Monday, May 18, 2015

Welcome Back/Uncrumpling the Blog

Let me explain...

Like many other undergraduates, the authors of this blog suffer from both inexperience and intense workloads. This was never more true than in the past two years, when we both created and proceeded to actively neglect this blog. Having experienced the phenomenon of failure induced by inactivity, or as they say in baseball, "a strikeout looking," I can say that I do not like the taste it leaves in the mouth. 

Now fortunately, many of us have recently become graduate students. Consider us upgraded, more seasoned, but still active and enthusiastic. We now have among us students of astrophysics, QFT, condensed matter, nonlinear dynamics, biophysics-- to be sure, we have a diverse and capable team. I, at least, will begin writing on this blog, for the exercise if nothing else.
For future reference, my current work will be related to mesoscopic systems, including granular media, elastic and elasto-plastic sheets, yarns, and just about anything in between that provides interesting physical problems for us to think about.

Now, for some content...

Today I want to briefly mention a topic (motivated from a review article, just email me if you want to know more) that is so obvious, so menial, that I am relatively confident that you won't expect it.

How does paper crumple?

How indeed, you say, it seems trivial. But when you consider the idea of a 2-D system, exposed to a uniform force, it becomes a curiously rich playground: why do ridges and vertices appear? Why does nature seem to want to concentrate all of that uniform stress into lower-dimensional structures, when it usually tries to distribute energy randomly?

You've only just begun to scratch the surface. These stress focusing phenomena are present in a wide variety of curious systems, from turbulent flows, to galactic accretion, to dielectric breakdown. When and why does nature decide to put all of its eggs in one basket? From the crumpling perspective, it's easy to tell.

Paper is very close to what we call an isometric sheet. This means that paper is much, much easier to bend than it is to stretch. As a consequence, paper tries to eliminate stretching, which requires that it only bend in one direction. Look at a piece of crumpled paper, and you'll realize that this is true. For the most part, shapes on the paper are only curved in one direction, and flat in the other. This is a very hard constraint to satisfy, and what you end up with when you try to confine a large piece of paper in a small area is the familiar network of ridges and vertices that you know as crumpled paper.

This post was a bit of a warm-up, and in the future I'll be bringing in more commentary-based posts on articles from various journals. I am breaking the arxiv trend, because I don't prefer to live by rules.


Wednesday, July 30, 2014

[hep-th] Unification of Inflation and Dark Energy from Spontaneous Breaking of Scale Invariance

Unification of Inflation and Dark Energy

This paper proposes a method using two independent non-Riemannian volume forms (integrand of a volume integral) to derive an effective potential for the scalar matter field that is capable of describing both the early universe expansion and dark energy in our universe today. In other words, they are able to derive an effective potential that produces accurate energy densities during the inflationary phase of the universe and our current universe containing dark energy.

This paper isn't very difficult, but does require some prerequisites to understand. You should be somewhat familiar with tensor calculus. If you understand what Lorena has been doing in her Numerical Relativity posts then you should be fine with the level of tensors here. Some familiarity with scale transformations would be nice so that you can understand their procedure. An understanding of manifolds would be great, but is not necessary for understanding the primary point of this paper.

Repeating the authors' calculations here would be redundant (especially due to the small size of the paper), so I will just mention specific things of interest and some information useful when reading the paper. However, I note that it is impossible for me to teach you all the material required to fully understand this paper. This would require a course in General Relativity.

The approach the authors use is to variate an action integral which is pretty standard. The peculiar thing here is that they use two independent non-Riemannian forms instead of one. Following standard procedure for the Lagrangians and measure densities, they obtain the following action 
$$ S = \int{d^{4}x \Phi_{1}(A)[R + L^{(1)}]} + \int{d^{4}x \Phi_{2}(B)[L^{(2)} + \frac{\Phi(H)}{\sqrt{-g}}]}  $$

Let's analyze this equation and gain some understanding of what it means. Typically in Classical Mechanics, one is dealing with an action of the form $ S = \int{dt L(\Phi,\partial_{\mu}\Phi)} $, where $L$ is the well known Lagrangian. In Special Relativity the Lagrangian may be rewritten as $ L = \int{d^{3}x \mathcal{L}(\Phi,\partial_{\mu}\Phi)} $, where $\mathcal{L}$ is the Lagrangian density. This allows us to generalize the action to $ S = \int{d^{4}x \mathcal{L}(\Phi,\partial_{\mu}\Phi)} $ where we are now integrating over space-time.

