x=vt Is Not the Full Space Time Description of a Particle Part 3

x=vt Is Not the Full Space Time Description of a Particle Part 3 Francesco R. Ruggeri Hanwell, N.B. Dec. 13, 2025 In Parts 1 and 2, we argued that the Newtonian equation x=vt is not the full description of spacetime for a particle moving with constant speed. In Part 1, we argued that the action = Lagrangian * t = -Et+px (for x/t=v) is more general and in Part 2, we argued that the Lorentz invariant A = -Et+px is more general. In other notes, we tried to show that the Lorentz invariant -Et+px must be the classical action -Et+px (x/t=v) for a free particle, thus the ideas of Parts 1 and 2 are consistent. Here we argue that the reason that x=vt is not sufficient to describe what happens in spacetime is because spacetime itself changes when viewed from a moving frame, i.e. x,t in a rest frame are not the same as x’,t’ seen from the constantly moving frame. This is something that is very well known already, but it is this fact that causes x,t to transform like p,E. Newtonian mechanics already shows that a particle at rest with rest mass mo obtains a kinetic energy .5movv and a momentum mov, but x and t remain the same as one coincides only the rest frame. In the special relativistic case, x and t do not remain the same, even in the v<<c limit because an x=0, t in the rest frame becomes x’ = g(v) v t and t’=g(v)t, where g(v)=1/sqrt(1-vv/cc). Even if v<<c and g(v)-->1, x’ = vt and not 0, so there is a jump in x. Thus, one must consider not only mo transforming to E,p, but x,t transforming to x’,t’ under the same transformation. Given the properties of the transformation and the minus metric which arises because one requires a negative to link E,p to mo as E>mocc, one finds that the Lorentz invariant A = -Et’+px’ exists and this allows for x,t and x’+hbar/p, t’+hbar/E to yield the same A value. It is the change from x,t to x’,t’ that ultimately leads to the physical x+hbar/p and t+hbar/E results which represent free particle quantum mechanics, we argue. We note that x’=vt’ for the center-of-mass, but that does not prevent a p impulse hit from being delivered at x’+hbar/p and an energy interaction from occurring at t’+hbar/E. In other words, the statement that the math form of physical equations must be the same in all constantly moving frames is independent of Newtonian mechanics (as seen by p=mov/sqrt(1-vv/cc)), and may give rise to physical properties in a given frame. In other words, special relativity is not simply about comparing results as seen in different frames, we argue. Newtonian Mechanics In Newtonian mechanics, a particle at rest with rest mass mo which undergoes a force acting over a time which causes the particle to move with v leads to: Kinetic energy = .5movv and momentum = mov ((1)) Mo transforms to a momentum and kinetic energy plus rest mass, but the x and t co-ordinate systems do not change as there is only one co-ordinate system described by x and t. The motion of the particle is then: x=vt ((2)) and in terms of movement, ((2)) seems to be the full description. Special Relativity Consider a particle with rest mass mo at rest at x=0 at t. If a person in a moving frame (moving at constant -v) sees this particle, he/she thinks it moves at v. At first guess, it should have Newton’s kinetic energy and momentum ((1)). The problem is that x and t from the rest frame (with an initial x=0, t=0) cannot apply to the moving frame. If a person in the moving frame has his/her ruler and clock designated by x’ and t’ and considers x’=0, t’=0 to be an initial point, then: x’ = g(v) v t and t’ = g(v) t where g(v) is an unknown function ((3)) Thus, x’,t’ and x,t as co-ordinate systems differ even though people in both frames recognize the math form: x=v1t or x’=v2t’ as applying ((4)) In the rest frame, v1=0 and in the moving v2 = v. This is a well-known result of special relativity, but it shows that something happens to x and t when viewed from a moving frame even though the math form x’= velocity t’ remains the same. Even though one might at first think that x=vt is the full description of motion in space, it is not the full description of spacetime. The question then becomes: Can the x’,t’ change in spacetime actually lead to physical consequence in a single frame? Special relativity seems to compare results as seen from different frames. For example, one notes that there may be Lorentz contraction, i.e. L seen in a rest frame might appear as L sqrt(1-vv/cc) to someone watching an object of L at rest move with v. The point is that a comparison of frames occurs. There is no consequence in a single frame. For example, a ruler of length L in a rest frame is accelerated to v. In the rest frame of the ruler, one still sees L even though x’,t’ change. The real issue then becomes whether a spacetime change can actually be associated with a physical consequence in a single frame. We argued in Part 2 that it can. We first note that mo transforming to E,p is expected from Newtonian mechanics whether a particle is accelerated via a force or seen by a frame moving with constant -v. One should be able to transform mo to E,p. If one uses Newtonian reasoning mo→ mov to create a momentum, but this does not transform mo to E. If one is willing to accept that .5movv might be a small change to moC, then E = moC + .5movv to first order. One may consider: | g(v) g(v) v | |0| ((4)) | vg(v) g(v0 |moC| where g(v) is undetermined Furthermore, E> moC (C is a constant to convert mo units to energy ones) because