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Einstein’s theory of diffusion in liquids, and my extension.

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In 1905 and 1908, Einstein developed two formulations for the diffusion of a small particle in a liquid. As a side-benefit of the first derivation, he demonstrated the visible existence of molecules, a remarkable piece of work. In the second formulation, he derived the same result using non-equilibrium thermodynamics, something he seems to have developed on the spot. I’ll give a brief version of the second derivation, and will then I’ll show off my own extension. It’s one of my proudest intellectual achievements.

But first a little background to the problem. In 1827, a plant biologist, Robert Brown examined pollen under a microscope and noticed that it moved in a jerky manner. He gave this “Brownian motion” the obvious explanation: that the pollen was alive and swimming. Later, it was observed that the pollen moved faster in acetone. The obvious explanation: pollen doesn’t like acetone, and thus swims faster. But the pollen never stopped, and it was noticed that cigar smoke also swam. Was cigar smoke alive too?

Einstein’s first version of an answer, 1905, was to consider that the liquid was composed of atoms whose energy was a Boltzmann distribution with an average of E= kT in every direction where k is the Boltzmann constant, and k = R/N. That is Boltsman’s constant equals the gas constant, R, divided by Avogadro’s number, N. He was able to show that the many interactions with the molecules should cause the pollen to take a random, jerky walk as seen, and that the velocity should be faster the less viscous the solvent, or the smaller the length-scale of observation. Einstein applied the Stokes drag equation to the solute, the drag force per particle was f = -6πrvη where r is the radius of the solute particle, v is the velocity, and η is the solution viscosity. Using some math, he was able to show that the diffusivity of the solute should be D = kT/6πrη. This is called the Stokes-Einstein equation.

In 1908 a French physicist, Jean Baptiste Perrin confirmed Einstein’s predictions, winning the Nobel prize for his work. I will now show the 1908 Einstein derivation and will hope to get to my extension by the end of this post.

Consider the molar Gibbs free energy of a solvent, water say. The molar concentration of water is x and that of a very dilute solute is y. y<<1. For this nearly pure water, you can show that µ = µ° +RT ln x= µ° +RT ln (1-y) = µ° -RTy.

Now, take a derivative with respect to some linear direction, z. Normally this is considered illegal, since thermodynamic is normally understood to apply to equilibrium systems only. Still Einstein took the derivative, and claimed it was legitimate at nearly equilibrium, pseudo-equilibrium. You can calculate the force on the solvent, the force on the water generated by a concentration gradient, Fw = dµ/dz = -RT dy/dz.

Now the force on each atom of water equals -RT/N dy/dz = -kT dy/dz.

Now, let’s call f the force on each atom of solute. For dilute solutions, this force is far higher than the above, f = -kT/y dy/dz. That is, for a given concentration gradient, dy/dz, the force on each solute atom is higher than on each solvent atom in inverse proportion to the molar concentration.

For small spheres, and low velocities, the flow is laminar and the drag force, f = 6πrvη.

Now calculate the speed of each solute atom. It is proportional to the force on the atom by the same relationship as appeared above: f = 6πrvη or v = f/6πrη. Inserting our equation for f= -kT/y dy/dz, we find that the velocity of the average solute molecule,

v = -kT/6πrηy dy/dz.

Let’s say that the molar concentration of solvent is C, so that, for water, C will equal about 1/18 mols/cc. The atomic concentration of dilute solvent will then equal Cy. We find that the molar flux of material, the diffusive flux equals Cyv, or that

Molar flux (mols/cm2/s) = Cy (-kT/6πrηy dy/dz) = -kTC/6πrη dy/dz -kT/6πrη dCy/dz.

where Cy is the molar concentration of solvent per volume.

Classical engineering comes to a similar equation with a property called diffusivity. Sp that

Molar flux of y (mols y/cm2/s) = -D dCy/dz, and D is an experimentally determined constant. We thus now have a prediction for D:

D = kT/6πrη.

This again is the Stokes Einstein Equation, the same as above but derived with far less math. I was fascinated, but felt sure there was something wrong here. Macroscopic viscosity was not the same as microscopic. I just could not think of a great case where there was much difference until I realized that, in polymer solutions there was a big difference.

Polymer solutions, I reasoned had large viscosities, but a diffusing solute probably didn’t feel the liquid as anywhere near as viscous. The viscometer measured at a larger distance, more similar to that of the polymer coil entanglement length, while a small solute might dart between the polymer chains like a rabbit among trees. I applied an equation for heat transfer in a dispersion that JK Maxwell had derived,

where κeff is the modified effective thermal conductivity (or diffusivity in my case), κl and κp are the thermal conductivity of the liquid and the particles respectively, and φ is the volume fraction of particles. 

