I should start this entry with a disclaimer like Wikipedia: “Caution! This article is written like an advertisement.” So it may seem, but what I am advertising can be readily had for a few hundred dollars from software company Mathematica and after reading this you may really want it.
First, I’m sure everyone reading this has had at least one of Martin Gardener’s “Aha” moments: that split second when hours of study have all clicked and the answer suddenly becomes clear. I can remember my first time. It was late at night and the college library was empty and closing. I had to retreat to a vacant classroom (left open all night in a small Ohio college town) to keep trying to figure out what all this epsilon/delta stuff in the calculus book was all about. When it finally hit I felt a brain-rush like I’d never experienced before. It was an intellectual high when your brain goes: “Aha, so that’s how it works!” After that, like a true addict, one can’t stop trying to experience it over and over again and, for some reason, mathematics seems to give more “Aha”s than many other endeavors.
Imagine then if one could have something that could give Aha-moments on demand? In mid-1988 software called “Mathematica” (a name recommended by Steve Jobs, as recently divulged by Stephen Wolfram the developer of Mathematica on the occasion of Steve Jobs passing) blossomed in the fledgling computer world with a program initially written to take advantage of Apple computer graphics capabilities. Suddenly almost every mathematical function known to man could be had for the cost of a few key strokes (albeit sometimes the syntax could be a bit obtuse) for unfailing execution, the results plotted in 3-D if so desired. The software was successful and the product grew and matured until today it stands at version 8.0. The reason why it offers ‘Aha’ moments is for the two-fold thrill of deciphering the commands to calculate your result and the production of the result itself. A quick look at the Mathematica web site will easily convince you of the extreme power and versatility. Today undergrads at MIT solve quantum mechanics problems of undreamed of complexity, in class, with their laptops running Mathematica. And the benefits reach all the way down to grade school. With version 8.0, Mathematica offers a web site of thousands of “Demonstrations” where some phenomena or other, from simple arithmetic to laser physics, is given on a screen with slide buttons: the effects of manually changing variables can be seen instantly. High school teachers love the ability to demonstrate a physics (or chemistry, math, astronomy...anything using math) principle projected on a white board with the ability to write on the board as well to help explain the concept under consideration.
Buy this software today!
And now for the odd connection, or not, again I have no answer, just conjecture. For quite a while, I don’t know when he started, but up to the day after the Challenger Space Shuttle disaster on January 28, 1986, Richard Feynman used to teach physics to Hughes employees (like me; I happened to have an office next door to the auditorium he used). Every other Wednesday afternoon he would hold forth in the auditorium on the first floor of building R1 on Imperial Blvd. in El Segundo; at that time it was all Hughes Aircraft. You can verify this astounding fact in his book “Surely You’re Joking, Mr. Feynman?” on p.330 (Actually, in the book, he says he “…used to teach…” And while the book was published in 1985 he was still teaching in 1986; I wonder if this wasn’t one of his famous ploys to keep the celebrity-curious from interfering with his enjoyment of lecturing on physics)?
OK, so I had Feynman as a teacher for 2 hours every other week. So, what didn’t I ask him? Well, I didn’t ask him what he saw in his mind’s eye when someone said “electron.” Wouldn’t that have been keen to hear? I asked him to draw me a copy of the very first Feynman Diagram he ever drew. He couldn’t; he didn’t remember what it was! I also didn’t ask him what his relationship was to Mathematica. But why would I think he had anything to do with Mathematica?
Now for another one of my completely unsubstantiated conjectures. In 1985 physicist and science historian Silvan Schweber published a paper in the “Reviews of Modern Physics,” (vol. 58, p. 452), discussing Feynman’s contributions to quantum mechanics. He mentions that while Feynman was still in high school in 1933 he assigned every symbol provided in the top row of keys on his typewriter a mathematical definition. This allowed him to type complex number equations using a sort of non-standard English language syntax unlike anything in use in mathematics at the time. The illustration provided looks sort of like what Mathematica uses today for entering functions. For instance in Mathematica you can type: “AgeOfUniverse” then a space to signify multiply and “SpeedOfLight.” Hit enter and you get 10 to the 26.1489 power meters; the distance light has traveled since the beginning of the universe. And Stephen Wolfram, the developer of Mathematica worked with Feynman at Caltech for about ten years right before it was released. And there is a famous photo of the two of them huddled together, Feynman writing, Wolfram watching, taken about the time Wolfram was developing Mathematica. And I can almost read Feynman’s lips saying: “Listen Wolfram, I think you need computer software that can be programmed to solve big complex problems by simply writing down the names of functions and giving the range of variables to evaluate!” And, you know what, that’s pretty much what we have!
