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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.

Arctic Ice has shrunk 1.5% since ’99 and Gore’s inconvenient truth. Is this bad?

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At the 1999 Copenhagen Climate Change Summit, Al Gore announced an inconvenient truth: “There is a 75 per cent chance that the entire north polar ice cap, during the summer months, could be completely ice-free within five to seven years.” It was a bold prediction, part of a campaign that got Mr Gore a Nobel Prize and motivated the US to devote billions to stopping global warming. Supposedly 98% of scientists agreed with Mr. Gore and his remedies. Prince Charles and Bill Gates too. Twenty three years later there is still arctic ice, 98.5% as much as in 1999. Two questions arise: 1. Is the ice loss bad? and 2. Why were those 98% of scientists so wrong?

Arctic sea ice extent 1999-2021
Arctic sea ice extent when Al Gore spoke (1999) and since. Not much change, nor clearly for the worse

The second question is far easier than the first: the 98% number was bogus, a lie, like many other climate lies that followed. it was effective at stopping argument, and could not be checked immediately. It bullied scientists who argued that global warming wasn’t bad, or wasn’t man-made, and it gave do-gooders the ability to label their opponents “liars” and “science deniers”. The claim of 98% was used to silence scientists with long, prominent careers. Deniers lost their funding and were no longer published. Other scientists learned to keep quiet. Twenty years later, when the arctic ice wasn’t gone and antarctic ice hit a record extent, the deniers’ careers largely were gone.

Scientists are not stupid, nor independently rich, for the most part. They are dependent on government funding and their employers, the universities are too. As a group they (we) are incapable of stemming the tide of public opinion. This week Biden signed a nearly 1 trillion dollar bill to stop climate change. Every scientist with a chance to get the money will go for it. Whether or not they think a colder earth is good, they will claim it is in their proposals, and imply that their work can stop the natural chaos that is climate. They will ask for their share of the $1T to study the appropriate things: solar cells, corn-based power, and wind turbines. The proposals will not mention the huge costs in mining or land use. Scientists already know they can not get funded for nuclear power, though it works and produces no CO2, nor should can scientists benefit by criticizing China, as the largest source of CO2. That is seen as undermine the green effort at home. When we stop manufacturing at home, BTW, we end up buying the same materials manufactured in China, where they really generate lots of pollution. When asked about this, Biden’s climate chief said not to worry about it, we had to do our part, and Biden would speak to the Chinese. The result is the biggest buildup in coal-fired power plants in the world, with more coming on line.

This second question is at least as important as the first one: is less arctic ice bad? Or, asking more generally, is a warm earth bad? It’s an opinion question; it’s in no way science, impossible to answer definitively. Cold weather is bad for food production, and that’s bad for people, in general. Most people prefer to live where it’s warm, I find. Supposedly polar bears prefer it cold, but I don’t know for sure. I’m not keen to go back to the climate of the ice ages, 10,000- 100,000 years ago when ice covered Canada and you could walk from France to England. I’m not convinced that life was better when the world was 1°C colder. The sea was lower in 1900, but had been higher in the year zero. Less arctic ice means easier shipping. For all I know we may want to make a Northwest Passage. More food and a easier shipping are the convenient truths about global warming.

Robert Buxbaum, August 19, 2022. If you believe any of what I said about Gore/Biden’s green energy, you may like a movie by Michael Moore, Planet of the Humans, see it here. The political greens are not saving energy or cooling the planet, and they know it. It’s a money maker.

IFL Science grows up.

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I used to follow an Australian science blog, called “I Fucking Love Science.” Elise Andrew and her crew scanned the literature with a keen eye for the interesting. They regularly posted to Facebook and alerted science nerds like me to all sorts of new science bits with minimal commentary, minimal advertisements, and no politics. On average they found 6 or 8 really interesting posts, per week, generally one or two on fundamental physics, one or two on materials, one or two on biology or medicine, one or two astronomy, perhaps a chemistry post. My post about the color of the sky on Mars was ignited by a picture of the Mars sky that I saw on IFL Science — the sky was yellow, and I had just written about why the sky on earth was blue, and not green.

But, as with all quirky things, this one matured. The name changed to “IFL Science” — a change that I suspect was designed to promote sharing. There were more advertisements, and click bait — “this starlet lost a ton of weight,” “you won’t believe what this famous person’s partner looks like now,” etc. And there was politics, vaguely presented as science. Ms Andrew wrote more and more of herself, making herself into a personality whose travels and speaking tours would interest us. And there were non-science, guest bloggers too: People telling you who to vote for and more importantly who to vote against. All for the good of the world, she said, but it was her opinion, and not what I’d gone to IFL for.

The science got less technical, too and more popular. More pretty pictures and misleading headlines. Currently there is no math, no equations, no chemical diagrams. A top story of this week told of a semi-interesting approach for women with constipation — something that “would change everything.” When you click on the story, you find that women put their finger in their vagina and work out the poop that way, something called “splinting.” It’s sort of science, but not the sort that made me love science. Another top story — the top one from today is as follows:

Top story from IFL science today, Feb 28, 2021. Is there really no fuel use? No. The fuel is a battery, and the speed in 4m/s (9mph), and the plane looks nothing like this.