Now we can make some comparisons between our standard special relativistic action integral and the one given in the paper/above. The terms $R$ and $g$ come from General Relativity. $R$ is the Ricci scalar which gives a measure of the curvature of the geometry in consideration. As a clarification to those who haven't encountered relativity or differential geometry of any sort; $g$ is not Earth's acceleration of gravity, rather it is $\|g_{\mu \nu}\|$ where $g_{\mu \nu}$ is the metric tensor for whatever geometry you would like to use. The measures $\Phi$ are the field strengths of typical gauge fields. This will tell you which field is the primary contributor for specific phenomena.

The most difficult part of this paper is where they pull out this gauge transformation that leaves the action invariant and therefore informs us of a symmetry. Gauge transformations can be a pain and aren't very intuitive. Moving along, they variate the action integral and discover two terms that are not invariant under the scale transformation and lead to spontaneous symmetry breaking.

In the end, what is obtained is an interesting effective Lagrangian and scalar field potential (see paper). The potential smoothly moves from one flat region at large negative values of the parameter to another flat region at large positive values, as can easily be seen in the figure at the end of the paper.

The key to the success of this calculation was the use of two volume forms instead of the typical one used by most physicists. Similar results can be obtained with one volume form, but this leads to nonlinear terms which are obviously ugly. It suffices to say that using two volume forms gets rid of this issue and presents a nice way of calculating the vacuum energy density for both the early universe inflationary phase and our present day dark energy phase.

That's about all I have to say about this paper. It's a great read that performs a quick calculation of some pretty cool stuff. I wouldn't get bogged down in the details of the paper if you aren't strong in General Relativity. Rather you should notice how awesome this result is! Hope you enjoyed the post as much as I did.

Monday, July 28, 2014

Numerical Relativity: Black Holes and Gravity - Part 3


The definition of spacetime comes from special relativity. It tells us that different inertial frames have different notions of time. Let's take three Cartesian coordinates (x, y, z). By moving these axes parallel to themselves from some origin, we create new values of x, y, and z. In addition, by moving these coordinates we are creating a notion of time, which is different in each of these inertial frames. This means that inertial frames are spanned by four coordinates (t, x, y, z).

Moving reference frames. Graphic from

Flat Spacetime and the Metric

Just like in geometry where we use the Pythagorean theorem, $  a^{2} + b^{2} =  c^{2} $ relativity uses something somewhat similar called a line element of flat spacetime $$  (\Delta s)^{2} = -(c\Delta t)^{2} + (\Delta x)^{2} + (\Delta y)^{2} + (\Delta z)^{2} $$ A line element, such as the one above, tells us the change in position vectors as time also changes. Combining these changes gives us the distance between points in spacetime - denoted by $ \Delta s $ . In fact, this is the one used in special relativity because it gives us flat spacetime (which is not true in general relativity). Since we have four coordinates that give us a quantity and a direction, they can be written as vectors and the  vectors can be written using index notation as shown below $$ \textbf{a} \cdotp \textbf{b} = (a^{\alpha} \textbf{e}_{\alpha}) \cdotp (b^{\beta} \textbf{e}_{\beta}) = (\textbf{e}_{\alpha} \cdotp \textbf{e}_{\beta}) a^{\alpha} b^{\beta} = \eta_{\alpha \beta} a^{\alpha} b^{\beta} $$ where $ \eta_{\alpha \beta} $ is the metric of flat space time and can be shown as a diagonal matrix

\(\eta_{\alpha \beta} =\begin{pmatrix}-c^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix} =\begin{pmatrix}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix} = diag(-1,1,1,1)\)

Here the speed of light is made equal to one in order to make further calculations easier. This metric, called a Minkowsky metric, gives us information of the geometry and structure of spacetime. Although the matrix notation looks messy, it can be very useful for understanding calculations. Instead of writing $  (\Delta s)^{2} = \cdots $ we can just say $$  ds^{2} = \eta_{\alpha\beta} dx^{\alpha} dx^{\beta} $$
The reason why we changed $ \Delta $ into $ d $ is to show that the values are now changing by infinitesimal amounts. Our $ \alpha 's $ and $ \beta 's $ sum over our four coordinates. I recommend learning about Einstein notation so that the formula above becomes clearer. This notation will prove crucial in the posts of gr-qc publications.

