Unicorns of Geometry: Getting Cross About Products

Or: Why  you should forget about the vector cross product!

So titled because while both spoken about in tales and legends, neither unicorns nor cross products actually exist. Forget what you think you know about vector geometry!

A question for the beginner physics students: consider two vectors $u = u_x \hat{x} + u_y \hat{y}+ u_z \hat{z}$ and $v = v_x\hat{x} + v_y \hat{y} + v_z \hat{z}$. The coefficients $u_i, v_i$ have units of length. You know the dot product

$u \cdot v = u_xv_x + u_yv_y + u_x v_z$,

which is a number, has units of area, and is the square of the length of one vector projected onto the other. You probably also know the other interesting operation, the cross product

$u \times v = (u_yv_z - u_zv_y)\hat{x} + (u_zv_x - u_xv_z)\hat{y} + (u_xv_y - u_yv_x)\hat{z}$

which is a vector. Its length is the magnitude of the parallelogram defined by $u$ and $v$. But its coefficients have units of area. How can it still be a vector in the same space as the other two if its coefficients have different units?

Predicting gravitational waves

There has been a lot of excitement recently with the news that physicists at LIGO have directly detected gravitational waves. Many wonderful people have done popular science blog posts or videos about what gravitational waves are, but I haven’t really seen anyone talk much about the mathematics of it. For example, why do Einstein’s equations mean that gravitational waves exist? How did Einstein predict gravitational waves?

It turns out that starting with Einstein’s equations and a few simplifying assumptions, it’s relatively easy to derive the necessity of gravitational waves. That’s what this post will try and do.

Extremal Black Holes and Cosmic Censorship

The Cosmic Censorship Hypothesis (CSH) was put forward by Roger Penrose in 1969, and (roughly) states

“there are no naked singularities”.

The hypothesis proposes that whenever a singularity occurs, such as in the center of a black hole, it must occur behind an event horizon. A singularity outside of an event horizon is termed a naked singularity, and the CSH says that such singularites do not exist. Cosmic Censorship has pretty profound implications for fundamental physics. For instance, a failure of the Cosmic Censorship Hypothesis leads to a failure of determinism in classical physics, since one can’t predict the behaviour of spacetime in the causal future of a singularity. On the other hand, if the Cosmic Censorship Hypothesis holds then (outside the event horizon), the singularity does not affect determinism. In 1991, Stephen Hawking made a wager with physicists Kip Thorne and John Preskill that the Cosmic Censorship Hypothesis is true. Six years later in 1997, Hawking conceded the bet “on a technicality”, after computer calculations showed that naked singularities could exist, albeit only in exceptional (not physically realistic) circumstances. In this post, we’re going to be looking at extremal black holes, and their relationship to the Cosmic Censorship Hypothesis.

Photon Spheres and Rotating Black Holes

This is another post on my current theme of black holes. Recently, I wrote a blog post about light orbiting a Schwarzchild black hole. These so called Photon spheres are sufficiently interesting to wonder whether they exist for more complicated black holes. They do, and we’re going to find out about them. This post is based on a fantastic paper [1] I read the other week. If you find this stuff interesting, and you’ve got an introductory level of GR you should definitely check out the paper here.

One of the coolest thing about black holes is you can describe any black hole with just three numbers: Mass $(M)$, Charge $(Q)$, and Angular Momentum $(J)$. Of course, if we’re being completely general we also need to describe where the black hole is and how fast it’s moving through space – but we can always perform a Lorentz transformation so that we’re in the rest-frame of the black hole and so we really only care about $M$, $Q$ and $J$. The fact that you can describe black holes with only three numbers is one of the main reasons I find them so interesting. Consider for example, two black holes, A and B. Black hole A is made entirely from matter – protons, neutrons, electrons and the like. On the other hand, black hole B is made entirely from anti-matter – anti-protons, anti-neutrons, anti-electrons and the like. Further, suppose A and B have the same charge, angular momentum and mass. How can we tell A from B? That is, what experiment can we do to tell the difference between the matter black hole or the anti-matter black hole? The answer is, we can’t tell the difference. For all intents and purposes, there is no way of telling whether a black hole is made of matter or anti-matter. Indeed, a third black hole C, with the same mass, charge and angular momentum, but made entirely from light, would be similarly indistinguishable from the first two.

