Spinor Info

Earlier today, I noticed something really strange. A lamp was radiating darkness. Or so it appeared.

Of course there was a mundane explanation. Now that the Sun is lower in the sky and the linden tree in front of our kitchen lost many of its leaves already, intense sunlight was reflecting off the hardwood floor in our dining area.

Still, it was an uncanny sight.

I live in a condominium townhouse. We’ve been living here for 25 years. We like the place.

Our unit, in particular, is the middle unit in a three-unit block. The construction is reasonably sound: proper foundations, cinderblock firewalls between the units, woodframe construction within, pretty run-of-the-mill by early 1980s North American standards. We have no major complaints.

Except that… for the past several years, every so often the house wobbled a bit. Almost imperceptibly, but still. At first, I thought it was a minor earthquake (not uncommon in this region because it is still subject to isostatic rebound from the last ice age; in fact we did live through a couple of notable earthquakes since we moved in here.) But no, it was no earthquake.

I thought perhaps it was related to the downtown light rail tunnel construction? But no, the LRT tunnels are quite some ways from here and in any case, that part of the construction has been finished long ago.

But then what the bleep is it? Could I be just imagining things?

Our phones have very sensitive acceleration sensors. Not for the first time, I managed to capture one of these events. A little earlier this afternoon, I heard the woodframe audibly creak as the house began to move again. I grabbed my phone and turned on a piece of software that samples the acceleration sensor at a reasonably high rate, about 200 times a second. Here is the result of the first few seconds of sampling:

The sinusoidal signal is unmistakably there, confirmed by a quick Fourier-analysis to be a signal just above 3 Hz in frequency:

Like Sheldon Cooper in The Big Bang Theory, I can claim that no, I am not crazy, and in this case not because my mother had me tested but because my phone’s acceleration sensor confirms my perception: Something indeed wobbles the house a little, enough to register on my phone’s acceleration sensor, measuring a peak-to-peak amplitude of roughly 0.05 m/s² (the vertical axis in the first graph is in g-units.) That wobble is certainly not enough to cause damage, but it is, I admit, a bit unnerving.

So what is going on here? A neighbor engaging in some, ahem, vigorous activity? Our current neighbors are somewhat noisier than prior residents, occasionally training their respective herds of pygmy elephants to run up and down the stairs (or whatever it is that they are doing). But no, the events are just too brief in duration and too regular. Underground work, perhaps a secret hideout for the staff of the nearby Chinese embassy? Speaking of which, I admit I even thought that this ~3 Hz signal might be related to the reported cases of illness by embassy staff at several embassies around the world, but I just don’t see the connection: even if those cases are real and have an underlying common cause (as opposed to just mere random coincidences) it’s hard to see how a 3 Hz vibration can have anything to do with them.

OK, so I have a pretty good idea of what this thing isn’t, but then, what the bleepety-bleep is it?

We have a new manuscript on arXiv. Its title might raise some eyebrows: Algebraic wave-optical description of a quadrupole gravitational lens.

Say what? Algebra? Wave optics? Yes. It means that in this particular case, namely a gravitational lens that is described as a gravitational monopole with a quadrupole correction, we were able to find a closed form description that does not rely on numerical integration, especially no numerical integration of a rapidly oscillating function.

Key to this solution is a quartic equation. Quartic equations were first solved algebraically back in the 16th century by Italian mathematicians. The formal solution is usually considered to be of little practical value, as it entails cumbersome algebra, and polynomial equations can be routinely and efficiently solved using numerical methods.

But in this case… The amazing thing is that the algebraic solution reveals so much about the physics itself!

Take this figure from our paper, for instance:

On the left is light projected by the gravitational lens, its so-called point-spread function (PSF) which tells us how light from a point source is distributed on an imaginary projection screen by the lens. On the right? Why, that’s the discriminant of the quartic equation

$$ x^4-2\eta\sin\mu \, x^3+\big(\eta^2-1\big)x^2+\eta\sin\mu \, x+{\textstyle\frac{1}{4}}\sin^2\mu=0, $$

in a plane characterized by polar coordinates \((\eta,\tfrac{1}{2}\mu)\), that is, \(\eta\) as a radial coordinate and \(\tfrac{1}{2}\mu \) as an azimuthal angle. When the discriminant is positive, the equation is expected to have four real (or four complex) roots; everywhere else, it’s a mix of real and imaginary roots. This direct connection between the algebra and the lensing phenomenon is unexpected and beautiful.

The full set of real roots of this equation can be shown in the form of an animation:

Of course one must read the paper in order for this animation to make sense, but I think it’s beautiful.

How good is this quartic solution? It is uncannily accurate. Here is a comparison of the PSF computed using the quartic solution and also using numerical integration, as well as some enlarged details from the so-called caustic boundary:

It’s only in the immediate vicinity of the caustic boundary that the quartic solution becomes less than accurate.

We can also use the quartic solution to simulate images seen through a telescope (i.e., the Einstein ring, or what survives of it, that would appear around a gravitational lens when we looked at the lens through a telescope with a point source of light situated behind the lens.) We can see again that it’s only in the vicinity of the caustic boundary that the quartic solution produces artifacts instead of accurately reproducing it when spots of light widen into arcs:

This paper was so much joy to write! Also, for the first time in my life, this paper gave us a legitimate, non-pretentious reason to cite something from the 16th century: Cardano’s 1545 treatise in which the quartic solution (as well as the cubic) are introduced, together with discussion on the meaning of taking the square root of negative numbers.