Another good-bye, this time from a space probe

 Space  No Responses »
Apr 302015
 

OK, I have had some sad good-byes in my blog this month, so here is a bittersweet one.

Earlier this afternoon, NASA’s Messenger probe, the first planetary probe to orbit Mercury, crashed into Mercury’s surface.

Although this means the end of Messenger, it also means that this particular probe fulfilled all expectations and then some: it worked flawlessly until it ran out of fuel and could no longer maintain a stable orbit around Mercury. The information it provided about the Solar System’s innermost planet will no doubt be studied for many years to come.

Good-bye, Messenger, and thanks for all the good work.

 Posted by at 4:50 pm

Emmy Noether’s birthday

 History, Internet, Physics  No Responses »
Mar 232015
 

Emmy Noether… not exactly a household name, at least outside of the community of theoretical physicists and mathematicians.

Which is why I was so surprised today when I noticed Google’s March 23 Doodle: a commemoration of Emmy Noether’s 133rd birthday.

Wow. I mean, thank you, Google. What a nice and deserving tribute to one of my heroes.

 Posted by at 11:36 pm

Two papers

 Personal, Physics  1 Response »
Mar 052015
 

Last month, something happened to me that may never happen again: I had not one but two papers accepted by Physical Review D in the same month, on two completely different topics.

The first was a paper I wrote with John Moffat, showing how well his scalar-tensor-vector gravity theory (STVG, also called MOG) fits an extended set of Milky Way rotational curve data out to a radius of nearly 200 kpc. In contrast, the archetypal modified gravity theory, MOND (Mordehai Milgrom’s MOdified Newtonian Dynamics) does not fare so well: as it predicts a flat rotation curve, its fit to the data is rather poor, although its advocates suggest that the fit might improve if we take into account the “external” gravitational field due to other galaxies.

The other paper, which I wrote together with an old friend and colleague, Eniko Madarassy, details a set of numerical simulations of self-gravitating Bose-Einstein condensates, which may form exotic stars or stellar cores. There has been some discussion in the literature concerning the stability of such objects. Our simulation shows that they are stable, which confirms my own finding, detailed in an earlier paper (which, curiously, was rejected by PRD), namely that the perceived instability arises from an inappropriate application of an approximation (the Thomas-Fermi approximation) used to provide a simplistic description of the condensate.

allcases

Oh, and we also had another paper accepted, not by Physical Review D, but by the International Journal of Modern Physics D, but still… it is about yet another topic, post-Galilean coordinate transformations and the analysis of the N-body problem in general relativity. Unlike the first two papers, this one was mostly the work of my co-author, Slava Turyshev, but I feel honored to have been able to contribute. It is a 48-page monster (in the rather efficient REVTeX style; who knows how many pages it will be in the style used by IJMPD) with over 400 equations.

All in all, a productive month insofar as my nonexistent second career as a theoretical physicist is concerned. Now I have to concentrate on my first job, the one that feeds the cats…

 Posted by at 3:21 pm

Quantum physics poll

 Physics  No Responses »
Feb 082015
 

cat-dead-aliveI have some half-baked ideas about the foundations of quantum physics (okay, who doesn’t.) When I say half-baked, I don’t mean that they are stupid (I sure hope not!) I simply mean I am not 100% sure about them, and there is more to learn.

But, I am allowed to have opinions. So when I came across this informal 2013 poll among (mostly) quantum physicists, I decided to answer the questions myself.

Question 1: What is your opinion about the randomness of individual quantum events (such as the decay of a radioactive atom)?

a. The randomness is only apparent: 9%
b. There is a hidden determinism: 0%
c. The randomness is irreducible: 48%
d. Randomness is a fundamental concept in nature: 64%

(“Jedenfalls bin ich überzeugt, daß [der Alte] nicht würfelt.”)

Question 2: Do you believe that physical objects have their properties well defined prior to and independent of measurement?

a. Yes, in all cases: 3%
b. Yes, in some cases: 52%
c. No: 48%
d. I’m undecided: 9%

(Note that the question does not say that “well-defined” is a synonym for “in an eigenstate”.)

