# B Why the speed of light squared?

#### johngalt47

In Einstein's famous equation, why did he use the speed of light squared? Why not some other constant?

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#### sophiecentaur

Gold Member
Welcome to PF.
Have you searched anywhere else for an answer to this question? Einstein's own monograph "Special Relativity' is widely available and probably free as an ebook if you shop around. He actually explains it all very well. If you start off assuming that the speed of light appear the same for measurements in any internal frame (not accelerating) then the Lorenz transformation leads to a c2 even if you don't do it rigorously. Try this link.

#### russ_watters

Mentor
In Einstein's famous equation, why did he use the speed of light squared? Why not some other constant?
It's an energy equation involving speed: the units demand that it be squared. It's not unlike the Newtonian KE equation, which also uses a speed squared.

• sophiecentaur

#### johngalt47

Thanks! I have just recently started to think about this and really don't know much. Also, I have not taken any formal classes in physics so ...

#### russ_watters

Mentor
Thanks! I have just recently started to think about this and really don't know much. Also, I have not taken any formal classes in physics so ...
Fair enough. So remember: when first learning arithmetic there are no units, but when used in science, equations are like sentences and the units are a big part of the meaning. Numbers are almost never just numbers.

• sophiecentaur

#### swampwiz

OK, I'll give you the pop science summary. Because light is observed by any observer to be constant, and that in a reference frame moving at some relative velocity, light being observed moving in a lateral direction to that would be observed by an observer in that other frame at the speed of light, the original observer would observe the light's component of velocity in that lateral direction as being slower - for the same reason that the hypotenuse of a triangle is always longer than any of the right-angle sides - and that the motion of light can be the basis of a proper clock (i.e., think of light bouncing between a pair of mirrors as propagating through time as "light clock" "ticks"), the original observer would observe any proper clock moving with that other reference frame as running slow (i.e., time dilation) - and vice-versa. And since once a proper clock has been established, such a "light clock" could be oriented in the direction of the relative motion of the 2 reference frames to be a "light ruler", for the same reason that a rectangle has its area optimized by being a square, length is observed to be length-contracted in the direction of the relative velocity. In essence, the speed of light is like the hypotenuse, with one of the other sides being the relative velocity, and the other being the square root of the difference of the square of the speed of light an the square of the relative velocity. The speed of light divided by that square root of that parameter would become the parameter typically referred to as "gamma" that describes the amount of time-dilation & length-contraction.

Now consider a pair of spaceships that can each fire out identical billiard balls at some small speed such that they collide. Since the laws of physics are the same in any inertial reference frame (i.e., a frame where something at rest would stay at rest instead of falling like on Earth), there would be conservation of momentum in that collision using the velocities and as well masses of those billiard balls as observed by an observer on either ship. The result of the 2 observers' observation of the other being time-dilated & length-contracted is that the mass of the other is observed to be more massive than that of the observer such that it is scaled by gamma, which means that an object that is moving has extra mass simply due to its motion. And since a mass that is originally at rest can be pushed with some force to change its velocity, the product of that push and the distance of the push is the (kinetic) energy added to that mass, there is a relationship between the additional observed mass of a moving object and the amount of work done on that object; this relationship is calculated as being such that the work done is equal to the product of the original observed mass (i.e., when it was rest, this being the "proper mass") and the square of the speed of light, with that original term for the hypotenuse carried through so as to result in this extraordinarily elegant relationship. From this, it must be that any observation of mass is equivalent to an observation of energy, and thus the rest mass of an object is equivalent to energy as per this relationship.

#### PeroK

Homework Helper
Gold Member
2018 Award
In Einstein's famous equation, why did he use the speed of light squared? Why not some other constant?

"Why the speed of light squared?"

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