 185. What powers them? The physicists are still arguing.

~~ Rich Olcott

Titan’s Atmosphere Is A Gas

On September 26, 2016November 28, 2025 By rolcottIn Astronomy: Planetary Science, Chemistry, UncategorizedLeave a comment

One year ago I kicked off these weekly posts with some speculations about how Life might exist on Saturn’s moon Titan. My surmises were based on reports from NASA’s Cassini-Huygens mission, plus some Physical Chemistry expectations for Titan’s frigid non-polar mix of liquid ethane and methane. Titan offers way more fun than that.

The environment on Titan is different from everything we’re used to on Earth. For instance, the atmosphere’s weird.Titan’s atmosphere is heavy-duty compared with Earth’s — 6 times deeper and about 1½ times the surface pressure. When I read those numbers I thought, “Huh? But Titan’s diameter is only 40% as big as Earth’s and its surface gravity is only 10% of ours. How come it’s got such a heavy atmosphere?”

Wait, what’s gravity got to do with air pressure? (I’m gonna use “air pressure” instead of “surface atmospheric pressure” because typing.) Earth-standard sea level air pressure is 14.7 pounds of force per square inch. That 14.7 pounds is the total weight of the air molecules above each square inch of surface, all the way out to space.

(Fortunately, air’s a hydraulic fluid so its pressure acts on sides as well as tops. Otherwise, a football’s shape would be even stranger than it is.)

Newton showed us that weight (force) is mass times the the acceleration of gravity. Gravity on Titan is 1/10 as strong as Earth’s, so an Earth-height column of air on Titan should weigh about 1½ pounds.

But Titan’s atmosphere (measured to the top of each stratosphere) goes out 6 times further than Earth’s. If we built out that square-inch column 6 times taller, it’d weigh only 9 pounds on Titan, well shy of the 22 pounds the Huygens lander measured. Where does the extra weight come from?

My first guess was, heavy molecules. If gas A has molecules that are twice as heavy as gas B’s, then a given volume of A would weigh twice as much as the same volume of B. An atmosphere composed of A will press down on a planet’s surface twice as hard as an atmosphere composed of B.

Good guess, but doesn’t apply. Earth’s atmosphere is 78% N₂ (molecular weight 28) and 21% O₂ (molecular weight 32) plus a teeny bit of a few other things. Their average molecular weight is about 29. Titan’s atmosphere is 98% N₂ so its average molecular weight (28) is virtually equal to Earth’s. So no, those tarry brown molecules that block our view of Titan’s surface aren’t numerous enough to account for the high pressure.

My second guess is closer to the mark, I think. I remembered the Ideal Gas Law, the one that says, “pressure times volume equals the number of molecules times a constant times the absolute temperature.” In symbols, P·V=n·R·T.

Visualize one gas molecule, Fred, bouncing around in a cube sized to match the average volume per molecule, V/n=R·T/P. If Fred goes outside his cube in any direction he’s likely to bang into an adjacent molecule. If Fred has too much contact with his neighbors they’ll all stick together and become a liquid or solid.

The equation tells us that if the pressure doesn’t change, the size of Fred’s cube rises with the temperature. Just for grins I calculated the cube’s size for standard Earth conditions: (22.4 liters/mole)×(1 cubic meter/1000 liters)×(1 mole/6.02×10²³ molecules)=37.2×10^-27 cubic meter/molecule. The cube root of that is the length of the cube’s edge — 3.3 nanometers, about 8.3 times Fred’s 0.40-nanometer diameter.^∗

Earth-standard surface temperature is about 300°K (absolute temperatures are measured in Kelvins). Titan’s surface temperature is only 94°K. On Titan that cube-edge would be 8.3*(94/300)=2.6 times Fred’s diameter — if air pressure were Earth-standard.