Question 3: Einstein’s view of quantum mechanics

a. Is correct: 0%
b. Is wrong: 64%
c. Will ultimately turn out to be correct: 6%
d. Will ultimately turn out to be wrong: 12%
e. We’ll have to wait and see: 12%

(Einstein’s views are dated, but I feel that he may nonetheless be vindicated because his reasons for holding those views would turn out to be valid. But, we’ll have to wait and see.)

Question 4: Bohr’s view of quantum mechanics

a. Is correct: 21%
b. Is wrong: 27%
c. Will ultimately turn out to be correct: 9%
d. Will ultimately turn out to be wrong: 3%
e. We’ll have to wait and see: 30%

(If I said “wait and see” on Einstein’s views, how could I possibly answer this question differently?)

Question 5: The measurement problem

a. A pseudoproblem: 27%
b. Solved by decoherence: 15%
c. Solved/will be solved in another way: 39%
d. A severe difficulty threatening quantum mechanics: 24%
e. None of the above: 27%

(Of course it’s a pseudoproblem. It vanishes the moment you look at the whole world as a quantum world.)

Question 6: What is the message of the observed violations of Bell’s inequalities?

a. Local realism is untenable: 64%
b. Action-at-a-distance in the physical world: 12%
c. Some notion of nonlocality: 36%
d. Unperformed measurements have no results: 52%
e. Let’s not jump the gun—let’s take the loopholes more seriously: 6%

(I don’t like how the phrase “local realism” is essentially conflated with classical eigenstates. Why is a quantum state not real?)

Question 7: What about quantum information?

a. It’s a breath of fresh air for quantum foundations: 76%
b. It’s useful for applications but of no relevance to quantum foundations: 6%
c. It’s neither useful nor fundamentally relevant: 6%
d. We’ll need to wait and see: 27%

(I wish there was another option: e. A fad. Then again, it does have some practical utility, so b is my answer.)

Question 8: When will we have a working and useful quantum computer?

a. Within 10 years: 9%
d. In 10 to 25 years: 42%
c. In 25 to 50 years: 30%
d. In 50 to 100 years: 0%
e. Never: 15%

(The threshold theorem supposedly tells us what it takes to avoid decoherence. What I think it tells us is the limits of quantum error correction and why decoherence is unavoidable.)

Question 9: What interpretation of quantum states do you prefer?

a. Epistemic/informational: 27%
b. Ontic: 24%
c. A mix of epistemic and ontic: 33%
d. Purely statistical (e.g., ensemble interpretation): 3%
e. Other: 12%

(Big words look-up time, but yes, ontic it is. I may have remembered the meaning of “ontological”, but I nonetheless would have looked up both, just to be sure that I actually understand how these terms are used in the quantum physics context.)

Question 10: The observer

a. Is a complex (quantum) system: 39%
b. Should play no fundamental role whatsoever: 21%
c. Plays a fundamental role in the application of the formalism but plays no distinguished physical role: 55%
d. Plays a distinguished physical role (e.g., wave-function collapse by consciousness): 6%

(Of course the observer is a complex quantum system. I am surprised that some people still believe this new age quantum consciousness bull.)

Question 11: Reconstructions of quantum theory

a. Give useful insights and have superseded/will supersede the interpretation program: 15%
b. Give useful insights, but we still need interpretation: 45%
c. Cannot solve the problems of quantum foundations: 30%
d. Will lead to a new theory deeper than quantum mechanics: 27%
e. Don’t know: 12%

(OK, I had to look up the papers, as I had no recollection of the word “reconstruction” used in this context. As it turns out, I’ve seen papers in the past on this topic and they left me unimpressed. My feeling is that even as they purport to talk about quantum theory, what they actually talk about are (some of) its interpretations. And all too often, people who do this leave QFT completely out of the picture, even though it is a much more fundamental theory than single particle quantum mechanics!)

