Physical Significance of the Transition from Classical to Non-classical Behavior in the Orbits of Stars

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We derive conditional expressions for a quantization crossover within Measurement Quantization. The crossover is defined by equality of effective and bound circular-orbit mass parameters at positive radius. The construction adopts a mass-to-length count bound and a dimensionless System Frame coefficient. Its historical Milky Way model comparison gives a residual population standard deviation of 1.394 km s⁻¹ over 84 nonzero-radius samples through 84,000 light-years. The reported crossing is 9.072419 × 10³ light-years, with 2.779712 kpc retained as the historical source quotation. These are model-dependent results. Because the effective velocity is calculated from the modelled velocity, their close agreement does not constitute an independent observational test.

Keywords: Star Velocity, Non-classical, Orbit, Dark Matter, Spacetime Curvature, Discrete Gravity

1. INTRODUCTION

The quantization crossover is the positive radial distance at which the effective and bound mass descriptions used here coincide. We denote radial distance in m by R. Its value for a galaxy follows from the adopted modelled orbital-velocity profile and the bound prescription. This operational definition sets the scope of the title’s classical-to-non-classical terminology. Identifying it with an observed change in galactic dynamics requires an independent physical test.

Measurement Quantization (MQ) expresses physical measures using discrete counts. Here n_L, n_M and n_T are the dimensionless radial, source-mass and elapsed-time counts of the fundamental measures l_f, m_f and t_f, whose International System of Units (SI) realizations have units m, kg and s, respectively. The radial realization is R=n_Ll_f. The construction draws on the speed of light, escape velocity and Heisenberg’s uncertainty principle [1] ([2], Appx. W).

The Internal Frame is the discrete count configuration domain. The System Frame is its non-discrete relational encoding, induced by the Frames mapping, which relates these descriptions while preserving realization consistency (historical framework context in [2], Appx. K). The fundamental measures provide MQ reference units, distinguished from the conventional Planck units [3] ([2], Appx. X). Let c be the speed of light in m s⁻¹ and n_Lc its dimensionless displacement count over the same elapsed interval. Then

c = \frac{n_{L c} l_{f}}{n_{T} t_{f}} = \frac{l_{f}}{t_{f}}

(1a)

, with the light-displacement count equal to the elapsed-time count. The radial count remains distinct. This count description supplies the setting for the MQ mapping of physical quantities [4].

The lower reference-count case has unit length and time counts. Reciprocal fundamental measures and their ratios also enter the bound constructions ([2], Appx. O). Their dimensions must be retained when constructing a rate or a physical quantity. A dimensionless count is obtained only after the reference measure or normalization interval has been specified. The mass-to-length count normalization used in this paper is stated explicitly before Eq. (2).

The galactic construction adopts a local mass-frequency bound, expressed here as a boundary on the mass-to-length count ratio for a specified configuration. It assigns a bound mass at each radius and an effective orbital mass from the modelled-to-bound velocity ratio. These quantities are defined in Section 3. The boundary is a reference for the response prescription, rather than a universal ceiling on accumulated galactic mass. In particular, the effective mass can lie above or below the bound mass as the velocity ratio changes.

The dimensionless System Frame coefficient enters the bound-velocity prescription. Combining that bound with the effective-mass relation gives a response curve for the adopted orbital-velocity profile.

The historical MQ comparison uses modelled Milky Way velocities in the context of McGaugh’s Galactic model studies [5, 6]. Section 3 presents the retained curve comparison and crossing.

2. METHODS

The calculations apply the MQ count description to classical escape and circular-orbit relations. Related applications are discussed in the publication record [4, 7–11]. Here the local mass-frequency bound and the System Frame coefficient are adopted premises. The linear effective-mass prescription is an additional model assumption; the bound-mass and crossover expressions follow conditionally from these choices.

The introductory paper Measurement Quantization [2] develops the underlying count description and its proposed physical interpretation. Its appendices supply the historical MQ context cited throughout this paper and a more extensive set of definitions. Online Resources links the related materials. The present count-normalized expressions state their own realization conventions and domains.

