# Knight in November video

I’ve recently released a new video for the song Knight in November off of the 2012 Agapanthus album Smug. I wrote the lyrics and music in the early 90s and recorded an ambient version of it circa 2002. This version was titled Night in November (not to be confused with the 1994 play by the same name) and also released on Smug. The original lyrics were about one particularly moody journey to Mount Hamilton’s Lick Observatory with a good friend. Driving up to Mount Hamilton was a frequent midnight pilgrimage in my youth while growing up in San Jose.

I recorded a heavier variant of the original song, now called Knight in November, in the summer of 2012. The lyrical verses sound like they just repeat the same phrase five times. But actually they form a set of (mostly) nonsensical homophones. Check out the video below to appreciate the effect. The tune can be downloaded for free from soundcloud (note the version on the album and soundcloud is a slightly different mix in the video).

Also below is the audio for the ambient piece, Night in November, from soundcloud. Hope you enjoy.

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# Thus spake Rankine: “U” for potential energy

Why is the symbol U often used to represent potential energy? This question came up recently in a faculty discussion.

Before you get too excited, this post won’t resolve the issue. However, the earliest use of the letter “U” for potential energy was in a paper from 1853 by William John Macquorn Rankine: “On the general law of the transformation of energy,” Proceedings of the Philosophical Society of Glasgow, vol. 3, no. 5, pages 276-280; reprinted in: (1) Philosophical Magazine, series 4, vol. 5, no. 30, pages 106-117 (February 1853). The wikipedia article on potential energy indicates that his article is the first reference to a modern sense of potential energy. Below is the original text (yellow highlighting mine). I think we will have to ask Bill Rankine why he chose the symbol “U”:

“Let U denote this potential energy.”
Thus spake Rankine.

# The field near a conductor

This post is directed primarily at physics students and instructors and stems from discussions with my colleague Prof. Matt Moelter at Cal Poly, SLO. In introductory electrostatics there is a standard result involving the electric field near conducting and non-conducting surfaces that confuses many students.

Near a non-conducting sheet of charge with charge density $\sigma$, a straightforward application of gauss’s law gives the result

$\vec{E}=\frac{\sigma}{2\epsilon_0}\hat{n}\ \ \ \ (1)$

While, near the surface of a conductor with charge density $\sigma$, again, an application of gauss’s law gives the result

$\vec{E}=\frac{\sigma}{\epsilon_0}\hat{n}\ \ \ \ (2)$

The latter result comes about because the electric field inside a conductor in electrostatic equilibrium is zero, killing off the flux contribution of the gaussian pillbox inside the conductor. In the case of the sheet of charge, this same side of the pillbox distinctly contributed to the flux. Both methods are applied locally to small patches of their respective systems.

Although the two equations are derived from the same methods, they mean different things — and their superficial resemblance within factors of two can cause conceptual problems.

In Equation (1) the relationship between $\sigma$ and $\vec{E}$ is causal. That is, the electric field is due directly from the source charge density in question. It does not represent the field due to all sources in the problem, only the lone contribution from that local $\sigma$.

In Equation (2) the relationship between $\sigma$ and $\vec{E}$ is not a simple causal one, rather it expresses self-consistentancy, discussed more below. Here the electric field represents the net field outside of the conductor near the charge density in question. In other words, it automatically includes both the contribution from the local patch itself and the contributions from all other sources. It is has already added up all the contributions from all other sources in the space around it (this could, in some cases, include sources you weren’t aware of!).

How did this happen? First, in contrast to the sheet of charge where the charges are fixed in space, the charges in a conductor are mobile. They aren’t allowed to move while doing the “statics” part of electrostatics, but they are allowed to move in some transient sense to quickly facilitate a steady state. In steady state, the charges have all moved to the surfaces and we can speak of an electrostatic surface distribution on the conductor. This charge mobility always arranges the surface distributions to ensure $\vec{E}=0$ inside the conductor in electrostatic equilibrium. This is easy enough to implement mathematically, but gives rise to the subtle state of affairs encountered above. The $\sigma$ on the conductor is responding to the electric fields generated by the presence of other charges in the system, but those other charges in the system are, in turn, responding to the local $\sigma$ in question. Equation (2) then represents a statement of self-consistency, and it breaks the cycle using the power of gauss’s law. As a side note, the electric displacement vector, $\vec{D}$, plays a similar role of breaking the endless self-consistency cycle of polarization and electric fields in symmetric dielectric systems.

Let’s look at some examples.

Example 1:
Consider a large conducting plate versus large non-conducting sheet of charge. Each surface is of area A. The conductor has total charge Q, as does the non-conducting sheet. Find the electric field of each system. The result will be that the fields are the same for the conductor and non-conductor, but how can this be reconciled with Equation (1) and (2) which, at a glance, seem to give very different answers? See the figure below:

For the non-conducting sheet, as shown in Figure (B) above, the electric field due to the source charge is given by Equation (1)

$\vec{E}_{nc}=\frac{\sigma_{nc}}{2\epsilon_0}\hat{n}$

where

$\sigma_{nc}\equiv\sigma=Q/A$

(“nc” for non-conducting) and $\hat{n}=+\hat{z}$ above the positive surface and $\hat{n}=-\hat{z}$ below it.

Now, in the case of the conductor, shown in Figure (A), Equation (2) tells us the net value of the field outside the conductor. This net value is expressed, remarkably, only in terms of the local charge density; but remember, for a conductor, the local charge density contains information about the entire set of sources in the space. At a glance, it seems the electric field might be twice the value of the non-conducting sheet. But no! This is because the charge density will be different than the non-conducting case. For the conductor, the charge responds to the presence of the other charges and spreads out uniformly over both the top and bottom surface; this ensures $\vec{E}=0$ inside the conductor. In this context, it is worth point out that there are no infinitely thin conductors. Infinitely thin sheets of charge are fine, but not conductors. There are always two faces to a thin conducting surface and the surface charge density must be (at least tacitly) specified on each. Even if a problem uses language that implies the conducting surface is infinitely thin, it can’t be.

