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Since 2019, the meter, kilogram, ampere, and kelvin, are defined by setting exact numerical values for the speed of light $c$, the Planck constant $h$, the elementary charge $e$, and the Boltzmann constant $k_B$.

The second still remains as being defined to be the duration of 9,192,631,770 periods of the caesium-133 atom, while the gravitational constant $G$ remains an imprecise number.

Why didn't the SI also fix $G$ to an exact value like $6.674 \text{m}^3 \text{kg}^{-1}\text{s}^{-2}$, such that all physical constants have an exact value, rather than the period of some random atom having an exact value?

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    $\begingroup$ Because $G$ is really hard to measure experimentally so that makes it not much use as a standard. The frequency of light emitted by a random atom is very easy to measure to high precision. $\endgroup$ Jul 14, 2020 at 11:09
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    $\begingroup$ Aside from the practical problem of measuring it, since we know that we don't yet have a model that can reconcile quantum mechanics and general relativity, do we even know what "G" is supposed to represent physically? It might turn out to be only a non-relativistic approximation to something else, for example. $\endgroup$
    – alephzero
    Jul 14, 2020 at 11:18
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    $\begingroup$ Caesium-133 isn't just "some random atom". Caesium is the heaviest stable alkali metal, and it has only 1 stable isotope, Cs-133. Natural caesium is almost entirely composed of Cs-133, with a tiny trace of the long-lived weakly radioactive isotope Cs-135. These properties make it a good choice for the basis of an atomic clock; OTOH, we do now have better clocks based on other atoms. $\endgroup$
    – PM 2Ring
    Jul 14, 2020 at 13:04
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    $\begingroup$ @alephzero To be fair, all constants could be only a non-relativistic approximation to something else. $\endgroup$
    – Javier
    Jul 14, 2020 at 14:59
  • $\begingroup$ @alephzero would that necessarily be a problem? $e$ is also the low energy limit of the electromagnetic coupling, but you can use it to define the Ampere $\endgroup$
    – fqq
    Jul 14, 2020 at 15:00

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A standard unit is useful if I can use it to make measurements in my lab. For example a caesium clock really does count the oscillations of the light (microwave) emitted by a caesium atom, and I can buy one off the shelf. That means the times I measure in my lab are as close to the standard unit as they could possibly be. Likewise given that the speed of light is a defined constant I can use my very accurate time measurements to very accurately measure lengths.

By contract Newton's gravitational constant is very hard to measure accurately. It is only known to an accuracy of about $0.002\%$, and that is from experiments far too complicated for me to routinely reproduce in my lab. Choosing $G$ as a primary standard would be of no use to me as an experimenter.

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  • $\begingroup$ 2nd issue: G also relates to the kilogram, so if it were used to define the second, the second would have depended on an artifact standard until very recently. $\endgroup$
    – The Photon
    Jul 14, 2020 at 15:38

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