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The closing paragraph of Julian Barbour's review of Penrose's Cycles of Time contains the following (emphasis mine):

Despite his great attraction to conformal geometry, Penrose still accords length a real physical role. But in fact we only ever observe angles, never lengths as such.

Would someone kindly explain what this means? (E.g., is using a yardstick not "observing a length as such"?)

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  • $\begingroup$ Length depends on the observer. Look at length contraction for example. Angles do not. $\endgroup$
    – Jan2103
    Commented Oct 11, 2020 at 15:44
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    $\begingroup$ @Jan2103 that is factually incorrect. Angles are not invariant under Lorentz transformations. $\endgroup$
    – Prof. Legolasov
    Commented Oct 11, 2020 at 16:14
  • $\begingroup$ That is true. My bad! $\endgroup$
    – Jan2103
    Commented Oct 11, 2020 at 16:15
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    $\begingroup$ I'm not sure how it is possible to answer your question. For once, I completely disagree with the claim. Length are definitely measurable – we all know what the distance from the Earth to the Sun is, what the diameter of the hydrogen atom is, the values of masses of elementary particles (in quantum field theory, mass acts as inverse length). $\endgroup$
    – Prof. Legolasov
    Commented Oct 11, 2020 at 16:18
  • $\begingroup$ I agree with @Prof.Legolasov's comment. Notice that Barbour's sentence just before the one quoted in the question starts with "Let me end provocatively..." Maybe Barbour has some kind of non-mainstream theory in mind, but then this might be a question that only Barbour can answer. If asking Barbour directly isn't an option, then maybe you can find some clues in Barbour's own research papers, like this one: arxiv.org/abs/1105.0183. $\endgroup$
    – Chiral Anomaly
    Commented Oct 11, 2020 at 16:50

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Physics is local; length is an intrinsically nonlocal concept. The only way to determine length is to take data from spacelike separated locations, combine it at a point, and infer the length from "afar". The laws of nature never deal with length directly.

Angles can in principle be measured locally. The dot product of any two unit four-vectors is the cosine or hyperbolic cosine of the angle between them. If all observables are Lorentz scalars then they're all vector lengths and angles, or products of them.

On the other hand it's a clear fact about the world that objects from protons to galaxies tend to maintain consistent sizes over long periods of time, though the reasons they do it are pretty complicated. Without context, I'm not sure what point Barbour was trying to make.

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  • $\begingroup$ The laws of nature may be local, but not all observables are. For example, the proper time along a geodesic is a valid observable with the dimension of time = length in GR. It depends on the fundamental variables (local fields) through an integral. $\endgroup$
    – Prof. Legolasov
    Commented Oct 12, 2020 at 9:55

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