Can't make a clock out of light

[bcrowell]

Someone a gazillion times smarter than me says:

Now, massless particles (of whatever spin) satisfy conformally invariant equations.[5] With such conformal invariance holding in the very early universe, the universe has no way of "building a clock".

--Roger Penrose, http://epaper.kek.jp/e06/PAPERS/THESPA01.PDF

Reference [5] is Penrose, R. (1965) "Zero rest-mass fields including gravitation: asymptotic behaviour," Proc. Roy. Soc. London, A284, 159-203, first page here:
http://www.jstor.org/pss/2415306

I would like to understand this statement better.

The most naive version of the argument is that an electromagnetic plane wave moves at c, so its elapsed proper time is always zero. However, this doesn't really do the whole job, because you can make superpositions of electromagnetic waves propagating in different directions. Such a field has a total energy and momentum E and p, and an equivalent rest mass m given by m²=E²-p². Therefore it has a well-defined center of mass frame, and the world-line of the center of mass has a nonvanishing elapsed proper time.

So if you want to make this more rigorous, then I think you have to do as Penrose does, and argue in terms of conformal invariance. This requires filling in the gap between the fact of conformal invariance and "no way of 'building a clock.'" This seems like a nontrivial gap, since Penrose doesn't actually define "clock."

One could argue that if a sinudoidal electromagnetic plane wave is sweeping over me, then I can count periods, and that makes a clock. One possible way of dealing with this argument is to say that it doesn't really qualify as a clock, because there is no way of verifying that such a clock runs at a constant rate. That is, if I put such a sine wave through a conformal transformation, I can make it not be a sine wave. Since there is no way to tell a periodic wave from a nonperiodic one, you can't count periods and call it a clock.

But that doesn't necessarily clear up my confusion in the case of a more complex interacting system of light waves. For instance, it's well known that colliding electromagnetic plane waves can create black holes. Once you've got black holes, you've got massive particles, which can then be used to create clocks. For instance, a pair of black holes orbiting around their common center of mass seems to be a perfectly valid clock. Why is this not a counterexample to Penrose's assertion?

There is a technical issue about whether this holds for all possible spins (as Penrose claims in the quote above). The massless Klein-Gordon equation is not conformally invariant unless you add an extra correction term φR (Wald, appendix D). We had a separate thread about this https://www.physicsforums.com/showthread.php?t=446425 , and it's not really my main concern in this thread.

 

[Bill_K]

Ben, Penrose likes to express deep ideas in a casual way. To understand what he's driving at, think what is missing in a conformally invariant world because it breaks conformal invariance. Electrons. Not because they have mass, but because they have a specific mass. A pair of photons has a center-of-mass mass, and a black hole created from just photons will have a mass. But it can have any mass. You can't measure anything with a mass that is arbitrary to begin with.

With electrons or any of the particles with mass, you can use their well-defined mass to start "building" things. A mass defines an energy scale, which defines a wavelength (Compton wavelength) which defines a time scale. Atoms are a certain size not because of the strength of electromagnetism but because of the specific integer charges that all particles have. The specific charge breaks conformal invariance. Using it you can set up measurements of time and space and say, for example, that the universe is X times as old as the period of a hydrogen alpha line. This is what Penrose means by a "clock".

He claims the early universe must have been conformally invariant. That is, self-similar. "Can't build a clock" means you can't tell how old it was, because even though it was expanding, it was expanding in a self-similar fashion and things could always be rescaled so that the universe (during this period) never looked any different.

 

[bcrowell]

Aha! That totally makes sense. Thanks, Bill!