Spherical Solutions due to the Exterior Geometry of a Charged Weyl Black Hole

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1 Spheical Solutions due to the Exteio Geomety of a Chaged Weyl Black Hole Fain Payandeh 1, Mohsen Fathi Novembe 7, 018 axiv:10.415v [g-qc] 10 Oct 01 1 Depatment of Physics, Payame Noo Univesity, PO BOX , Tehan, Ian Depatment of Physics, Islamic Azad Univesity, Cental Tehan Banch, Tehan, Ian Abstact Fistly we deive peculia spheical Weyl solutions, using a geneal spheically symmetic metic due to a massive chaged object with definite mass and adius. Aftewads, we pesent new analytical solutions fo elevant cosmological tems, which appea in the metics. Connecting the metics to a new geometic definition of a chaged Black Hole, we numeically investigate the effective potentials of the total dynamical system, consideing massive and massless test paticles, moving on such Black Holes. 1 Intoduction Among genealized theoies of gavity, Weyl gavity is emakable, since it leads to consideable desciptions of cosmological paametes elevant to Dak Enegy poblem. It is known that Weyl theoy, contibutes in the R theoies of gavity. This means that this theoy is govened by field equations, combined of second ode diffeentiations of the Ricci scala [1]: W αβ = ρ α R βρ + ρ β R αρ R αβ g αβ ρ λ R ρλ R ρβ Rα ρ + 1 g αβr ρλ R ρλ 1 α β R g αβ R RR αβ + 1 g αβr ) = 1 4π T αβ. 1) In the above equation, W αβ notates the components of the Weyl equations, and T αβ is the enegy momentum tenso, coesponding to the souce. The vacuum equations, geneally have been solved and a spheically symmetic metic has been deived []. In this pape, we ae not concening about the geneal solution. Instead, we conside a peculia one, that coesponds to a massive chaged spheically symmetic souce. Such souce has been consideed fo the Reissne-Nodstöm solutions RN) of Weyl gavity [], but hee we use the backgound field method and linea appoximation to deive othe RN-like solutions, also descibing a chaged Black Hole. The method is like the one, which has done in [4], analytically investigating the constants egading the Dak Enegy poblem. Howeve hee, constant values of chage and mass will be consideed, to eaange the metic to a new fom to descibe a Black Hole. Finally, the effective potentials will be numeically plotted to illustate the geometic behavio of the dynamical system, affected by such Black Hole. A spheical solution of Weyl field equations due to a chaged massive spheical souce Let us conside the following geneal metic: ds = 1 GM 1 )dt f) + 1 GM 1 ) 1d f) + dω, ) f payandeh@pnu.ac.i Coesponding autho: mohsen.fathi@gmail.com 1

2 in which f) is an abitay -dependent function, ought to be obtained. The Ricci tenso components, due to the spacetime, defined by metic ) will be: R 00 = + Gm + 1 f) f + f) f 1 1 f f + f) + 1 f ) 6 1 f R 11 = + f) 1 f + 1 f 1 f + f) f ) 6 + Gm + 1 f) 1, R = 1 f ) 1 f + f) + 1 f f, ) f +4 f + f) 1 f ) f +4 f + f) 1 f f + f ) + 1 f ) 6, f + f ) + 1 f ) R = R sin θ). ) Also the Ricci scala will be deived as: R = 1 f + 4 f + f. 4) Employing these values in the components of Weyl equations in 1), one obtains: As we expect, W 00 = f 7 f) 4 Gmf + 88 Gmf 6 Gmf 4 4 G m f 60 Gmf + 60 G m f 1 f + 4 f + 6 f 6 5 f f f 5 10 Gmf f) 1 4 Gmf f 1 f) Gmf + 96 Gm f) f 48 4 Gm f) f 4 Gmf f 5 f) f f + 96 f Gm 144 G m f f Gm 4 f) 4 f f + 1 f 4 f f f) f) f 4 5 f f 1 4 f f + 4 Gmf + 4 f) f 8 f f + 48 f f) 144 Gm f) + 6 f 5 f 4 f) f + f) 5 f 4G m f ), W 11 = Gm+f)) 4 f f) + 4 f + 1 f 4 f 4 f 4 f + 8 f f) + f 4 f 7 f Gm +4 f f 4 f) f 6 f Gm f Gmf ), W = f 4 f + 8 f f) + 7 Gmf 4 f) f + 4 f) f + 4 f f) + 4 f f + f 4 f + 1 f 4 f f ) f 1 f 7 f Gm + 1 f Gm + 4 f = W sin θ). All the components in 5), have a vacuum solution like: Substituting 6) in ) yields: ds = c + 1 c 1 ) dt + W α α = 0. f) = c 1 c 6GM. 6) c + 1 c 1 ) 1 d + dω. 7) Now to evaluate the included constants in 6), we shall use the backgound field method in the weak field limit. The zeo-zeo component of the metic ) can be ewitten as: g 00 = η 00 + h 00, 5)

