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

Gravity and other fictitious forces

On July 25, 2016November 28, 2025 By rolcottIn Physics: Fundamentals, Physics: Newton and Gravity, Uncategorized3 Comments

In this post I wrote, “gravitational force is how we we perceive spatial curvature.”
Here’s another claim — “Gravity is like centrifugal force, because they’re both fictitious.” Outrageous, right? I mean, I can feel gravity pulling down on me now. How can it be fictional?

Fictitious triangle — A fictitious triangle

“Fictitious,” not “fictional,” and there’s a difference. “Fictional” doesn’t exist, but a fictitious force is one that, to put it non-technically, depends on how you look at it.

Newton started it, of course. From our 21st Century perspective, it’s hard to recognize the ground-breaking impact of his equation F=m·a. Actually, it’s less a discovery than a set of definitions. Its only term that can be measured directly is a, the acceleration, which Newton defined as any change from rest or constant-speed straight-line motion. For instance, car buffs know that if a vehicle covers a ~~one-mile~~ half-mile (see comments) track in 60 seconds from a standing start, then its final speed is 60 mph (“zero to sixty in sixty”). Furthermore, we can calculate that it achieved a sustained acceleration of 1.47 ft/sec².

Both F and m, force and mass, were essentially invented by Newton and they’re defined in terms of each other. Short of counting atoms (which Newton didn’t know about), the only routes to measuring a mass boil down to

compare it to another mass (for instance, in a two-pan balance), or
quantify how its motion is influenced by a known amount of force.

Conversely, we evaluate a force by comparing it to a known force or by measuring its effect on a known mass.

Once the F=m·a. equation was on the table, whenever a physicist noticed an acceleration they were duty-bound to look for the corresponding force. An arrow leaps from the bow? Force stored as tension in the bowstring. A lodestone deflects a compass needle? Magnetic force. Objects accelerate as they fall? Newton identified that force, called it “gravity,” and showed how to calculate it and how to apply it to planets as well as apples. It was Newton who pointed out that weight is a measure of gravity’s force on a given mass.

Incidentally, to this day the least accurately known physical constant is Newton’s G, the Universal Gravitational Constant in his equation F=G·m₁·m₂/r². We can “weigh” planets with respect to each other and to the Sun, but without an independently-determined accurate mass for some body in the Solar System we can only estimate G. We’ll have a better value when we can see how much rocket fuel it takes to push an asteroid around.

But there are other accelerations that aren’t so easily accounted for. Ever ride in a car going around a curve and find yourself almost flung out of your seat? This little guy wasn’t wearing his seat belt and look what happened. The car accelerated because changing direction is an acceleration due to a lateral force. But the guy followed Newton’s First Law and just kept going in a straight line. Did he accelerate?

This is one of those “depends on how you look at it” cases. From a frame of reference locked to the car (arrows), he was accelerated outwards by a centrifugal force that wasn’t countered by centripetal force from his seat belt. However, from an earthbound frame of reference he flew in a straight line and experienced no force at all.

Suppose you’re investigating an object’s motion that appears to arise from a new force you’d like to dub “heterofugal.” If you can find a different frame of reference (one not attached to the object) or otherwise explain the motion without invoking the “new force,” then heterofugalism is a fictitious force.

Centrifugal and centripetal forces are fictitious. The “force” “accelerating” one plane towards another as they both fly to the North Pole in this tale is actually geometrical and thus also fictitious So is gravity.

In this post you’ll find a demonstration of gravity’s effect on the space around it. Just as a sphere’s meridians give the effect of a fictitious lateral force as they draw together near its poles, the compressive curvature of space near a mass gives the effect of a force drawing other masses inward.

~~ Rich Olcott

What’s that funnel about, really?

On July 18, 2016November 28, 2025 By rolcottIn Physics: Newton and Gravity, UncategorizedLeave a comment

If you’ve ever watched or read a space opera (oh yes, you have), you know about the gravity well that a spacecraft has to climb out of when leaving a planet. Every time I see the Museum’s gravity well model (photo below), I’m reminded of all the answers the guy gave to, “Johnny, what can you make of this?”

The model’s a great visitor-attracter with those “planets” whizzing around the “Sun,” but this one exhibit really represents several distinct concepts. For some of them it’s not quite the right shape.

The simplest concept is geometrical. “Down” is the direction you move when gravity’s pulling on you.