Question 12: What is your favorite interpretation of quantum mechanics?

a. Consistent histories: 0%
b. Copenhagen: 42%
c. De Broglie–Bohm: 0%
d. Everett (many worlds and/or many minds): 18%
e. Information-based/information-theoretical: 24%
f. Modal interpretation: 0%
g. Objective collapse (e.g., GRW, Penrose): 9%
h. Quantum Bayesianism: 6%
i. Relational quantum mechanics: 6%
j. Statistical (ensemble) interpretation: 0%
k. Transactional interpretation: 0%
l. Other: 12%
m. I have no preferred interpretation 12%

(OK, this is the big one: which camp is yours! And the poll authors themselves admit that it was a mistake to leave out n. Shut up and calculate. I am disturbed by the number of people who opted for Everett. Information-based interpretations seem to be the fad nowadays. I am surprised by the complete lack of support for the transactional interpretation, and also by the low level of support for Penrose. I put myself in the Other category, because my half-baked ideas don’t precisely fit into any of these boxes.)

Question 13: How often have you switched to a different interpretation?

a. Never: 33%
b. Once: 21%
c. Several times: 21%
d. I have no preferred interpretation: 21%

(I am not George W. Bush. I don’t “stay the course”. I change my mind when I learn new things.)

Question 14: How much is the choice of interpretation a matter of personal philosophical prejudice?

a. A lot: 58%
b. A little: 27%
c. Not at all: 15%

(I put my mark on a. because that’s the way it is today. If you asked me how it should be, I’d have answered c.)

Question 15: Superpositions of macroscopically distinct states

a. Are in principle possible: 67%
b. Will eventually be realized experimentally: 36%
c. Are in principle impossible: 12%
d. Are impossible due to a collapse theory: 6%

(Of course it’s a. Quantum physics is not about size, it’s about the number of independent degrees of freedom.)

Question 16: In 50 years, will we still have conferences devoted to quantum foundations?

a. Probably yes: 48%
b. Probably no: 15%
c. Who knows: 24%
d. I’ll organize one no matter what: 12%

(Probably yes but do I really care?)

OK, now that I answered these poll questions myself, does that make me smart? I don’t feel any smarter.

 Posted by at 2:52 pm

Needle in a Martian haystack

 Space  No Responses »
Jan 162015
 

Beagle 2 has been found.

Beagle 2 was the British lander component of the European Space Agency’s Mars Express mission. It was supposed to land on Mars on Christmas Day, 2003; however, no radio signal was ever received from the spacecraft. Beagle 2 was considered lost, its fate unknown.

But now, it has been found. Beagle 2, together with its parachute and rear cover, have been spotted by the High Resolution Imaging Science Experiment (HiRISE) camera on board the Mars Reconnaissance Orbiter (MRO) spacecraft, which itself has been orbiting Mars since March 10, 2006.

Imagine: a spacecraft orbiting another planet was able to spot an object barely more than a square meter in size, on that planet’s surface.

We may not yet have humans walking on Mars, but nonetheless, we live in amazing times. Now if only we somehow managed to stop murdering and hating each other, I might even begin to believe that there is hope for us yet…

 Posted by at 11:27 pm

The Paks nuclear incident

 Hungary, Physics  No Responses »
Jan 042015
 

Courtesy to a two-part article (part 1 and part 2, in Hungarian) of the Hungarian satirical-liberal magazine Magyar Narancs (Hungarian Orange), I now have a much better idea of what happened at Hungary’s sole nuclear generating station, the Paks Nuclear Power Plant, in 2003. It was the most serious nuclear incident to date in Hungary (the only INES level 3 incident in the country.)

At the root of the incident is a characteristic issue with these types of Soviet era nuclear reactors leading to magnetite contamination of the fuel elements and control rods. To deal with this contamination and prolong the life of fuel elements, cleaning ponds are installed next to the reactor blocks, where under roughly 30 feet of water, in a specially designed cleaning tank, fuel bundles can be cleaned.

As the problem of contamination became increasingly acute, the power plant ordered a new type of cleaning tank. On April 10, 2003, this cleaning tank was used for the first time on fuel bundles that were freshly removed from the reactor. The cleaning of the fuel bundles was completed successfully by 5 PM in the afternoon; however, the crane that was supposed to replace the fuel bundle in the reactor was used for another task and was not going to be available before midnight. The situation was complicated by language issues, as the technicians attending the new cleaning tank were from Germany and could not speak Hungarian. Nonetheless, the German crew assured the plant’s management that the delay would not represent a problem and that cooling of the fuel bundle inside the cleaning tank was adequate.

