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

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 $D\le 4$, 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

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

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:

$T^{\alpha\beta}_{;\beta}=0$

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

z8_GND_5296. Credit: V. Tilvi (Texas A&M), S. Finkelstein (UT Austin), the CANDELS team, and HST/NASA

Sorry for such a lengthy title, but the subject came up because of a widely circulated announcement of the discovery of “The Most Distant Galaxy Yet Seen,” a title that’s being constantly revived (Matt Francis over at Universe Today has a nice discussion of why this galaxy really is a big deal).

The galaxy, which for now goes by z8_GND_5296 was discovered by the CANDELS collaboration at a distance of about 30 billion light-years.

“But wait!” my friends said on facebook and twitter, “how can we even see a galaxy 30 billion light years away when the universe is only 13.7 billion years old? Isn’t light the ultimate speed limit in the universe?”

Yes. It is. But the universe was smaller in the past.

What does “size of the universe” mean?

To begin with, the CANDELS team didn’t actually find that the galaxy was 30 billion light-years away directly. Rather, they found that it had a “redshift” of about 7.5. Or, to put it another way, the universe was about
$a_{emitted}=\frac{1}{1+z}=\frac{1}{1+7.5}=\frac{1}{8.5}=11.7\%$
the size it is now, and we measure that fact by noting that light that left this distant galaxy has grown by a factor of 8.5 in the time it takes to reach us. That’s what redshift is all about. Long wavelengths of light are “redder” than blue ones.

Now for the misconception. Naively you might suppose that if the universe were 11.7% the size it is now at some point in the past, that must mean that the bounds of the universe were 11.7% smaller than they are now. But the universe has no bounds!

Instead, the standard picture is that the universe is much like a balloon, and as it inflates, the distances between galaxies becomes larger by a fixed rate. “Doubling in size” is shorthand for “galaxies getting twice as far apart from one another,” as well as “gas and dark matter becoming eight times as diffuse.” (8 times, because the universe increases in each of 3 dimensions). Here’s a particularly crude version of the whole shebang that Jeff drew for the User’s Guide:

Light and Space

Imagine that, rather than beaming light to us directly, our pal z8_GND_5296 sent us a signal by means of post-stations or whisper down the lane. It sent a signal to a galaxy a few million light years away from it. Galaxy 1 sent a signal to galaxy 2, and so on, until we got the message. Further, imagine that each of those galaxies are currently 10 million light-years apart from one another. This, by the way, is known as their “comoving distance” if you want the technical term.

But the first signal took much less than 10 million years to transmit. Because the universe was so much smaller then than now, it took about 1.1 million years. The next 10 million light-years of comoving distance might be traversed by a light beam in 1.3 million years, and so on, so that after 13 billion years of whispering down the lane, the total distance now between us and z8_GND_5296 is about 30 billion light years.

I warn you, though, even this 30 billion light year number isn’t terribly useful. After all, if I say that it’s 2 miles to the drug store, the implication is that you’ll have to traverse 2 miles to get there. On the other hand, a galaxy 30 billion light years away will be significantly farther by the time you try to reach it. Indeed, there are galaxies out there that, even traveling at the speed of light, we couldn’t ever reach. That’s just a consequence of living in an expanding universe.

What is the maximum distance?

Though we can see to a distance of more than 13.8 Billion light-years, we can’t see infinitely far. The ultimate limit is what’s known as the particle horizon, which for us is a (comoving) distance of about 45 billion light-years. Anything further than that and we have no hope of seeing it.

On the other hand, the universe also has an “event horizon” (yes, just like a black hole), the maximum distance we ever could reach. Because our universe is accelerating, the limit maxes out at a certain point, and there are regions of space that are forever inaccessible to us. Those are systems more than about 60 billion light years from here.

Beyond that is anyone’s guess.

-Dave

ps If there’s interest, I may do a followup on this as to why it is that cosmologists claim there must be cosmic inflation. It’ll even use Zeno’s paradox to talk about how the smallness of the early universe was trumped by its youngness.

First question: when co

Good news, everyone! I was on the Professor Blastoff Podcast this week talking about “The Universe” (which may win the prize for best episode name) with Tig, David and Kyle. We talk about time travel, symmetry, antimatter (and Uncley matter, a joke that I failed to get for about 10 minutes), and much more.

Check it out, and don’t forget to subscribe via iTunes. Oh, and a warning for the faint of heart. There’s a fair bit of saucy language.

-Dave

Francois Englert and Peter Higgs

In a move that surprised almost no one in the physics community, the Nobel Prize in Physics was awarded this morning to François Englert and Peter W. Higgs “for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS and CMS experiments at CERN’s Large Hadron Collider”

This is a huge deal, and very well deserved. I won’t go on and on about the Higgs right now, in large part because I’ve already written a ton on the subject, including:

Also, of course, there’s a whole chapter about the Higgs and the Standard Model in my new book, The Universe in the Rearview Mirror.

Congratulations again to Drs. Englert and Higgs!

-Dave

Just a quicky. I have a new, somewhat navel-gazing post up at io9 on what you really need to know about the universe, and more generally, why people embrace pop-sci at all. I’m curious to hear what you think, and of course, I’m always looking for more questions for the column.

-Dave

Hi. Would you like to see me give a talk on symmetry? Of course you would. Here’s a professionally produced video of my presentation to the good folks at Google entitled, “Why Symmetry Matters.” Enjoy!

-Dave

Greetings, true believers. I have a piece up at Slate today: Four Reasons You Shouldn’t Exist: Physics says you’re an impurity in an otherwise beautiful universe.. The headline might be a bit over the top, but it’s a fun little piece about the importance of symmetry breaking. Take a look, and then when you’re done, be sure to check out a piece I did a few years ago: A Physicist Looks at the Time-Traveler’s Wife.

-Dave