To convert this to diffusion, I replaced κl by Dl, and κp by Dp where

Dl = kT/6πrηl

and Dp = kT/6πrη.

In the above ηl is the viscosity of the pure, liquid solvent.

The chair of the department, Don Anderson didn’t believe my equation, but agreed to help test it. A student named Kit Yam ran experiments on a variety of polymer solutions, and it turned out that the equation worked really well down to high polymer concentrations, and high viscosity.

As a simple, first approximation to the above, you can take Dp = 0, since it’s much smaller than Dl and you can take Dl to equal Dl = kT/6πrηl as above. The new, first order approximation is:

D = kT/6πrηl (1 – 3φ/2).

We published in Science. That is I published along with the two colleagues who tested the idea and proved the theory right, or at least useful. The reference is Yam, K., Anderson, D., Buxbaum, R. E., Science 240 (1988) p. 330 ff. “Diffusion of Small Solutes in Polymer-Containing Solutions”. This result is one of my proudest achievements.

R.E. Buxbaum, March 20, 2024

Relativity’s twin paradox explained, and why time is at right angles to space.

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One of the most famous paradoxes of physics is explained wrong — always. It makes people feel good to think they understand it, but the explanation is wrong and confusing, and it drives young physicists in a wrong direction. The basic paradox is an outgrowth of the special relativity prediction that time moves slower if you move faster.

Thus, if you entered a spaceship and were to travel to a distant star at 99% the speed of light, turn around and get here 30 years, you would have aged far less than 30 years. You and everyone else on the space ship would have aged three years, 1/10 as much as someone on earth.

The paradox part, not that the above isn’t weird enough by itself, is that the person in the spaceship will imagine that he (or she) is standing still, and that everyone on earth is moving away at 99% the speed of light. Thus, the person on the spaceship should expect to find that the people on earth will age slower. That is, the person on the space ship should return from his (or her) three year journey, expecting to find that the people on earth have only aged 0.3 years. Obviously, only one of these expectations can be right, but it’s not clear which (It’s the first one), nor is it clear why.

The wrong explanation appears in an early popular book, “Mr Tompkins in Wonderland”, by Physicist, George Gamow. The book was written shortly after Relativity was proposed, and involves a Mr Tompkins who falls asleep in a physics lecture. Mr. Tompkins dreams he’s riding on a train going near the speed of light, finds things are shorter and time is going slower. He then asks the paradox question to the conductor, who admits he doesn’t quite know how it works (perhaps Gamow didn’t), but that “it has something do do with the brakeman.” That sounds like Gamow is saying the explanation has to do with deceleration at the turn around, or general relativity in general, implying gravity could have a similarly large effect. It doesn’t work that way, and the effect of 1G gravity is small, but everyone seems content to explain the paradox this way. This is particularly unfortunate because these include physicists clouding an already cloudy issue.

In the early days of physics, physicists tried to explain things with a little legitimate math to the lay audience. Gamow did this, as did Einstein, Planck, Feynman, and most others. I try to do this too. Nowadays, physicists have removed the math, and added gobbledygook. The one exception here are the cinematographers of Star Wars. They alone show the explanation correctly.

The explanation does not have to do general relativity or the acceleration at the end of the journey (the brakeman). Instead of working through some acceleration, general relativity effect, the twin paradox works with simple, special relativity: all space contracts for the duration of the trip, and everything in it gets shorter. The person in this spaceship will see the distance to the star shrink by 90%. Traveling there thus takes 1/10th the time because the distance is 1/10th. There and back at 99% the speed of light, takes exactly 3 years.

The equation for time contraction is: t’ = v/x° √(1-(v/c)2) = t° √(1-(v/c)2) where t’ is the time in the spaceship, v is the speed, x° is the distance traveled (as measured from earth), and c is the speed of light. For v/c = .99, we find that √1-(v/c)2 is 0.1. We thus find that t’ = 0.1 t°. When dealing with the twin paradox, it’s better to say that x’ = 0.1x° where x’ is the distance to the star as seen from the spaceship. In either case, when the people on the space ship accelerate, they see the distance in front of them shrink, as shown in Star Wars, below.

Star Wars. The millennium falcon jumps to light speed, and beyond.

That time was at right angles to space was a comment in one of Einstein’s popular articles and books; he wrote several, all with some minimal mathematics Current science has no math, and a lot of politics, IMHO, and thus is not science.