HS cone — Gravitational potential energy change
for small height differences

A gravity well model for that concept would be just a straight line between you and the neighborhood’s most intense gravity source.

You learned the second concept in high school physics class. Any object has gravitational potential energy that measures the amount of energy it would give up on falling. Your teacher probably showed you the equation GPE = m·g·h, where m is the mass of the object, h is its height above ground level, and g is a constant you may have determined in a lab experiment.

If the width of the gravity well model at a given height represents GPE at that level, the model is a simple straight-sided cone.

Newton energy cone — Gravitational potential energy change
for large height differences
The h indicates
an approximately linear range
where the HS equation could apply.

But of course it’s not that simple. Newton’s Law of Gravity says that the potential energy at any height r away from the planet’s center is proportional to 1/r.

Hmm… that looks different from the “proportional to h” equation. Which is right?

Both equations are valid, but over different distance scales. The HS teachers didn’t quite lie to you, but they didn’t give you the complete picture either. Your classroom was about 4000 miles (21,120,000 feet) from Earth’s center, whereas the usual experiments involve height differences of at most a dozen feet. Even the 20-foot drop from a second-story window is less than a millionth of the way down to Earth’s center.

Check my numbers:

Height h	1/(r+h) × 10⁸	Difference in 1/(r+h) × 10¹⁴
0	4.734,848,484	0
20	4.734,844,001	4.48
40	4.734,839,517	8.97
60	4.734,835,033	13.45
80	4.734,830,549	17.93
100	4.734,826,066	22.42

Sure enough, that’s a straight line (see the chart). Reminds me of how Newton’s Law of Gravity is valid except at very short distances. The HS Law of Gravity works fine for small spans but when the distances get big we have to use Newton’s equation.

We’re not done yet. That curvy funnel-shaped gravity well model could represent the force of gravity rather than its potential energy. Newton told us that the force goes as 1/r² so it decreases much more rapidly than the potential energy does as you get further away. The gravity force well has a correspondingly sharper curve to it than the gravity energy well.

Newton force cone — The *force* of gravity
or an embedding diagram

The funnel model could also represent the total energy required to get a real spacecraft off the surface and up into space. Depending on which sci-fi gimmickry is in play, the energy may come from a chemical or ion rocket, an electromagnetic railgun, or even a tractor beam from some mothership way up there.

No matter the technology, the theoretical energy requirement to get to a given height is the same. In practice, however, each technology is optimal for some situations but forbiddingly inefficient in others. Thus, each technology’s funnel has its own shape and that shape will change depending on the setting.

In modern physics, the funnel model could also represent Einstein’s theory of how a mass “bends” the space around it. (Take a look at this post, which is about how mass curves space by changing the local distance scale.) Cosmologists describe the resulting “shapes” with embedding diagrams that are essentially 2D pictures of 3D (or 4D) contour plots. The contours are closest together where space is most compressed, just as lines showing a steep hillside on a landscape contour map are close together.

The ED around a non-spinning object looks just like the force model picture above. No surprise — gravitational force is how we we perceive spatial curvature.

~~ Rich Olcott

A Summertime Slice of π

On July 11, 2016March 17, 2020 By rolcottIn Cosmology, UncategorizedLeave a comment

So you think you’re standing still? Let’s run some circles, all variations on the theme of 2πR…

Circles in circles — The Earth rotates on its axis,
as it and the moon revolve around their barycenter,
as the barycenter revolves around the Sun.
Not to scale, of course.

The Earth’s radius is 4,000 miles and it completes one rotation every 24 hours. Its circumference at the Equator (2πR) is 25,000 miles, so if you’re reading this in Ecuador you’re doing 25000/24 = 1041 miles per hour.

I’m writing this in Denver, at 39.75^oN, where the circumference perpendicular to the axis of rotation is only 19,200 miles. Sitting here I’m circling the Earth at 800 miles per hour. But that’s not all.

The Earth and the Moon both revolve around their common center of gravity (their barycenter). The barycenter is inside the Earth, offset from its center by 2881 miles. The center of the Earth runs a circle around the barycenter once every month (27.3 days), at a relatively piddly 27.6 miles per hour. But that’s not all.

Earth’s orbit is (nearly) a circle. The orbit’s radius is 93 million miles so its circumference is 584 million miles. If you ran that many miles in a year you’d have to hit a pace of 66,600 miles per hour (no rest stops). But that’s not all.