Oh, yeah, and maybe this isn’t completely unsubstantiated. In 2005 Wolfram wrote: “We [Feynman] talked a lot about how it should work. He was keen to explain his methodologies for solving problems: for doing integrals, for notation, for organizing his work. I even managed to get him a little interested in the problem of language design. Though I don't think there's anything directly from Feynman that has survived in Mathematica.” Yet it’s hard not to think: use Mathematica and get a little help from Feynman!
And, if any further advertising were required to encourage you to try Mathematica, I give you John Forbes Nash (remember the movie “A Beautiful Mind?”): “I spoke on how I had been using MATHEMATICA in my work on the game models…I used the talk materials as prepared for my talk earlier in 2003 in Napoli (which was the latest lecture on the topic of game theory and economic interest). (It happened that progress since then had been slow, because of difficulties in actually finding solutions by computational means.)…were included to illustrate the key topic of the applicability of the…MATHEMATICA software "to complex problems in game theory"”. (July 25, 2003).
Buy this software and get Feynman and Jobs!
Wednesday, November 2, 2011
How Einstein Didn’t Flunk High School Science
How many times have we been at a party and someone asks what we do for a living and finding out we’re in science they say: “Wow! Science! That’s hard, even Einstein flunked high school science (or math)!” As we fight to stifle a scream we look at our shoes and mumble: “Well…gee…uh...” Because we really don’t know the story of how he didn’t flunk we’re sort of stuck agreeing from ignorance. Well, the real story is one of the best April Fools-type jokes ever!
It turns out Einstein left his German high school early without his diploma, but with a letter of recommendation from his teachers to the ETH in Zurich, Switzerland, saying that he was an outstanding mathematician and should be allowed to take their entrance exam a year early. Although he was still two years younger than the minimum entrance age he passed the science and math parts with such high grades the head of the physics department, Professor Weber, requested Einstein be allowed entrance immediately! However, due to some rather more mundane grades in history and foreign languages, and lack of a high school diploma, it was decided he should take a senior year of Swiss high school just to even out his total intellectual ‘package’ and pick up the diploma required for entrance by the ETH.
His first semester at the Aargau Kantonsschule high school was accomplished with a GPA of 2. Not too good right? Wrong! They were using a grading system of “1” as superior through “6,” a total failure. So he was doing B-level work (although his math and physics grades were “A”s his foreign language and liberal arts were still “C” level).
Now it gets interesting. After the first semester the Swiss educational bureaucracy issued a fiat that for the next semester the grading system was going to be turned on its head: 6 for superior and 1 for the less advantaged. I’m sure you can see where this is going!
Of course! The second semester he failed everything miserably by the grading standards of the previous semester! Although his second semester GPA was 5.4 out of 6, a B+ level. So he actually got smarter the second semester.
And at the end of the year he was welcomed into the ETH with open arms and as they say: “The rest is history.”
However, as time passed and the hoary mists of history enveloped the finer details of a foreign secondary school system, all that was remembered was that, somehow, inexplicably, because even the great Einstein failed his high school science, science must be really, really hard, so why study it?
Well, now you know how he didn’t fail. I wonder if Einstein ever thought about the irony of his senior year high school grades? And the nameless Swiss bureaucrats who made the switch would have pulled it off without notice if they hadn’t had one certain brilliant young man in their school system that year: 105 years later Time magazine’s “Man of the Century.”
All the above information can be easily had by reading “The Collected Papers of Albert Einstein,” volume 1, page 17.
The lesson learned is that if you are going to use a system (e.g. alphabet letters, numbers, emoticons, etc.) to indicate goodness it should be set-up to leverage seemingly pre-conceived human expectations of a top score (first letters, biggest numbers, smiley face). In this respect the American tradition of assigning “A”s coupled with 4 GPs is a stroke of genius combining the best of both of the Swiss grading systems that bedeviled Einstein!
It can be reasonably inferred after reviewing the standard Einsteinian biographies that Einstein himself never gave his high school grades a second thought; that’s what publishing articles in Annalen der Physik will do!