But really Titan’s air pressure is 1.5 times higher because its column is so tall and contains so much gas. The additional pressure squeezes Fred’s cube-edge down to 2.6*(1/1.5)=1.8 times his diameter. Still room enough for Fred to feel well-separated from his neighbors and continue acting like a proper gas.

The primary reason Titan’s atmosphere is so dense is that it’s chilly up there. Also, there’s a lot of Freds.

~~ Rich Olcott

^∗ – For the technorati… The cube-root of the Van der Waals volume for N₂. And yeah, I know I’m almost writing about Mean Free Path but I think the development’s simpler this way.

Superluminal Superman

On September 19, 2016November 28, 2025 By rolcottIn Mathematics: Dimensions, Physics: Light and Relativity, UncategorizedLeave a comment

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?

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=√(x²+y²+z²) 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

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
start Operation Squish at some time in his own future to push towards the past.

Nice trick. Would Superman buy in?

~~ Rich Olcott

Superman flying at lightspeed? Umm… no

On September 12, 2016November 28, 2025 By rolcottIn Mathematics: Dimensions, Physics: Light and Relativity, UncategorizedLeave a comment

Back when I was in high school I did a term paper for some class (can’t remember which) ripping the heck out of Superman physics. Yeah, I was that kind of kid. If I recall correctly, I spent much of it slamming his supposed vision capabilities — they were fairly ludicrous even to a HS student and that was many refreshes of the DC universe ago.But for this post let’s consider a trope that’s been taken off the shelf again and again since those days, even in the movies. This rendition should get the idea across — Our Hero, in a desperate effort to fix a narrative hole the writers had dug themselves into, is forced to fly around the Earth at faster-than-light speeds, thereby reversing time so he can patch things up.

So many problems… Just for starters, the Earth is 8000 miles wide, Supey’s what, 6’6″?, so on this scale he shouldn’t fill even a thousandth of a pixel. OK, artistic license. Fine.

Second problem, only one image of the guy. If he’s really passing us headed into the past we should see two images, one coming in feet-first from the future and the other headed forward in both space and time. Oh, and because of the Doppler effect the feet-first image should be blue-shifted and the other one red-shifted.

Of course both of those images would be the wrong shape. The FitzGerald-Lorentz Contraction makes moving objects appear shorter in the direction of motion. In other words, if the Man of Steel were flying just shy of the speed of light then 6’6″ tall would look to us more like a disk with a short cape.

Tall-Dark-And-Muscular has other problems to solve on his way to the past. How does he get up there in the first place? Back in the day, DC explained that he “leaped tall buildings in a single bound.” That pretty much says ballistic high-jump, where all the energy comes from the initial impulse. OK, but consider the rebound effects on the neighborhood if he were to jump with as much energy as it would take to orbit a 250-lb man. People would complain.

Remember Einstein’s famous E=mc²? That mass m isn’t quite what most people think it is. Rather, it’s an object’s rest mass m₀ modified by a Lorentz correction to account for the object’s kinetic energy. In our hum-drum daily life that correction factor is basically 1.00000… When you get into the lightspeed ballpark it gets bigger.

Here’s the formula with the Lorentz correction in red: m=m₀/√[1-(v/c)²]. The square root nears zero as Superman’s velocity v approaches lightspeed c. When the divisor gets very small the corrected mass gets very large. If he got to the Lightspeed Barrier (where v=c) he’d be infinitely massive.

So you’ve got an infinite mass circling the Earth about 7 times a second — ocean tides probably couldn’t keep up, but the planet would be shaking enough to fracture the rock layers that keep volcanoes quiet. People would complain.

Of course, if he had that much mass, Earth and the entire Solar System would be orbiting him.

On the E side of Einstein’s equation, Superman must attain that massive mass by getting energy from somewhere. Gaining that last mile/second on the way to infinity is gonna take a lot of energy.

