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Archive for the ‘Physics’ Category

Recently Virgin Galactic suffered a horrible setback: their SS2 “spaceplane” crashed, killing one and injuring another. My deepest sympathies go out to their families; this blog post is not meant to disrespect these brave men in any way.

My beef is with the graphic-design bozos at Virgin Galactic, who give us this laughable graphic:

ss2

It looks nifty, sure. But the science (as represented by this travesty) is weak to say the least. In fact, I’ll say more: the science in this graphic is laughable.

First of all, notice how there’s a dotted line that says “edge of space”. It’s like the Mason-Dixon line: on one side, you can buy sweet tea, on the other side, you can’t. It’s nice how they colored space “black” and colored “not space” blue. Thanks. That clarifies things.

In point of fact, of course, there is no “Edge of space”. The atmosphere decreases gradually as you move away from the Earth. Where do you draw the line? Should it be the upper limit of human survivability, around 10,000 meters, or maybe the upper limit of commercial airline flights, at around 18,000 meters? The Fédération Aéronautique Internationale (FAI) puts “space” at 100,000 meters, but that is arbitrary. Nothing special “happens” at that height.

Secondly, notice how the graphic says that there’s “zero gravity” at that height. Sigh. Don’t they go over this in 6th grade?

There’s plenty of gravity in space; at least, where satellites orbit. (I discuss this at greater length in an earlier post.) At 100,000 meters, the acceleration due to gravity g has the value of 9.5 m/s2, compared to 9.8 m/s2 at sea level. That’s not “zero gravity.”

I’m sure what they meant was that the plane is traveling in some parabolic arc, and at that the top of that arc the plane is in free fall, so (momentarily) people on the plane experience the absence of any normal force, otherwise known as a state of “apparent weightlessness”. Oh, who am I kidding. They didn’t mean that…they meant what they said, and what they said was nonsense.

I’m not pointing any fingers for the SS2 disaster, and anyway, the NTSB will figure it out eventually. Until then, don’t rely on the Virgin Galactic design team to know anything beyond 6th grade physics.

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Magnus Carlsen is the current world chess champion. He’s the best in the world at something. Not that many people can make that claim, can they?

Magnus Carlsen at FIDE World Chess Championship

Then again, there are lots of things in the world that you could be best at. Whistling, lemur training, lemon-pie-making, juggling, lying, rock climbing, sleepwalking. Somewhere in the world, there is “the best in the world” at each of these pursuits. Maybe my chances of being best at something are not so bad, after all? Maybe I just have to find the right thing…

Consider the modern pentathlon. In this sport, athletes compete in five events—fencing, shooting, swimming, running, and horse jumping—to achieve the overall best combined score. The winner need not be the best at any one specific event, but must have proficiency in all five.

Let’s say I am in the 99th percentile in all five events: very good, but not world class. [Here I am assuming that I’m in the 99th percentile of all humans, not just people who fence.] Taken individually, I wouldn’t have a prayer of making the Olympics. For example, the 99th percentile in épée fencing would still mean that there are

(0.01)^1 * 7,000,000,000 = 70,000,000

people with a similar proficiency around the world. Doesn’t seem that impressive, does yet? But I’m in the 99th percentile in all five events, right? So in reality there are only

(0.01)^5 * 7,000,000,000 = 0.7

people like me. That is, there’s just me. I’m probably the best at this combination of events. I should medal in the modern pentathlon.

And this brings me to my broader point. If you can think of five events in which you are in the 99th percentile individually, then in all likelihood you would be world champion if these events were combined into a single composite event. For those scoring at home, here’s where the number five comes from:

(0.01)^N * 7,000,000,000 = 1 (a single champion)

N ln (0.01) = ln [1/(7 x 10^9)]

N = [–ln (7 x 10^9)] / [ln (0.01)] = 4.9 ≈ 5

Let’s take my own skill set and see how I would do. I am certainly in the 99th percentile when it comes to physics. (Remember, I am comparing myself to the general population, not just physicists. I would never claim to be in the 99th percentile of people with physics PhD’s.) I am probably in the 99th percentile when it comes to chess (considering that I am in the 85th percentile for tournament players based on an 1800 rating). But am I good, really good, at anything else?

