Posts Tagged ‘Einstein’

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:


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:


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:


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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Somewhere in the wilderness of Toelek is a store.  Inside the store are clerks.  The clerks sell bags of Wavy Lay’s potato chips:

Wavy Lays Original

Now, the thing is, if you give money to clerks to buy some chips, the clerks never give any money back: they’re greedy.  What’s more, they leave the store immediately with whatever money they have left over from the transaction.  However, the clerks don’t always get very far, because out behind the store is a bridge that the clerks have to cross, and the bridge is guarded by a troll named Voltar.  The troll is greedy, too.  He demands a toll, and if a clerk can’t pay up then he can’t cross the bridge.

Toelek is a weird place.  Its citizens have a very rigid society, and people are required to wear differently colored clothing depending upon how much money they have in their pocket.  The money is always in multiple of 50 cents.  For example, if you carry $0.50 then you must wear red; if you carry $1.00 you must wear orange; if you carry $1.50 you must wear yellow.

Look, over there—I see a line of people approaching the store.  They’re all wearing red.  They enter the store…but I don’t see any clerks exiting out the back, and consequently no clerks cross the bridge.  More and more reds go into the store, and a faster and faster rate, but it doesn’t matter.  There are no clerks coming out.  The troll doesn’t get any business.

I conclude that a bag of chips costs more than $0.50.

Later, I see a line of oranges go into the store.  I now observe some clerks coming out, but none of these can cross the bridge.  I conclude that chips cost $1.00, but since the clerks have no money left over, they can’t pay the troll’s toll.

Even later, I see a line of yellows go into the store.  Clerks are coming out, and these can cross the bridge.  The troll must be demanding a toll of $0.50 or less.

There are many quantities which are important in analyzing this situation: the amount of money a person has before entering the store (use E for entering), the price of a bag of Wavy Lay’s potato chips (let’s call this price W), the toll that Voltar the troll demands (let’s call this V), and the amount of money a clerk has, K (they speak Dutch in Toelek, so the clerks are called klerks) upon exiting the store.  It should be obvious that

K = E – W,

since the amount of money a clerk has upon leaving the store is just the amount of money a person has upon entering the store minus the cost of some chips.  Additionally, for a clerk to cross the bridge, it must be true that

E – W ≥ V

so that the clerk has enough to pay the toll.  If the clerk barely makes it, this inequality is an equality and

E – W = V.

The V at which this happens (for a given E and W) is called the cutoff toll Vo;  if Voltar were to increase the toll by any amount at all, the clerks wouldn’t get to cross the bridge.

It’s interesting to graph the cutoff toll Vo vs. money that customers have upon entering the store E.  You get something like this:


Notice that the cutoff value of E is $1.00, which is the price of a bag of chips.  At or below this value the troll need not charge any toll at all, since no clerk will have any money to pay him.  That is, when V = Vo = 0,  then E = W.


The physicists reading this blog have already guessed the game I’m playing: I have presented an analogy for Einstein’s explanation of the photoelectric effect (hence Toelek, from fotoelektrisch).  Make the following transformations:

Customers = photons;

Color of customer = frequency of photon;

Money = energy;

Store = photoelectric material;

Price of chips = work function;

Clerks = electrons;

Clerk’s money = kinetic energy;

Bridge = potential difference;

Voltar’s toll = kinetic energy required to jump the gap.

With these transformations, you can re-write the story as follows:

There is a photoelectric material, a metal such as platinum.  Inside the metal are electrons.  The electrons can be liberated if enough energy is added.

Now, the thing is, if you give energy to electrons to liberate them, the electrons don’t give any energy back: they’re greedy.  What’s more, they leave the metal immediately with whatever energy they have left over from the transaction.  However, the electrons don’t always get very far, because out behind the metal is a potential difference that the electrons have to cross in order for current to be observed.  Jumping this gap requires a certain amount of kinetic energy, without which the electrons don’t produce current.

How do we add energy to the metal?  Well, by shining light on it.  Light energy is quantized, in chunks called photons.  The energy of a single photon is proportional to its frequency, by Einstein’s formula E = hf.

Look, over there—I see some red light (E = 1.9 eV) approaching the metal.  Unfortunately, no electrons exit out the back, and consequently there is no current.  More and more red photons hit the metal, and a faster and faster rate, but it doesn’t matter.  There are no electrons coming out.

I conclude that the amount of energy need to liberate an electron (called the work function) is greater than 1.9 eV.

Later, I see orange light (E = 2.1 eV) go into the metal.  I now observe some electrons coming out, but none of these produce current.  I conclude that the work function is W = 2.1 eV.

Even later, I see yellow light (E = 2.18 eV) go in.  Electrons are coming out, and these can cross the potential difference.  The potential difference must be 0.08 volts or less.

