xTras for xAct

I'm a big fan of xAct, a tensor algebra package for Mathematica. While there are other tensor packages for Mathematica on the market, xAct is by far the best. It canonicalizes tensorial expressions blazingly fast, and its perturbation capabilities are state-of-the-art. If you've always wanted to do second-order perturbations of four-derivative curvature tensors but were afraid the actual calculation might take you some weeks, xAct is your man. It does it in a few seconds, and more importantly, doesn't mess up minus signs.

In fact, I like xAct so much, I wrote an additional package for. It's aptly called xTras, and its available over on www.xact.es/xtras. It brings some functionality that I found missing in xAct, like computing equations of motion for the metric, perturbations around AdS spaces, and Young projectors (yes, this also includes Bianchi identities).

I'll demonstrate some of the new functionality by computing the linearization of the Einstein tensor around AdS spaces. Here goes!
First, open up Mathematica, and enter the following line:

   In:    <<xAct`xTras`

This loads the xTras package (assuming you're managed to download and install it). Next we'll define a manifold and a metric:

   In:    DefConstantSymbol[dimension,PrintAs->"D"]
   In:    DefManifold[M,dimension,IndexRange[a,f]]
   In:    DefMetric[-1,g[-a,-b],CD]

We'll be doing stuff on AdS spaces, which has a constant curvature. Hence we need to define a constant symbol to indicate that curvature:

   In:    DefConstantSymbol[L]

Now we're ready to define the standard Lagrangian for gravity with a cosmological constant:

   In:    lagrangian = RicciScalarCD[] -(dimension-2)(dimension-1)L

We'd like to compute the equations of motion that follow from this Lagrangian. To do so, we first write the command

   In:    DefMetricVariation[g,h,eps]

This command makes it possible to do covariant metric variations. It also registers the command VarL, which varies Lagrangians:

   In:    eom = VarL[g[a,b]][lagrangian] //TensorCollect
Out:    \( \frac{1}{2} (2 - 3 D + D^2) L g_{ab} + R_{ab} - \frac{1}{2} g_{ab} R\)

The equations of motion should allow for AdS spaces. To check this, we first generate a list of replacement rules for curvature tensors of the covariant derivative CD on symmetric spaces:

   In:    AdSrules = SymmetricSpaceRules[CD,L]

 And indeed, the equations of motion are zero for this background:

   In:    eom /. AdSrules // ToCanonical
Out:    \( 0 \)

This means we can perturb around this solution. So without any further ado, here's the linear perturbation of the Einstein tensor:

   In:    ExpandBackground[eom, BackgroundSolution -> AdSrules] // TensorCollect
Out:    \( (l -  D l) h^{1}{}_{ab} + \frac{1}{2} (-1 + D) l g_{ab} h^{1c}{}_{c} - \frac{1}{2} \triangledown_{a}\triangledown_{b}h^{1c}{}_{c} + \frac{1}{2} \triangledown_{c}\triangledown_{a}h^{1}{}_{b}{}^{c} + \frac{1}{2} \triangledown_{c}\triangledown_{b}h^{1}{}_{a}{}^{c} \)
                                      \(- \frac{1}{2} \triangledown_{c}\triangledown^{c}h^{1}{}_{ab} - \frac{1}{2} g_{ab} \triangledown_{d}\triangledown_{c}h^{1cd} + \frac{1}{2} g_{ab} \triangledown_{d}\triangledown^{d}h^{1c}{}_{c} \)

 And that's it! Granted, we could also have done this by hand. But the power of xAct is that it can do much more complicated calculations without breaking a sweat. If we would like to know the second order pertubation of the Einstein tensor, we can simply replace the above input by ExpandBackground[eom,2,BackgroundSolution->AdSrules]. Pretty cool, right?

Symmetry in and of nature

In part three of "What on earth has Teake been doing for the last four years?" I'll talk a little on how symmetry appears in nature. When considering this subject, it seems natural to think of symmetric things that appear in nature. Snowflakes, like the one above, are good examples, but also flowers, the wings of butterflies, and sea stars (to name a few) all show some form of symmetry. They fall into the category "things in nature that look kind of symmetric", but in fact have little or nothing to do with the physicist's concept of the symmetry of nature.

One definition of symmetry in physics is as follows: the symmetries of nature are those transformations that do not change the laws of physics. To illustrate this concept, have a look at the following four clocks:

The first clock obviously satisfies the laws of physics. We now apply two different transformations, namely a mirror operation and a time reversal operation. Both give a clock that is distinct from the original one -- you can see the dial moving in the opposite direction. So the clock is not invariant under these transformations, and thus they are not symmetries of the clock. However, there's nothing physically wrong with a clock running in the opposite direction, apart from the fact that it's broken. Therefore parity (a fancy word for a mirror operation) and time reversal are symmetries of the laws of physics of this particular system. In general parity and time reversal are broken, but that's another story.

As a side note, you can see that the combined actions of parity (C) and time reversal (T) do leave the clock invariant, so CT is a symmetry of the clock.

This concept of symmetry is quite powerful. For instance, Special Relativity can be derived by demanding that the laws of physics are invariant under translations, rotations, and boosts (which together make the Poincaré group). Modern physics is to a large extent build on similar considerations of symmetry. So the study of group theory is not just a nice mathematical exercise; it actually has some useful applications throughout the field of physics. In the upcoming blogposts on "What on earth has Teake been doing for the last four years?" I'll try to explain how I applied some fancy group theory to even fancier things like supergravity. Stay tuned!