Shortly before 10 PM, an alarm system detected increased radiation and noble gas levels in the hall housing the cleaning pond. Acting upon the suspicion that a fuel rod assembly was leaking (the German crew suggested that the fuel bundles may have been incorrectly placed in the cleaning tank) the crew proceeded with a plan to open the cleaning tank. When the lid of the cleaning vessel was unlocked, a large steam bubble was released, and radiation levels spiked. Indeed, the crane operator received a significant dose of radiation contamination on his face and arms. The hall was immediately evacuated and its ventilation system was turned on. However, as the system had no adequate filtering systems installed (despite a regulation that six years prior mandated their installation) some radiation was released into the environment.

As it turns out, the culprit was the new type of cleaning tank. A model that, incidentally, was approved using an expedited process, due to the urgency of the situation at the power plant. The fact that the supplier was a proven entity also contributed to a degree of complacency.

Both the new and the old tank had a built-in pump that circulated water and kept the fuel bundle cool. However, in the old tank, the water inlet was at the bottom, whereas the outlet was near the top. This was not the case in the new tank: both inlet and outlet were located at the bottom, which allowed the formation of steam inside the cleaning vessel near the top. Combined with the lack of instrumentation, and considering that the fuel bundle released as much as 350 kW of heat, this was a disaster in the making.

And that is exactly what happened: due to the delay with the crane, there was enough time for the heat from the fuel bundle to cause most of the water inside the vessel to turn into steam, and the fuel elements heated to 1,000 degrees Centigrade. This caused their insulation to crack, which led to the initial detection of increased radiation levels. When the cleaning tank’s lid was opened, a large bubble of steam was released, while cold water rushed in causing a minor steam explosion and breaking up the fuel elements inside, contaminating the entire pond.

It took another ten years before the last remaining pieces of broken-up fuel elements were removed from the power plant, taken by train through Ukraine to a reprocessing plant in Russia. The total cost of the incident was in the $100 million range.

As nuclear incidents go, Paks was by no means among the scariest: after all, no lives were lost, there was only one person somewhat contaminated, and there was negligible environmental damage. This was no Chernobyl, Fukushima or Three Mile Island. There was some economic fallout, as this reactor block remained inoperative for about a year, but that was it.

Nonetheless, this incident is yet another example how inattention by regulatory agencies, carelessness, or failure to adhere to regulations can lead to catastrophic accidents. Despite its reputation, nuclear power remains one of the safest (and cleanest!) ways to generate electricity but, as engineers are fond of saying, there are no safeguards against human stupidity.

 Posted by at 4:25 pm

Merry Christmas

 Personal, Space  No Responses »
Dec 242014
 

Year after year, I can find no better way to wish Merry Christmas to all my family, my friends, and all good people on Earth, than with the immortal words of Apollo 8 astronaut Frank Borman from 46 years ago: “And from the crew of Apollo 8, we close with good night, good luck, a Merry Christmas and God bless all of you – all of you on the good Earth.”

 Posted by at 3:57 pm

Philae has landed

 Space  No Responses »
Nov 122014
 

Judging by the enthusiastic reaction I just saw moments ago on CBC Newsworld, the lander Philae, part of the Rosetta mission to the comet 67P/Churyumov-Gerasimenko, has landed successfully.

This is big. This is the first time a man-made device landed on a comet. It is called “primary exploration”.

It is also big for the European Space Agency. Rosetta is a major deep space mission: the spacecraft spent ten years traveling to this comet.

All in all, wonderful news.

 Posted by at 11:12 am

The parameters of the Standard Model

 Physics  1 Response »
Nov 042014
 

Standard_Model_of_Elementary_Particles.svgMany popular science books and articles mention that the Standard Model of particle physics, the model that unifies three of the fundamental forces and describes all matter in the form of quarks and leptons, has about 18 free parameters that are not predicted by the theory.

