Scientists at CERN have found apparent evidence that neutrinos can travel faster than light.

Suppose that tomorrow historians at Harvard find apparent evidence that the South won the American Civil War — not in some metaphorical “they accomplished their goals” sense, but in the literal sense that it was actually Grant who handed his sword to Lee at Appomatox and not the other way around.

Question: Of which conclusion would you be more skeptical?

Of course your answer might depend on exactly what this new “apparent evidence” consists of. So let me reword: As of this moment, which do you think is more likely — that neutrinos can travel faster than light, or that the South won the Civil War?

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You win if your answers agree — unless you were both asked the “cats” question, in which case you win if your answers disagree.

A little reflection should convince you that if you are allowed to meet with your partner and plot strategy before the game, then the best you can do is agree to always agree — say by both always answering “yes”. That way, you win 75% of the time, and there’s no way to do better. In particular, there’s nothing to be gained by randomizing your answers.

That, at least, is true, in a world governed by the laws of classical physics and probability theory. But in a world governed by the laws of quantum mechanics — which is to say, in the world we live in — you can in principle do better. Namely: You each carry with you one of a pair of entangled “quantum coins” (actually elementary particles, but I prefer to think of them as coins, since you’re going to use them as randomizing devices).

Because these coins are very small, you need special apparatus to see whether they’re heads-up or tails-up. Before making your measurement, you can rotate your apparatus through either of two angles — call them C and D. The two coins agree 85% of the time — unless both you and your partner’s apparatus have been rotated through angle C, in which case they disagree 85% of the time.

Now you and your partner can each adopt this strategy: If you get the cat question, rotate your apparatus through angle C; if you get the dog question, rotate through angle D. Then examine your coin, and answer “yes” if it’s heads, or “no” if it’s tails. If you both follow this strategy, you’ll win 85% of the time.

Moral of the story: Quantum technology can improve your performance in strategic situations.

(You can read more about this in Chapter 15 of The Big Questions.)

Some time ago, Gordon Dahl and I wrote a paper that explores the implications of quantum mechanics for the cat/dog game and similar (more economically interesting) strategic interactions. Early versions of this paper have floated around for a while, but we’ve just completed a substantial rewrite that I both hope and believe is substantially more readable. We’ll be very glad for feedback from readers who have a taste for this sort of thing. Click here to read it!

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Lugo’s original post is so good it seems almost superflous to paraphrase it, but I can’t resist the temptation.

Drill a tunnel through the earth, from anywhere to anywhere — New York to Maine, or New York to Australia, or wherever else you like. Like so:

Now drop the object of your choice (Lugo suggests a burrito, but you might prefer a gravity-driven train) into the tunnel entrance and wait till it comes out the other side. It’s a standard calculus problem to calculate how long you’ll have to wait: The answer is 42 minutes, regardless of the length of the tunnel. I’m sure I once found it surprising that the tunnel length doesn’t matter, but I’ve known it long enough that I now take it in stride. So that’s not how Lugo surprised me.

The surprise is that if you change the size of the earth (while maintaining its density), the answer is still 42 minutes. Whether the earth is the size of a pea or the size of the solar system, it’s a 42 minute trip from one end of the tunnel to the other. (We’re — quite reasonably — ignoring the effects of relativity here. For an earth that was half the size of the universe, we’d have to make some corrections.)

Why so? You could, of course, discover this through a direct calculation. But Lugo provides a much slicker argument, namely:

Once you know that the travel time is independent of the tunnel length, it’s clear there are only three things it can depend on: The earth’s density, its radius, and the gravitational constant. Neither density nor radius have anything to do with time, so we’re really going to have to use that gravitational constant, which is measured in units of

(The first factor is the 32 feet per second squared that you learned about in elementary school; the second factor depends on distance from the center of the earth and the mass of the earth, neither of which varies in those elementary school problems, so it’s never explicitly mentioned.)

If you multiply that gravitational constant by density (which is pounds per cubic foot), you’ll get a bunch of cancellation, leaving you with

Now you’ve got something that depends only on time! To turn this into an actual number of seconds, you can just invert it and take the square root. At that point, you’re done! (Except maybe for multiplying by some dimensionless constant.) Any attempt to bring the radius of the earth into the formula is going to introduce units of length, which have no business popping up in a formula for time. Therefore the earth’s radius must be irrelevant.

