Three Frames of Reference Prerequisite to Deriving the Physical Constants

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Phenomena are often described using two frames of reference, that of the observer and that of the phenomenon being observed. To this, there is an implicit third frame such that all expressions are defined relatively as a function of the speed of light. We approach this anew. Using the Pythagorean theorem we describe observations as a function of a Reference, Measurement, and Target Frame. The discrete reference and Measurement frames internal to the universe and the non-discrete Target Frame of the universe are initially mapped to a right-angle triangle. This establishes a correlated framework with which we can then resolve expressions and values for the physical constants using only the fundamental measures. Notably, the fundamental measures are more precise versions of Planck’s unit expressions that account for the effects of length contraction associated with discrete measure. They are physically significant, discrete, and countable. As a demonstration, we then resolve expressions and values for the gravitational, fine structure and Hubble constants.

Keywords: Frames of Reference, Frames, Discrete, Measure, Measurement Frame, Target Frame

1. INTRODUCTION

The importance of frames of reference has become more evident with the introduction of Special and General Relativity (SR/GR) [1]. The notion of frames has been one where we consider measure and the laws of nature relatively as a function of inertial and/or gravitational masses. In the case of a gravitational mass, we consider measure relatively between observers and their relation to a center of mass. To simplify this discussion, all such examples are referred to as two-frame systems of reference.

What have we learned as a community? For one, we find both nomenclature and approach essential to resolve physically precise descriptions of phenomena. But also, we gain insight into the underlying structure of reality as a function of those principles best described by relativity.

This paper does not discuss two-frame systems. Rather, we discuss descriptions of the physical constants as a difference between the discrete internal Measurement Frame of the universe and the non-discrete Target Frame of the universe. The Measurement Frame is physically assessed as a function of measures resolved with respect to the discrete Reference Frame of the observer ([2], Appx. K). Therein are resolved the reference measures l_f, m_f, t_f and θ_si (i.e., more precise solutions to the Planck units [3] that account for a length contraction effect associated with discrete measure ([2], Appx. D, W)).

Notably, θ_si has a value equal to 3.26239030392(48) ([2], Appx. AD) and can be resolved as either a momentum or an angle, the angular measure resolved as modelled with respect to specific Bell states, demonstrated with entanglement experiments by Shwartz, et. al. [4-6] ([2], Appx. A, Q-S). Also describing half of the Planck momentum, θ_si is further clarified by Po-Ning Chen, et. al. in their paper entitled, Conserved quantities in general relativity – the view from null infinity [7] whereby angle and mass are clarified as conserved quantities supertranslation invariant. This corresponds to our own observations of equivalence at the lower count bound ([1], Appx. S). Therein, the focus of this paper starts with physical support for discreteness and countability of measure as established in the paper entitled, Measurement Quantization [2].

With respect to the Measurement Quantization (MQ) approach, the Measurement Frame coincides with the modern definition of inertial frames but is shown to be a discrete function of physically significant reference measures. And such that the universe has no external reference, the Target Frame of the universe is shown to be non-discrete.

Notably, drawing attention to frames of reference is in itself physically insignificant. Frames are merely tools used to describe nature. This paper advances prior observations demonstrating that tools introduce bounds to what we can understand about nature. Using the MQ approach, we demonstrate that the difference between the discrete and non-discrete frames is what provides the physical substructure that gives rise to the physical constants and the laws of nature.

As a demonstration of these claims, we resolve expressions and values for the gravitational, fine structure and Hubble constants ([2], Appx. C, AA, and BA). Our calculations match the measurement data with up to 12 digits of correspondence. We also present expressions that correlate quantum and cosmological phenomena. Importantly, the MQ approach is distinguished from prior considerations such as Loop Quantum Gravity (LQG) [8-10]. That is, MQ does not find discreteness to be a property of all frames.

2. METHODS

This paper uses classical expressions in a way that account for new properties of measure resolved using discrete expressions describing gravitational curvature, the speed of light, escape velocity and Heisenberg’s uncertainty principle [11]. We identify this approach with the name Measurement Quantization. Notably, several examples in the publication record exist [2, 12-17] whereby MQ is applied to problems in modern theory.

While not essential to understanding this presentation, this paper rests on physical evidence supporting the discreteness and countability of measure. Derivations predicting these properties and supporting measurement data are presented in the introductory paper entitled, Measurement Quantization [2]. All referenced appendices, along with a more extensive list of term and keyword definitions are included. A pre-print can also be found in Online Resources.

https://www.researchgate.net/publication/361148724_OnlineResources

Also available are two spreadsheets. Suppl. 1 contains calculations of all expressions in the introductory paper, which include calculations for expressions listed in the referenced appendixes. Suppl. 2 contains calculations of all expressions in this and 49 other papers that implement the MQ approach. Importantly, MQ is predicated on the physical support established in these resources.