Curved Spacetime and the Coordinate Problem

**NOTE: The gravitational constant, G, and the speed of light, c, will be equal to one from now on.** 

Inertial frames are useful because of the special symmetries of flat spacetime. However, in general relativity spacetime is curved and, generally, not symmetric. This means that we cannot take one coordinate system (and I am not talking only about Cartesian or spherical) to simplify the laws; instead, we have to chose a specific system for a specific problem. For example, if we use Schwarzschild coordinates (below)
$$ ds^2 = - \left( 1 - \frac{2M}{r} \right)dt^2 + \left(1 - \frac{2M}{r} \right)^{-1}dr^2 + r^2(d \theta ^2 + sin^2{\theta} d \phi^2) $$
we will find that it breaks down at r = 2M, 0. If instead we decide to use Eddington-Finkelstein coordinates (below)
$$ ds^2 = - \left( 1 - \frac{2M}{r} \right)dv^2 + 2dvdr + r^2(d \theta ^2 + sin^2{\theta} d \phi^2) $$ we will only get infinity at the physical singularity - when at r = 0. This shows us that sometimes, we just need to use a different coordinate system, and there is not one that can be used to solve all problems.

Just as we use Maxwell's equations for electromagnetic fields, we use Einstein's equation for gravitational fields. Einstein's equation is a set of ten second-order partial differential equations; they are also nonlinear. This set of equations tell us how a measure of local spacetime curvature is related to a measure of matter energy density. To find a measure of spacetime curvature, we must come up with a thought experiment on the motion of at least two test particles. Imagine two rings of test particles, one is in space and the other is falling towards the Earth. The particles in space feel no force, so they stay in the shape of a ring. However, the ones experiencing the Earth's gravitational force will start forming an ellipse. This occurs because the particles closer to the Earth feel a stronger pull than the ones further away, and the particles on the sides of the falling ring will begin to come closer together since they are pulled towards the center of the Earth. This change of shape in the rings is a measure of spacetime curvature. This is exactly what Newtonian gravity dictates and the derivation is short.

Unlike the Newtonian view, Einstein saw that the relative accelerations of these particles are not caused by changes in the gravitational force, but instead, they are caused by the fact that particles simply follow geodesics, or straight lines, in spacetime. And because we are seeing these particles accelerate, their geodesics must be curving relative to each other. This shows us that spacetime is actually being curved. To follow the derivation of the equation of geodesic deviation in general relativity and understand the implications in-depth, you must understand the use and meaning of Einstein notation, tensors, and Christoffel symbols. You will be able to follow the math without knowing about Christoffel symbols. In the derivations linked below, I will explain as much as I can without going off topic. While the first link is easier to follow, the second link will be mathematically rigorous.

Links from above:

Derivation of the Newtonian Deviation Equation

In this derivation, I will use "A General Relativity Workbook" by Thomas A. Moore as a reference. Below are some basic equations that will help guide us towards the derivation.
$$ \textbf{F}_{grav} = m \textbf{g} = - \left( \frac{GmM}{r^2} \right) \textbf{e}_r = -m \boldsymbol{\nabla} \Phi (\textbf{x}) $$
where $ \textbf{e}_r $ is the unit vector pointing from the first particle to the other and $ \Phi (\textbf{x}) $ is the gravitational potential from the first particle. Solve for the acceleration in terms of the gravitational potential and you get $ \textbf{a} \equiv \textbf{g} \equiv - \boldsymbol{\nabla} \Phi $ . You can rewrite this using index notation to make it more compact.  It should then look like this
$$ \frac{d^2 x^i}{dt^2} = - \eta^{ij} \frac{\partial \Phi}{\partial x^j} = - \eta^{ij} [\partial_j \Phi]_\textbf{x} $$
Here $ x^i $ is the position of the particle in our frame of reference and $ \eta^{ij} $ represents the spatial components of flat spacetime. This doesn't change the equation but only helps us in balancing out our indices. The equation above tells us how one of our particles falls, but now we need to write a similar equation for the second particle. However, this second particle is at some distance $ \textbf{n} (t) $ away from the first particle; we call this a separation vector. This means that the position of the second particle is at $ x^i(t) + n^i(t) $. With this information we can now write the equation of motion for the second particle, which is
$$ \frac{d^2 (x^i + n^i)}{dt^2} =  - \eta^{ij} [\partial_j \Phi]_\textbf{x + n} $$
Now we must expand the right side in a Taylor series in order to get an estimate of the derivative. This is what the expansion around the reference looks like
$$ [\partial_j \Phi]_\textbf{x + n} = [\partial_j \Phi]_\textbf{x} + n^k \left( \frac{\partial}{\partial x^k} [\partial_j \Phi] \right)_\textbf{x} + \cdots $$
This works assuming that the quantity of $ \textbf{n} $ is small, so we ignore higher order terms. Now, we are able to substitute this expansion into our equation of motion for the second particle, and then subtract the equation of motion of the first particle from that of the second particle.