Isn’t that cool? And amazing?

(And yes, there are hidden assumptions, e.g. that the solution must be not just a function of mass, length and the gravitational constant, but a rational function — so this is not quite a proof, but a highly convincing heuristic argument that does give the correct answer. That, I think, in no way detracts from its coolness.)

Lugo’s post is both more concise and more carefully worded, but I thought this slightly longer and less formal version might be helpful to at least a reader or two. Besides, I felt like writing it.

Do you have an equally amazing argument to share?

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(Hopes had been raised a few years back, when a pair of German physicists claimed to have broken the light speed barrier by propagating waves that effectively arrived before they departed. I said at the time that I could do just as well without all the fancy lab equipment — all I need is a yardstick and an axe. If I chop off the last 12 inches, then the center of the yardstick moves from the 12″ mark to the 18″ mark — a six inch advance in exactly zero time. Of course a sane person might argue that this six inch “advance” involves no actual forward motion, but if I correctly understood the German paper, the same sort of objection would apply there as well.)

It’s being reported that the results from Hong Kong doom all hope for time travel. That’s true or false depending on how restrictively you interpret the phrase “time travel”. Back in 1938, Kurt Goedel (yes, the great logician Kurt Godel — though this particular work has nothing to do with his work in mathematical logic) constructed an example of a universe — that is, a structure that obeys all the laws of physics as laid down by Einstein — that is completely filled with closed timelike curves. If you lived in that universe, you’d be able to travel only forward in time, but still eventually come back to your starting point in both time and space — just as a bug can go in just one direction around a circle and still come back to its starting point. Just like Groundhog Day.

(Well, not just like Groundhog Day, because Bill Murray never returns to the exact same place in space and time — he always returns to a slightly altered version of that place, altered, for example, by the memories he brings along.)

To be more precise: If you lived in Godel’s Universe, you’d always have the option of traveling back into your own past (while always traveling forward in time), but you’d never choose to exercise that option (because while these forward-into-the-past paths exist, no matter ever travels along them). Conceivably there is some variant of Godel’s Universe in which that option does get exercised. Maybe some reader knows more about this.

We (almost surely) don’t live in anything like Godel’s Universe. (Thought question: How would we know?) But the existence of that Universe tells us that the laws of physics can’t rule out Godel-style time travel — and more generally that the laws of physics sometimes allow far more than we expect them to. There are a lot of universes out there, and some of them are pretty weird.

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1. A “mathematical object” consists of abstract entities (that is, “things” with no intrinsic properties) together with some relations among them. For example, the euclidean plane that you studied in high school geometry consists of points, together with certain relations among them (such as “points A, B and C are collinear”). Mathematical objects can be very complicated. Mathematical objects can have “substructures”, which is a fancy name for “parts”. A line in the plane, for example, is a substructure of the plane.

2. Every modern theory of physics says that our universe is a mathematical object, and that we are substructures of that object. Theories differ only with regard to which mathematical object we happen to be a part of. Particles, forces and energy are not just described by equations; they are the equations (together with abstract, purely mathematical relations among those equations).

3. If you want to think of the universe as something other than a mathematical object — say, something that is controlled by mathematics, or described by mathematics, as opposed to made of mathematics — then you’re up against the fact that nobody has the slightest idea how to construct a useful physical theory along those lines. It’s not just that science rejects all the alternatives; it’s that no scientist (as far as I know) has even been able to imagine a useful alternative. (Perhaps you can find solace in religion.)

4. So — at least if you accept modern physics — at least one mathematical object (namely the one we live in, and are part of) exists physically. This naturally raises the question of which other mathematical objects exist physically.

5. Before you can answer that, you’ve got to ask what it could even mean for one mathematical object to exist physically while another doesn’t. Our universe is a certain mathematical object. Surely there are other mathematical objects that differ from it only in detail. They contain substructures as complicated as we are, in more or less the same ways, and therefore as well equipped as we are to perceive their surroundings as physically real. If you want to claim that our universe is in fact “real” and theirs is not, then you’ve got to explain what that reality consists of.

