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Posts Tagged ‘special relativity’

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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sazon

If you’re ever in Cullowhee, NC, give Sazón a try.

Suppose I agree to meet my wife for dinner at 8 pm.  She goes to El Pacífico (a local Mexican restaurant) whereas I go Sazón (another Mexican restaurant).  The restaurants are a kilometer apart.  I’ve made an error, of course.

The next week, we agree to meet at El Pacífico.  She arrives at 7 pm, I get there at 8 pm.  Oops, I’ve made another error, this time not in location, but in time.

Which error is worse?

Any student of special relativity will be familiar with the terms spacelike, timelike, and lightlike interval.  Surprisingly, these terms are perfect for discussing my dinner date woes.  But what do these terms mean, on an intuitive level?  Are they even comprehensible in the realm of low, non-relativistic velocities?

Imagine two events, such as me clapping my hands, and you clapping yours.  The events are separated in space by a distance ΔR and separated in time by a duration Δt.  It turns out that if you think of the cosmos as being 4-dimensional, there is then a relationship between ΔR and Δt.

This relationship is the 4D distance formula, Δs2 = ΔR2 – c2Δt2.  Think about it as a sort of “Pythagorean theorem” for 4D spacetime.  That is, its square root (Δs) gives the “distance” between two events in spacetime, given that any event has coordinates (x,y,z,ct).  (Unfortunately, sometimes Δs2 is negative, in which case you cannot take the square root.  But that’s OK; we just talk about Δs2 and don’t even worry about what Δs “means”.)

OK, so why is c (the speed of light) in there?  Well, for two reasons.  One, there has to be some velocity as a conversion constant, so that the fourth coordinate ct has dimensions of distance (just like x, y, or z).  Secondly, the 4D distance formula is constructed explicitly so that if you’re travelling at speed c, then your speed will always be c in every other reference frame.  This is done to match experiment, but whether it’s justified or not depends upon whether the final result (the formula for Δs2) also has merit.  In this case it does, and it turns out that the quantity Δs2 is an invariant: it stays the same regardless of the reference frame you’re in.  Distance is relative; time is relative; but the unholy combination of distance and time, Δs2, is not.

But we’re here to talk about intervals Δs, which represent spacetime distances between events.  Let’s call our first event P1;  our second event is P2.  Now, P1 will be “I throw a ball.”  P2 will be “You catch the ball.”  Let’s say we’re separated by ΔR = 10 m and you catch the ball Δt  = 0.4 s later.  Then:

ΔR2 = 102 = 100 m2,

c2Δt2 = (3 x 108)2 (0.4)2 = 1.44 x 1016 m2.

Wow!  The “time term” (in the Δs2 formula) dominates, so that Δs2=100 – 1.44 x 1016 = – 1.44 x 1016m2, which is negative.  In plain English, the events are separated more by time than they are by space.  When this happens, the invariant interval Δs2 is negative.  We call such an interval timelike.  Another way to think of this is that P1 and P2 can influence each other by cause and effect.

Why is the time term so big?  Basically, because light is so fast.  Remember that we had to multiply time by the speed of light to make the fourth 4D spacetime coordinate have units of distance.  So unless events are very, very far apart—or the time difference is very, very small—then Δs2 will be negative and you will have a timelike separation.

Now, in your everyday life, distances are never that great.  The biggest distance you will ever have between you and a friend is the diameter of the Earth; namely, ΔR = 12,700,000 m.  Even at that distance, events will have a timelike separation unless Δt is below a certain threshold:

ΔR2 > c2Δt2

Δt < ΔR/c = (12,700,000 m)/ (3 x 108 m/s) = 0.042 s = 42 ms.

I do something Pin North Carolina; something else Phappens to my friend Rick in Perth, Australia.  In order for these events to be causally connected, at least 42 ms must past between P1 and P2.  If Δt is less than 42 ms, then light doesn’t have enough time to get from me to Rick; the events cannot be causally connected and we have a spacelike interval instead.  In such a case Δs2 is always greater than zero.  In plain English, the events are separated more by space than they are by time.

(Note: 42 ms is a good value to keep in mind.  On the Earth, for an interval between events to be spacelike, they have to be almost simultaneous: Δt has to be less than 42 ms.)

Let’s apply our new terminology to the example that started this post.  The events P1 and P2 are “I arrive at my destination” and “my wife arrives at her destination”.  We arrive at the same time, 8 o’clock, but the restaurants are a kilometer apart.  So:

Δt = 0,

ΔR = 1000 m,

Δs2 = ΔR2 – c2Δt2 = 10002 = 1,000,000 m2.