The Sun’s not just standing still all alone in space. It’s part of the Milky Way Galaxy, which rotates once per 230 million years. The Sun is about 26,000 light-years (152.8 quadrillion miles) from the center of the galaxy, so in one cycle it travels some 960 quadrillion miles. That’s a rate of 476,000 miles per hour. But that’s not all.

The Milky Way is one of about 50 galaxies in the Local Group. The galaxies move with respect to each other and the whole assembly undoubtedly rotates. Unfortunately, the astronomers are just now devising technology that can measure all that motion. Expect large numbers for the net speeds when they figure them out. But that’s not all.

The entire Local Group is flying towards a point in the constellation Centaurus. Our flight speed has been measured at about 1,430,000 miles per hour. The astronomers think the flight is linear, but on a larger scale it may be part of yet another rotation.

Feeling a bit dizzy? Have a frosty glass of iced tea with your delicious π and just let the Earth spin along.

~~ Rich Olcott

A Little Summertime Monkey Business

On July 4, 2016November 14, 2018 By rolcottIn UncategorizedLeave a comment

Surely you’ve heard of The Infinite Monkey Theorem. You probably don’t believe it. No way could that monkey accidentally type out anything meaningful, much less the complete works of Shakespeare. Well…

In several of his Discworld books, author Terry Pratchett featured something called Library-space, L-space for short. It’s defined as “a dimension that connects every library and book depository in the universe. L-Space is portrayed as a natural outgrowth of the fact that knowledge = power = energy = matter = mass and mass warps space, and therefore, libraries in the Discworld universe are a very dangerous place indeed for the unprepared”.

Somewhere, Pratchett wrote that L-space contains all the books that have been written, all those that will be written, and all those that would have been written but the author thought better of it. Well, how big is L-space?

To over-estimate, suppose L-space contains a billion (10⁹) books, each book is 500 pages long, each page contains 4000 characters, and the characters are chosen from an “alphabet” of 500 marks (upper- and lower-case letters, numbers and punctuation marks, all in normal, bold and italic forms in a several different fonts). One book would then contain two million marks.

Now, how many possible books are there, including ‘impossible’ character combinations like “zqzqzqzq”? We can construct a “possible” book by choosing some random one of the 500 marks as the first character, the same or a different one as the second character (500×500 = 500² = 250,000 possibilities so far) and so on, until we’ve built (or our monkey has typed) a two-million-character book. It could be a book that contains nothing but a string of a million copies of “zq” — but that’s OK, it’s still a possible book. So is the book that contains all the works of Shakespeare and so is a typo version that inconsistently misspells “Romeo.”

On this basis there are some 500^2,000,000 = 10^5,397,940 different possible books. L-space with only a billion books is thus very small indeed compared to the number of possible books. Put another way, the set of all possible books (which we can call B-space) could hold 10^5,397,931 versions of the L-space that initially seemed so immense.

Note that there are two distinct operations involved in the Monkey Theorem’s process

Generate a string of characters, and
Identify a meaningful substring within that.

The monkey* has no clue what it’s typing. Any given random string might or might not be intelligible to someone who reads English, or German, or Cherokee. The string might be a computer program in FORTRAN or JavaScript, or maybe a sequence of DNA icons for a gene mutation that makes green hair — or it (probably) would have no valid interpretation in any context.

The monkey doesn’t care, it’s just typing.

In the second step of the process someone has to recognize Macbeth or The Tempest buried in all the nonsense. If we were walking through the stacks of B-space and pulled a book off the shelf, what are the odds that the book we grabbed belongs to L-space?

The answer is one in 10^5,397,931. That’s a very small probability, BUT IT’S NOT ZERO. By construction, we’re guaranteed that all the L-space books are in B-space – but we have a vanishingly small chance of finding one of them.

Now for our extremely patient monkey who has been typing for a really, really long time. It’s been at it long enough to produce many, many copies of B-space. After all, even 10^5,397,940 is a very small number compared to infinity.

The core of the Infinite Monkey Theorem is that with so much opportunity for duplication, we are guaranteed that there exists at least one complete and perfect copy of B-space and so at least one good copy of L-space and so at least one good copy of all the works of Shakespeare. Also there’s at least one copy of “zqzqzqzq”.