He showed that time and space are at right angles by analogy from Pythagoras. Pythagoras showed that distance on a diagonal, d between two points at right angles, x and y is d = √(x2 + y2). Another way of saying this is d2 =x2 + y2. The relationship is similar for relativistic distances. To explain the twin paradox, we find that the square of the effective distance, x’2 = x°2 (1 – (v/c)2) = x°2 – (x°v)2/c2 = x°2 – (x°v/c)2 = x°2 – (t°2/c2). Here, x°2 is the square of the original distance, and it comes out that the term, – (t°2/c2) behaves like the square of an imaginary distance that is at right angles to it. It comes out that co-frame time, t° behaves as if it were a distance with a scale factor of i/c.

For some reason people today read books on science by non-scientist ‘explainers.’ I These books have no math, and I guess they sell. Publishers think they are helping democratize science, perhaps. You are better off reading the original thinkers, IMHO.

Robert Buxbaum, July 16, 2023. In his autobiography, Einstein claimed to be a fan of scientist -philosopher, Ernst Mach. Mach derived the speed of sound from a mathematical analysis of thermodynamics. Einstein followed, considering that it must be equally true to consider an empty box traveling in space to be one that carries its emptiness with it, as to assume that fresh emptiness comes in at one end and leaves by the other. If you set the two to be equal mathematically, you conclude that both time and space vary with velocity. Similar analysis will show that atoms are real, and that energy must travel in packets, quanta. Einstein also did fun work on the curvature of rivers, and was a fan of this sail ship design. Here is some more on the scientific method.

Rotating sail ships and why your curve ball doesn’t curve.

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The Flettner-sail ship, Barbara, 1926.

Sailing ships are wonderfully economic and non-polluting. They have unlimited range because they use virtually no fuel, but they tend to be slow, about 5-12 knots, about half as fast as Diesel-powered ships, and they can be stranded for weeks if the wind dies. Classic sailing ships also require a lot of manpower: many skilled sailors to adjust the sails. What’s wanted is an easily manned, economical, hybrid ship: one that’s powered by Diesel when the wind is light, and by a simple sail system when the wind blows. Anton Flettner invented an easily manned sail and built two ships with it. The Barbara above used a 530 hp Diesel and got additional thrust, about an additional 500 hp worth, from three, rotating, cylindrical sails. The rotating sales produced thrust via the same, Magnus force that makes a curve ball curve. Barbara went at 9 knots without the wind, or about 12.5 knots when the wind blew. Einstein thought it one of the most brilliant ideas he’d seen.

Force diagram of Flettner rotor (Lele & Rao, 2017)

The source of the force can be understood with help of the figure at left and the graph below. When a simple cylinder sits in the wind, with no spin, α=0, the wind force is essentially drag, and is 1/2 the wind speed squared, times the cross-sectional area of the cylinder, Dxh, and the density of air. Multiply this by a drag coefficient, CD, that is about 1 for a non-spinning cylinder, and about 2 for a fast spinning cylinder. FD= CDDhρv2/2.

A spinning cylinder has lift force too. FL= CLDhρv2/2.

Numerical lift coefficients versus time, seconds for different ratios of surface speed to wind speed, a. (Mittal & Kumar 2003), Journal of Fluid Mechanics.

As graphed in the figure at right, CL is effectively zero with sustained vibrations at zero spin, α=0. Vibrations are useless for propulsion, and can be damaging to the sail, though they are helpful in baseball pitching, producing the erratic flight of knuckle balls. If you spin a cylindrical mast at more than α=2.1 the vibrations disappear, and you get significant lift, CL= 6. At this rotation speed the fast surface moves with the wind at 2.1 times the wind speed. That is it moves significantly faster than the wind. The other side of the rotor moves opposite the wind, 1.1 times as fast as the wind. The coefficient of lift lift, CL= 6, is more than twice that found with a typical, triangular, non-rotating sail. Rotation increases the drag too, but not as much. The lift is about 4 times the drag, far better than in a typical sail. Another plus is that the ship can be propelled forward or backward -just reverse the spin direction. This is very good for close-in sailing.

The sail lift, and lift to drag ratio, increases with rotation speed reaching very values of 10 to 18 at α values of 3 to 4. Flettner considered α=3.5. optimal. At this α you get far more thrust than with a normal sail, and you can go faster than the wind, and far closer to the wind than with any normal sail. You don’t want α values above 4.2 because you start seeing vibrations again. Also more rotation power is needed (rotation power goes as ω2); unless the wind is strong, you might as well use a normal propeller.