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

The supplementary resources cited here are spreadsheets. Suppl. 1 presents calculations associated with the introductory paper and its appendices. Suppl. 2 presents calculations for this and related MQ applications. The historical galactic comparison and the current calibration of the MQ quantities have different roles. The former retains the source-model curves; the latter uses the 2022 calibration identified in Appendix A [12].

3. RESULTS

3.1. Calculations

We begin with the Newtonian point-mass escape-velocity relation in MQ count notation ([2], Appx. H). In Eq. (1), v_e is escape speed in m s⁻¹, M=n_Mm_f is source mass in kg, and G is the upper-count-limit gravitational coupling in m³ kg⁻¹ s⁻². The substitution

G = \frac{c^{3} t_{f}}{m_{f}}

(1b)

and the fundamental length-to-time ratio introduced above give the count form. This reference escape relation does not establish the potential of a flattened galaxy. Its subsequent use in the orbital bound is a separate prescription.

O1. The orbital mass-bound prescription uses the escape-velocity count relation at the adopted mass-frequency bound, with the System Frame coefficient applied as specified below.

This starting expression keeps the mass and radial counts explicit. Its kinetic-energy context is discussed in [2], Appx. AJ. We retain the factor of two in the escape relation when applying the mass-frequency prescription ([2], Appx. AF).

v_{e} = \sqrt{2 \frac{G M}{R}} = \sqrt{2 \frac{l_{f}^{3}}{t_{f}^{3}} \frac{t_{f}}{m_{f}} \frac{n_{M} m_{f}}{n_{L} l_{f}}} = \sqrt{2 \frac{l_{f}^{2}}{t_{f}^{2}} \frac{n_{M}}{n_{L}}} = c \sqrt{2 \frac{n_{M}}{n_{L}}} .

(1)

From this point onward, v_e denotes effective orbital velocity rather than the escape velocity used in Eq. (1). We introduce

n_{M_{b}}

(1c)

as the dimensionless reference count associated with the fixed 1 kg normalization. This reference count is distinct from the count of the radius-dependent bound mass. At the adopted boundary, the mass-to-length count ratio is its reciprocal, as specified in Eq. (2).

n_{M_{b}} = \frac{1 kg}{m_{f}}, \frac{n_{M}}{n_{L}} = \frac{1}{n_{M_{b}}} = \frac{m_{f}}{1 kg} .

(2)

The kilogram reference makes the historical numerical normalization explicit. It does not impose a one-kilogram ceiling on a galaxy. It is a fixed physical reference within the adopted prescription; choosing a different reference mass would change the boundary. A change of measurement units alone leaves that physical reference unchanged. The count ratio in Eq. (2) gives the unscaled boundary velocity v_bc, in m s⁻¹ ([2], Appx. AF).

v_{b c} = c \sqrt{2 \frac{n_{M}}{n_{L}}} = c \sqrt{\frac{2}{n_{M_{b}}}} .

(3)

The bound velocity v_b, in m s⁻¹, includes the dimensionless coefficient θ_si through multiplication of the unscaled boundary velocity. This scaled boundary prescription is adopted here ([2], Appx. AF).

v_{b} = θ_{s i} c \sqrt{\frac{2}{n_{M_{b}}}} = 204.054 km s^{- 1} .

(4)

We obtain the bound mass M_b, in kg, by equating the boundary prescription to the circular-orbit relation at the same positive radius ([2], Appx. AG). The mass is an equivalent circular-orbit response parameter. Interpreting it as the actual mass enclosed by that radius would additionally require suitable assumptions about the gravitational potential and geometry. The Galactic disk calculations of McGaugh [5] retain the disk geometry. Substitution of the defined upper-count coupling gives Eqs. (6)–(7), and Eqs. (8)–(9) give the conditional linear-radius law.

θ_{s i} c \sqrt{\frac{2}{n_{M_{b}}}} = \sqrt{\frac{G M_{b}}{R}},

(5)

M_{b} = \frac{2 {θ_{s i}}^{2} R c^{2}}{G n_{M_{b}}} = \frac{2 {θ_{s i}}^{2} R c^{2} m_{f}}{c^{3} t_{f} n_{M_{b}}},

(6)

M_{b} = \frac{2 {θ_{s i}}^{2} R m_{f}}{c t_{f} n_{M_{b}}} = \frac{2 {θ_{s i}}^{2} R m_{f}}{l_{f} n_{M_{b}}},

(7)

\frac{M_{b}}{R} = \frac{2 {θ_{s i}}^{2} m_{f}}{l_{f} n_{M_{b}}},

(8)

M_{b} = \frac{2 {θ_{s i}}^{2} m_{f}}{l_{f} n_{M_{b}}} R .