It turns out Einstein left his German high school early without his diploma, but with a letter of recommendation from his teachers to the ETH in Zurich, Switzerland, saying that he was an outstanding mathematician and should be allowed to take their entrance exam a year early. Although he was still two years younger than the minimum entrance age he passed the science and math parts with such high grades the head of the physics department, Professor Weber, requested Einstein be allowed entrance immediately! However, due to some rather more mundane grades in history and foreign languages, and lack of a high school diploma, it was decided he should take a senior year of Swiss high school just to even out his total intellectual ‘package’ and pick up the diploma required for entrance by the ETH.
His first semester at the Aargau Kantonsschule high school was accomplished with a GPA of 2. Not too good right? Wrong! They were using a grading system of “1” as superior through “6,” a total failure. So he was doing B-level work (although his math and physics grades were “A”s his foreign language and liberal arts were still “C” level).
Now it gets interesting. After the first semester the Swiss educational bureaucracy issued a fiat that for the next semester the grading system was going to be turned on its head: 6 for superior and 1 for the less advantaged. I’m sure you can see where this is going!
Of course! The second semester he failed everything miserably by the grading standards of the previous semester! Although his second semester GPA was 5.4 out of 6, a B+ level. So he actually got smarter the second semester.
And at the end of the year he was welcomed into the ETH with open arms and as they say: “The rest is history.”
However, as time passed and the hoary mists of history enveloped the finer details of a foreign secondary school system, all that was remembered was that, somehow, inexplicably, because even the great Einstein failed his high school science, science must be really, really hard, so why study it?
Well, now you know how he didn’t fail. I wonder if Einstein ever thought about the irony of his senior year high school grades? And the nameless Swiss bureaucrats who made the switch would have pulled it off without notice if they hadn’t had one certain brilliant young man in their school system that year: 105 years later Time magazine’s “Man of the Century.”
All the above information can be easily had by reading “The Collected Papers of Albert Einstein,” volume 1, page 17.
The lesson learned is that if you are going to use a system (e.g. alphabet letters, numbers, emoticons, etc.) to indicate goodness it should be set-up to leverage seemingly pre-conceived human expectations of a top score (first letters, biggest numbers, smiley face). In this respect the American tradition of assigning “A”s coupled with 4 GPs is a stroke of genius combining the best of both of the Swiss grading systems that bedeviled Einstein!
It can be reasonably inferred after reviewing the standard Einsteinian biographies that Einstein himself never gave his high school grades a second thought; that’s what publishing articles in Annalen der Physik will do!
Friday, October 7, 2011
CERN LHC Luminosity Milepost
What with all the newsy swirl the last few days this went completely unreported:
“Last night, [October 3, 2011] while most of the collaboration was sleeping, LHC reached the 2011 milestone of delivering one inverse femtobarn of luminosity to LHCb.” http://www.quantumdiaries.org/2011/10/03/one-inverse-femtobarn-lhcb/
“Last night, [October 3, 2011] while most of the collaboration was sleeping, LHC reached the 2011 milestone of delivering one inverse femtobarn of luminosity to LHCb.” http://www.quantumdiaries.org/2011/10/03/one-inverse-femtobarn-lhcb/
Tuesday, September 27, 2011
The Fermata Joke
A high school neighborhood friend recently gave me what has to be the most esoteric joke I’ve ever seen, this cartoon created by her friend Thomas Stumpf:
But my knowledge of music is legendary for its nonexistence so I had to ask Linda, an accomplished pianist and piano teacher to explain the in-joke:
“'fermata' in Italian, means 'stop'; stop signs in Italy look just like ours, but say 'fermata'! In music, it means to just wait on the note, for as long as seems to be the best length, in context of the music. The fermata sign is the top half a circle with a dot inside. The fermata is over a rest,(which is silence), which would mean to be silent for more time than the time value of the rest, which is 2 beats.”
So, there are no notes on the score which means there is no sound and the fermata above means that one should keep doing what is written below for as long as necessary, which is nothing or: “shut up.”
But my knowledge of music is legendary for its nonexistence so I had to ask Linda, an accomplished pianist and piano teacher to explain the in-joke:
“'fermata' in Italian, means 'stop'; stop signs in Italy look just like ours, but say 'fermata'! In music, it means to just wait on the note, for as long as seems to be the best length, in context of the music. The fermata sign is the top half a circle with a dot inside. The fermata is over a rest,(which is silence), which would mean to be silent for more time than the time value of the rest, which is 2 beats.”
