Is E=mc² the Formula for the Speed of Light?

[TaurusSteve]

TL;DR Summary

Speed Of Light Formula

What is the equation/formula for the Speed Of Light? E=mc²? I thought it would be simple to find in a search.

 

[Ibix]

It's defined. There's no formula for it.

Historically it was measured by timing its flight over a known distance, just as you'd measure the speed of anything else. But modern understanding lead to the idea that you measure time and define the speed of light to be a specific value. Then distance units are defined in terms of how far light travels in a certain time.

 

[PeroK]

Summary:: Speed Of Light Formula

What is the equation/formula for the Speed Of Light? E=mc²? I thought it would be simple to find in a search.

The speed of light is, nowadays, defined to be exactly 299,792,458 metres per second; hence defining the metre.

You can also get it from the electromagnetic constants: $c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$

 

[TaurusSteve]

It's defined. There's no formula for it.

Historically it was measured by timing its flight over a known distance, just as you'd measure the speed of anything else. But modern understanding lead to the idea that you measure time and define the speed of light to be a specific value. Then distance units are defined in terms of how far light travels in a certain time.

Ah ok cheers!

 

[Ibix]

Just to add - for any wave, its speed is its wavelength times its frequency, so $c=f\lambda$. For electromagnetic waves, also $c=1/\sqrt{\epsilon_0\mu_0}$, where the two constants in the square root are the permittivity and permeability of free space. These appear to be ways to calculate $c$, and there are probably other formulae that didn't occur to me just now. But all of them have one thing in common - if you track through the measurements and calculations you make you will find that somewhere you used the defined value of $c$, most probably when you measured a distance or length. So they are just complicated ways of hiding the definition.

 

[PeterDonis]

for any wave, its speed is its wavelength times its frequency

More precisely, its phase velocity is its wavelength times its frequency.

 

[Ibix]

More precisely, its phase velocity is its wavelength times its frequency.

True. But it's a distinction without a difference for light in vacuum as far as I'm aware, so I glossed over it in a B level thread.

 

[Omega0]

Summary:: Speed Of Light Formula

What is the equation/formula for the Speed Of Light? E=mc²? I thought it would be simple to find in a search.

Physics just describes how things are. c is a constant as far as we know. That's all. No equation.

 

[haushofer]

The speed of light equals 1.

 

[Omega0]

The speed of light is, nowadays, defined to be exactly 299,792,458 metres per second; hence defining the metre.

Correct.

You can also get it from the electromagnetic constants: $c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$

This is correct in a sense of theoretical physics but it is not correct in the sense of measurement technology. $\epsilon_0$ or $\mu_0$ are meanwhile things we have to measure. They aren't given anymore as defined constants.

 

[vanhees71]

True, it also shows that $c$ is just a conversion factor defining the system of units. In theoretical physics we set all the inconvenient conversion constants ($c$, $\hbar$, $k_{\text{B}}$,...) to 1. Exactly the same is done since last year in the definition of the SI units, only that one chooses somewhat more convenient values, defining practical units such that you get convenient values for physical quantities relevant in everyday life and engineering.

 

[Mister T]

This is correct in a sense of theoretical physics but it is not correct in the sense of measurement technology. $\epsilon_0$ or $\mu_0$ are meanwhile things we have to measure. They aren't given anymore as defined constants.

I was under the impression that $\mu_0$ has the exact value of $4 \pi \times 10^{-7}\ \mathrm{N/A^2}$, and since $c$ has an exact value so does $\epsilon_0$.

 

[Vanadium 50]

I was under the impression that μ0\mu_0 has the exact value of 4π×10−7 N/A24 \pi \times 10^{-7}\ \mathrm{N/A^2},

It used to, but in the latest SI revision it is no longer exact.

 

[vanhees71]

In fact the electrodynamic quantities got the "largest" redefinition. If I remember right units like the Ohm got redefinitions at the order of $10^{-9}$. If needed, I can try to find the citations for this. I think it can be found at NIST and other national metrological institutes.

 

[Mister T]

NIST gets its authority from BIPM. That's the best place to look ...

https://www.bipm.org/en/publications/si-brochure/

 

[vanhees71]

Great! That's the link I've been after. First read the Brochure, which gives the formal definitions of all units (except the second) in terms of fixing fundamental constants. Only the second uses a "material constants", namely the hyperfine transition of Cs atomic states, i.e., fixing $\nu_{\text{Cs}}$. All the units of course are linked together via the definition of the fundamental constants (which are in fact becoming just conversion factors from natural units to SI units), $h$, $e$, $k_{\text{B}}$, $N_{\text{A}}$, starting from the definition of the second.

To see how the definitions are realized in practice and about the contemporary accuracy, see

https://www.bipm.org/en/publications/mises-en-pratique/

There they give the uncertainty $\mu_0$ has now on p. 6 of the mises en pratique for the Ampere.

 

[Omega0]

In fact the electrodynamic quantities got the "largest" redefinition. If I remember right units like the Ohm got redefinitions at the order of $10^{-9}$. If needed, I can try to find the citations for this. I think it can be found at NIST and other national metrological institutes.

A nice thing for you to read (written in German) is the following: https://www.ptb.de/cms/presseaktuel...taebe/massstaebe-heft-14-masse-fuer-alle.html
In the printed version there is a nice extra brochure explaining the relations between the constansts in nature and the measurement process. I am not a collector but this edtion of the "Maßstäbe" is something I really recommend because of the big change in 2019. Sounds enthusiastic but for everyone able to read German: Get it!

PS: The extra brochure is especially awesome, it shows the dependency of the natural constants and the measurement process!

 

[Frodo]

The speed of light may be thought of as the "conversion factor between distance and time".

If we take the speed of light to be 3 x 10^8 metres/second this means that in spacetime, 3 x 10^8 metres of distance is equal to 1 second of time.