Very few popular accounts actually tell you what these parameters are.

So here they are, in no particular order:

  1. The so-called fine structure constant, \(\alpha\), which (depending on your point of view) defines either the coupling strength of electromagnetism or the magnitude of the electron charge;
  2. The Weinberg angle or weak mixing angle \(\theta_W\) that determines the relationship between the coupling constant of electromagnetism and that of the weak interaction;
  3. The coupling constant \(g_3\) of the strong interaction;
  4. The electroweak symmetry breaking energy scale (or the Higgs potential vacuum expectation value, v.e.v.) \(v\);
  5. The Higgs potential coupling constant \(\lambda\) or alternatively, the Higgs mass \(m_H\);
  6. The three mixing angles \(\theta_{12}\), \(\theta_{23}\) and \(\theta_{13}\) and the CP-violating phase \(\delta_{13}\) of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which determines how quarks of various flavor can mix when they interact;
  7. Nine Yukawa coupling constants that determine the masses of the nine charged fermions (six quarks, three charged leptons).

OK, so that’s the famous 18 parameters so far. It is interesting to note that 15 out of the 18 (the 9 Yukawa fermion mass terms, the Higgs mass, the Higgs potential v.e.v., and the four CKM values) are related to the Higgs boson. In other words, most of our ignorance in the Standard Model is related to the Higgs.

Beyond the 18 parameters, however, there are a few more. First, \(\Theta_3\), which would characterize the CP symmetry violation of the strong interaction. Experimentally, \(\Theta_3\) is determined to be very small, its value consistent with zero. But why is \(\Theta_3\) so small? One possible explanation involves a new hypothetical particle, the axion, which in turn would introduce a new parameter, the mass scale \(f_a\) into the theory.

Finally, the canonical form of the Standard Model includes massless neutrinos. We know that neutrinos must have mass, and also that they oscillate (turn into one another), which means that their mass eigenstates do not coincide with their eigenstates with respect to the weak interaction. Thus, another mixing matrix must be involved, which is called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. So we end up with three neutrino masses \(m_1\), \(m_2\) and \(m_3\), and the three angles \(\theta_{12}\), \(\theta_{23}\) and \(\theta_{13}\) (not to be confused with the CKM angles above) plus the CP-violating phase \(\delta_{\rm CP}\) of the PMNS matrix.

So this is potentially as many as 26 parameters in the Standard Model that need to be determined by experiment. This is quite a long way away from the “holy grail” of theoretical physics, a theory that combines all four interactions, all the particle content, and which preferably has no free parameters whatsoever. Nonetheless the theory, and the level of our understanding of Nature’s fundamental building blocks that it represents, is a remarkable intellectual achievement of our era.

 Posted by at 2:49 pm

A glimpse of the Sun

 Space  No Responses »
Oct 302014
 

Spacecraft sometimes catch a glimpse of the Sun as it reflects off a sea or an ocean. Here is an example:

Except that this example was not captured by Earth-orbiting spacecraft. The sea here is not a terrestrial ocean. It is a hydrocarbon sea of Saturn’s largest moon, Titan.

Just to clarify, the reflection of the Sun is in the upper left of the image, where the outline of the sea is also clearly visible. The redder, arrow-shaped object closer to the center is a cloud formation.

 Posted by at 10:37 pm

Mathematical fiction

 Books, Mathematics  No Responses »
Oct 132014
 

Science fiction has a subgenre: mathematical fiction. Stories of this nature are rare; good stories are even rarer. One memorable story that I recall from ages ago was A Subway Named Moebius, written by A. J. Deutsch in 1950. There was another story more recently: Luminous by Greg Egan, which I read in Asimov’s SF magazine shortly before I stopped reading (and eventually, stopped subscribing to) said magazine. (Nothing wrong with the magazine; it’s just that I found many of the stories unsatisfying, and I found I had less and less time to read them. The genre is just not the same as it was back in the Golden Age of Science Fiction.)

So recently, I found out that Egan wrote a sequel: Dark Integers, published in the same magazine in 2007. I now had a chance to read it and I was not disappointed.