For example, the following Figure for an “infinitely thin conducting surface with charge density $\sigma$“, which then applies Equation (2) to the setup to determine the field, makes no sense:

This application of Equation (2) cannot be reconciled with Equation (1). We can’t have it both ways. An “infinitely thin conductor” isn’t a conductor at all and should reduce to Equation (1). To be a conductor, even a thin one, there needs to be (at least implicitly) two surfaces and a material medium we call “the conductor” that is independent of the charge.

Back to the example.

If the charge Q is spread out uniformly over both sides of the conductor in Figure (A), the charge density for the conductor is then

$\sigma_c=Q/2A=\sigma_{nc}/2=\sigma/2$

(“c” for conducting). The factor of 2 comes in because each face has area A and the charge spreads evenly across both. Equation (2) now tells us what the field outside the conductor is. This isn’t just for the one face, but includes the net contributions from all sources

$\vec{E}_{c}=\frac{\sigma_c}{\epsilon_0}\hat{n}=\frac{\sigma_{nc}}{2\epsilon_0}\hat{n}=\vec{E}_{nc}$.

That is, the net field is the same for each case,

$\vec{E}_{c}=\vec{E}_{nc}$.

Even though Equations (1) and (2) might seem superficially inconsistent with each other for this situation, they give the same answer, although for different reasons. Equation (1) gives the electric field that results directly from $\sigma$ alone. Equation (2) gives a self consistent net field outside the conductor, which uses information contained in the local charge density. The key is understanding that the surface charge density used for the sheet of charge and the conductor are different in each case. In the case of a charged sheet, we have the freedom to declare a surface with a fixed, unchanging charge density. With a conductor, we have less, if any, control over what the charges do once we place them on the surfaces.

It is worth noting that each individual surface of charge on the conductor has a causal contribution to the field still given by Equation (1), but only once the surface densities have been determined — with one important footnote. The net field in each region can be determined by adding up all the (shown) individual contributions in superposition only if the charges shown are the only charges in the problem and were allowed to relax into this equilibrium state due to the charges explicitly shown. This last point will be illustrated in an example at the end of this post. It turns out that you can’t just declare arbitrary charge distributions on conductors and expect those same charges you placed to be solely responsible for it. There may be “hidden sources” if you insist on keeping your favorite arbitrary distribution on a conductor. If you do, you must also account for those contributions if you want to determine the net field by superposition. However, all is not lost: amazingly, Equation (2) still accounts for those hidden sources for the net field! With Equation (2) you don’t need to know the individual fields from all sources in order to determine the net field. The local charge density on the conductor already includes this information!

Example 2:
Compare the field between a parallel plate capacitor with thin conducting sheets each having charge $\pm Q$ and area A with the field between two non-conducting sheets of charge with charge $\pm Q$ and area A. This situation is a standard textbook problem and forms the template for virtually all introductory capacitor systems. The result is that the field between the conducting plates are the same as the field between the non-conducting charge sheets, as shown in the figure below. But how can this be reconciled with Equations (1) and (2)? We use a treatment similar to those in Example 1.

Between the two non-conducting sheets, as shown in Figure (D), the top positive sheet has a field given by Equation (1), pointing down (call this the $-\hat{z}$ direction) . The bottom negative sheet also has a field given by Equation (1) and it also points down. The charge density on the positive surface is given by $\sigma=Q/A$. We superimpose the two fields to get the net result

$\vec{E}=\vec{E}_{1}+\vec{E}_2=\frac{+\sigma}{2\epsilon_0}(-\hat{z})+\frac{-\sigma}{2\epsilon_0}(+\hat{z})+=-\frac{\sigma}{\epsilon_0}(\hat{z})$.

Above the top positive non-conducting sheet the field points up due to the top non-conducting sheet and down from the negative non-conducting sheet. Using Equation (1) they have equal magnitude, thus the fields cancel in this region after superposition. The fields cancel in a similar fasshion below the bottom non-conducting sheet.

Unfortunately, the setup for the conductor, shown in Figure (C), is framed in an apparently ambiguous way. However, this kind of language is typical in textbooks. Where is this charge residing exactly? If this is not interpreted carefully, it can lead to inconsistencies like those of the “infinite thin conductor” above. The first thing to appreciate is that, unlike the nailed down charge on the non-conducting sheets, the charge densities on the parallel conducting plates are necessarily the result of responding to each other. The term “capacitor” also implies that we start with neutral conductors and do work bringing charge from one, leaving the other with an equal but opposite charge deficit. Next, we recognize even thin conducting sheets have two sides. That is, the top sheet has a top and bottom and the bottom conducting sheet also has a top and bottom. If the conducting plates have equal and opposite charges, and those charges are responding to each other. They will be attracted to each other and thus reside on the faces that are pointed at each other. The outer faces will contain no charge at all. That is, the $\sigma=Q/A$ from the top plate is on that plate’s bottom surface with none on the top surface. Notice, unlike Example 1, the conductor has the same charge density as its non-conducting counterpart. Similar for the bottom plate but with the signs reversed. A quick application of gauss’s law can also demonstrate the same conclusion.

With this in mind, we are left with a little puzzle. Since we know the charge densities, do we jump right to the answer using Equation (2)? Or do we now worry about the individual contributions of each plate using Equation (1) and superimpose them to get the net field? The choice is yours. The easiest path is to just use Equation (2) and write down the results in each region. Above and below all the plates, $\sigma=0$ so $\vec{E}=0$; again, Equation (2) has already done the superposition of the individual plates for us. In the middle, we can use either plate (but not both added…remember, this isn’t superposition!). If we used the top plate, we would get

$\vec{E}=\frac{\sigma}{\epsilon_0}(-\hat{z})=-\frac{\sigma}{\epsilon_0}\hat{z}$

and if we used the bottom plate alone, we would get

$\vec{E}=\frac{-\sigma}{\epsilon_0}\hat{z}=-\frac{\sigma}{\epsilon_0}\hat{z}$.

