You’re taking your morning shower and a thought occurs to you. “In classical electrodynamics, an accelerating charge radiates. In general relativity, acceleration is equivalent to a gravitational field. Therefore a stationary charge should radiate simply by virtue of being in a gravitational field. What’s up with that?”
You wonder about what the radiated power would be for a given gravitational field. You figure maybe you could use the Larmor formula with the Stefan-Boltzmann law to estimate the equivalent thermal radiation but you don’t remember either one of those equations exactly and you’re pretty sure you’d have to finagle some spatial factors anyway (the Stefan-Boltzmann law has a factor of surface area).
One alternative is to try to construct a quantum field theory in curved spacetime, but this is ludicrously tough even if you’re not in the shower without pen and paper. But we might be able to just juggle some constants around and get an estimate. We’ll start off with the assumption that the effective temperature of the radiation by the charge is proportional to the local acceleration due to gravity. In other words, that the temperature is just the acceleration multiplied by some constants that we don’t know:
Well, what is that stuff? In terms of units, we’ve got:
We need to pick the “stuff” so that this works out. Remember that the kelvin is the unit of temperature, 0 K being absolute zero with room temperature (roughly) around 300 K. Well, we need to pick a fundamental constant involving temperature. Boltzmann’s constant k fits the bill. Its units are J/K (ie, energy per temperature), so our expression needs to have k in the denominator so that the kelvin comes out on top:
That meters/seconds needs to be killed off. There’s pretty much only one universal constant involving speed, and that’s the speed of light c. Again, it needs to be in the denominator to cancel out the meters and seconds:
There’s still a “seconds” that we need to get rid of, and there’s also that “joules” that we introduced when we decided to drag Boltzmann’s constant into things. Are there any constants involving joules and seconds that might help us out? There is indeed a doozy of a constant: Planck’s constant h. We end up with units that cancel properly:
With the units written explicitly:
That might reassure us that the radiation emitted by a charge accelerating at 9.8 m/s^2 is pretty small, since h is so tiny and c is so large. k is pretty tiny as well, but the tininess of h is much greater. Or less, depending on how you look at it. In any case in our estimate T is around 10^-19 K for an acceleration equivalent to the earth’s surface gravity.
So you get out of the shower and Google the Unruh effect, you’ll see that the actual equation derived with much pain and suffering by Bill Unruh is:
We’re off by a factor of 4 pi^2, or about 40, but in terms of order of magnitude that’s pretty good. We didn’t even have to get out of the shower.