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Every so often,* we get a question about what would happen should there be a change in a physical constant that contains dimensional information, such as $\hbar$, $c$, $G$, or often "the scale of the universe", and we often wind up reinventing the wheel to explain why this is not a well-posed question. I would therefore like to set this up as a canonical question to refer to later and save some duplication of effort. So:

Why is it meaningless to speak about changes to a physical constant that contains dimensional information?

* These are some examples but there's more .

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This sums up the answer pretty concisely... youtube.com/… Bam. –  tpg2114 Sep 26 '13 at 18:19
See Duff, "Comment on time-variation of fundamental constants," arxiv.org/abs/hep-th/0208093 . –  Ben Crowell Sep 27 '13 at 4:46

1 Answer 1

It's not always meaningless. For example, we may talk about the change of the mass of a human (in kilograms) after a diet.

But the point is that the value of a dimensionful quantity – and its change or constancy – depends on the magnitude of the units and they're matter of conventions which may change, too. So an increasing numerical value of $\hbar$ could be just to a decreasing value of what we call one kilogram. We define e.g. one kilogram using the international prototype and consider it a "constant mass" while the mass of a human on diet is more variable.

In this comparison, the constants $\hbar,c,G$ are even more naturally constant than the international prototype. In fact, "adult" physicists use units where $\hbar=c=1$ (quantum relativistic units) and sometimes also $G=1$ (general relativistic units or Planck units when all conditions are imposed). So $\hbar,C,G$ can't really change when natural units are used because they're always equal to one.

Even if we use more everyday units in which the numerical value isn't one, it's still better to define such units in such a way that $\hbar,c,G$ are constant. In fact, one meter is already defined so that $c=299,792,458$ m/s at all times. It's not quite the case for $\hbar$ and $G$ yet but this may change in the future.

Because $\hbar=c=k_B=1$ etc. are so natural, it makes sense to consider the constancy or variability of all other dimensionful constants by the constancy or variability of these constants converted to the natural units.

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