If you follow the links to here, it turns out that the plane (unmanned) looks nothing like this. It uses electric energy from a battery to move ionized air rearward at an efficiency far lower than with a propeller. The forward speed is 4 m/s (9 mph) and the maximum distance covered was 55m, half a Canadian football field. As presented in IFL science, it’s a misleading, non-math clickbait for something that’s interesting engineering — sort of. As for being Star Trek like, no. To move this plane, you need air.

I’m sorry, you can not make quantum mechanics for dummies. No dummy will understand it. You can make a book that’s not quantum mechanics for dummies, or a quantum book not for dummies. Just saying.

In the treatment of the work of the recent Noble laureates, IFL Science didn’t talk about the work so much as the biographies of the people, and their struggles, and that two of the people who won Nobels for their work in biology were women — for an advance related to CRISPERS– but that wasn’t science. I’d prefer to know what the advance was, and how it works. I’d prefer to figure out that these were women from their names or from the pronouns like, “she” or “her”. There was also no information about other two researchers (males, I assume, or perhaps females who had less-interesting biographies?). It was the same with the Physics Nobel except that I already knew there was a black hole at the center of the galaxy, and that those who found it are long dead. Instead the Note Prize was being awarded for a photograph of the black hole. Interesting (sure doesn’t look like a black hole to me). Is there something they learn from the photo. I’ve noted that we are likely within a black hole, and I show why this is using some, not too difficult math.

Having griped along this way, I have to say that that IFL isn’t that bad, it’s just non-mathy, popular, and a little grown up. That’s sad, but it’s not toxic. Grownups make money, and please customers, and that’s how it goes. To quote a wonderful book, The outsiders, “Nothing gold can stay.” In my own blog, I try to be more math-y, and more science-y. My model is Isaac Asimov, a writer who excited me to love science from when I was 8 to when I entered college, nine years later (1972). He would die of AIDS from a transfusion, 20 years after that.

Robert Buxbaum, February 28, 2021

Sweden v Michigan: different approaches, same outcome.

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Sweden has scientists; Michigan has scientists. Sweden’s scientists said to trust people to social distance and let the COVID-19 disease run its course. It was a highly controversial take, but Sweden didn’t close the schools, didn’t enforce masks, and let people social distance as they would. Michigan’s scientists said to wear masks and close everything, and the governor enforced just that. She closed the schools, the restaurants, the golf courses, and even the parks for a while. In Michigan you can not attend a baseball game, and you can be fined for not wearing a mask in public. The net result: Michigan and Sweden had almost the same death totals and rates, as the graphs below show. As of July 28, 2020: Sweden had 5,702 dead of COVID-19, Michigan had 6,402. That’s 13 more dead for a population that’s 20% smaller.

Sweden’s deaths pre day. There are 5,702 COVID dead since the start, out of a population of 10.63 million. There are 79,494 confirmed COVID cases, but likely a lot more infected.

Sweden and Michigan are equally industrial, with populations in a few dense cities and a rural back-country. Both banned large-scale use of hydroxy-chloroquine. Given the large difference in social distance laws, you’d expect a vastly different death rate, with Michigan’s, presumably lower, but there is hardly any difference at all, and it’s worthwhile to consider what we might learn from this.

Michigan’s deaths pre day. There are 6,426 COVID dead since the start, out of a population of 9.99 million. There are 88,025 confirmed COVID cases, but likely a lot more infected.

What I learn from this is not that social distance is unimportant, and not that hand washing and masks don’t work, but rather it seems to me that people are more likely to social distance if they themselves are in control of the rules. This is something I also notice comparing freezer economies to communist or controlled ones: people work harder when they have more of a say in what they do. Some call this self -exploitation, but it seems to be a universal lesson.

Both Sweden and the US began the epidemic with some moderate testing of a drug called hydroxychloroquine (HCQ)and both mostly stopped in April when the drug became a political football. President Trump recommended it based on studies in France and China, but the response was many publications showing the didn’t work and was even deadly. Virtually ever western country cut back use of the drug. Brazil’s scientists objected — see here where they claim that those studies were crooked. It seems that countries that continued to use the drug had fewer COVID deaths, see graph, but it’s hard to say. The Brazilians claim that the anti HCQ studies were politically motivated, but doctors in both Sweden and the US largely stopped prescribing the drug. This seems to have been a mistake.

US hospitals stopped using HCQ in early April almost as soon as Trump recommended it. Sweden did the same.

In July, Henry Ford hospitals published this large-scale study showing a strong benefit: for HCQ: out of 2,541 patients in six hospitals, the death rate for those treated with HCQ was 13%. For those not treated with HCQ, the death rate was more than double: 26.4%. It’s not clear that this is cause and effect. It’s suggestive, but there is room for unconscious bias in who got the drug. Similarly, last week, a Yale researcher this week used epidemiological evidence to say HCQ works. This might be proof, or not. Since epidemiology is not double-blind, there is more than common room for confounding variables. By and large the newspaper experts are unconvinced by epidemiology and say there is no real evidence of HCQ benefit. In Michigan and Sweden the politicians strongly recommend continuing their approaches, by and large avoiding HCQ. In Brazil, India and much of the mideast, HCQ is popular. The countries that use HCQ claim it works. The countries that don’t claim it does not. The countries with strict lock-down say that science shows this is what’s working. The countries without, claim they are right to go without. All claim SCIENCE to support their behaviors, and likely that’s faulty logic.