6. Whatever that reality consists of, it must be some purely mathematical property, because our universe is a purely mathematical object, so that all its properties are purely mathematical.

7. I want to stress that: If, as physics tells us, the universe is “made of math”, then its physical reality is part of that math. We should therefore expect similar mathematical structures to share the property of physical reality.

8. Which structures count as “sufficiently similar” to ours that we should expect them to be physically real? That’s up to us, really, since we’re in the process of defining the term “physical reality”. My own instinct is to call a mathematical object “physically real” if it contains what Max Tegmark calls “self-aware substructures”, i.e. structures that are complicated in a way that makes them aware of their own existence. Tegmark’s own preference is to say that there’s simply no point in discriminating among mathematical objects, and we should simply call all of them physically real.

9. But why bother arguing? Why struggle to come up with a definition for some vaguely imagined notion of “physical existence”? Ockham’s Razor cautions us not to burden ourselves with unnecessary metaphysical baggage. There are mathematical objects. Some have self-aware substructures. Our universe is one of those. The business of science is to figure out which one. What more needs to be said?

10. Finally: I never cease to be amazed by people who uncritically accept the reality of rocks, geese and butterflies but want to deny the reality of mathematical objects. Science tells us that rocks, geese and butterflies are mathematical objects. What else could they be?

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To understand the universe at the deepest level, we need to know not only how the universe behaves, but why.

• Why is there something rather than nothing?
• Why do we exist?
• Why this particular set of laws and not some other?

So say Stephen Hawking and Leonard Mlodinow in their book The Grand Design, and so say I.

The Big Big Question is the first one: Why is there something rather than nothing? Hawking’s answer: The laws of physics — and especially the form of the law of gravity — allow for the spontaneous creation of universes out of nothing at all. We live in one of those spontaneously created universes. But this, of course, only serves to raise a new Big Big Question, namely: Why are the laws of physics as they are? Hawking’s answer: The laws of physics must be consistent and must predict finite results for the quantities we can measure. It turns out that those criteria pretty much dictate the form of the laws of physics.

So unless I’ve misunderstood him, here is Hawking’s position: In order for us to be able to measure the things that we measure, the laws of physics must have a certain form, and in order for them to have that form, universes must be able to arise from nothing. Therefore our universe was able to arise from nothing. But this does not seem to answer the question of why things couldn’t have been very different. Why couldn’t there have been no us, no measurements, no laws of physics and no anything?

I know of only one satsfying (to me) answer to this question, and Hawking comes tantalizingly close to it without ever quite going there. He spends a lot of pages reviewing current physical theories but never mentions the one glaring feature they all share: Every modern physical theory, taken literally, predicts that our universe is a mathematical object. For example, the simplest version of special relativity posits that we live in a four-dimensional geometric object called “spacetime”. More sophisticated theories posit that spacetime is part of some larger geometric object whose properties we perceive as “forces” or “particles”. According to modern physics, everything is made of math.

Now you might say that physical theories aren’t meant to be taken that literally; that instead they describe mathematical objects with properties that are analogous to the properties of the physical universe. But it seems to me that if, like Hawking, you trust in theories to explain the mystery of creation itself, then you ought, at least provisionally, to take those theories literally. Otherwise, what you’ve got is not a theory. It’s a theory plus a bunch of ad hoc and arbitrary choices about which parts of that theory you choose to believe.

Once you believe the universe is a mathematical object, its existence ceases to be a mystery—at least if you believe, along with most mathematicians, that mathematical objects can’t help but exist. Hawking embraces M-theory, which tells us that the universe is a particular 11-dimensional object (with a whole bunch of additional geometric curlicues that appear to our senses as everything from stars to bacteria. M-theory also says there are a whole bunch of other 11-dimensional universes, all of which were spontaneously created, and we just happen to live in this one.

What I’m suggesting is that the universes of M-theory are only a tiny fraction of the universes out there, because anything that exists mathematically is a universe, though most of them (like most of the universes of M-theory) are far too simple to contain anything like sentience. This is essentially the view of cosmologists like Max Tegmark of MIT.

