DOCSLIB.ORG

Roads to the Planck Scale

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Roads to the Planck Scale Open Journal of Mathematics and Physics | Volume 2, Article 138, 2020 | ISSN: 2674-5747 https://doi.org/10.31219/osf.io/2cdwb | published: 11 Aug 2020 | https://ojmp.org FI [white paper] Diamond Open Access Roads to the Planck scale Open Quantum Collaboration∗† August 15, 2020 Abstract A summary of a few roads exploring the Planck scale is presented. We consider both Newtonian and relativistic systems. keywords: Planck scale, quantum gravity The most updated version of this paper is available at https://osf.io/2cdwb/download How to co-author this paper? 1. Send us a complete review report and answer the following questions. 2. Which discussions would you add to this paper? 3. Did you find any mistakes or inconsistent arguments? Give a detailed justification. ∗All authors with their affiliations appear at the end of this paper. †Corresponding author: [email protected] | Join the Open Quantum Collaboration 1 Introduction 4. This white paper is a summary of the quantum roads that lead to the investigation of the Planck scale physics. 5. This paper is not meant to be pedagogical; the purpose is to present a global view of important results of the ultramicroscopic world. Feynmann units 6. h h so that 2π 1. ≈ ̵ ≈ The Planck units 7. There is only one combination of G, c, h for length, time, and mass/energy [1]. 8. In the following, we summarize the mathematical steps needed to ob- tain some of the Planck physical quantities. 9. ℓp Planck length 10. tp = Planck time 11. Mp= Planck mass 12. Ep =Planck energy 13. X =means the units of X. 14. [G] m3 kgs2 15. [c ] =m s# 16. [h] = Js# kgm2 s m5 17. [h]G= s3= # [ ] = 2 2 hG 18. m c3 19. = hG ℓ p $ c3 hG 2 = 20. c5 s 21. % & = hG t p $ c5 kgm3 = 22. hc s2 [ 2] = (hc ) 23. kg G 24. = ) ) hc M p $ G 5 kgm7 = 25. hc s6 [ 2 ] =hc(5 ) 26. J G 27. = ( ) hc5 E p $ G = Spacetime distortion 28. [1] 29. Prerequisites (a) Quantum mechanics: E hν (b) Special relativity: E mc2 = = 3 GM (c) Newtonian potential: φ r (d) General relativity Newtonian limit (weak gravitation field = + slowly moving bodies): ds2 1 2φ c2 cdt 2 1 2φ c2 dx 2 + + 30. Let δ spacetime distortion. = ( + # )( ) +( + # )( ,) 31. = GM δ ℓc2 32. For M Mp and ℓ ℓp in (31), ! G = = δ λ , c2 p where λp Mp ℓp is the linear de!nsity of spacetime in the Planck scale. 33. It’s quite =intere#sting that the spacetime distortion depends on a single dimension. Gravitational Uncertainty Principle 34. [1] 35. Prerequisites (a) Uncertainty principle (b) Gravity (29.c) 36. The gravitational uncertainty principle is given by h ∆p ∆x ℓ2 . ∆p p h ! - . + - . 37. The uncertainty in position has a minimum at ℓp h ∆p, then ∆xmin 2ℓp, ! # which corresponds to a photon of w=avelength ℓp and energy Ep. 4 Light ranging (with a classical clock) 38. [1] 39. Prerequisites (a) Classical optics (b) Gravity (29.c) 40. Since light is a wave, the uncertainty in the measurement of the length ℓ is about its wavelength, ∆ℓw λ. 2 41. The spatial distortion due to th!e energy of the light is ∆ℓg ℓ λ. 42. The uncertainty in position, considering the wave nature of!ligh#t and gravity, is then ℓ2 ∆ℓ ∆ℓ ∆ℓ λ p . w g λ 43. The expression in (42) has!a min+imum!at λ+ ℓp, ∆ℓmin 2ℓp 2λ. = ! = A Planck size black hole 