Hydroxychloroquine and COVID-19 fatality rates in different countries as of early June 2020. This isn’t enough to prove HCQ effectiveness, but it’s promising, and suggests that increased use is warranted, at least among those without heart problems.

Given my choice, I’d like to see more use of HCQ. I’m not sure it works, but I’m ,sure there’s enough evidence to put it into the top tier of testing. I’d also prefer the Sweden method, of nor enforced lockdown, or a very moderate lockdown, but I live I’m Michigan where the governor claims she knows science, and I’m willing to live within the governor’s lockdown.There is good, scientific evidence that, if you don’t you get fined or go to jail.

Robert Buxbaum, July 29, 2020. As I side issue, I think iodine hand wash is a good thing. I may be wrong, but here’s my case.

Brazilian scientists speak out for hydroxychloroquine

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Brazil has decided to go its own route in response to the Corona virus pandemic. They’re using minimal social distancing with a heavy reliance on hydroxychloroquine (HCQ), a cheap drug that they claim is effective (as has our president). Brazil has been widely criticized for this, despite so far having lower death rate per million than the US, Canada, or most of Europe. In an open letter, copied in part below, 25 Brazilian scientists speak out against the politicalization of science, and in favor of their approach to COVID-19. The full letter (here). The whole letter is very worth reading, IMHO, but especially worthwhile is their section on hydroxychloroquine (HCQ), copied below.

….. Numerous countries such as the USA, Spain, France, Italy, India, Israel, Russia, Costa Rica and Senegal use the drug (HCQ) to fight covid-19, whereas other countries refrain from using HCQ as one of the strategies to contain the pandemic, betting on other controversial tactics.

Who then speaks here in the name of “science”? Which group has a monopoly on reason and its exclusive authorization to be the spokesperson of “science”? Where is such authorization found?One can choose an opinion, and base his strategy on it, this is fine, but no one should commit the sacrilege of protecting his decision risking to tarnish with it the “sacred mantle of science”.

Outraged, every day I hear mayors and governors saying at the top of their lungs that they “have followed science”. Presidents of councils and some of their advisers, and of academies and deans in their offices write letters on behalf of their entire community, as if they reflect everyone’s consensual position. Nothing could be more false.Have they followed science? Not at all! They have followed the science wing which they like, and the scientists who they chose to place around them. They ignore the other wing of science, since there are also hundreds of scientists and articles that oppose their positions and measures.

Worse, scientists are not angels. Scientists are people, and people have likes and dislikes, passions and political party preferences. Or wouldn’t they? There are many scientists, therefore, who do good without looking at whom, I know and admire many of them. But there are also pseudoscientists who use science to defend their opinion, their own pocket, or their passion. Scientists have worked and still work hard and detached to contribute to the good of humanity, many of whom are now in their laboratories, risking their lives to develop new methods of detecting coronavirus, drugs and vaccines, when they could stay “safe at home”. But, to illustrate my point, I know scientists who have published articles, some even in major journals such as “Science” or “Nature”, with data they have manufactured “during the night”; others who have removed points from their curves, or used other similar strategies. Many scientists were at Hitler’s side, weren’t they? Did they act in the name of “science”? Others have developed atom bombs. Others still develop chemical and biological weapons and illicit drugs, by design.

The Manaus’ study with chloroquine (CQ) performed here in Brazil and published in the Journal of the American Medical Association (JAMA) [1], is emblematic to this discussion of “science”. Scientists there used, the manuscript reveals, lethal doses in debilitated patients, many in severe conditions and with comorbidities. The profiles of the groups do not seem to have been “randomized”, since a clear “preference” in the HIGH DOSE group for risk factors is noted. Chloroquine, which is more toxic than HCQ, was used, and it seems that they even made “childish mistakes” in simple stoichiometric calculations, doubling the dosage with the error. I’m incapable of judging intentions, but justice will do it. The former Brazilian Health Minister Luiz Henrique Mandetta quoted this study, supported it, and based on it, categorically stated: “I do not approve HCQ because I am based on ‘science, science, science’!”.

Another study published by Chinese researchers in the British Medical Journal (BMJ) and which is still persistently used against HCQ was also at least revolting [2]. In it, the authors declared: “we administer 1,200 mg for 3 days, followed by 800 mg for 12 to 21 days, in patients with moderate to severe symptoms”. In other words, they gave a huge dosage of the drug that could reach the absurdity of 20 grams in the end, and it given was too late to patients (HCQ should be administered in the first symptoms or even earlier). And even worse, overdosing on HCQ or any other drug for severe cases is poisonous. What do you think, was it good science? The recommended dosage in Brazil, since May 20th, 2020, by the new Ministry of Health, for mild symptoms is 2 times 400 mg in the first day (every 12 hours) and 400 mg for 5 days for a total of 2.8 grams.

In other published studies, also in these internationally renowned journals such as The New England Journal of Medicine, JAMA and BMJ [3-5], once again, “problems” are clearly noted, since or the patients were randomized in irregular ways, placing older, more susceptible or most severe and hypoxemic patients in the higher (lethal) dose groups, or more men (almost 3 times more deadly by covid than women), or more black people (in the USA, black people have displayed higher mortality) and more smokers, and where most of the deaths occurred in the first days of the studies (signs that were deaths of critically ill patients, who at this stage would be more “intoxicated” than “treated” with HCQ), or they administered HCQ isolated, when it is known that it is necessary to associate HCQ at least with azithromycin. One of these studies [5] administered HCQ only on the sixteenth day of symptoms (for really early treatment, HCQ administration should be started up to fifth day), in other words, at the end of the disease, when the drug can do little good or nothing to the patient.