The driving force is always at right angles to the perceived wind, called the “fair wind”, and the fair wind moves towards the front as the ship speed increases. Controlling the rotation speed is somewhat difficult but important. Flettner sails were no longer used by the 1930s because fuel became cheaper and control was difficult. Normal sails weren’t being used either for the same reasons.

In the early 1980s, there was a return to the romantic. Famous underwater explorer, Jacques Cousteau, revived a version of the Flettner sail for his exploratory ship, the Alcyone. He used aluminum sails, and an electric motor for rotation. He claimed that the ship drew more than half of its power from the wind, and claimed that, because of computer control, it could sail with no crew. This claim was likely bragging, but he bragged a lot. Even with today’s computer systems, people are needed to steer and manage things in case something goes wrong. The energy savings were impressive, though, enough so that some have begun to put Flettner sails on cargo ships, as a right. This is an ideal use since cargo ships go about as fast as a typical wind, 10- 20 knots. It’s reported that, Flettner- powered cargo ships get about 20% of their propulsion from wind power, not an insignificant amount.

And this gets us to the reason your curve ball does not curve: it’s likely you’re not spinning it fast enough. To get a good curve, you want the ball to spin at α =3, or about 1.5 times the rate you’d get by rolling the ball off your fingers. You have to snap your wrist hard to get it to spin this fast. As another approach, you can aim for α=0, a knuckle ball, achieved with zero rotation. At α=0, the ball will oscillate. It’s hard to do, but your pitch will be nearly impossible to hit or catch. Good luck.

Robert Buxbaum, March 22, 2023. There are also Flettner airplane designs where horizontal, cylindrical “wings” rotate to provide high lift with short wings and a relatively low power draw. So-far, these planes are less efficient and slower than a normal helicopter. The idea could bear more development work, IMHO. Einstein had an eye for good ideas.

Einstein’s theory of happiness

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Note for a talk in Tokyo: Einstein's theory of happiness.

Note for a talk in Tokyo: Einstein’s theory of happiness.

In 1922, Einstein was in Tokyo to give a speech, and had just recently been informed that he would win the Nobel prize. He knew that he’d be more famous than he had been, and everyone else did too. The prize money and more had already been contracted out to his wife for his divorce, but most people didn’t know that, and the few who did, didn’t realize that even after receiving the prize, he’d remain as poor as he had been. Anyway, shortly after the announcement a bell boy delivered something to his room, but Einstein had no money available. Instead he gave the bell-boy two scraps of thoughts for the talk, one of them on the Tokyo hotel stationery. The more famous one, “his theory of happiness” says, In German:

“A calm and modest life brings more happiness than the pursuit of success combined with constant restlessness.” The note is signed, Albert Einstein, dated November 1922 Tokyo. It sold at action October, 2018 for $1.56 million, not a bad tip, in both senses of the word. Einstein told the bell-boy that this note would probably be worth more than the usual tip. It was, and is.

In general Einstein told people to avoid academia, and instead go into something productive that you can do well for an income. Do your creative work, he advised, in your spare time, he advised; it ruins the enjoyment of creativity to always have to discover something new for your income, “always have to pull a rabbit out of your hat.” Einstein’s happiest time, and his most productive were his years working at the patent office in Bern, Switzerland, while doing physics in his spare time at home. Einstein produced relatively little of permanent physics value in the years following 1922. The discovery that Einstein’s theories predicted gravitational waves was not Einstein’s, nor was the discovery that his equations suggested an expanding universe. The former was the suggestion of, Howard Robertson, a reviewer of a paper by Einstein, and the latter was made by a Belgian scientist-priest named Georges Lemaitre. it was only after Hubble observed an expanding universe in 1929 that Einstein realized that Lemaitre had been right, and only in 1936 that he came to accept gravitational waves. Gravitational waves were finally observed in 2016. The observation earned Rainer Weiss, Barry Barish, and Kip Thorne the 2017 Nobel Prize in physics.

I’ve written about Einstein a few times. He seems to have been among the few creative people who lived a happy, productive life and died well liked by all. Here are some life lessons, and some thoughts on how you tel a genius from a nut. You can find out more about Einstein’s love letters and his divorce here, including about the divorce settlement.

Robert Buxbaum, November 2, 2018. The essence of a nice gift is in the note.

Our expanding, black hole universe

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In a previous post I showed a classical derivation of the mass-to-size relationship for black -holes and gave evidence to suggest that our universe (all the galaxies together) constitute a single, large black hole. Everything is inside the black hole and nothing outside but empty space — We can tell this because you can see outside from inside a black hole — it’s only others, outside who can not see in (Finkelstein, Phys Rev. 1958). Not that there appear to be others outside the universe, but if they were, they would not be able to see us.

