—————————— read pre-print on research gate (new window) ——————————

There are no specific inputs needed to resolve this expression.

• l_f, m_f and t_f are the fundamental measures, more precise expressions for Planck’s units – length, mass, and time – that consider the effects of length contraction associated with discrete measure.
• θ_si can be measured as the polarization angle of quantum entangled X-rays at the degenerate frequency of a maximal Bell state. As an angle θ_si=3.26239 rad ± 2 μrad; as a momentum θ_si=3.26239030392(48) kg m s^-1 and with respect to the Target Frame, θ_si has no units. The relation of angle and mass is mathematically demonstrated, as well, by No-Ping Chen, et. al.
• v is velocity between an observer and a target.
• c is the speed of light which may also be written as c=n_Ll_f/n_Tt_f=299,792,458 m/s such that n_L=n_T=1 is physically significant.
• n_L, n_M and n_T are physically significant discrete counts of l_f, m_f and t_f respectively.
• n_Lr describes the count of l_f representative of the position of an observable with respect to the frame of a center of mass.
• E is the energy of a gravitational mass with a given velocity.
• M and m are the observed mass of a system.

We present an MQ description of the orbital velocity of stars for the Milky Way as a function of the upper count bound to units of fundamental mass per increment of fundamental time. The purple line in the graph below describes this bound, unadjusted for the mass profile of a galaxy. When account for a profile of the Milky Way we resolve the curved charted in red. The source data derives from McGaugh's MOND data, which is charted in green. By comparison, the blue line describes Newton's expression.

Importantly, the data demonstrates the physical significance of count bounds and this provides a physical foundation for correlation of the above expressions which resolve also the expression for kinetic energy.

There are many approaches to a resolution of the expression for kinetic energy. What distinguishes this approach from often seen classical implementations is use of the Measurement Quantization (MQ) expression for rotational star velocity. The derivation also reveals cross disciplinary applications, for instance, new approaches to a description of the upper bound velocity to the velocity of stars in a galaxy and how that bound distinctly identifies the relation of energy to mass. We also describe the count relation of fundamental mass to that of length and time, which are also present in Einstein's expression E=mc². Consider then a review.

MQ is an expansion of the existing classical nomenclature. Its most notable feature is a separation of measure - the fundamental measures l_f, m_f, and t_f - from counts counts of those measures. When applied to a description of [Heisenberg's uncertainty principle][5] we discover, that the reference measures cancel out leaving us with only counts.

The MQ approach is physically important, therein, in its ability to separate physical measurement from the geometry associated with an observed phenomenon. With this overview, we turn our attention to the derivation.

The rotational velocity of stars can be described as a function of c times the square root of fundamental unit of mass over a count of fundamental unit of length, each correlated with time. The count ratio is then 2m_f, a description of the upper count bound to m_f with respect to the discrete Measurement Frame of the observer. Resolved with respect to the non-discrete Target Frame of the universe - which has no external reference - the expression is scaled by θ_si, the radial rate of universal expansion.

Continuing the reduction, without the expansion parameter (that is keeping with the Measurement Frame), we resolve the self-referencing form of the fundamental expression under the root. This can then be reduced to terms of energy equal to one unit of fundamental mass, E_f=2θ_sic, giving us just 2E_f. In other words, the energy of a fundamental unit of mass is E_f=2v² and if we were to consider a mass m_f where the smallest count of m_f is n_M=1/2, then that mass moving at the speed of light is just E_f=2(1/2)m_fc². If generalized for any energy and any mass we multiply both sides by n_E=n_M to get E=mc², a simple demonstration of the considerable flexibility and directness of MQ when working with physical expressions.

A continuation of the derivation, we move m_f to the denominator and then recognize that 1/m_f is the upper count bound frequency of a fundamental unit of mass. We will need to keep in mind that this is not any count, but specifically that count at the upper count bound corresponding to mass frequency.

Finally, we multiply by the fundamental expression (2θ_si/cm_f)=1 and then reduce with the energy expression E_f=2θ_sic. But, this time we take the generalized form as noted above in our derivation of Einstein's energy equation to find that we are left with just 2E/m under the root. Finally, squaring both sides we resolve the expression for kinetic energy.

The reduction incorporates several physical disciplines: the rotational velocity of stars, the upper count bound to fundamental units of mass which in turn constrain gravity and lead to the dark matter phenomenon, Einstein's expression for energy and finally the expression for kinetic energy. We also discover several other relations that are not well known. For one, in the expression we correlate mass and velocity. That is, squaring the first and last term of the first line we get v²=c²(n_M/n_Lr). This can then be organized into the form of a relativistic speed parameter such that (v²/c²)=(n_M/n_Lr) giving us the relation between motion and mass which constitutes gravitational curvature at the upper count bound to the mass frequency. This expression leads to the expression for motion as a product of the Pythagorean theorem and demonstrates that equivalence is a demonstratable outcome of the geometry.