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ProceedingsofRAGtime?/?,?–?/?–? September,????/????, Opava,CzechRepublic 1 S.Hled´ıkandZ.Stuchl´ık,editors,SilesianUniversityinOpava,????,pp.1–6 Stress-energy tensor of a radiating sphere inclosing black hole Maciek Wielgus1,2,a and Marek Abramowicz3,1,4,b 1Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka18, 00-716 Warszawa, Poland 5 1 2Instituteof Micromechanics and Photonics, ul. A. Boboli 8, 02-525 Warszawa, Poland 0 3Instituteof Physics, Faculty of Philosophy and Science, Silesian University in Opava, 2 Bezru˘covo nám. 13, 746 01 Opava,Czech Republic 4PhysicsDepartment, GothenburgUniversity,412-96 Göteborg, Sweden n [email protected] a [email protected] J 7 ] c ABSTRACT q Weconsider a uniformly luminousradiating sphereand a static black hole - locatedinthecenterofthatsphere. Wegiveanalyticformulasforradiation r g stress-energy tensor components in such a configuration, for the observer [ located at an arbitrary distance from thestatic black hole horizon. 1 Keywords: black hole – radiation in general relativity – stress-energy v tensor 0 4 5 1 INTRODUCTION 1 0 In Abramowicz et al. (1990) an analytic formula for the stress-energy tensor of . 1 uniformly radiating static relativistic star was found, for ZAMO (zero angular mo- 0 mentumobserver)atthearbitrarydistancefromthestarsurface. Theresultisvery 5 usefulforinvestigationsofthetestparticlemotioninacurvedspacetime,underthe 1 : radiation four-force influence and several groups pursue this branch of research in v recent years, see, e.g., Bini and Geralico (2010), Sok Oh et al. (2011), Stahl et al. i X (2012), Wielgus et al. (2012), Stahl et al. (2013). Here we consider the extended r problem, i.e., the observer may be located inside the uniformly radiating luminous a sphere of radius R. In that way we complete the solution of Abramowicz et al. (1990), allowing for any ratio of the radiating sphere’s and observer’s radii in the Schwarzschildspacetime(itisassumedthattheradiatingspheredoesnotcontribute to the spacetime curvature). But the problemis not only interesting for the reason of the mathematical completeness of this solution. Such a solution is particularly well suited to describe the interactions between black hole and the Cosmic Back- ground Radiation, which corresponds to the uniformly luminous sphere located at infinity. The extended solution is also physically relevant when we consider the NO-ISBN-SET-X (cid:13)c ???? – SU in Opava. All rights reserved. 2 M. Wielgus and M. Abramowicz radiation from numerous, randomly spread sources, that can be approximated by a homogeneousluminosity. That canbe the case ofa black hole locatedin the center ofasphericalgalaxy,forinstance, orthe radiativelyefficientsphericalaccretion on the black hole. 2 CASE OF THE LUMINOUS INNER SPHERE Let us first review the solution for an observer located above the static luminous sphere in Schwarzschild spacetime, given by Abramowicz et al. (1990). We follow a slightly different approach, giving some more general formulas, in order to make a very smooth extension to the case of r < R (observer located at radius r inside the luminous sphere of radius R) in the next section. Assuming homogeneous, isotropic radiation flux in the emitter’s rest frame we conclude that the luminosity observed by any static observer located at a given radial location is constant over the observed sphere surface. Thus, the problem of calculating the radiation stress- energy tensor components in the static ZAMO frame reduces to the calculation of the constant specific intensity moments. Hence, radial dependence of the following two quantities need to be established (1) specific intensity I(r) (value to be integrated over the observer’s local sky), (2) sphere viewing angle α (boundary for the specific intensity integration). 