diff --git "a/MNAyT4oBgHgl3EQfgfgN/content/tmp_files/load_file.txt" "b/MNAyT4oBgHgl3EQfgfgN/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/MNAyT4oBgHgl3EQfgfgN/content/tmp_files/load_file.txt" @@ -0,0 +1,609 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf,len=608 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='00358v1 [gr-qc] 1 Jan 2023 Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (will be inserted by the editor) Nonsingular Black Holes in Higher dimensions Bikash Chandra Paul Received: date / Accepted: date Abstract We present a class of new nonsingular black holes in higher dimensional theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Assuming a specific form of the stress energy tensor exact an- alytic solutions of the field equation are generated in general theory of relativity (GR) and Rastall theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The non-singular black hole solutions are obtained with a finite pressure at the centre in D = 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For D > 4 the transverse pres- sure is found finite at the centre for a set of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In the later case the transverse pressure is more than that in the usual four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The exact ana- lytic solution of the field equations in higher dimensions for large r coincides with the Schwarzschild black hole solution in the usual four and in higher dimensions which is singularity free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The different features of the generalized non-singular black hole in GR and modified GR are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' A new vacuum nonsingular black hole is found in Rastall gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We also study the motion of massive and massless particles around the black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 1 Introduction The idea that spacetime dimensions should be extended from four to higher dimen- sions came from the seminal work of Kaluza and Klein [1,2] who first tried to unify gravity with electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Kaluza-Klein approach has been revived and considerably generalized after realizing that many interesting theories of particle interactions need spacetime dimensions more than four for their formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Dur- ing the last few decades considerable research activities in progress to understand the quantum properties of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The investigation seems to lead some people to believe that a consistent theory of quantum gravity cannot be obtained within the framework of point-field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For example, superstring theory is considered B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Paul E-mail: bcpaul@nbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='in Department of Physics, University of North Bengal, Siliguri, Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' : Darjeeling 734 013, West Bengal, India and IUCAA Centre for Astronomy Research and Development, North Bengal 2 Bikash Chandra Paul to be the promising candidate which may unify gravity with the other fundamen- tal forces in nature which requires ten dimensions for consistent formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The advent of string theory has opened up new and interesting possibilities in this con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The discovery that a supergravity theory coupled to Yang-Mills fields with a gauge group SO(32) or E8 ×E8 is anomaly-free in ten dimensions had inspired con- siderable activities in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Although the expected breakthrough has not yet come, worldwide hectic activities have served to focus on a number of issues which need further investigations in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The present ideas in dimensional reduction suggest that our cosmos may be a 3-brane evolving in a D-dimensional spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Cadeau and Woolgar [3] addressed this issue in the context of black holes which led to homogeneous but non- FRW-braneworld cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Classical General Relativity (GR) in more than the usual four dimensions is thus a sub- ject of increasing attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' A successful development of counting of the five dimensional black hole entropy [4] and the AdS/CFT correspondence relates the properties of a D-dimensional black hole with those of a quantum field theory in (D − 1) dimensions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' There has been a growing interest to investigate the physics of higher-dimensional black holes [6] which is markedly different, and much richer in structure compared to four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In connection with localized sources, higher dimensional generalization of the spherically symmetric Schwarzschild, Reisner-Nordstr¨om black holes, Kerr black holes can be found in the literature [7,8,9,10,11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The generalization of the rotating Kerr black hole [8,9,13,14] and black holes in compactified spacetime [7, 12] are also found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The linearized stability of the black holes [15], no hair theorems [16], black hole thermodynamics and Hawking