They both give the same individual result, which is the same result as the non-conducting sheet case above where we added individual contributions.

If were were asked “what is the force of the top plate on the bottom plate?” we actually do need to know the field due to the charge on the single top plate alone and apply it to the charge on the second plate. In this case, we are not just interested in the total field due to all charges in the space as given by Equation (2). In this case, the field due to the single top plate would indeed be given by Equation (1), as would the field due to the single bottom plate. We could then go on to superimpose those fields in each region to obtain the same result. That is, once the charge distributions are established, we can substitute the sheets of non-conducting charge in place of the conducting plates and use those field configurations in future calculations of energy, force, etc.

However, not all charge distributions for the conductor are the same. A strange consequence of all this is that, despite the fact that Example 1 gave us one kind of conductor configuration that was equivalent to single non-conducting sheet, this same conductor can’t be just transported in and made into a capacitor as shown in the next figure:

On a conductor, we simply don’t have the freedom to invent a charge distribution, declare “this is a parallel plate capacitor,” and then assume the charges are consistent with that assertion. A charge configuration like Figure (E) isn’t a parallel plate capacitor in the usual parlance, although the capacitance of such a system could certainly be calculated. If we were to apply Equation (1) to each surface and superimpose them in each region, we might come to the conclusion that it had the same field as a parallel plate capacitor and conclude that Figure (E) was incorrect, particularly in the regions above and below the plates. However, Equation (2) tells us that the field in the region above the plates and below them cannot be zero despite what a quick application of Equation (1) might make us believe. What this tells us is that there must unseen sources in the space, off stage, that are facilitating the ongoing maintenance of this configuration. In other words, charges on conducting plates would not configure themselves in such a away unless there were other influences than the charges shown. If we just invent a charge distribution and impose it onto a conductor, we must be prepared to justify it via other sources, applied potentials, external fields, and so on.

So, even though plate (5) in Figure (E) was shown to be the same as a single non-conducting plate, we can’t just make substitutions like those in this figure. We can do this with sheets of charge, but not with other conductors. Yes, the configuration in Figure (E) is physically possible, it just isn’t the same as a parallel plate capacitor, even though each element analyzed in isolation makes it seem like it would be the same.

In short, Equations (1) and (2) are very different kinds of expressions. Equation (1) is a causal one that can be used in conjunction with the superposition principle: one is calculating a single electric field due to some source charge density. Equation (2) is more subtle and is a statement of self-consistency with the assumptions of a conductor in equilibrium. An application of Equation (2) for a conductor gives the net field due to all sources, not just the field do to the conducting patch with charge density sigma: it comes “pre-superimposed” for you.

# Quick 4-dimensional visualization

How can you visualize a 4th spatial dimension? There has been much written and discussed on this topic; I won’t pretend that this post will compete with the vast resources available online. However, I do feel that I can contribute one small visualization trick for hypercubes that, for some reason, has not been emphasized very much elsewhere (although it is out there), which helped me get a foothold into the situation.

My first exposure as a kid to the topic of visualizing higher dimensions was given by Carl Sagan on the original Cosmos. In it, he introduces a hypercube called a tesseract:

While Cosmos is an inspirational introduction, it isn’t very complete. Still, there are many great resources on the web to help appreciate and understand the tesseract on many levels from rotations to inversions and beyond. They are part of a larger class of very cool objects known as polytopes. You are one google search away from vast resources on this topic. I won’t even bother compiling links.

What I hope to accomplish is to give you an intellectual foothold into the visualization, which will help considerably as you delve further into the topic.

Below is an image of a tesseract taken from the Wikipedia page on tesseracts

In what sense is this object a hypercube? Well, strictly speaking, this object is not a tesseract or hypercube. Technically, it is a two-dimensional projection (i.e. it is on this web page) of a three-dimensional shadow (the wire frame object if it were in 3D) of a 4D hypercube.

But how exactly can this object help us see into a 4th spatial dimension? Here is a visualization trick I’ve found most helpful for me:

Let’s start with something familiar. I can draw two parallelograms, one larger than the other, then connect the corresponding vertices. One’s mind will quickly interpret this as a cube as viewed from some angle, although it is just a two dimensional thing on a page. Your mind naturally views the (slightly) smaller parallelogram as being the same actual size as the larger one. It just looks smaller because we interpret it as farther away, thanks to perspective. Furthermore, all the angles, although drawn otherwise, are interpreted as right angles. The description makes it sound more complex than it is; it is just the representation of an ordinary cube viewed from some angle outside the page:

In the drawing, the parallelograms are almost the same size, so it is easy to flip back and forth between which one is the “front” face and which is the “back” face, generating weird distortions if it is viewed “incorrectly.”

Now, I rotate the cube so we are looking directly down one face. Think of this drawing as looking down a crude wirefame corridor:

However, on the page it is really just two nested squares with connected vertices. Still, one’s brain fills in the three dimensional details pretty naturally. Viewed this way, the smaller square is just further away and the angles are all right angles. If the smaller square were made smaller, even going to a point, you could imagine that the end of the corridor was just very far away.

The tesseract projection really is not really much different:

The visualization tool to remember is that the smaller cube only looks smaller because of perspective: the two cubes are actually the same size but the smaller cube only looks smaller because it is farther away. Further away in what direction? Into a 4th spatial dimension! When looking at the tesseract projection, think of it as looking “down” a kind of wireframe corridor directed such that the farthest point is actually at the mutual center of the cubes. This is the same sense that an ordinary long corridor drawn in two dimensions would have the far point (at infinity) located at the center of the squares. This mutual center is then interpreted as pointing in a direction not in ordinary three dimensional space; indeed, all six faces of the larger cube look “down” this corridor toward the other end. If you had such a hypercube in your living room, each of the six faces would act as a separate corridor directed towards the far point in a fourth spatial dimension. If your friend walked into the cube and continued down the corridor, they would not exit on the other side of the cube in your living room but rather would get smaller and smaller walking toward the center of the cube.

