x=vt Is Not the Full Space Time Description of a Particle Part 2

x=vt Is Not the Full Space Time Description of a Particle Part 2 Francesco R. Ruggeri Hanwell, N.B. Dec. 12, 2025 In Part 1, we argued that x=vt, although a correct description of motion of free particle with constant v, it is not the full space time description of such a particle. We suggested that a free particle is associated with additional physics which should be experimentally measurable. Here we suggest that the extra physics is conceptually associated with an x,t,E, p point being linked with a number which is invariant in all frames which move relative to each other with a constant speed. One may use the relative result v =p/E (for c=1) and write x= p/E t, but this equation always relates to a rest frame for a particle with rest mass. What one needs is a form which allows one to link one moving frame to another and that does not come from x=vt or x=p/E t. Another expression in x,t,E, p is needed suggesting that something beyond x=vt is associated with this Lorentz invariance. This must be linked to a physical result as argued. Given that x=vt is not sufficient for a full description, we suggest that there may be variations to x and t which still leave the number A unchanged. Thus, A is consistent with x=vt for each x,t,E,p point, but there is another pair x+hbar/p and t+hbarE which leaves A unchanged. In fact, as noted in Part 1, there are many such possibilities, but only one which leads to a probability, associated with x,t fluctuations, which conserves momentum and energy, i.e. exp(-iEt+ipx). Free Particle Motion In Newtonian mechanics, free particle motion with constant speed is described by: x= vt ((1)) One may introduce a constant kinetic energy = .5movv and nonrelativistic momentum, p=mov, but that does not change the motion ((1)). There is no notion of frames moving at constant speeds with respect to each other. There is, however, no physical restriction in considering such frames and suggesting that the math form of the physical equations seen from each frame should be the same. Such ideas were already proposed by Einstein. This viewing from different frames and seeing the same equations of motion means that one must see ((1)) in each frame, except x, t and v differ. Special relativity yields: p=mov/sqrt(1-vv/cc) and E = mocc /sqrt(1-vv/cc) ((2)) Thus: x=vt -> x = p/E t for c=1 ((3)) The point, however, is that p/E yields a velocity relative to the frame in which there is a rest mass mo at rest. It does not allow for a relationship between two frames which may be moving with respect to the rest frame. We suggest that one should have an invariance between any two frames as long as they move at constant speeds with respect to each other. These two frames move at different constant speeds with respect to a rest frame, but one does not need to focus on that. As a result, one requires a different kind of equation to describe the situation, i.e. ((3)) does not suffice because it links everything to the rest frame. This suggests that there is more physics present than what is constrained in x=vt. Given that x=vt focuses on x and t, we suggest that one should expect some possible changes to x and t in a more general scheme. As x=vt is still the correct form of motion, such changes may be in the form of fluctuations, but these must appear in the formalism and be experimentally verifiable (i.e. we speculate that x and t may fluctuate from the trajectory x=vt values). To allow for consideration between any two constantly moving frames, we suggest that an x,t,E,p point in one frame corresponds to an x’.t’,E’,p’ as seen in another. Physically, these points are the same, they are just viewed from different frames. We postulate that there should be a constant number A for each x,t,E,p point. Such a point represents a single x,t pair in the trajectory x=vt (or x’=v’t’ in another frame). This leads to the following relation: A = constant = f(x,t,E,p) where f is a fixed function and x,t,E,p are the values seen by a particular frame. Regardless of the frame, A is the same. ((4)) One may now ask: Does ((4)) introduce new physical information that goes beyond x=vt? If it does, then there must be some physical manifestation of such information. After all, ((4)) is a relation between what is seen by different frames. What does that have to do with what is seen in a single frame? Is x=vt the only thing seen in a single frame? To answer this question, one must quantitatively find f(x,t,E,p). From the results of special relativity, A = -Et+px ((5)) The interesting feature of ((5)) is that if one uses a given x,t on a trajectory to find A, then: x+hbar/p and t+hbar/E ((6)) leaves A unchanged. Furthermore, ((6)) is not the only way to leave A unchanged. The key idea is that a single A value is linked to an x,t on an x=vt trajectory, but also different x1,t1’ values ((6)) that are not on the trajectory. Thus, there is more information involved in space time than simply the trajectory of Newtonian mechanics, we argue. Physical motion is still along an x=vt trajectory so we suggest that ((6)) may represent fluctuations about this trajectory. These fluctuations would be linked with unusual (non-Newtonian) interaction patterns (e.g. 2-slit interference etc). First, however, we try to justify why one should use ((6)). We discussed this in Part 1 and noted that it is ((6)) which leads to a probability pattern of: exp(-iEt+ipx) ((7)) ((7)) creates the hbar/p and hbar/E intervals, but as a probability also conserves momentum and energy. In other words, if one has a 2-body collision: exp(-iE1t) exp(-iE2t) = exp(-iE3t)exp(-iE4t) if E1+E2=E3+E4 ((8)) and a similar expression for p. As a result, we suggest that the notion of viewing the same physical equations (i.e. math forms) from frames moving at constant speeds with respect to each other and a rest frame leads to space time effects which go beyond x=vt. As a result, there is extra spacetime physics involved in Lorentz invariance, we argue. Conclusion In conclusion, in Part 1 we argued that there is physics beyond x=vt in the classical action, which happens to be the Lorentz invariant -Et+px (for both the relativistic and nonrelativistic cases). This extra physics is associated with the notion that physical equations retain the same math form when viewed from different frames. We suggest that this is also linked with each x,t,E.p point being associated with a fixed number A regardless of the frame. X,t would normally be considered to be trajectory points of x,t and such a pair give rise to a specific A value. If there were no other physical information in spacetime present, then only the x,t pair would yield the value of A (assuming a starting point of x=0, t=0). The fact that special relativity (which is needed to describe behaviour as seen from different frames) yields A = -Et+px shows that x,t and x+hbar/p and t=hbar/E yield the same A value. This suggests that there is more than x=vt associated with the constant A idea (i.e. the Lorentz invariance idea) and physics seen from different frames), we argue. We note, as in Part 1, that x+hbar/p and t+hbar/E is only one way to change the x,t values to keep A constant. This choice, however, leads to exp(-iEt+ipx) which is a probability which is consistent with hbar/p and hbar/E and also conserves energy and momentum, which are physical requirements.