Appendix. A.1 Painlevé-Gullstrand Coordinates for General Spherically Symmetric Metrics

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1 Aendix I have been imressed with the urgency of doing. Knowing is not enough; we must aly. Being willing is not enough; we must do. Leonardo da Vinci A. Painlevé-Gullstrand oordinates for General Sherically Symmetric Metrics Beginning from the metric given by Eq. (.94) and following the notations of Ref. [], we search for a new time coordinate (Painlevé-Gullstrand time). The transformation t!.t; R/ @R dr and @R dr ; which transforms the line element (.94) into ds e. M=R/.@=@t/ d e "e M ddr # dr R d./ : M=R (A.) We now imose that the new time coordinate is such that g, which imlies that Sringer International Publishing Switzerland 05 V. Faraoni, osmological and Black Hole Aarent Horizons, Lecture Notes in Physics 907, OI 0.007/

2 94 e M=R : (A.) Then, the metric comonent g 0 in the new coordinates is r g 0 e R (A.3) and the line element assumes the form (.95). A. Kodama Vector in FLRW Sace Here we comute the comonents of the Kodama vector in FLRW sace in seudo- Painlevé-Gullstrand and in comoving coordinates. A.. Pseudo-Painlevé-Gullstrand oordinates In these coordinates the -metric h ab of Eq. (.7) and its inverse are given by.h ab / 0 0 h HR.H R kr =a / kr =a HR kr =a HR kr =a kr =a A ; HR A H R kr =a by decomosing the metric (3.5). The volume form on the normal -sace is (A.4) (A.5) ab jhj.dt/ a ^.dr/ b kr =a.ı a0ı b ı a ı b0 / ; (A.6) while ab g ac g bd.ı c0ı d ı c ı d0 / kr =a h a0 h b h a h b0 kr =a :

3 A. Kodama Vector in FLRW Sace 95 The Kodama vector is h a0 K a ab h b h a h b0 r b R kr =a ı b h a0 h h a h 0 kr =a and, therefore, K 0 kr =a H R kr =a H R kr =a kr =a kr =a ; K h 0 h h h 0 kr =a 0: To conclude, we have K kr =a ;0;0;0 (seudo-painlevé-gullstrand coordinates): (A.7) A.. omoving oordinates In comoving coordinates the FLRW line element is ds dt a.t/ kr dr R d./ h ab dx a dx b R d./ ; (A.8) where R a.t/r is the areal radius. The volume form on the -sace.t; r/ has comonents ˇ jhj.dt/ ^.dr/ˇ a ı 0 ıˇ ı ıˇ0 kr while ˇ g gˇı ı a g gˇı ı 0 ı ı ı ı ı0 kr a g 0 gˇ g gˇ0 : kr

4 96 Aendix The comonents of the Kodama vector in comoving coordinates are K ˇrˇR ˇ Parıˇ0 aıˇ a Parh 0 h 0 ah 0 h Parh h 00 ah h 0 kr a ah 0 h Parh h 00 : kr Now, K 0 K a kr ah00 h a kr a kr Parh h 00 Paar kr kr a kr a ; kr a Hr kr ; and the comonents of the Kodama vector are K kr ; Hr kr ;0;0 (comoving coordinates): (A.9) The norm squared of K a is K a K a g 00.K 0 / g.k /. kr / a kr H r. kr / Pa r kr P r =rah I (A.0) it vanishes at the aarent horizon r AH Pa k : (A.) Reference. Nielsen, A.B., Visser, M.: Production and decay of evolving horizons. lass. Quantum Grav. 3, 4637 (006)

5 Index Symbols f.r/ gravity, 05, 5, 8, 83, 87, 88 A advanced time, 7, 8 anti-de Sitter, 08 anti-traed surface, 30 aarent horizon,, 4, 37 40, 47, 49, 50, 53, 59, 63, 64, 66, 69 7, 78, 79, 8 86, 9 96, 98 0, 07, 08, 7, 3 7, 33, 34, 36 38, 4, 45, 49 53, 55 58, 67, 7 76, 80 8, 84, 86 89, 96 aarent horizon tube, 56 areal radius, 5 7, 4, 47, 49, 53, 60, 6, 64, 73, 78, 84, 87, 95, 07, 0,, 8,, 7, 33, 35, 4, 4, 47 50, 55, 70, 74, 78, 80, 8, 83, 95 areal volume, 7, 95 B binary system,, 35 Boyer-Linquist coordinates, 6, 8 Brans-icke arameter, 68, 69, 7, 74, 75, 80 Brans-icke theory, 38, 68, 69, 88 auchy horizon,, 4, 39, 4 conformal anomaly, 43 conformal factor, 39 4, 43, 45, 46, 57 conformal Killing equation, 39 conformal Killing horizon, 37 conformal Killing vector, 39 conformal time, 40, 53, 55 conformal transformation, 39, 4, 43 45, 57, 76, 77 de Sitter sace,,, 6, 63, 77, 78, 8, 8, 87 9, 98, 0 deviation vector, 6, 7 dominant energy condition, 9 dragging of inertial frames, 7 dynamical horizon,,, 38, 40, 8 E Eddington-Finkelstein coordinates, 8,, 46 effective action, 3, 05 energy suly vector, 96 ergoshere, 7 event horizon,,, 5, 6, 8, 9,, 7, 33, 35 39, 43, 44, 49, 6, 70, 73 79, 8, 85, 88 9, 94, 99 0, 08,, 4 6, 4, 56, 68, 85 extremal horizon,, 4, 7, 50, 9, 5,, 4 6 F Fisher sacetime, 46 fluid-gravity duality, 08 future aarent horizon, 37 future inner traing horizon, 39 future null infinity,, 35, 36 future outer traing horizon, 39 Sringer International Publishing Switzerland 05 V. Faraoni, osmological and Black Hole Aarent Horizons, Lecture Notes in Physics 907, OI 0.007/