radiation have also been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Mandelbrot [17] investigted the problem on the variability of dimensions and describe how a ball of thin thread is seen as an observer changes scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' An object which looks like a point object from a very large distance becomes a three-dimensional ball visible at a closer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Therefore at various scales the ball appears to change shape as an observer moves down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' While the embedding dimensions for the ball has not changed, the effective dimensions of the contents however also remains same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is possible that there are compact [18] or non- compact [19] dimensions present at a certain point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In this case the (3 + 1) metric is simply not true, although one obtains a valid description with general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We also probe the black hole solution in Rastall theory [20] which is prescribed by a modification of GR to accommodate the present accelerating universe [21,22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' non-singular) black holes have been initiated by Bardeen [24] and thereafter a number of black hole solutions in four dimensions have been obtained [25,26,27,28,29,30,31,32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In this case one can find metrics which are spher- ically symmetric, static, asymptotically flat, with regular centres, and for which the resulting Einstein tensor is physically reasonable, satisfying the weak energy condition and having components which are bounded and fall off appropriately at large distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Dymnikova [25] obtained nonsingular Schwarzsclid black hole solu- tion in vacuum and thereafter extended to obtain nonsingular cosmological black hole [26] solutions to include the de Sitter solution in the usual four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Formation and evaporation of non-singular black hole is also discussed [34] from an initial vacuum region accommodating Bardeen-like static region supported by finite density and pressures, subsequently its pressure vanishes rapidly at large ra- dius which however behaves as a cosmological constant at a small radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In 2019, Nonsingular Black Holes in Higher dimensions 3 Event Horizon Telescope group captured the first ever image of a supermassive black hole at the centre of the M87 galaxy which triggers the various possibili- ties for the state of the compact object and opened up new horizon in theoretical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The motivation of the paper is to obtain a nonsingular black hole solution in higher dimensions and investigate the different features of such black holes in GR and beyond GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For this we consider Higher dimensional Einstein gravity (GR) and Rastall gravity for a comparative study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The paper is organised as follows: In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 2, the Einstein field equation in a static higher dimensional metric is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 3, non-singular black holes are obtained in Rastall theory with extension of spacetime dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 4, we present analytical set up of the non-singular black hole solution to investigate the shadow of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The effective potential and the shadow behaviour of the black holes are analyzed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Finally we summarize in sec 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 2 Einstein Field Equation in Higher Dimensions We consider a higher dimensional gravitational action which is given by I = − 1 16πGD � √−g dDx R + Im (1) where R is the Ricci scalar, GD is the D-dimensional gravitational constant and Im represents the matter action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Einstein field equation is given by RAB − 1 2gABR = κ2TAB (2) where A, B = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='D −1 and T A B = (−ρ, Pr, P⊥, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=') the energy-momentum tensor, ρ the energy density, Pr the radial pressure, P⊥ transverse pressure, κ2 = 8πGD c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We consider D dimensional spacetime metric given by ds2 = −eνdt2 + eλ dr2 + r2dΩ2 D−2 (3) where ν and λ are functions of radial coordinate r and ΩD−2 is for unit sphere in SD−2 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The components of the Einstein equations and the metric given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (26) are given by T t t = (D − 2) 2 � e−λ � D − 3 r2 − λ′ r � − D − 3 r2 � , (4) T r r = (D − 2) 2 � e−λ � D − 3 r2 + ν′ r � − D − 3 r2 � , (5) T θ1 θ1 = −(D − 3)(D − 4) 2r2 + e−λ 2 × � ν′′ + ν′2 2 − λ′ν′ 2 + (D − 3)(ν′ − λ′) r + (D − 3)(D − 4) r2 � , (6) T θ2 θ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. = T θD−2 θD−2 = T θ1 θ1 (7) for simplicity we have taken κ2 = 8πGD c2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The radial null vector lA can be selected to have the components lt = eλ/2, lr = ±eν/2 and li = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The two 4 