I will claim without proof that I am also in the 99th percentile (among the general population) in the following additional skills:

  • Knowledge of classical music
  • Playing the recorder
  • Geometry

Remember, I am not claiming any particularly high proficiency in any of these things. I just claim a 99th percentile rank in the general population. And individually, any one of these skills would only put me in the company of some 70 million others.

But now: make a hybrid event, where competitors have to take a battery of tests on physics, geometry, and classical music, then perform on the recorder, and then play chess… I believe I may do well in such an event. I might even be world champion.

Of course, nothing is that simple. I have ignored the fact that some of these skills may be correlated. Anyone who can play the recorder will probably also know about classical music. And many physicists will also be good at geometry. This means that my competition will be stiffer than I suppose, since if the events aren’t mutually exclusive then I’ve calculated the probabilities incorrectly. But I can improve my chances by making the five events as disparate as possible. I might change “Geometry” to “Movie Trivia”, for example.  My chances of becoming world champion would thereby be increased.

If you think that “99th percentile” is too high a bar, we could lower it to 90th percentile. Most people are in the top 10% at several things. Redoing our calculation, we get N = 9.8 in this case. So if you can find ten things you’re fairly good at and combine them, you too can be a world champion.

Of course, you also have to convince the Olympic governing body that that particular concatenation of events is worthy of a medal. But hey, that’s your problem.

I have some geometry to do.

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Screen Shot 2014-07-22 at 2.00.36 PM

Photo credit: brusspup

Recently I came across the following “optical illusion” on the normally good website I Fucking Love Science:

http://www.iflscience.com/brain/cycloid-optical-illusion-will-boggle-your-mind

I encourage you to read the article and think about the video (otherwise the rest of this post will be less than illuminating).

Supposedly, there is an “optical illusion” in the video because you think there’s a wheel when in “reality” the dots are moving linearly.

This is bullshit.

This is not an illusion. The dots are moving linearly, that’s true. But there is also a wheel. If this causes you cognitive dissonance, so be it. It is not a paradox, however. It is the case that some wheels, when spinning inside other wheels, have points on them which travel linearly.

I admit, this is a field of mathematics close to my heart. Much of my recent work has involved geometric phase, which has connections to Spirographs, epitrochoids and hypotrochoids as mentioned in the IFLS article. I have battered notebooks with over 500 pages of algebra devoted to such things. This is something I know something about.

One way to see that this isn’t an illusion at all is to watch it being drawn. Go to the following Spirograph applet:

http://www.personal.psu.edu/dpl14/java/parametricequations/spirograph/

Play around a little bit. Then create the following specific Spirograph (which is exactly the one in the IFLS “illusion”):
Radius1 = 60
Radius2 = -30
Position = 29
Velocity = 8
If you need to, CLEAR the picture, input the above parameters, and hit DRAW. Hit DRAW again to watch it all over again.

There’s no illusion. A circle is rotating inside another circle. Simultaneously, a particular part of that circle is traversing a straight line. This comes as a surprise to many people: there’s a frisson of incredulity from the idea that your motion can be simultaneously linear, but curved as well. But there’s an easy explanation. Your motion is different in different frames of reference.

Consider an ant on the wheel. With respect to the center of that wheel, he just orbits in a circular manner. But with respect to us, outside of the contraption, he moves back and forth along a single line. This is not an optical illusion. It’s the relativity of geometric shapes.  And I think that’s even cooler than some cognitive trickery.

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The big news of late is the discovery of gravitational waves from the very earliest time after the Big Bang.  What hasn’t been widely reported is that this represents a huge bit of indirect evidence that multiple universes really do exist.

Here’s more:

Big Bang Discovery Opens Doors to the “Multiverse”.

la-sci-sn-gravitational-waves-inflation-big-ba-001

(Harvard University / EPA)

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The conventional wisdom among people who know a little bit of quantum mechanics is that quantum mechanics is weird.