0 Specificintensityasseenbytheobserverlocatedatradialdistancer correspondsto the surface intensity I(R) of the radiation source, gravitationallyredshifted by the presenceofthecentralmass. Thequantitiesthatareconservedalongtheparticular light ray traveling through the curved spacetime are the photon energy E =pαη =0=const., (1) α for the photon four-momentum pα and Killing vector ηα = δα, and the redshifted t intensity I(r) I = =const., (2) 0 (pαv )4 α forthestationaryobserver’sfour-velocityvα =ηα(g ) 1/2,seeMisneretal.(1973) tt − | | section 4.22 for some more details. Hence, we find 2 1 2M/R I(r)=I(R) − , (3) (cid:18)1 2M/r(cid:19) − so clearly I(r) < I(R) for r > R. Viewing angle α corresponds to the largest 0 possiblevalueofangleαintheFig. 1,whichoccursforthelargestpossibleemission angle δ for which the photon can be observed. For R 3M δ = π/2, i.e., all 0 0 ≥ emitted photons are able to escape from the star vicinity. This is not true for 2M <R<3M, where sin2δ =27(1 2M/R)/R2. (4) 0 − Stress-energy tensor of a radiating sphere inclosing black hole 3 δ u(r) EM u(φ) EM α u(φ) r OB R u(r) OB Figure 1. Photon emitted from thestar surface R with theemission angle δ. This result will be explained a little further. The relevant angles can be defined using the photon four-velocity in the local orthonormal frame u(α), i.e., u(ϕ) g (uϕ)2 1/2 u(ϕ) g (uϕ)2 1/2 ϕϕ ϕϕ tanα= = ; tanδ = = , (5) (cid:20)u(r)(cid:21) (cid:20) g (ur)2 (cid:21) (cid:20)u(r)(cid:21) (cid:20) g (ur)2 (cid:21) OB rr OB EM rr EM lower subscripts OB and EM denoting the location of the photon emission and observation,respectively. Using the normalizationuαu =0 itis easyto show that α the following equation always holds r2sin2γ u ϕ = =ℓ=const., (6) 1 2M/r −u t − where tanγ =u(ϕ)/u(r), γ =α and γ =δ, from which we find that OB EM R 2 1 2M/r sin2α = − sin2δ (7) 0 0 (cid:18)r(cid:19) (cid:18)1 2M/R(cid:19) − and since α (r) must decrease with radius, the only solution is 0 1/2 arcsin R 1−2M/r for 3M R r  (cid:20)r (cid:16)1−2M/R(cid:17) (cid:21) ≤ ≤ α0(r)=arcsin 3√3M(1−r2M/r)1/2 for R<3M <r (8) π arhcsin 3√3M(1−2M/ri)1/2 for R<r <3M  − h r i 4 M. Wielgus and M. Abramowicz Finally,havingcalculatedthe I(r) andα (r) distributions,the stress-energytensor 0 ZAMO components are found by the integration over the observer’s local sky T(α)(β)(r)=I(r) n(α)n(β)dΩ ; n(α) =p(α)/p(t) (9) Z to give T(t)(t) =2πI(r)(1 cosα ) , (10) 0 − T(t)(r) =πI(r)sin2α , (11) 0 2 T(r)(r) = πI(r)(1 cos3α ) , (12) 0 3 − 1 T(θ)(θ) =T(ϕ)(ϕ) = πI(r)(2 3cosα +cos3α ) . (13) 0 0 3 − All other components are simply equal to zero. 3 LUMINOUS OUTER SPHERE The case of the observer located inside the luminous sphere is similar to certain extent,sinceinthestaticspacetimephotonsmaytravelalongthesamenullgeodesic trajectory in both directions. Hence, the relation (3) holds all the same, only difference being that I(r)>I(R) for R>r, i.e., radiationis now blueshifted in the static observer’s frame. Equation (3) also ensures that the observer’s local sky is uniformly bright with exception of the part occluded by the black hole. Hence, the remaining part is to calculate the angular diameter of the black hole as a function of radius. For the analogy with the previous case, we denote this quantity with 2α ,seeFig. 