There are many quantities which are important in analyzing this situation: the amount of energy a photon has before entering the metal (E), the work function (W), the voltage that electrons have to jump (V), and the amount of energy an electron has upon exiting the metal, K.  It should be obvious that

K = hf – W,

since the amount of energy an electron has upon exiting the metal is just the amount of energy a photon has upon entering the metal minus the cost of liberating an electron.  Additionally, for an electron (with charge e) to jump the gap, it must be true that

hf – W ≥ eV

so that an electron has enough kinetic energy to overcome the potential difference.  If the electron barely makes it, this inequality is an equality and

hf – W = eV.

The V at which this happens (for a given f and W) is called the stopping potential Vo.

It’s interesting to graph stopping potential Vo vs. energy of incoming photons E.  You get something like this:


Notice that the cutoff value of E is 2.1 eV, which is W.  At or below this value there need not be any potential difference at all, since no electron will be liberated.  That is, when V = Vo = 0,  then E = hf= W.

I hope you find this analogy useful.  As for me, I need to go to the store: all this talk of potato chips has made me hungry.

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“I am your probability density”

In an earlier post I discussed my philosophy of teaching special relativity.  My main idea was that physics professors should keep the “weird stuff” at bay, and start with non-controversial statements; once students are on board, you can push under the grass and show them the seething Lynchian bugs beneath.

Well, what about quantum mechanics?  Does the same philosophy apply?

My answer is yes, of course.  Don’t start with Schrödinger’s cat.  Don’t mention the Heisenberg uncertainty principle, or wave collapse, or the EPR experiment, or Bell’s theorem, or the double slit experiment, or quantum teleportation, or many worlds, or Einstein’s dice.  Start with the problems of physics, circa 1900, and how those problems were gradually solved.  In working out how physicists were gradually led to quantum mechanics, students will build up the same mental framework for understanding quantum mechanics.  At least, that’s how it works in theory.

Now, my perspective is from the point of view of a professor who teaches only undergraduates.  I only get to teach quantum mechanics once a year: in a course called Modern Physics, which is sort of a survey course of 20th century physics.  (If I were to teach quantum mechanics to graduate students, my approach would be different; I’d probably start with linear algebra and the eigenvalue problem, but that’s a post for another day.)  As it is, my approach is historical, and it seems to work just fine.  I talk about the evidence for quantized matter (i.e. atoms), such as Dalton’s law of multiple proportions, Faraday’s demonstration in 1833 that charge is quantized, Thomson’s experiment, Millikan’s experiment, and so on.  Then I explain the ultraviolet catastrophe, show how Planck was able to “fix” the problem by quantizing energy, and how Einstein “solved” the problematic photoelectric effect with a Planckian argument.  Next is the Compton effect, then the Bohr model and an explanation of the Balmer rule for hydrogen spectra…

We’re not doing quantum mechanics yet.  We’re just setting the stage; teaching the student all the physics that a physicist would know up until, say, 1925.  The big breakthrough from about 1825-1925 is that things are quantized.  Things come in lumps.  Matter is quantized.  Energy is quantized.

The big breakthrough of 1925-1935 is, strangely, the opposite: things are waves.  Matter is waves.  Energy is waves.  Everything is a wave.

So then, quantum mechanics.  You should explain what a wave is (something that is periodic in both space and time, simultaneously).  Here, you will need to teach a little math: partial derivatives, dispersion relations, etc.  And then comes the most important step of all: you will show what happens when two (classical!) wave functions are “averaged”:

ψ1 = cos(k1x – ω1t)

ψ2 = cos(k2x – ω2t)

Ψ(x,t) = (1/2) cos(k1x – ω1t)  + (1/2) cos(k2x – ω2t)

Ψ(x,t) = cos(Δk·x – Δω·t) · cos(k·x – ω·t)

where Δk ≡ (k1 – k2)/2, k ≡ (k1 + k2)/2, etc.

[Here I have skipped some simple algebra.]

This entirely classical result is crucial to understanding quantum mechanics. In words, I would say this: “Real-life waves are usually combinations of waves of different frequencies or wavelengths.  But such ‘combination waves’ can be written simply as the product of two wave functions: one which represents ‘large-scale’ or global oscillations (i.e. cos(Δk·x – Δω·t)) and one which represents ‘small-scale’ or local oscillations (i.e. cos(k·x – ω·t)).

This way of looking at wave functions (remember, we haven’t introduced Schrödinger’s equation yet, nor should we!) makes it much easier to introduce the concept of group velocity vs. phase velocity: group velocity is just the speed of the large-scale wave groups, whereas phase velocity is the speed of an individual wave peak.  They are not necessarily the same.