But it’s worse. Even at less than lightspeed, the Kryptonian isn’t flying in a straight line. He’s circling the Earth in an orbit. The usual visuals show him about as far out as an Earth-orbiting spacecraft. A GPS satellite’s stable 24-hour orbit has a 26,000 mile radius so it’s going about 1.9 miles/sec. Superman ‘s traveling about 98,000 times faster than that. Physics demands that he use a powered orbit, continuously expending serious energy on centripetal acceleration just to avoid flying off to Vega again.

The comic books have never been real clear on the energy source for Superman’s feats. Does he suck it from the Sun? I sure hope not — that’d destabilize the Sun and generate massive solar flares and all sorts of trouble.

Not even the DC writers would want Superman to wipe out his adopted planet just to fix up a plot point.

~~ Rich Olcott

Light’s hourglass

On September 5, 2016November 28, 2025 By rolcottIn Cosmology, Mathematics: Dimensions, Physics: Light and Relativity, UncategorizedLeave a comment

Terry Pratchett’s anthropomorphic character Death (who always speaks in UPPER CASE with a voice that sounds like tombstones falling) has a thing about hourglasses. So do physicists, but theirs don’t have sand in them. And they don’t so much represent Eternity as describe it. Maybe.

The prior post was all about spacetime events (an event is the combination of a specific (x,y,z) spatial location with a specific time t) and how the Minkowski diagram divides the Universe into mutually exclusive pieces:

“look but don’t touch” — the past, all the spacetime events which could have caused something to happen where/when we are
“touch but don’t look” — the future, the events where/when we can cause something to happen
“no look, no touch” — the spacelike part that’s so far away that light can’t reach us and we can’t reach it without breaching Einstein’s speed-of-light constraint
“here and now” — the tiny point in spacetime with address (ct,x,y,z)=(0,0,0,0)

Last week’s Minkowski diagram was two-dimensional. It showed time running along the vertical axis and Pythagorean distance d=√(x²+y²+z²) along the horizontal one. That was OK in the days before computer graphics, but it loaded many different events onto the same point on the chart. For instance, (0,1,0,0), (0,-1,0,0), (0,0,1,0) and (0,0,0,1) (and more) are all at d=1.

This chart is one dimension closer to what the physicists really think about. Here we have x and y along distinct axes. The z axis is perpendicular to all three, and if you can visualize that you’re better at it than I am. The xy plane (and the xyz cube if you’re good at it) is perpendicular to t.

That orange line was in last week’s diagram and it means the same thing in this one. It contains events that can use light-speed somehow to communicate with the here-and-now event. But now we see that the line into the future is just part of a cone (or a hypercone if you’re good at it).

If we ignite a flash of light at time t=0, at any positive time t that lightwave will have expanded to a circle (or bubble) with radius d=c·t. The circles form the “future” cone.

Another cone extends into the past. It’s made up of all the events from which a flash of light at time at some negative t would reach the here-and-now event.

The diagram raises four hotly debated questions:

Is the pastward cone actually pear-shaped? It’s supposed to go back to The Very Beginning. That’s The Big Bang when the Universe was infinitesimally small. Back then d for even the furthest event from (ct,0,0,0) should have been much smaller than the nanometers-to-lightyears range of sizes we’re familiar with today. But spacetime was smaller, too, so maybe everything just expanded in sync once we got past Cosmic Inflation. We may never know the answer.
What’s outside the cones? You think what you see around you is right now? Sorry. If the screen you’re reading this on is a typical 30 inches or so distant, the light you’re seeing left the screen 2½ nanoseconds ago. Things might have changed since then. We can see no further into the Universe than 14 billion lightyears, and even that only tells us what happened 14 billion years ago. Are there even now other Earth-ish civilizations just 15 billion lightyears away from us? We may never know the answer.
How big is “here-and-now”? We think of it as a size=zero mathematical point, but there are technical grounds to think that the smallest possible distance is the Planck length, 1.62×10^-35 meters. Do incidents that might affect us occur at a smaller scale than that? Is time quantized? We may never know the answers.
Do the contents of the futureward cone “already” exist in some sense, or do we truly have free will? Einstein thought we live in a block universe, with events in future time as fixed as those in past time. Other thinkers hold that neither past not future are real. I like the growing block alternative, in which the past is real and fixed but the future exists as maybes. We may never know the answer.

