## Best. Class. Ever.

Credit: me. A simple 1-d simulation of a complex scalar field in an electric field. You’ll find this more instructive after you’ve read the lecture notes. Click on the link if it doesn’t auto play in your browser.

This term, I’m teaching a new class on The Standard Model. It is, bar none, the most fun I’ve had with a course in just about forever. What’s especially novel about it is that we’re tackling just about everything classically, with only occasional recourse to quantum fields.

We’re doing everything from elementary group theory to GUTs, from the Higgs mechanism to neutrino oscillation, with just about everything in between.

Why am I telling you this (besides the fact that I can’t contain my excitement)? Because in designing this class, I couldn’t find exactly the right text, so I’m thinking of transforming my lecture notes into a book. As a first stab, here are:
The First 4 weeks of lectures!
Please excuse any typos, and the verbiage wasn’t originally meant for human consumption. This is in the same spirit as my previous lecture notes on probability. Like my previous notes, these are extremely technical, and the course itself is designed for an advanced undergraduate/grad audience.

The notes themselves are a work in progress, but if you think any big topics are missing, please let me know.

BTW, part 4 of the notes is on scalar fields and inflation, and I am especially indebted to this excellent post by John Preskill for providing such a nice demonstration of dimensional analysis.

-Dave

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## Cosmos

By now, you’ve either seen the first episode of the new Cosmos or you’re not going to. The nerdverse has been breathless with anticipation for months now, and now that it’s here, the reviews are… pretty good. I don’t make it a point to be a culture critic, but as the Drexel University Relations department asked for my take on the new show, I thought I may as well give a reasonably thoughtful one. Below are my somewhat expanded comments.

There’s a lot to admire about the Cosmos reboot. Neil Tyson, for one, is one of the best evangelists for science we have right now. He’s incredibly knowledgeable, personable, passionate, and a dedicated public educator. That said, while the science in the show was generally very accurate, I do have a fear that there is an element of preaching to the choir. Much is made in the introduction about the rigors of science, and yet there’s no discussion of how the science is done. Dedicated fans of science will already know much of what is presented, and those who doubt the scientific method, or evolution, or the Big Bang model of cosmology, are unlikely to be persuaded by Tyson’s assurances.

Instead, there’s a clear effort to overwhelm the audience with special effects. To a certain sort of fan, Tyson’s authoritative intonation of the scale of the cosmos (the focus of the first episode) will genuinely evoke excitement. It is also potentially a great entryway for kids who are just learning about science for the first time. Sagan’s original packed a wallop in large part because people like me were kids at the time. It evoked curiosity and an excitement about science. There is the very real hope that Tyson’s version will do the same.

As of the first episode, most of the updates (apart from the host) are cosmetic. Certainly, we know the timescale of the universe a bit better than we did in 1980, but the biggest changes center around using computer graphics to project the earth into the future, or to draw Tyson’s spaceship. Some of these effects (and the rather gratuitous use of lens flares) are a little cheesy. But there is also some excellent footage from modern spacecraft that is integrated seamlessly into the narrative, especially of Mars.

The most significantly new discussion centers around Giordano Bruno, a Franciscan Friar who proposed the possibility of many worlds (and thus, many aliens, each with their own savior), who was ultimately killed as a heretic. Bruno is a somewhat more complicated figure than Cosmos makes out, but it was refreshing to see a usually overlooked thinker included in the discussion.

There were a few instances in which the visualization obscures scientific accuracy. For instance, as Tyson flies through the asteroid belt in his “Ship of the Imagination” (a ray-traced, Prius-esque reimagining from the original series), the animators make the typical science fiction mistake of putting the field so crowded that Tyson is barely able to squeeze through. In reality, the typical distance between asteroids is about a million miles, roughly 4 times the distance to the moon. While a realistic asteroid field would look dull, I feel as though the show missed a number of opportunities to realistically portray the true scope and emptiness of space.

There were several other occasions where “what if?” scenarios were played interchangeably with “This is what happened.” The most glaring of these occurred during a sequence in which we’re shown a particular asteroid nudged gravitationally in its orbit and then later that same asteroid is seen to commit mass dinocide. This is meant to be a hypothetical, but to the uninitiated (including, apparently, the folks on the Culture Gabfest at Slate), it isn’t clear that is not, in fact, a detailed model rather than a very vague guess.

