diff --git "a/CdAyT4oBgHgl3EQfePjS/content/tmp_files/load_file.txt" "b/CdAyT4oBgHgl3EQfePjS/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/CdAyT4oBgHgl3EQfePjS/content/tmp_files/load_file.txt" @@ -0,0 +1,793 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf,len=792 +page_content='Quantum Hairy Black Hole Formation and Horizon Quantum Mechanics R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Cavalcanti∗ and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Hoff da Silva† Departamento de F´ısica, Universidade Estadual Paulista (Unesp), Guaratinguet´a 12516-410, Brazil After introducing the gravitational decoupling method and the hairy black hole recently derived from it, we investigate the formation of quantum hairy black holes by applying the horizon quan- tum mechanics formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' It enables us to determine how external fields, characterized by hairy parameters, affect the probability of spherically symmetric black hole formation and the generalized uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' INTRODUCTION Given their intrinsic connection with intense gravitational fields, solid theoretical basis [1–3], and several observa- tional results corroborating their existences, black holes play a central role in contemporary high-energy physics and astrophysics [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Despite the characterization of the horizon of stationary black hole solutions being well-known within general relativity [3, 8], the nature of the horizons of non-stationary or stationary solutions beyond general relativity is still a source of extensive research [9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The investigation of black holes is not restricted to astrophysical objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' they are also expected to be formed whenever a high concentration of energy is confined to a small region of spacetime, producing so-called quantum black holes [7, 13–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' However, the precise formation mechanism of classical and quantum black holes is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Although we do not have a theory of quantum gravity, phenomenology suggests that some features of quantum black holes are expected to be model-independent [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From a certain scale, candidate theories should modify the results of general relativity, giving birth to some alternatives to Einsteins’s theory of gravity [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Examples could allow for the presence of non-minimal coupled fundamental fields or higher derivative terms during the action, which directly affects the uniqueness theorems of black holes in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The famous no-hair theorem is not preserved outside the general relativity realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' These solutions lead to effects that are potentially detectable near the horizon of astrophysical black holes [20–22], or in quantum black holes’ formation [23, 24], and may provide hints for the quantum path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' One of the major challenges in general relativity is finding physically relevant solutions to Einstein’s field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' On the other hand, deriving new solutions from other previously known ones is a widespread technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' This approach is precisely what the so-called gravitational decoupling (GD) method intends to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' It has recently commanded the community’s attention due to its simplicity and effectiveness [25–27] in generating new, exact analytical solutions by considering additional sources to the stress-energy tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The recent description of anisotropic stellar distribu- tions [28, 29], whose predictions might be tested in astrophysical observations [30–33], as well as the hairy black hole solutions by gravitational decoupling, are particularly interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The latter describes a black hole with hair sourced by generic fields, possibly of quantum nature, surrounding the vacuum Schwarzschild solution [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Exciting results have been found during investigation of this solution [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From the quantum side, one of the key features of quantum gravity phenomenology is the generalized uncertainty principle (GUP), which modifies the Heisenberg uncertainty principle accordingly ∆x∆p ≳ ℏ � 1 + ϵ(∆p)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (1) This expression of the GUP, which stems from different approaches to quantum gravity [37–46], characterizes a minimum scale length ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' This feature emerges quite naturally in the horizon quantum mechanics formalism (HQM) [16, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' In addition to the GUP, HQM also provides an estimation of the probability of quantum black hole formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' In a scenario of extra-dimensional spacetimes, the HQM gave an explanation for the null results of quantum black hole formation in current colliders [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Could it also tell us something about a mechanism for decreasing the fundamental scale to something near the scale of current colliders?