In several ways having a private, black hole universe is a gratifying thought. It provides privacy and a nice answer to an easily proved conundrum: that the universe is not infinitely big. The black hole universe that ends as the math requires, but not with a brick wall, as i the Hitchhiker’s guide (one of badly-laid brick). There are one or two problems with this nice tidy solution. One is that the universe appears to be expanding, and black holes are not supposed to expand. Further, the universe appears to be bigger than it should be, suggesting that it expanded faster than the speed of light at some point. its radius now appears to be 40-46 billion light years despite the universe appearing to have started as a point some 14 billion years ago. That these are deeply disturbing questions does not stop NASA and Nova from publishing the picture below for use by teachers. This picture makes little sense, but it’s found in Wikipedia and most, newer books.

Standard picture of the big bang theory. Expansions, but no contractions.

Standard picture of the big bang theory: A period of faster than light expansion (inflation) then light-speed, accelerating expansion. NASA, and Wikipedia.

We think the creation event occurred some 14 billion years ago because we observe that the majority of galaxies are expanding from us at a rate proportional to their distance from us. From this proportionality between the rate of motion and the distance from us, we conclude that we were all in one spot some 14 billion years ago. Unfortunately, some of the most distant galaxies are really dim — dimmer than they would be if they were only 14 billion light years away. The model “explains this” by a period of inflation, where the universe expanded faster than the speed of light. The current expansion then slowed, but is accelerating again; not slowing as would be expected if it were held back by gravity of the galaxies. Why hasn’t the speed of the galaxies slowed, and how does the faster-than-light part work? No one knows. Like Dr. Who’s Tardis, our universe is bigger on the inside than seems possible.

Einstein's preferred view of the black-hole universe is one that expands and contracts at some (large) frequency. It could explain why the universe is near-uniform.

Einstein’s oscillating universe: it expands and contracts at some (large) frequency. Oscillations would explain why the universe is near-uniform, but not why it’s so big or moving outward so fast.

Einstein’s preferred view was of an infinite space universe where the mass within expands and contracts. He joked that two things were infinite, the universe and stupidity… see my explanation... In theory, gravity could drive the regular contractions to an extent that would turn entropy backward. Einstein’s oscillating model would explain how the universe is reasonably stable and near-uniform in temperature, but it’s not clear how his universe could be bigger than 14 billion light years across, or how it could continue to expand as fast as it does. A new view, published this month suggests that there are two universes, one going forward in time the other backward. The backward in time part of the universe could be antimatter, or regular matter going anti entropy (that’s how I understand it — If it’s antimatter, we’d run into the it all the time). Random other ideas float through the physics literature: that we’re connected to other space through a black hole/worm hole, perhaps to many other universes by many worm holes in fractal chaos, see for example, Physics Reports, 1992.

The forward-in-time expansion part of the two universes model.

The forward-in-time expansion part of the two universes model. This drawing, like the first, is from NASA.

For all I know, there are these many black hole  tunnels to parallel universes. Perhaps the universal constant and all these black-hole tunnels are windows on quantum mechanics. At some point the logic of the universe seems as perverse as in the Hitchhiker guide.

Something I didn’t mention yet is the Higgs boson, the so-called God particle. As in the joke, it’s supposed to be responsible for mass. The idea is that all particles have mass only by interaction with these near-invisible Higgs particles. Strong interactions with the Higgs are what make these particles heavier, while weaker – interacting particles are perceived to have less gravity and inertia. But this seems to me to be the theory that Einstein’s relativity and the 1919 eclipse put to rest. There is no easy way for a particle model like this to explain relativistic warping of space-time. Without mass being able to warp space-time you’d see various degrees of light bending around the sun, and preferential gravity in the direction of our planet’s motion: things we do not see. We’re back in 1900, looking for some plausible explanation for the uniform speed of light and Lawrence contraction of space.As likely an explanation as any the_hitchhikers_guide_to_the_galaxy

Dr. r µ ßuxbaum. December 20, 2014. The  meaning of the universe could be 42 for all I know, or just pickles down the worm hole. No religion seems to accept the 14 billion year old universe, and for all I know the God of creation has a wicked sense of humor. Carry a towel and don’t think too much.

Einstein failed high-school math –not.

