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.