3. RESULTS

A quantized view of the universe is achieved using three frames of reference. While these frames partially coincide with modern approaches to physical description, they also differ from those frames so often used with relativity. This is not to say that relativity is not relevant. Rather, the three-frame approach is used for different purposes. The approach we present is used to derive the physical constants, their values, and the laws of nature using only the fundamental measures l_f, m_f, t_f and θ_si. The approach can also be used to derive expressions for relativity, in discrete or classical form ([2], Appx. AU, AV).

To clarify, classical theory considers frames as having equivalent properties. MQ considers the properties of measure – its discreteness and countability – with respect to their instantiation, their measure and whether that measure is within the Target Frame or with respect to the Target Frame. Notably, discrete measure implies that some part of that measure has no physical significance (i.e., the fractional count n_L of l_f resulting from a more precise calculation). The effect is distinctly different than that described by relativity. Thus, descriptions of constants and phenomena must consider the effects of length contraction associated with discrete measure in addition to the effects described by relativity.

The process is as follows. We resolve the fundamental form of an expression defined with respect to the non-discrete Target Frame. We apply a frame transform – the metric differential – to resolve its Planck-like form with respect to the discrete Measurement Frame. And finally, we account for the length contraction effects of discrete measure – the Informativity differential – with respect to that frame. This provides us the classical form of the expression, what is measured in the laboratory.

The three frames – also represented by sides a, b, and c in Fig. 1 – are described by the ([2], Appx. K):

• Reference Framework—This is the framework of the observer where properties of the reference AB are observed. With respect to the standard understanding, this framework differs only in that measure is a count function of discrete length measures equal to one.

• Measurement Framework— This framework shares properties with the Reference Framework. It is characterized as some known count of the reference describing where count properties of the reference BC are observed.

• Target Framework—This framework is characterized by the property of measure of non-discreteness, that being the framework of the universe that contains the phenomenon AC.

We can apply these definitions in several ways to resolve descriptions of observed phenomena. By example, we can present solutions to the gravitational constant, the fine structure constant, and Hubble’s constant. We can also discuss the relation of quantum and cosmological phenomena.

Notably, the metric differential ([2], Appx. T) is a function of the phenomenon being described and its relation to the Target Frame. Such that the Target Frame can be described as a dimensionless count of θ_si, we find the Planck-like form of an expression will differ from its fundamental form – its Target Frame definition – depending on its dimensional composition. For gravitation, that relation is described by c/θ_si (i.e., m_f^-1). For the fine structure constant that relation is described as a function of the remainder of a whole unit multiple of θ_si. For the Hubble constant, it is a factor of 2.

1. Gravitational Constant

We begin with an expression that describes the loss of the non-discrete immeasurable count Q_L of l_f per increment of fundamental time t_f ([2], Appx. C). The observer/observable system is anchored relative to the Target Frame with the scaling relation c/θ_si. The speed of light c describes the rate of expansion at the system’s leading edge relative to that edge. And θ_si describes the radial rate of universal expansion ([2], Appx. AZ). We combine these terms to describe gravitational curvature (Fig. 2).

As noted in [2], an italicized G differs from its non-italicized form such that G is a measure of the gravitational constant at the upper count limit of n_L. Conversely, italicized G is any measure less than that limit, but most often meant to describe counts close to the demarcation associated with electromagnetic phenomena ([2], Appx. AB, AE, AN). If we had presented the expressions above using G, we would also need to replace the equality symbol with an approximately equal symbol. Accounting for the effects of length contraction is a significant component of the MQ approach ([2], Appx. M).

As discussed earlier, the CODATA collaboration has been challenged to resolve the most appropriate measure for G. MQ offers a physical explanation of recent selections (Table 1).

Table 1. Matching CODATA values for ℏ, G and the Planck Units with MQ Calculated Values, reprinted from [2].

^a CODATA publications for 2010, 2014 and 2018 [18-20].

^b Using precise G and ℏ, we account for the Informativity differential at the blackbody demarcation n_Lr= 84.6005496647(07) (i.e., italicized), the upper count limit (i.e., lim_nLr→∞) (not italicized), or a mix of the two.