As promised in the title, we’re going to be looking at rotating black holes. A rotating black hole is, like the Schwarzchild black hole, uncharged ($Q=0$). Unlike the Schwarzchild solution however, the rotating black hole has a non-zero angular momentum $J$. Rotating black holes are also called Kerr black holes, after Roy Kerr who was the first to write down an exact solution for one. We will consider an uncharged black hole which is rotating with constant angular momentum. For convenience, we define the angular momentum per unit mass $a = \frac{J}{M}$. Since the black hole is rotating, we lose the spherical symmetry we have with the Schwarzchild metric, so we expect the solution to be a little more complicated. We do, however, still have axial-symmetry, i.e. rotational symmetry around the axis of rotation.

Photon Spheres

Ok, now that I have finished the preamble over here, I can finally start talking about what I originally intended to blog about: Photon spheres.

The defining feature of a black hole is that the gravitational attraction beneath the event horizon is so strong that even light can’t escape. That is, the curvature of spacetime in the vicinity of the black hole is so intense that there are no geodesics which are able to leave. This is what makes black holes so interesting, since anything (including light) which is dropped into a black hole is lost forever (we’re talking classical black holes at the moment, so no Hawking radiation). On the other hand, light (or matter) travelling near a black hole can escape, provided it stays outside of the event horizon.

Light from the blue object is deflected towards the grey massive object

General relativity predicts that light passing near a heavy object will be deflected by the gravitational field of that object. Equivalently, light will follow a geodesic in the curved spacetime around the heavy object.

This situation strongly resembles the situation we have in orbital mechanics, where a small object like a satellite or asteroid is moving near a large object, like a planet. Depending on the velocity of the object, it either escapes the planet, falls into the planet, or starts orbiting the planet. It seems like a natural question, then, to ask:

Can light orbit a black hole? Continue reading

Preamble to Black Holes

I’ve been reading a lot about General Relativity this past week and so I thought I would do a couple of posts on Black Holes, since they are well and truly one of the most interesting things about General Relativity. This first post is really just a (very) brief introduction to General Relativity. My main goal is to write about some of the cool things I’ve come across lately, so think of this as setting the theme for the next few posts.

General Relativity (GR) is a geometric theory of gravitation proposed by Einstein. In GR, the flat background spacetime of special relativity is replaced by a curved spacetime. The curvature of spacetime is greater around objects with a higher mass, and particles (including light) travel along paths called geodesics (essentially the ‘shortest’ path between two points) in this curved spacetime.

The standard analogy here is bowling ball on a rubber sheet: If you take a big rubber sheet stretched flat and put a bowling ball in the middle, the sheet stretches and curves around the bowling ball, but as you get further away from the ball the sheet becomes less curved.

The rubber sheet analogy – heavy objects stretch the spacetime around them

Now imagine a tiny little marble moving on the rubber sheet. As the marble gets closer to the bowling ball, the curvature of the sheet causes the marble to accelerate towards the bowling ball. The marble feels this acceleration as the gravitational force of the bowling ball – gravity is just the curvature of spacetime. The path that the marble traces out on the rubber sheet is called a geodesic. In flat space, a geodesic is simply a straight line, but on a curved surface geodesics can be more complicated. Archibald Wheeler described GR rather poetically, saying “Spacetime tells matter how to move, matter tells spacetime how to curve”

Light following a geodesic through curved spacetime

The rubber sheet analogy is all well and good, but let’s get into some honest mathematics!

General Relativity is described, rather succinctly, with the Einstein Field Equations (EFE).

$R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4}T_{\mu \nu}$

The EFE relates the local curvature of spacetime with the local energy and momentum in that spacetime. Solutions to the EFE are metrics $g_{\mu \nu}$ which describe the geometry of spacetime. It should be noted that the apparent simplicity of this equation is incredibly misleading. The Einstein Field Equations are actually a set of 10 coupled, non-linear, hyperbolic-elliptic, partial differential equations – solving them is a highly non-trivial task and exact solutions are only known for the simplest cases.

The EFE determine how a given distribution of matter/energy influences the geometry of spacetime. To describe the motion of freely falling matter through curved spacetime, we also need the geodesic equation.