These studies indicate that some scientists either forgot how “science” is done or that there is a huge effort to disprove, whatever it takes, that HCQ works. How can someone or even Councils and Academies of Medicine cite such studies as the “science” of their decisions? How can that be?

On the contrary, the study published – and today with more than 3 thousand patients tested – and carried out by Dr. Didier Raoult in France [6], using the correct dosage and at the right time, with a very low mortality rate (0.4%), and the Prevent Senior’s clinical experience in Brazil – also quite encouraging – are disqualified with very “futile” arguments such as: “Didier Raoult is a controversial and unworthy researcher”, “At Prevent Senior Clinic they were not sure of the diagnosis” (but none of the hospitalized patients with clear COVID symptoms died), “Placebo effect” (what a supernatural power of inducing our mind that reduces mortality from 40% to zero, I want this placebo!), “Study performed by a health plan company” (I do not doubt that this people indeed want to save lives, because the patients were their customers who pay their bills), and similar ephemeral arguments.…

The Brazilian scents who signed the letter. Read the whole letter here.

I admire the spunk of these fellows going agains the doctors, WHO. Beyond being a critique of bad research on a particular drug, it is a defense of science. Science is a discussion, a striving for truth. It is not supposed to demand blind allegiance to a few politically appointed experts. They’ve convinced me that the tests sponsored by the world health organization seem designed to show failure, and reminded me that there is rarely a one-size-fits-all for problems and all times.

I also find striking the highly critical response of my local newspapers and TV reporters. While they both like to highlight efforts by South America as they try entering the first world, with help from Bill gates and leftist politicians, they have been uniformly condemned Brazil for its non-left approach and now for use of HCQ. They want Sous Americans to think, but only if their conclusions are no different from those of their favorite, liberal thinkers.

Robert Buxbaum, May 28, 2020. Check out my notes on how to do science right. And by the way, you might want to use iodine hand wash to minimize your chance of getting or spreading COVID and other diseases.

What drives the gulf stream?

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I’m not much of a fan of todays’ kids’ science books because they don’t teach science IMHO. They have nice pictures and a few numbers; almost no equations, and lots of words. You can’t do science that way. On the odd occasion that they give the right answer to some problem, the lack of math means the kid has no way of understanding the reasoning, and no reason to believe the answer. Professional science articles on the web are bad in the opposite direction: too many numbers and for math, hey rely on supercomputers. No human can understand the outcome. I like to use my blog to offer science with insight, the type you’d get in an old “everyman science” book.

In previous posts, I gave answers to why the sky is blue, why it’s cold at the poles, why it’s cold on mountains, how tornadoes pick stuff up, and why hurricanes blow the way they do. In this post, we’ll try to figure out what drives the gulf-stream. The main argument will be deduction — disproving things that are not driving the gulf stream to leave us with one or two that could. Deduction is a classic method of science, well presented by Sherlock Holmes.

The gulf stream. The speed in the white area is ≥ 0.5 m/s (1.1 mph.).

The gulf stream. The speed in the white area is ≥ 0.5 m/s (1.1 mph.).

For those who don’t know, the Gulf stream is a massive river of water that runs within the Atlantic ocean. As shown at right, it starts roughly at the end of Florida, runs north to the Carolinas, and then turns dramatically east towards Spain. Flowing east, It’s about 150 miles wide, but only about 62 miles (100 km) when flowing along the US coast. According to some of the science books of my youth this massive flow was driven by temperature according to others, by salinity (whatever that means), and yet other books of my youth wind. My conclusion: they had no clue.

As a start to doing the science here, it’s important to fill in the numerical information that the science books left out. The Gulf stream is roughly 1000 meters deep, with a typical speed of 1 m/s (2.3 mph). The maximum speed is the surface water as the stream flows along the US coast. It is about 2.5 metres per second (5.6 mph), see map above.

From the size and the speed of the Gulf Stream, we conclude that land rivers are not driving the flow. The Mississippi is a big river with an outflow point near the head waters of the gulf stream, but the volume of flow is vastly too small. The volume of the gulf stream is roughly

Q=wdv = 100,000 x 1000 x .5 =  50 million m3/s = 1.5 billion cubic feet/s.

This is about 2000 times more flow than the volume flow of the Mississippi, 18,000 m3/s. The great difference in flow suggests the Mississippi could not be the driving force. The map of flow speeds (above) also suggest rivers do not drive the flow. The Gulf Stream does not flow at its maximum speed near the mouth of any river.  We now look for another driver.

Moving on to temperature. Temperature drives the whirl of hurricanes. The logic for temperature driving the gulf stream is as follows: it’s warm by the equator and cold at the poles; warm things expand and as water flows downhill, the polls will always be downhill from the equator. Lets put some math in here or my explanation will be lacking. First lets consider how much hight difference we might expect to see. The thermal expansivity of water is about 2x 10-4 m/m°C (.0002/°C) in the desired temperature range). To calculate the amount of expansion we multiply this by the depth of the stream, 1000m, and the temperature difference between two points, eg. the end of Florida to the Carolina coast. This is 5°C (9°F) I estimate. I calculate the temperature-induced seawater height as:

∆h (thermal) ≈ 5° x .0002/° x 1000m = 1 m (3.3 feet).