Hawking is 90% of the way there. The many universes of M-theory are mathematical objects, and all are pieces of a bigger mathematical object called the multiverse. “Spontaneous creation” means that the multiverse is structured in such a way that it must contain these universes. But why is there a multiverse and why is it structured in that way? That’s the part Hawking seems not to address. Proposed answer: The multiverse itself is only one of many multiverses. They all exist for the same reason the natural numbers exist: The laws of mathematics require it. And unlike the laws of physics, which differ from multiverse to multiverse, the laws of mathematics, which live outside any universe, could not have been otherwise.

(For more on this subject, read Chapter 1 of The Big Questions !)

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]]> http://www.thebigquestions.com/2010/09/27/the-grand-design/feed/ 41 http://www.thebigquestions.com/2010/02/08/the-secret-of-attraction/ http://www.thebigquestions.com/2010/02/08/the-secret-of-attraction/#comments Mon, 08 Feb 2010 07:01:06 +0000 Steve Landsburg http://www.thebigquestions.com/?p=2195 This will be old news to the physics geeks, but I still remember what a revelation it was, back in grad school, when the physicist Gary Horowitz told me why an electric current exerts a magnetic force on a moving charged particle. (This is the source of all magnetism; those magnets on your refrigerator have little electric currents flowing through them all the time.)

So imagine a wire, made of protons that stay still and electrons that drift rightward; that drift is what we call a current. And imagine a nearby charged particle—call it Fred—also traveling rightward.

Now relativity tells us that Fred is allowed to think of himself as stationary, and the protons (along with you and me) as drifting off to the left. Relativity also tells us that if passengers on a moving train say the cars are 100 feet apart, then an observer at the station will say they’re closer than that. In this case (according to Fred) you and I are the passengers moving with the train of protons, and if we say they’re an angstrom apart, then Fred says they’re closer. That means Fred sees more positive charge per inch of wire than we do. If Fred himself happens to be negatively charged, he’ll be drawn toward the wire.

As far as Fred is concerned, that’s a purely electrical force, but it’s a force that you and I can’t account for on electrical grounds. So you and I call it magnetism.

At the same time, Fred sees the electrons in the wire as slower-moving, and therefore farther apart, than you and I do, so he sees less negative charge per inch of wire than you and I do. According to Fred, then, the gap between positive and negative charge in the wire is even greater, which means he’s pulled in even harder, which you and I call even more magnetism.

If you’re geeky enough to care, it’s a nice exercise in relativity theory to show that the magnetic force is proportional both to the current (that is, the number of electrons per inch, times the speed of the electrons, as measured by you and me) and to Fred’s velocity. It just now took me three tries to get this right, but it’s very nice when it finally works.

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]]> http://www.thebigquestions.com/2010/02/08/the-secret-of-attraction/feed/ 28 http://www.thebigquestions.com/2010/01/25/the-big-answers-2/ http://www.thebigquestions.com/2010/01/25/the-big-answers-2/#comments Mon, 25 Jan 2010 07:01:20 +0000 Steve Landsburg http://www.thebigquestions.com/?p=1993 Last week, I posed some brain teasers and a riddle about special relativity.

The brain teasers were all solved by multiple commenters; I’ll summarize their answers at the end of this post. The special relativity problem proved trickier; here it is again:

A circular train (front of the locomotive attached to the rear of the caboose) sits on a circular track. At some point, the train accelerates and starts traveling around the track. Because the train is moving, I (an observer stationary relative to the track) should see it shrink. But the track doesn’t shrink. So the train can’t stay on the track, and gets pulled inward, ending up inside the track. On the other hand, the passengers say the track has shrunk, so they should expect to get pushed outside the track. How can everyone be right?

Now to the answer.

First, “At some point, the train accelerates” is ambiguous. Presumably it means that each part of the train accelerates at the same time, but of course “at the same time” means something different to a train passenger than it does to you and me (the observers stationary relative to the track).