Bikash Chandra Paul radial null-null components of the Ricci tensor are equal, and given by RABlAlB = eλRtt + eνRrr = (D − 2) (eλ+ν)′ 2eλ+ν , which vanishes if and only if (λ + ν) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' A rescaling of the time coordinate can be set to make the sum of the terms equal to zero for black hole solution and we write λ + ν = 0 (8) Now substituting λ = − ln f(r) (9) we obtain the following components of energy momentum tensors in terms of f(r), which are given by T t t = T r r = D − 2 2 � f(r) � D − 3 r2 + f′ rf(r) � − D − 3 r2 � (10) T θ1 θ1 = f(r) 2 � f′′ f + 2(D − 3)f′ rf(r) + (D − 3)(D − 4) r2 � − (D − 3)(D − 4) 2r2 (11) T θ2 θ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. = T θD−2 θD−2 = T θ1 θ1 (12) where the prime denotes the derivative with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The source term satis- fying T t t = T r r , and T θ2 θ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. = T θD−2 θD−2 = T θ1 θ1 (13) and the equation of state, T A B ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='A = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Assume the density profile in higher dimensions T t t = −ρ as ρ = −T t t = ρ0 e − rD−1 rD−1 ∗ (14) where r∗ is a dimensional constant connected with a constant density ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The density ρ0 also permits a D dimensional de Sitter solution with its size given by r2 0 = (D − 1)(D − 2) 2Λ , (15) where Λ = ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Using the density profile given eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (14) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (10) we integrate and obtain the metric potential which yields f(r) = 1 − rD−3 g rD−3 + 2ρ0rD−1 ∗ (D − 1)(D − 2) 1 rD−3 e − rD−1 rD−1 ∗ (16) where rD−3 g = � 2ρ0 (D−1)(D−2) � rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The higher dimensional metric is now can be written as ds2 = − � 1 − Rs(r) rD−3 � dt2 + dr2 � 1 − Rs(r) rD−3 � + r2dΩ2 D−2 (17) where we denote Rs(r) = rD−3 g � 1 − exp � −rD−1 rD−1 ∗ �� (18) Nonsingular Black Holes in Higher dimensions 5 and rD−1 ∗ = r2 0 rD−3 g , (19) where r2 0 = (D−1)(D−2) 2ρ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' This is an exact spherically symmetric solution of the Einstein field equations in D-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For D = 4 the solution given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (17) reduces to the solution obtained by Dymnikova [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The other components of energy momentum tensor can be obtained using the Einstein’s field equations which are given by T θ2 θ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' = T θD−2 θD−2 = � D − 1 D − 2 � r r∗ �D−1 − 1 � ρ0e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (20) It is evident that in the usual 4 dimensions Dymnikova [25] black hole solutions recovered with anisotropic fluid distribution when r = r∗, which is true also in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The nonsingular black hole (NSBH) solutions are permitted with anisotropic fluid distributions in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The energy density and radial pressure follow the vacuum configuration but the tangential pressures do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The tangential pressure is non-zero which remains positive definite for r > r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' At the center the tangential pressure is negative indicating existence of exotic mat- ter (P⊥ < 0) at the center of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The nonsingular black hole solution obtained by Dymnikova can not be described in lower dimension D = 2 + 1, how- ever, we can extend the concept of NSBH in more than the usual four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The generalization of the black hole solution in higher dimensions accommodates a new class of NSBH solutions where the tangential pressure increases to a large extent inside the non-singular black hole with a different feature but away from the centre of the black hole it decreases exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The mass of a massive object in higher dimension is given by m(r) = AD−2 � r 0 r′D−2ρ(r′)dr′ (21) where AD−2 = 2π D−1 2 Γ ( D−1 2 ) which at r → ∞ is connected to the whole mass M con- nected with rD−3 g by the Schwarzschild relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The modulus difference between Rs(rg) and rD−3 g is rD−3 g e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The measure of the difference between the higher dimensional Schwarzschild mass m(r) ∼ rD−3 g in a singular black hole and Rs of a non-singular black hole is given by M − m(r) M = exp � −rD−1 rD−1 ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (22) Here m(r) becomes M at infinite distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is found that the mass difference decreases as the dimension in which black hole embedded increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The metric has two event horizons located at r+ = rg � 1 − O � exp � −r2g r2 0 ��� , r− = r0 � 1 − O � exp � −r0 rg ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (23) 6 Bikash Chandra Paul Here r+ is the external event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The metric evaluated at gtt(r+) = 0 de- scribes an object with the similar properties properties of a black hole by a dis- tant observer, it does not send light signals outside