Measurements are matched digit-for-digit with corresponding MQ calculations (last four lines) to resolve the physical conditions of the selected experiments. That is, we carry out calculations of G, G, ħ, and ℏ respective of their values at either the electromagnetic demarcation or the upper count limit. We also account for those effects described by the Informativity differential. Matches are identified with a line surrounding the value in the same column. Despite the unexpected consistency of such a high-level approach, it should be noted that an unbounded correspondence is achieved by looking at the physical details of each experiment and calculating results consistent with the demarcations of each measurement.

2. Fine Structure Constant

With respect to the fine structure constant ([2], Appx. AA), we again look at the frame difference beginning with the count n_Lr associated with a phenomenon. We offer three source measures, the blackbody demarcation, the charge coupling demarcation and the elementary charge demarcation ([2], Appx. AB, AE, and AN). In each case, the associated demarcations are the same to seven digits (Table 2).

Table 2. Demarcations associated with physical constants.

We then look to the fundamental expression ([2], Appx. V) to resolve the relation of counts n_Lr of l_f with respect to the Measurement Frame to counts n_θ of θ_si with respect to the Target Frame.

Such that the relation is described by n_Lr≙2n_θ, we resolve, in terms of blackbody radiation, that

And this we recognize as the fundamental form of the inverse fine structure constant (i.e., α_f⁻¹=42θ_si). Notably, demarcations are resolved as a function of the predicted measure. This is why we do not use the demarcation associated with the fine structure constant but must depend on a comparison phenomenon.

We then apply the metric differential 42θ_si-RND(42θ_si) to resolve the Planck-like form of the expression. And lastly, we apply the Informativity differential to resolve its classical form.

This is then compared with the 2018 CODATA (Table 3).

Table 3. Fundamental, Planck-like and Classical solutions to the inverse fine structure constant, adapted from [17].

^a 2018 CODATA publication [20].

Notably, there is no difference with Planck’s expression for the ground state orbital of an atom ([2], Appx. BL). There is a small difference with respect to the CODATA, α_c⁻¹ in the thirteenth digit. The difference is expected. We account for length contraction in ℏ, but do not do so for ɛ₀ or e. Moreover, the difference follows the twelfth digit, beyond the physical significance of the component measures.

3. Hubble’s Constant

Some may agree that the expansion of the universe is a property of the universe. Given this line of thought, we can state that expansion is not a physically significant property of astronomical units such as the parsec. So why is the Hubble constant defined in terms of parsecs?

We present expansion in terms of the diameter D_U and age A_U of the universe ([2], Appx. AZ). Moreover, the same can be used to define the quantum measure m_f.

The advantage of this approach is clarity; the radial rate of expansion θ_si is constant. The best means to approach physical verification is a comparison of quantum epoch calculations ([2], Appx. BF-BH) with expansionary epoch measurements (Table 4). Importantly, the calculation does not predict an inflationary period, is free of variables, and finds a constant rate of mass accretion that becomes the CMB following initial expansion. Comparing each dimension l_f, m_f, t_f and θ_si with measurement we validate that an expansion of the fundamental expression ([2], Appx. V) is physically significant across epochs.

Table 4. Tests extending the fundamental expression across the measurement domain, adapted from [2].

^a A consolodated study of CMB temperature measurements prior to 2009 [21].

^b Astrophysical constants [22].

^c 2018 Planck Collaboration Results [23].

And with this we resolve the Hubble constant as a function of the calculated age of the universe, the CMB temperature and the rate of expansion as defined by 2θ_si ([2], Appx. BA),

The result falls in the center of the Hubble tension debate [24], and notably similar to measures made by Freedman using the TRGB method, H₀=69.8±0.8(stat) ±1.7(sys) km s^-1 Mpc^-1 [25].

Finally, we remind the reader that this calculation describes the metric expansion of the universe, not the usual measure of the expanding distance between galaxies (i.e., it’s stellar expansion). While the two rates are similar, their difference is a function of forces accumulated over the history of the universe with respect to the observed mass.

4. Correlating Quantum and Cosmological Frames

For our last consideration we offer two expressions. Each correlates the relation of measure with respect to the Measurement Frame (i.e., l_f, m_f, and t_f) to the Target Frame (i.e., θ_si). The fundamental expression offers the simplest description of that relation ([2], Appx. V).

The expression also demonstrates that a change in expansion which differs from 2θ_si would also affect the relation of the fundamental measures to one another. Such that the speed of light c=n_Ll_f/n_Tt_f constrains the relation, this leaves only the count n_M of m_f, which we know increases with increasing n_T of t_f ([2], Appx. BE).

