Can the speed of light be 1 ?

 We know that the speed of light is an universal constant that has a value 3.8×10⁸m/s and is denoted by 𝑐. It is so fundamental that it is used in many important revolutionary scientific equations. It is okay for scientists to use the normal SI unit system (i.e. meter, second, kilogram etc.) to measure properties of slow moving objects. But in the case of objects moving with huge speed (comparable with the speed of light), we have to use the ridiculous value of 𝑐 every time. Since it is so much useful and important in science, can’t we have a new system of measurement so that we don’t have to deal with the huge value of 𝑐. We certainly can and it makes the calculations and equations much more readable and easier.  Even this unit system is very popular and useful in the Special theory of relativity.

In this new system, time and distance both are measured in meter, energy and momentum in 𝑘𝑔, acceleration in 𝑚⁻¹ etc. Looking weird right? Don’t worry, you will get familiar with this if you practice it and it will seem to be “natural” for you. Now let’s move to the definition of the new unit system.

Here, the new unit of time is meter. One meter is the time taken by light to travel one meter (You may have heard of “lightyears” which is the distance covered by light in one year. It is the reverse approach). So the speed of light becomes,

                        𝑐 = ( 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑙𝑖𝑔𝒉𝑡 𝑡𝑟𝑎𝑣𝑒𝑙𝑠 𝑖𝑛 𝑎 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑖𝑚𝑒 ) ⁄ ( 𝒕𝒉𝒆 𝑔𝑖𝑣𝑒𝑛 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 )

                  ⟹ 𝑐 = 1𝑚  ∕ ( 𝑡𝒉𝑒 𝑡𝑖𝑚𝑒 𝑙𝑖𝑔𝒉𝑡 𝑡𝑎𝑘𝑒𝑠 𝑡𝑜 𝑡𝑟𝑎𝑣𝑒𝑙 1𝑚 )

                  ⟹ 𝑐 =  1𝑚  ∕  1𝑚

                  ⟹ 𝑐 = 1

You may wonder how the last line can be derived as we are dividing distance by time. But note that meter, second in SI unit system are not fundamental. These are convenient for human use.  We measure distance and time with respect to a frame. The one that is fundamental, is the speed of light. We measure the speed of an object with respect to the speed of light. Suppose you have a graph paper. Now draw a point on the graph. How will you measure the position of the point? By drawing x and y axis right? So, here you define a measurement system and take the point of intersection of the axes as the origin and then calculate the position of the point. Now think, you can take another point as the origin point and calculate the position of the point. This time you will get different values. This means that measurement of a property depends only on the measurement system. Similarly, here we have introduced a new system where 𝑐 is exactly 1.

So you can see the speed of light is exactly 1. You may notice one more thing: 𝑐 is dimensionless! This implies that velocity is dimensionless in this “natural” system. Now you can easily derive the units of other physical properties: Energy is defined by the expression  𝐸 = ½ 𝑚𝑣²   ,but velocity is dimensionless in this system .So the unit of energy becomes kg (only the unit of mass). Similarly the unit of momentum is kg (Again the unit of mass only). Again acceleration is defined by  𝑎  =  𝑣  ∕  𝑡, velocity 𝑣 is dimensionless so we consider the unit of time only. This implies that the unit of acceleration is  𝑚⁻¹ .

One more thing to discuss, the relation between second and meter in this “natural” system:

Light has a speed 𝑣 = 3.8×10⁸ 𝑚 ∕ 𝑠 ( SI unit). It means that light travels 3.8×10⁸  m     in one second. This implies that light takes 1/(3.8×10⁸ ) 𝑠  to travel one meter. Therefore,  1𝑚 (in new system) = 1/(3.8×10⁸ ) 𝑠  ( in SI system).