The Importance of Newtonian KE and P on Special Relativity

The Importance of Newtonian KE and P on Special Relativity Francesco R. Ruggeri Hanwell, N.B. Oct. 6, 2025 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. Special Relativity Newtonian mechanics was developed in the 1600s, but special relativity did not really appear until the late 1800s with studies of properties of transformations of Maxwell’s electromagnetic equations. In 1905. Einstein introduced thought experiments using light (c being considered the same as seen in all frames and a maximum speed) which treated frames moving at constant speeds with respect to each other as being invariant. As a result, one might wonder if special relativity could have been developed earlier with no notion of photon speed or Maxwell’s electromagnetic equations. We also give up here the notion of invariant frames, even though this idea is purely physical and could have been considered at Newton’s time. The Importance of Describing a Free Particle by Both Kinetic Energy and Momentum Newton’s laws seem to focus on momentum, which is a vector. Even the second law: dp/dt = F ((1)) is based on momentum. Momentum is physically linked to impulse and so one might think that it is a complete description of a free particle, although one might want to also include mo as m1v1=m2v2. It is true that one may obtain a second quantity: Mo dv = Fdt = Fdx/v —> Integral F dx = .5movv (if v initial =0) ((2)) This second quantity, however, is used by Newton in scenarios in which there exists a potential. For a free particle, momentum seems to almost always suffice and may be used to describe collisions with a target. There is, however, a case in Newtonian mechanics, where both momentum and kinetic energy are used together and that is in the case of elastic scattering. Although not all two body scattering cases are elastic, the Maxwell-Boltzmann ideal gas is based on these elastic collisions and so elastic collisions are important. The key idea we wish to stress is that even though kinetic energy usually appears together with potential energy V(x), it is still part of the description of a free particle. In other words, we suggest that Newton’s work implies that a free particle cannot be described by momentum alone, but should be described by: Mo, .5movv and p = mov (nonrelativistic) ((3)) A particle at rest has mo, .5movv=0 and p=0 ((4)). If v is very tiny, to a first approximation, one has mo. Thus, we suggest that for small v, mo and .5movv as well as as p vector should describe a free particle. This implies that one should have: mo bb + .5movv and p vector ((4)) Here b is a constant with units of speed which has the same value as viewed from any frame. Presumably it should represent a physical object, but for the present, its existence suffices. We generalize ((4)) and postulate that any free particle state is represented by Mo f(v/b) bb and p vector = mov h(v/b) where f and h are unknown functions ((5a)) Given that one only has mo in the rest frame, it seems that for one -dimensional motion, one may create the vector: (p=0, mobb) ((5b)) and argue that there exists a 2x2 matrix which transforms it to create p’b and mo f(v/b) bb. We do not assume any invariance between a rest frame and one moving at a constant speed. In the Newtonian picture, a particle at rest differs from a moving one (constant speed) by the condition that work has been applied. Nevertheless, the fact that ((5)) is linear in mo and that one has only mo in the rest frame, one may consider: | g(v/b) v g(v/b) | ((6)) | vg(v/b) g(v/b) | At this point, however, g(v/b) is completely unknown, and one has also taken the liberty of introducing some a priori symmetry into ((6)) which may or may not exist. A key idea, however, is that this transformation, like mobb, requires the existence of constant speed b which is the same as viewed in all frames. This seems to imply an object with no rest mass, but other than that, it is unknown. The very fact that b exists and has this property, however, implies that clocks must be different in a rest and moving frame. Clocks in a Rest and Moving Frame Given the existence of a constant speed b (for an object with zero rest mass), the usual arguments of special relativity show that a clock cannot tell the same time in both frames. To see this, consider an object with speed b moving along the y axis and reflecting between two plates separated by distance L. In the moving frame, the overall speed of the object must still be b and it will travel a distance L in the y direction, but also a distance vt’ along the x direction. Thus: bbt’t’ = b ( vvt’t’ + btbt) → t’ = t/ sqrt(1-vv/bb) ((7)) As a result, special relativity follows from the existence of b which has the same value as seen in all constant speed frames, but the existence of b follows from the idea that one may characterize a free particle state by mo f(v/b) and p=mov h(v/b). Thus, it is ultimately Newton’s idea that there must be both .5movv and p=mov, extended to the case of mobb +.5movv for v very small which ultimately leads to special relativity, we argue. Given ((7)), one may return to ((6)) and show that: g(v/b) = 1/sqrt(1-vv/bb) ((8)) At no point have we argued that there is invariance between two frames moving at constant speed with respect to each other. We have simply argued that a free particle must be characterized by a number proportional to mo which must become mobb for v=0 and a vector proportional to mov (and multiplied by some function h(v/b)). One may, however, show mathematically that ((6)) ultimately leads to different frames moving at a constant speed relative to each other being invariant. In particular, given ((7)) and ((6)) one may show that: X’x’ - bt’bt’ = xx - btbt ((9)) As a result, the two frames must be invariant, which matches the physical observation that one cannot tell if one is in a moving frame or a rest frame if one sees an object move at constant speed. This idea was already presented by Einstein. Using ((6)) one finds that: mobb/sqrt(1-vv/bb)) and p = mo v /sqrt(1-vv/bb) ((10)) yield the two values (number and vector) which describes a free particle. Conclusion In conclusion, we argue that Newton’s idea that a free particle may be described by both p dot p /2m and p vector is the key idea which leads to the development of special relativity. In a free particle case, p may be used to describe the impulse hit against a target, but one should really also have the number mo because m1v1=m2v2 yield the same impulse. P dot p/2m is usually used when there is a potential V(x), i.e. one does not have a free particle, but in the case of an elastic collision, both p dot p/2m and p are key to describe a particle, We suggest that one generalize this idea and argue that any free particle be characterized by mo f(v/b) and mov h(v/b), where f and h are unknown functions. This requires introducing a constant speed b which has the same value as seen in all frames moving at constant speed. B presumably represents an object with zero mo. Given that one only has mobb in the rest frame, one may create a 2x2 matrix to transform it into mo f(v/b) and mov h(v/b) without any notion of the frames being invariant. To find the actual values of the matrix elements, one may note that the existence of b requires that clocks in a rest and moving frame to differ by: t’ = t /sqrt(1-vv/bb) with t’initial = t initial = 0. This allows one to construct the Lorentz 2x2 matrix and find that mobb becomes: mobb/sqrt(1-vv/bb) and p= mov/sqrt(1-vv/bb). For v<<b, one has mobb +.5mvv and p=mov, the generalized Newtonian results. Thus, we argue that Newton’s idea that one must have p dot p /2m and p=mov vector both describe a free particle is the seed which leads to special relativity.