and could not interact with its surroundings by the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In four dimensions it is found that both r+ and r− are removable singularities of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The singularities can be eliminated by an appropriate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 3 Higher Dimensional Rastall gravity In this section we explore NSBH solution in the Rastall theory of gravity for D ≥ 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Rastall theory [20] is based on the modification of the Einstein field equation for a spacetime with Ricci scalar filled by an energy momentum source as follows: T AB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' A = λRB (24) where λ is the Rastall parameter which is a measure for deviation from the stan- dard GR conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Consequently the Rastall field equation can be written as GAB + κ2λgABR = κ2TAB (25) where κ2 is the Rastall gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The above field equation reduces to that of GR in the limit λ → 0 and κ2 = 8πG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' However, for a vanishing trace of the energy-momentum tensor, for example the electrovacuum solution can be obtained when λ = 1 4 or R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is important to note that the former possibility is not physically acceptable as the trace of the energy momentum tensor vanishes T = 0 for any scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Consequently the matter configuration where the energy- momentum tensor has null trace, the relativistic solution obtained in Rastall theory is same as that one obtains in the general theory of relativity (GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' This feature of Rastall theory which is a modified GR led us to look for black holes solutions in a background of matter/energy with non-vanishing trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It may be pointed out here that the Rastall gravity is widely used to accommodate acceptable explanation for the current acceleration of the universe which has no solution in GR and for this it is interesting to explore NSBH in Rastall theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We consider the metric for black hole solution in higher dimensions D ≥ 4: ds2 = −f(r)dt2 + dr2 f(r) + r2dΩ2 D−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (26) Using the metric, we obtain the non-vanishing components of the Rastall tensor HAB = GAB + λgABR and κ2 = 1, Ht t = D − 2 2r2 � rf′ − (D − 3) + (D − 3)f� + λR, (27) Hr r = D − 2 2r2 � rf′ − (D − 3) + (D − 3)f� + λR, (28) Hθi θi = r2f′′ + (D − 3)(2rf′ + (D − 4)(f − 1) 2r2 + λR (29) where i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=', (D − 2), and the Ricci scalar in D dimensions is given by R = − 1 r2 � r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1) � (30) Nonsingular Black Holes in Higher dimensions 7 in the above we denote ()′ to represent derivative with respect to the radial co- ordinate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We solve the field equation to obtain higher dimensional non-singular black holes in Rastall theory and for this Ht t = T t t and Hrr = T rr yield Pr = D − 2 2r2 � rf′ − (D − 3) + (D − 3)f� − λ r2 � r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1) � , (31) and also we consider Hθ1 θ1 = T θ1 θ1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' and HθD−2 θD−2 = T θD−2 θD−2 which yield P⊥ = 1 2r2 � r2f′′ + 2(D − 3)rf′ + (D − 3)(D − 4)(f − 1) � − λ r2 � r2f′′ + 2(D − 2)rf′ + (D − 2)(D − 3)(f − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (32) In this case we explore the non-singular Black hole obtained in higher dimensional Rastall gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' the general solution of the metric is f(r) = 1 − rD−3 g rD−3 + 2ρ0rD−1 ∗ (D − 1)(D − 2) 1 rD−3 e − rD−1 rD−1 ∗ (33) The energy density and radial pressure are ρ = \uf8eb \uf8ed D − 2 − 2λD + 2(D − 1)λ rD−1 rD−1 ∗ D − 2 \uf8f6 \uf8f8 ρ0e − rD−1 rD−1 ∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (34) Pr = − \uf8eb \uf8ed D − 2 − 2λD + 2(D − 1)λ rD−1 rD−1 ∗ D − 2 \uf8f6 \uf8f8 ρ0e − rD−1 rD−1 ∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (35) the tangential pressure is given by P⊥ = � (1 − 2λ)D − 1 D − 2 rD−1 rD−1 ∗ − D − 2 − 2λD D − 2 � ρ0e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (36) The energy density and the transverse pressure in Rastall gravity framework ob- tained in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (34) and (36) reduces to the eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (14) and (20) in GR for λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The modification introduced in GR by Rastall admits nonsingular Dymnikova [25] black hole (NSBH) with normal matter while the radial pressure corresponds to vacuum equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' At the center of the NSBH the energy density is ρ = (D − 2 − 2λD)ρ0, which increases as the.