The challenge is in laying hands on that one good copy. From a physicist’s perspective, it’s such a low-probability event that it can be ignored. On the other hand, the probability of Life arising on Earth was pretty low, too, but I’m glad it happened.

~~ Rich Olcott

* – I had a great “Monkeys typing” graphic, but they were chimps. Pratchett’s Diskworld Librarian would object, quite firmly, because apes aren’t monkeys.

** – I also had a pretty good “feature image,” a collage of many different monkey faces, but it seems at least one of them has a copyright lawyer. Now the feature image is a picture of my library prior to the down-sizing.

Throwing a Summertime curve

On June 27, 2016November 28, 2025 By rolcottIn Cosmology, Mathematics: Dimensions, Physics: Black Holes and Relativity, UncategorizedLeave a comment

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.

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.

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 2π, 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.

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.

Reflections in Einstein’s bubble

On June 20, 2016November 28, 2025 By rolcottIn Mathematics: Dimensions, Physics: Black Holes and Relativity, Physics: Light and Relativity, Physics: Quantum Mechanics, UncategorizedLeave a comment

There’s something peculiar in this earlier post where I embroidered on Einstein’s gambit in his epic battle with Bohr. Here, I’ll self-plagiarize it for you…

Consider some nebula a million light-years away. A million years ago an electron wobbled in the nebular cloud, generating a spherical electromagnetic wave that expanded at light-speed throughout the Universe.

Last night you got a glimpse of the nebula when that lightwave encountered a retinal cell in your eye. Instantly, all of the wave’s energy, acting as a photon, energized a single electron in your retina. That particular lightwave ceased to be active elsewhere in your eye or anywhere else on that million-light-year spherical shell.

Suppose that photon was yellow light, smack in the middle of the optical spectrum. Its wavelength, about 580nm, says that the single far-away electron gave its spherical wave about 2.1eV (3.4×10^-19 joules) of energy. By the time it hit your eye that energy was spread over an area of a trillion square lightyears. Your retinal cell’s cross-section is about 3 square micrometers so the cell can intercept only a teeny fraction of the wavefront. Multiplying the wave’s energy by that fraction, I calculated that the cell should be able to collect only 10^-75 joules. You’d get that amount of energy from a 100W yellow light bulb that flashed for 10^-73 seconds. Like you’d notice.

But that microminiscule blink isn’t what you saw. You saw one full photon-worth of yellow light, all 2.1eV of it, with no dilution by expansion. Water waves sure don’t work that way, thank Heavens, or we’d be tsunami’d several times a day by earthquakes occurring near some ocean somewhere.

Here we have a Feynman diagram, named for the Nobel-winning (1965) physicist who invented it and much else. The diagram plots out the transaction we just discussed. Not a conventional x-y plot, it shows Space, Time and particles. To the left, that far-away electron emits a photon signified by the yellow wiggly line. The photon has momentum so the electron must recoil away from it.

The photon proceeds on its million-lightyear journey across the diagram. When it encounters that electron in your eye, the photon is immediately and completely converted to electron energy and momentum.

Here’s the thing. This megayear Feynman diagram and the numbers behind it are identical to what you’d draw for the same kind of yellow-light electron-photon-electron interaction but across just a one-millimeter gap.

It’s an essential part of the quantum formalism — the amount of energy in a given transition is independent of the mechanical details (what the electrons were doing when the photon was emitted/absorbed, the photon’s route and trip time, which other atoms are in either neighborhood, etc.). All that matters is the system’s starting and ending states. (In fact, some complicated but legitimate Feynman diagrams let intermediate particles travel faster than lightspeed if they disappear before the process completes. Hint.)

Because they don’t share a common history our nebular and retinal electrons are not entangled by the usual definition. Nonetheless, like entanglement this transaction has Action-At-A-Distance stickers all over it. First, and this was Einstein’s objection, the entire wave function disappears from everywhere in the Universe the instant its energy is delivered to a specific location. Second, the Feynman calculation describes a time-independent, distance-independent connection between two permanently isolated particles. Kinda romantic, maybe, but it’d be a boring movie plot.

As Einstein maintained, quantum mechanics is inherently non-local. In QM change at one location is instantaneously reflected in change elsewhere as if two remote thingies are parts of one thingy whose left hand always knows what its right hand is doing.