We can also describe the three measures using the unity expression ([2], Appx. AX) in the form of the Pythagorean Theorem. Such that n_L describes the increasing space between an observer and the perimeter of the universe and such that n_Lc describes the motion of light in vacuum with respect to that observer, we find that,

In each case, the expression correlates the discrete and non-discrete frames. These examples differ from the traditional description of frames, as are considered when working with relativity. Relativity describes the distortion of measure as a function of motion and/or gravitation, all which are relative to the Measurement Frame.

4. DISCUSSION

Frames of reference are a cornerstone to the MQ approach. But what is less obvious is that MQ is an approach, a few additional insights as to how to describe phenomena and how to interpret those descriptions. For all other concerns, we haven’t changed classical mechanics. We don’t need to go back to the drawing board to demonstrate a new claim because a new claim is not made. We present to the reader that any observation made was there all along. Measurement quantization is an interpretation that had not been considered.

That there are other aspects of MQ (i.e., classical mechanics) is attractive. For one, there are no hidden or free variables, newly proposed particles, or forces. MQ does not employ fitting, modeling, approximations, or approaches such as additional dimensions, inflation theory [26], LQG [8-10], String Theory [27] or Supersymmetry. Predictions resolved with MQ are conservative, or from a physical perspective, classical. By example, early universe history, CMB formation, dimensionality, and the unification of gravitation with electromagnetism ([2], Appx. AT), are all examples of the success of classical theory, when interpreted from an MQ perspective. For all these insights, we call to the reader’s attention that the non-discrete Target Frame is a cornerstone property to physical theory. Without this, there would be no means to understand why the physical constants, or the laws of nature exist [17].

5. APPENDICES

APPENDIX A: TERMS

Complete set of term and keyword definitions in the paper entitled, Measurement Quantization [2].

• 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 [3-5]; as a momentum θ_si=3.26239030392(48) kg m s^-1; and with respect to the Target Frame, θ_si has no units. Notably, the relation of angle and momentum are clarified as conserved quantities supertranslation invariant, by Po-Ning Chen, et. al. ([7], Sec. 4.4.2) consistent with our own solution of equivalence at the lower count bound.

• c is the speed of light which may also be written as c=n_Ll_f/n_Tt_f =299,792,458 m/s [20] such that n_L=n_T=1 is physically significant.

• G is the gravitational constant, 6.6740779428(56) 10^-11 m³ kg^-1 s^-2 such that its value considers the effects of length contraction associated with discrete measure at the upper count limit. Italicized G identifies a measure not at the limit (e.g., G=6.6738448362(53) 10^-11 m³ kg^-1 s^-2 at the blackbody demarcation).

• 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.

• l_p, m_p and t_p are Planck’s units for length, mass, and time.

• n_L, n_M and n_T are physically significant discrete counts of l_f, m_f and t_f respectively.

• n_L describes the count of l_f representative of a change in position of an observable measured with respect to the observer’s frame of reference.

• 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.

• n_Lc describes the count of l_f representative of a change in position of light measured with respect to the observer’s frame of reference.

• n_θ is the Target Frame count of θ_si resolved as a function of the Measurement Frame demarcation adjusted for the metric differential (i.e., RND(84.60055/2)=42).

• RND(n) is a function; round to the nearest whole-unit value.

• Q_L is the fractional portion of a count of l_f when engaging in a more precise calculation.

• Q_Ln_Lr, also known as the Informativity differential, describes the contraction of length associated with discrete measure.

• ℏ is the reduced Planck constant, 1.054571817 10^-34 m² kg s^-1. When accounting for the Informativity differential at the upper count bound, this term is not italicized (i.e., ħ=1.0545349844(45) 10^-34 m² kg s^-1).

• m_e is the mass of an electron.

• a₀ is the ground state orbital of an atom.

• α_f^-1, α_p^-1, and α_c^-1 are the fundamental, Planck-like, and classical forms of the inverse fine structure constant. α_f^-1=42θ_si is defined with respect to the Target Frame; the remaining two against the Measurement Frame such that α_c^-1 differs from α_p^-1, accounting for the length contraction associated with discrete measure.

• γ is a collection of terms that describe the geometry between the Target and Measurement frames.

STATEMENTS AND DECLARATIONS

The author has no relevant financial or non-financial interests to disclose.

The author has no competing interests to declare that are relevant to the content of this article.

DATA AVAILABILITY

The manuscript has data included as electronic supplementary material.

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