1) Plug in the Taylor series expansion into the equation of our second particle as follows
$$ \frac{d^2 (x^i + n^i)}{dt^2} =  - \eta^{ij} [\partial_j \Phi]_\textbf{x} - \eta^{ij} n^k \left( \frac{\partial}{\partial x^k} [\partial_j \Phi] \right)_\textbf{x} $$
$$ \frac{d^2 (x^i + n^i)}{dt^2} = \frac{d^2 x^i}{dt^2} + \frac{d^2 n^i}{dt^2} $$
2) Subtract the equation of the first particle from that of the second like this
$$ \frac{d^2 x^i}{dt^2} + \frac{d^2 n^i}{dt^2} - \frac{d^2 x^i}{dt^2} = - \eta^{ij} [\partial_j \Phi]_\textbf{x} - \eta^{ij} n^k [\partial_k \partial_j \Phi]_\textbf{x} + \eta^{ij} [\partial_j \Phi]_\textbf{x} $$
3) Simplify by canceling out terms to get
$$ \frac{d^2 n^i}{dt^2} \approx - \eta^{ij} [\partial_k \partial_j \Phi]_\textbf{x} n^k \equiv - \eta^{ij} \left( \frac{\partial^2 \Phi}{\partial^k \partial^j} \right) n^k $$
This is the Newtonian Deviation Equation. By knowing the separation of the two particles at one time and assuming the separation vector remains small, we can determine the deviation of the particles at any given time.

Derivation of the Equation of Geodesic Deviation

Because this derivation is often given as homework in classes that teach relativity, I will not show step-by-step derivations. Instead, I will only show the steps that books tend to give. If you are confident in your use of tensor notation, you shouldn't have a problem filling in the steps. However, it can get extremely messy when it calls for a change in indices. I will again reference "A General Relativity Workbook" by Thomas A. Moore.