I don’t know quite why people persist in claiming that Einstein failed high school math. Perhaps it’s to put down teachers –who clearly can’t teach or recognize genius — or perhaps to stake a claim to a higher understanding that’s masked by ADHD — a disease Einstein is supposed to have had. But, sorry to say, it ain’t true. Here’s Einstein’s diploma, 1896. His math and physics scores are perfect. Only his English seems to have been lacking. He would have been 17 at the time.

Einstein's high school diploma

Albert Einstein’s high school diploma, 1896.

Robert Buxbaum, December 16, 2014. Here’s Einstein relaxing in Princeton. Here’s something on black holes, and on High School calculus for non-continuous functions.

A simple, classical view of and into black holes

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Black holes are regions of the universe where gravity is so strong that light can not emerge. And, since the motion of light is related to the fundamental structure of space and time, they must also be regions where space curves on itself, and where time appears to stop — at least as seen by us, from outside the black hole. But what does space-time look like inside the black hole.

NASA's semi-useless depiction of a black hole -- one they created for educators. I'm not sure what you're supposed to understand from this.

NASA’s semi-useless depiction of a black hole — one they created for educators. Though it’s sort of true, I’m not sure what you’re supposed to understand from this. I hope to present a better version.

From our outside perspective, an object tossed into a black hole will appear to move slower as it approaches the hole, and at the hole horizon it will appear to have stopped. From the inside of the hole, the object appears to just fall right in. Some claim that tidal force will rip it apart, but I think that’s a mistake. Here’s a simple, classical way to calculate the size of a black hole, and to understand why things look like they do and do what they do.

Lets begin with light, and accept, for now, that light travels in particle form. We call these particles photons; they have both an energy and a mass, and mostly move in straight lines. The energy of a photon is related to its frequency by way of Plank’s constant. E = hν, where E is the photon energy, h is Plank’s constant and ν is frequency. The photon mass is related to its energy by way of the formula m=E/c2, a formula that is surprisingly easy to derive, and often shown as E= mc2. The version that’s relevant to photons and black holes is:

m =  hν/c2.

Now consider that gravity affects ν by affecting the energy of the photon. As a photon goes up, the energy and frequency goes down as energy is lost. The gravitational force between a star, mass M, and this photon, mass m, is described as follows:

F = -GMm/r2

where F is force, G is the gravitational constant, and r is the distance of the photon from the center of the star and M is the mass of the star. The amount of photon energy lost to gravity as it rises from the surface is the integral of the force.

∆E = – ∫Fdr = ∫GMm/r2 dr = -GMm/r

Lets consider a photon of original energy E° and original mass m°= E°/c2. If ∆E = m°c2, all the energy of the original photon is lost and the photon disappears. Now, lets figure out the height, r° such that all of the original energy, E° is lost in rising away from the center of a star, mass M. That is let calculate the r for which ∆E = -E°. We’ll assume, for now, that the photon mass remains constant at m°.

E° = GMm°/r° = GME°/c2r°.

We now eliminate E° from the equation and solve for this special radius, r°:

r° =  GM/c2.

This would be the radius of a black hole if space didn’t curve and if the mass of the photon didn’t decrease as it rose. While neither of these assumptions is true, the errors nearly cancel, and the true value for r° is double the size calculated this way.

r° = 2GM/c2

r° = 2.95 km (M/Msun).

schwarzschild

Karl Schwarzschild 1873-1916.

The first person to do this calculation was Karl Schwarzschild and r° is called the Schwarzschild radius. This is the minimal radius for a star of mass M to produce closed space-time; a black hole. Msun is the mass of our sun, sol, 2 × 1030 kg.  To make a black hole one would have to compress the mass of our sun into a ball of 2.95 km radius, about the size of a small asteroid. Space-time would close around it, and light starting from the surface would not be able to escape.

As it happens, our sun is far bigger than an asteroid and is not a black hole: we can see light from the sun’s surface with minimal space-time deformation (there is some seen in the orbit of Mercury). Still, if the mass were a lot bigger, the radius would be a lot bigger and the density would be less. Consider a black hole the same mass as our galaxy, about 1 x1012 solar masses, or 2 x 1042  kg. This number is ten times what you might expect since our galaxy is 90% dark matter. The Schwarzschild radius with the mass of our galaxy would be 3 x 1012 km, or 0.3 light years. That’s far bigger than our solar system, and about 1/20 the distance to the nearest star, Alpha Centauri. This is a very big black hole, though it is far smaller than our galaxy, 5 x 1017 km, or 50,000 light years. The density, though is not all that high.