But let’s resolve this ambiguity in the natural way by assuming that the entire train starts moving at the same time as measured by you and me. In that case, we do not see the train shrink. How could we? The front and back ends of each car have, at every moment (as measured by our watches) been moving forward at identical speeds. Given that, the distance between those front and back ends (a measured by our meter sticks) cannot change. Ditto for any couplings between the cars.

What just became of relativity? If the cars are in motion shouldn’t they appear smaller to us than to our friend Jeeter, who’s riding on the train? Sure. But that doesn’t mean we have to see Jeeter’s train car get smaller. In this case, it means that Jeeter sees his car get bigger—because by his watch, the front of his car started moving before the back did, so his train car got stretched out.

But that doesn’t mean Jeeter sees the entire train get bigger. Yes, his car got stretched when the front started moving before the back. But the car opposite him (180 degrees around the track) got shrunk when its back end started moving (according to Jeeter’s watch) before its front.

So nobody has to see the train change size and nobody has to believe the train leaves the track—which is good, because the train doesn’t leave the track.

I have ignored the fact that Jeeter is not in an inertial frame, which complicates the calculation of exactly what he experiences, but I’m nearly sure that the above captures everything important. If you want, replace the circular track with a nearly square track (with slightly rounded corners if you like) so that most of the train passengers are in inertial frames at any given moment.

*****************************

Now to the brain teasers: For #1, choose a number randomly c from (say) a normal distribution on the real numbers and compare it to the number x that I’ve just revealed to you. If x is greater than c, guess that x is the larger of my numbers; if x is less than c, guess that x is the smaller of my numbers. Your chance of winning is 1 if c is between my two numbers and 1/2 otherwise; this makes your overall chance of winning greater than 1/2.

For #2, Jon Shea’s answer is perfect.

For #3, New Mexico/Colorado is one of many good answers.

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]]> http://www.thebigquestions.com/2010/01/25/the-big-answers-2/feed/ 93 http://www.thebigquestions.com/2010/01/22/geek-or-dork/ http://www.thebigquestions.com/2010/01/22/geek-or-dork/#comments Fri, 22 Jan 2010 07:01:22 +0000 Steve Landsburg http://www.thebigquestions.com/?p=295 There are a bazillion alleged “paradoxes” in special relativity, all based on exactly the same fallacy, but I might have just invented a brand-new one—-where “invented” is shorthand for “confused the hell out of myself for a while”. When I finally got up and drew a picture (as opposed to lying in bed with my eyes closed doing something that felt like thinking), it became clear that, sure enough, it was the same old fallacy again (how could it not have been?), but in a new enough guise that someone reading this might find it amusing.

So: A circular train (front of the locomotive attached to the rear of the caboose) sits on a circular track. At some point, the train accelerates and starts traveling around the track. Because the train is moving, I (an observer stationary relative to the track) should see it shrink. But the track doesn’t shrink. So the train can’t stay on the track, and gets pulled inward, ending up inside the track. On the other hand, the passengers say the track has shrunk, so they should expect to get pushed outside the track. How can everyone be right?

Yes, the train is moving in a circle and is therefore not in an inertial frame. No, that has nothing to do with resolving this. I won’t spoil the fun by posting the answer just yet. I had a very “d’oh!” moment when I got it.

I’m not sure whether I’m a geek for figuring this out or a dork for not seeing it instantly.

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]]> http://www.thebigquestions.com/2010/01/22/geek-or-dork/feed/ 17 http://www.thebigquestions.com/2009/12/02/life-the-universes-and-everything/ http://www.thebigquestions.com/2009/12/02/life-the-universes-and-everything/#comments Wed, 02 Dec 2009 07:01:03 +0000 Steve Landsburg http://www.thebigquestions.com/?p=1186 As I mentioned the other day, I’ve recently (at the direction of my old friend Deirdre McCloskey) been reading some of the work of John Polkinghorne, the physicist-turned-theologian who seems to write about a book a week attempting to reconcile his twin faiths in orthodox science and orthodox Christianity.

Although Belief in God in an Age of Science is a very short book, it is too long to review in a single blog post. Fortunately, though, much of the non-lunatic content is concentrated in roughly the first ten pages, so I’ll comment here only on those.