6 98 Index G Gauss-Bonnet gravity, 67, 68 generalized Raychaudhuri equation, 3 geodesic deviation equation, 6 geodesic equation, 5, 6, 8, 40 Gibbons-Hawking entroy, 9 H Hawking radiation,,, 34, 39, 45, 59, 93, 06, 6, 56, 67, 86 Hawking temerature, 43, 93, 4, 43, 45, 89 Hawking-Hayward quasi-local energy,, 47, 50, 68 Ho rava-lifschitz gravity, 38, 86 Horndeski theory, 67, 86, 88 Hubble horizon, 78, 7 Hubble arameter, 9, 60, 6, 63, 07,, 3, 6, 7, 4, 55 hyersherical coordinates, 6, 64, 7, 74, 76, 84 I inflation,, 0, 59, 73, 78, 00, 57 isolated horizon, 40, 46 isotroic radius, 6, 09, 0, 7,, 8 30, 33, 40, 69, 73, 83 J Jebsen-Birkhoff theorem, 68 M marginal surface, 30 marginally outer traed tube, 30 marginally traed surface, 35, 44 marginally traed tube, 40 Misner-Shar-Hernandez mass,, 46, 47, 50, 53, 63, 65, 66, 70, 7, 78, 9, 95, 98, 0, 08, 0,, 7, 34, 4, 45, 67, 68, 88 N naked singularity, 4, 7, 08, 3, 7, 5, 36, 46, 50, 5, 56 58, 7, 75, 76, 8, 86 Nolan gauge, 63 Nolan interior solution, 7 9 normal surface, 9 null curvature condition, 38 null dominant energy condition, 9 null energy condition, 9, 30, 38, 50, 0 P Painlevé-Gullstrand coordinates, 0, 46, 48, 49, 6 65, 67, 68, 70, 87, 90, 93 article creation, 43, 44 article horizon, 70, 7 76, 79, 85, 88, 00 ast inner traing horizon, 83 hantom energy, 9, 06, 3, 87 hantom field, 54 hantom fluid, 8, 00, 6, 7 hantom universe, 77, 6, 7, 3 ositive curvature condition, 8 K Kerr-Schild coordinates,, 3 Kerr-Schild metric,, 3 Kerr-Schild transformation, 57 Killing equation, 36, 43, 44, 89, 90 Killing horizon, 36, 37, 40, 43, 44, 5, 89, 90, 6, 7, 39 Killing vector, 0, 7, 34, 36, 37, 40 44, 46, 5 53, 89 9, 6 Kodama vector, 37, 4, 66, 67, 9, 93, 96, 67, 68, Kretschmann scalar, 7 Kruskal-Szekeres coordinates, 7, 8,, 4, 5, 6 L Lemaître-Tolman-Bondi model, 06, 38, 56, 58 Q quantum gravity, 40, 6, 38 R Raychaudhuri equation, 8, 9 retarded time, 7, 8 Ricci tensor, 4 Riemann tensor, 4 Rindler horizon,, 3, 30, 3 34, 5, 5, 79 Rindler observer, 3, 5 S S-curve, 5, 56, 58, 75, 76, 78, 8, 86, 88 slowly evolving horizon,, 4 sacetime singularity, 5, 6, 6, 0,, 4,, 3, 4, 47, 56, 7, 88

7 Index 99 static limit, 7 strong energy condition, 9, 77 suergravity, 6, 67 suernovae, 3 Synge aroach, 7 U uniform acceleration, 30, 3, 34, 5 Unruh effect, 33, 34 Unruh temerature, 33, 5 untraed surface, 30, 49 T thermodynamics of sacetime, 3, 5, 00 timelike membrane, 40 Tolman-Oenheimer-Volkoff equation, 9 tortoise coordinate, 7, 5 traed surface,, 5, 30 traing horizon,,, 37 39, 44, 45, 50, 70, 83, 88, 95, 05, 56, 58, 68, 88, 89 traing horizon tube, 88 V Vaidya sacetime,, 35, 38, 56 W weak energy condition, 9, 8, 95, 0, 6 white hole, 8, 9,, 37, 39, 74, 86 wormhole, 9, 37, 48, 5