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Our aim is to investigate the quantitative and qualitative effects of black hole hair, regarding the probability of black hole formation and the GUP by applying the horizon quantum mechanics formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' This paper is organized as follows: Section II is dedicated to reviewing the gravitational decoupling procedure, the metric for GD hairy black holes, and an approximation for the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' In Section III, we apply the horizon ∗Electronic address: rogerio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='cavalcanti@unesp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='br †Electronic address: julio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='hoff@unesp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='br arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='00319v1 [gr-qc] 1 Jan 2023 2 quantum mechanics formalism to the hairy black hole solution of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' We compare the probability of quantum black hole formation and the GUPs of hairy black holes for a range of hair parameters, unveiling the effects of the hair fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Finally, Section IV is dedicated to conclusions and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' HAIRY BLACK HOLES AND HORIZON RADIUS Starting from Einstein’s field equations Gµν = 8π ˇTµν, (2) where Gµν = Rµν − 1 2Rgµν denotes the Einstein tensor, the gravitational decoupling (GD) [25] method takes the energy–momentum tensor decomposed as ˇTµν = Tµν + Θµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (3) Here, Tµν is the source of a known solution to general relativity, while Θµν introduces a new field or extension of the gravitational sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From ∇µ Gµν = 0, we also have ∇µ ˇT µν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The effective density and the tangential and radial pressures can be determined by examining the field equations ˇρ = ρ + Θ 0 0 , (4a) ˇpt = p − Θ 2 2 , (4b) ˇpr = p − Θ 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (4c) The idea is to deform a known solution to split the field equations in a sector containing the known solution with source Tµν and a decoupled one governing the deformation, encompassing Θµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' In fact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' assuming a known spherically symmetric metric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' ds2 = −eκ(r)dt2 + eζ(r)dr2 + r2dΩ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (5) and deforming κ(r) and ζ(r) as κ(r) �→ κ(r) + αf2(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (6a) e−ζ(r) �→ e−ζ(r) + αf1(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (6b) the resulting decoupled field equations read 8π Θ 0 0 = α �f1 r2 + f ′ 1 r � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (7a) 8π Θ 1 1 − α e−ζ f ′ 2 r = α f1 � 1 r2 + κ′(r) + αf ′ 2(r) r � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (7b) 8πΘ 2 2 −αf1Z1(r) =αf ′ 1 4 � κ′(r) + αf ′ 2(r)+ 2 r � +αZ2(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (7c) where [25] Z1(r) = α2f ′ 2 (r)2 + 2 α � f ′ 2 (r) κ′ (r) + f ′ 2 (r) r + f ′′ 2 (r) � + κ′ (r)2 + 2 κ′ (r) r + 2 κ′′ (r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (8a) Z2(r) = αe−ζ � 2f ′′ 2 + f 2′ 2 + 2f ′ 2 r + 2κ′f ′ 2 − ζ′f ′ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (8b) The above equations state that if the deformation parameter α goes to zero, then Θµν must go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' It is worth mentioning that for extended geometric deformation, that is, for f2 ̸= 0, the sources are not individually conserved in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' However, as discussed in [26], in this case, the decoupling of the field equations without an exchange of energy is allowed in two scenarios: (a) when Tµν is a barotropic fluid whose equation of state is T00 = T11 or (b) for vacuum regions of the first system Tµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' When minimal geometric deformation is applied, on the other hand, the sources are shown to be individually conserved [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Assuming the Schwarzschild solution to be the known one and requiring a well-defined horizon structure [27], from grr = − 1 gtt follows � 1 − 2M r � � eαf2(r) − 1 � = αf1(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (9) 3 Therefore, one is able to write ds2 = − � 1 − 2M r � eαf2(r)dt2+ � 1 − 2M r �−1 e−α f2(r)dr2 + r2 dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (10) Further, assuming strong energy conditions, ˇρ + ˇpr + 2 ˇpt ≥ 0, (11a) ˇρ + ˇpr ≥ 0, (11b) ˇρ + ˇpt ≥ 0, (11c) and managing the field equations, a new hairy black hole solution was found [27] ds2 = −f(r)dt2 + 1 f(r)dr2 + r2dΩ2, (12) where f(r) = 1 − 2GM + αℓ r + αe− r GM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (13) The dimensionless parameter 0 ≤ α ≤ 1 tracks the deformation of the Schwarzschild black hole, e is the Euler constant, and ℓ is the direct effect of the nonvanishing additional font Θµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Notice that by taking α = 0, the Schwarzschild solution is restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Further, the ℓ parameter is limited to 2GM/e2 ≤ ℓ ≤ 1 due to the assumption of a strong energy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' In extreme cases, ℓ = 2GM/e2 and fe(r) = 1 − 2GM r + α � e− r GM − 2GM e2 r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (14) The hairy black hole has a single horizon, located at r = rH, such that � 1 + αe− rH GM � rH = 2GM + αℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (15) Such an equation has no analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Nevertheless, a very accurate analytical approximation is found by Taylor expanding it around the Schwarzschild horizon radius rS = 2GM, rH GM ≈ 4 � αℓe2/GM − 3 α + e2� αℓe2/GM − 4 α + 2 e2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (16) Figure 1 shows a comparison between the exact and approximated horizon radii for different values of the hairy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' In the following section, we are going to use Equation (16) for the analytical expression of the hairy black hole’s horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='0 ℓ GM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='8 rH GM