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Open Journal of Mathematics and Physics | Volume 2, Article 78, 2020 | ISSN: 2674-5747 https://doi.org/10.31219/osf.io/js7rf | published: 7 Feb 2020 | https://ojmp.wordpress.com DA [microresearch] Diamond Open Access

A hole in the hole

Open Physics Collaboration∗† April 19, 2020

Abstract Supposedly, falls inside the whenever it reaches its horizon. The Planck scale, however, imposes a limit on how much matter can occupy the center of a black hole. It is shown here that the of matter exceeds Planck density in the singularity, and as a result, tears apart. After the black hole is formed, matter flows from its center to its border due to a topological force, namely, the increase on the tear of spacetime due to its limit until it reaches back to the , generating the phenomenon. We conclude that there is no spacetime inside black holes. We propose a solution to the black hole information paradox.

keywords: black hole information paradox, singularity, firewall, , topology,

Introduction

1. Black holes are controversial astronomical objects [1] exhibiting such a strong gravitational field that nothing–not even –can escape from inside it [2].

∗All authors with their affiliations appear at the end of this paper. †Corresponding author: [email protected] | Join the Open Physics Collaboration

1 2. A black hole is formed when the density of matter exceeds the amount supported by spacetime.

3. It is believed that at or near the event horizon, there are high-energy quanta, known as the black hole firewall [3].

4. At the center of the black hole, there is a singularity, i.e., a mathe- matical infinite.

5. Neither nor can actually account for what is going on inside those extreme objects.

6. However, both theories are robust, and they will guide us in the pro- cesses described here.

7. The main motivation for this work is that the entropy of a black hole is area-dependent which might give us a clue that all matter lies within its surface.

Limits of the Planck scale

8. The Planck scale provides us with the limits of space and time accord- ing to the quantum theories.

9. The Planck density is given by

Mp 94 3 ρp 3 10 g cm , lp

= ∼ 3 where Mp hc G is the Planck mass, and lp hG c is the . √ √ = ̵ = ̵ 10. A has typically a mass M 1034 g, volume 106 cm3, which gives a density of 1028 g cm3. ∼ ∼ 11. If all this mass is to be a∼ttracted to the minimal Planck length, lp 10 33 cm, the density of the stellar black hole at the Planck volume − ∼ 2 becomes 10133 g cm3, which exceeds by far the limit on the density of spacetime. ∼ 12. We conjecture that (11) causes spacetime to tear apart inside the black hole, starting at its singularity.

13. The line element for the in the is

2M dr2 dτ 2 1 dt2 r2dφ2. (1) r 2M 1 r = ( − ) − − 14. In the singularity, r 0, dτ dt , w(hi−ch ca)n be interpreted as being the end of time for all external observers, considering that dt 0. = → ∞ 15. In addition, since mass “slows” time from an outsider perspect→ive, the maximum mass density ultimately vanishes time.

16. Another evidence for our claim is that the Bekenstein-Hawking en- tropy of the black hole [4,5],

kB A Sbh 2 , 4lp depends solely on the area of the=black hole (A), and not in its “vol- ume”.

17. kB is the Boltzmann’s constant.

18. The black-hole entropy is the maximal entropy of spacetime [6].

19. The black hole entropy (16) is related to the microstates (microscopic configurations) of a physical system, and it depends exclusively on the area of its event horizon; therefore, it is natural to think that after the black hole is formed, all matter lies within its surface, namely, the event horizon.

2 20. The term A lp in the entropy Sbh represents the total number of Planck 2 cells (A lp) within the surface of the event horizon of the black hole. 3 The birth of a black hole and the tear of spacetime

21. We believe that the following steps take place during the birth of a black hole.

22. Let’s define g-hole to be the tridimensional hole formed due to the void of spacetime inside the black hole.

23. g stands for genus from topology.

24. Step 1. The star becomes maximally compacted, i.e., it reaches the Planck density.

25. Step 2. Spacetime tears apart in the singularity.

26. Step 3. The mass previously in the singularity remains coupled with space and goes to the border of the g-hole.

27. Step 4. The new incoming mass reaching the surface of the g-hole sur- passes the Planck density limit, and the radius of the g-hole increases in size.

28. Step 5. The process continues so on and so forth until it reaches an equilibrium, namely, the event horizon.

29. We are considering that spacetime has finite “elasticity”.

Discussion

30. The singularity attracts huge amounts of mass and energy, so it ex- ceeds the maximum amount supported by spacetime.

31. The result is that the fabric of spacetime tears apart.

32. Matter and energy are still bounded with space and time.

4 33. Particles previously in the singularity move forward to the border of the hole.

34. Again, the limit is reached; the hole increases; and the process contin- ues so on and so forth.

35. It summarily ends with all matter and energy occupying the surface of the black hole, resulting in the firewall phenomenon [3].

36. One can think on the fabric of spacetime as a rubber band.

37. It will eventually tear apart when the star reaches the Planck density, typically with a radius of 10 20 cm for M 1034 g. − 38. The topological force o∼riginated from th∼e puncture of spacetime pushes all matter to the event horizon like an accelerated small in reverse.

39. Therefore, from the premise that spacetime elasticity is finite, we can conclude that there is NO spacetime inside the black holes.

40. We anticipate our essay to be a starting point for a complete under- standing of black holes since everything there is to exist lies within the event horizon, avoiding for example, the black hole information paradox [7].

41. In this scenario, no information is lost whatsoever.

42. The discovery of black holes radically affected our understanding of general relativity.

43. The verification–both mathematical and observational (directly or indirectly)–for this conjecture might shed light on the quantum nature of gravity.