(9)

We next express the same bound mass in radial-count form using the radial realization introduced above.

M_{b} = \frac{2 {θ_{s i}}^{2} (n_{L} l_{f}) m_{f}}{l_{f} n_{M_{b}}},

(10)

M_{b} = \frac{2 n_{L} {θ_{s i}}^{2} m_{f}}{n_{M_{b}}} .

(11)

The reference count and the coefficient are dimensionless. The remaining mass factor in Eq. (11) is m_f, so the result has units kg. No substitution of a momentum realization for the dimensionless coefficient is required. The kinetic-energy context remains that of [2], Appx. AJ.

The effective mass M_e, in kg, determines the effective orbital velocity already distinguished from escape speed, through

{v_{e}}^{2} = \frac{G M_{e}}{R}

(11a)

. Let v_m be modelled orbital velocity in m s⁻¹. Equation (12) defines the dimensionless fractional deviation

Δ_{f (v_{m})}

(11b)

from the bound velocity. The MQ response prescription in Eq. (13) then gives Eq. (14) ([2], Appx. AK). We require

R > 0, M_{b} > 0, v_{b} > 0, v_{m} \geq \frac{v_{b}}{2}

(11c)

on the real nonnegative orbital branch. The effective mass exceeds, equals or falls below the bound mass according as the modelled velocity exceeds, equals or falls below the bound velocity within this domain.

Δ_{f (v_{m})} = \frac{(v_{m} - v_{b})}{v_{b}} = \frac{v_{m}}{v_{b}} - 1,

(12)

M_{e} = M_{b} (2 Δ_{f (v_{m})} + 1) = M_{b} (2 \frac{v_{m}}{v_{b}} - 2 + 1),

(13)

M_{e} = M_{b} (2 \frac{v_{m}}{v_{b}} - 1) .

(14)

At a positive-radius crossover the effective and bound masses are equal. Dividing Eq. (14) by the positive bound mass gives Eq. (15), and hence Eq. (16). The effective velocity has the same value on the nonnegative orbital branch.

1 = (2 \frac{v_{m}}{v_{b}} - 1) .

(15)

v_{m} = v_{b} = v_{e} .

(16)

Milky Way masses with quantization crossover inset — **Fig.** 1. Historical Milky Way mass comparison. Classical mass is blue, effective mass is red [5, 6], and bound mass is purple. Here M_⊙ denotes one solar mass, a mass reference with SI unit kg.

At the crossover, Eq. (16) gives equality of the modelled, bound and effective velocities. The historical MQ calculation used the boundary velocity reported as 204.054 km s⁻¹ ([2], Appx. AF) and reported a crossing at 9.072419 × 10³ light-years, historically quoted as 2.779712 kpc, in the Galactic model context [5, 6]. The displayed digits identify the retained numerical interpolation, rather than a measured distance precision. Figure 1 preserves the historical classical, effective and bound mass curves from [2], Fig. AL.1. Their positive-radius effective–bound intersection represents the defined crossover ([2], Appx. AL).

The count normalization in Eq. (2) is distinct from an unrestricted total-mass ceiling. To examine the related length–mass–time constraint, we use the count-scaled fundamental relation, which links the fundamental length, mass and time through the momentum realization ([2], Appx. V). The momentum realization p_si, in kg m s⁻¹, is distinct from the dimensionless coefficient already introduced. For the count family obeying n_Ln_M=n_T, rearrangement of that relation at positive radial count gives Eq. (17).

n_{M} m_{f} = 2 p_{s i} \frac{n_{T} t_{f}}{n_{L} l_{f}} .

