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We use the speed of light to define the length of the meter, but we also use the speed of light to count the number of clicks on our clocks (because all the electromagnetic events on the smallest subatomic level known interfere with the speed of light).
So we have space = $f(c)$ and time = $g(c)$. Now we say that the speed of light is $c = f(c)/g(c)$. This looks more like a ratio between our definitions of space and time other than a constant. What do I miss?

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    $\begingroup$ I disagree with “we also use the speed of light to count the number of clicks on our clocks”. There is electromagnetic radiation used, but we don’t use its speed. We only use its frequency, and the speed is not used. Just because something is present doesn’t mean that it is being used. $\endgroup$
    – Dale
    Apr 20, 2021 at 0:35
  • $\begingroup$ Isn’t the electromagnetic radiation frequency a dependent of the speed of electromagnetic radiation which is actually the speed light? $\endgroup$ Apr 20, 2021 at 11:50
  • $\begingroup$ No. For example, you could put a material with a high index of refraction (reduce the speed of light) between the caesium atoms and the oscillator and it would not affect the operation of the clock. The wavelength and speed would be reduced but the frequency would be unaffected. $\endgroup$
    – Dale
    Apr 20, 2021 at 16:34
  • $\begingroup$ The frequency is determined only at the source of the radiation. At that source is the speed of causality (or speed of light) on the subatomic events inside caesium that is determine its frequency. So even if you manipulate the speed of light outside the atom the frequency you measure still depends on the speed of light in vacuum. $\endgroup$ Apr 20, 2021 at 21:29
  • $\begingroup$ The units for speed are m/s, so yes, it is the ratio and will obviously change with our choice of units. Are you actually trying to ask if it is a conversion factor between "distances" in space and "distances" in time in some metaphysical sense? $\endgroup$ Apr 21, 2021 at 6:06

4 Answers 4

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The numerical value of the speed of light is not a universal constant. It's the quantity that is a universal constant.

As an analogy, in Newtonian physics, the distance between any two points in space, or the time interval between any two events, is a universal constant.

Pick any two points in space in a Newtonian universe. Say, one observer measures the distance between them to be 1m. Another observer observes the same distance to be 100cm. The numerical values of the measurements, i.e. 1 and 100, are different. But both observers are perceiving the same amount of distance between the points. It's just that they use different units. The distance between the points is a universal constant.

Similarly in relativity, the speed of light is a universal constant. Different observers will perceive light to be moving at the same "degree of fastness". The numerical value of the speed of light is not a universal constant, as different observers can use different units to measure the numerical value.

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    $\begingroup$ In the off-chance you don't know this and anyone ever pushes you on what you mean when distinguish a "quantity" from a "numerical value," you can say that the speed of light is a torsor. Terry Tao has an excellent explanation of that here. $\endgroup$
    – Andrew
    Aug 30, 2021 at 23:07
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First: While it is technically true to say that we use the speed of light to define the value of the meter, or the value of a second, it is also slightly misleading. From a more practical standpoint, our human records of "how long a meter is" or "how long a second is" must be rooted in specific measurements of the speed of light, taken by specific scientists at specific places and times.

Suppose, for the sake of argument, that the speed of light in a vacuum was not constant, and tomorrow the Earth moved into a region of space where light moved 10% faster. Would we notice? Of course we would, because all our instruments are built on the units that were based on the old speed of light, not the new one. The ruler on your desk would not suddenly become 10% longer. So it is always possible to detect whether $c$ is remaining constant or not.

Second: The value of the second, and even the value of the meter, are not functions of the speed of light alone. There are other, independent physical constants in the universe; these include the Planck constant $h$, the elementary charge $e$, and the vacuum permittivity $\varepsilon_0$. All of these constants help to determine how excited atoms emit radiation and which wavelengths they emit, which shapes the operation of lasers and atomic clocks and so forth, which are in turn used to define the SI units.

We express the relationships between these constants in terms of the fine-structure constant $\alpha$, which is itself apparently unchanging. If we changed the speed of light without changing any of the other physical constants, the fine-structure constant would also change, and excited atoms would start emitting different spectra of light (among other things). It would be very hard to miss.


Sources:

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  • $\begingroup$ I am not sure your statement is true any longer. We now define the metre as "the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second". If in oldthink the speed of light actually increased and a standard meter bar now appeared shorter than previously thought, we would have to say it has become shorter than when it was previously measured $\endgroup$
    – Henry
    Apr 20, 2021 at 13:37
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    $\begingroup$ As a continuation of your last point. If c changed and something else changed so that $\alpha$ remained unchanged then the change in c would be undetectable. $\endgroup$
    – Dale
    Apr 20, 2021 at 16:42
  • $\begingroup$ @Henry I went back and edited the first paragraph in response to your comment. $\endgroup$
    – MJ713
    Aug 30, 2021 at 20:03
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We exist along wordlines in 4D space. We like to measure three of those dimensions using one unit and the fourth using another, but we could use the same unit for all four and the constant $c$ would vanish from all calculations. We would then notice, for example, that nothing other than light can include both (0, 0, 0, 0) and (1, 0, 0, 1) in its wordline (in the usual Cartesian coordinate system in flat spacetime), making its speed a dimensionless 1.

The speed of light is a conversion factor between meters and light-seconds, and there’s nothing more fundamental about it than there is in the figure of 40.8233133 kg/firkin.

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You actually describe, why the speed of light is necessarily assumed to be the fundamental constant with the definitions you mention. Although of course its actual numerical value in the end depends on OUR choice of definition (for example on which part of the distance traveled by light in one second defines a $m$.).

Let's use an analogy: Let's assume the universal constant we want to use would not be the speed of light, but, as it has been in the past, the rotation of the earth (which is of course NOT a fundamental constant, not even a constant, but it was close enough to one for centuries, so people treated it as fundamental constant). Let's take that constant to be the speed of the earth's surface at the equator $v_{\mathrm{earth/equator}}$, so we do not have to think about angular velocity.

Let's define the second as $$1s\equiv\frac{1}{3600\cdot24}\mathrm{time\,for\,one\,revolution}$$ (so here we have to, additionally to looking at the speed, keep track of a certain 'frequency', that is related to $v_{\mathrm{earth/equator}}$, kind of analogous to the actual definition of a second by counting transitions between certain atomic states) and with that $$1m\equiv\frac{1}{463}\mathrm{distance\,traveled\,in\,1}s\mathrm{\,at\,surface\,at\,equator}.$$ Now we know what one second and one meter is, defined with the help of the ''fundamental constant'' given by the earth's rotation. Then the numerical value, in $m$ and $s$, of the speed of earth's surface at the equator is $$v_{\mathrm{earth/equator}}=\frac{436m}{s},$$ so it is necessarily a ratio of the quantities we defined, since we defined these quantities by our assumed fundamental constant.

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