Those concerns aside, we would have a far more science literate audience if the public at large internalized the content of the show. In many ways, the reboot is long overdue. There’s a great deal of mistrust and misunderstanding of science. One need only look at the public views of evolution, global warming, and the anti-vax community to be aware of the backlash against science — real science, as opposed to “geek culture,” like in The Big Bang Theory and elsewhere. This is why it is so important that Cosmos not only get the next generation excited, but also to explain not only what we know, but how we know it.

I want to close with an observation about the final scene — one which most reviewers found quite touching. Tyson shows us Carl Sagan’s appointment book from when he (Tyson) was a boy, and that Sagan made an entire day for him, signed a copy of his book, and generally took Tyson under his wing and mentored him. It took me a little while to articulate what bothered me about this scene, but ultimately I feel as though it smacks of predestination. It reminds me of that famous photograph in which a fresh-faced Bill Clinton met JFK as a boy. The implication, in both cases, is that there was a symbolic passing of the torch. The problem, of course, is that most kids aren’t going to have such an experience. If a kid isn’t marked by some great scientist or mentor very early on, do they simply lack the spark?

I realize that this isn’t what Cosmos was trying to say in the sequence. Tyson and the show were simply trying to convey the generosity of Sagan’s spirit, and I appreciate that. But given the cult of personality that has grown up around Tyson in the last decade, it’s very easy to see it in another light.

-Dave

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## Does it make sense?

My dad always told me, after finishing a math problem for homework, ‘look at the answer and ask your self ‘does this answer make sense'” So, my question is: do the answers we get from the Wave Function make sense? And, if they don’t, shouldn’t folks like you consider finding a function that renders sensible answers?

My response:

I take it from the question that you feel that it doesn’t. The universe, unfortunately, doesn’t always conform to our common sense. We evolved in a world where knowledge on macroscopic (non-quantum) scales, non-relativistic speeds, and weak gravity were the norm, so time dilation or wave-particle duality are not part of the intuition wired into our brain. One of the amazing things about scientific discovery is that we’re able to overcome our natural intuition. The wave-function makes perfect sense in that it predicts (with astonishing accuracy) the behavior of all elements on the periodic table, astrophysical objects like white dwarves and neutron stars, the double slit experiment, radioactive decay and on and on. “Sensible,” in the case, should mean “correct,” and by that standard quantum mechanics (wave functions and all) is the single most successful theory in the history of science.

-Dave

## Probability for Students

I’ve had a lot of students over the years, and I’ve realized that even those who’ve taken a probability and statistics class often lack a gut feeling intuition about how to deal with data. I’ve written a (first draft, admittedly) version of a “Probability for Science Students” (pdf).

Comments are welcome (especially for straight-up errors, although comments from fellow physicists who might suggest additional topics are welcome, too). A bit of warning, it is extremely technical, and is designed for undergraduates or grad students in physics or closely related disciplines. It is not even for mathematicians who might find my lack of rigor disturbing.

-Dave

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## Goings on

Greetings, internet. I’ve got a few posts in the queue on the potential dangers of colliding with a star in your warp drive and such like, but I wanted to give update on a couple of really exciting interviews this coming week.

Last week was pretty exciting, too. On Monday, I did a “Science on Tap” talk at National Mechanics. The turnout was nuts:

Next month promises to be even better. On February 10th, my pal Ken Lacovara will be talking about “Next Generation Paleontology.”

On Tuesday, I gave a really fun double-header talk with Max Tegmark at the Free Library. If you couldn’t come out in the rain, don’t fret. The good folks at the library have made the talks (or at least the audio content) available online.

That’s all for now, but I’ve been getting a lot of fun quickie questions lately that will likely turn into blog posts. As always, feel free to send me more.

-Dave

## Talks around town

I hope everyone has enjoyed the holidays. I’ve been hibernating, but I’m back now, and I’ve got a bunch of talks and events over the next few weeks, in addition to a new term.

In other news, the paperback edition of the “User’s Guide to the Universe is finally out and available. I just got my copy today.

There are lots of other surprises in the new year. I’m not going to ruin them now, but I think you’ll enjoy them. Hope everyone is keeping nice and warm.

-Dave

## Christmas Time is Here!

And what better time to pick up one of New Scientist’s Top 10 (12?) Science Books of the Year for yourself or for someone you love. I’ve prefer you choose mine, but they’re all good.

Symmetry Magazine also has a list of top books to read in the post-Higgs era, and both of mine made the cut, with a short review for each. I particularly enjoyed this gem: “Goldberg is a Gift.” Words to live by. Hint, hint.