Seems like a big error!  But the next week I get the time wrong (not the location) and we find:

Δt = 1 hr. = 3600 s,

ΔR = 0,

Δs2 = ΔR2 – c2Δt2 = – c2 (3600)2 = –1,166,400,000,000,000,000,000,000 m2.

That’s worse.  A lot worse.  It represents a much bigger “interval” in spacetime.  And, truth be told, our culture reflects this; it is much worse to be an hour late for dinner then to show up on time but 1 km away.  And this makes sense: if you go to the wrong restaurant you can correct your error fairly easily; it wouldn’t take that long (maybe 1 minute?) to drive 1 km.  If instead you’re an hour late, there’s not much you can do…

[Here’s the website for Sazón: http://www.sazoncullowhee.com/wordpressinstall/]

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Should we pledge to enact sanctions against ancient Carthage?

It’s time to start marginalizing Grover Norquist.

Haven’t heard of him?  That’s because he hasn’t really done anything noteworthy.  Sure, he got an M.B.A. from Harvard, and he did write speeches for the U.S. Chamber of Commerce for one year in the 1980’s, but other than that he’s done nothing except be a lobbyist.  He’s never had an elected position.  His reputation is based on lobbying.

Have I said he’s just a lobbyist?

Now, to the guy’s credit, he’s good at his job, and he wields power through his personal instrument Americans for Tax Reform.  That’s the lobbying group he founded.  Its only purpose is to advocate for Norquist’s world view.  Part of that world view is to lower tax rates in America, and I won’t comment on whether that’s a good idea or not…that’s a problem for economists to sort out.  But part of that world view is getting politicians (by scare tactics and intimidation) to commit to a “no tax raises” pledge.

I can’t think of anything sillier than a politician making such a pledge.  What is this, the days of Hamilton and Burr at Weehawken?

First of all, tactically, it’s always better to have options than to not have options.  If you pledge to never raise taxes, ever, then you’re a fool, plain and simple.  You’re locking yourself into a position that might make no sense at some point in the future.  When taking such a pledge, you’re saying, basically, the following: “I don’t think raising taxes is a good idea.  In fact, I feel strongly that it’s a bad idea.  But I am also convinced that I will never change my mind; I will never let new data change my mind; even if the circumstances change, it is logically inconceivable that I will ever change my mind; and even if I want to change my mind I won’t be able to because I am locked into a pledge.”  By taking a pledge, you are thumbing your nose at a future self (and potential wiser self) and forcing them down a path they might not agree with.

[Of course, there’s another reason to take such a pledge: you may not agree with it, but you take the pledge anyway in order to get elected.  Anyone who falls into that category is beneath contempt.]

What if scientists took pledges?  Newtonian physics was on very firm footing in 1904.  What if every physicist signed a pledge saying that Newtonian physics was 100% correct and was never to be doubted ever again?  What, then, would have happened with patent clerk Einstein in 1905?

Suppose everyone in Congress took the Norquist pledge.  And then suppose that aliens visited Earth, and offered to give us an unlimited source of clean energy.  The catch is, we have to raise taxes on upper incomes by, say, 1%, in order to pay for distribution costs.  I guess we’d have to say, “Sorry, we all took a ‘pledge’ so we can’t do it.  Fealty to Grover Norquist and his 18th century ‘pledge’ takes precedence over the country, over science, over common sense, and over anything else you can think of.  Have fun with your infinite energy, rest of the world.”

My point has nothing to do with the merits (or lack thereof) of the pledge.  I have a problem with the idea of such a pledge itself.  A pledge is indicative of an anti-science mentality; a tendency towards dogmatism; a lack of mental flexibility—and those are not traits I want to see in our country’s leaders.  Leaders need to keep everything on the table.  You have to decide based on current data what the best course for the country is.  You cannot let a decision made 20 years ago affect your thinking today.  I’m sure that 2200 years ago I might have been in favor of sanctions against Carthage; I may have even signed a pledge to that effect.  Today, though, that pledge wouldn’t mean very much…

Let’s all agree to never mention Grover Norquist again.  He’s irrelevant.  He’s a lobbyist, and his only purpose is to push his own agenda.  His tax foundation doesn’t do scientific research, doesn’t create jobs, doesn’t build things, doesn’t design things, doesn’t contribute to science, or culture, or human knowledge, or service, or humanity.  Norquist himself is not a super villain.  He’s just a random dude with a loud megaphone.  Luckily, we have the ability to ignore him if we like.  Maybe then he’ll just go away.

Then again, probably not.  After all, he is a lobbyist.