3 fo small fluctuations h 00 = GM + 1 f). The -component of the Poisson s equation implies that: h 00 d d + d ) d )h 00 = 8π T 00 + E 00, 8) in which T 00 is the stess-enegy tenso due to the mass of the souce. And hee, associated to a chaged, spheically symmetic massive souce, the tenso is the volume density: T 00 = ρ 0 = m ) π 0 In 9), m 0 is the mass of the spheical body and 0 is its known adius. In elation 8), E 00 is the stess-enegy tenso, associated to the chage amount of the massive object. Hee, since the souce is assumed to be static, we take the vecto potential A µ = Φ), 0, 0, 0), whee Φ) is the electic potential at point in the exteio geomety of the total chage, distibuted in a cetain volume Φ) = q0 ). We have [5]: E 00 = 1 q0 ) 1 + 8π Φ) q ) 0 4π = 1 q0 8π 4. 10) Now consideing the expession 6), and the values in 9), 10) in 8), and solving fo c 1 o c yields: consideing 1) we get: c 1 = m 0 1 ) c. 11) c = 9 m 0 ) 0 c 1. 1) g 1) 00 = 1 m 0 1 ) 0 c 1 ). 1) The geneal spheically symmetic solution to Weyl gavity, has been deived to be []: g W 00 = 1 β βγ) βγ + γ k ), 14) in which, as it has been mentioned by the authos, the paametes γ and k have been consideed to be elevant to the Dak Enegy theoy. The tem c 1 in 1), theefoe can be coesponded to the tem k in 14). To estimate a value fo c 1, we use 11). We conside the chaacteistics of the obsevable univese, m 0 = m obs = g, and 0 = obs = cm, which ae espectively, the estimated mass and adius of the obsevable univese. We also take = q obs = 0, because it is assumed that a finite univese must have a zeo net chage [6]. Taking c = 0 in 11) one obtains: c 1 = m obs obs g/cm, which is compaable to the estimated value fo the cosmological constant see [4] and Ref.s theein). Now let us conside 11) to obtain: g ) 00 = 1 m 0 1 ) c ). 15) Looking at 15), leads us to coespond the tem 9 c to γ in 14). Once moe we use the chaacteistics of the obsevable univese in 1). Taking c 1 = 0 and = obs we obtain: c = 9M obs 0.76 g/cm 10 8 cm 1. And this is exactly the value fo γ which has been pesented in [], elated to the Dak Enegy theoy. In compaison with the Reissne-Nodstöm-de Sitte metic g00 RN d = 1 m 0 + ) 1 Λ ), 16)