It is also easy at this point to show that if you combine more and more wave functions, you get something that looks more and more like a wave “packet”.  In the limit as the number of wave functions goes to infinity, the packet becomes localized in space.  And then it’s simple to introduce the classical uncertainty principle: Δk·Δx > ½.  It’s not simple to prove, but it’s simple to make plausible.  And that’s all we want at this point.

We’re still not doing quantum mechanics, but we’re almost there.  Instead, we’ve shown how waves behave, and how uncertainty is inherent in anything with a wave-like nature.  Of course now is the time to strike, while the iron is hot.

What if matter is really made from waves?  What would be the consequences of that?  [Enter de Broglie, stage right]  One immediately gets the Heisenberg relations (really, this is like one line of algebra at the most, starting from the de Broglie relations) and suddenly you’re doing quantum mechanics!  The advantage of this approach is that “uncertainty” seems completely natural, just a consequence of being wave-like.

And whence Schrödinger’s equation?  I make no attempt to “prove” it in any rigorous way in an undergraduate course.  Instead, I just make it imminently plausible, by performing the following trick.  First, introduce complex variables, and how to write wave functions in terms of them.  Next, make it clear that a partial derivative with respect to x or t can be “re-written” in terms of multiplication:

d ψ /dx  →  ik ψ

d ψ /dt  →  –iω ψ

Then “proving” Schrödinger’s equation in a non-rigorous way takes 4 lines of simple algebra:

E = p2/2m

E ψ = (p2/2m)ψ

Now use the de Broglie relations E = ħω and p = ħk…

ħw ψ = (ħ2k 2/2m) ψ

iħ(∂ψ/∂t) = (–ħ2/2m) ∂2ψ/∂x2

There’s time enough for weirdness later.  Right now, armed with the Schrödinger equation, the student will have their hands full doing infinite well problems, learning about superposition, arguing about probability densities.  As George McFly said, “I am your density.”  And as Schrodinger said, probably apocryphally, “Don’t mention my cat till you see the whites of their eyes.”

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As a physics professor, I have certain pet peeves.  For example, I cringe when someone says that “gravity” is 9.8 m/s2 when they mean the acceleration due to gravity.  I’m annoyed if someone says that an object “weighs” 7 kg.  And I stifle a laugh if someone says that a roller coaster is exciting because it goes so “fast”—humans can only detect acceleration, not speed, which is why we don’t notice that we’re traveling something like 67,000 mph right now in our orbit around the sun.


“I feel the need for acceleration!”

But my biggest pet peeve may be students doing algebra with numbers.

Fellow physics professors will know exactly what I’m talking about, but for the uninitiated, here’s an example:

If you drop an object from a height of 20 m, how long will it take to hit the ground?

A student knows that a kinematics equation is needed, hits upon the correct one, Δyvi Δt + (1/2) a Δt2, and then correctly identifies Δy = –20 m, a = –9.8 m/s2, and vi = 0.  So far, so good.  They’ve studied their physics, right?  What happens next is sheer madness:



Over and over again I tell students, “don’t plug numbers in until the end.”  But students love plugging in numbers.  They feel they’re actually getting closer to the answer if they’re manipulating numbers.  On some level, they still feel uncomfortable with letters—as if manipulating letters isn’t really “math”.

How does this problem look in my answer key?  Like this:

algebra 2

You can now plug in values if you like…and get Δt = √[2(-20)/-9.8] = 2.02 s.

Which of these approaches is more beautiful, more powerful?  The approach you pick indicates whether you “get” algebra or not.  If you do algebra with numbers, the answer you get is very narrow and very specific, even if you do it correctly.  That hypothetical student could have gotten 2 seconds as an answer, and I would have given them full credit.  But their answer would have been ugly.

The second approach is beautiful, because it is completely general and applicable to multiple situations.  I try to tell students “Look!  You found the time to fall a certain distance.  You now know the answer no matter what the height is, and even no matter what planet you’re on, since g doesn’t have to be 9.8 m/s2.”  This is usually followed by a blank open-mouthed stare, much like Kristen Stewart in a Twilight movie.

There is a more practical reason to avoid doing algebra with numbers.  It’s simply that when you do algebra with numbers, other people cannot follow your work as easily.  And then, if you make a mistake, it’s harder for someone else to spot.  Quick: what algebra error did the student make above?  It takes a while to find the mistake.

My ultimate point is that students need experience seeing the power of algebra.  It’s all well and good that algebra classes stress real-world applications—else, why teach algebra in the first place?  But real-world doesn’t only mean with numbersE=mc2 is certainly a real-world application of algebra, and it’s a lot more elegant than saying that 378,000,000,000,000 Joules is released when a teaspoon of sugar with mass 4.2 grams  is converted to pure energy, given that the speed of light is 300,000,000 m/s.  The hard part, for us physics professors, is to help this spoonful of algebra go down.

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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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