Both stories are very good. Both stories are based on the notion that as yet unproven mathematical theorems can go either way; that the Platonic book of all math has not only not yet been written, but that there is no unique book, and multiple versions of mathematics may coexist, with an uneasy boundary.

Now imagine that you perform innocent mathematical experiments on your computer, using, say, computer algebra to probe ever more exotic theorems in a subfield few non-mathematicians ever heard about. And imagine how you would feel if you realized that by doing so, you are undermining the very foundations of another universe’s existence, literally threatening to wipe them out.

OK, if you start poking holes in that idea, there are many, but the basic notion is not completely stupid, and the questions that the stories raise are worth contemplating. And Egan writes well… the stories are fun, too!

Incidentally, this was the first decent (published) science fiction story I ever came across that contained a few lines of C++ code.

 Posted by at 4:00 pm

Why didn’t I go to Calgary?

 Canada, Personal, Weather and Climate  No Responses »
Sep 102014
 

I arrived in Ottawa in mid-July, 1987 as a landed immigrant. I was sponsored by my aunt and her husband András. It was András who awaited me at the airport on the evening of my arrival. (No, I did not arrive by air. My connecting flight from Montreal was canceled, so Air Canada put me in a limo along with another passenger. As the limo driver was not from Ottawa, and I knew nothing about the layout of the city, he dropped me off at the airport instead of taking me directly to my aunt’s house.)

I spent some time in the old (since decommissioned) airport building waiting for András to arrive. (In the pre-cellphone days, I first had to exchange some currency, then get some change, then find a payphone in order to be able to notify them about my whereabouts.) After a wait of a half hour or so, András did arrive. We only ever met once before, briefly, when they were visiting Hungary and I spent a few hours at my parents’ home, on leave from my mandatory military service. So when András saw me, he was not sure if I was the right person… as he approached me, he asked, “So you are Viktor?”

“Yes,” I answered, to which András replied with a second question: “Why did you come here, why didn’t you go to Calgary instead?”

Yes, András had a weird sense of humor. Not everyone appreciated it, but I did. I really grew to like him.

Earlier this week, it was Nature’s turn to be funny, while also providing me with a perfectly good answer to András’s question from 27 years ago. This is why, András:

Yes, András, I am a wimp. I can tolerate winter, but I really don’t like late summer snow storms.

Alas, András is no longer among us to hear my response. He passed away many years ago, after losing his battle with pancreatic cancer.

 Posted by at 5:26 pm

Feynman’s lectures online

 Physics  No Responses »
Sep 042014
 

Richard Feynman’s Lectures on Physics remains a classic to this day.

Its newest edition has recently (I don’t know exactly when, I only came across it a few days ago) been made available in its entirety online, for free. It is a beautifully crafted, very high quality online edition, using LaTeX (MathJax) for equations, redrawn scalable figures.

Perhaps some day, someone will do the same to Landau’s and Lifshitz’s 10-volume Theoretical Physics series, too?

 Posted by at 10:25 am

Einstein’s field equations… in a Subway sandwich commercial?

 Physics, Television  No Responses »
Aug 132014
 

Last night, as I was watching the latest episode of Tyrant (itself an excellent show about a fictitious Middle Eastern dictatorship and its ruling family), I happened to glance at the TV during a commercial break just at the right split second to see this:

This was part of an ad, a Subway sandwich commercial, with an animated monkey handing this exam sheet back to a student (also a monkey). What caught my eye was the equation on this sheet. What??? Einstein’s field equations?

Yup, that’s exactly what I saw there, the equation \(G_{\alpha\beta}=\dfrac{8\pi G}{c^4}T_{\alpha\beta}\). General relativity.

Other, easily recognizable equations on the sheet included an equation of the electrostatic Coulomb force, the definition of the quantum mechanical probability amplitude, and the continuity equation.

What struck me was that all these are legitimate equations from physics, not gibberish. And all that in a silly Subway commercial. Wow.

 Posted by at 4:48 pm

Double Rainbow

 Local Ottawa, Weather and Climate  1 Response »
Jul 102014
 

Two days ago, I was driving south on Bank Street when I saw this:

Yes, a double rainbow. The last time I saw a double rainbow like this was nearly 20 years ago, when my wife and I were driving through the Rocky Mountains on our way to California.