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' number of spacetime dimension in- creases for a given range of Rastall parameter λ < D−2 2D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is evident that for a given dimension, NSBH admits greater mass for lower values of λ and the lower limiting value for λ < D−2 2D and |λ| > D−2 2D (for negative λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The corresponding tangential pressure at the center P⊥ = (2Dλ + 2 − D)ρ0 is finite but negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In D = 4 dimensions, at the center of the black hole, ρ(r = 0) = 2(1 − 4λ)ρ0 and tangential pressure P⊥ = −(1−4λ)ρ0 which indicates existence of exotic matter at the centre in GR (as λ = 0) as well as in Rastall theory for λ > − 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus NSBH can be realized with both central radial pressure and tangential pressure negative and equal but an anisotropy in pressure develops away from the center in Rastall 8 Bikash Chandra Paul gravity, normal matter exists when r > � D−2−2λD (D−1)(1−2λ �1/(D−1) r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The tangential pressure indicates black hole surrounded by exotic matter in Rastall gravity [35] for the range 1 4 < λ < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. For r → ∞, the energy density and pressure vanishes asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' When λ = D−2 2D , we get the following : ρ = ρ0 � D − 1 D rD−1 rD−1 ∗ � e − rD−1 rD−1 ∗ , (37) Pr = −ρ = ρ0 � D − 1 D rD−1 rD−1 ∗ � e − rD−1 rD−1 ∗ , (38) the tangential pressure is given by P⊥ = 2ρ0 � D − 1 D(D − 2) rD−1 rD−1 ∗ � e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (39) one obtains NSBH with ρ > 0, ρ + Pr = 0 and P⊥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For D = 4 dimensions, λ = 1 4 and the NSBH can be realized in Rastall gravity with normal matter which however is not permitted in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Also we note that at the centre of the NSBH the tangential pressure vanishes, admitting a perfect vacuum NSBH in the usual four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The result obtained in this case is also applicable in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' This is a new result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 4 Analytical set up The modified Schwarzschild metric for a non-singular black hole is given by ds2 = −f(r) dt2 + f(r)−1 dr2 + r2dΩ2 D−2 (40) where f(r) = 1 − � rg r �D−3 + � rg r �D−3 exp � − � r r∗ �D−1� , making use of the as- sumption κ2 = 8π made earlier, we write rg = � 16πM (D−2)AD−2 � 1 D−3 and the area of D dimensional sphere AD−2 = 2π D−1 2 Γ( D−1 2 ), where M represents the mass of the non-singular Black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The metric function gtt = f(r), whose sign determines gravitational trapping [34], we plot to draw a sketch to study the existence of black hole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The metric potential f(r) is plotted with r in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (1) for D = 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (2) for D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is evident that both the extreme black hole and non-extreme black holes can be obtained for a given set of values of rg and r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We note that extreme black hole exists for rg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 and r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57 in D = 4 and rg = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 and r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57 in D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In the first case no black hole exist for rg < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 and the later case for r0 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The photon radii are tabulated in Table-I for D = 4 and Table-II for D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is found that for D = 4, it increases with decrease of ρ0 ∼ 1/r2 0 for a given mass but for a given ρ0, photon radius is found to increase with mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In D = 10 dimensions as ρ0 is decreases the photon radius decreases then increases and decreases once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In Fig (3) dimensional varia- tion of the photon radius for M = 1 is plotted for non-singular black holes with Nonsingular Black Holes in Higher dimensions 9 1 2 3 4 5 6 r �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 f�r� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 1 Radial variation of f(r) for rg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8 (Red), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 (Black), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 (Blue) for r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57 in D = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 1 2 3 4 5 6 r �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 f�r� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 2 Radial variation of the metric function f(r) in D = 10 for r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='1 (Blue), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='57 (Black) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 (Red) with rg = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The photon radius is maximum at D = 4 and then decreases sharply as the dimensions is increases and remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Lagrangian is given by L = 1 2gAB ˙xA ˙xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (41) 2 4 6 8 10 �1 0 1 2 3 r V Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 3 Radial variation of the potential for J= 5 (Blue), 6 (Green), 8 (Dashed), 10 (Red) in D = 4 for non-singular BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 10 Bikash Chandra Paul where ˙() = d dτ and τ is the affine parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Expanding eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (41) we get 2L = −f(r)˙t2 + 1 f(r) ˙r2 + (r2 ˙θ1 2 + sin2θ1 ˙θ2 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='.) (42) To obtain trajectory of light path, we set θi = π 2 where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=', D − 3 and θD−2 is a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The momenta are given by Pt = ∂L ∂ ˙t = −f(r)˙t, Pr = ∂L ∂ ˙r = 1 f(r) ˙r, Pθ1 = ∂L ∂ ˙θ1 = r2 ˙θ1, Pθ2 = ∂L ∂ ˙θ2 = r2sin2θ1 ˙θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='. (43) Now as defined above, θi = π 2 , and at the equatorial plane θ1 = π 2 , ∂L ∂ ˙t = constant (44) and we determine the energy (E) and angular momentum (J) at r → ∞ as f(r)˙t = E, PθD−2 = r2 ˙ θD−2 = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (45) The Hamilton Jacobi equation is the most general method to find the geodesic equation of motion around black hole or a compact object, we adopt the technique to obtain the photon orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In higher dimensions we get ∂S ∂τ = H = −1 2gAB ∂S ∂xA ∂S ∂xB (46) where gAB is the inverse of the metric and S is the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Jacobian is given by S = 1 2m2τ − E + JθD−2 + Sr(r) + D−3 � i=1 Sθi(θi) (47) where Sr(r) and Sθi(θi) are functions of r and θi respectively and m is the mass of the test particle, it is zero for photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The Hamilton-Jacobi eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (46) can be written as r4 � 1 − Rs rD−3 �2 � ∂S ∂τ � = E2r4 − r2 � 1 − Rs rD−3 � (K + J2) (48) D−3 � i=1 1 Πi−1 n=1sin2θn �∂Sθi ∂θi �2 = K − ΠD−3 i=1 J2cot2θi (49) where K is the Carter constant [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Using the above eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (43) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (46) we get the following ˙t = E f(r), ˙θD−2 = J r2ΠD−3 i=1 sin2θi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' r2 ˙r = ± √ R, r2 D−3 � i=1 Πi−1 n=1sin2θn ˙θi = ± � Θi (50) Nonsingular Black Holes in Higher dimensions 11 in the above ”+” and ”-” sign corresponds to motion of photon in outgoing and incoming radial direction and over dot represents derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='t to the affine parameterτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For the null curves the eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (49) can be expressed as R(r) = E2r4 − r2f(r)(K2 + J2), (51) Θi(θi) = K − ΠD−3 i=1 J2cot2θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (52) The characteristics of photon near the black hole can be defined by two impact parameters, which are functions of the constants E, J and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For general orbit we define the impact parameters ξ = J E and η = K E2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The boundary of the shadow of a black hole can be estimated from the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The radial null geodesic from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (48) and (50) is given by � dr dτ �2 + Veff = 0, (53) where Veff is the effective potential, for radial motion we obtain Veff = f(r) r2 (K + J2) − E2 = 1 r2 � 1 − �rg r �D−3 � 1 − e−( r r∗ ) D−1�� (K + J2) − E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (54) The effective potential is identical to the classical equation describing the motion of a massless particle in a 1-dimensional potential V (r) provided its energy is 1 2E2 (of course the true energy should be E), but we use this form to obtain an expression for potential in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We plot radial variation of V (r) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (4) in a four dimensional universe for singular as well as non-singular black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' As the angular velocity increases the photons heading towards the black hole are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 2, it is found that there is no difference of the behaviour of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (4) we plot radial variation of V (r) for different angular momentum, for a non-singular BH, it is evident that as the angular momentum increases the photon can approach near to the BH unbounded The photon orbits are circular and unstable for a maximum value of the effec- tive potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The unstable circular orbit determines the boundary of the apparent shape and can be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The maximal value of the effective potential corre- sponds to the circular orbits and the unstable photons satisfies Veff ��� r=rp = dVeff dr ��� r=rp = 0, R(r) = dR(r) dr ��� r=rc = 0 (55) The impact parameters are now related as Using eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (54) and (55), we get f(rp) r2p (K + J2) − E2 = 0 rpf′(rp) − 2f(rp) r3p (K + J2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (56) In four dimensions the potential V (r) is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (4) with different angular momentum (J) for rg = 2 and E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The particles are bounded for a radius r < rmin and unbounded for the range rmin < r < rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The range of values 12 Bikash Chandra Paul rp in rp in rp in r0 M = 2 M = 5 M = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8757 