-Dave

## A Technical Post – On more than 3 dimensions

I’ve made the occasional passing comment about how fortunate we are to live in a 3-dimensional universe. In an old ““Ask a Physicist” column (in a precursor to a fairly important discussion in my new book) I dropped a little anthropic truth on the world:

You may remember, dimly, something about gravity being an inverse square law. The idea is that if you double the distance between two objects, their force of gravity drops by a factor of four. The same rule holds for electromagnetism.

The inverse square law isn’t an accident. It turns out that it’s entirely a function of the fact that we live in a three-dimensional universe. If we lived a four-dimensional one then we’d have an inverse cube law.

I then concluded with:

It turns out, though, that an inverse square law is very special. Higher dimensional universes (with their inverse cube or inverse-fourth gravity laws or whatever) don’t have any stable orbits. In other words, in a 4-dimensional universe, the earth would either spiral in toward the sun or fly away. We wouldn’t get to enjoy the five billion or so years of nearly constant sunlight that we do in our universe.

That statement of fact wasn’t enough for one of our Drexel physics grad students who asked why, exactly, $D > 3$ matters so much. To answer that, I’m going to give a fairly technical answer (though there are some nice figures to help illustrate the point). If you’re scared off by equations, and haven’t seen any intermediate-level undergraduate mechanics, this may not be the post for you.

Have the faint of heart left? Good. Let’s get started.

Consider a particle of mass $m$ in an equatorial orbit at radius, $r$, in a central potential $U(r)$. The choice of an equatorial orbit is arbitrary, since in a central potential, we can always rotate the coordinates until the particle is moving around in a fixed plane. For an attractive force, there’s always a perfectly balanced circular orbit:

(though we’ll derive the properties of that orbit in a moment).

The Lagrangian

Generalizing the orbit requires a little bit of work, and introduction of a few physical principles that you may or may not remember. Unfortunately, this isn’t the time for a full course in variational mechanics, but a good book like Marion and Thornton or even the wikipedia page on Lagrangian Mechanics is a good place to start.

But here’s the 10 second version.

In ordinary particle mechanics, the trajectory of a particle can be computed based on the kinetic and potential energies via a quantity called the Lagrangian:

$L(q_i,\dot{q}_i,t)=K-U$

That’s the difference (rather than the sum) of the kinetic and potential energy. According to Hamilton’s Principle the Lagrangian is important because particles will travel along paths that minimize a quantity known as the “Action” (which is traditionally given by an S):

$S=\int_{t_1}^{t_2} L dt$

How do we figure out what path works? We use the Euler-Lagrange Equations

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right)=\frac{\partial L}{\partial q_i}$

where $q_i$ is some coordinate of the system (in our case, $r$ or $\phi$) and $\dot{q}_i$ is the time derivative of that coordinate.

For a particle in orbital motion around a central potential, the Lagrangian can be written as:

$L=\frac{1}{2}m\dot{r}^2+\frac{1}{2}mr^2\dot{\phi}^2-U(r)$

The Euler-Lagrange equation for $\phi$ produces:

$\frac{d}{dt}\left( mr^2\dot{\phi}\right)=0$

which is just the conservation of angular momentum. Thus, we could define:

$l\equiv mr^2\dot{\phi}$

as a constant of motion.

Our second E-L equation yields:

$m\ddot{r}=mr\dot{\phi}^2-\frac{\partial U}{\partial r}$

You may recognize that last term on the right as the radial component of the gravitational force, which would (in a 3-d universe) be:

$F_r=-\frac{\partial U}{\partial r}=-\frac{GMm}{r^2}$

but in a D-dimensional universe, a D-Sphere has a surface area proportional to $r^{D-1}$ so we’ll get a result like:

$F_r=-\frac{\partial U}{\partial r}=-\frac{C}{r^{D-1}}$

where $C$ contains the equivalent of the gravitational constant and the mass of the central body.

We’ll define an “effective” radial force by combining both terms on the RHS of the numbered equation above:

$F_{eff}\equiv mr\dot{\phi}^2-\frac{\partial U}{\partial r}$

or, writing it in terms of the fixed angular momentum:

$F_{eff}\equiv \frac{l^2}{mr^3}-\frac{\partial U}{\partial r}$

The circular and nearly circular orbit

At any given radius, there is clearly a solution that yields a circular orbit, supposing that $U(r)$ is a monotonically increasing function. That is, the circular orbit solution is given by:

$l_{\circ}=\sqrt{mr^3 \frac{\partial U}{\partial r}}$

and thus the radial force is identically zero.