If you were the one doing the walking, it would be just like walking down a corridor into another room, albeit one that was entirely embedded — from all directions — within another one in three dimensions.

This is basically the idea behind Dr. Who’s tardis, as explained by the Doctor himself (although in his usual curt and opaque way):

You could think of the outer cube a crude 2 x 2 x 2 meter exterior to a tardis. The inner cube might be a 2 x 2 x 2 meter room inside the tardis (the same shape as the outside) 2 meters away into the 4th dimension. However, the tardis isn’t a mere hypercube. It has rooms inside that are bigger than the outside of the tardis. But to get them to fit inside the outer cube, you just put them farther away into the extra dimension. That is, you can visualize the inner cube as being a 100 x 100 x 100 meter room inside a 2 x 2 x 2 meter exterior box — except imagine you are 1000 meters away looking “down” the corridor of the 4th dimension, so the giant room looks small and thus fits fine into the exterior. This is exactly the point the Doctor is trying to make in the clip.

This idea was also a part of the plot of Stranger in a Strange Land by Heinlein. Valentine Michael Smith can make things vanish into a fourth dimension. The effect, as viewed by all observers in our own three dimensions, is to see the object get smaller and smaller from all angles until it vanishes. This is akin to walking down the corridor of the tesseract towards the center. The object appears to get smaller only because it is further away in this other direction outside of our usual three.

In my opinion, visualizing the tesseract as looking down a corridor into another spatial dimension with added perspective is the best first step in appreciating higher dimensional thinking. Here is a neat looking game that emphasizes the perspective approach and gives some practical practice with these ideas.

# Sexual harassment in NYC measuring mental illness?

An upsetting video (SFW) by Rob Bliss shows a woman being repeatedly verbally harassed as she walks the streets of New York City. The video is an edited sample of a 10 hour experiment. The actress, Shoshana B. Roberts, and Bliss were working on a project for Hollaback, an advocacy group trying to end street harassment. According to Bliss, who used a hidden camera and discreetly walked several paces in front of Roberts, the actress was harassed about 100 times during her 10 hour walk around the City (not all are shown in the video). In the video, one can clearly see Roberts simply walking and looking forward, minding her own business, not engaging or inviting conversation or interaction. Yet various men constantly vie for her attention, sometimes very aggressively, using a spectrum of nearly universally inappropriate strategies. This included many expressions like unsolicited neutral comments, catcalls, inappropriate remarks (usually about her looks), aggressive talking, shouting, following, and so on. The Washington Post has a good article summarizing the project and players. Here is the original video

A similar project was done on The Daily Show by comedian and correspondent Jessica Williams

I personally found the videos very disturbing and significant on many levels. They have helped me appreciate the issues women face while just walking from point A to B. Yes, as a man I have to navigate the occasional nuisance while walking along the street, but nothing like those shown in the videos. If these projects represent typical experiences for women, this represents a serious social problem. Even if it is atypical, a notion these videos do not support (the women in the videos seem “typical” — for example, no one is a recognizable popular celebrity whose presence might be especially socially disruptive), it is still upsetting. No one should need to experience interactions like that just walking around (including celebrities).

While emotionally impactful, it is important to realize the videos in no way represent a scientific experiment. There is no baseline measurement or control group. However, the video below might be a pretty decent effort as a control experiment:

In all seriousness, despite a lack of scientific rigor, I am willing to accept that the videos are broadly representative of the experiences many women have walking around. They demonstrate to me that the harassment is real, unsolicited, annoying, and occasionally terrifying. No one should have to put up with behavior like that and it is a terrible thing to be subjected to. We, as a society, need to figure out how to understand and manage this.

Other than the fact that all the harassers were men, one rather conspicuous thing jumped out at me while watching these videos: the men in the video seemed to be mentally and/or emotionally ill individuals. This in no way justifies their behavior and the harassment is clearly real. But seriously, what kind of person just starts randomly talking to another person about ANYTHING as they walk down the street, with no other context, demanding all of their attention? Someone who is mentally ill, practically by definition. Sure, talking to someone randomly on the street is occasionally appropriate. The annoying sales person can be given a legitimate excuse, even if frustrating. A panhandler is perhaps also in a special category (panhandling is not necessarily acceptable, but it is understood to a degree). Yes, the occasional “hello” or “have a good day” to a stranger might work when it is natural — which it usually isn’t while just walking down the street minding your own business. That they were mostly non-white men in Bliss’s video is likely a selection bias on the part of the editor. That they were men shows a clear testosterone connection.

In the videos, the perpetrators seem to be men who lack self control, who genuinely can’t manage their own impulses, physical and verbal, who don’t understand social conventions and basic etiquette. Self evidently, they are men who lack empathy or understanding of another person’s physical and emotional space. It is as if they have some kind of aggressive nervous tick they can’t control. The adult human mind is full of noise; there are impulses coming from many sectors of the psyche. However, most people, emotionally and mentally healthy adults, men and women alike of all walks of life, learn how to manage those internal impulses. Adults who can’t do that usually have some kind of brain damage, perhaps to the frontal lobe where impulse control is seated, or are not emotionally or mentally healthy in some other way.

A back-of-the-envelope calculation is worth doing. How many people does one expect to be in “interaction range” during a 10 hour excursion in New York City and what percentage is the observed 100 harassments of that number? This will help set the scale for the fraction of individuals harassing these women.