4 the metic 1) shows impotant diffeences. Specially when we notice its attactive invese squae potential due to the chaged body, instead of the epulsive one in 16). Both of these metics, have the vacuum enegy tem, elated to the acceleated expansion of the univese. Howeve, the tem in 16) would be exactly the cosmological constant, and the one in 1) will have the same value in some limits. On the othe hand, the metic in 15) appeas to contain anothe tem, which has been not included by common spheically symmetic solutions to Einstein field equations. This tem, also by imposing some limits, leads to the same values fo the Dak Enegy tem, in the geneal spheically symmetic solutions to Weyl gavity. We will continue ou discussion, investigating the effective potentials fo the geometical Black Holes, defined by metics 1) and 15), without the Dak Enegy tems. Effective potentials fo a massive chaged object aound a chaged Weyl Black Hole Consideing a test paticle, having the chaacteistics, m fo mass and q fo chage, which is moving on a Weyl Black Hole, one can deive the effective potential, using the Hamilton-Jacobi equation of wave cests [7, 8]: g µν P µ + qa µ )P ν + qa ν ) + m = 0. 17) P µ is the momentum 4-vecto 1 P µ = g µσp σ dx σ = g µσ dλ. 18) whee λ is the geodesics affine paamete. The metic components g µν ae deived fom the exteio geomety of the souce, namely metic 1) and 15) without the cosmological tems. Also the vecto potential A µ fo ou static chaged souce has been peviously defined to be: whee Φ) = q0 as: A µ = Φ), 0, 0, 0), 19) is the scala electical potential, outside the Black Hole. One can define the two conseved quantities which is the test-paticle s enegy, and E = P 0, 0) L = P φ L 0). 1) which is its angula momentum. We choose θ = π, fo which the paticle s motion is confined to the equatoial otations. Theefoe P θ = dθ dλ = 0. Consideing 1) in 17) yields: o Equation ) can be ewitten as [8]: d dλ ) = [E q qq0 E ) 1 m q + 1 m ) 1 d dλ ) + L + m = 0, d dλ ) = E q ) 1 m )m + L ). ) 1 m fom which we define the effective potential as: V 1) eff = q + q 0 )m + L ) ) ][E q + 1 m 0 1 q 0 0 )m + L ) ) ], 1 m )m + L ), ) 1 Hee we use the notation in Ref. [7], in which in 5. the 4-momentum has been defined like Eq. 18). 4

5 which is the effective potential due to metic 1). We took the positive pat to be assued that a positive potential is available. Using the same pocedue fo 15) yields: V ) eff = q + 1 m )m + L ). 4) Figue 1 shows illustations fo these effective potentials fo diffeent values of angula momentums. As chaged Black Holes, eithe of the Weyl Black Holes must have two event hoizons fo g 1) 00 = 0 and g) 00 = 0. Fo a Black Hole with a constant adius 0 and mass m 0, one obtains: 1) ± = 1 6 ) ± = 1 6 6m 0 0 ± 04 6 m 0 0 ) 0 m 0, 5) 6m 0 0 ± m 0 0 ) 0 m 0, 6) espectively fo 1) and 15). Note that, fo a Reissne-Nodstöm Black Hole we have: In the next section, we estict ou discussion to massless paticles. RN) ± = m 0 ± m 0. 7) 4 Effective potentials fo massless paticles tavelling on a Weyl Black Hole Fo massless paticles, namely Photon, Neutino o Gaviton, the chaacteistics of the test paticle changes due to this fact that the concepts of mass and angula momentum, will beak down. We shall intoduce the atio [7, 9]: Peviously we defined: b = lim m 0 g φφ P φ = L dφ dλ = L, theefoe, accoding to 8), fom Eq. ) fo a massless neutal paticle we have: We take L. 8) E m ) 1 ) 1 d dφ ) 1 m = b. 9) 1 = 1 m 0 C 0 1 ), fo the Weyl Black Hole, which is defined by 1), and also we take = 1 m 0 C ), fo the one, defined by 15). Note that, fo b C, the paticle can get to any point. Hee, C is consideed to be the effective potential fo the massless paticles, moving aound a Weyl Black Hole. This means that the maximum, o the citical value fo b we call b cit ) is the minimum value fo C. One can deive this citical value fo eithe of Weyl Black Holes. Fo fist type Black Holes, this citical value appeas at =. We have: b 1) cit = [C 1] min = 1 q0 m 0 0 ). 0) 5