So, there are no notes on the score which means there is no sound and the fermata above means that one should keep doing what is written below for as long as necessary, which is nothing or: “shut up.”
Monday, September 26, 2011
Columnar Basalt
Just about anybody who lives in California, and many others who don’t, know about Devil’s Post Pile Park located on the west side of Mammoth Mountain ski area. It is a spectacular west-facing wall of basalt columns several feet wide rising 60 feet straight up. When I visited there in 1999 one scary fact impressed on the casual trekker by a Park Service sign was that the whole thing was earlier on considered a nuisance and a prime source of building material for a dam on the adjacent river for gold mining purposes or some such. The whole display of the post piles is large, but well within the purview of the visitor, so obviously this site could have been easily destroyed. As much as I wanted a rock sample I realized it would be totally inappropriate to take a piece of this magnificent structure! Not to mention that the very smallest samples would have weighed in at a significant fraction of a ton!
Much to my surprise and delight then, while shopping for flagstone pavers at the local rock and brick construction supply yard, I spotted a box labeled: “Small columns,” and inside were fragmented pieces of columnar basalt, supposedly from Washington State. I was able to find a nice piece for my collection, a few inches long, see photo. Notice it has four well developed faces that could be extrapolated out to give six faces total, although since columnar basalt is not a single crystal, the number of faces are not determined by crystal structure considerations and it could possibly have only 5 faces. Plus one end has been broken off showing the basic mineral structure and the other end has been roughly sawed showing a cross section view.
Much to my surprise and delight then, while shopping for flagstone pavers at the local rock and brick construction supply yard, I spotted a box labeled: “Small columns,” and inside were fragmented pieces of columnar basalt, supposedly from Washington State. I was able to find a nice piece for my collection, a few inches long, see photo. Notice it has four well developed faces that could be extrapolated out to give six faces total, although since columnar basalt is not a single crystal, the number of faces are not determined by crystal structure considerations and it could possibly have only 5 faces. Plus one end has been broken off showing the basic mineral structure and the other end has been roughly sawed showing a cross section view.
Friday, September 23, 2011
Faster than light expert opinion
The experts are starting to publish opinions on the neutrino-speed of light issue. Here is Sean Carroll, Caltech professor of theoretical physics:
“Faster-Than-Light Neutrinos?
by Sean
Probably not. But maybe! Or in other words: science as usual.
For the three of you reading this who haven’t yet heard about it, the OPERA experiment in Italy recently announced a genuinely surprising result. They create a beam of muon neutrinos at CERN in Geneva, point them under the Alps (through which they zip largely unimpeded, because that’s what neutrinos do), and then detect a few of them in the Gran Sasso underground laboratory 732 kilometers away. The whole thing is timed by stopwatch (or the modern high-tech version thereof, using GPS-synchronized clocks), and you solve for the velocity by dividing distance by time. And the answer they get is: just a teensy bit faster than the speed of light, by about a factor of 10-5. Here’s the technical paper, which already lists 20 links to blogs and news reports.
The things you need to know about this result are:
It’s enormously interesting if it’s right.
It’s probably not right.
By the latter point I don’t mean to impugn the abilities or honesty of the experimenters, who are by all accounts top-notch people trying to do something very difficult. It’s just a very difficult experiment, and given that the result is so completely contrary to our expectations, it’s much easier at this point to believe there is a hidden glitch than to take it at face value. All that would instantly change, of course, if it were independently verified by another experiment; at that point the gleeful jumping up and down will justifiably commence.
This isn’t one of those annoying “three-sigma” results that sits at the tantalizing boundary of statistical significance. The OPERA folks are claiming a six-sigma deviation from the speed of light. But that doesn’t mean it’s overwhelmingly likely that the result is real; it just means it’s overwhelmingly unlikely that the result is simply a statistical fluctuation. There is another looming source of possible error: a “systematic effect,” i.e. some unknown miscalibration somewhere in the experiment or analysis pipeline. (If you are measuring something incorrectly, it doesn’t matter that you measure it very carefully.) In particular, the mismatch between the expected and observed timing amounts to tens of nanoseconds; but any individual “event” takes the form of a pulse that is spread out over thousands of nanoseconds. Extracting the signal is a matter of using statistics over many such events — a tricky business.
