Is speed of light a ratio or a universal constant?

Question

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?

Answer 1

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.

Answer 2

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.

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Sources:

• https://scienceblogs.com/builtonfacts/2010/02/24/has-the-speed-of-light-changed
• https://www.forbes.com/sites/startswithabang/2019/05/25/ask-ethan-what-is-the-fine-structure-constant-and-why-does-it-matter/
• https://en.wikipedia.org/wiki/Physical_constant
• https://en.wikipedia.org/wiki/Fine-structure_constant

Answer 3

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.

Answer 4

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.