Someone a gazillion times smarter than me says:(adsbygoogle = window.adsbygoogle || []).push({});

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

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 m^{2}=E^{2}-p^{2}. 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 isnotconformally 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.

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# Can't make a clock out of light

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