This is a fair amount of height. It’s only about 1/100 the height driving the Mississippi river, but it’s something. To see if 1 m is enough to drive the Gulf flow, I’ll compare it to the velocity-head. Velocity-head is a concept that’s useful in plumbing (I ran for water commissioner). It’s the potential energy height equivalent of any kinetic energy — typically of a fluid flow. The kinetic energy for any velocity v and mass of water, m is 1/2 mv2 . The potential energy equivalent is mgh. Combine the above and remove the mass terms, and we have:

∆h (velocity) = v2/2g.

Where g is the acceleration of gravity. Let’s consider  v = 1 m/s and g= 9.8 m/s2.≤ 0.05 m ≈ 2 inches. This is far less than the driving force calculated above. We have 5x more driving force than we need, but there is a problem: why isn’t the flow faster? Why does the Mississippi move so slowly when it has 100 times more head.

To answer the above questions, and to check if heat could really drive the Gulf Stream, we’ll check if the flow is turbulent — it is. A measure of how turbulent is based on something called the Reynolds number, Re#, it’s the ratio of kinetic energy and viscous loss in a fluid flow. Flows are turbulent if this ratio is more than 3000, or so;

Re# = vdρ/µ.

In the above, v is velocity, say 1 m/s, d is depth, 1000m, ρ = density, 1000 kg/m3 for water, and  0.00133 Pa∙s is the viscosity of water. Plug in these numbers, and we find a RE# = 750 million: this flow will be highly turbulent. Assuming a friction factor of 1/20 (.05), e find that we’d expect complete mixing 20 depths or 20 km. We find we need the above 0.05 m of velocity height to drive every 20 km of flow up the US coast. If the distance to the Carolina coast is 1000 km we need 1000*.05m/20 = 1 meter, that’s just about the velocity-head that the temperature difference would suggest. Temperature is thus a plausible driving force for 0.5 m/s, though not likely for the faster 2.5 m/s flow seen in the center of the stream. Turbulent flow is a big part of figuring the mpg of an automobile; it becomes rapidly more important at high speeds.

World sea salinity

World sea salinity. The maximum and minimum are in the wrong places.

What about salinity? For salinity to work, the salinity would have to be higher at the end of the flow. As a model of the flow, we might imagine that we freeze arctic seawater, and thus we concentrate salt in the seawater just below the ice. The heavy, saline water would flow down to the bottom of the sea, and then flow south to an area of low salinity and low pressure. Somewhere in the south, the salinity would be reduced by rains. If evaporation were to exceed the rains, the flow would go in the other direction. Sorry to say, I see no evidence of any of this. For one the end of the Gulf Stream is not that far north; there is no freezing, For two other problems: there are major rains in the Caribbean, and rains too in the North Atlantic. Finally, while the salinity head is too small. Each pen of salinity adds about 0.0001g/cc, and the salinity difference in this case is less than 1 ppm, lets say 0.5ppm.

h = .0001 x 0.5 x 1000 = 0.05m

I don’t see a case for northern-driven Gulf-stream flow caused by salinity.

Surface level winds in the Atlantic.

Surface level winds in the Atlantic. Trade winds in purple, 15-20 mph.

Now consider winds. The wind velocities are certainly enough to produce 5+ miles per hour flows, and the path of flows is appropriate. Consider, for example, the trade winds. In the southern Caribbean, they blow steadily from east to west slightly above the equator at 15 -20 mph. This could certainly drive a circulation flow of 4.5 mph north. Out of the Caribbean basin and along the eastern US coat the trade winds blow at 15-50 mph north and east. This too would easily drive a 4.5 mph flow.  I conclude that a combination of winds and temperature are the most likely drivers of the gulf stream flow. To quote Holmes, once you’ve eliminated the impossible, whatever remains, however improbable, must be the truth.

Robert E. Buxbaum, March 25, 2018. I used the thermal argument above to figure out how cold it had to be to freeze the balls off of a brass monkey.

Yogurt making for kids

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Yogurt making is easy, and is a fun science project for kids and adults alike. It’s cheap, quick, easy, reasonably safe, and fairly useful. Like any real science, it requires mathematical thinking if you want to go anywhere really, but unlike most science, you can get somewhere even without math, and you can eat the experiments. Yogurt making has been done for centuries, and involves nothing more than adding some yogurt culture to a glass of milk and waiting. To do this the traditional way, you wait with the glass sitting outside of any refrigeration (they didn’t have refrigeration in the olden days). After a few days, you’ll have tasty yogurt. You can get taster yogurt if you add flavors. In one of my most successful attempts at flavoring, I added 1/2 ounce of “skinny syrup” (toffee flavor) to a glass of milk. The results were most satisfactory, IMHO.

My latest batch of home-made flavored yogurt, made in a warm spot behind this urn.

My latest batch of home-made flavored yogurt, made in a warm spot behind this coffee urn.

Now to turn yogurt-making into a science project. We’ll begin with a hypothesis. I generally tell people to not start with a hypothesis, (it biases your thinking), but here I will make an exception as I have a peculiarly non-biased hypothesis to suggest. Besides, most school kids are told they need one. My hypothesis is that there must be better ways to make yogurt and worse ways. A hypothesis should be avoided if it contains any unfounded assumptions, or if it points to a particular answer — especially an answer that no one would care about.