(17)

At fixed elapsed-time count, Eq. (17) requires an increase in mass count to be accompanied by a decrease in radial count within this constrained family. The corresponding count-defined speed

\frac{n_{L} l_{f}}{n_{T} t_{f}}

(17a)

therefore decreases. This does not imply an increase in the speed of light, which remains

c = \frac{l_{f}}{t_{f}}

(17b)

. The separate classical escape condition gives

\frac{n_{M}}{n_{L}} < \frac{1}{2}

(17c)

outside its equality boundary ([2], Appx. H, O). The adopted galactic bound in Eq. (2) specifies the smaller boundary ratio used in Eqs. (3)–(11); it is not proved solely by rearranging Eq. (17).

In the historical mass profile, the classical curve in Fig. 1 differs from the effective and bound curves. The nonzero crossover refers to the intersection of effective and bound mass at positive radius. Their common zero-radius intercept does not supply a second instance of the positive-radius condition used in Eqs. (5)–(16).

The crossing is a property of the adopted profile and boundary prescription ([2], Appx. AK). A different profile can have no crossing or several crossings of the boundary velocity. The bound alone supplies neither a universal radius nor a uniqueness result.

3.2. Historical model comparison

The historical Milky Way calculation places the crossover near the quoted 2.78 kpc. This lies within the region excluded from McGaugh’s 2008 circular-velocity fits [5] because the bulge–bar motions were interpreted as non-circular. The retained crossing must therefore be interpreted within the adopted circular-orbit model. Figure 2 shows the modelled and effective curves below the bound over part of the interval from 4 to 9 thousand light-years; that feature of the input-dependent comparison does not independently locate a physical threshold.

Equation (18) gives

(v_{e} - v_{m}) (v_{e} + v_{m}) = - {(v_{m} - v_{b})}^{2}

(17d)

. On the stated nonnegative branch, v_e≤v_m, with equality only when v_m=v_b. The squared deviation on the right suppresses the difference near the crossing. The historical comparison uses modelled velocity as its input, with McGaugh’s study of modified Newtonian dynamics (MOND) [5] supplying part of its source context. Close agreement near the crossing is consequently built into the transformation and is not independent evidence for the bound.

McGaugh’s 2018 research note [6] updates a Galactic mass model using a specified Galactocentric distance and the radial acceleration relation. Its azimuthally averaged component-potential description distinguishes a model circular speed from an individual star’s observed velocity. This is the source-model context of the retained comparison.

Fig. 2. Historical Milky Way velocity comparison. Modelled velocity is green [5, 6], classical velocity is blue, effective velocity is red, and bound velocity is purple.

For Fig. 2, the historical standard deviation of effective-minus-modelled velocity is 1.394 km s⁻¹. This is the population standard deviation over the 84 samples from 1 to 84 thousand light-years, excluding the plotted origin (using the historical curve in [2], Appx. AK). It removes the mean residual and therefore differs from a root-mean-square error about zero. It describes the dispersion of the transformation residuals and supplies no observational uncertainty. The figure retains the historical velocity curves from [2], Fig. AK.1. The assigned zero velocity at the origin is a plotting convention outside the real-velocity domain

v_{m} \geq \frac{v_{b}}{2}

(17e)

. Equation (18) follows from Eq. (14) and the effective orbital identity stated before Eq. (12), under the positive-radius and nonnegative-velocity conditions.

{v_{e}}^{2} = {v_{b}}^{2} (2 \frac{v_{m}}{v_{b}} - 1) .

(18)

4. DISCUSSION

The positive-radius equality of effective and bound mass gives the velocity condition in Eq. (16). It is a defined crossover of response parameters, not an intersection of every curve in Fig. 2 or an established abrupt transition in Galactic dynamics. Interpreting the response as a new physical effect requires evidence beyond the equality imposed by the model.

The distinction between realization within one count configuration and accumulated mass over an elapsed interval remains essential ([2], Appx. O). Bounds between the length, mass and time counts apply in their stated domain. The reference normalization in Eq. (2) fixes the boundary ratio used here, while Eq. (17) constrains its specified length–mass–time family. Neither relation alone imposes a universal ceiling on the total mass observed over successive intervals.

The frame distinction introduced above is necessary to interpret these count expressions. Observable separations are realized through the Frames mapping of the discrete configurations (historical framework context in [2], Appx. K). The dimensionless coefficient and its momentum realization must retain their separate roles when translating count relations into SI quantities.