But is that orbit stable?

Consider what happens if the particle gets a small, positive radial kick. In a 3D universe, this is not a problem. As $r$ increases, the restoring force is negative. If $r$ decreases, the restoring force is positive.

This is a force corresponding to a Simple Harmonic Oscillator. (You would also find, if you choose to plug in the numbers that the frequency of radial oscillation is the same as the frequency of the orbit. Thus, the orbit is closed and produces an ellipse, a result first observed by Kepler, and shown mathematically by Newton.)

In general, the effective force is not quite so simple:

$F_{eff}\equiv \frac{l^2}{mr^3}-\frac{1}{C r^{D-1}}$

or, pulling out factors of $r$:

$F_{eff}=\frac{1}{r^3}\left( \frac{l^2}{m}-\frac{1}{C} r^{4-D}\right)$

It’s the last term that you should pay attention to. If $Dle 4$, then the “restoring force” gets larger for perturbations away from the circular orbit. In other words, for a 3.1 dimensional universe (which is essentially the sort of thing you get near strong gravitational fields) you might get:

High Dimensions

For $D ge 4$, the problem is even worse. By inspection, the effective restoring force gets weaker and weaker for larger $r$. As a result, an angular momentum even a small amount greater than $l_{circ}$ causes a particle to fly off to infinity:

Because it matters, here I’m using $l=1.01 l_{circ}$ in a 4-D universe. The green dot indicates the starting point of the orbit. As you can see, the planet quickly spirals outward.

For a planet orbiting ever so slightly less than the circular velocity, the fate is even worse:

Although a sun can’t even exist in a universe with no stable orbits.

If you think this is just about macroscopic scales, you’re wrong. As I noted in the original article:

Because electromagnetism also obeys an inverse square law, it turns out that atoms wouldn’t be stable. They’d all spontaneously collapse. It’s really hard to imagine complex life without atoms, and even tougher to imagine having this conversation without the existence of life.

A note to the experts. Somebody is likely to point out in the comments section that electrons don’t “orbit” atoms in the same way that planets do the sun. True enough, but if you grind through the equations in quantum mechanics and do the problem correctly, you hit the same problem. No stable atoms. Sorry.

Finally, if you’d like to play around with non inverse-square laws on your own (and you have the vpython library installed on your machine), feel free to download my source code.

Thanks for indulging a bit of mathematical excess.

-Dave

## Ask a Physicist: When will the Universe End?

Just in time for Halloween, I bring you my eschatology special: a detailed discussion of the future timeline of the universe, and how neither we nor our robot or photonic progeny are going to get to enjoy it indefinitely. It’s a real pick-me-up. Enjoy!

-Dave

## I get mail: Dark Energy, The Expanding Universe, and Noether’s Theorem

You know I love getting mail. I like getting questions, brain-teasers, compliments, or in rare cases, all three. Not every question is right for my “Ask a Physicist” column over at io9, either because it’s a bit too specific or a bit too technical, but even so, sometimes the answer is so much fun that I have to share it with you.

I got this email from a satisfied customer named Stephen:

This question comes from reading your excellent book “The Universe in the Rearview Mirror.”

If Dark Energy is in fact “energy”, and it is somehow vacuum energy, then since the universe is expanding, the amount of vacuum is increasing over time and thus the amount of dark energy is increasing over time. Doesn’t this violate the conservation of energy and thus the time translation symmetry?

To Stephen: Thanks so much for the kind words. I’m glad you’re enjoying the book, and it seems as though have stumbled upon exactly the right recipe for getting a prompt response: flattery.

You’ve also stumbled on an extremely good question, and one that makes the idea of a conservation law a little tricky to describe in an expanding universe.

Noether’s Theorem

To the rest of you: For those of you not already in the know, Stephen is referring to a consequence of Noether’s Theorem which says, in short, that every continuous symmetry in the laws of physics give rise to a conservation law. The laws are unchanged at all places in the universe, and therefore, according to Noether’s Theorem, we get conservation of momentum. The laws are unchanged over time, and therefore, we get conservation of energy. There are others, but you get the gist. Noether is the patron saint of symmetry, and the hero of my book.

Conservation laws are the bread and butter of physics. Conservation of momentum yields the wobbles of stars from the orbits of the hidden planets around them, as well as the principles behind rocket science. Conservation of energy gives us the principles behind gravity and the atomic bomb. Conservation of angular momentum ultimately explains, among much else, why our solar system is spinning in a nice, flat plane.