As with all science you’ll want to take numerical data of cause and effect. I’d suggest that temperature data is worth taking. The yogurt-making bacteria is called lactose thermophillis, and this suggests that warm temperatures will be good (lact = milk in Latin, thermophilic = loving heat). Also making things interesting is the suspicion that if you make things too warm, you’ll cook your organisms and you won’t get any yogurt. I’ve had this happen, both with over-heat and under-heat. My first attempt was to grow yogurt in the refrigerator, but I got no results. I then tried the kitchen counter and got yogurt, and then I heated things a bit more by growing next to a coffee urn, and got better yogurt; yet more heat and nothing.

For a science project, you might want to make a few batches of yogurt, at least 5, and these should be made at 2-3 different temperatures. If temperature is a cause for the yogurt to come out better or worse, you’ll need to be able to measure how much “better”? You may choose to study taste, and that’s important, but it’s hard to quantify, so that should not be the whole experiment. I would begin by testing thickness, or the time to a get some fixed degree of thickness; I’d measure thickness by seeing if a small weight sinks. A penny is a cheap, small weight, and I know it sinks in milk, but not in yogurt. You’ll want to wash your penny first, or no one will eat the yogurt. I used hot water from the urn to clean and sterilize my pennies.

Another thing that is worth testing is the effect of using different milks: whole milk, 2%, 1% or skim; goat milk, or almond milk. You can also try adding stuff to it, or starting with different starter cultures, or different amounts. Keep numerical records of these choices, then keep track of how they effect how long it takes for the gel to form, and how the stuff looks or tastes to you. Before you know it, you’ll have some very good product at half the price of the stuff in the store. If you really want to move forward fast, you might apply semi-random statistics to your experimental choices. Good luck.

Robert Buxbaum, March 2, 2018. My latest observation: what happens if you leave the yogurt to mold too long? It doesn’t get moldy, perhaps the lactic acid formed kills germs (?), but the yogurt separated into curds and whey. I poured off the whey, the unappealing, bitter yellow liquid. The thick white remainder is called “Greek” yogurt. I’m not convinced this tastes better, or is healthier, BTW.

How Tesla invented, I think, Tesla coils and wireless chargers.

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I think I know how Tesla invented his high frequency devices, and thought I’d show you, while also explaining the operation of some devices that develop from in. Even if I’m wrong in historical terms, at least you should come to understand some of his devices, and something of the invention process. Either can be the start of a great science fair project.

physics drawing of a mass on a spring, left, and of a grounded capacitor and inception coil, right.

The start of Tesla’s invention process, I think, was a visual similarity– I’m guessing he noticed that the physics symbol for a spring was the same as for an electrical, induction coil, as shown at left. A normal person would notice the similarity, and perhaps think about it for a few seconds, get no where, and think of something else. If he or she had a math background — necessary to do most any science — they might look at the relevant equations and notice that they’re different. The equation describing the force of a spring is F = -k x  (I’ll define these letters in the bottom paragraph). The equation describing the voltage in an induction coil is not very similar-looking at first glance, V = L di/dt.  But there is a key similarity that could appeal to some math aficionados: both equations are linear. A linear equation is one where, if you double one side you double the other. Thus, if you double F, you double x, and if you double V, you double dI/dt, and that’s a significant behavior; the equation z= at2 is not linear, see the difference?

Another linear equation is the key equation for the motion for a mass, Newton’s second law, F = ma = m d2x/dt2. This equation is quite complicated looking, since the latter term is a second-derivative, but it is linear, and a mass is the likely thing for a spring to act upon. Yet another linear equation can be used to relate current to the voltage across a capacitor: V= -1/C ∫idt. At first glance, this equation looks quite different from the others since it involves an integral. But Nicola Tesla did more than a first glance. Perhaps he knew that linear systems tend to show resonance — vibrations at a fixed frequency. Or perhaps that insight came later. 

And Tesla saw something else, I imagine, something even less obvious, except in hindsight. If you take the derivative of the two electrical equations, you get dV/dt = L d2i/dt2, and dV/dt = -1/C i . These equations are the same as for the spring and mass, just replace F and x by dV/dt and i. That the derivative of the integral is the thing itself is something I demonstrate here. At this point it becomes clear that a capacitor-coil system will show the same sort of natural resonance effects as shown by a spring and mass system, or by a child’s swing, or by a bouncy bridge. Tesla would have known, like anyone who’s taken college-level physics, that a small input at the right, resonant frequency will excite such systems to great swings. For a mass and spring,

Basic Tesla coil. A switch set off by magnetization of the iron core insures resonant frequency operation.

Basic Tesla coil. A switch set off by magnetization of the iron core insures resonant frequency operation.

resonant frequency = (1/2π) √k/m,

Children can make a swing go quite high, just by pumping at the right frequency. Similarly, it should be possible to excite a coil-capacitor system to higher and higher voltages if you can find a way to excite long enough at the right frequency. Tesla would have looked for a way to do this with a coil capacitor system, and after a while of trying and thinking, he seems to have found the circuit shown at right, with a spark gap to impress visitors and keep the voltages from getting to far out of hand. The resonant frequency for this system is 1/(2π√LC), an equation form that is similar to the above. The voltage swings should grow until limited by resistance in the wires, or by the radiation of power into space. The fact that significant power is radiated into space will be used as the basis for wireless phone chargers, but more on that later. For now, you might wish to note that power radiation is proportional to dV/dt.