44. [2–4] 45. Prerequisites 2GM (a) Schwarzschild radius: rs c2 (b) Planck units: M , ℓ p p = 46. The minimum size black hole with a Schwarzschild radius rs ℓp has mass M Mp. = = 5 The Photon Black Hole 47. [5] 48. Prerequisites 2GM (a) Schwarzschild radius: rs c2 (b) Energy of a photon: E hν = (c) Relativistic energy: E mc2 = (d) c λν = 49. The Sc=hwarzschild radius for a single photon is rs 2ℓp. ≈ A Planck mass black hole 50. Prerequisites 2GM (a) Schwarzschild radius: rs c2 (b) Planck units: M , ℓ p p = 51. The Schwarzschild radius for a black hole with the Planck mass Mp is rs 2ℓp. ≈ When gravity meets electromagnetism 52. [1] 53. Prerequisites GM2 (a) Gravitational Newtonian force: Fg r2 (b) Electric force: F q1q2 e r2 = α e2 (c) The fine structure≈constant: hc ≈ 6 54. The gravitational force becomes comparable with the electric force for two objects of mass M with electron charge e if GM 2 e2 . r2 r2 55. Then M 2 e2 G. ≈ 56. ≈ # M M p 12 ! Final Remarks 57. Some of the roads to the Planck scale were presented in a summarized fashion which we believe can help scientists to gain intuition into the ultramicroscopic reign of the Planck scale. Open Invitation Review, add content, and co-author this paper [6,7]. Join the Open Quantum Collaboration (https://bit.ly/ojmp-slack). Send your contribution to [email protected]. Open Science The latex file for this paper together with other supplementary files are available [8]. Ethical conduct of research This original work was pre-registered under the OSF Preprints [9], please cite it accordingly [10]. This will ensure that researches are con- ducted with integrity and intellectual honesty at all times and by all means. 7 Acknowledgement Center for Open Science https://cos.io + Open Science Framework https://osf.io + References [1] Adler, Ronald J. “Six easy roads to the Planck scale.” American Jour- nal of Physics 78.9 (2010): 925-932. https://arxiv.org/pdf/1001.1205 [2] Misner, Charles W., Kip S. Thorne, and John Archibald Wheeler. Gravitation. Macmillan, 1973. [3] Wald, Robert M. General relativity. University of Chicago press, 2010. [4] Faraoni, Valerio. “Three new roads to the Planck scale.” American Journal of Physics 85.11 (2017): 865-869. https://aapt.scitation.org/doi/pdf/10.1119/1.4994804 [5] Santos, Caio M. F., Silva, João M. C., Gomes, Érica C., & Lobo, Matheus P. “Uma proposta didático-matemática para o uso da escala de Planck: dos fótons aos buracos negros.” Revista Brasileira de Ensino de Física 42 (2020): e20190350. https://dx.doi.org/10.1590/1806-9126-rbef-2019-0350 [6] Lobo, Matheus P. “Microarticles.” OSF Preprints, 28 Oct. 2019. https://doi.org/10.31219/osf.io/ejrct [7] Lobo, Matheus P. “Simple Guidelines for Authors: Open Jour- nal of Mathematics and Physics.” OSF Preprints, 15 Nov. 2019. https://doi.org/10.31219/osf.io/fk836 8 [8] Lobo, Matheus P. “Open Journal of Mathematics and Physics (OJMP).” OSF, 21 Apr. 2020. https://doi.org/10.17605/osf.io/6hzyp [9] COS. Open Science Framework. https://osf.io [10] Lobo, Matheus P. “Roads to the Planck Scale.” OSF Preprints, 11 Aug. 2020. https://doi.org/10.31219/osf.io/2cdwb The Open Quantum Collaboration Matheus Pereira Lobo (lead author, [email protected])1,2 https://orcid.org/0000-0003-4554-1372 1Federal University of Tocantins (Brazil) 2Universidade Aberta (UAb, Portugal) 9.