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0192 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='1532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0221 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='1851 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='3389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='3079 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5053 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='6958 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5880 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8142 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0383 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='1150 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='3702 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='990 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='9921 Table 1 The variation of the photon radius (rp) in D = 4 with r0 = � (D−1)(D−2) 4ρ0 and the mass of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' rp in rp in r0 M = 5 M = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='9369 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0035 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0002 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='8426 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='0105 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='5942 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='9106 Table 2 The variation of the photon radius (rp) in D = 10 with r0 = � (D−1)(D−2) 4ρ0 and the mass of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 3 4 5 6 7 8 9 10 D 2 4 6 8 10 Rp [t] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 4 Dimensional variation of the photon radius for M = 1 for a non-singular BH can be determined from the sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is found that rmin decreases as angular momentum (J) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We note that the potentials for Schwarzschild black hole (singular) and that for non-singular black holes overlaps for a set of similar values of D, J and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We draw the shadow contour of non-singular black hole in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (8) for a given value of ρ0 (say, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='16 unit) in all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is shown that as the spacetime dimensions increases the radius of the shadow decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Nonsingular Black Holes in Higher dimensions 13 �10 �5 0 5 10 �10 �5 0 5 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 5 Contour plot for an object having M = 2M⊙ with ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='16 unit for D = 4 (Red), D = 5 (Dashed) and D = 6 (Green), D = 8 (Thick) �40 �20 0 20 40 �40 �20 0 20 40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 6 Contour plot for an object for ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='04 unit in D = 4 with M = 2M⊙ (Black), M = 4M⊙ (Green), M = 6M⊙ (Red), M = 10M⊙ (Blue) 5 Effective potential and shadow behaviour The effective potential of the Schwarzschild-Tangherlini black holes exhibits a max- imum for the photon sphere radius rp corresponding to the real and the positive solution of the constraint obtained from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (56), rpf′(rp) − 2f(rp) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (57) Defining impact parameters η and ξ that are functions of the energy E, angular momentum J and the Carter constant K as ξ = J E , η = K E2 (58) we get from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (55) corresponding to Veff E2 = 0 and R E2 = 0, the following η + ξ2 = r2p f(rp), η + ξ2 = 4r2p rf′(rp) + 2f(rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (59) 14 Bikash Chandra Paul Now we obtain η + ξ2 = 5r2p rpf′(rp) + 3f(rp), (60) where the right hand side corresponds to r2 p f(rp), the observer’s frame the shadow can be described properly making use of the celestial coordinates α and β as introduced earlier [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Following the definition introduced by Subrahmanyan as follows α = lim rp→∞ � rpP θD−2 P t � , βi = lim rp→∞ � rpP θi P t � , (61) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='(D − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For an observer on the equatorial plane, these equations reduced to η + ξ2 = α2 + β2 = r2p f(rp) (62) the radius of the shadow is Rbhs = rp √ f(rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The form of f(r) is complex and therefore we study numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The photon radius depends on the dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The photon radius is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 4, it is evident that as the mass of the black hole increases the radius decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is maximum in D = 4 but decreases sharply as the dimension increases but almost constant with the increase in dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The fig (9) shows that as the mass increases the radius of the shadow also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 6 Discussion We obtain non-singular black hole (NSBH) solutions in the higher dimensional Einstein’s general theory of gravity (GR) and found that the methods in GR can be adopted also in Rastall gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Considering