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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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    The speed of light in a vacuum isn't roughly $3\times 10^8 :\rm m/s$; it's $3.00\times 10^8 :\rm m/s$ exactly (to within the precision given). – Williham Totland Sep 06 '18 at 00:53
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    @WillihamTotland Only because you chose to display two decimals. – Mr Lister Sep 06 '18 at 06:32
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    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 $3.00 \times 10^8\ m/s$. – Mick Sep 06 '18 at 07:36
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    Why bother with approximations? It only takes a few more characters to write the exact value of 299792458 m/s. – PM 2Ring Sep 06 '18 at 08:16
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    @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. – luk32 Sep 06 '18 at 10:06
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    The speed of light in vacuum is exactly $c = 1$. – Danijel Sep 06 '18 at 14:03
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    It is cool to say that c = 1 but it is just circular reasoning. The units used to conclude this already use the value of c in SI units to obtain their value. –  Sep 06 '18 at 15:20
  • @William so does the definition of 1 meter – Cedric H. Sep 06 '18 at 18:13
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    So, I guess, based off the answers given, the answer to the question is "Yes" – The Anathema Sep 06 '18 at 20:37
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    Is “hard vacuum” a technical term? I've only ever heard it used in sci-fi contexts because it adds a certain rough and palpable, albeit cliché, tone to narrative or dialogue. “Prepare to suck hard vacuum, fool!” and the like. – can-ned_food Sep 07 '18 at 04:21
  • @can-ned_food hard vacuum is a thing, yes, but it's still not the same as a perfect vacuum with no particles at all. – Mr Lister Sep 07 '18 at 06:30
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    I'd like to point out (as an absolute non-expert with a healthy interest in the subject) that absolute speed does not exist, it's all relative. I have no idea if an individual photon could travel faster than c because there's as I understand no way of knowing that, the only thing we can know is that no matter how fast something moves, relative to a measurer its speed will never exceed c. Even if two rockets each flying at light speed would fly towards each other they'll both measure the other's speed as exactly light speed when passing each other. Special relativity is weird. – kevin Sep 07 '18 at 11:51
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    So taking that into account, if a planet flies through the solar system at c /2, from it someone fires a rocket at c /2, then the rocket would fly at approximately c relative to the solar system. Now, if that rocket were to have a headlight, at what speed would the photons shoot out of that light? The rocket is already flying at c, so anything launched from it in the same direction would naturally go faster than c, right? Nope, relative to the solar system it'll still only be c, even though we're firing photons at light speed from a device already traveling at light speed. – kevin Sep 07 '18 at 11:56
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    No physical object in theory could go faster tan c, because it would violate causality principle. If such things could happen, then you could see how you was borning, hehe. There may be some strange things that may happen in a time t < l/c in some two events at a distance l in space,- see quantum entanglement. BUT in quantum entanglement no information is transfered between A and B points – Agnius Vasiliauskas Sep 07 '18 at 12:17
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    @kevin the rocket wouldn’t be at c. Actually it would be at a measurably lower speed than c, because in above 10% of c velocities you shouldn’t use the standard relative motion equation from Galilean relativity, two objects at c/2 flying towards each other would measure the incoming impact at less than c. The reason the solar system and the rocket both watch the photon moving at c in vacuum instead of different speeds is because of time dilation and lenght contraction. –  Sep 07 '18 at 23:55
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    @PM2Ring Said no physicist ever ... pi is 3; g is 10 and that will suffice for any back of the envelope calculation. – UKMonkey Sep 10 '18 at 12:33
  • @kevin What William said. Relative to the solar system, that rocket would be moving with a speed of 4c/5. – PM 2Ring Sep 10 '18 at 12:45

5 Answers5

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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.

John Rennie
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    Since $\Delta c/c$ is surely dimensionless maybe change the $= 3\times 10^{-4} P$ to $\sim 3\times 10^{-4} P$? – ZeroTheHero Sep 05 '18 at 17:13
  • And if we know the effect of the medium on our measurement then we can correct for that anyway, yes? – John Bollinger Sep 05 '18 at 18:35
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    @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 $$\frac{\Delta c}{c} = 3 \times 10^{-4} \frac{P}{P_\mathrm{atm}}.$$ – Emilio Pisanty Sep 05 '18 at 19:36
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    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 $1 + 3 \times 10^{-4}$. – Leif Willerts Sep 06 '18 at 11:31
  • @LeifWillerts yes, true, I'm being a little careless about the wording. I'll have a look at tidying that up. – John Rennie Sep 06 '18 at 11:39
  • There isn’t even any need to use a high-vacuum system. Simply plot the speed of light as a function of pressure, and extrapolate to zero! We did this in a freshman physics lab using a Michelson interferometer (a sensitive speed measurement) to obtain the refractive index of air. – Gilbert Sep 07 '18 at 13:33
  • Unless what we perceive to be a vacuum actually has some stuff in it, e.g. the CMB, or even something else, which if removed might perhaps transmit light faster. – it's a hire car baby Sep 08 '18 at 18:46
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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.

Brick
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  • 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. – can-ned_food Sep 07 '18 at 04:23
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    @can-ned_food and the kilogram still is, slowly changing weight. – Tim Sep 07 '18 at 20:58
  • @Tim The mass of the standard kilogram can’t change, also by definition. – Mike Scott Sep 09 '18 at 01:13
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    @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. – aekmr Sep 09 '18 at 01:46
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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.

Stéphane Rollandin
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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.

Count Iblis
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    +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... – AnoE Sep 07 '18 at 09:05
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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.

safesphere
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    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... – hyper-neutrino Sep 07 '18 at 15:56
  • The speed of light in water is approximately 225,000,000 m/s (experimental result). I think the statement should be qualified. – Peter Mortensen Sep 09 '18 at 21:43