Load more

Recommended publications
  • Planck Mass Rotons As Cold Dark Matter and Quintessence* F
    Planck Mass Rotons as Cold Dark Matter and Quintessence* F. Winterberg Department of Physics, University of Nevada, Reno, USA Reprint requests to Prof. F. W.; Fax: (775) 784-1398 Z. Naturforsch. 57a, 202–204 (2002); received January 3, 2002 According to the Planck aether hypothesis, the vacuum of space is a superfluid made up of Planck mass particles, with the particles of the standard model explained as quasiparticle – excitations of this superfluid. Astrophysical data suggests that ≈70% of the vacuum energy, called quintessence, is a neg- ative pressure medium, with ≈26% cold dark matter and the remaining ≈4% baryonic matter and radi- ation. This division in parts is about the same as for rotons in superfluid helium, in terms of the Debye energy with a ≈70% energy gap and ≈25% kinetic energy. Having the structure of small vortices, the rotons act like a caviton fluid with a negative pressure. Replacing the Debye energy with the Planck en- ergy, it is conjectured that cold dark matter and quintessence are Planck mass rotons with an energy be- low the Planck energy. Key words: Analog Models of General Relativity. 1. Introduction The analogies between Yang Mills theories and vor- tex dynamics [3], and the analogies between general With greatly improved observational techniques a relativity and condensed matter physics [4 –10] sug- number of important facts about the physical content gest that string theory should perhaps be replaced by and large scale structure of our universe have emerged. some kind of vortex dynamics at the Planck scale. The They are: successful replacement of the bosonic string theory in 1.
    [Show full text]
  • Dark Energy and Dark Matter in a Superfluid Universe Abstract
    Dark Energy and Dark Matter in a Superfluid Universe1 Kerson Huang Massachusetts Institute of Technology, Cambridge, MA , USA 02139 and Institute of Advanced Studies, Nanyang Technological University, Singapore 639673 Abstract The vacuum is filled with complex scalar fields, such as the Higgs field. These fields serve as order parameters for superfluidity (quantum phase coherence over macroscopic distances), making the entire universe a superfluid. We review a mathematical model consisting of two aspects: (a) emergence of the superfluid during the big bang; (b) observable manifestations of superfluidity in the present universe. The creation aspect requires a self‐interacting scalar field that is asymptotically free, i.e., the interaction must grow from zero during the big bang, and this singles out the Halpern‐Huang potential, which has exponential behavior for large fields. It leads to an equivalent cosmological constant that decays like a power law, and this gives dark energy without "fine‐tuning". Quantum turbulence (chaotic vorticity) in the early universe was able to create all the matter in the universe, fulfilling the inflation scenario. In the present universe, the superfluid can be phenomenologically described by a nonlinear Klein‐Gordon equation. It predicts halos around galaxies with higher superfluid density, which is perceived as dark matter through gravitational lensing. In short, dark energy is the energy density of the cosmic superfluid, and dark matter arises from local fluctuations of the superfluid density 1 Invited talk at the Conference in Honor of 90th Birthday of Freeman Dyson, Institute of Advanced Studies, Nanyang Technological University, Singapore, 26‐29 August, 2013. 1 1. Overview Physics in the twentieth century was dominated by the theory of general relativity on the one hand, and quantum theory on the other.
    [Show full text]
  • An Atomic Physics Perspective on the New Kilogram Defined by Planck's Constant
    An atomic physics perspective on the new kilogram defined by Planck’s constant (Wolfgang Ketterle and Alan O. Jamison, MIT) (Manuscript submitted to Physics Today) On May 20, the kilogram will no longer be defined by the artefact in Paris, but through the definition1 of Planck’s constant h=6.626 070 15*10-34 kg m2/s. This is the result of advances in metrology: The best two measurements of h, the Watt balance and the silicon spheres, have now reached an accuracy similar to the mass drift of the ur-kilogram in Paris over 130 years. At this point, the General Conference on Weights and Measures decided to use the precisely measured numerical value of h as the definition of h, which then defines the unit of the kilogram. But how can we now explain in simple terms what exactly one kilogram is? How do fixed numerical values of h, the speed of light c and the Cs hyperfine frequency νCs define the kilogram? In this article we give a simple conceptual picture of the new kilogram and relate it to the practical realizations of the kilogram. A similar change occurred in 1983 for the definition of the meter when the speed of light was defined to be 299 792 458 m/s. Since the second was the time required for 9 192 631 770 oscillations of hyperfine radiation from a cesium atom, defining the speed of light defined the meter as the distance travelled by light in 1/9192631770 of a second, or equivalently, as 9192631770/299792458 times the wavelength of the cesium hyperfine radiation.