We use tensors in this derivation so that it is valid in arbitrary coordinates. Let's go back to our particles and define their positions by $ x^{\alpha}(\tau) $ and $ \bar{x}^{\alpha}(\tau) \equiv x^{\alpha}(\tau) + n^{\alpha}(\tau) $, where $ \tau $ is our proper time and $ \textbf{n} $ is our infinitesimal separation four-vector. Keep in mind that although this derivation is similar to that in Newtonian mechanics, this derivation uses relativistic concepts. Just as with the Newtonian derivation, we must write expressions that govern the motion of both particles along their geodesics.
$$ 0 = \frac{d^2 x^{\alpha}}{d \tau^2} + \Gamma^{\alpha}_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau} , \qquad 0 = \frac{d^2 \bar{x}^{\alpha}}{d \tau^2} + \bar{\Gamma}^{\alpha}_{\mu \nu} \frac{d \bar{x}^{\mu}}{d \tau} \frac{d \bar{x}^{\nu}}{d \tau} $$
Here, we express the particles' coordinate accelerations in terms of the Christoffel symbols instead of having them in terms of the gravitational potential. Christoffel symbols are combinations of first derivatives of the metric that describe effects of parallel transport in manifolds. Just like we did in the Newtonian derivation, we can use Taylor series to expand Christoffel symbol of the second particle around the value of the first particle since the separations are infinitesimal (shown below). 
$$ \bar{\Gamma}^{\alpha}_{\mu \nu} (at \: \bar{x}^{\alpha} (\tau)) \approx \Gamma^{\alpha}_{\mu \nu} (at \: x^{\alpha} (\tau)) + n^{\sigma} [ \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu}] (at \: x^{\alpha} (\tau)) $$
Now, we know $ \bar{x}^{\alpha}(\tau) \equiv x^{\alpha}(\tau) + n^{\alpha}(\tau) $ so we will plug this and the expansion above into the equation of motion of the second particle which gives us
$$ 0 = \frac{d^2 (x^{\alpha} + n^{\alpha})}{d \tau^2} + [\Gamma^{\alpha}_{\mu \nu} + n^{\sigma} ( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu})] \frac{d (x^{\mu} + n^{\mu})}{d \tau} \frac{d (x^{\nu} + n^{\nu})}{d \tau} $$
$$  \qquad \qquad \; \; = \frac{d^2 x^{\alpha}}{d \tau^2} + \frac{d^2 n^{\alpha}}{d \tau^2} + [\Gamma^{\alpha}_{\mu \nu} + n^{\sigma} ( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu})] \left( \frac{d x^{\mu}}{d \tau} + \frac{d n^{\mu}}{d \tau} \right) \left( \frac{d x^{\nu}}{d \tau} + \frac{d n^{\nu}}{d \tau} \right) $$
We want to simplify this more by multiplying out everything and using our geodesic equations. Doing this will give us
$$ ** \quad 0 = \frac{d^2 n^{\alpha}}{d \tau^2} + 2 \Gamma^{\alpha}_{\mu \nu} u^{\mu} \frac{d n^{\nu}}{d \tau} + n^{\sigma} ( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} u^{\mu}) u^{\mu} u^{\nu} $$
where $ u^{\mu} \equiv \frac{d x^{\mu}}{d \tau} $ is the first particle's four-velocity. Because this is  a four-vector we know that its derivative with respect to $ \tau $ is
$$ \left( \frac{d \textbf{n}}{d \tau} \right)^{\alpha} = \frac{d n^{\alpha}}{d \tau} + \Gamma^{\alpha}_{\mu \nu} u^{\mu} n^{\nu} $$
and so we take the second derivative and find it to be
$$ \left( \frac{d^2 \textbf{n}}{d \tau^2} \right)^{\alpha} = \left( \frac{d}{d \tau} \left[ \frac{d \textbf{n}}{d \tau} \right] \right)^{\alpha} = \frac{d}{d \tau} \left( \frac{d n^{\alpha}}{d \tau} + \Gamma^{\alpha}_{\mu \nu} u^{\mu} n^{\nu} \right) + \Gamma^{\alpha}_{\sigma \nu} u^{\sigma} \left( \frac{d n^{\nu}}{d \tau} + \Gamma^{\nu}_{\beta \gamma} u^{\beta} n^{\gamma} \right) $$
$$ = \frac{d^2 n^{\alpha}}{d \tau^2} + \frac{d \Gamma^{\alpha}_{\mu \nu}}{d \tau} u^{\mu} n^{\nu} + \Gamma^{\alpha}_{\mu \nu} \frac{d u^{\mu}}{d \tau} n^{\nu} + \Gamma^{\alpha}_{\mu \nu} u^{\mu} \frac{d n^{\nu}}{d \tau} + \Gamma^{\alpha}_{\sigma \nu} u^{\sigma} \frac{d n^{\nu}}{d \tau} + \Gamma^{\alpha}_{\sigma \nu} \Gamma^{\nu}_{\beta \gamma} u^{\sigma} u^{\beta} n^{\gamma} $$
Now, this is getting really messy so let us simplify by noticing two key things:
$$ 1. \frac{d \Gamma^{\alpha}_{\mu \nu}}{d \tau} = \frac{d x^{\sigma}}{d \tau} \frac{\partial \Gamma^{\alpha}_{\mu \nu}}{\partial x^{\sigma}} = u^{\sigma} \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} $$
$$ 2. \; Changing \; \sigma \; to  \; \mu \; will \; make \; two \; of \; the \; terms \; the \; same. $$
Using both of these in the messy equation we have so far simplifies it to
$$ \left( \frac{d^2 \textbf{n}}{d \tau^2} \right)^{\alpha} = \frac{d^2 n^{\alpha}}{d \tau^2} + \left( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} \right) u^{\sigma} u^{\mu} n^{\nu} + \Gamma^{\alpha}_{\mu \nu} \frac{d u^{\mu}}{d \tau} n^{\nu} + 2 \Gamma^{\alpha}_{\mu \nu} u^{\mu} \frac{d n^{\nu}}{d \tau} + \Gamma^{\alpha}_{\sigma \nu} \Gamma^{\nu}_{\beta \gamma} u^{\sigma} u^{\beta} n^{\gamma} $$
Ok, so now the tricky part comes. We need to do three things: use the geodesic equation of the first particle to eliminate $ \frac{d u^{\mu}}{d \tau} $, use the equation labeled with ** in order to eliminate $ \frac{d^2 n^{\alpha}}{d \tau^2} $, and rename indices to pull out a common factor of $ u^{\sigma} u^{\mu} n^{\nu} $. This will take a while. The trickiest part is definitely changing the indices because there are so many of them. Try to see what you can cancel out and remember the symmetries of the Christoffel symbols. After this, we will get
$$ \left( \frac{d^2 \textbf{n}}{d \tau^2} \right)^{\alpha} = \left( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} - \partial_{\nu} \Gamma^{\alpha}_{\mu \sigma} + \Gamma^{\alpha}_{\sigma \gamma} \Gamma^{\gamma}_{\mu \nu} - \Gamma^{\alpha}_{\nu \gamma} \Gamma^{\gamma}_{\mu \sigma} \right) u^{\sigma} u^{\mu} n^{\nu} $$
Since the left side is a tensor and everything outside the parenthesis on the right side is a tensor, then the stuff inside the parenthesis must be a tensor. Not surprisingly, the stuff inside is actually the Riemann tensor, but with the signs flipped. So the Riemann tensor, which is the measure of spacetime curvature, is defined as
$$ R^{\alpha}_{\mu \nu \sigma} \equiv \partial_{\nu} \Gamma^{\alpha}_{\mu \sigma} - \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} + \Gamma^{\alpha}_{\nu \gamma} \Gamma^{\gamma}_{\mu \sigma} - \Gamma^{\alpha}_{\sigma \gamma} \Gamma^{\gamma}_{\mu \nu} $$
Therefore, we can rewrite the last equation using the Riemann definition and we get
$$ \left( \frac{d^2 \textbf{n}}{d \tau^2} \right)^{\alpha} = - R^{\alpha}_{\mu \nu \sigma} u^{\sigma} u^{\mu} n^{\nu} $$
This is the Geodesic Deviation Equation. This is extremely useful because it allows us to see if spacetime is flat or curved, which we can't do by looking at the metric. So if the any of the components of the Riemann tensor are non-zero, then spacetime is curved.