The experimenters and their colleagues at other experiments know this perfectly well, of course. As Adrian Cho reports in Science, OPERA’s spokesperson Antonio Ereditato is quick to deny that they have overturned Einstein. “I would never say that… We are forced to say something. We could not sweep it under the carpet because that would be dishonest.” Now there’s a careful and honest scientist for you, I wish we were all so precise and candid. Cho also quotes Chang Kee Jung, a physicist not on the experiment, as saying, “I wouldn’t bet my wife and kids because they’d get mad. But I’d bet my house.” A careful and honest husband and father.
Scientists do difficult experiments all the time, of course, and yet we believe their results. That’s simply because it’s proper to be extra skeptical when the results fly in the face of our expectations: extraordinary claims require extraordinary evidence, as someone once paraphrased Bayes’s Theorem. When the supernova results in 1998 suggested that the universe is accelerating, most cosmologists hopped on board fairly quickly, both because we had a simple theoretical model in hand (the cosmological constant) and because the result helped explain several other nagging observational problems (such as the age of the universe). Here that’s not quite true, although we should at least mention that Fermilab’s MINOS experiment also saw evidence for faster-than-light neutrinos, albeit at a woefully insignificant level. More relevant is the fact that we have completely independent indications that neutrinos do travel at the speed of light, from Supernova 1987A. If the OPERA results are naively taken at face value, the SN 87A should have arrived a couple of years before we saw the explosion using good old-fashioned photons. But perhaps we should resist being naive; the SN 87A events were electron neutrinos, not muon neutrinos, and they were at substantially lower energies. If neutrinos do violate the light barrier, it’s completely consistent to imagine that they do so in an energy-dependent way, so the comparison is subtle.
Which brings up a crucial point: if this result is true (which is always a possibility), it is much more surprising than the acceleration of the universe, but it’s not as if we don’t already have ways to explain it. The most straightforward idea is to violate Lorentz invariance, a strategy of which I’m quite personally fond (although I’ve never applied the idea to neutrino physics). Lorentz invariance says that everyone measures the speed of light to be the same; if you violate it, it’s easy enough to imagine that someone (like, say, a neutrino) measures something different. Once you buy into that idea, neutrinos are an interesting place to apply the idea, since our constraints on their properties are relatively weak. It’s an interesting enough topic that there are review articles, and even a Wikipedia page on the idea.
And there are more way-out possibilities. Graininess in spacetime from quantum gravity might affect the propagation of nearly-massless particles; extra dimensions might provide a shortcut through space. This experimental result will probably give a boost to theorists thinking about these kinds of things, as well it should — there’s nothing disreputable about trying to come up with models that fit new data. But it’s still a long shot at this time. I hate to keep saying it over and over in this era of tantalizing-but-not-yet-definitive experimental results, but: stay tuned.”
http://blogs.discovermagazine.com/cosmicvariance/2011/09/23/faster-than-light-neutrinos/
“Faster-Than-Light Neutrinos?
by Sean
Probably not. But maybe! Or in other words: science as usual.
For the three of you reading this who haven’t yet heard about it, the OPERA experiment in Italy recently announced a genuinely surprising result. They create a beam of muon neutrinos at CERN in Geneva, point them under the Alps (through which they zip largely unimpeded, because that’s what neutrinos do), and then detect a few of them in the Gran Sasso underground laboratory 732 kilometers away. The whole thing is timed by stopwatch (or the modern high-tech version thereof, using GPS-synchronized clocks), and you solve for the velocity by dividing distance by time. And the answer they get is: just a teensy bit faster than the speed of light, by about a factor of 10-5. Here’s the technical paper, which already lists 20 links to blogs and news reports.
The things you need to know about this result are:
It’s enormously interesting if it’s right.
It’s probably not right.
By the latter point I don’t mean to impugn the abilities or honesty of the experimenters, who are by all accounts top-notch people trying to do something very difficult. It’s just a very difficult experiment, and given that the result is so completely contrary to our expectations, it’s much easier at this point to believe there is a hidden glitch than to take it at face value. All that would instantly change, of course, if it were independently verified by another experiment; at that point the gleeful jumping up and down will justifiably commence.
This isn’t one of those annoying “three-sigma” results that sits at the tantalizing boundary of statistical significance. The OPERA folks are claiming a six-sigma deviation from the speed of light. But that doesn’t mean it’s overwhelmingly likely that the result is real; it just means it’s overwhelmingly unlikely that the result is simply a statistical fluctuation. There is another looming source of possible error: a “systematic effect,” i.e. some unknown miscalibration somewhere in the experiment or analysis pipeline. (If you are measuring something incorrectly, it doesn’t matter that you measure it very carefully.) In particular, the mismatch between the expected and observed timing amounts to tens of nanoseconds; but any individual “event” takes the form of a pulse that is spread out over thousands of nanoseconds. Extracting the signal is a matter of using statistics over many such events — a tricky business.
