… Data aequatione quotcunque fluentes quantitates involvente fluxiones invenire et vice versa …

Category: Has references (not self contained)

Snowflakes

It took me two tries to get out of my flat this morning. I really ought to get into the habit of looking out the window in the morning; too often do I open the front door, ready myself to step out, only to turn back to fetch my umbrella. The annoying thing about snow is that I can’t hear it, unlike the pitter-patter of rain.

Somehow or another I ended up looking at Wilson Bentley’s micro-photographs of snow crystals. And a question forms in my mind, “Why are they all so symmetrical?” If all snowflakes were to look alike, then perhaps the dynamics leading to the formation of snow crystal is stable, and the global convergence unto a completely symmetrical pattern would not be surprising. But not all snowflakes look alike. In fact, colloquially we speak of them as each completely different from every other. This implies that the dynamics of snow crystal growth should be at least somewhat sensitive to atmospheric conditions in a local scale (and perhaps to the nucleus that seeds the crystal) so that the seemingly random to-and-fro dance as the snowflake falls from the sky can effect different shapes and branches.

Now, much experimental evidence has gone to show that the formation of ice crystals tends to by catalyzed by impurities. Pure water can be supercooled, in normal pressure conditions, to temperatures below 273 Kelvin. But in these situations a single mite of impurity dropped into the water can cause the entire beaker to freeze over suddenly. Similarly, ice crystals in the upper atmosphere tend to form around impurities: bacterium floating in the air, dust or ash, or perhaps particles introduced artificially. So one may surmise that the fact that all 6 branches of a snowflake grows in the same way because, somehow, the eventual shape of the snowflake is already encoded in the original formation of the central nucleus. Let me try to explain why this hypothesis is not very convincing. I’ll make one a priori assumption, that the growth of a crystal structure is purely local, and not due to some long-range interaction.

To draw an analogy, consider a large group of acrobats. They are trying to bring themselves into a formation around a leader. Disallowing long-range interaction can be thought of requiring that the leader cannot shout out orders to individual troupe members. But we can allow passing of information by short-range interactions, i.e. whispering instructions to the people already in formation. So the leader stands alone at the start. Then he grabs on a few people nearby to form the nucleus. Then he tells each of the people he grabbed a set of instructions on how to grab more people, where to put them, and what instructions to pass on to those people (included in these instructions are instructions to be passed on to the even more remote layer of acrobats and so on). Then if the instructions were passed correctly, a completely ordered pattern will form. But as anyone who has played the game of telephone can testify, in these scenarios some errors will always work its way into the instructions. In the physical case of snowflakes, these are thermodynamical fluctuations. So some irregularities should happen. Now, if the instructions the leaders were trying to pass down were very short and easy to remember, the errors tend not to build up, and the formation will, for the most part, be correct. But keeping the message short has the drawback that the total number of formations one can form is fewer. In the snowflake case, one can imagine somehow each small group of molecules in the snow crystal can encode some fixed amount of information. If the encoding is very redundant (so the total number of shapes is small), then the thermodynamical fluctuations will not be likely to break the symmetries between the arms. But considering the large number of possible shapes of snowflakes, such encoding of information should be taxed to the limit, and small fluctuation (errors in the game of telephone) should be able to lead one arm to look drastically different from the others. One possible way to get around this difficulty would be to use some sort of self similarity principle. But this will suggest the snowflakes are true fractals, which they are not. Read the rest of this entry »

The “Hoop Conjecture” of Kip Thorne and Spherically Symmetric Space-times

Abstract. (This being a rather long post, I feel the need to write one.) In the post I first gather some miscellaneous thoughts on what the hoop conjecture is and why it is difficult to prove in general. After this motivation, I show also how the statement becomes much easier to state and prove in spherical symmetry: the entire argument collapses to an exercise in ordinary differential equations. In particular, I demonstrate a theorem that is analogous, yet slightly different, from a recent result of Markus Khuri, using much simpler machinery.

The Hoop conjecture is a proposed criterion for when a black-hole will form under gravitational collapse. Kip Thorne, in 1972 [see Thorne, Nonspherical Gravitational Collapse: a Short Review in Magic without Magic] made the conjecture that (I paraphrase here)

Horizons form when and only when a mass $M$ gets compacted into a region whose circumference $C$ in EVERY direction is bounded by $C \lesssim M$.

This conjecture, now widely under the name of “Hoop conjecture”, is deliberately vague. (This seemed to have been the trend in physics, especially in general relativity. Conjectures are often stated in such a way that half the effort spent in proving said conjectures are used to find the correct formulation of the statement itself.) Read the rest of this entry »

Cartan’s Spiral Staircase

I’ve just spent much too long puzzling over the geometric formalism of a paper of Lazar and Hehl, so I figure I’ll write a little something about Riemann-Cartan geometry here. Note: I will only discuss geometry, and from a very limited perspective at that. For physical applications please see the paper cited above.

(pseudo/semi-)Riemannian geometry
First we recall what a semi-Riemannian geometry is (see also Chapter 3 of B.O’Neill, Semi-Riemannian Geometry). To begin, let us recall some definitions.

