K2    Natural Units of Measurement in the STR and GTR


If one consistently thinks in space-time units, then it is clear that one must assign the speed of light the unit value 1: One space-time unit per space-time unit. One light-second per second, when expressed in time units. Thus energy, mass and momentum automatically receive the same units, E = m • c2, for example, yields simply E = m. This approach is the basis for all of the following proposals. Study the simplifications that result for the relationships we presented in E5!

Carl Friederich Gauss had already proposed eliminating the fundamental electromagnetic constants ε0 and μ0 by choosing the unit system such that they receive the value 1. This greatly simplifies the theory of electromagnetism. Given Maxwell's equation c2 = 1 / (ε0 • μ0) it thus also follows that c = 1, which Gauss could not have known. According to this proposal by Gauss, the electric and magnetic fields are measured in the same units. And it makes sense: According to STR, these two fields can be directly transformed into one another!

One can go even further: Setting the gravitational constant G to the unit-free value of 1, eliminates the kilogram and allows masses to be measured in time or length units. Especially nice is the use of length units, a mass is then just as heavy as its Schwarzschild radius RS!

Setting even the Boltzmann constant k to the value 1 yields a new temperature scale and allows all physical quantities to be expressed, for example, in cm! This unit system is consistently used in [29].

One need not necessarily take such a radical approach. In any case it makes sense to apply ε0 = μ0 = c = 1. Study for each of the following, how the units of time, length, mass, acceleration, force, momentum and energy convert into those of our conventional MKS system. Determine in each case the corresponding values of the ‘fundamental constants’. This should give you a whole new feeling for the dependence of these values on each other!

  1. Times and lengths in seconds, masses in kilograms. What value does the gravitational constant G have? 
  2. Times and lengths in nanoseconds and the value of the gravitational constant 1. What would the mass of one kilogram be? 
  3. Times and lengths in centimeters and the value of the gravitational constant 1. What is a kilogram equivalent to? How heavy is the sun? 
  4. Same as 3 but also the Boltzmann constant is assigned the value 1. What is one degree Kelvin equivalent to?

Perhaps you may add an appropriate definition of electric charge so that the electric and magnetic units are also included!


“What makes Einstein’s theory of relativity remarkable is its ability to unify various ideas in physics that had previously been treated independently. It unifies electricity with magnetism, materials [perhaps better: mass] with energy, gravity with acceleration and space with time.”     [45-25]