The experimenters and their colleagues at other experiments know this perfectly well, of course. As Adrian Cho reports in Science, OPERA’s spokesperson Antonio Ereditato is quick to deny that they have overturned Einstein. “I would never say that… We are forced to say something. We could not sweep it under the carpet because that would be dishonest.” Now there’s a careful and honest scientist for you, I wish we were all so precise and candid. Cho also quotes Chang Kee Jung, a physicist not on the experiment, as saying, “I wouldn’t bet my wife and kids because they’d get mad. But I’d bet my house.” A careful and honest husband and father.
Scientists do difficult experiments all the time, of course, and yet we believe their results. That’s simply because it’s proper to be extra skeptical when the results fly in the face of our expectations: extraordinary claims require extraordinary evidence, as someone once paraphrased Bayes’s Theorem. When the supernova results in 1998 suggested that the universe is accelerating, most cosmologists hopped on board fairly quickly, both because we had a simple theoretical model in hand (the cosmological constant) and because the result helped explain several other nagging observational problems (such as the age of the universe). Here that’s not quite true, although we should at least mention that Fermilab’s MINOS experiment also saw evidence for faster-than-light neutrinos, albeit at a woefully insignificant level. More relevant is the fact that we have completely independent indications that neutrinos do travel at the speed of light, from Supernova 1987A. If the OPERA results are naively taken at face value, the SN 87A should have arrived a couple of years before we saw the explosion using good old-fashioned photons. But perhaps we should resist being naive; the SN 87A events were electron neutrinos, not muon neutrinos, and they were at substantially lower energies. If neutrinos do violate the light barrier, it’s completely consistent to imagine that they do so in an energy-dependent way, so the comparison is subtle.
Which brings up a crucial point: if this result is true (which is always a possibility), it is much more surprising than the acceleration of the universe, but it’s not as if we don’t already have ways to explain it. The most straightforward idea is to violate Lorentz invariance, a strategy of which I’m quite personally fond (although I’ve never applied the idea to neutrino physics). Lorentz invariance says that everyone measures the speed of light to be the same; if you violate it, it’s easy enough to imagine that someone (like, say, a neutrino) measures something different. Once you buy into that idea, neutrinos are an interesting place to apply the idea, since our constraints on their properties are relatively weak. It’s an interesting enough topic that there are review articles, and even a Wikipedia page on the idea.
And there are more way-out possibilities. Graininess in spacetime from quantum gravity might affect the propagation of nearly-massless particles; extra dimensions might provide a shortcut through space. This experimental result will probably give a boost to theorists thinking about these kinds of things, as well it should — there’s nothing disreputable about trying to come up with models that fit new data. But it’s still a long shot at this time. I hate to keep saying it over and over in this era of tantalizing-but-not-yet-definitive experimental results, but: stay tuned.”
http://blogs.discovermagazine.com/cosmicvariance/2011/09/23/faster-than-light-neutrinos/
Thursday, September 22, 2011
Speed of Light Exceeded?
"A total of 15,000 beams of neutrinos -- tiny particles that pervade the cosmos -- were fired over a period of 3 years from CERN towards Gran Sasso 730 (500 miles) km away, where they were picked up by giant detectors.
"Light would have covered the distance in around 2.4 thousandths of a second, but the neutrinos took 60 nanoseconds -- or 60 billionths of a second -- less than light beams would have taken.
"It is a tiny difference," said Ereditato, who also works at Berne University in Switzerland, "but conceptually it is incredibly important. The finding is so startling that, for the moment, everybody should be very prudent."
Obviously this overturns Einstein's axiom that the speed of light is the maximum allowable velocity in the universe, one of the foundational laws of modern physics.
"Light would have covered the distance in around 2.4 thousandths of a second, but the neutrinos took 60 nanoseconds -- or 60 billionths of a second -- less than light beams would have taken.
"It is a tiny difference," said Ereditato, who also works at Berne University in Switzerland, "but conceptually it is incredibly important. The finding is so startling that, for the moment, everybody should be very prudent."
Obviously this overturns Einstein's axiom that the speed of light is the maximum allowable velocity in the universe, one of the foundational laws of modern physics.
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