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It is well established that the light speed in a perfect vacuum is roughly $3\times 10^8 \:\rm m/s$. But it is also known that outer space is not a perfect vacuum, but a hard vacuum. So, is the speed limit theoretically faster than what we can measure empirically, because the hard vacuum slows the light down? Is this considered when measuring distances with light?

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    $\begingroup$ @WillihamTotland Only because you chose to display two decimals. $\endgroup$ – Mr Lister Sep 6 '18 at 6:32
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    $\begingroup$ I think rounding (the already rounded) $2.998 \times 10^8\ m/s$ to $3 \times 10^8\ m/s$ is better than stating it as $3.00 \times 10^8\ m/s$. $\endgroup$ – Mick Sep 6 '18 at 7:36
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    $\begingroup$ Why bother with approximations? It only takes a few more characters to write the exact value of 299792458 m/s. $\endgroup$ – PM 2Ring Sep 6 '18 at 8:16
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    $\begingroup$ @mick technically it's not, the .00 is way more precise. $3x10^8$ could even be $3.4$. Saying that it's $3.00x10^8$ is not stating, it's a correct rounding with precise information conveyed. That's what the original comment was about. $\endgroup$ – luk32 Sep 6 '18 at 10:06
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    $\begingroup$ The speed of light in vacuum is exactly $c = 1$. $\endgroup$ – Danijel Sep 6 '18 at 14:03
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If we take air, then the refractive index at one atmosphere is around $1.0003$. So if we measure the speed of light in air we get a speed a factor of about $1.0003$ too slow i.e. a fractional error $\Delta c/c$ of $3 \times 10^{-4}$.

The difference of the refractive index from one, $n-1$, is proportional to the pressure. Let's write the pressure as a fraction of one atmosphere, i.e. the pressure divided by one atmosphere, then the fractional error in our measurement of $c$ is going to be about:

$$ \frac{\Delta c}{c} = 3 \times 10^{-4} \, P $$

In high vacuum labs we can, without too much effort, get to $10^{-10}$ torr and this is around $10^{-13}$ atmospheres or 10 nPa. So measuring the speed of light in this vacuum would give us an error:

$$ \frac{\Delta c}{c} \approx 3 \times 10^{-17} $$

And this is already smaller than the experimental errors in the measurement.

So while it is technically correct that we've never measured the speed of light in a perfect vacuum, the vacuum we can generate is sufficiently good that its effect on the measurement is entirely negligible.

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    $\begingroup$ Since $\Delta c/c$ is surely dimensionless maybe change the $= 3\times 10^{-4} P$ to $\sim 3\times 10^{-4} P$? $\endgroup$ – ZeroTheHero Sep 5 '18 at 17:13
  • $\begingroup$ And if we know the effect of the medium on our measurement then we can correct for that anyway, yes? $\endgroup$ – John Bollinger Sep 5 '18 at 18:35
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    $\begingroup$ @ZeroTheHero John explicitly calls for the pressure to be measured in atmospheres, so he gets off on a technicality. But frankly, that should really be expressed as $$\frac{\Delta c}{c} = 3 \times 10^{-4} \frac{P}{P_\mathrm{atm}}.$$ $\endgroup$ – Emilio Pisanty Sep 5 '18 at 19:36
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    $\begingroup$ The phrase "a speed a factor of $3 \times 10^{-4}$ too slow" seems to imply that the unmeasured perfect vacuum speed was about $3333.\overline{3}$ times the speed in air, the actual factor is of course the above $1 + 3 \times 10^{-4}$. $\endgroup$ – Leif Willerts Sep 6 '18 at 11:31
  • $\begingroup$ @LeifWillerts yes, true, I'm being a little careless about the wording. I'll have a look at tidying that up. $\endgroup$ – John Rennie Sep 6 '18 at 11:39
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The answer by John Rennie is good so far as the impacts of the imperfect vacuum go, so I won't repeat that here.

As regards the last part of your question about whether this should be accounted when measuring distance, it's worth noting that the standard defines the speed of light to be a specific value and then, using also the definition of the second, derives the meter as a matter of measurement. So as the standards are currently written, the speed of light is exact by definition.

Your question, as written, implicitly assumes that the meter and the second are given by definition and the speed of light a question of measurement.

So from that perspective, your question really should be written to ask whether the impact of imperfect vacuum impacts our definition of the meter. The answer to that, is that it probably does, as was approximately quantified by John Rennie. Whether or not it is important depends on what method is used and what other experimental uncertainties are inherent to that method.

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  • $\begingroup$ In the old days, the length of a meter was defined from a certain rod of iridium-platinum alloy which is now kept under glass. $\endgroup$ – can-ned_food Sep 7 '18 at 4:23
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    $\begingroup$ @can-ned_food and the kilogram still is, slowly changing weight. $\endgroup$ – Tim Sep 7 '18 at 20:58
  • $\begingroup$ @Tim The mass of the standard kilogram can’t change, also by definition. $\endgroup$ – Mike Scott Sep 9 '18 at 1:13
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    $\begingroup$ @MikeScott Strictly speaking, it is the value of the standard's mass in units of kg which can't change (since it is fixed to 1 by definition, which I assume is what you meant to say). The mass itself can (and does) change. Just wanted to point out that there is an important difference between a physical quantity and its numerical value in some specific unit. $\endgroup$ – aekmr Sep 9 '18 at 1:46
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There is a constant in physics called $c$ that is the "exchange rate" between space and time. One second in time is in some sense "equivalent" to $c$ times one second (which then gives a distance in space). Light is taken to travel at $c$. Note that $c$ isn't the speed of light, but rather the speed of light is $c$, which is a subtle distinction ($c$ being what it is causes light to travel at that speed, rather than light traveling at that speed causes $c$ to be that value). $c$ has been measured by looking at how fast light travels, but there are also several other ways of finding $c$. For instance, $c^2$ is equal to the reciprocal of the product of the vacuum permittivity and the vacuum permeability. So not only is the effect of imperfect vacuums negligible in measuring $c$ by looking at the speed of light, but there are multiple other observables that depend on it.

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  • $\begingroup$ I liked this change of perspective quite a bit :) $\endgroup$ – Helen Sep 30 '18 at 8:29
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An experiment designed to measure some physical quantity such as the speed of light will take into account any perturbing effects. If, for whatever reasons, actually performing speed of light measurements in near vacuum would be impossible, we could still measure it under different air densities and extrapolate the results to zero air density. This extrapolation can be done accurately by fitting the known theoretical dependence of the speed of light on the air density, but we can just as well proceed in a model independent way and not use any theoretical input when doing the extrapolation to a perfect vacuum.

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    $\begingroup$ +1 This should be the accepted answer. The actually accepted answer is correct as well, boiling down to "it's an unmeasurable difference", but the fact is that the scientists who (can) do these kinds of measurements would certainly think of any residual matter in outer space and factor that in into their calculations... and this is the aspect the question is shooting for, as far as I can tell... $\endgroup$ – AnoE Sep 7 '18 at 9:05
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The speed of light is by definition exactly 299,792,458 m/s. If the vacuum was not perfect during our measurements only our definition of a meter would change.

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    $\begingroup$ changing the definition of a metre would still change the speed of light. this doesn't actually answer the question because it's just deflecting the apparent effect... $\endgroup$ – HyperNeutrino Sep 7 '18 at 15:56
  • $\begingroup$ The speed of light in water is approximately 225,000,000 m/s (experimental result). I think the statement should be qualified. $\endgroup$ – Peter Mortensen Sep 9 '18 at 21:43

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