Saturday, July 26, 2014

Review and application of group theory to molecular systems biology [part 1]

Edward A Rietman, Robert L Karp, and Jack A Tuszynski; Theor Biol Med Model. 2011; 8: 21.


My opinion: I love it when physicists stick mathematics where it doesn't belong. This paper is a bit old I know (Elwin wanted recent papers), but I really wanted to read it, and I figured I'd summarize it too while I'm at it for this blog. I'm going to break this up into a few pieces since I want to explain concepts as we go for the reader new to abstract algebra.

Any arguments with opinions or analyses in this summary are warmly welcomed.

Ratings for part 1 only:

General: 9/10
My familiarity with subject: 7/10
Style: 9/10
Ease (for layman scientist with no algebra background): 7/10?
Length: 7pgs [total paper: 27pgs] (excluding references and images)


According to general biology textbooks, life is characterized by

  • metabolism
  • self-maintenance
  • duplication involving genetic material
  • evolution by natural selection

But many complex details of life are overlooked or unaccounted for. The universe may be considered a Riemannian resonator and "life can be thought of as some sort of machinery the universe uses to diminish energy gradients" (page 1). The foundations of physics is based in group theory: \(SU(3)\) quarks and \(SU(2)\times U(1)\) for electro-weak interactions.

Group Theory

For the reader not well versed in group theory, I suggest following Nick's post on particle physics. But I will go over the general concepts anyway.

A group is a set of objects that behave in a certain way under some operation (addition, multiplication, etc.). More formally, a group \(G\)  is a nonempty set with a binary operation, \(\cdot\), which satisfies:
  1. Closure -- any two elements of \(G\) combined return another element in \(G\).
  2. Associativity -- although the positions of elements usually matters, the order in which you combine juxtaposed elements does not.
  3. Identity -- there exists an element that causes no effect when combined with any other element
  4. Inverses -- For every element, there exists another that may be combined with it to return the identity.
Or stated even more formally,
  1. \(\forall g,g' \in G\ \ g\cdot g' \in G\).
  2. \(\forall a,b,c \in G\ \ a\cdot(b \cdot c) = (a\cdot b)\cdot c\).
  3. \(\exists e \in G\ \forall g \in G\ g\cdot e = e\cdot g = g\).
  4. \(\forall g \in G \ \exists g^{-1} \in G\ gg^{-1} = g^{-1}g = e\).
However the authors of this paper suspiciously leave out condition 1. Which in my mind is absolutely terrifying when considering a group.

"Isomorphic" is a concept that basically means that two groups behave the same way, and if the elements of one group were relabeled, they would in fact be identical. This is important because "Any group [...] is isomorphic to a subgroup of matrix groups" (page 4). And a subgroup, just so you know, is exactly what you'd want it to be -- a group inside of another group.

