The Importance of Newtonian KE and P on Special Relativity

   Special relativity is often thought of as arising as a math transform linked to Maxwell’s electromagnetic equations, or through Einstein’s thought experiments using light and the invariance of frames moving at constant speeds relative to each other.

   Here, we argue that even though Newton’s equations seem to be largely focused on the notion of momentum, which is a vector, the second law dp/dt = F leads to a scalar (under rotation), namely p dot p /2m which is not simply  p dot p. In other words, we argue that  Newton’s notion of both a p (vector) and p dot p /2m (scalar under rotations) as describing a free particle state actually leads to special relativity.  In other words, a single object, either p or p dot p /2m is not sufficient for describing a free particle. We note that an object at rest is also described by a number mo and p=0. For a particle moving at a very low v, the state is still mo to first approximation even though there is mov and .5movv. As a result, we suggest that Newton’s idea of a vector p and scalar .5movv describing a state actually be extended to a particle at rest, and that one must consider mof(v/b)bb and mov h(v/b) where f and h are unknown functions. One may see that a constant b must be introduced and b should have the same value as seen from any constantly moving frame, i.e. should represent a particle with mo=0. We use the idea of a scalar under rotation and a vector as describing a state as characterizing a free particle state. We note that for the rest state p=0 and so one only mobb. This means that one may introduce a linear, i.e. matrix transformation, because mobb, mov and .5movv are all linear in the factor mo. We suggest that given the presence of b, which has velocity units, it must be a constant and have the same value as seen in all frames. This, we argue, implies that a clock in a rest frame cannot have the same period as one in a moving frame.

    As a result, we suggest that special relativity follow from Newton’s idea of characterizing a free particle state by a dual, mo, .5movv and p vector and that one may obtain special relativity from this idea alone because it requires the introduction of a speed b which is the same as seen in all frames moving at constraint v.