$\frac{\text{d}^2 x^{\mu}}{\text{d}\lambda^2} + \Gamma^{\mu}_{\alpha \beta} \frac{\text{d} x^{\alpha}}{\text{d} \lambda} \frac{\text{d} x^{\beta}}{\text{d} \lambda} = 0$

The simplest solution to the Einstein Field Equations is the Minkowski metric – that is, flat space. Traditionally denoted by $\eta_{\mu \nu}$ rather than $g_{\mu \nu}$, the Minkowski metric describes spacetime with no matter, no energy, no curvature – nothing. The metric has, in cartesian coordinates, the following form

The metric is related to the line element (spacetime interval) in the following way

$ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}$

where the repeated indices are implicitly summed over. For the Minkowski metric in cartesian coordinates $(ct, x, y, z)$, this becomes

$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$

A metric can either be specified by its components in a particular coordinate system $g_{\mu \nu}$, or the line element $ds^2$. Note that I’m using physicists terms like ‘line element’ and ‘spacetime interval’ in this article. There are two reasons for this. Firstly, I don’t want to go into the mathematics of differential geometry in this post – my goal is to describe some cool physics, and the clearest way to do that is to use physics notation. Secondly, this is the notation used in the literature.

I’m going to finish off this post by describing another solution to the Einstein Field Equations – the Schwarzchild metric. This metric describes spacetime around a spherically symmetric, uncharged, non-rotating massive object. In polar coordinates $(ct, r, \theta, \phi)$, the metric takes the form

$ds^2 = -(1-\frac{R_s}{r})c^2 dt^2 + (1-\frac{R_s}{r})^{-1}dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2$

where $R_s$ is the Schwarzchild radius, a constant related to the mass of the object.

If any spherically symmetric, uncharged, non-rotating massive object has a radius less than its Schwarzchild radius, the object forms a black hole and is called a Schwarzchild black hole. The physics of black holes is an incredibly cool area of General Relativity, and there are a number of surprising results that you can get. In the next blog post I will describe one of the lesser known results – the photon sphere.

Quantum Cryptography

[Accompanying notes for Physsoc (University of Canterbury) seminar on 2/8/2013]

Cryptography consists of making and breaking methods to communicate information without third parties which are able to intercept communications being able to gain knowledge of this information.

1. Cryptography before Quantum Mechanics

In the traditional example Alice and Bob wish to communicate privately despite Eve (who wishes to know what Alice and Bob are communicating) being able to intercept their communications. However in reality the situations in which cryptography has been used is for spies when letters are being intercepted, for the military — especially after the invention of radio, and today on the internet when we access online bank accounts and complete other online transactions.

In order to do this Alice and Bob could try to disguise their message (for example invisible ink or microdots), or they can employ a cipher scheme. The problem with a disguised message is that if a third party discovers the method of hiding the message they can immediately read all following messages. A cipher can be seen as an invertible function mapping the plain text to the enciphered text. The recipient can then apply the inverse function to discover this message. (Note: Strictly speaking $f$ does not have to be a function, $f^{-1}$ needs to be a function so it gives a unique message that was sent however the enciphering process can give multiple possible encrypted messages from which one can be randomly chosen.)

However this seems to have the same disadvantage — as soon as the inverse function is discovered messages can immediately be read, so in general we use a function of both the message and a key.  Thus even if the function in question is known, without the key which can be regularly changed it may still be possible to maintain security. However as the next example will show security of the key is not sufficient as this may be derivable from the message.

Example 1: Substitution Cipher

Alice and Bob map each letter of the alphabet to another letter. As there are $26!$ such bijective mappings this is too many for Eve to go through in any reasonable time. However in the English alphabet not all letters are used in equal frequencies in normal text. Hence if we look at the statistical distribution of the various letters in a large section of cipher text, we can identify this with the statistical distribution of letters in normal English, thus allowing us to find the key (the specific bijection used). Also the frequency of combinations of two or more letters in consecutive order can be used. The English language (or any other natural language) only uses a tiny proportion of all possible sequences of characters, the unique properties of the subspace of character sequences can be used in breaking several encryption schemes.