~~ Rich Olcott

You can’t get there from here

On August 29, 2016November 28, 2025 By rolcottIn Mathematics: Dimensions, Physics: Light and Relativity, UncategorizedLeave a comment

In this series of posts I’ve tried to get across several ideas:

By Einstein’s theories, in our Universe every possible combination of place and time is an event that can be identified with an “address” like (ct,x,y,z) where t is time, c is the speed of light, and x, y and z are spatial coordinates
The Pythagorean distance between two events is d=√[(x₁-x₂)²+(y₁-y₂)²+(z₁-z₂)²]
The Minkowski interval between two events is √[(ct₁-ct₂)²– d²]
When i=√(-1) shows up somewhere, whatever it’s with is in some way perpendicular to the stuff that doesn’t involve i

That minus sign in the third bullet has some interesting implications. If the time term is bigger then the spatial term, then the interval (that square root) is a real number. On the other hand, if the time term is smaller, the interval is an imaginary number and therefore is in some sense perpendicular to the real intervals.

We’ll see what that means in a bit. But first, suppose the two terms are exactly equal, which would make the interval zero. Can that happen?

Sure, if you’re a light wave. The interval can only be zero if d=(ct₁-ct₂). In other words, if the distance between two events is exactly the distance light would travel in the elapsed time between the same two events.

In this Minkowski diagram, we’ve got two number lines. The real numbers (time) run vertically (sorry, I know we had the imaginary line running upward in a previous post, but this is the way that Minkowski drew it). Perpendicular to that, the line of imaginary numbers (distance) runs horizontally, which is why that starscape runs off to the right.
The smiley face is us at the origin (0,0,0,0). If we look upwards toward positive time, that’s the future at our present location. Looking downward to negative time is looking into the past. Unless we move (change x and/or y and/or z), all the events in our past and future have addresses like (ct,0,0,0).

What about all those points (events) that aren’t on the time axis? Pick one and use its address to figure the interval between it and us. In general, the interval will be a complex number, part real and part imaginary, because the event you picked isn’t on either axis.

The two orange lines are special. Each of them is drawn through all the events for which the distance is equal to ct. All the intervals between those events and us are zero. Those are the events that could be connected to us by a light beam.

The orange connections go one way only — someone in our past (t less than zero) can shine a light at us that we’ll see when it reaches us. However, if someone in our future tried to shine a light our way, well, the passage of time wouldn’t allow it (except maybe in the movies). Conversely, we can shine a light at some spatial point (x,y,z) at distance d=√(x²+y²+z²) from us, but those photons won’t arrive until the event in our future at (d,x,y,z).

The rest of the Minkowski diagram could do for a Venn diagram. We at (0,0,0,0) can do something that will cause something to happen at (ct,x,y,z) to the left of the top orange line. However, we won’t be able to see that effect until we time-travel forward to its t. That region is “reachable but not seeable.”

Similarly, events to the left of the bottom orange line can affect us (we can see stars, for instance) but they’re in our past and we can’t cause anything to happen then/there. The region is “seeable but not reachable.”

Then there’s the overlap, the segment between the two orange lines. Events there are so far away in spacetime that the intervals between them and us are imaginary (in the mathematical sense). To put it another way, light can’t get here from there. Neither can cause and effect.

Physicists call that third region space-like, as opposed to the two time-like regions. Without a warp drive or some other way around Einstein’s universal speed limit, the edge of “space-like” will always be The Final Frontier.

~~ Rich Olcott

Does a photon experience time?

On August 22, 2016November 28, 2025 By rolcottIn Cosmology, Mathematics: Dimensions, Physics: Fundamentals, Physics: Light and Relativity, Uncategorized2 Comments

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?