2. Notethatitfollowsfromthe Eq. (6)thatphotonsalways crossthe 0 horizon r =2M with angle γ =0, i.e., perpendicularly to the horizon surface. Let usnowinvestigatetherelation(6)inmoredetails. Onemaynoticethatforaphoton trajectory to extend from r 2M to the black hole horizon it is necessary that 0 ≫ we are able to define a meaningful photon radialfour-velocity component for every r >r >2M, i.e., if 0 g uru >0 = u uϕ+u ut <0 = ℓ2 ϕϕ 27. (14) r ϕ t ⇒ ⇒ ≤− g ≤ tt 27isavalueofaglobalmaximumof g /g thatoccursforr =3M. Thismeans ϕϕ tt − that only photons with ℓ<3√3 fall into the black hole and putting the maximum value of ℓ into Eq. (6), we find Eq. (4) (remember that the outgoing trajectory in thecaseofinnerluminousspherecorrespondstotheingoingtrajectoryinthe outer Stress-energy tensor of a radiating sphere inclosing black hole 5 luminous sphere case). Considering that α must decrease monotonously with r, 0 the only solution for α (r) that satisfies Eq. (6) is 0 arcsin 3√3M(1−2M/r)1/2 for r 3M α0(r)=π arhcsin 3√3Mr (1−2M/ri)1/2 for r <≥3M (15) − h r i  which is the same as the Eq. (8) in the case of inner radiating sphere radius R < 3M. Note that the formula for α(r) does not depend on luminous sphere radius R. Finally, after the local sky integration, we find T(t)(t) =2πI(r)(1+cosα ) , (16) 0 T(t)(r) = πI(r)sin2α , (17) 0 − 2 T(r)(r) = πI(r)(1+cos3α ) , (18) 0 3 1 T(θ)(θ) =T(φ)(φ) = πI(r)(2+3cosα cos3α ) . (19) 0 0 3 − This system is quite similar to the result of Abramowicz et al. (1990), yet the flux is of a different sign, and the angle is substituted α π α . It is interesting to 0 0 → − observe,thatα π asr 2M,sotheareaofintegration(brightskyregion)goes 0 → → to zero as the observer approaches the horizon. On the other hand, I(r) given by δ u(φ) OB u(r) OB u(φ) α EM u(r) R 2M EM r Figure 2. Photon emitted from theouter spheresurface R with theemission angle α. 6 M. Wielgus and M. Abramowicz Eq. (3) divergesin sucha limit. So does the result ofintegration,the stress-energy tensor components, vanish in the limit of the horizon, diverge or have some finite limit? The answer to this question and its implications are discussed in details by Wielgus et al. (2014). 4 CONCLUSIONS We presented the extension of the classic analytic calculation of a static luminous star radiation stress-energy tensor to the case of a luminous sphere observed from the inside. We found out that because of the symmetries involved, such a problem has a very similar solution. Analysis of the Cosmic Background Radiation field properties close to the black hole horizon is one example of application of the presented formulas. ACKNOWLEDGEMENTS This work was supported by the Czech Grant CZ.1.07/2.3.00/20.0071 (Synergy, Opava) as well as the Polish NCN grant UMO-2011/01/B/ST9/05439. We thank George F. R. Ellis and Frederic Vincent for many illuminating discussions about the subject of radiation treatment in general relativity. REFERENCES Abramowicz, M. A., Ellis, G. F. R. and Lanza, A. (1990), Relativistic effects in superlu- minal jets and neutron star winds, Astrophysical Journal, 361, pp.470–482. Bini, D. and Geralico, A. (2010), Spinning bodies and the Poynting-Robertson effect in the Schwarzschild spacetime, Classical and Quantum Gravity, 27(18), 185014, arXiv: 1107.2793. Misner, C. W., Thorne, K. S. and Wheeler, J. A. (1973), Gravitation, Freeman, San Francisco. SokOh,J.,Kim,H.andMokLee,H.(2011),FinitesizeeffectsonthePoynting-Robertson effect: A fully general relativistic treatment, New Astronomy, 16, pp. 183–186, arXiv: 1011.3104. Stahl, A., Klu´zniak, W., Wielgus, M. and Abramowicz, M. (2013), Escape, capture, and levitation of matter in Eddington outbursts, Astronomy & Astrophysics, 555, A114, arXiv: 1306.6556. Stahl, A., Wielgus, M., Abramowicz, M., Klu´zniak, W. and Yu, W. (2012), Eddington capture sphere around luminous stars, Astronomy & Astrophysics, 546, A54, arXiv: 1208.2231. Wielgus, M., Ellis, G. F. R., Vincent, F. H. and Abramowicz, M. A. (2014), Cosmic background radiation in the vicinity of a Schwarzschild black hole: No classic firewall, Physical Review D, 90(12), 124024, arXiv: 1406.6551. Wielgus, M., Stahl, A., Abramowicz, M. and Klu´zniak, W. (2012), Oscillations of the Eddington capture sphere, Astronomy & Astrophysics, 545, A123, arXiv: 1208.2939.