    [Show full text]
  • Quantum Vacuum Energy Density and Unifying Perspectives Between Gravity and Quantum Behaviour of Matter
    Annales de la Fondation Louis de Broglie, Volume 42, numéro 2, 2017 251 Quantum vacuum energy density and unifying perspectives between gravity and quantum behaviour of matter Davide Fiscalettia, Amrit Sorlib aSpaceLife Institute, S. Lorenzo in Campo (PU), Italy corresponding author, email: [email protected] bSpaceLife Institute, S. Lorenzo in Campo (PU), Italy Foundations of Physics Institute, Idrija, Slovenia email: [email protected] ABSTRACT. A model of a three-dimensional quantum vacuum based on Planck energy density as a universal property of a granular space is suggested. This model introduces the possibility to interpret gravity and the quantum behaviour of matter as two different aspects of the same origin. The change of the quantum vacuum energy density can be considered as the fundamental medium which determines a bridge between gravity and the quantum behaviour, leading to new interest- ing perspectives about the problem of unifying gravity with quantum theory. PACS numbers: 04. ; 04.20-q ; 04.50.Kd ; 04.60.-m. Key words: general relativity, three-dimensional space, quantum vac- uum energy density, quantum mechanics, generalized Klein-Gordon equation for the quantum vacuum energy density, generalized Dirac equation for the quantum vacuum energy density. 1 Introduction The standard interpretation of phenomena in gravitational fields is in terms of a fundamentally curved space-time. However, this approach leads to well known problems if one aims to find a unifying picture which takes into account some basic aspects of the quantum theory. For this reason, several authors advocated different ways in order to treat gravitational interaction, in which the space-time manifold can be considered as an emergence of the deepest processes situated at the fundamental level of quantum gravity.
    [Show full text]
  • Chapter 5 Dimensional Analysis and Similarity
    Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences.
    [Show full text]
  • Units and Magnitudes (Lecture Notes)
    physics 8.701 topic 2 Frank Wilczek Units and Magnitudes (lecture notes) This lecture has two parts. The first part is mainly a practical guide to the measurement units that dominate the particle physics literature, and culture. The second part is a quasi-philosophical discussion of deep issues around unit systems, including a comparison of atomic, particle ("strong") and Planck units. For a more extended, profound treatment of the second part issues, see arxiv.org/pdf/0708.4361v1.pdf . Because special relativity and quantum mechanics permeate modern particle physics, it is useful to employ units so that c = ħ = 1. In other words, we report velocities as multiples the speed of light c, and actions (or equivalently angular momenta) as multiples of the rationalized Planck's constant ħ, which is the original Planck constant h divided by 2π. 27 August 2013 physics 8.701 topic 2 Frank Wilczek In classical physics one usually keeps separate units for mass, length and time. I invite you to think about why! (I'll give you my take on it later.) To bring out the "dimensional" features of particle physics units without excess baggage, it is helpful to keep track of powers of mass M, length L, and time T without regard to magnitudes, in the form When these are both set equal to 1, the M, L, T system collapses to just one independent dimension. So we can - and usually do - consider everything as having the units of some power of mass. Thus for energy we have while for momentum 27 August 2013 physics 8.701 topic 2 Frank Wilczek and for length so that energy and momentum have the units of mass, while length has the units of inverse mass.