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.

One 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

Calvin And Hobbes And i

On August 15, 2016November 28, 2025 By rolcottIn General: Fun Stuff, Mathematics: Dimensions, UncategorizedLeave a comment

I so miss Calvin and Hobbes, the wondrous, joyful comic strip that cartoonist Bill Watterson gave us between 1985 and 1995. Hobbes was a stuffed toy tiger — except that 6-year-old Calvin saw him as a walking, talking man-sized tiger with a sarcastic sense of humor.

So many things in life and physics are like Hobbes — they depend on how you look at them. As we saw earlier, a fictitious force disappears when viewed from the right frame of reference. There’s that particle/wave duality thing that Duc de Broglie “blessed” us with. And polarized light.

In an earlier post I mentioned that light is polar, in the sense that a single photon’s electric field acts to vibrate an electron (pole-to-pole) within a single plane.
In this video, orange, green and blue electromagnetic fields shine in from one side of the box onto its floor. Each color’s field is polar because it “lives” in only one plane. However, the beam as a whole is unpolarized because different components of the total field direct recipient electrons into different planes giving zero net polarization. The Sun and most other familiar light sources emit unpolarized light.

When sunlight bounces at a low angle off a surface, say paint on a car body or water at the beach, energy in a field that is directed perpendicular to the surface is absorbed and turned into heat energy. (Yeah, I’m skipping over a semester’s-worth of Optics class, but bear with me.) In the video, that’s the orange wave.

At the same time, fields parallel to the surface are reflected. That’s what happens to the blue wave.

Suppose a wave is somewhere in between parallel and perpendicular, like the green wave. No surprise, the vertical part of its energy is absorbed and the horizontal part adds to the reflection intensity. That’s why the video shows the outgoing blue wave with a wider swing than its incoming precursor had.

The net effect of all this is that low-angle reflected light is polarized and generally more intense than the incident light that induced it. We call that “glare.” Polarizing sunglasses can help by selectively blocking horizontally-polarized electric fields reflected from water, streets, and that *@%*# car in front of me.

Wave_Polarisation — David Jessop’s brilliant depiction of plane and circularly polarized light

Things can get more complicated. The waves in the first video are all in synch — their peaks and valleys match up (mostly). But suppose an x-directed field and a y-directed field are headed along the same course. Depending on how they match up, the two can combine to produce a field driving electrons along the x-direction, the y-direction, or in clockwise or counterclockwise circles. Check the red line in this video — RHC and LHC depict the circularly polarized light that sci-fi writers sometimes invoke when they need a gimmick.

Physicists have several ways to describe such a situation mathematically. I’ve already used the first, which goes back 380 years to René Descartes and the Cartesian x, y,… coordinate system he planted the seed for. We’ve become so familiar with it that reading a graph is like reading words. Sometimes easier.

In Cartesian coordinates we write x– and y-coordinates as separate functions of time t:
x = f₁(t)
y = f₂(t)
where each f could be something like 0.7·t²-1.3·t+π/4 or whatever. Then for each t-value we graph a point where the vertical line at the calculated x intersects the horizontal line at the calculated y.

But we can simplify that with a couple of conventions. Write √(-1) as i, and say that i-numbers run along the y-axis. With those conventions we can write our two functions in a single line:
x + i y = f₁(t) + i f₂(t)
One line is better than two when you’re trying to keep track of a big calculation.

But people have a long-running hang-up that’s part theory and part psychology. When Bombelli introduced these complex numbers back in the 16th century, mathematicians complained that you can’t pile up i thingies. Descartes and others simply couldn’t accept the notion, called the numbers “imaginary,” and the term stuck.

Which is why Hobbes the way Calvin sees him is on the imaginary axis.

~~ Rich Olcott

Is cyber warfare imaginary?

On August 8, 2016March 17, 2020 By rolcottIn General: Fun Stuff, Mathematics: Dimensions, UncategorizedLeave a comment

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.)