Now let’s consider a black hole comprising 15 billion galaxies, the mass of the known universe. The folks at Cornell estimate the sum of dark and luminous matter in the universe to be 3 x 1052 kg, about 15 billion times the mass of our galaxy. This does not include the mass hidden in the form of dark energy, but no one’s sure what dark energy is, or even if it really exists. A black hole encompassing this, known mass would have a Schwarzschild radius about 4.5 billion light years, or about 1/3 the actual size of the universe when size is calculated based on its Hubble-constant age, 14 billion years. The universe may be 2-3 times bigger than this on the inside because space is curved and, rather like Dr. Who’s Tardis it’s bigger on the inside, but in astronomical terms a factor of 3 or 10 is nothing: the actual size of the known universe is remarkably similar to its Schwarzschild radius, and this is without considering the mass its dark energy must have if it exists.

Standard picture of the big bang theory. Dark energy causes the latter-stage expansion.

Standard picture of the big bang theory. Dark energy causes the latter-stage expansion.

The evidence for dark energy is that the universe is expanding faster and faster instead of slowing. See figure. There is no visible reason for the acceleration, but it’s there. The source of the energy might be some zero-point effect, but wherever it comes from, the significant amount of energy must have significant mass, E = mc2. If the mass of this energy is 3 to 10 times the physical mass, as seems possible, we are living inside a large black hole, something many physicists, including Einstein considered extremely likely and aesthetically pleasing. Einstein originally didn’t consider the possibility that the hole could be expanding, but a reviewer of one of his articles convinced him it was possible.

Based on the above, we now know how to calculate the size of a black hole of any mass, and we now know what a black hole the size of the universe would look like from the inside. It looks just like home. Wait for further posts on curved space-time. For some reason, no religion seems to embrace science’s 14 billion year old, black-hole universe (expanding or not). As for the tidal forces around black holes, they are horrific only for the small black holes that most people write about. If the black hole is big, the tidal forces are small.

 Dr. µß Buxbaum Nov 17, 2014. The idea for this post came from an essay by Isaac Asimov that I read in a collection called “Buy Jupiter.” You can drink to the Schwarzchild radius with my new R° cocktail.

Political tensegrity: the west is best

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We are regularly lectured about the lack of kindness and humility of the western countries. Eastern and communist leaders in Russia, Iran, Saudi Arabia point to Western pollution, consumerism, unemployment as prof you need a strong leader and central control to do good by regulation, thought policing, and wealth redistribution.

Let me point out that the good these leaders provide is extracted from the populace, and the advantage of central control is rarely as clear to the populous as to the leadership. When leaders redistribute wealth or place limits on the internet, movies or books, the leaders are generally exempted, and the populous are not made more moral or generous either. One does not say a prisoner or slave-worker is more generous of moral than one on the outside despite the prisoner working for free. The leaders feel certain they are protecting their people from thought and greed, but it isn’t clear outside of the leadership that these dangers are as great as the danger of despotism or rule by whim.

Authors and thoughts are blocked in the East by the whim of a supreme leader who also determines who is an infidel or enemy, or friend, and which businesses should flourish, and who should be rich (his buddies). By contrast, two fundamentals of western society — things that lead to purported immorality, are citizen rights and the rule of law: that citizens can possess things and do things for their own reasons, or no reason at all, and that citizens may stand as equals before a bar of law, to be judged by spelled-out laws or freed, with equal believability and claim.

In Russia or Iran, the Commissar and Imam have special rights: they can take possessions from others at whim, shut down businesses at whim; imprison at whim  — all based on their own interpretation of God’s will, the Koran, or “the good of the state.” Only they can sense the true good, or the true God well enough to make these decisions and laws. And when they violate those laws they are protected from the consequences; the masses can be prosecuted to the full extent of the law, and then some, not even requiring a trial in many places (Gaza, for example) if the leader feels speed is needed. The rule of law with equal treatment is a fundamental of western civilization (republicanism). It is commanded by Moses in the Bible at least seven times: Numbers 15:15, Numbers 15:16, Numbers 15:29, Exodus 12:49, Numbers 9:14, and Leviticus 24:22, “One law and one ordinance you should have, for the home-born, and the foreigner who dwells among you.”

The equal treatment under the law: for rich and poor, king and commoner, citizen and foreigner is a revolutionary idea of the west; that justice is blind. Another idea is personal possessions and freedoms. There is no concept of equality under business law unless there is a business that you can own, and personal possessions and rights. These are not in place in eastern theocracies: they tend to treat the preachers (imams) better than non preachers because they are presumed smarter and better; similarly men are treated better than women, who have few rights, and the state religion is treated better than infidels. In communist countries and dictatorships the dictator can get away with anything. Admittedly, in capitalistic states the rich and powerful find loopholes while the poor find prison, but not always (our Detroit’s ex-mayor is in prison) and it’s not the law. A feature of Eastern theocracies and dictatorships is that they lack a free press, and thus no forum for public exposure of legal mischief.