Definition 1 (Affine connection)
An affine connection $\nabla$ on a smooth manifold $M$ is a map $\displaystyle C^\infty(M,TM)\times C^\infty(M,TM) \to C^\infty(M,TM)$ taking two smooth vector-fields $V,W$ to a third smooth vector-field denoted $\nabla_VW$ satisfying the following three properties:

• ($\mathbb{R}$-linearity in $W$) $\nabla_V(cW) = c \nabla_VW$ for any real number $c$,
• ($C^\infty(M,\mathbb{R})$-linearity in $V$) $\nabla_{fV}W = f\nabla_VW$ for any smooth, real-valued function $f$ on $M$, and
• (Leibniz rule for $C^\infty(M,\mathbb{R})$-multiplication in $W$) $\nabla_V(fW) = V(f) + f \nabla_VW$ for any smooth, real-valued function $f$.

Intuitively an affine connection affords us a way of identifying the tangent space at two points $T_pM, T_qM$, but the identification depends on the path taken from $p \to q$. This is the notion of a parallel transport (perhaps someday I will write more on these fundamental things about geometry; but I’ll just assume that the reader is familiar with it for now). Read the rest of this entry »

How to derive the Kerr metric by cheating quite a bit. Part 3

( … continued from Part 2)

3. Deriving the Kerr metric

In this section we show how the Kerr metric may be (in a large part) derived by studying Problem 2.23.

The main result that we rely on is a lemma given in Mars’ 1999 paper (If you actually look at the paper, you’d see that there are some factors of 2 differences in a lot of the statements. These are related to the fact that our definitions of anti-self-dual forms differs by a factor of 2, and that our definitions of the Ernst two-form and the Ernst potential also differ by a factor of 2).

Lemma 3.1
We can define the real-valued function $y$ and $z$ by $-\sigma^{-1} = y + i z$. Then there exists a non-negative real number $B$ such that $B \geq z^2$, and
$\displaystyle (\nabla y)^2 = \frac{y^2 - 2y + B}{M^2(y^2+z^2)}$
$\displaystyle (\nabla z)^2 = \frac{B-z^2}{M^2(y^2+z^2)}$

How to derive the Kerr metric by cheating quite a bit. Part 2

(… continued from Part 1)

The anti-self-dual fields and complexification

For ease of algebraic manipulations, often we consider the anti-self-dual versions of two-forms. Observe that on a four-dimensional Lorentzian manifold, the Hodge star operator takes two-forms to two-forms, and squares to -1. This implies that its eigenvalues can only be $\pm i$. So we complexify our geometry by $\otimes_\mathbb{R}\mathbb{C}$ linearly (so in particular $(X+iY)^2 = X^2 + 2i g(X,Y) - Y^2$ and not the Hermitian product). It is clear that (via a little bit of linear algebra) that the space of two-forms $\Lambda^2T^*M$ splits after complexification

Equation 2.10
$\Lambda^2T^*M\otimes_\mathbb{R}\mathbb{C} = \Lambda_+ \oplus\Lambda_-$

where $\Lambda_\pm$ are spaces of complex-valued two-forms that have eigenvalues $\pm i$ under * respectively. It is also clear that there is a natural isomorphism from $\Lambda^2T^*M$ to each of $\Lambda_\pm$ (they all have real dimension 6). Read the rest of this entry »

How to derive the Kerr metric by cheating quite a bit. Part 1

This came from a lecture I gave to MAT 451 at Princeton University on April 23, 2009. I have originally written this up as a LaTeX document; since I won’t be publishing this in any conventional way (and indeed, the material covered is rather unconventional), I figure I’ll use this as an experiment for the first posts on this blog. The original is a 10 page paper, which is why I am splitting this into several installments.

MAT 451 is a senior level mathematics course in which the instructor has great leeway in deciding what to teach. This year my thesis advisor was in charge, and focused the discussion on mathematical aspects of general relativity. This first post will, therefore, be rather on the technical side: the reader is assumed to have familiarity with basic pseudo-Riemannian geometry and with various aspects of general relativity. I will, however, be happy to answer any questions left in the comments.

The Kerr metric I refer to is, of course, the rotating black hole solution in Einstein’s theory of general relativity. For a brief history surrounding its discovery, see Dautcourt’s survey “Race for the Kerr field”.

1. Introduction and the first ansatz

In this note we give a heuristic derivation of the Kerr metric, in a way quite significantly different from the classical methods. This is in no way a formal write-up, so for a more rigorous derivation, and for references, please see the wonderful article by Roberto Bergamini and Stefano Viaggiu, “A novel derivation for Kerr metric in Papapetrou gauge,” Class. Quantum Grav. 21 4567–4573 (2004). The method described herein is inspired by Marc Mars’ paper “A spacetime characterization of the Kerr metric,” Class. Quantum Grav. 16 2507–2523 (1999), and also by my 2009 PhD dissertation. Read the rest of this entry »