α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='00 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='20 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='40 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='60 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='80 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='0 Exact FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 1: The radius of the hairy black hole horizon rH as a function of ℓ for different values of the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The colored dashed lines represent the approximated radius, and the gray lines are the exact ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' It shows how the hairy horizon deviates from the Schwarzschild horizon for an increasing α and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The ranges for α and ℓ were fixed due to the assumption of a strong energy condition [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' THE HORIZON QUANTUM MECHANICS FORMALISM Horizon quantum mechanics (also known as horizon wave function formalism) is an effective approach capable of providing the signatures of black hole physics to the Planck scale [48–51] (see [47] for a comprehensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The main idea is to extend quantum mechanics and gravity further than the current experimental limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' In such an approach, we face the conceptual challenge of consistently describing classical and quantum mechanical objects, such as horizons and particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' This is achieved by assigning wave functions to the quantum black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' This association allows the use of quantum mechanical machinery to distinguish between particles and quantum black holes and to estimate the GUPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Nevertheless, first, we must choose a model describing the particle wave function to derive the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Due to the previous results’ simplicity and efficiency, we shall use the Gaussian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From classical general relativity, we know that the horizons of black holes are described by trapping surfaces, whose locations are determined by gij∇ir∇jr = 0 , (17) where ∇ir is orthogonal to the surfaces of the constant area A = 4πr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' A trapping surface then exists if there are values of r and t such that the gravitational radius RH satisfies RH(r, t) ≥ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (18) Considering a spinless point-particle of mass m, an uncertainty in the spatial particle localization of the same order of the Compton scale λm ≃ ℏ/m = lp mp/m follows from the uncertainty principle, where lp and mp are the Planck length and mass, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Arguing that quantum mechanics gives a more precise description of physics, RH makes sense only if it is larger than the Compton wavelength associated with the same mass, namely RH ≳ λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Thus, for the Schwarzschild radius RS = 2Gm = 2 lp mp m, lp m/mp ≳ lp mp/m =⇒ m ≳ mp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (19) This suggests that the Planck mass is the minimum mass such that the Schwarzchild radius can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From quantum mechanics, the spectral decomposition of a spherically symmetric matter distribution is given by the expression |ψS⟩ = � E C(E) |ψE⟩ , (20) with the usual eigenfunction equation ˆH |ψE⟩ = E |ψE⟩ , (21) regardless of the specific form of the actual Hamiltonian operator ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Using the energy spectrum and inverting the expression of the Schwarzschild radius, we have E = mp rH 2lp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (22) Putting it back into the wave function, one can define the (unnormalized) horizon wave function as ψH(rH) = C � mp rH 2lp � (23) whose normalization is fixed, as usual, by the inner product ⟨ψH | φH⟩ = 4π � ∞ 0 ψ∗ H(rH)φH(rH)r2 HdrH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (24) However, the classical radius RH is thus replaced by the expected value of the operator ˆRH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From the uncertainty of the expectation value, it follows that the radius will necessarily be “fuzzy”, similar to the position of the source itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The next aspect one has to approach to establish a criterion for deciding if a mass distribution does or does not form a black hole is if it lies inside its horizon of radius r = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From quantum mechanics, one finds that it is given by the product P<(r < rH) = PS(r < rH)PH(rH), (25) 5 where the first term, PS(r < rH) = 4π � rH 0 |ψS(r)|2r2dr, (26) is the probability that the particle resides inside the sphere of radius r = rH, while the second term, PH(rH) = 4πr2 H|ψH(rH)|2 (27) is the probability density that the value of the gravitational radius is rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Finally, the probability that the particle described by the wave function ψS is a BH will be given by the integral of (25) over all possible values of the horizon radius rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Namely, PBH = � ∞ 0 P<(r < rH)drH, (28) which is one of the main outcomes of the formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Gaussian Sources The previous construction can be made explicit by applying the Gaussian model for the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' To implement this idea, let us recall that spectral decomposition is also assumed to be valid for momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Therefore, from (20), ⟨p |ψS⟩ = C(p) ≡ ψH(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The Gaussian wave function for ψS scales as r2 in the position space and leads to a Gaussian wave function