Einstein on freedom producing good. I'd say freedom is also a good in itself

Einstein on freedom producing good. I’d say freedom is also a good in itself.

The strongest arguments for socialist dictatorship and theocracy is that this is needed to protect the weak. Clement Attlee (labor socialist British Prime Minister, 1945 -56) explained his government’s take over of almost all British business: “There was a time when employers were free to work little children for sixteen hours a day… when employers were free to employ sweated women workers on finishing trousers at a penny halfpenny a pair. There was a time when people were free to neglect sanitation so that thousands died of preventable diseases. For years every attempt to remedy these crying evils was blocked by the same plea of freedom for the individual. It was in fact freedom for the rich and slavery for the poor. Make no mistake, it has only been through the power of the State, given to it by Parliament, that the general public has been protected against the greed of ruthless profit-makers and property owners.”  (Quotes from Spartacus.edu). it’s a brilliant speech, and it taps into the government’s role in the common defense, but it’s not at all clear that a chinless bureaucrat will be a better boss than the capitalist who built the firm. Nor is it clear that you help people by preventing them from work at a salary you decide is too low. 

England suffered a malaise from public ownership and the distribution of profit by those close the liberal party. Under Attlee there was lack of food and coal while the rest of Europe, and particularly Germany prospered, and passed England in productivity. Germany had no minimum wage, and  still doesn’t have one. In eastern countries, ingenuity is deadened by the knowledge that whatever a genius or worker achieves is taken by the state and redistributed. A cute joke exchange: Churchill and Attlee are supposed to have found themselves in adjoining stalls of the men’s room of Parliament. Churchill is supposed to have moved as far as possible from Attlee. “Feeling standoffish, Winston” Attlee is supposed to have said. “No. Frightened. “Whenever you see something large you try to nationalize it.” Perhaps more telling is this Margret Thatcher’s comment, and exchange. Making everyone’s outcome equal does more to penalize those with real pride in their ideas and work than it does to help the truly needy.

While there is a need for government in regards to safety, roads, and standards, and to maintain that equality of law. It seems to me the state should aid the poor only to the extent that it does not turn them into dependents. There is thus a natural tension between private good and public service similar to the tensegrity that holds cells together. Capitalists can only make money by providing desired goods and services at worthwhile rate, and paying enough to keep workers; they should be allowed to keep some of that, while some must be taken from them to get great things done. I’ve related the tensegrity of society to the balance between order and disorder in a chemical system.

Robert E. Buxbaum,  August 27, 2014. This essay owes special thanks to a Princeton chum, Val Martinez. Though my training is in engineering, I’ve written hobby pieces on art, governance, history, and society. Check out the links at right.

Einstein’s fuzzy slippers — and a fetish lawyer joke

First, the joke about the fetishistic lawyer: He got off on a technicality.

It’s funny because  ….  it’s a double entendre, a multi-word, sexual homophone (no insult  to the homophone community). It also relates to a fact as true and significant as any in life. What a person considers enjoyable, fun (or not) depends mostly on what’s in his mind. Whether judging sexy or scary; pleasant or disagreeable, it has relatively little to do with a physical reality, and is mostly in the imagination of the person. As a result, the happiest people seem to be those who embrace their inner weirdness. They try to find jobs that they are good at, that allow them to take perverse pleasure in their own weird way within the bounds of a civil society.

Take pleasure in your own weirdness.

Einstein in fuzzy slippers outside of his Princeton home; take pleasure in your own weirdness.

Einstein, at left, seems to have enjoyed doing physics, playing the violin, and wearing odd clothes: sweaters, and these (pink) fuzzy slippers. the odd clothes didn’t detract from his physics, and may have even helped him think. Boris Spassky (the Russian chess champion) was asked which he preferred: sex or chess, he said: “it very much depends on the position.” Do what you like, and like what you do. As the old joke goes, “I don’t suffer from insanity: I enjoy every moment.”

Robert Buxbaum. April 1, 2014; I mostly blog about science and hydrogen, but sometimes, like here, about personal relations, or last week economics (dismal). Here’s a thermodynamic look at life. And a picture of an odd sculpture I made. I take my own advice, by the way: this blog doesn’t get me any money but it’s fun, and maybe I’ll help some day — e.g. maybe it’ll spark my creativity. Here’s a bit about Einstein’s mustache, and the universe being curved in.