Over the last several lectures, I’ve been covering special relativity in my freshman majors class. We derived the Lorentz transforms on Friday, and then spent yesterday showing how our previous work leads inevitably to E=mc^{2}.

As a technical aside, if any other physicists out there have particularly good derivations of relativistic energy (based on freshman level physics and math) please let me know. The best I’ve seen so far was given to me by Rich Gott, who imagined a particle decaying into two photons, and using only E=pc (known to Maxwell), conservation of momentum, and the derivation of Doppler quickly shows that the photons must have decayed from a particle of mass m=E/c^{^2}.

But I digress… I bring all of this up because about a quarter of the class came back to my office afterwards (if any of you guys are reading this, hello!) and we spent the next two hours talking — arguing, really — about the sort of brain-bending questions that arise when you really start to get a handle on special relativity. Among others, I got, “What happens if you’re going the speed of light and you turn on your headlights?” Now this is a *very* good question. It’s pretty much the same one we use as our jumping off point in Chapter 1 of our book.

The problem is that the correct answer is essentially, “You can’t, so quit asking.” This is a deeply dissatisfying answer. Even when a student has the physics (aka the equations, and the physical principles from which they arise) under his/her belt, I can explain it somewhat better by pointing out that time would be infinitely dilated, and it would require an infinite amount of work to get any massive particle up to the speed of light.

As a consolation (I suppose) we *can* explain what happens when you 99.99999999999% the speed of light. The answer is that your headlights look perfectly normal. In fact, everything appears perfectly normal, so long as you’re traveling at a constant speed. Even if students appreciate this fact, they still want to know, “what if?”

I’d be interested in hearing how others deal with counterfactual questions. Once we start asking (or answering, really) what happens in circumstances where the laws of physics are clearly violated, what possible meaning can our answers have. I finally settled on, “ponies and rainbows,” and the students seemed somewhat satisfied with that. Still, it seems like something of a cheat.

Yes, I’ll grant that workaday physicists may be happy with mathematical proof as a substitute for physical intuition (especially in situations like relativity and quantum mechanics, where our intuition breaks down), but it makes students mad. They start flailing around, knocking books of the shelves. So without simply asserting, “The equations say so,” (even to students who *know* the equations) how do you handle this one? Or do you have other good questions that are physically ill-posed, but still enlightening?

**-Dave**

So in my thermo class, we were going over an isolated system, who’s subsystems… leading to a statement about thermal efficiency: d!W_w_maximum = ( 1- T_H / T )*(-d!Q) – dE. The inexact differential of the maximal amount of work an adiabatic secondary subsystem can do is limited by the thermal efficiency, (1-T_h/T), of the heat transfer to an isometric tertiary subsystem and the initial decrease in energy of the primary system. Something like that. Where T_H is the (absolute) temperature of the secondary subsystem (heat bath) and T is the (abs.) temperature of the primary subsystem.

Immediately after my Professor finished writing this up on the board with a caveat that the thermal efficiency must be positive and less than one, I asked him two, not so obvious, but important questions.

1) Does that mean you can’t have negative temperatures?

2) In something like a spin system where we can use Stat. Mech. to compute a negative temperature, that an efficiency greater than one is possible?

After studying the situation some more, and revisiting the initial setup of the problem, these are a bit more obvious to answer.

Lesson learned. Intuition is good. Math is rigorous. Intuition can certainly be wrong and Math is quite useless without some outside constraints guiding the understanding.

Ans. 1) No. There are no negative “absolute” temperatures.

Ans. 2) From 1., maximal efficiency would have T_H = 0, thereby making the primary systems absolute temperature irrelevant.

I still think turning on the headlights already at v=c is possible- photons don’t have mass. I don’t have to move electrons to generate light. I could just have them lower in energy level. Right? ðŸ˜›