The conventional wisdom is wrong.  Quantum mechanics is not weird.  Interpretations of quantum mechanics are weird.

My thinking on this has changed over the years.  In high school I read everything I could about the “weirdness” of our universe: Schrödinger’s cat, wave-particle duality, the collapse of the wave function, many-worlds theory, the Heisenberg uncertainty principle.

Then a strange thing happened: I went to college.  I studied physics.  And guess what?  None of that stuff gets more than the briefest mention in the physics classroom.  Why?

Because those things are beside the point.  Quantum mechanics works.  How you interpret quantum mechanics is your problem.

There’s a dichotomy here which is the source of most people’s confusion.  Theories are different from interpretations of theories.  A theory is a mathematical model that allows us to make predictions.  An interpretation is a philosophical construct that allows us to sleep at night; it is a squishy heuristic that helps us unimaginative humans make sense of the math before us.  Theories get things done.  Interpretations never helped anybody, not really.

sr_atlas_2b-015

An abandoned shack.

Let’s say that in an abandoned shack you discovered a notebook with the word “PHYSICS” written by hand over and over, thousands of times, apparently filling every page.  You haven’t looked at the last few pages, but your theory is that these pages will also have the word “PHYSICS” written out.  Each time you turn a page, your theory is validated: “PHYSICS” is there, as predicted.

Next to this notebook is another that looks just like it.  You open the first page, and are not surprised to see “PHYSICS PHYSICS PHYSICS” again.  What’s going on?  Did some crazy person live in this shack?  Such speculation doesn’t really matter, since you can still hypothesize that “PHYSICS” fills this notebook as well.  In fact, you have a stronger theory: every notebook in this shack is filled with “PHYSICS”.

You perform an experiment: you turn the page.  “PHYSICS PHYSICS PHYSICS”.  The experiment supports your theory.  You find more notebooks; same results.  Every notebook in the shack is filled, apparently, with “PHYSICS”.  But guess what?  There are dozens of possible interpretations.  And in the absence of further data, you can never know which one is “correct”.

Maybe the shack was once inhabited by a crazy person, who wrote “PHYSICS” precisely 250,001 times in a futile attempt at summoning Cthulhu from his ancient slumber.

Maybe a student misspelled “physics” on a test, and her cruel teacher punished her in the most depraved way possible.

Maybe Matt Damon filled the notebooks, in a method-acting attempt to get into the mindset of an OCD scientist.

Which of these interpretations is the “truth”?  Without further data you cannot really say.  Arguing about which is right and which is wrong is futile at best, and annoying at worst.

Of course, new data may turn up.  We might find out that the notebooks are 75 years old, ruling out our Matt Damon interpretation.  That interpretation is no longer a valid interpretation of the data.

Which brings me to my next point: there is no official arbiter of what constitutes a theory versus what constitutes an interpretation.  Different philosophers and scientists have used the words differently at different times.  All you can hope for is that a particular author is consistent in his/her use of the terms.  I personally use the word “interpretation” to describe competing theories that cannot currently be differentiated by any known scientific experiment.  If two different interpretations make different, testable predictions, then they are promoted to being totally different theories.  (Caveat: others use the words slightly differently.  Deal with it.)

So what does this have to do with quantum mechanics?

Quantum mechanics is an entirely mathematical theory.  Its postulates are logical, concise, and powerful.  We can use quantum mechanics to invent cell phones, computers, lasers, and iPods.  Quantum mechanics doesn’t care if you “understand what it really means”, or not.  It is arguably the most successful and powerful theory to come out of the 20th century.

Now, the mathematics of quantum mechanics are abstract and hard to visualize.  Nevertheless, people insist on trying to visualize anyway.  And the result is all kinds of weirdness: Schrödinger’s cat, wave-particle duality, the collapse of the wave function, many-worlds theory, the Heisenberg uncertainty principle.  These ideas are all mental hoops that people have jumped through to explain some unambiguous, concrete, abstract linear algebra.  The math is just math, and it works; what it means is anyone’s guess.