The broader MQ treatment distinguishes the metric differential, which concerns the representation mismatch between the frames, from the Informativity differential, which concerns the realized count-dependent length contraction ([2], Appx. T, D). The curvature and expansion discussions supply further context ([2], Appx. C, BA). Here the established result is the conditional bound-mass law and its crossover condition. Testing its physical interpretation requires independent kinematic and mass constraints with appropriate treatment of Galactic geometry and uncertainty. The present model comparison alone establishes neither discrete galactic structure nor the absence of dark matter.

5. APPENDICES

APPENDIX A: TERMS

The introductory paper Measurement Quantization [2] gives the historical definition context. The entries below retain the realization conventions stated here.

The coefficient θ_si used in Section 3 has the calculated value 3.26239030504(25) under the 2022 inverse fine-structure calibration [12]. The previously introduced momentum realization p_si has the same numerical coefficient in kg m s⁻¹ under the stipulated SI normalization; its angular realization Θ_si has that coefficient in rad. These are distinct dimensional realizations. Shwartz and Harris [13], Table I, give 3.26239 rad as a calculated signal and idler angle for their specified X-ray down-conversion geometry. That tabulated angle is not an independent measurement of the coefficient or momentum realization used here. The related X-ray studies [14, 15] and the discussion of relativistic conserved quantities [16] are contextual references, rather than inputs to this calibration.

For the light-speed count relation introduced in Section 1, the exact SI value of c is 299,792,458 m s⁻¹ [17]. Equality of the light-displacement and elapsed-time counts preserves this value for any common positive interval, including unit counts.

For the MQ coupling G, the upper-count calculation gives 6.6740779383(10) × 10⁻¹¹ m³ kg⁻¹ s⁻² under the 2022 calibration [12]. The finite-count value at this calibration’s electromagnetic demarcation, its stipulated interaction separation, is 6.6738448317(10) × 10⁻¹¹ m³ kg⁻¹ s⁻² and uses G. These are MQ realization calculations, rather than the recommended constants in that adjustment or direct measurements. Both uncertainties propagate the same inverse fine-structure input conditional on the MQ relations; they do not include a test of model discrepancy.

The fundamental measures introduced in Section 1 are MQ reference units. They are kept distinct from conventional Planck units in the following background notation.

The conventional Planck units l_p, m_p and t_p denote length in m, mass in kg and time in s, respectively.

The three fundamental-measure counts retain the radial, source-mass and elapsed-time meanings assigned in Section 1.

n_Lm 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.

The radial count introduced in Section 1 locates the observable relative to the chosen center of mass. It is distinct from the displacement counts used for motion and light.

The light-displacement count retains the same elapsed interval as the time count in the light-speed relation of Section 1.

For a dimensionless real argument x, the mixed bracket pair ⌊x⌉ denotes rounding to the nearest whole-unit value.

n_θ is the dimensionless System Frame count of θ_si resolved from the Internal Frame count at that demarcation with the metric differential. The retained rounded example uses the nearest-whole-unit convention

⌊ \frac{84.60055}{2} ⌉ = 42

(18a)

Q_L is the dimensionless geometric residual in the right-triangle count construction. At positive radial count it obeys

Q_{L} + n_{L} = \sqrt{1 + {n_{L}}^{2}}

(18b)

. In this finite-count normalization the radial count is evaluated at the interaction separation; at the electromagnetic demarcation it refers to that separation rather than to a galactic radius.

The finite-count length-normalization factor 2Q_Ln_L encodes the Informativity differential. The departure of this factor from unity describes the non-Lorentz length contraction in realized spatial separation under the Frames mapping. The factor tends to unity at the upper-count limit.

The conventional reduced Planck constant, in J s, obeys

ℏ = \frac{h}{2 π}

(18c)

and is exact in SI. Here h is Planck’s constant with exact SI value 6.62607015 × 10⁻³⁴ J s, and π is the dimensionless circle constant [12]. The distinct MQ upper-count realization ħ has value 1.0545349844417293(56) × 10⁻³⁴ J s under the same calibration. Parenthetical digits give one standard uncertainty in the last quoted digits and do not include unquantified model discrepancy.

m_e is the electron mass in kg.

a₀ is the Bohr radius, a length in m.

