# The value of $c$ in $E=mc^2$

In Einstein’s formula $$E = m c^2$$, $$c$$ is the speed of light. However the speed of light is not a unique number. It can be expressed in kilometres per second or miles per second for example, which give different numbers. I realise it is a relationship of distance travelled against time however both are surely human generated values so how can the equation have a real value? If not, is the equation meaningless? Additionally, the $$c$$ represents the speed of visible light in vacuum. Why would the universal conversion of mass to energy and vice versa depend on the speed of visible light only and not, say, ultraviolet or X-ray?

• You can say the same about $F=ma$ or $V=IR$ or anything else with units.... Commented Mar 15, 2018 at 8:06

First of all, you have got yourself completely wrong in saying that visible light is different than ultraviolet or X-rays. They are all part of a larger electromagnetic spectrum, only our eyes have developed to be sensitive to a part of the spectrum, the so called visible light spectrum.

Now, the electromagnetic spectrum is just a collection of different frequencies (or wavelengths) of electromagnetic waves, and all electromagnetic waves travel with the same speed $c$ in vacuum. And $c$ is a constant, the numerical value changes depending upon your units. If you use $m/s$ (and $kg$ for mass), you get $E$ in Joules. Similarly, any other unit will land you with a different unit of $E$.

And that's why the law is universal. (Actually even this is not universal. The actual law is $E^2=p^2c^2+m^2c^4$).

• Thanks Yuzuriha. Much appreciated. That has been bugging me for years and I just found out about stackexchange. My question answered in minutes 👍 Commented Mar 15, 2018 at 8:01
• Its cool. You are always welcome Commented Mar 15, 2018 at 8:02

The units you use for the speed $c$ determine the units of the energy. If you express $c$ in metres per second and the mass in kilogrammes then the energy will be measured in joules. You are free to use different units if you want, and this will produce an energy measured in different units.

The units don't change the quantity, just the way we represent it. For example $3 \times 10^8$ m/sec and $6.706 \times 10^8$ miles per hour are both the same speed $c$ written in different units. The energy that our equation $E=mc^2$ gives us is the same energy regardless of what units we use.

Re your last question: $c$ is not just the speed of light It is the speed of any massless particle. So infra-red and X-rays move at $c$ just like visible light.

Your first question regarding the units (km/s or miles/s) is not specific to this particular formula, but could apply to other formula in physics.

True, if you put the speed of light in km/s you will get a different number in the end than if you put in the speed of light in miles/s. However your result, the energy, also has a unit, and this will depend on the units you use for mass and speed of light. If you use (kg, km/s) as units your energy will be in $\frac{kg~km^2}{s^2}$, if you use (kg, miles/s) your energy will be in $\frac{kg~miles^2}{s^2}$, if you use SI units (kg, m/s), your energy will be in $\frac{kg~m^2}{s^2}=J$ which is also called a Joule.

So, generally all fundamental equations in physics work this way. You can put in any units you like, as long as you keep track of those units (i.e. you don't cancel terms that are km/miles or some such.

Occasionally (often in experimental physics or engineering) you might see equations which state that you have to input things in a specific unit. However these are usually non-fundamental equations where (in order to simplify/shorten things) one got for instance rid of constant prefactors that would also have a unit.

To your second question. The $c$ in this equation is the speed of light (electromagnetic wave) in vacuum which is independent of the frequency of light and is actually a number that has been fixed to a specific value: $c=299,792,458 m/s$ exact (not a measured value).

The equation is valid in any system of units you choose to use. If you use $\rm mi/s$ over $\rm m/s$, then $E$ and/or $m$ will have correspondingly different units.

$c$ represents the speed of all light, not just that of visible light. X-rays, UV rays, and visible light all have the same speed.

All electromagnetic waves propagate at the same speed in vacuum. Light in the narrow sense is mostly only visible light, but radio waves, microwaves, X-rays and gamma rays are exactly the same thing: coupled oscillating electric and magnetic fields. Their propagation speed is a fundamental constant that depends on the permittivity and permeability of free space (also two constants).

$c$, just like $E$ and $m$, has units. Energy has the unit $\frac{kg m^2}{s^2}$ (SI), mass has $kg$ and $c$ obviously $\frac{m}{s}$. As you see, these correspond to each other. You are free to go to another unit system, which changes the numbers involved, but their relation is still the same.

Basically, $c$ is more than just the speed of some light, it is a very fundamental constant that links electric and magnetic fields, and also shows up in special relativity.