    [Show full text]
  • Estimation of Forest Aboveground Biomass and Uncertainties By
    Urbazaev et al. Carbon Balance Manage (2018) 13:5 https://doi.org/10.1186/s13021-018-0093-5 RESEARCH Open Access Estimation of forest aboveground biomass and uncertainties by integration of feld measurements, airborne LiDAR, and SAR and optical satellite data in Mexico Mikhail Urbazaev1,2* , Christian Thiel1, Felix Cremer1, Ralph Dubayah3, Mirco Migliavacca4, Markus Reichstein4 and Christiane Schmullius1 Abstract Background: Information on the spatial distribution of aboveground biomass (AGB) over large areas is needed for understanding and managing processes involved in the carbon cycle and supporting international policies for climate change mitigation and adaption. Furthermore, these products provide important baseline data for the development of sustainable management strategies to local stakeholders. The use of remote sensing data can provide spatially explicit information of AGB from local to global scales. In this study, we mapped national Mexican forest AGB using satellite remote sensing data and a machine learning approach. We modelled AGB using two scenarios: (1) extensive national forest inventory (NFI), and (2) airborne Light Detection and Ranging (LiDAR) as reference data. Finally, we propagated uncertainties from feld measurements to LiDAR-derived AGB and to the national wall-to-wall forest AGB map. Results: The estimated AGB maps (NFI- and LiDAR-calibrated) showed similar goodness-of-ft statistics (R­ 2, Root Mean Square Error (RMSE)) at three diferent scales compared to the independent validation data set. We observed diferent spatial patterns of AGB in tropical dense forests, where no or limited number of NFI data were available, with higher AGB values in the LiDAR-calibrated map. We estimated much higher uncertainties in the AGB maps based on two-stage up-scaling method (i.e., from feld measurements to LiDAR and from LiDAR-based estimates to satel- lite imagery) compared to the traditional feld to satellite up-scaling.
    [Show full text]
  • Improving the Accuracy of the Numerical Values of the Estimates Some Fundamental Physical Constants
    Improving the accuracy of the numerical values of the estimates some fundamental physical constants. Valery Timkov, Serg Timkov, Vladimir Zhukov, Konstantin Afanasiev To cite this version: Valery Timkov, Serg Timkov, Vladimir Zhukov, Konstantin Afanasiev. Improving the accuracy of the numerical values of the estimates some fundamental physical constants.. Digital Technologies, Odessa National Academy of Telecommunications, 2019, 25, pp.23 - 39. hal-02117148 HAL Id: hal-02117148 https://hal.archives-ouvertes.fr/hal-02117148 Submitted on 2 May 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Improving the accuracy of the numerical values of the estimates some fundamental physical constants. Valery F. Timkov1*, Serg V. Timkov2, Vladimir A. Zhukov2, Konstantin E. Afanasiev2 1Institute of Telecommunications and Global Geoinformation Space of the National Academy of Sciences of Ukraine, Senior Researcher, Ukraine. 2Research and Production Enterprise «TZHK», Researcher, Ukraine. *Email: [email protected] The list of designations in the text: l
    [Show full text]
  • Guide for the Use of the International System of Units (SI)
    Guide for the Use of the International System of Units (SI) m kg s cd SI mol K A NIST Special Publication 811 2008 Edition Ambler Thompson and Barry N. Taylor NIST Special Publication 811 2008 Edition Guide for the Use of the International System of Units (SI) Ambler Thompson Technology Services and Barry N. Taylor Physics Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899 (Supersedes NIST Special Publication 811, 1995 Edition, April 1995) March 2008 U.S. Department of Commerce Carlos M. Gutierrez, Secretary National Institute of Standards and Technology James M. Turner, Acting Director National Institute of Standards and Technology Special Publication 811, 2008 Edition (Supersedes NIST Special Publication 811, April 1995 Edition) Natl. Inst. Stand. Technol. Spec. Publ. 811, 2008 Ed., 85 pages (March 2008; 2nd printing November 2008) CODEN: NSPUE3 Note on 2nd printing: This 2nd printing dated November 2008 of NIST SP811 corrects a number of minor typographical errors present in the 1st printing dated March 2008. Guide for the Use of the International System of Units (SI) Preface The International System of Units, universally abbreviated SI (from the French Le Système International d’Unités), is the modern metric system of measurement. Long the dominant measurement system used in science, the SI is becoming the dominant measurement system used in international commerce. The Omnibus Trade and Competitiveness Act of August 1988 [Public Law (PL) 100-418] changed the name of the National Bureau of Standards (NBS) to the National Institute of Standards and Technology (NIST) and gave to NIST the added task of helping U.S.