6 a) b) Figue 1: a) The effective potentials fo a test-paticle, having diffeent angula momentums, moving on a chaged Weyl Black Hole defined by metic 1) and b) by metic 15). c) The effective potentials fo a test-paticle moving on a Reissne- Nodstöm RN) Black Hole. c) 6

7 Also fo the second type Weyl Black Holes, the citical point would be at = q0, and: b ) cit = [C ] min = 1 1 ). 1) q 0 m 0 0 In Figue, the values of effective potentials fo a massless paticle fo both types of Weyl Black Holes, has been plotted. When b fo eithe of the massless paticles on eithe of Black Holes, exceeds the b cit, then the paticle a) b) Figue : a) The effective potentials fo a massless test-paticle, moving on a fist type and b) second type Weyl Black Hole. t would be the tuning point. Fo enegies highe than the b cit), the massless paticle is captued by the Black Hole, while fo enegies lowe than this, we have a eflection fom the potential well, having the minimum distance t. appoaches the Black Hole having the minimum distance t, and then goes to infinity. On the maximum of the effective potential, whee b = b cit, the paticle will have unstable cicula obits. Fo b < b cit, the test paticle which is coming fom infinity, falls into the Black Hole hoizon. Moe details can be found on text books, fo example see [10]. 5 Conclusion While Einstein theoy of elativity, illustates a finite classical univese, with a positive acceleation in time like coodinates, some othe gavitational theoies, ae pesenting solutions fo some unexplained featues. In this aticle one of the most well known ones, namely the Weyl theoy of gavity has been consideed. Though this theoy, we pesented some analytical expessions fo the coefficients, elevant to Dak Enegy theoy, and deived thei numeic values, which have been in good ageement with thei measued values. Also, we specialized the spheically symmetic metics, explaining the exteio geomety of chaged spheical massive souce, into two shapes of metic potentials. These time like metics, having coesponding singulaities, descibed two types of chaged Black Holes, in analogy to the Reissne-Nodstöm metic. Consideing them, we calculated the effective potentials fo massive and massless test paticles and compaed them though numeical illustations. 7

8 Acknowledgments This wok was suppoted unde a eseach gant by Payame Noo Univesity. Refeences [1] D. Kazanas and P.D. Mannheim, Geneal stuctue of the gavitational equations of motion in confomal Weyl gavity, Astophysical Jounal Supplement Seies 76: ). [] P.D. Mannheim and D. Kazanas, Exact vacuum solution to confomal Weyl gavity and galactic otation cuves, Astophysical Jounal 4: ). [] P.D. Mannheim and D. Kazanas, Solutions to the Reissne-Nodstöm, Ke and Ke- Newman poblems in fouth ode confomal Weyl gavity, Phys. Rev. D 44, ). [4] M.R. Tanhayi, M. fathi, M.V. Takook, Obsevable Quantities in Weyl Gavity, Mod. Phys. Lett. A 6, 011). [5] J.D. Jackson 1998), Classical Electodynamics d ed.), Wiley. ISBN X. [6] L.D. Landau, E.M. Lifshitz, The Classical Theoy of Fields. Vol. 4th ed.), Buttewoth-Heinemann 1975). ISBN [7] C.W. Misne, K.S. Thone and J.A. Wheele, Gavitation, Feeman 197). [8] Maco Olivaes, Joel Saaveda, J.R. Villanueva and Calos Leiva, Motion of chaged paticles on the Reissne- Nodstöm Anti)-de Sitte black holes, axiv: v. [9] Benad F. Schutz, A Fist Couse in Geneal Relativity, Second Edition, Cambidge Univesity Pess 009). [10] S. Chandasekha, The Mathematical Theoy of Black Holes, Oxfod Univesity Pess, New Yok 198). 8