A version of the above excited by AC current. In this version, you achieve resonance by adjusting the coil, capacitor and resistance to match the forcing frequency.

A more -modern version of the above excited by AC current. In this version, you achieve resonance by adjusting the coil, capacitor and resistance to match the forcing frequency.

The device above provides an early, simple way to excite a coil -capacitor system. It’s designed for use with a battery or other DC power source. There’s an electromagnetic switch to provide resonance with any capacitor and coil pair. An alternative, more modern device is shown at left. It  achieves resonance too without the switch through the use of input AC power, but you have to match the AC frequency to the resonant frequency of the coil and capacitor. If wall current is used, 60 cps, the coil and capacitor must be chosen so that  1/(2π√LC) = 60 cps. Both versions are called Tesla coils and either can be set up to produce very large sparks (sparks make for a great science fair project — you need to put a spark gap across the capacitor, or better yet use the coil as the low-voltage part of a transformer.

power receiverAnother use of this circuit is as a transmitter of power into space. The coil becomes the transmission antenna, and you have to set up a similar device as a receiver, see picture at right. The black thing at left of the picture is the capacitor. One has to make sure that the coil-capacitor pair is tuned to the same frequency as the transmitter. One also needs to add a rectifier, the rectifier chosen here is designated 1N4007. This, fairly standard-size rectifier allows you to sip DC power to the battery, without fear that the battery will discharge on every cycle. That’s all the science you need to charge an iPhone without having to plug it in. Designing one of these is a good science fair project, especially if you can improve on the charging distance. Why should you have to put your iPhone right on top of the transmitter battery. Why not allow continuous charging anywhere in your home. Tesla was working on long-distance power transmission till the end of his life. What modifications would that require?

Symbols used above: a = acceleration = d2x/dt2, C= capacitance of the capacitor, dV/dt = the rate of change of voltage with time, F = force, i = current, k = stiffness of the spring, L= inductance of the coil, m = mass of the weight, t= time, V= voltage, x = distance of the mass from its rest point.

Robert Buxbaum, October 2, 2017.

summer science: a toad or turtle terrarium

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Here’s an easy summer science project, one I just made: a toad habitat. It’s similar to a turtle terrarium (I’ll show how to make that too). I’d made the turtle terrarium ten years ago for my 8-year-old daughter (here’s some advice I gave her on her 16th birthday).

For this project you’ll need: a large flower-pot, fish tank, or plastic clothes bin. You’ll need some dirt for the bottom, and a small plastic bin, jar, or Tupperware for toad (or turtle) transport. You’ll also need a smallish plastic dish or tub (~6″ by 1″ deep) to serve as a lake in the toad habitat. For the turtle version you don’t need the lake, but will need a rock or brick. And that’s all, besides your toad or turtle. The easy way to get your pet is to find one by a river. If that doesn’t work, go to a pet-store and get one that is native to your area of the country. Local fauna (fauna= animals) will be heartier and cheeper, and will allow you to keep your terrarium outside if you choose. Keeping my toad outside means he (or she) can catch bugs without me having to buy them all the time. It also seems more “natural” to study animals in their natural temperature cycles. I caught my toads three weeks ago, in mid April after the last frost — I plan to set one free in the fall –the other I gave away.

For my toad habitat, I used a large, old flower-pot that I had sitting outside my house. It is 21″ across at the top and 18″ tall. I put 6″ of dirt in it. six inches is deep enough for the toad to dig in, and it left 12″ of airspace — I don’t think the toad can jump a foot in the air to get out. I made sure the soil was muddy, and had worms. Toads seem to like mud and they eat worms. Toads drink water through their skin, and may not like chlorinated water. I also added some leaves and a small flower pot for shade, and put in some bits of fruit and some bugs, and planted a single plant. My hope was to develop a colony of ants and bugs for the toads to eat. I buried my plastic water bowl, my mini-lake, slightly below ground level with the top 1/2″ above. I then went off with my toad transport to catch a toad or three in the wetlands areas near me (I live in Oak Park, MI).

Some good toad hunting spots in Keego Harbor MI

Some good toad hunting spots in Keego Harbor MI

The first place I went was the banks of the Rouge river near Lawrence Tech. Sorry to say, the area showed no signs of toads, frogs, turtles, or even fish. There was an illegally connected drain, though — not good. I plan to bring the illegal grain up with the “Friends of the Rouge” (good group). I then went to an oak swamp on the Rouge. The area was beautiful and scenic, but there was no oxygen in the water and so no fish or toads; oxygen is important for the health of a river; without it, you’ve got  a swamp. I finally hit pay-dirt in Keego Harbor, MI, see map, a rural community 10 miles away from my home. In Keego harbor I found American toads aplenty: jumping all over, and big, hollow toad-mounds by the river. The locals were friendly too. Toad catching is a good conversation starter. I put two toads in my bin with some lake water and took them home to the terrarium, see movie.

My neighbor got the other toad and put him/her in a fish-tank terrarium in his bathroom. His terrarium has a screen on top with holes small enough to keep the toad and his food from escaping. He is feeding his toad meal worms, but I don’t have a movie. Apparently they like it.