Symmetry

This is the first post of what eventually should become the "What on earth has Teake been doing for the last four years?"-series. Brace yourself: it's about maths and physics. Run while you still can!

I'll try to keep things simple by starting of with a concept that is as mundane as it is fascinating: symmetry. Not only makes it our world round, but it’s also what makes it go round. From the perfect circular wheels on our bikes and cars that deliver an enjoyable ride, to the error-correction protocols that keep e-mails from turning into junk; it’s literally all around us.

So what is symmetry exactly? A symmetry is an action on an object that, once you’re done performing it, does not change that object. It's a somewhat abstract definition, but take for example the triangle, which has 6 symmetries.  There are two different rotations (over 120˚ and 240˚), three reflections, and finally the action of doing nothing at all (yes, that's also a symmetry). You can try them out in the following applet. Clicking on the arrows causes the triangle to rotate and reflect.

This is all pretty straightforward, right? But things start to get interesting we you keep track of the effect of the different rotations and reflections. Let's paint the corners so we can see where they end up:

One thing you'll notice is that doing twice a clockwise rotation is equal to doing one counter-clockwise rotation. The same is true for any other combination of actions -- it will always yield the net effect of one single reflection or rotation. It might also happen that the triangle ends up in the original configuration, but remember that doing nothing is also a symmetry.

The combined actions describe what mathematicians call a group. A group is a set of elements plus a rule of multiplying those elements. Let's call the set G and the multiplication rule "•". Then the precise definition of a group (in which a, b are elements of the set G) is the validity of the following four statements:

• Closure.
  The result of the operation a • b is also in G.
• Identity element.
  There exists an element 1 in G, such that for all elements a in G, the equation 1 • a = a • 1 = a holds.
• Inverse element.
  There exists an element a^-1 in G such that a • a^-1 = a^-1 • a = 1.
• Associativity.
  The equation (a • b) • c = a • (b • c) holds.
  

These four set of rules are called the group axioms. They might sound a bit abstract, but in fact, they're not. Let's have a look at our triangle again.

The symmetry actions are labeled as follows:

• 1: identity element ("doing nothing at all").
• y: counter-clockwise rotation by 120˚.
• p: clockwise rotation by 120˚.
• R: reflection in the top vertex.
• G: reflection in the lower-right vertex.
• B: reflection in the lower-left vertex.

We already noticed that the closure axiom holds. The existence of the identity element is also pretty obvious. What about the identity axiom? This indeed also holds: every action has an inverse. The reflections are their own inverse, whereas the rotations are each other's. The last thing to check is associativity. It's a bit harder to verify, but believe me, it holds too.

Just for completeness sake, here's one last version of the triangle applet. This one includes the group multiplication table, which keeps track of what happens when you first do the action in the first row, followed by the action in the first column. (If you didn't believe me on the validity of associativity you can use this table to check it.)

What I've shown you so far is that the symmetries of the triangle can be described in terms of the mathematical concept of a group. The importance of group theory lies in the fact that any symmetry you can think of can be described as a group, and that conversely all groups describe a symmetry.

By now the answer to the question "What on earth has Teake been doing for the last four years?" will hardly come as a surprise: it's group theory. More on that in part two of this series!

Poster on my research

Don't worry, there's nothing wrong with your eyes. The text on the poster above is indeed too small to read. The poster is supposed to be A0-sized (which is a whopping 0.84 by 1.18 meter), but on screen it's a tad smaller. There's a bigger version with readable text on my Picasa album -- just click on the image to get there.

I made this poster back in April this year for a conference, but then I tore it to shreds while trying to remove it from a wall. Double-sided tape can be a bitch sometimes. To avoid further incidents with tape I decided to order a new version on foam. After some trouble getting it to our institute (you can't fold foam, and A0 catches a lot of wind when you're on a bike) it now decorates a previous empty spot on the wall.


So, what's this poster all about, you might ask. Short answer: my research. Long answer: unfortunately the long answer is so long I'll have to spend another post or two on it. So stay tuned ...

PhD-day

As a theoretical physicist working in the Netherlands, I'm a member of the DRSTP (the Dutch Research School for Theoretical Physics). I'm also a member of its students council, and as such I'm involved in the organization of the second DRSTP PhD-day.
The first one was back in April 2008 and was a lot of fun -- I'm pretty confident this year's edition will be the same. We're sticking to the same format: 6 speakers, 5 of which PhD students from the different Dutch universities and 1 former PhD student. Topics range from the gauge / gravity duality to petrophysics, and should be quite interesting (admittedly not for the non-physicist perhaps).

And can you spot the differences in the posters? I made both, and can't really decide which one I like best. It's funny how small changes can change the look quite dramatically. But perhaps I should focus less on graphical design and more on writing my PhD thesis. The bugger is due at the end of the year. Better start cracking.

East coast vs. West coast in particle physics

Last year saw the jaw-dropping Large Hadron Rap, explaining the LHC at CERN as best as possible to the layman in five minutes. For those of you forgot, here's the video:

This year Fermilab strikes back! It looks like science rapper funky49 made a proper gangsta-rap about the Tevatron. There's no video yet, but he's posted the lyrics on his website:

"(...) Tevatron, OG atom smasher
say hello to CERN’s party crasher, the
new “Lord of the Rings” LHC, hear me, this
be competitive collaboration baby (...)"

There has been some rivaly between CERN and Fermilab on who discovers the Higgs first. It looks like the rivalry now has entered a whole new domain ... will we see the likes of the East Coast vs. West Coast feud for particle physics?