work is done. If one wishes to have a constant modulus squared as usually done for linear algebra, one cannot use the usual EE + ppcc = momo ccccc (with C=c from special relativity theory) ((5a)) One must use instead: -EE + ppccc = -momo ccccc ((5b) A key idea is now that x,t transform according to ((4)) as well and one may show that g(v)=1/sqrt(1-vv/cc). Such a transformation on x, t is not anticipated in Newtonian mechanics. This leads to the Lorentz invariant: A = -Et’+px’ ((6)) E and p are fixed values because they were created through work and changing them, even due to a fluctuation, would violate conservation of energy and momentum. One might argue that x’=vt’ still applies and so x and t are fixed values, i.e. only values from x’=0, t’=0 to x’,t’ which satisfy x’=vt’ are allowable. This is true for center-of-mass motion, yet at the same time, ((6)) mathematically allows: x1’=x’+hbar/p and t1’=t’+hbar/E leaving A unchanged ((7)) We reiterate that ((6)) would not exist in the first place unless x,t → x’,t’. Thus, it is the spacetime change from x,t to x’,t’ which allows for ((7)) to exist. If x’=vt’ because equations of motion appear identical in math form in any constantly moving frame, how can one possibly justify an x’+hbar/p and t’+hbar/E? One cannot justify it in terms of center-of-mass motion, but one can have off center-of-mass hits of p in space and E interactions of the center-of-mass time. This is unusual, but does not violate any of the ideas of physics discussed above. There is no violation of x’=vt’ for the center-of-mass and because E and p do not change, there is no violation of conservation of energy and momentum. We suggest that the x,t-> x’,t’ transformation allows for the possibility of x fluctuations for the delivery of p impulses and t fluctuations for the interaction of E. This is an effect which follows because x,t transforms to x’,t’ and can be tested through a 2-slit experiment. In other words, the assumption that physical equations have the same math form in any constantly moving frame is an idea which is independent of Newtonian mechanics. There is no reason why this cannot lead to new physical results in a given frame and not simply a recipe for comparing values as seen from different frames. For instance, momentum is mov/sqrt(1-vv/cc) which is a physically measurable result in a given frame and differs from Newtonian mechanics, becoming the Newtonian result mov in the v<< c limit. Similarly, E→ mocc + .5movv in the same limit. There is not reason that there may not be more physical consequences than this, in particular to space and time, not through center-of-mass motion x’=vt’, but through the relationship of interactions of E and p in x,t, i.e. not at the center-of-mass x’,t’ points. In particular, A = -Et’+px’ is a relationship between E, p and x’,t’. We ask: Why would one need such a relationship in the first place if x’=vt’ describes everything? Again, one might simply use A=constant to compare different frames, but in any given frame, it is a relationship between E and p which goes beyond x’=vt’ and so should describe some physics in the given frame, we argue. Conclusion In Parts 1 and 2, we argued that x=vt cannot be a full description of spacetime effects for a particle moving with constant speed. Certainly, the center-of-mass should move as x=vt. This is even the assumption made by special relativity. An equation of motion in one constantly moving frame has the same math form as seen in a frame moving with a different constant speed. It is also known from special relativity that x,t -> x’,t’. This is something that would not be anticipated by Newtonian mechanics, but one might argue that this simply means that one may compare results in different frames. This is the usual procedure, e.g. Lorentz contraction of a ruler etc. One does not usually argue that the x,t transformation to x’,t’ can actually cause spacetime effects in a single frame as seen in that frame. We argue that because x,t→x’,t’, one may create the Lorentz invariant A = -Et’+px’. Then, x1’=x’+hbar/p and t1’ = t’ +hbar/E both yield the same A. The center-of-mass must move as x’=vt’ and with x’ initial =0, t’initial =0, one would expect that only x’,t’ values satisfying x’=vt’ to be allowable. This is true for the center-of-mass motion. E and p are constant values in the A expression due to conservation of energy and momentum, but there is no reason why a p impulse hit may not occur at x1’ and an energy interaction at t1’ instead of x’,t’. This we argue is the case of free particle quantum mechanics and it arises because x,t transforms to x’,t’ in the same manner that mo transforms to E,p. It is the spacetime change which is usually associated with a comparison of two frames which actually leads to changes in a single frame. For example, even in a rest frame, one has: A = -mo t + p1 x where p1->0 so x-> infinite and t1=t+hbar/mocc. As a result, a probability which describes this behaviour is exp(-iEt+ipx). This is measurable physics which goes beyond x=vt and is directly due to x,t→x’,t’ in the same manner as mo→p,E. It is not simply a comparison of values in different frames, but an actual new spacetime phenomenon in a single frame. This phenomenon is a manifestation of the constraint that the math form of equations in all frames moving at constant speeds be the same. This is not an assumption of Newtonian mechanics, but a new constraint and there is no reason that it should not yield new physics, i.e. free particle quantum mechanics.