 Posted by at 8:30 am

A strange plume…

 Space  No Responses »
May 312014
 

Yesterday, around 7:17 AM in the morning Eastern time, I took a look at the new NASA site that is streaming Earth-observing video live from the ISS.

While I looked, I noticed a strange plume. It was barely visible, but it was definitely there. As I watched, it was quickly fading away/disappearing behind the horizon, so I was barely able to get a screen capture.

An asteroid impact? A secret nuclear test? Alien invasion? Who knows.

 Posted by at 6:33 pm

Planck length physics

 Physics  No Responses »
Apr 042014
 

A physics meme is circulating on the Interwebs, suggesting that any length shorter than the so-called Planck length makes “no physical sense”.

Which, of course, is pure nonsense.

The Planck length is formed using the three most fundamental constants in physics: the speed of light, \(c = 3\times 10^8~{\rm m}/{\rm s}\); the gravitational constant, \(G = 6.67\times 10^{-11}~{\rm m}^3/{\rm kg}\cdot{\rm s}^2\); and the reduced Planck constant, \(\hbar = h/2\pi = 1.05\times 10^{-34}~{\rm m}^2{\rm kg}/{\rm s}\).

Of these, the speed of light just relates two human-defined units: the unit of length and the unit of time. Nothing prevents us from using units in which \(c = 1\); for instance, we could use the second as our unit of time, and the light-second (\(= 300,000~{\rm km}\)) as our unit of length. In other words, the expression \(c = 300,000,000~{\rm m}/{\rm s}\) is just an instruction to replace every occurrence of the symbol \({\rm s}\) with the quantity \(300,000,000~{\rm m}\).

If we did this in the definition of \(G\), we get a new value: \(G’ = G/c^2 = 7.41\times 10^{-28}~{\rm m}/{\rm kg}\).

Splendid, because this reveals that the gravitational constant is also just a relationship between human-defined units: the unit of length vs. the unit of mass. It allows us to replace every occurrence of the symbol \({\rm kg}\) with the quantity \(7.41\times 10^{-28}~{\rm m}\).

So let’s do this to the reduced Planck constant: \(\hbar’ = \hbar G/c^3 = 2.61\times 10^{-70}~{\rm m}^2\). This is not a relationship between two human-defined units. This is a unit of area. Arguably, a natural unit of area. Taking its square root, we get what is called the Planck length: \(l_P = 1.61\times 10^{-35}~{\rm m}\).

The meme suggests that a distance less than \(l_P\) has no physical meaning.

But then, take two gamma rays, with almost identical energies, differing in wavelength by one Planck length, or about \(10^{-35}~{\rm m}\).

Suppose these gamma rays originate from a spacecraft one astronomical unit (AU), or about \(1.5\times 10^{11}~{\rm m}\) from the Earth.

The wavelength of a modest, \(1~{\rm MeV}\) gamma ray is about \(1.2\times 10^{-12}~{\rm m}\).

The number of full waves that fit in a distance of \(1.5\times 10^{11}~{\rm m}\) is, therefore, is about \(1.25\times 10^{23}\) waves.

A difference of \(10^{-35}~{\rm m}\), or one Planck length, in wavelength adds up to a difference of \(1.25\times 10^{-12}~{\rm m}\) over the \(1~{\rm AU}\) distance, or more than one full wavelength of our gamma ray.

In other words, a difference of less than one Planck length in wavelength between two gamma rays is quite easily measurable in principle.

In practice, of course we’d need stable gamma ray lasers placed on interplanetary spacecraft and a sufficiently sensitive gamma ray interferometer, but nothing in principle prevents us from carrying out such a measurement, and all the energy, distance, and time scales involved are well within accessible limits at present day technology.

And if we used much stronger gamma rays, say at the energy level of the LHC (which is several million times more powerful), a distance of only a few thousand kilometers would be sufficient to detect the interference.

So please don’t tell me that a distance less than one Planck length has no physical meaning.

 Posted by at 11:09 am
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