Suppose 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

The question Newton couldn’t answer

On August 1, 2016November 28, 2025 By rolcottIn Physics: Black Holes and Relativity, Physics: Newton and Gravity, UncategorizedLeave a comment

250 years ago, when people were getting used to the idea that the planets circle the Sun and not the other way around, they wondered how that worked. Isaac Newton said, “I can explain it with my Laws of Motion and my Law of Gravity.”

The first Law of Motion is that an object will move in a straight line unless acted upon by a force. If you’re holding a ball by a string and swing the ball in a circle, the reason the ball doesn’t fly away is that the string is exerting a force on the ball. Using Newton’s Laws, if you know the mass of the ball and the length of the string, you can calculate how fast the ball moves along that circle.

Newton said that the Solar System works the same way. Between the Sun and each planet there’s an attractive force which he called gravity. If you can determine three points in a planet’s orbit, you can use the Laws of Motion and the Law of Gravity to calculate the planet’s speed at any time, how close it gets to the Sun, even how much the planet weighs.

Astronomers said, “This is wonderful! We can calculate the whole Solar System this way, but… we don’t see any strings. How does gravity work?”

Newton was an honest man. His response was, “I don’t know how gravity works. But I can calculate it and that should be good enough.”

And that was good enough for 250 years until Albert Einstein produced his Theories of Relativity. This graphic shows one model of Einstein’s model of “the fabric of space.” According to the theory, light (the yellow threads) travels at 186,000 miles per second everywhere in the Universe.

As we’ve seen, the theory also says that space is curved and compressed near a massive object. Accordingly, the model’s threads are drawn together near the dark circle, which could represent a planet or a star or a black hole. If you were standing next to a black hole (but not too close). you’d feel fine because all your atoms and the air you breathe would shrink to the same scale. You’d just notice through your telescope that planetary orbits and other things in the Universe appear larger than you expect.

This video shows how a massive object’s space compression affects a passing light wave. The brown dot and the blue dot both travel at 186,000 miles per second, but “miles are shorter near a black hole.” The wave’s forward motion is deflected around the object because the blue dot’s miles are longer than the miles traveled by the brown dot.

When Einstein presented his General Theory of Relativity in 1916, his calculations led him to predict that this effect would cause a star’s apparent position to be altered by the Sun’s gravitational field.

An observer at the bottom of this diagram can pinpoint the position of star #1 by following its light ray back to the star’s location. Star #2, however, is so situated that its light ray is bent by our massive object. To the observer, star #2’s apparent position is shifted away from its true position.

In 1919, English physicist-astronomer Arthur Eddington led an expedition to the South Atlantic to test Einstein’s prediction. Why the South Atlantic? To observe the total eclipse of the sun that would occur there. With the Sun’s light blocked by the Moon, Eddington would be able to photograph the constellation Taurus behind the Sun.

Sure enough, in Eddington’s photographs the stars closest to the Sun were shifted in their apparent position relative to those further way. Furthermore, the sizes of the shifts were almost embarrassingly close to Einstein’s predicted values.

Eddington presented his photographs to a scientific conference in Cambridge and thus produced the first public confirmation of Einstein’s theory of gravity.

Wait, how does an object bending a light ray connect with that object’s pull on another mass? Another piece of Einstein’s theory says that if a light ray and a freely falling mass both start from the same point in spacetime, both will follow the same path through space. American physicist John Archibald Wheeler said, “Mass bends space, and bent space tells mass how to move.”

~~ Rich Olcott

Hard Science Ain't Hard

Author: rolcott

The Titanic Winds of Titan (And Venus)

Titan’s Atmosphere Is A Gas

Superluminal Superman

Superman flying at lightspeed? Umm… no

Light’s hourglass

You can’t get there from here

Does a photon experience time?

Calvin And Hobbes And i

Is cyber warfare imaginary?

The question Newton couldn’t answer