There’s no crying in baseball, and there’s no philosophy in quantum mechanics.

leagueoftheirown

There’s no philosophy in quantum mechanics!

Don’t like the many-worlds interpretation?  Fine.  Be a Copenhagenist.  Don’t like pilot waves?  Great.  Stick to your pet idea about superluminal communication.  Just remember that all of these competing interpretations make the exact same predictions, so for all practical purposes they are the same.  Some people go so far as to say, just shut up and calculate.  [Note added 3-19-14: there are problems with pilot wave theories that in my view rule them out as being a valid interpretations of quantum mechanics.  But there are hoops people can jump through to try and “force” pilot wave theories to be consistent with, say, Bell’s theorem.  My broader point is that there are multiple interpretations of QM and that all have followers to this day, but that none of the interpretations really have any distinct implications for our lives.]

I don’t usually go that far.  I actually think that the many-worlds interpretation is a testable theory, not an interpretation (hence the name of this blog).  I think many-worlds is falsifiable.  (If we ever observe a wave function collapsing, then many-worlds will have to be discarded.)  But I don’t think that will happen: many-worlds is too elegant, and too powerful, to not be true.

But we’ll see.

If you think it’s absurd that a cat can be alive and dead at the same time…if you think that it’s crazy to hypothesize other universes…if you think that God does not play dice with the universe…don’t blame quantum mechanics.  Blame the philosophers who try to interpret it.

Quantum mechanics works.  Otherwise, you’d be reading this on an actual piece of paper.

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I was watching Dr. Who the other day and came across a physics mistake so common I thought I’d address it here.  The mistake is this:

Black holes suck you in like a vacuum cleaner!

The setup: in Dr. Who [2.8] “The Impossible Planet”, the good Doctor and Rose meet the crew of a ship who are on “an expedition [to] the mysterious planet Krop Tor, impossibly in orbit around a black hole.” [Wikipedia]  That phrase “impossibly in orbit” made me almost spit out my drink while watching the show.

Black holes have event horizons.  I get it.  Even light cannot escape.  I get that, too.  But why does that mean I cannot orbit a black hole?

OK, time for a little general relativity.  Einstein figured out, between 1905 and 1915, that gravity is “just” a warping of space-time.  Matter causes the space-time around it to curve; the curvature of space-time determines how matter moves (insofar as objects in the absence of gravitational forces follow geodesics).  The formulas that link the distribution of matter to the curvature of space are Einstein’s equations:

einstein_equation

This expression is compact and might seem relatively simple, but it’s not.  Gαβ and Tαβ are components of tensors, which are like vectors, but worse; they’re really 4×4 matrices.  So this equation is not one equation, but 16 different equations, since α and β can take on any of four values each.

What do all those letters stand for?  Gαβ is a component of the Einstein tensor, which tells you about how space-time is curved; the indices α and β can be any of four values in a 4D space-time.  (If you’re mathematically inclined, the Einstein tensor can be related to the Ricci scalar, the Ricci tensor, and the Riemann tensor.)  Tαβ is a component of the stress-energy tensor, which basically describes how matter/momentum/energy/stress/strain is distributed in a region of space-time.  So here’s another way to visualize Einstein’s equations:

einstein_explained

The cause (mass) is on the right; the effect (the curvature of space-time) is on the left.

So what does this have to do with black holes?

One of the first solutions discovered to the Einstein equations is called the Schwarzschild solution, which applies to a spherically symmetric gravitational source.  The solution gives you a “metric” (essentially, a geometry) that is almost the same as “flat” space-time, except for a pesky (1–2GM/c2r) term.  But that pesky term has a strange implication: when that term equals zero, the solution “blows up” (i.e. becomes infinite).  Space becomes so curved that you essentially have a hole in the fabric of space-time itself.

When does this happen?  It happens when R = 2GM/c2, as one line of algebra will show.  This is called the Schwarzschild radius.  The Einstein equations predict that something weird and horrifying happens when a mass is squeezed down to the size of its Schwarzschild radius.  Current understanding is that the mass would then keep going, and squeeze itself into a point of zero radius.  Literally, zero.  (I did say it was weird and horrifying).