I left my pot outside, as I mentioned, so my toad can catch insects that fly by, and spiders. My toad seems to like spiders. I also tried putting in wax-worms ($1 for 12). The good thing about wax worms is they move slowly, unlike crickets (crickets cost more and can jump out). My toad ate all 12 worms in 2 days. I have not put a lid on my pot yet. Perhaps that’s a mistake. My colony of bugs seems to be breeding fast enough to make up for escapees and eating, but perhaps that’s because the toad doesn’t eat many. A fellow at the pet store sold me ten small crickets for $3.00, but I don’t think the toad ate any before they escaped. See what your toad eats; it’s science. I think my toad is a female: it doesn’t vibrate or croak at night. Male toads vibrates and croak. Toads can be gender fluid, though; somethings two “female” toads will breed. Your job is to watch, enjoy, and perhaps learn something.

The main difference between this project, and the turtle terrarium I’d made is that the turtle terrarium was mostly water, with a brick, and this is mostly mud with a lake. I made the turtle terrarium in a laundry bin, a bigger environment, and flooded it except for the brick. I bought the turtles (a red-ears and a snapping) and fed it chicken bits and dandelion leaves. As with this terrarium, I kept the turtles outside through the spring, summer, and fall, but I brought the turtles in the winter. They lasted that way for about 8 years. Toads only live for 2-3 years, and mime may be a year or two old already. I won’t be too surprised if it croaks on my watch. For now, she seems safe and hoppy.

Robert Buxbaum, May 3, 2017. Here are some other science fair projects, chemical, and biological.

if everyone agrees, something is wrong

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I thought I’d try to semi-derive, and explain a remarkable mathematical paper that was published last month in The Proceedings of the Royal Society A (see full paper here). The paper demonstrates that too much agreement about a thing is counter-indicative of the thing being true. Unless an observation is blindingly obvious, near 100% agreement suggests there is a hidden flaw or conspiracy, perhaps unknown to the observers. This paper has broad application, but I thought the presentation was too confusing for most people to make use of, even those with a background in mathematics, science, or engineering. And the popular versions press versions didn’t even try to be useful. So here’s my shot:

Figure 2 from the original paper. For a method that is 80% accurate, you get your maximum reliability at the third to fifth witness. Beyond that, more agreement suggest a flaw in the people or procedure.

Figure 2 from the original paper. For a method that is 80% accurate, you get your maximum reliability at 3-5 witnesses. More agreement suggests a flaw in the people or procedure.

I will discuss only on specific application, the second one mentioned in the paper, crime (read the paper for others). Lets say there’s been a crime with several witnesses. The police line up a half-dozen, equal (?) suspects, and show them to the first witness. Lets say the first witness points to one of the suspects, the police will not arrest on this because they know that people correctly identify suspects only about 40% of the time, and incorrectly identify perhaps 10% (the say they don’t know or can’t remember the remaining 50% of time). The original paper includes the actual factions here; they’re similar. Since the witness pointed to someone, you already know he/she isn’t among the 50% who don’t know. But you don’t know if this witness is among the 40% who identify right or the 10% who identify wrong. Our confidence that this is the criminal is thus .4/(.4 +.1) = .8, or 80%.

Now you bring in the second witness. If this person identifies the same suspect, your confidence increases; to roughly (.4)2/(.42+.12) = .941,  or 94.1%. This is enough to make an arrest, but let’s say you have ten more witnesses, and all identify this same person. You might first think that this must be the guy with a confidence of (.4)10/(.410+.110) = 99.99999%, but then you wonder how unlikely it is to find ten people who identify correctly when, as we mentioned, each person has only a 40% chance. The chance of all ten witnesses identifying a suspect right is small: (.4)10 = .000104 or 0.01%. This fraction is smaller than the likelihood of having a crooked cop or a screw up the line-up (only one suspect had the right jacket, say). If crooked cops and systemic errors show up 1% of the time, and point to the correct fellow only 15% of these, we find that the chance of being right if ten out of ten agree is (0.0015 +(.4)10)/( .01+ .410+.110) = .16%. Total agreement on guilt suggests the fellow is innocent!

The graph above, the second in the paper, presents a generalization of the math I just presented: n identical tests of 80% accuracy and three different likelihoods of systemic failure. If this systemic failure rate is 1% and the chance of the error pointing right or wrong is 50/50, the chance of being right is P = (.005+ .4n)/(.01 +.4n+.1n), and is the red curve in the graph above. The authors find you get your maximum reliability when there are two to four agreeing witness.

Confidence of guilt as related to the number of judges that agree and your confidence in the integrity of the judges.

Confidence of guilt as related to the number of judges that agree and the integrity of the judges.

The Royal Society article went on to a approve of a feature of Jewish capital-punishment law. In Jewish law, capital cases are tried by 23 judges. To convict a super majority (13) must find guilty, but if all 23 judges agree on guilt the court pronounces innocent (see chart, or an anecdote about Justice Antonin Scalia). My suspicion, by the way, is that more than 1% of judges and police are crooked or inept, and that the same applies to scientific analysis of mental diseases like diagnosing ADHD or autism, and predictions about stocks or climate change. (Do 98% of scientists really agree independently?). Perhaps there are so many people in US prisons, because of excessive agreement and inaccurate witnesses, e.g Ruben Carter. I suspect the agreement on climate experts is a similar sham.

Robert Buxbaum, March 11, 2016. Here are some thoughts on how to do science right. Here is some climate data: can you spot a clear pattern of man-made change?