What Conservation Is

Consider what a conservation law means under normal (Euclidean) circumstances, in which we imagine the universe as a large (fixed) room. I can divide that room into many smaller boxes. In that case, there are two ways of thinking about a conserved quantity, but they amount to the same thing:

1. The local definition. Suppose we have a conserved quantity (electric charge, perhaps). The rule is that individually, the only way for the charge within any given box to change is for charges to pass through the boundaries between an adjacent box, either in or out. This is like money. The change balance in your account is directly related to the money put in (by deposits and interest) less the amount you spend (and fees).
2. The global definition. Add up the amount of charge in ALL of the boxes now and at some point in the future. They should be unchanged from the first count to the last.

Common sense dictates that the two of these definitions should be the same, but in an expanding universe, they aren’t – or at least the definition is a little more complicated. Indeed, in Einstein’s universe, time gets wonky in lots of ways, which means that it’s not even obvious how to count up all of the stuff in the universe at one particular time. In other words, it’s only the local definition that matters.

Imagine we construct a large (many light years on a side) impermeable box in space. Imagine now, that if the universe doubles in size, then the dimensions of the box increase by the same ratio. For ordinary matter, this isn’t a problem. Double the box, the volume increases by 8, thus the density goes down by a factor of 8, exactly as it does in the real universe.

Credit: The good folks at wikipedia.

But what about photons? Photons carry energy, and the universe is filled with them. Early on (in the first 70,000 years or so), they dominated the energy of the universe.

But now think about them in our box. The universe doubles in scale, the density of photons goes down by a factor of 8 (there are a fixed number in total, after all), but also, the energy of each individual photon drops off by a factor of 2 (that’s what’s illustrated up at the top). The energy density drops by 16!, and thus the total energy density in photons has gone down. What happened?

Energy and Pressure

The secret is that photons have pressure, and in a relativistic universe, we don’t simply consider energy or momentum separately. Rather, we consider the entirety of the “stress-energy,” which includes pressure and momentum and energy and turbulent flows. This is really the conserved quantity. After all, since space and time are coupled, you didn’t really think that momentum and energy weren’t also coupled, did you?

Instead, think of what happens to the box as it expands. The photons and the box act like a piston. They apply pressure on the box as it expands, and thus, they do work (Pressure*change in volume is work). If you do work, you lose that amount of your energy. And numerically, it works out perfectly.

Credit: splung.com

In general relativity, we say that “It’s the covariant derivative of the stress-energy tensor that’s conserved.” If you’d like to see the equation, it looks like this:

$T^{alphabeta}_{;beta}=0$

which is just a fancy way of saying that you have to include all of the contributions of energy, pressure, etc., and calculate the changes over time, space, and the curvature of the universe itself and that is equal to zero.

If that’s too much of a mouthful, you can just as easily say that the photons did work in the expansion and transferred some of their energy into the universe itself – really the gravitational field. It’s very much like energy being used to overcome the gravitational attraction of the earth. Yes, it really is rocket science.

Dark Energy and the Expanding Universe

Now how about Dark Energy? Dark energy is weird because it has negative pressure, what we’d call tension. If positive pressure (like with photons) does positive work, then negative pressure does negative work. Dark energy PULLS energy out of the universe as it expands. What makes a cosmological constant special is that it’s exactly the right pressure to keep the density the same no matter how much you expand it.

The magic number — the ratio of pressure to energy density — is known as “w,” and to be a true cosmological constant, $w=-1$. Most cosmologists, myself included, pretty much assume that w is identically -1, but we still need to measure it to be sure. A perfect cosmological constant has a lot of interesting properties. For instance, unlike a normal gas, where you can feel that you’re moving relative to it by air resistance, a cosmological constant appears exactly the same no matter what your state of motion. Moreover, it behaves exactly the same way that the so-called “vacuum energy” of the universe is expected to behave, except for one not so small difference; it’s a factor of $10^{120}$ times lower density.

BUT… we still need to measure it to be sure. And interestingly, recent results (ones with fairly large systematic errorbars, to be sure), actually suggest that the best fit value is $w=-1.186$. As I said, between the systematic and random errors, it’s entirely possible that this is just a random fluctuation, but the universe would be more curious still if our dark energy weren’t a cosmological constant.

But that’s a story for another day.

-Dave