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 =interesting 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≈light 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