    [Show full text]
  • How Are Units of Measurement Related to One Another?
    UNIT 1 Measurement How are Units of Measurement Related to One Another? I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind... Lord Kelvin (1824-1907), developer of the absolute scale of temperature measurement Engage: Is Your Locker Big Enough for Your Lunch and Your Galoshes? A. Construct a list of ten units of measurement. Explain the numeric relationship among any three of the ten units you have listed. Before Studying this Unit After Studying this Unit Unit 1 Page 1 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. High School Chemistry: An Inquiry Approach 1. Use the measuring instrument provided to you by your teacher to measure your locker (or other rectangular three-dimensional object, if assigned) in meters. Table 1: Locker Measurements Measurement (in meters) Uncertainty in Measurement (in meters) Width Height Depth (optional) Area of Locker Door or Volume of Locker Show Your Work! Pool class data as instructed by your teacher. Table 2: Class Data Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Width Height Depth Area of Locker Door or Volume of Locker Unit 1 Page 2 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction.
    [Show full text]
  • Download Full-Text
    International Journal of Theoretical and Mathematical Physics 2021, 11(1): 29-59 DOI: 10.5923/j.ijtmp.20211101.03 Measurement Quantization Describes the Physical Constants Jody A. Geiger 1Department of Research, Informativity Institute, Chicago, IL, USA Abstract It has been a long-standing goal in physics to present a physical model that may be used to describe and correlate the physical constants. We demonstrate, this is achieved by describing phenomena in terms of Planck Units and introducing a new concept, counts of Planck Units. Thus, we express the existing laws of classical mechanics in terms of units and counts of units to demonstrate that the physical constants may be expressed using only these terms. But this is not just a nomenclature substitution. With this approach we demonstrate that the constants and the laws of nature may be described with just the count terms or just the dimensional unit terms. Moreover, we demonstrate that there are three frames of reference important to observation. And with these principles we resolve the relation of the physical constants. And we resolve the SI values for the physical constants. Notably, we resolve the relation between gravitation and electromagnetism. Keywords Measurement Quantization, Physical Constants, Unification, Fine Structure Constant, Electric Constant, Magnetic Constant, Planck’s Constant, Gravitational Constant, Elementary Charge ground state orbital a0 and mass of an electron me – both 1. Introduction measures from the 2018 CODATA – we resolve fundamental length l . And we continue with the resolution of We present expressions, their calculation and the f the gravitational constant G, Planck’s reduced constant ħ corresponding CODATA [1,2] values in Table 1.
    [Show full text]
  • Rational Dimensia
    Rational Dimensia R. Hanush Physics Phor Phun, Paso Robles, CA 93446∗ (Received) Rational Dimensia is a comprehensive examination of three questions: What is a dimension? Does direction necessarily imply dimension? Can all physical law be derived from geometric principles? A dimension is any physical quantity that can be measured. If space is considered as a single dimension with a plurality of directions, then the spatial dimension forms the second of three axes in a Dimensional Coordinate System (DCS); geometric units normalize unit vectors of time, space, and mass. In the DCS all orders n of any type of motion are subject to geometric analysis in the space- time plane. An nth-order distance formula can be derived from geometric and physical principles using only algebra, geometry, and trigonometry; the concept of the derivative is invoked but not relied upon. Special relativity, general relativity, and perihelion precession are shown to be geometric functions of velocity, acceleration, and jerk (first-, second-, and third-order Lorentz transformations) with v2 coefficients of n! = 1, 2, and 6, respectively. An nth-order Lorentz transformation is extrapolated. An exponential scaling of all DCS coordinate axes results in an ordered Periodic Table of Dimensions, a periodic table of elements for physics. Over 1600 measurement units are fitted to 72 elements; a complete catalog is available online at EPAPS. All physical law can be derived from geometric principles considering only a single direction in space, and direction is unique to the physical quantity of space, so direction does not imply dimension. I. INTRODUCTION analysis can be applied to all.
    [Show full text]