(Photo credit: http://en.wikipedia.org/wiki/File:CarthageElectrumCoin250BCE.jpg)

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Poor Einstein.  Is there anyone else who is misquoted more often?  Is there anyone else to whom more nonsense is attributed?

I have no desire to rehash things that Einstein said about “God”.  Einstein was by all accounts an atheist, an agnostic, or a pantheist—depending upon your definitions—and various religious apologists have been trying to co-opt the man for years by misquoting him.  Others have already discussed this at length.

My goal today is to tackle that old chestnut, “Imagination is more important than knowledge,” as seen on T-shirts, bumper stickers, and even on the packaging of the Albert Einstein action figure.  Did Einstein really say this, and if so, what did he mean?

Here’s the quote in context:

“At times I feel certain I am right while not knowing the reason.  When the [solar] eclipse of 1919 confirmed my intuition, I was not in the least surprised.  In fact I would have been astonished had it turned out otherwise.  Imagination is more important than knowledge.  For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.”  [From A. Einstein, Cosmic Religion: With Other Opinions and Aphorisms, p. 97 (1931).]

So Einstein did say this.  However, I maintain that the full quote in context has a different feel to it than the quote in isolation.

When I see “Imagination is more important than knowledge” on a bumper sticker, I think this: “Flights of fancy and imagination are more important than learning stuff.  So why should I study?  Einstein didn’t study.  He just sat around and daydreamed and came up with the most remarkable breakthroughs about the workings of our universe.  Imagination is more important than learning all the proofs and figures ranged in columns before me.  So I am going to follow good ol’ Einstein and daydream about being Batman.”

The New Age meaning of the quote is this: “I’d rather daydream than study.”  It’s Walt Whitman’s “learn’d astronomer” nonsense all over again.

In context, it’s clear that Einstein was talking about doing science.  Imagination is more important in making scientific breakthroughs than knowledge, but that doesn’t mean that knowledge is not important.  Einstein worked very, very hard to learn an awful lot of physics.  By all accounts, it took him almost 10 years to flesh out general relativity, during which time he had to acquire a lot of mathematical knowledge about Riemannian geometry and tensor analysis.  The “intuition” that Einstein developed during this time frame is what allowed him to be so confident of the results of Eddington’s expedition.  What Einstein calls “intuition” is just knowledge that has become so ingrained that you are no longer cognizant of it.

Einstein may have been more famous than most of his contemporaries, and it was probably due to his superior imagination.  But take Einstein’s imagination today and give it to a twenty-five year old high school dropout, and he’d be lost in obscurity, stocking shelves at Wal-Mart.  Imagination is more important than knowledge.  But only slightly more.

[Note: my book Why Is There Anything? is now available for download on the Kindle!]

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Einstein circa 1905

There are a lot of people who, to this day, deny the truth of Einstein’s special relativity (SR).  I’m not even referring to the OPERA superluminal neutrino debacle—an anomaly that was eventually found to be caused by a misconnected fiber optic cable.  No, I’m referring to laymen who deny SR because it goes against common sense.

A Google search will find such people readily.  Most of the time, their arguments aren’t even worth refutation, since it’s obvious in most cases that they haven’t mastered even the simplest algebra, much less sophomore-level physics.  (I am planning to use this gem in my Modern Physics class in the spring as a homework problem: for 20 pts., find the elementary flaw in this person’s logic.)  However, as a working physicist, I sometimes find myself dismissing such people too readily: it is easy, and self-gratifying, to call such people cranks.

A person who doubts SR is not necessarily a crank.  After all, relativity is very counter-intuitive, and our brains have been exquisitely fine-tuned by natural selection to perceive the world as inherently classical.  In fact I will go so far as to say that if you accept SR whole-cloth, without any mathematical or scientific background, then you’re basically showing a blind faith in science in the same way that Iotians have a blind faith in “The Book”.  I would rather beginning physics students showed some skepticism; it makes their final “conversion” that much more intellectually pleasing.

I think the main problem with perceptions of SR is the way it is normally presented.  My thesis is this: most physicists are teaching it wrong.  And as a consequence, many people who have studied SR come away with a misguided notion of what SR is all about.

The old way to teach SR begins with Einstein’s two postulates.  The first is that the laws of physics should be the same, in any inertial reference frame.  The second is that the speed of light is the same for all inertial observers.  There is then an obligatory picture of a train and lightning bolts, and talk about how simultaneity isn’t preserved in SR.  This leads (usually after a lengthy derivation) to time dilation and length contraction.  And then, out of the blue, there might be talk of the twin paradox and the obligatory pole vaulter in the barn.