Incidentally, the Schwarzschild radius is exactly the radius you’d get if you set the escape speed for an object equal to the speed of light.  So this means that not even light can escape this super-squeezed object.

And here’s where various misconceptions start to creep in.

Another name for the Schwarzschild radius is the event horizon.  It’s a boundary of no return:  if you cross it, you can never go back.  But that’s all it is: a boundary.  There is not necessarily anything physical at the event horizon.  You might never know that you had crossed it.  Remember, all the mass is at the center.

Here’s how I “picture” a black hole:

black hole

Now, if I am outside the event horizon, what would I see?  Well, nothing from inside the event horizon could reach me (hence the term “black”) but I might see Hawking radiation.  I would certainly see gravitational lensing: the bending of distant light around a black hole.  Here’s a cool picture of gravitational lensing in action (artists conception only!) from Wikipedia:

225px-Black_hole_lensing_web

Let’s say the Sun were a black hole.  Its event horizon would be around 3 km.  As long as we never got closer than 3km, we could do what we like.  We could fly in, fly out, orbit the black hole as we please.

Would the black hole “suck us in”?  Sure, in the same way that the Sun sucks us in already.  There is a strong pull of the Sun on the Earth.  And there would be a strong pull on our hypothetical spaceship.  But change the Sun to a black hole, and the pull would not get any stronger.  That is the key point that most people miss: black hole gravity is not somehow “stronger” than ordinary gravity.  There is just gravity; that’s it.  Change the Sun to a black hole, and the Earth would continue in its orbit, and nothing would be any different.  Except for, maybe, the lack of light.

Why was the planet Krop Tor’s orbit impossible?  Astronomical black holes (created by stellar collapse) have a lot of mass; when there’s a lot of mass hanging around, things tend to orbit them.  That’s what you’d expect.  It would only be impossible if somehow the orbit crossed the event horizon multiple times during its trajectory.  But of course, the show didn’t mention this.

I want to end my rant on GR with a suggestion: that there are two kinds of sci-fi: science fiction, and “sciency” fiction.  The first kind tries to get the science right, and makes an effort to be possible (if not plausible).  The second kind throws sciency words around in an effort to appeal to a certain demographic.  Basically, “sciency” fiction is fantasy, set in outer space.  When seen in this light, Dr. Who has more in common with Lord of the Rings than it does with 2001.

Don’t get me wrong: I love Lord of the Rings, and I love Dr. Who.  Just don’t call it science fiction.

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In an earlier post I talked about events in spacetime, and about how an error in time is usually more grievous than an error in space.

Let’s now talk about the coincidence of spacetime coordinates.  Specifically, how significant is it if you share one, two, three, or even four coordinates with a famous person?

Gettysburg Address

First, some preliminary discussion.  An event is a point (x,y,z,ct) in spacetime.  Technically, you are not an event; you are a series of (unfortunate?) events smoothly snaking its way forward in time.  As you sit there, reading this post, your x, y, and z are probably staying constant while ct is continually increasing.  (Of course if you are reading this on the bus, then x, y, and z may be changing as well.)  Note that I will use a relative coordinate system where x and y are measured with respect to the Earth (they are effectively longitude and latitude) and z is height above sea level.  This way, we don’t have to deal with the annoying detail that the Earth is spinning, and orbiting the Sun, and that the solar system is hurtling through space.

Now the act of you reading this is an event; let’s say it has the coordinates (x,y,z,ct) in spacetime.  But let’s also suppose that when you read that word, Matt Damon was eating a bagel with cream cheese.  That event had the coordinate (X,Y,Z,cT), say.  Unless you happened to have been with Matt Damon just then, your spatial coordinates did not coincide.  However, it should be obvious that t=T.  This means that it is no big deal to share a time coordinate with a celebrity.  You currently share a time coordinate with every living celebrity.  Right now, as you read this, Quentin Tarantino is doing something.  So is cricketer Michael Clarke.  So is chess grandmaster Magnus Carlsen.

michael-clarke

What are the spacetime coordinates of the Ashes?