44. Additionally, at the singularity of a black hole, the entanglement of spacetime undergoes a sudden death due to its strong gravity [8], i.e.,

5 entanglement breaks down, and therefore, spacetime tears apart [9]; the hole (void of spacetime) increases, reaching the event horizon, resulting in the firewall phenomenon, and having no information lost in the black hole.

45. The results proposed here state that there is no spacetime inside black holes.

46. There are at least three huge evidence for this claim, namely, Bekenstein- Hawking entropy, the firewall phenomenon, and the spacetime infor- mation threshold, i.e., the limit of information that spacetime can hold.

The role of the singularity

47. The singularity (at r 0) exists only during the black hole formation.

48. After the black hole is=formed, there is no singularity since there is no spacetime in its interior.

Experimental evidence

49. Are the Universal interferometric signatures of a black hole’s pho- ton ring [10] evidence of the firewall phenomenon and the hollow black holes [11]?

Final Remarks

50. In summary, spacetime tears apart at its center, on its very formation, causing matter to move forward to its border.

51. This is a “topological force” in nature, and it is somehow related to topological changes [12].

6 52. There are a number of topologies that can be defined in a mathematical space.

53. The birth of a black hole implies a change in the original topology of the underlying spacetime.

54. Changing a topology means changing the collection of open sets; this, in turn, changes which functions are continuous and which subsets are compact or connected [13].

55. According to our conjecture, the black hole has the topology of a hollow sphere.

56. We dubbed (55) the Hollow Black Holes [11].

57. The change in the topology of spacetime that originates the black hole from the inside (singularity) to the outside (event horizon) is in accor- dance with the [6,14], which makes the informa- tion correspondence from a volume to a lower-dimensional boundary.

58. This work is a foundational investigation and it agrees with ’s proposal that “the information is stored not in the interior of the black hole as one might expect, but in its boundary, the event horizon,” presented at the KTH Institute of Technology in Stockholm [15].

59. Regarding that all of the black hole mass is in the event horizon, the physics of the black holes will reach new levels of inquiries, discoveries, and understanding.

60. Analogies between black holes and bidimensional materials might shed light to .

61. Some suggestions include to search on topological changes [12], both in applied and pure mathematical approach, in order to understand this topological mechanism more profoundly.

7 62. Other interesting questions arise, such as, what is the impact of this result in the black hole evaporation?

63. What is the “width” of the firewall?

Open Invitation

Review, add content, and co-author this paper [16,17]. Join the Open Physics Collaboration. Send your contribution to [email protected].

Ethical conduct of research

This original work was pre-registered under the OSF Preprints [18], please cite it accordingly [19]. This will ensure that researches are con- ducted with integrity and intellectual honesty at all times and by all means.

References

[1] Curiel, Erik. “The many definitions of a black hole.” Nature Astron- omy 3.1 (2019): 27.

[2] Wald, Robert M. General relativity. Chicago, University of Chicago Press, 1984, 504 p (1984).

[3] Almheiri, Ahmed, et al. “Black holes: complementarity or firewalls?.” Journal of High Energy Physics 2013.2 (2013): 62.

[4] Wald, Robert M. Quantum field theory in curved spacetime and black hole . University of Chicago Press, 1994.

[5] Carlip, Steve. “Black hole thermodynamics.” International Journal of Modern Physics D 23.11 (2014): 1430023.

8 [6] Bousso, Raphael. “The holographic principle.” Reviews of Modern Physics 74.3 (2002): 825.

[7] Mathur, Samir D. “The information paradox: a pedagogical introduc- tion.” Classical and Quantum Gravity 26.22 (2009): 224001.

[8] Doukas, Jason, et al. “Entanglement and discord: Accelerated obser- vations of local and global modes.” Physical Review A 87.1 (2013): 012306.

[9] Van Raamsdonk, Mark. “Building up spacetime with quantum entan- glement.” General Relativity and Gravitation 42.10 (2010): 2323-2329.

[10] Johnson, Michael D., et al. “Universal Interferometric Signatures of a Black Hole’s Photon Ring.” arXiv preprint arXiv:1907.04329 (2019). https://arxiv.org/pdf/1907.04329

[11] Lobo, Matheus P. “Hollow Black Holes.” OSF Preprints, 12 Aug. 2019. https://doi.org/10.31219/osf.io/rewub

[12] Borde, Arvind. “Regular black holes and topology change.” Physical Review D 55.12 (1997): 7615.

[13] Munkres, James. Topology. Pearson Education, 2014.

[14] Susskind, Leonard. “The world as a hologram.” Journal of Mathe- matical Physics 36.11 (1995): 6377-6396.

[15] Hawking, Stephen W. “Information preservation and weather fore- casting for black holes.” arXiv preprint arXiv:1401.5761 (2014).

[16] Lobo, Matheus P. “Microarticles.” OSF Preprints, 28 Oct. 2019. https://doi.org/10.31219/osf.io/ejrct

[17] 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

9 [18] COS. Open Science Framework. https://osf.io

[19] Lobo, Matheus P. “A Hole in the Black Hole.” OSF Preprints, 18 Apr. 2019. https://doi.org/10.31219/osf.io/js7rf

The Open Physics Collaboration

Matheus Pereira Lobo (lead author, [email protected]),1,2 Tiago Sousa Moraes,3 Jonas Gomes da Silva,1 Ronaldo Silva Rêgo4

1Federal University of Tocantins (Brazil); 2Universidade Aberta (UAb, Portugal); 3Open Physics Collaboration; 4Universidade Estadual da Região Tocantina do Maranhão

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