The dimensionless quantities

{α_{f}}^{- 1}

(18d)

{α_{p}}^{- 1}

(18e)

and

{α_{c}}^{- 1}

(18f)

denote the fundamental, Planck-like and classical inverse fine-structure forms. The fundamental form obeys

{α_{f}}^{- 1} = 42 θ_{s i}

(18g)

. The Planck-like form includes the metric differential, and the classical form additionally includes the finite-count Informativity differential. The adopted physical inverse fine-structure calibration is 137.035999177(21) from the 2022 adjustment [12]. These forms describe successive realization steps, rather than independent measurements.

The historical electromagnetic gamma approach mentioned in the cited MQ literature is distinct from its later fundamental gamma and is not used in this galactic calculation.

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 numerical velocity series used for the historical comparison is preserved in the embedded chart cache of Fig. 2. The supplementary resources cited in Methods identify the original calculation context.

REFERENCES

[1] Heisenberg, W., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43 (3–4): 172–198 (1927), Bibcode:1927ZPhy.43.172H, doi:10.1007/BF01397280, S2CID 122763326.

[2] J.A. Geiger: Measurement Quantization, International Journal of Geometric Methods in Modern Physics, 20(4), 2350069 (2023), doi:10.1142/S021988782350069X.

[3] M. Planck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu, Berlin, 5, p. 480 (1899).

[4] J.A. Geiger: Measurement Quantization Describes the Physical Constants, International Journal of Theoretical and Mathematical Physics, 11(1), 29–59 (2021), doi:10.5923/j.ijtmp.20211101.03.

[5] McGaugh, S.S.: Milky Way Mass Models and MOND (2008) ApJ, 683, 137-148, doi:10.1086/589148.

[6] McGaugh, S.S.: A Precise Milky Way Rotation Curve Model for an Accurate Galactocentric Distance (August 2018) Res. Notes AAS, 2, 156, doi:10.3847/2515-5172/aadd4b.

[7] J.A. Geiger: Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture, Journal of High Energy Physics Gravitation and Cosmology. Vol. 5, No. 2 (Apr 2019), doi:10.4236/jhepgc.2019.52028.

[8] J.A. Geiger: Measurement Quantization Unites Classical and Quantum Physics, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 2, 262–311 (Apr 2018), doi:10.4236/jhepgc.2018.42019.

[9] J.A. Geiger: Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 4 (Oct 2018), doi:10.4236/jhepgc.2018.44038.

[10] J.A. Geiger: Physical Significance of Measure, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 1 (Jan 2020), doi:10.4236/jhepgc.2020.61008.

[11] J.A. Geiger: Measurement Quantization Describes History of Universe – Quantum Inflation, Transition to Expansion, CMB Power Spectrum, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 2 (Apr 2020), doi:10.4236/jhepgc.2020.62015.

[12] P.J. Mohr, D.B. Newell, B.N. Taylor and E. Tiesinga: CODATA Recommended Values of the Fundamental Physical Constants: 2022, Reviews of Modern Physics, 97, 025002 (2025), doi:10.1103/RevModPhys.97.025002.

[13] S. Shwartz and S. E. Harris, Polarization Entangled Photons at X-Ray Energies, Phys. Rev. Lett. 106, 080501 (2011), arXiv: 1012.3499, doi:10.1103/PhysRevLett.106.080501.

[14] S. Shwartz, R. N. Coffee, J. M. Feldkamp, et. al., X-ray Parametric Down-Conversion in the Langevin Regime, Physical Review Letters, 109, 013602 (July 6, 2012), doi:10.1103/PhysRevLett.109.013602.

[15] T. E. Glover et al., X-ray and optical wave mixing, Nature, 488, 7413, 603–608 (2012), doi:10.1038/nature11340.

[16] Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang, Shing-Tung Yau, Conserved quantities in general relativity -- the view from null infinity (Apr. 8, 2022), doi:10.48550/arXiv.2204.04010.

[17] NIST: CODATA Recommended Values of the Fundamental Physical Constants: 2018, NIST SP 961 (May 2019), https://physics.nist.gov/cuu/pdf/wall_2018.pdf.