But how significant is it if one spatial coordinate (x, y, or z) coincides with a celebrity?  Or two spatial coordinates?  Can we sort this out?

Here are some other possible cases:

x or y (and t) coincide: this is not likely to be true for you at this instant, but it happens with great frequency.  It means that either your longitude or latitude is the same as a celebrity, such as Christopher Walken.  Let’s say you’re currently in Jacksonville, FL whereas Walken is in Los Angeles.  Obviously, your x’s are very different and your y’s, although close, are also different.  But you decide to drive to Raleigh, NC for a friend’s wedding.  At some point along your drive on I-95 your y-coordinate will be the same as Walken’s, as the line of your latitude sweeps through 34 degrees North.  (If you’re curious, it will happen a little before you stop for lunch at Pedro’s South of the Border.)  On a flight from Seattle to Miami, your lines of x and y will coincide (at different times) with a majority of celebrities in the USA.

z (and t) coincide: this is also quite common.  It means that you and a celebrity (such as chess grandmaster Hikaru Nakamura) share an altitude.  I am currently at z = 645 m (2116 ft.) in elevation…well, scratch that, I am three floors up, so it’s closer to z = 657 m.  Anyway, if Nakamura drives from Saint Louis (Z = 142 m) to Denver (Z = 1600 m) on I-70 then our elevations will coincide at some point along his drive (presumably a little bit past Hays, KS).

x or y, with z and t: this is much rarer, but does happen.  For this to occur, your line of longitude or latitude would have to sweep through a celebrity (such as quarterback Cam Newton), but you would also have to coincidentally be at the same altitude.  Now, if you live in the same city as the celebrity (in this case, Charlotte, NC) then a simple trip across town to visit Trader Joe’s would probably be sufficient to achieve x=X (or y=Y) along with z=Z and t=T.  However, for someone like me who lives at an arbitrary (and uncommon) elevation such as 645 m, this does not happen often.

x, y, z….but not t: this means that you have visited the exact location that a famous person has visited, but not at the same time.  This probably happens hundreds of times in your life.  An obvious example is when you go to a famous location: maybe Dealey Plaza in Dallas, maybe the Blarney Stone, maybe the location of Lincoln’s Gettysburg address.  (By the way, today is the 150th anniversary of that speech!)  A not-so-obvious example (but much more common) is when you drive along a much-used road.  I have driven I-95 for huge stretches, for example, and I am sure many celebrities have driven that highway as well.  At some point along my drives, I will have “visited” the same location as another celebrity (Tina Fey, let’s say) when she decided to drive down to Savannah for the weekend.  I’m sure she stopped at Pedro’s South of the Border, and so have I.

pedro's

Proof that I went there.

x,y,z and t: this is the holy grail of celebrity coincidence.  It means you met the person.  Now, of course, humans are not bosons, so the spatial coordinates cannot be exactly the same, but if you meet the person I will say that the coordinates are close enough.  My (x,y,z,ct) were once the same as Al Gore.  My (x,y,z,ct) were once the same as Alan Dershowitz.  My (x,y,z,ct) were once (almost) the same as Hikaru Nakamura.  That’s about it.

I have left out several cases (such as x and/or z coinciding, without t) because they are trivial and uninteresting.  Imagine the entire world line of a celebrity such as Winston Churchill, who traveled all over the world.  If his spatial coordinates were projected onto the ground (painted bright yellow, say) then this looping curvy line would be a huge mess, spanning the globe, and covering huge swaths of England like spaghetti.  As I live my life, at any given instant I am pretty sure that one or two of my coordinates match some part of this snaky line.  No big deal.

It’s not like he was Matt Damon or anything.

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If you enjoyed this post, you may also enjoy my book Why Is There Anything? which is available for the Kindle on Amazon.com.

sargasso

I am also currently collaborating on a multi-volume novel of speculative hard science fiction and futuristic deep-space horror called Sargasso Nova.  Publication of the first installment will be January 2015; further details will be released on Facebook, Twitter, or via email: SargassoNova (at) gmail.com.

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