Some important groups with associated general properties:
  • Orthogonal groups \(O(n)\) -- group of rotations in \(n\)-dimensional Euclidean space including reflections.
  • Special orthogonal groups \(SO(n)\) -- groups of rotations in \(n\)-dimensional Euclidean space excluding reflections.
  • Unitary groups \(U(n)\) -- The inverse of an element is its complex conjugate transpose.
  • Special unitary groups \(SU(n)\) -- Inverses are complex conjugate transposes, and the determinant is also \(\pm 1\).

Genetic Code

Ribosomes take in tRNA (nucleic acids) and output proteins (amino acids). This translation of information is done by "codons", which are sections of three nucleic acids. Each codon codes for a specific amino acid. Mathematically, we may regard a codon as the direct product of the set of nucleic acids \(S = \{U, C, A, G\}\) with itself thrice which yields \(4^3 = 64\) possible codons. Now hold on, because this is going to get fun.

Since there are not 64 amino acids, many codons code for the same amino acids. We can compile two lists:
  • \(M_1 = \{ AC, CC, CU, ...\}\)
  • \(M_2 = \{ CA, AA, AU, ... \}\)
The first set \(M_1\) corresponds to doublets whose third nucleic acid doesn't matter. Any nucleic acid following the two in an element from \(M_1\) will not change the resulting amino acid. The second set \(M_2\) corresponds to doublets whose third nucleic acid absolutely matters. Without the third nucleic acid, a sequence from \(M_2\) will not code anything.

With these sets we may define an operation. Let this operation be switching one letter for another. We have a few possibilities:
  • \(\alpha: A \leftrightarrow C\) and \(U \leftrightarrow G\).
  • \(\beta: A \leftrightarrow U\) and \(C \leftrightarrow G\).
  • \(\gamma: A \leftrightarrow G\) and \(U \leftrightarrow C\).
or written in permutation notation,
  • \(\alpha =\begin{pmatrix}A & U & G & C \\ C & G & U & A\end{pmatrix}\)
  • \(\beta =\begin{pmatrix}A & U & G & C \\ U & A & C & G\end{pmatrix}\)
  • \(\gamma =\begin{pmatrix}A & U & G & C \\ G & C & A & U\end{pmatrix}\)
With this operation defined, we have a group among our sets. In fact, this group we have defined is isomorphic to something known as the Klein four group, the group that preserves the symmetries of a rectangle in two dimensions. Two scientists extended this representation to a 4-dimensional hypercube, which looks absolutely crazy:

A subgroup \(N\) of \(G\) is called a normal subgroup if any element in \(N\) may be multiplied on the left by some element in \(G\), multiplied on the right by that element's inverse, and still be an element of \(N\), or stated formally, \(N \unlhd G \iff \forall n\in N,\ \forall g \in G\ gng^{-1}\in N\). It's just a fancy subgroup that has absolutely terrible philosophical implications upon closer inspection. The four normal subgroups for the \((K4 \times K4)\) representation shown above are written on page 6.

With all this work done, we may now see some meaning. We may develop a 64-dimensional hypercube of general genetic code (as stated above) by \(D = \{A, U, G, C\} \otimes \{A, U, G, C\} \otimes \{A, U, G, C\}\). The symmetry operations on this space are the codons. Multiple vertices code for the same amino acid, so our mapping is surjective.

If we continue even further and include time evolution, then we have a 65-dimensional differentiable information space manifold \(M[X]\). It is actually postulated that evolution is a geodesic in this information spacetime. Holy shit, right? You should rather be thinking bullshit, but let's continue.

We may define a metric between species (polynucleotide trajectories) statistically by
$$ d = \left[ \sum\limits_{\mu} \left(x'^{\mu} - x^{\mu}\right)^2\right]^{\frac{1}{2}} $$
the regular ol' Euclidean metric. From here we may "see regions of the information-spacetime that have not been explored by evolution" (page 7).

We may actually analyze our system in terms of symmetry breaking within a higher dimensional Lie algebra. From the symplectic group \(sp(6)\) we may break its symmetry to result in our system.
  1. \(sp(6) \supset \left[sp(4) \otimes su(2)\right]\)
  2. \(\left[sp(4) \otimes su(2)\right] \supset \left[su(2)\otimes su(2) \otimes su(2)\right]\)
  3. \(\left[su(2)\otimes su(2) \otimes su(2) \right]\supset\left[ su(2)\otimes u(1) \otimes su(2)\right]\)
  4. \(\left[su(2)\otimes u(1) \otimes su(2)\right] \supset \left[su(2) \otimes u(1)\right]\)
  5. \(\left[su(2) \otimes u(1)\right] \supset u(1)\)
We end on page 7 of 29 (the end of this discussion on codons).