a specific exponential form of the energy density we obtain NSBH which reduces to the Dymnikova NSBH solution [25] obtained in the usual four dimensions (D = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In 2+1 dimensions no black hole solution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' However, a non-rotating NSBH solutions obtained by Dymnikova in four dimensional GR can be accommodated in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' We obtained NSBH in a vacuum described by T t t + T rr = 0 with p⊥ < 0 (where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=', D − 2) near the center indicating requirement of exotic matter which however extends up to certain height thereafter p⊥ > 0 for r > � D−2 D−1 � 1 D−1 r∗ in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' But in the Rastall gravity p⊥ > 0 for r > � 2−D+2λD (D−1)(2λ−1) � 1 D−1 r∗ when λ ̸= D−2 2D and thereafter at a large distance it vanishes because the pressure decreases rapidly i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=', exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Both in GR and modified gravity it indicates existence of exotic matter near the center of the NSBH but in the later case the Rastall parameter plays an important role in determining the distance from the centre where the normal matter exists in the tangential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In the usual four dimensions at the center of the NSBH in the modified theory we get the following estimations ρ(r = 0) = 2(1 − 4λ)ρ0 and tangential pressure P⊥ = −2(1 − 4λ) which are determined by the Rastall parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' It is evident that exotic matter at the center of the black hole requires both in GR (as λ = 0) as well as in Rastall theory with the lower limiting value of the Rastall parameter λ < 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The tangential pressure is negative it indicates Nonsingular Black Holes in Higher dimensions 15 NSBH surrounded by exotic matter in Rastall gravity, existence of BH with exotic matter also reported in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus NSBH is realized with both the central radial pressure and tangential pressure negative and equal initially but an anisotropy in pressure develops away from the center in Rastall gravity with normal matter thereafter when r > � D−2+2λD (1−2λ)(D−2) � 1 D−1 r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' However, we note a new and interesting result in Rastall theory that permits a NSBH with normal matter in the usual four and in higher dimensions when λ = D−2 2D , which however is not permitted in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' In It is also noted that away from the center at a large distance, the tangential pressure remains positive definite at a maximum radial distance which is P⊥ = (1−2λ) D−1 D−2 rD−1 rD−1 ∗ ρ0e − rD−1 rD−1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus one gets a physically realistic NSBH for 1 4 < λ < 1 2 in D = 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' For r → ∞, both the energy density and pressure vanishes asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus the Rastall gravity has rich structure which unearth the structure of non-singular black hole even with normal matter for a restricted domain of the Rastall parameter depending on the embedding spacetime dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Thus, we see that for λ ̸= 0 the Rastall theory plays an important role leading to distinct solutions relative to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The sketch of the potentials permissible in the theory are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (1) and (2), which show that both extreme and non-extreme black holes exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The contour plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (5) for NSBH shows that the circular shadow radius decreases as the spacetime dimension is increased for a given mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The circular shadow radius in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' (6) show that the radii increases with the mass of the compact objects for a given dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' The rotating NSBH will be taken up elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Acknowledgment The author would like to thank IUCAA , Pune and IUCAA Centre for Astronomy Research and Development (ICARD), NBU for extending research facilities and North Bengal University for a research grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' BCP acknowl- edge the suggestions and constructive criticism of the anonymous Referee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' References 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Hayward, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 96 , 031103 (2006) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Morais Graca, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Lobo, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' C 78, 101 (2018) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Carter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' 174, 1559 (1968) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Vazquez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} +page_content=' Esteban, Nuovo Cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfgfgN/content/2301.00358v1.pdf'} 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