Speculation on Describing All Uniform Motion by the Same Constant
Francesco R. Ruggeri · Zenodo (CERN European Organization for Nuclear Research) · 2022
It is well known that uniform classical motion may be described by a constant velocity ie. one which does not change in time. A different constant number, however, is needed for each velocity so uniform motion in general is not being represented by a single number in this scenario. In (1) we noted how a single number such as energy E could represent systems linked to different physical theories. For example, a classical particle may move according to .5mvv + V(x) = E where v is velocity. On the one hand a single number E describes each point (x,t), but at each point there is different detailed motion such as potential energy, velocity etc. Similarly in a Maxwell-Boltzmann gas exp(- ( .5mvv + V(x))/T) is proportional to probability. For a given E1 = .5mvv + V(x) there are many v,x points which have the same E1 and thus probability. Finally for a quantum bound state there is a single En (energy) even though there is a potential energy V(x) and average kinetic energy -1/2m d/dx dW/dx / W (W=wavefunction) at each x. In this note we ask whether a single number may apply to all uniform motion with this constant being equal then to different details in each system. The reason we ask this question is that we think a given constant velocity, say 25 m/s, does not characterize uniform motion. The problem we argue is one of resolution of x and t. For example, 25 m/s may characterize very erratic motion from x=0 to x=25 m in a period of 1 sec. This simply means the average speed is 25 m/s, not that the motion is uniform. To bypass this argument one may suggest perfect x, t resolution for all speeds. In such a case erratic motion may only exist in an extremely tiny dx or dt which approaches 0. This suggests there exists a spacetime (x,t) which may analyze all uniform motion with infinitesimal precision i.e is all-seeing. It is not clear that there exists any physical mechanism for this. We argue instead that velocity is a property of a moving object, not of spacetime. An object undergoes force over distance and receives a velocity which is one of its properties. In such a case infinitesimal resolution of t and x may not apply to all velocities. In fact if v velocity is a property of the object, x,t resolution may also be associated properties. Thus details of a particular uniform motion are not simply v (which yields p and E=pp/2m nonrelativistically), but also x and t resolution. Is it possible that these details may combine to create a single constant representing all uniform motion? We argue that this is possible and that this constant is in fact the free particle action -Et+px which holds both relativistically and nonrelativistically. For t proportional to 1/E and p proportional to 1/p (same proportionality constant) one has a single constant, Action=0, describing all uniform motion. We however, arrive at the form -Et+px without using notions of action or Lagrangian.