1) The population density of New York City: 26403 people per square mile ~ 0.01 people per square meter ~ 1 person per 100 square meters

2) 100 square meters might be regarded as a sensible “interactions zone” around a typical person walking around: +/- 5 meters in each direction

3) The typical walking speed of a person is around: 1.5 m/s

4) Imagine breaking New York City into a grid of 10 x 10 meter squares

5) The time to transverse 10 meters and move to one unique 10 square meter cell: about 6.67 seconds

6) There are 3600 seconds in 1 hour, so a 10 hour walk in NYC will sample about 5400 unique people on average in New York

7) If there were 100 harassment events/5400 persons during the walk in the video, this is about a 1.8% or 2% effect

That is, about 2% of the people Roberts interacted with during her excursion with Bliss harassed her to various degrees, violating her personal mental and emotional state. Again, this obviously isn’t scientific, but rather just a back-of-the-envelope. If I had to guess, I would say I underestimated the number of unique people per square meter one encounters on the street during the day in NYC. In other words, 2% is probably high.

If you asked me in advance “what fraction of people in New York City have mental problems involving a pathological lack of self control?” I would likely have guessed something like 10%. So, I could easily believe that the 2% number is looking at a subset of that that group, representing adult men whose mental illness, emotional illness, and excessive lack of self control is particularly aggressive and directed towards women. This 2% number then represents about 4% of the male population. This, I believe, is what these videos are measuring: a mental health problem specific to some men. It also explains the relative uniformity of the distribution across New York City, a point emphasized in William’s video.

The good news is, if it is a specific kind of mental health problem intrinsic to some population of men, and not some completely ill-defined problem, then perhaps this points to a strategy to help organizations like Hollaback end the awful street harassment many women experience.

Let me clarify that:
I’m in no way claiming that all harassment directed toward women across all social and cultural modes is due to mental illness alone; the causes of harassment are surely complex, perhaps involving trained dysfunctional socialized behaviors from early childhood, personality disorders, and other extensions of “healthy” mental states — but which are not a form of mental illness per se. I also hope that I have not given the impression that am I rationalizing away the effect or removing the element of personal responsibility from perpetrators. I’m merely proposing that one contribution to the problem — particularly in the context of aggressive street harassment of the sort shown in the video — may be a particular form of mental illness. I’m suggesting that scientifically exploring this contribution, by trained professionals, may be worthwhile.

# Newton’s First Law is not a special case of his Second Law

When teaching introductory mechanics in physics, it is common to teach Newton’s first law of motion (N1) as a special case of the second (N2). In casual classroom lore, N1 addresses the branch of mechanics known as statics (zero acceleration) while N2 addresses dynamics (nonzero acceleration). However, without getting deep into concepts associated with Special and General Relativity, I claim this is not the most natural or effective interpretation of Newton’s first two laws.

N1 is the law of inertia. Historically, it was asserted as a formal launching point for Newton’s other arguments, clarifying misconceptions left over from the time of Aristotle. N1 is a pithy restatement of the principles established by Galileo, principles Newton was keenly aware of. Newton’s original language from the Latin can be translated roughly as “Law I: Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.” This is attempting to address the question of what “a natural state” of motion is. According to N1, a natural state for an object is not merely being at rest, as Aristotle would have us believe, but rather uniform motion (of which “at rest” is a special case). N1 claims that an object changes its natural state when acted upon by external forces.

N2 then goes on the clarify this point. N2, in Newton’s language as translated from the Latin was stated as “Law II: The alteration of motion is ever proportional to the motive force impress’d; and is made in the direction of the right line in which that force is impress’s.” In modern language, we would say that the net force acting on an object is equal to its mass times its acceleration, or

$\vec{F}_{\rm net}=m\vec{a}$

In the typical introductory classroom, problems involving N2 would be considered dynamics problems (forgetting about torques for a moment). A net force generates accelerations.

To recover statics, where systems are in equilibrium (again, modulo torques), students and professors of physics frequently then back-substitute from here and say something like: in the case where $\vec{a}=0$ clearly we recover N1, which can now be stated something like:

$\vec{a}=0$
if and only if
$\vec{F}_{\rm net}=0$

This latter assertion certainly looks like the mathematical formulation of Newton’s phrasing of N1. Moreover, it seemed to follow from the logic of N2 so, ergo, “N1 is a special case of N2.”

But this is all a bit too ham-fisted for my tastes. Never mind the nonsensical logic of why someone as brilliant as Newton would start his three laws of motion with special case of the second. That alone should give one pause. Moreover, Newton’s original language of the laws of motion is antiquated and does’t illuminate the important modern understanding very well. Although he was brilliant, we definitely know more physics now than Newton and understand his own laws at a deeper level than he did. For example, we have an appreciation for Electricity and Magnetism, Special Relativity, and General Relativity, all of which force one to clearly articulate Newton’s Laws at every turn, sometimes overthrowing them outright. This has forced physicists over the past 150 yeas to be very careful how the laws are framed and interpreted in modern terms.

So why isn’t N1 really a special case of N2?

I first gained an appreciation for why N1 is not best thought of as a special case of N2 when viewing the famous educational film called Frames of Reference by Hume and Donald Ivey (below), which I use in my Modern Physics classes when setting up relative motion and frames of reference. Then it really hit home later while teaching a course specifically about Special Relativity from a book by the same name by T.M by Helliwell.

A key modern function of N1 is that it defines inertial frames. Although Newton himself never really addresses inertial frames in his work, this modern interpretation is of central importance in modern physics. Without this way of interpreting it, N1 does functionally become a special case of N2 if you treat pseudoforces as actual forces. That is, if “ma” and the frame kinematics are considered forces. In such a world, N1 is redundent and there really are only two laws of motion (N2 and the third law, N3, which we aren’t discussing here). So why don’t we frame Newton’s laws this way. Why have N1 at all? One might be able to get away with this kind of thinking in a civil engineering class, but forces are very specific things in physics and “ma” is not amongst them.

So why is “ma” not a force and why do we care about defining inertial frames?

Basically an inertial frame is any frame where the first law is obeyed. This might sound circular, but it isn’t. I’ve heard people use the word “just” in that first point: “an inertial frame is just any frame where the first law is obeyed.” What’s the big deal? To appreciate the nuance a bit, the modern logic of N1 goes something like this:

if
$\vec{F}_{\rm net}=0$
and
$\vec{a}=0$
then you are in an inertial frame.