Polkinghorne begins in awe. He is awestruck by the extent to which our Universe seems to have been fine-tuned to support life; this is the subject matter of the much-discussed anthropic cosmological principle. To take just one example (which Polkinghorne does not mention): The very existence of elements other than hydrogen and helium depends on the fact that it’s possible, in the interior of a star, to smoosh three helum atoms together and make a carbon atom; everything else is built from there. But it’s not enough to make that carbon atom; you’ve also got to make it stick together long enough for a series of other complicated reactions to occur. Ordinarily, that doesn’t happen, but now and then it does. And the reason it happens even occasionally is that the carbon atom happens to have an energy level of exactly 7.82 million electron volts. In fact, this energy level was predicted (by Fred Hoyle and Edwin Salpeter) before it was observed, precisely on the basis that without this energy level, there could be no stable carbon, no higher elements, and no you or me.

That energy level is only one of many (apparent) cosmic coincidences that make us possible; change any of the fundamental physical constants (like, say, the strength of gravity) by a little bit in either direction, and the Universe would, as far as we can tell, become completely inhospitable to life. So one does tend to feel that there’s something here that needs explaining.

Some have attempted to dismiss the issue by turning the direction of causality on its head: Here we are, so of course the laws of physics must allow for our existence. Case closed. Douglas Adams, for example, offers this brief and brilliant parable:

Imagine a puddle waking up one morning and thinking, ‘This is an interesting world I find myself in, an interesting hole I find myself in, fits me rather neatly, doesn’t it? In fact it fits me staggeringly well, must have been made to have me in it!’

But I have some sympathy for Professor Polkinghorne’s refusal to accept this dismissal. Instead, he takes his stand with the philosopher John Leslie:

The fine tuning is evidence, genuine evidence, of the following fact: that God is real, and/or there are many and varied universes.

I agree with that (with the proviso that evidence is not proof). I agree with it to exactly the same extent that I agree with this:

The fine tuning is evidence, genuine evidence of the following fact: Either invisible pink bunny rabbits, created at the time of the Big Bang, fine tuned the physical constants in order to make the Universe hospitable to lettuce, and/or there are many and varied universes.

Or, more succinctly:

The fine tuning is evidence, genuine evidence of the following fact: There are many and varied universes.

Polkinghorne wants to reject this second horn of Leslie’s dilemma, but he manages to do so, I think, only by taking too crabbed a view of what those many and varied Universes might be. First, we have the parallel worlds promised to us by the many-worlds interpretation of quantum theory; Polkinghorne is absolutely right to say these can’t be the worlds we’re looking for, because they all obey the same basic laws of nature. Higher on what Polkinghorne calls the “scale of bold speculation” we have suggestions from quantum cosmology that Universes are bubbling up all the time as quantum fluctuations in some universal substrate. But again, Polkinghorne is right to say that this only pushes the mystery back a bit—why do those fluctuations obey laws that have even a chance of producing a habitable Universe? Where do the laws come from?

This is the point where Polkinghorne gives up and falls back on God. But it seems to me that he has given up just one level of abstraction too soon. A Universe is fundamentally a mathematical object—it’s an abstract pattern that might or might not contain subpatterns that might or might not be sufficiently complex in just the right away to achieve an awareness of their surroundings, and might or might perceive those surroundings as physical objects. And of course there are many Universes, because there are many mathematical patterns, including, as just one of a dazzling infinity of examples, the Universe in which we live.

That, in any event, is the best explanation I can come up with, and it’s an explanation that feels completely right to me (which admittedly proves nothing). In The Big Questions, I’ve elaborated on what I mean by all this, how it can be true, and why it is entirely consistent with mainstream physics and the stated views of many mainstream physicists.

Now, Professor Polkinghorne might or might not buy this vision, but my point is that he never even contemplates it. He makes the leap to theism by considering and rejecting all of the weakest alternatives, but ignoring the only one that makes sense. This oversight is all the more remarkable because Polkinghorne devotes his closing pages to a rousing defense of the independent reality of mathematical objects, in clear and convincing language that had me wishing I’d written these pages myself.

The rest of the book is far worse. I might come back to that in a later post.

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