So we also need a function such that there is no such method to determine the key that is implementable in a the length of time we require the message to be secure for given the computing power at the disposal of Eve.

For simplicity we write the message as a string of zeros and ones. As a key we have a random string of zeros and ones of equal length which we only use for this message and never reuse. Then we apply the AND operation (addition modulo 2) to the nth digits of the key and the message to give the nth digit of the encrypted message. The reason this is unbreakable is that given an encrypted message there is a key such that any message of the correct length could be encrypted to give this message. As the key is random and never reused we can not use the keys that would correspond to given messages to determine which is the actual key used, and hence the actual message sent.

The problem is this requires Alice and Bob to have met before in secret and have exchanged an equal amount of data to that which they will in the future wish to communicate, not to mention the added complexity of generating random numbers — which by definition can not be done algorithmically — all random number generators on computers without specific hardware that uses electronic noise or other physical phenomena to generate random numbers, are only using some algorithm to generate pseudorandom numbers. For application such as internet banking this is clearly not highly feasible.

However in the last century there was a revolution in cryptography — public key cryptography. There are two approaches to this. In this Alice and Bob both create a secret number, transmit partial information about this and use this to create a key known to both parties which can’t be created from the partial information. Continue reading

Quantum Chaos

This is a discussion of the work that I did over the summer of 2012-13 at the Quantum Science department of ANU as part of the Summer Research Scholarships program there. I did a project on the computational modelling of a chaotic quantum mechanical system, focusing on how the chaos in the system changes as a function of its energy scale – from the near-classical to the fully quantum. We intend to publish our results, so I can’t post them here; however I will give you a run-down on the ideas behind it.

A little bit of chaos theory

Chaos is a ‘sexy’  kind of word in mathematics and physics alike. It is easy to vigorously hand-wave about the butterfly effect and sensitivity to initial conditions and other such things but the fact is that there’s no easy way to quantify or measure ‘unpredictability’ – due to, not surprisingly, its unpredictable nature. What we need instead is some rigorous handwaving.

So how do we decide just what is a chaotic system? All of our good quantitative descriptions of chaotic systems come from studying phase space behaviour – the vector space spanned by $\left\{x_1, x_2, \dots, \frac{dx_1}{dt}, \frac{dx_2}{dt}, \dots \right\}$, where each $x_i$ is a physical degree of freedom. We consider points in phase space, each corresponding to a state of the system, and their time evolution. These produce trajectories through phase space that reflect the dynamics of the system.

Imagine a simple harmonic oscillator, the paragon of an elementary dynamical system -and something which is definitely not chaotic. This system has one variable, the position $x$, which is governed by the differential equation $\frac{d^2x}{dt^2} + k^2x = 0$ for some constant $k$. Now, if we pull the oscillator out to some particular distance $d_0$ and then let it go (i.e. set the system up with some particular set of initial conditions), the trajectory through phase space is simply an ellipse in the plane, centered on the origin. The aspect ratio of the ellipse is given by the ratio of the physical parameters of the system, the mass of the oscillator and the strength of the spring. For any given parameter set, with appropriate change of variables we can always make this ellipse a circle. The square of the radius gives the total energy in the system, which for us would be proportional to $d_0^2$).

Now suppose we were to repeat the experiment with the simple harmonic oscillator – we begin by pulling our oscillator out to distance $d_0$, but instead of letting go here we pull an additional amount $\delta$ further. This means that our initial states for this oscillator are separated in phase space (under the Euclidean metric) by an amount $\delta$ and that our initial energies differ by a constant proportional to $(d_0 + \delta)^2 - d_0^2 = 2d_0\delta + \delta^2$. But given that we know the energy gives the radius of the circular trajectory through the appropriately scaled phase space, which is constant in our lossless system, we know that at all times $t$, these two trajectories are separated by a constant amount.

We see that the simple harmonic oscillator is predictable in the sense that, given a small difference $\delta$ between the initial conditions, the trajectories of two points follow essentially same curve. It is this which separates chaotic systems from their non-chaotic brethren: chaotic systems are extremely sensitive to small changes of initial state and as such become dynamically unpredictable given small perturbations. We may state:

A system is considered chaotic if each pair of phase space trajectories, initially separated by a differential amount, eventually become dissimilar.