Note, this is NOT the same as a special case of N2 as stated above in the “if and only if” phrasing

$\vec{a}=0$
if and only if
$\vec{F}_{\rm net}=0$

That is, N1 is a one-way if-statement that provides a clear test for determining if your frame is inertial. The way you do this is you systematically control all the forces acting on an object and balance them, ensuring that the net force is zero. A very important aspect of this is that the catalog of what constitutes a force must be well defined. Anything called a “force” must be linked back to the four fundamental forces of nature and constitute a direct push or a pull by one of those forces. Once you have actively balanced all the forces, getting a net force of zero, you then experimentally determine if the acceleration is zero. If so, you are in an inertial frame. Note, as I’ve stated before, this does not include any fancier extensions of inertial frames having to do with the Principle of Equivalence. For now, just consider the simpler version of N1.

With this modern logic, you can also use modus ponens and assert that if your system is non-inertial, then you have can have either i) accelerations in the presence of apparently balanced forces or ii) apparently uniform motion in the presence of unbalanced forces.

The reason for determining if your frame is inertial or not is that N2, the law that determines the dynamics and statics for new systems you care about, is only valid in inertial frames. The catch is that one must use the same criteria for what constitutes a “force” that was used to test N1. That is, all forces must be linked back to the four fundamental forces of nature and constitute a direct push or a pull by one of those forces.

Let’s say you have determined you are in an inertial frame within the tolerances of your experiments. You can then go on to apply N2 to a variety of problems and assert the full powerful “if and only if” logic between forces and accelerations in the presence of any new forces and accelerations. This now allows you to solve both statics (no acceleration) and dynamics (acceleration not equal to zero) problems in a responsible and systematic way. I assert both statics and dynamics are special cases of N2. If you give up on N1 and treat it merely as a special case of N2 and further insist that statics is all N1, this worldview can be accommodated at a price. In this case, statics and dynamics cannot be clearly distinguished. You haven’t used any metric to determine if your frame is inertial. If you are in a non-inertial frame but insist on using N2, you will be forced to introduce pseudoforces. These are “forces” that cannot be linked back to pushes and pulls associated with the four fundamental forces of nature. Although it can be occasionally useful to use pseudoforces as if they were real forces, they are physically pathological. For example, every inertial frame will agree on all the forces acting on an object, able to link them back to the same fundamental forces, and thus agree on its state of motion. In contrast, every non-inertial frame will generally require a new set of mysterious and often arbitrary pseudoforces to rationalize the motion. Different non-inertial frames won’t agree on the state of motion and won’t generally agree on whether one is doing statics or dynamics! As mentioned, pseudoforces can be used in calculation, but it is most useful to do so when you actually know a priori that you are in a known non-inertial frame but wish to pretend it is inertial for practical reasons (for example, the rotating earth creates small pseudoforces such as the Coriolis force, the centrifugal force, and the transverse force, all byproducts of pretending the rotating earth is inertial when it really isn’t).

Here’s a simple example that illustrates why it is important not to treat N1 as special case of N2. Say Alice places a box on the ground and it doesn’t accelerate; she analyzes the forces in the frame of the box. The long range gravitational force of the earth on the box pulls down and the normal (contact) force of the surface of the ground on the box pushes up. The normal force and the gravitational force must balance since the box is sitting on the ground not accelerating. OR SO SHE THINKS. The setup said “the ground” not “the earth.” “The ground” is a locally flat surface upon which Alice stands and places objects like boxes. “The earth” is a planet and is a source of the long range gravitational field. You cannot be sure that the force you are attributing to gravity really is from a planet pulling you down or not (indeed, the Principle of Equivalence asserts that one cannot tell, but this is not the key to this puzzle).

Alice has not established that N1 is true in her frame and that she is in an inertial frame. This could cause headaches for her later when she tries to launch spacecraft into orbit. Yes, she thinks she knows all the forces at work on the box, but she hasn’t tested her frame. She really just applied backwards logic on N1 as a special case of N2 and assumed she was in an inertial frame because she observed the acceleration to be zero. This may seem like a “difference without a distinction,” as one of my colleagues put it. Yes, Alice can still do calculations as if the box were in static equilibrium and the acceleration was zero — at least in this particular instance at this moment. However, there is a difference that can indeed come back and bite her if she isn’t more careful.

How? Imagine that Alice was, unbeknownst to her or her ilk, on a large rotating (very light) ringworld (assuming ringworlds were stable and have very little gravity of their own). The inhabitants of the ringworld are unaware they are rotating and believe the rest of the universe is rotating around them (for some reason, they can’t see the other side of the ring). This ringworld frame is non-inertial but, as long as Alice sticks to the surface, it feels just like walking around on a planet. For Bob, an inertial observer outside the ringworld (who has tested N1 directly first), there is only once force on the box: the normal force of the ground that pushes the box towards the center of rotation and keeps the box in circular motion. All other inertial observers will agree with this analysis. This is very clearly a case of applying N2 with accelerations for the inertial observer. The box on the ground is a dynamics problem, not a statics problem. For Alice, who believes she is in an inertial frame by taking N1 to be a special case of N2 (having not tested N1!), she assumes there are two forces keeping the box in static equilibrium — it appears like a statics problem. Is this just a harmless attribution error? If it gives the same essential results, what is the harm? Again, in an engineering class for this one particular box under these conditions, perhaps this is good enough to move on. However, from a physics point of view, it introduces potentially very large problems down the road, both practical and philosophical. The philosophical problem is that Alice has attributed a long range force where non existed, turning “ma” into a force of nature, which is isn’t. That is, the gravity experienced by the ringworld observer is “artificial”: no physical long range force is pulling the box “down.” Indeed “down,” as observed by all inertial observers, is actually “out,” away from the ring. Gravity is a pseudoforce in this context. There has been a violation of what constitutes a “force” for physical systems and an unphysical, ad hoc, “force” had to be introduced to rationalize the observation of what appears to be zero local acceleration. Again, let us forgo any discussions of the Equivalence Principle here where gravity and accelerations can be entwined in funny ways.