Bohr didn’t care but Einstein did because relativity theory is based on geometry which is all about location. In relativity, change here can influence what happens there only by way of light or gravitational waves that travel at lightspeed.

In his book Spooky Action At A Distance, George Musser describes several non-quantum examples of non-locality. In each case, there’s no signal transmission but somehow there’s a remote status change anyway. We don’t (yet) know a good mechanism for making that happen.

It all suggests two speed limits, one for light and matter and the other for Einstein’s “deeper reality” beneath quantum mechanics.

~~ Rich Olcott

Gargh, His Heirs, and the AAAD Problem

On June 13, 2016November 28, 2025 By rolcottIn Physics: Fundamentals, Physics: Newton and Gravity, UncategorizedLeave a comment

Gargh, proto-humanity’s foremost physicist 2.5 million years ago, opened a practical investigation into how motion works. “I throw rock, hit food beast, beast fall down yes. Beast stay down no. Need better rock.” For the next couple million years, we put quite a lot of effort into making better rocks and better ways to throw them. Less effort went into understanding throwing.

There seemed to be two kinds of motion. The easier kind to understand was direct contact — “I push rock, rock move yes. Rock stop move when rock hit thing that move no.” The harder kind was when there wasn’t direct contact — “I throw rock up, rock hit thing no but come back down. Why that?”

Gargh was the first but hardly the last physicist to puzzle over the Action-At-A-Distance problem (a.k.a. “AAAD”). Intuition tells us that between pusher and pushee there must be a concrete linkage to convey the push-force. To some extent, the history of physics can be read as a succession of solutions to the question, “What linkage induces this apparent case of AAAD?”

Most of humanity was perfectly content with AAAD in the form of magic of various sorts. To make something happen you had to wish really hard and/or depend on the good will of some (generally capricious) elemental being.

Aristotle wasn’t satisfied with anything so unsystematic. He was just full of theories, many of which got in each other’s way. One theory was that things want to go where they’re comfortable because of what they’re made of — stones, for instance, are made of earth so naturally they try to get back home and that’s why we see them fall downwards (no concrete linkage, so it’s still AAAD).

Unfortunately, that theory didn’t account for why a thrown rock doesn’t just fall straight down but instead goes mostly in the direction it’s thrown. Aristotle (or one of his followers) tied that back to one of his other theories, “Nature hates a vacuum.” As the rock flies along, it pushes the air aside (direct contact) and leaves a vacuum behind it. More air rushes in to fill the vacuum and pushes the rock ahead (more direct contact).

We got a better (though still AAAD) explanation in the 17th Century when physicists invented the notions of gravity and inertia.

Newton made a ground-breaking claim in his Principia. He proposed that the Solar System is held together by a mysterious AAAD force he called gravity. When critics asked how gravity worked he shrugged, “I do not form hypotheses” (though he did form hypotheses for light and other phenomena).

Inertia is also AAAD. Those 17th Century savants showed that inertial forces push mass towards the Equator of a rotating object. An object that’s completely independent of the rest of the Universe has no way to “know” that it’s rotating so it ought to be a perfect sphere. In fact, the Sun and each of its planets are wider at the equator than you’d expect from their polar diameters. That non-sphere-ness says they must have some AAAD interaction with the rest of the Universe. A similar argument applies to linear motion; the general case is called Mach’s Principle.

The ancients knew of the mysterious AAAD agents electricity and its fraternal twin, magnetism. However, in the 19th Century James Clerk Maxwell devised a work-around. Just as Newton “invented” gravity, Maxwell “invented” the electromagnetic field. This invisible field isn’t a material object. However, waves in the field transmit electromagnetic forces everywhere in the Universe. Not AAAD, sort of.

It wasn’t long before someone said, “Hey, we can calculate gravity that way, too.” That’s why we now speak of a planet’s gravitational field and gravitational waves.

But the fields still felt like AAAD because they’re not concrete. Some modern physicists stand that objection on its head. Concrete objects, they say, are made of atoms which themselves are nothing more than persistent fluctuations in the electromagnetic and gravitational fields. By that logic, the fields are what’s fundamental — all motion is by direct contact.

Einstein moved resolutely in both directions. He negated gravity’s AAAD-ness by identifying mass-contorted space as the missing linkage. On the other hand, he “invented” quantum entanglement, the ultimate spooky AAAD.

~~ Rich Olcott

Hard Science Ain't Hard

Author: rolcott

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