Shudder.  Such a pedagogically confusing approach!  No wonder very few first-time SR students “get it” at all.

The original train picture from Einstein’s 1916 book

This approach has a long history.  In Relativity: The Special and General Theory (1916), by Einstein himself (!) the discussion begins with the two postulates, and there is then a diagram of a train and a discussion of simultaneity (see above).  Seriously?  I’m not blaming Einstein, mind you; I’m blaming the textbook authors today who can’t let go of that stupid train.  It’s been almost 96 years.  Get over it.  Hop off that train, please.  There are more intuitive approaches that are easier for the layman to grasp.

Here’s the approach I use in my classes.  This is not the only approach, of course, nor do I claim it is the best approach.  However in my experience (admittedly, just one data point) this approach is a better way to get students to gradually accept SR.  The trick is to present information one plausible chunk at a time, and then only gradually derive all the weird stuff.  Thus, without realizing it, the students have been convinced of the truth of SR despite themselves.  If you start with simultaneity and time dilation and length contraction then half of the students will get turned off immediately (because their common-sense alarms will be blaring full-force).

  1. Talk about classical (Galilean) relativity.  That is, discuss how the laws of physics should be the same in any (inertial) reference frame you choose.
  2. Talk about coordinate transforms: how you can take the spatial coordinates of an object (x,y,z) and find what the coordinates (x’,y’,z’) would be in a different coordinate system.
  3. Talk about how some coordinate transforms are “good” and some are “bad”.  For example, a translation in space such as x’ = x – L preserves distance, but a rescaling transform like x’ = ax does not.
  4. Mention how the good ol’ Pythagorean theorem s2 = x2 + y2 + z2 gives you an invariant quantity (s2) that is preserved under “good” transforms.
  5. Mention that the (experimental) behavior of light throws a monkey wrench into this analysis.  For whatever reason, all observers measure the same speed c for light, and this actually makes things a little harder.  (Don’t do any math at this point!)
  6. Here you should start talking about time as being a 4th dimension.  The earlier you introduce the idea of an event P as a point P=(x,y,z,ct) in space-time, the better.
  7. State Einstein’s postulate about the speed of light.
  8. Show that the light postulate implies that s2 = x2 + y2 + z2 is no longer an invariant quantity, when talking about transforms as applied to space-time.
  9. If s2 = x2 + y2 + z2 is no longer invariant, can we modify the formula in any way so as to make s2 invariant, while still preserving the light postulate?  The answer is yes; and so you should derive the 4D version of the distance formula, s2 = x2 + y2 + z2c2t2.

To me, this is the core idea of SR.  Everything else follows from the invariant interval s2.  One should no longer think of our existence as being 3D; time represents another “direction”.  And it turns out that the time you perceive depends upon your vantage point (time is “relative”), just like position.

For example, suppose you are looking at a row of trees.  From one location, the trees are lined up in front of you (they all share the same x-coordinate).  From another vantage point, they are separated by 1 m each (x=0, 1m, 2m, 3m, etc.)  No one, not even Galileo, would find this controversial.

But now imagine that you think of time as just another “direction”.  Why is it so hard to believe that your time coordinate could have one value in one reference frame, and another value in a different frame?  Why is it so hard to believe that events that are simultaneous in one frame are not simultaneous in another?

Time dilation and length contraction follow from this in a straightforward way.  And they are much easier to visualize if you buy into the paradigm (I’ll say it again) that time is another “direction”, and therefore relative just like position.

[Caveat: I do know that time is special, in the sense that there’s a minus sign in the s2 = x2 + y2 + z2c2t2 formula.  That minus sign is crucial.  But discussing its importance should be deferred to a later (pun-intended) time.]

If you’re interested, here’s the rest of my SR program:

10.  Discuss space-like, time-like, and light-like intervals, and the ideas of proper length and proper time.
11.   Show how the Galilean boost (which is a “good” transform in classical relativity) must be modified into the Lorentz boost in order to preserve s2.
12.   Show how the Lorentz boost implies length contraction and time dilation.
13.   Discuss relative velocity in SR.
14.   Discuss so-called paradoxes like the twin paradox and the pole vaulter paradox.

The discussion can then go into advanced topics: momentum, energy, E=mc2, forces, etc.  However, with the foundation I have described, I believe these topics are much easier to present.

I’m sure there are some great professors out there who have had great success with the “traditional” program of SR instruction.  I’m sure Feynman could teach circles around me, even with the train and lightning bolts.  But I prefer this different approach, as I have presented it, and I hope others will realize that there’s more than one way to explain special relativity.

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