This still might seem harmless at first. But image that Alice and her team on the ring fire a rocket upwards normal to the ground trying to exit or orbit their “planet” under the assumption that it is a gravitational body that pulls things down. They would find a curious thing. Rockets cannot leave their “planet” by being fired straight up, no matter how fast. The rockets always fall back and hit the ground and, despite being launched straight up with what seems to be only “gravity” acting on it, yet rocket trajectories always bend systematically in one directly and hit the “planet” again. Insisting the box test was a statics problem with N1 as a special case of N2, they have no explanation for the rocket’s behavior except to invent a new weird horizontal force that only acts on the rocket once launched and depends in weird ways on the rocket’s velocity. There does not seem to be any physical agent to this force and it cannot be attributed to the previously known fundamental forces of nature. There are no obvious sources of this force and it simply is present on an empirical level. In this case, it happens to be a Coriolis force. This, again, might seem an innocent attribution error. Who’s to say their mysterious horizontal forces aren’t “fundamental” for them? But it also implies that every non-inertial frame, every type of ringworld or other non-inertial system, one would have a different set of “fundamental forces” and that they are all valid in their own way. This concept is anathema to what physics is about: trying to unify forces rather than catalog many special cases.

In contrast, you and all other inertial observers, recognize the situation instantly: once the rocket leaves the surface and loses contact with the ringworld floor, no forces act on it anymore, so it moves in a straight line, hitting the far side of the ring. The ring has rotated some amount in the mean time. The “dynamics” the ring observers see during the rocket launch is actually a statics (acceleration equals zero) problem! So Alice and her crew have it all backwards. Their statics problem of the box on the ground is really a dynamics problem and their dynamics problem of launching a rocket off their world is really a statics problem! Since they didn’t bother to sysemtically test N1 and determine if they were in an inertial frame, the very notions of “statics” and “dynamics” is all turned around.

So, in short, a modern interpretation of Newton’s Laws of motional asserts that N1 is not a special case of N2. First establishing that N1 is true and that your fame is inertial is critical in establishing how one interprets the physics of a problem.

# Coldest cubic meter in the universe

My collaborators in CUORE, the Cryogenic Underground Observatory for Rare Events, at the underground Gran Sasso National Laboratory in Assergi, Italy, have recently created (literally) the coldest cubic meter in the universe. For 15 days in September 2014, cryogenic experts in the collaboration were able to hold roughly one contiguous cubic meter of material at about 6 mK (that is, 0.006 degrees above absolute zero, the coldest possible temperature).

At first, a claim like “this is the coldest cubic meter in the [insert spacial scale like city/state/country/world/universe]” may sound like an exaggeration or a headline grabbing ruse. What about deep space? What about ice planets? What about nebulae? What about superconductors? Or cold atom traps? However, the claim is absolutely true in the sense that there are no known natural processes that can reliable create temperatures anywhere near 6 mK over a contiguous cubic meter anywhere in the known universe. Cold atom traps, laser cooling, and other remarkable ultracold technologies are able to get systems of atoms down to the bitter pK scale (a billionth of a degree above absolute zero). However, the key term here is “systems of atoms.” These supercooled systems are indeed tiny collections of atoms in very small spaces, nowhere near a cubic meter. Large, macroscopic superconductors can operate at liquid nitrogen or liquid helium temperatures, but those are very warm compared to what we are talking about here. Even deep space is sitting a at a balmy 2.7 K thanks to the cosmic microwave background radiation (CMBR). Some specialized thermodynamic conditions, such as those found the the Boomerang Nebula, may bring things down to a chilly 300-1000 mK because of the extended expansion of gases in a cloud over long times. The CMB cold spot is only 70 micro-kelvin below the CMBR.

However, the only process capable of reliably bringing a cubic meter vessel down to 6 mK are sentient creatures actively trying to do so. While nature could do it on its own in principle, via some exotic process or ultra-rare thermal fluctuation, the easiest natural path to such cold swaths of space, statistically sampled over a short 13.8 billion years, is to first evolve life, then evolve sentient creatures who then actively perform the project. So the only other likely way for there to be another competing cubic meter sitting at this temperature somewhere in the universe is for there to be sentient aliens who also made it happen. The idea behind the news angle “the coldest cubic meter” was the brainchild of my collaborator Jon Ouelett, a graduate student in physics at UC Berkeley and member of the CUORE working group responsible for achieving the cooldown. His take on this is written up nicely in his piece on the arXiv entitled The Coldest Cubic Meter in the Known Universe.

I’ve been member of the CUORE and Cuoricino collaborations since 2004 when I was a postdoc at Lawrence Berkeley Laboratory. I’m now a physics professor at California Polytechnic State University in San Luis Obispo and send undergraduate students to Gran Sasso help with shifts and other R&D activities during the summers through NSF support. Indeed, my students were at Gran Sasso when the cooldown occurred in September, but were working on another part of the project doing experimental shifts for CUORE-0. CUORE-0 is a precursor to CUORE and is currently running at Gran Sasso. It is cooled down to about 10 mK and is perhaps a top-10 contendeder for the coldest contiguous 1/20th of a cubic meter in the known universe.

I will write more about CUORE and its true purpose in coming posts.

On a speculative note, one must naturally wonder if this kind of technology could be utilized in large scale quantum computing or other tests of macroscopic quantum phenomenon. While there are many phonon quanta associated with so many crystals at these temperatures (and so the system is pretty far from the quantum ground state, and has likely decohered on any time scales we could measure) it is still intriguing to ask if some carefully selected macroscopic quantum states of such a large system could be manipulated systematically. Large-mass gravitational wave antennae, or Weber bars, have been cooled to a point where the system can be considered in a coherent quantum state from the right point of view. Such measurements usually take place with sensative SQUID detectors looking for ultra-small strains in the material. Perhaps this new CUORE technology, involving macroscopic mK-cooled crystal arrays, can be utilized in a similar fashion for a variety of purposes.