Thursday, July 24, 2014

Numerical Relativity: Black Holes and Gravity - Part 2

Einstein's Theory of General Relativity

You have probably heard about Einstein's Theory of Relativity. His work on special relativity was published in 1905. This introduced a new framework in physics. He showed that the laws of physics are the same in all non-accelerating reference frame and that the speed of light is constant. This revolutionized different branches of physics and led to his 1915 Theory of General Relativity, or his theory of gravity. Einstein's view was that space and time could be distorted by objects with mass. This spacetime would be able to be stretched and bent. A classic example of this is to imagine a heavy object in the middle of a trampoline. Spacetime is distorted just like the fabric of the trampoline. This would change the way we perceive space and time. Perhaps the most outstanding predictions of this theory are black holes and gravitational waves.

Black Holes (BHs)

In the media, black holes are erroneously described as giant vacuum cleaners in space. Black holes do not suck in things in the way vacuum cleaners do. Physicist instead think of black holes as regions in space with extreme curvature. Black holes warp space and time in such an extreme manner that nothing - not even light - can escape once it is caught into orbit. Since light can't escape it, we cannot look up at the sky and actually point at a black hole. Instead we use different methods to find them. For example, we look for an object orbiting some dark region of space. We will discuss this further in a future post.

Black Hole Anatomy

Let's look at the anatomy of a black hole. Although a black hole is not an actual object we can touch, let's think of it as a black sphere; we choose black since we can't see black holes in the sky. First, let's call the surface of the sphere the event horizon. Once anything passes this point, it cannot return outside. In fact, we don't even know what happens inside the event horizon. All that we know is that there is a point inside which we call a singularity. A singularity is when the gravitational field is at infinity. This is where physics breaks down.

How Do Black Holes Form?

There are three main ways in which black holes form: implosion, high-energy collision, and binary black hole (BBH) collisions. Implosion, or gravitational collapse, is perhaps the most talked about way of black hole formation. This occurs when a star that is a few times larger than the size of our sun collapse onto itself due to a greater gravitation (or inward) force and smaller internal pressure (outward force). I will mention that for a star to implode, there is a minimum size, which is about 3 solar masses. This Tolman-Oppenheimer-Volkoff limit, however, is not well known because it depends on the equations of state for matter this dense, i.e. the relationship between the volume, pressure, temperature, and internal energy of a star.

Diagram of gravitational force vs. internal pressure on a star .
I won't go into details on the formation by high-energy collisions, since this could only happen in very special conditions. This type of event has never been detected, so it is purely theoretical. If it actually ever happened, the black hole would be so small that it would evaporate extremely fast.

Formation by binary black hole collisions is in a way the "easiest" to understand. Unlike implosion where there is an actual birth of a black hole, this type of collisions involve two already formed black holes that collide in order to form a bigger black hole. A very cool thing about this is that the final mass of this black hole is not equal to that of the sum of the two initial black holes. You might be wondering why! The next section gives us the answer.

Gravitational Waves (GWs)

Fantastic! So in the paragraph above we asked why the mass of the final black hole is not the sum of the two initial black holes that collided. This is actually my research area now, so I will try to explain it very thoroughly.

Gravitational waves, or gravitational radiation, are a result of acceleration; it is like light only in the sense that both types of radiation carry energy away. A great example of this is a system of two black holes. When two massive bodies orbit around each other, they accelerate. Because our system must conserve energy, gravitational radiation must be given off as our BHs accelerate, and this draws our BHs closer together which causes them to orbit faster. This radiation distorts spacetime, which is why you might have heard of the definition "Gravitational waves are ripples in spacetime." The closer they are, the more strongly radiate.

Binary neutron stars orbiting each other and radiating. Photo taken from NASA.

Below, you can see a picture of this radiation as the black holes get closer to each other and eventually collide. The height is the amplitude of the gravitational waves. the length shows the passing of time. Evidently, gravitational waves are given off as the BHs orbit each other. As they get closer the radiation increases by large factor. The highest point you see in the wave above, shows the time at which the black holes collided to become one. This releases an incredible amount of energy, making a BBH collision the most energetic phenomena in space. After they collide, the final BH will ring like a bell (radiating more) and eventually dying down to a quiescent state. When a black hole is not moving or "eating" anything, it will not radiate. 

Gravitational wave from a BBH.

I will go into more detail on how to detect this waves and what we can learn about them in a future post.
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