in the momentum space, scaling as p2, naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Finally, since the dispersion relation relates p2 with energy, we are able to have ⟨p |ψS⟩ = ψH(rH) via (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Hence, starting with a Gaussian wave function, we can describe a spherically symmetric massive particle at rest, such as ψS(r) = e− r2 2 l2 (l √π)3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (29) The corresponding function in momentum space is thus given by ˜ψS(p) = 4π � ∞ 0 sin(rp) √ 8π3rp e− r2 2 l2 (l √π)3/2 r2dr = e− p2 2 ∆2 (∆ √π)3/2 , (30) where ∆ = mp lp/l is the spread of the wave packet in momentum space, whose width l the Compton length of the particle should diminish, l ≥ λm ∼ mp lp m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (31) In addition to the straightforward handling of a Gaussian wave packet, it is also relevant to recall that the Gaussian wave function leads to a minimal uncertainty for the expected values computed with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Had we used another wave function, it would certainly imply a worsening uncertainty, eventually leading to unnecessary extra difficulties relating to the HQM and GUP (see next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Back to our problem, assuming the relativistic mass-shell relation in flat space [48] p2 = E2 − m2 , (32) the energy E of the particle is expressed in terms of the related horizon radius rH = RH(E), following from Equation (16), E = αmpℓe2 + � α − e2� mprH 2 (2 α − e2)lp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (33) 6 Thus, from Equations (30) and (33), one finds the the horizon wave function of the hairy black hole ψH(rH) = NHΘ(rH − RH) e(C2r2 H+C1rH+C0), where C0 = − α2l2m2 pℓ2e4 8 (2 α − e2)2l2p , C1 = − � α − e2� αl2m2 pℓe2 4 (2 α − e2)2l2p , C2 = − � α − e2�2l2m2 p 8 (2 α − e2)2l2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (34) The Heaviside step function Θ appears above due to the imposition E ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The normalisation factor NH is fixed according to N −2 H = 4π � ∞ 0 |ψH(rH)|2 r2 H drH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The normalized horizon wave function is thus given as follows ψH(rH) = − 2 C 3 2 2 e A(rH ) 2 √π � 4 C1C2eA(RH) − � 2 √ 2C2Γ � 3 2 , −A(RH) � + √ 2πC2 1 � erf � √ 2(2 C2RH+C1) 2 √−C2 � − 1 ��√−C2 , (35) A(x) = 4 C2 2x2 + 4 C1C2x + C2 1 2 C2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Here, Γ(s, x) denotes the upper incomplete Euler–Gamma function and erf(x) the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The expression above has two classes of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Two of these, α and ℓ, are related to the hairy black hole, and two are non-fixed a priori: the particle mass m, encoded in RH, and the Gaussian width l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The resulting probability PBH = PBH(l, m, ℓ, α) will also depend on the same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' According to the previous discussion, before finding the probability distribution, we have first to find the probability that the particle resides inside a sphere with the radius r = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From Equations (26) and (29), one obtains PS(r < rH) = 4π � rH 0 |ψS(r)|2r2dr = 2 √π γ �3 2, r2 H l2 � , with γ(s, x) = Γ(s) − Γ(s, x), the lower incomplete Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Equations (27) and (35) yield PH(rH), as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 0 1 2 3 4 5 mprH lpm 0 1 PH(rH) l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='50 l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='0 l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='5 l = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 2: The probability density for the value of the gravitational radius is rH for α = ℓ/(GM) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='5 and different values of the Gaussian width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Combining the previous results, one finds that the probability density for the particle resides within its own gravitational radius P<(r < rH) = 8√πγ �3 2, r2 H l2 � r2 H|ψH(rH)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 7 The probability of the particle described by the Gaussian to be a black hole is finally given by PBH(l, m, ℓ, α) = 8√π � ∞ RH γ �3 2, r2 H l2 � r2 H|ψH(rH)|2, (36) which has to be calculated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Assuming the Gaussian width has the same order as the particle Compton length, we could set l ∼ m−1 on Equation (36) and find the probability depending on either l or m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' On the other hand, by departing again from Equation (31), we may set values for m in terms of the Planck mass and find the probability in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Applying l ∼ m−1 yields PBH(l, ℓ, α) = 8√π � ∞ RH γ �3 2, r2 H l2 � r2 H|ψH(rH)|2, (37) or PBH(m, ℓ, α) = 8√π � ∞ RH γ �3 2, r2 Hm2 � r2 H|ψH(rH)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (38) The resulting probabilities are shown in Figure 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Figure 4 displays the probability for m given as a fraction of the Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 0 1 2 3 4 5 l lp 0 1 PBH ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='00 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='30 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='60 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='90 1 2 m mp 0 1 PBH ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='00 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='30 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='60 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='90 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 3: The probability of a ”particle” being a black hole depending on the Gaussian width or mass, assuming l ∼ m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 1 2 3 4 5 l lp 0 1 PBH ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='00 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='30 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='60 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='90 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 4: The probability of a ”particle” being a black hole depending on the Gaussian width and mass m given as a fraction of the Planck mass, with m = mp (solid), m = 3mp/4 (dashed), and m = mp/2 (dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' HQM and GUP Since the horizon quantum mechanics formalism applies the standard wave function description for particles, a natural question is whether it affects the Heisenberg uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' As mentioned, it produces a GUP similar to that produced by Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' In quantum mechanics, the uncertainty principle may be derived by calculating the uncertainty associated with the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Here, we start from the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From the Gaussian wave function (29), the particle size uncertainty is given by ∆r2 0 = ⟨r2⟩ − ⟨r⟩2 = 4π � ∞ 0 |ψS(r)|2r4dr − � 4π � ∞ 0 |ψS(r)|2r3dr �2 = 3π − 8 2π l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (39) One might find the uncertainty of the horizon radius in an analogous way,1 ∆r2 H = ⟨r2 H⟩ − ⟨rH⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' (40) The total radial uncertainty can now be taken as a linear combination of the quantities calculated above, ∆r = ∆r0 + ϵ∆rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' For the uncertainty in momentum, we have ∆p2 = ⟨p2⟩ − ⟨p⟩2 = 3π − 8 2π m2 pl2 p l2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Note that the momentum uncertainty and the width l are related such that ∆p ∼ 1/l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Using this fact in ∆r = ∆r0 + ϵ∆rH, one is able to find ∆r lp = 3π − 8 2π mp ∆p + ϵ∆H �∆p mp � , (41) which is similar to the GUP discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The function ∆H also depends on the wave function and hairy black hole parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Figure 5 shows the behavior of the GUP as a function of the momentum uncertainty, taking ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' There, we can see a minimum ∆r placed around the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' From the GUP expression, it is straightforward to see that a larger ϵ means significant correction to the quantum mechanics’ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The hairy parameters, however, have a small qualitative effect on fixing the minimum scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' As shown in Figure 5, their effects become prominent for a large ∆p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 1 2 ∆p mp 1 2 ∆r lp ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='00 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='30 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='60 ℓmp lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='90 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 5: GUP profile emerged from the horizon wave function formalism for ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' The dotted line represents the particle size uncertainty ∆r0, the dashed line represents the uncertainty of the horizon radius ∆rH, and the solid lines describe the GUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 1 The analytical expression of ∆r2 H is huge and little enlightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 9 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' DISCUSSION A few years ago, effective theories suggested lowering the scale of quantum black hole formation to TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Thus, in principle, it became experimentally accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' In spite of no quantum black holes being detected, solid theoretical results point out that such objects should exist in nature [7, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' They could give us valuable hints about quantum gravity features [7, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' One of this paper’s motivating questions was whether a generic black hole hair could significantly change the scale of quantum black hole formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' However, regarding the analysis carried out here, the hairy black holes look qualitatively similar to the Schwarzschild one, with a probability PBH of a similar shape and a related GUP, leading to the existence of a minimum length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Nevertheless, one of the main results of the present paper is that the existence of hair increases the probability PBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' This is indeed a point to be stressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Its explanation rests upon the fact that the hairy black hole radius is slightly larger than the one for Schwarzschild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' This implies that, although the scale of quantum black hole formation is still beyond the current experimental scale, additional fields may lower such scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Those results might impact future colliders’ estimations of quantum black holes coming from alternative theories of gravity and potentially stimulate investigations of specific models of quantum hairy black holes [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' Acknowledgements R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' thanks Unesp—AGRUP for the financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' thanks CNPq (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' 303561/2018-1) for the financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} +page_content=' [1] Hawking, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'} 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