# Boltzmann Brains by Agapanthus on Ultima Thule

One of my ambient music pieces, Boltzmann Brains (inspired by the weird physics idea of Boltzmann Brains), was just featured on the Australian radio show and podcast Ultima Thule. It was fun to see the tagline “The suspendedly animated sounds of Peter Challoner, Thomas D Gutierrez and Brian Eno.” (Oh, yeah, and that other guy we almost forgot “Brian Eno”, whoever he is). The Ultima Thule podcast is one of my favorite podcasts and long ago replaced Hearts of Space as my go-to ambient and atmospheric fix, so having my own music appear on the show was a real treat.

# RHIC/AGS User’s Meeting Talk and Yeti Pancakes

I was recently invited to give a talk on neutrinoless double beta decay at the RHIC/AGS User’s Meeting at at Brookhaven National Laboratory. The talk was entitled “Neutrinoless Double Beta Decay: Tales from the Underground” and was a basic overview (for other physicists, targeted primarily at graduate students) of neutrino physics and the state of neutrinoless double beta decay. The talk was only 20+5, so there wasn’t time to get into a lot of detail.

It was great to be back at BNL and see some of my old friends and colleagues. It was particularly nice to see my mentor and friend Professor Dan Cebra again and meet his recent crew of graduate students.

Being asked to give a neutrinless double beta decay talk at a meeting entirely focused on the details of heavy ion physics is a little like a yeti a pancake: they are terms not usually used in the same sentence, but somehow it works. Their motivation was noble. At these meetings, the organizers typically pick a couple topics in nuclear physics that are outside their usual routine and have someone give them a briefing on it. This was exactly in that spirit.

To download the Standard Model Lagrangian I used in the talk, visit my old UC Davis site where you can find pdf and tex versions of it for your own use. If you are interested in investigating the hadron spectra I show in the talk, you can download my demonstration available in CDF format from the Wolfram Demonstration Project. The Feynman diagram for neutrionless double beta decay was taken from wikipedia. Most of the other figures are standard figures used in neutrinoless double beta decay talks. As a member of the CUORE collaboration, I used vetted information regarding our data and experiment.

Enjoy

# Cal Poly Open House, All That Glitters Green and Gold 2014, Faculty Address

I was asked to give the 2014 Cal Poly Open House, All That Glitters Green and Gold 3 minute faculty address to about 600+ prospective students and parents for the College of Agriculture, Food, and Environmental Sciences, College of Architecture and Environmental Design, and the Orfalea College of Business. I somehow managed to get “quantum”, “atheist”, “delocalize”, and “live long and prosper” in there. In hindsight, asking someone from physics to do this for these Colleges is a bit like asking Snape to give the opening address to Hufflepuff. Still, it was great fun and a true honor. Here is a transcript of the speech.

Thank you President Armstrong. Welcome and good morning! I’m Tom Gutierrez, a professor in the Physics Department here at Cal Poly. I’m also currently the advisor for the Society of Physics Students, Sigma Pi Sigma (the physics honor society), and student club AHA (the Alliance of Happy Atheists).

How many of you watch or have seen the TV show The Big Bang Theory? Sadly, in my department it’s basically considered a documentary. I don’t watch it regularly, but to appreciate where I’m coming from: understand that I when I first saw it mistook it for a NOVA special on how physicists can actually improve their social skills. With that awkward introduction…

Why am I, a physics professor, speaking to you today? I’m here to give you a brief faculty perspective of Cal Poly. Cal Poly is a comprehensive polytechnic university that embraces a Learn-By-Doing philosophy. And physics, the most fundamental of all sciences, is at the very core of this mission. For a comprehensive polytechnic university in the 21st century, physics is the technical analog to the “liberal arts.” All technical majors across all Colleges at the University must take physics and almost all majors allow physics as an elective or as a general education course. This frequently puts my department at the nexus of the University and gives me the pleasure of interacting with a large cross section of our students on a regular basis.

I teach a wide spectrum of courses in the physics department. While it’s true most of my students are from engineering and the College of Science and Math, some of the most hard working and thoughtful students I’ve had have come from the Colleges represented in the session this morning, which include business, animal science, architecture, and forestry majors to name a few. To facilitate the Learn-By-Doing philosophy in practical terms, Cal Poly fosters amongst faculty what is known as the Teacher-Scholar Model. Faculty across all Colleges are carefully selected 1) for their passion for teaching and working with students and 2) for being engaged with active work in their fields. In my own experience, most educational institutions choose one or the other focus for faculty: a professor is either a teacher or a scholar. While there are many fine examples of each amongst today’s universities, one vocation typically suffers at the expense of the other. However, Cal Poly celebrates both forms of professional expression for individual faculty — and this generates a powerful and singular learning environment for the students who come here. Faculty engaged in their fields can bring real-world knowledge and research into the classroom. Conversely, teachers can bring their students and pedagogical wisdom into the real world.

My own work in particle physics, sponsored by the National Science Foundation, has allowed me to bring students to work at an underground lab in Italy and experience the joys of doing cutting-edge science. Students then bring this experience to their jobs and graduate programs. The message I’m getting from my colleagues at other institutions and in industry? “Send us more Cal Poly students!” Faculty at Cal Poly are allied with the student. We want you to graduate as lifelong learners who find a productive career and make a difference in the world. At Cal Poly, we want you to grow as a person and to challenge your pre-existing assumptions about how the world works. We want you to discover your Personal Project; think big, make collaborations, and not just dream, but discover how to translate those dreams into actions.

Anyway, enjoy the rest of your stay and come visit the Physics Department and CoSaM open house in the Baker Science building if you get a chance. May your quantum wave function always remain delocalized. Live long and prosper. Thank you!