Why $c$ is $3\times10^8$ times faster than a $1\ \mathrm{m/s}$ car? [duplicate]

Question

The value of a dimensional constant like $c$ is often regarded as unimportant since it can be arbitrarily changed to any desired value by changing our units. For example, $c=3\times10^8$ in $\mathrm{m/s}$, and $c=1$ in $\mathrm{ft/ns}$. The value of a dimensionless constant on the other hand, is significant since it's independent of our metric system (e.g. the ratio between the mass of a proton to that of an electron). My question is not the typical "Why is $c=3\times10^8\ \mathrm{m/s}$?" one, but the dimensionless version of it.

Say you're working in SI, you measure a certain car speed with respect to the ground of the earth to be $v=1\ \mathrm{m/s}$, now: $c/v=3\times10^8$. This ratio between $c$ and the speed of this particular car is dimensionless, therefore it's independent of our metric system. Now someone might argue that this ratio is not important since we can change it to any value we desire by changing our frame of reference. Indeed you can change the value of the ratio by changing our frame of reference, However the particular value $c/v=3\times10^8$ remains the same (independent of any metric system) if we stick to one frame (with respect to the ground of earth).

In this sense I ask why $c$ (the upper limit to the speed of any physical object) is $3\times10^8$ times faster than that car (confining ourselves to one frame of reference)? Why $c$ is not some other $x$ times faster or slower? Is there some fundamental reason behind this or is it just an empirical fact, so that it's possible that we can have infinite universes just like ours but with different ratio ($c/v$)?

[Edit]: to put it another way, the crux of my inquiry is this: it's quite conceivable that we could have been living in another universe that is identical with ours, and it has some $c$ as their upper limit, However the value of $c_\text{our universe}$ is different from $c_\text{another universe}$ (in the same metric system), so why we live in a universe with $c_\text{our universe}$ but not another value?

Answer 1

If you ask why $c/v$ is equal to 299,792,458 (this is the right value!), the question obviously depends both on $c$ and $v$. We know that the speed of light is 299,792,458 m/s because that follows from the modern definition of one meter adopted 30+ years ago: one meter is exactly so long that the speed of light in the vacuum is 299,792,458 m/s. This random numerical value was fixed to the integer value approximately measured by 1980 when the error margin was 1.2 m/s. Previously, the numerical value came from the definitions of one second through the solar day, and one meter defined as a fraction of the circumference of a meridian (40,000 km).

One second is defined through a periodic process used in some atomic clocks.

However, your question also depends on $v$, the speed of "the" car, and it's the much less well-defined aspect of your question. Because $c$ is such a robust, well-known, universal constant of physics, your question is basically equivalent to the question "Why you chose the car and its $v$ to be 1 m/s?" This is a particularly good question because if a company produced cars whose speed is just 1 m/s, it wouldn't sell any (except if they were toys for children).

There exists no "canonical car" in the laws of physics whose speed would be $v=1$ m/s and to pretend that this value of the speed has a canonical meaning is equivalent, as dmckee pointed out, to pretending that the dimensionful quantity $c$ is dimensionless. Well, $c$ is not dimensionless, and the ludicrous implicit claims that all cars should have $v=1$ m/s is just another way to see the big difference between dimensionful and dimensionless quantities. 1 m/s is a dimensional quantity, a nontrivial unit, and calling it "the speed of the car" doesn't make it any more dimensionless.

One may ask a similar question. Let's fix the description of 1 m/s as the speed of a walking child instead of a "car", to be more realistic. So the question is why the speed of walking children is about $c/(3\times 10^8)$? Again, the answer depends on the definition of "walking children". What exactly "children" are depends on all the details of life on Earth and the whole evolution of species and the definition of humans, a particular species, as one of the branches of evolution. Different species' infants have significantly different speeds.

So the corrected question is a messy interdisciplinary question in biology, astrophysics, geology, and other fields. There exists absolutely no reason why there should exist a simple answer or even an "exact calculation" of the ratio – obviously, different children have different speeds, anyway. Physics may say that the ratio $c/v$ must be greater than one (special relativity). And it must be substantially greater than one if the walking children don't ionize each other when they hit other children. The speed of light is about 137 times greater than the speed of the electron in the hydrogen atom – a more interesting ratio. It is no coincidence that 137 is the fine-structure constant.

But one must understand that the speed of light $c$ is the normal, canonical, universal speed among the two, while the speed of the walking child or the non-existent superslow hypothetical car $v$ is the contrived part of the ratio, one that doesn't really belong to physics, at least not fundamental physics.