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Obviously some things, such as the speed of light in a vacuum, are defined to be a precise value. The kilogram was recently defined to have a specific value by fixing Plank's constant to $6.62607015\cdot 10^{−34}\frac{m^2 kg}{s}$.

In particular, in the case of the latter, we held off on defining this value until the two competing approaches for measuring the kilogram agreed with each other within the error bounds of their respective measurements.

Which leads me to wonder, what is the most precisely measured (not defined) value that the scientific community has measured. I am thinking in terms of relative error (uncertainty / value). Before we defined it, Plank's constant was measured to a relative error of $10^-9$. Have we measured anything with a lower relative error?

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Carrying the fame of being one of the most precisely verified propositions in physics, the ratio of the gravitational to inertial mass was verified to be unity within $1$ in $10^{15}$ by the MICROSCOPE satellite in $2017$. The earlier best precision was $5\times10^{-14}$, obtained by Baessler, et al. in $1999$.

References:

  1. https://en.wikipedia.org/wiki/MICROSCOPE_(satellite)
  2. https://en.wikipedia.org/wiki/Equivalence_principle#Tests_of_the_weak_equivalence_principle
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The magnetic moment of the electron has been measured to a few parts in $10^{13}$. (Source) This provides an exquisite test of quantum electrodynamics, and calculating the relevant Feynman diagrams has been a Herculean effort over decades.

Note that the more precise tests cited in other answers are basically null results: no difference between gravitational and inertial mass; no difference in magnitude of charge between proton and electron; no mass of photon. So I believe the magnetic moment of the electron is the most precise measurement that is non-null and thus “interesting”.

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    $\begingroup$ Yeah that’s the answer I was looking for but somehow only found $\alpha$. $\endgroup$ Aug 16 '19 at 5:35
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    $\begingroup$ While I'm not going to change the question to invalidate answers, your note is a good one that I'd edit in if I had a chance. While those null results are indeed interesting, I have the intuitive sense that it's easier to measure them with arbitrarily high precision. It is interesting, however, how few orders of magnitude separate this result from some of the equivalency tests listed in other answers. $\endgroup$
    – Cort Ammon
    Aug 16 '19 at 16:18
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    $\begingroup$ For full clarity, this isn't the most accurate or precise measurement known, and the fact that @CortAmmon has accepted this answer (when there are tighter uncertainties reported in other answers) is a good example of why this thread is problematic. If it comes to "non-null" results, though, this paper presents a measurement with considerably more precision than the one in this answer (but I'm not going to take the (pretty arrogant) tack that others here have taken in asserting that the first thing I found is "the" most precise experiment). $\endgroup$ Aug 21 '19 at 10:18
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A few more candidates:

  • The quantized Hall resistance is a great and surprising example, since it's an emergent property of rather complicated, "dirty" systems. As stated here (2013) one can measure the resistance to one part in $3 \times 10^{10}$. For this reason this effect is now used to define the Ohm.
  • Equivalence principle tests using torsion balances reached precisions of about one part in $10^{11}$ in $1964$, see here. Another existing answer gives a more precise result from a more modern experiment. These can be thought of as either verifying the equality of gravitational and inertial mass, or placing bounds on the strength of long-range fifth forces.
  • The electrical neutrality of bulk matter follows because the electron and proton have exactly opposite charges. Treating the charge of the electron as given, tests of neutrality effectively measure the charge of the proton, with one experiment achieving a sensitivity of one part in $10^{21}$ in $1973$. Arguments from cosmology can be used to set much larger bounds, though they require more assumptions.
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  • $\begingroup$ Can you suggest me some reference where I can look up how the Equivalence Principle tests can be interpreted as putting bounds on a long-range fifth force? $\endgroup$
    – Dvij D.C.
    Aug 16 '19 at 4:59
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    $\begingroup$ Another example: Earth-Moon distance measured with a laser. Distance about 3e10 cm, measured with a few cm accuracy I think. Should investigate more to be sure. $\endgroup$ Aug 16 '19 at 8:47
  • $\begingroup$ Electrical neutrality of bulk matter: That only proves the equality of the charges if the number of electrons and protons in bulk matter is the same - but if the charge on the electron was 1.1 times that on the proton, I would expect bulk matter to have about 10% more electrons and still be neutral. $\endgroup$ Aug 16 '19 at 13:03
  • $\begingroup$ Update: the ohm is now defined in terms of the elementary charge $\endgroup$ Aug 21 '19 at 6:11
  • $\begingroup$ @gen-z Not particularly - in actual metrological practice, it's rather more accurate to say that the coulomb is defined in terms of the ohm. $\endgroup$ Aug 21 '19 at 10:20
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A good candidate is the measurement of the fine structure constant $\alpha$. This wiki article on precision tests of QED states that:

The agreement found this way is to within ten parts in a billion ($10^{−8}$), based on the comparison of the electron anomalous magnetic dipole moment and the Rydberg constant from atom recoil measurements...

There is also an upper bound on the mass of the photon, which is in the range of $10^{-27}eV/c^2$ although that's an upper bound rather than a measurement since the expected value is $0$.

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    $\begingroup$ The ratio of the gravitational to inertial mass was verified to be unity within $1$ in $10^{15}$ in $2017$ by the MICROSCOPE satellite: en.wikipedia.org/wiki/MICROSCOPE_(satellite) $\endgroup$
    – Dvij D.C.
    Aug 16 '19 at 4:04
  • $\begingroup$ I think it could be added as a possible answer but I am not sure if there exist more precise measurements in particle physics, maybe some electroweak precision measurements? $\endgroup$
    – Dvij D.C.
    Aug 16 '19 at 4:08
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To answer the question from a different angle, LIGO has measured gravitational waves several times over the last few years. To do so they have to observe a distortion in a 4km arm, in the order of $10^{-18}$m. In other words, it has to detect a fractional imprecision in its length of ~$2.5 * 10^{-23}$. That's pretty accurate.

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    $\begingroup$ LIGO is extremely sensitive, but pretty inaccurate, its calibration has an uncertainty of a few percent. $\endgroup$ Aug 16 '19 at 13:05
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    $\begingroup$ Thank you for the answer! What Bas Swinckles mentions is something I was trying to avoid when crafting the criteria of the question. The 4km long arm is not what LIGO actually measures, but rather the 10^-18m number is what is being measured. LIGO was one heck of a feat, but I was trying to avoid high-sensitivity/low-accuracy measurements because we can always come up with a "higher" value by comparing it against something bigger. $\endgroup$
    – Cort Ammon
    Aug 16 '19 at 15:22
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Another quantity that is well known is the frequency of the 21 cm line of Hydrogen hyperfine splitting that has been measured to about 13 significant figures, 1420405751.7667±0.0009 Hz. Unlike the electron magnetic moment, mentioned in another answer, the QED calculation for hyperfine splitting is not as accurate. This calculation also depends on other fundamental constants, including electron/proton mass ratio and proton magnetic moment. which are not known to equivalent precision.

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The accuracy of the latest atomic clocks is a good candidate for the prize, with an accuracy of less than 1 part in 10^20. One example has been used to confirm gravitational time dilation over a vertical height of a single millimeter, with even more precise confirmation possible with the latest atomic clocks.

An atomic clock measured how general relativity warps time across a millimeter

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Aren't clocks the most precise devices, relatively speaking? Their uncertainty is 1 second in 33.7 billion years, or better than 10^-18. Refs: https://link.aps.org/doi/10.1103/PhysRevLett.123.033201 https://en.wikipedia.org/wiki/Quantum_clock#Accuracy

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