On the shadows of black holes and of other compact objects

Transcription

1 On the shadows of black holes and of other compact objects Volker Perlick ( ZARM, Univ. Bremen, Germany) 1. Schwarzschild spacetime mass m photon sphere at r = 3m shadow ( escape cones ): J. Synge, Kerr spacetime mass m, spin a photon region of spherical geodesics shadow: J. Bardeen, Plebański-Demiański spacetime mass m, spin a, electric charge q e, magnetic charge q m, NUT parameter l, cosmological constant Λ photon region of spherical geodesics shadow: A. Grenzebach, 2013

2 1. Schwarzschild spacetime ( g = 1 r ) S c 2 dt 2 + r ( 1 r S r ) 1 dr 2 + r 2( dϑ 2 + sin 2 ϑdϕ 2), r S = 2m Unstable photon sphere at r = 3m = 1.5r s

3 δ r O Angular radius δ of the shadow : sin 2 δ = 27r2 S (r O r S ) 4r 3 O Escape cones : J. Synge, Mon. Not. R. Astron. Soc. 131, 463 (1966) r O = 1.05r S r O = 1.3r S r O = 3r S /2 r O = 2.5r S r O = 6r S S. Virbhadra and G. Ellis, Phys. Rev. D, 65, (2002) A naked singularity is called weakly naked if surrounded by an unstable photon sphere, and strongly naked if not.

4 2. Kerr spacetime What happens to the photon sphere at r = 3m and with the shadow if the spin a is switched on? Shadow of a Kerr black hole: J. Bardeen in C. DeWitt and B. DeWitt (eds.): Black Holes Gordon and Breach (1973), p. 215 Shadow of a Kerr(-Newman) naked singularity: A. devries, Class. Quantum Grav. 17, 123 (2000) C. Bambi and K. Freese, Phys. Rev. D 79, (2009) If a is switched on, the photon sphere at r = 3m turns into a photon region filled with spherical null geodesics.

5 Kerr metric in Boyer Lindquist coordinates (r, ϑ, ϕ, t): g = (r,ϑ) 2 ( dr 2 (r) + dϑ2 ) + sin2 ϑ ( ) 2 adt (r 2 + a 2 (r) ( ) 2 )dϕ dt asin 2 ϑdϕ (r,ϑ) 2 (r,ϑ) 2 (r,ϑ) 2 = r 2 + a 2 cos 2 ϑ, (r) = r 2 2mr + a 2. Lightlike geodesics: (r,ϑ) 2 ṫ = a ( L Ea sin 2 ϑ ) + (r2 + a 2 ) ( (r 2 + a 2 )E al ) (r), (r,ϑ) 2 ϕ = L Ea sin2 ϑ sin 2 ϑ + (r2 + a 2 )ae a 2 L, (r) (r,ϑ) 4 ϑ2 = K (L Ea sin2 ϑ) 2 sin 2 ϑ =: Θ(ϑ), (r,ϑ) 4 ṙ 2 = K (r) + ( (r 2 + a 2 )E al ) 2 =: R(r).

6 Lightlike geodesics: (r,ϑ) 2 ṫ = a ( L Ea sin 2 ϑ ) + (r2 + a 2 ) ( (r 2 + a 2 )E al ) (r), (r,ϑ) 2 ϕ = L Ea sin2 ϑ sin 2 ϑ + (r2 + a 2 )ae a 2 L, (r) (r,ϑ) 4 ϑ2 = K (L Ea sin2 ϑ) 2 sin 2 =: Θ(ϑ), ϑ (r,ϑ) 4 ṙ 2 = K (r) + ( (r 2 + a 2 )E al ) 2 =: R(r). Spherical lightlike geodesics exist in the region where R(r) = 0, R (r) = 0, Θ(ϑ) 0. a L E = r2 + a 2 2r (r) r m, K E 2 = 4r2 (r) (r m) 2. ( 2r (r) (r m) (r,ϑ) 2 ) 2 4a 2 r 2 (r) sin 2 ϑ (unstable if R (r) 0)

7 Pictures of spherical lightlike geodesics: E. Teo, Gen. Rel. Grav. 35, 1909 (2003) Pictures of the photon region: W. Hasse and VP, J. Math. Phys. 47, (2006) VP: Living Rev. Relativity 7, (2004), 9 For pictures of the photon region, use e r as the radial coordinate e r r red solid: singularity red dashed: throat

8 a = 0.15m blue: unstable spherical lightlike geodesics green: stable spherical lightlike geodesics orange: causality violation black: region between horizons red solid: singularity red dashed: throat

9 a = 0.75 m blue: unstable spherical lightlike geodesics green: stable spherical lightlike geodesics orange: causality violation black: region between horizons red solid: singularity red dashed: throat

10 a = m blue: unstable spherical lightlike geodesics green: stable spherical lightlike geodesics orange: causality violation black: region between horizons red solid: singularity red dashed: throat

11 a = 1.15m blue: unstable spherical lightlike geodesics green: stable spherical lightlike geodesics orange: causality violation black: region between horizons red solid: singularity red dashed: throat

12 a = 1.25m blue: unstable spherical lightlike geodesics green: stable spherical lightlike geodesics orange: causality violation black: region between horizons red solid: singularity red dashed: throat

13 The shadow is determined by light rays that approach an unstable spherical lightlike geodesic. Choose observer at r O and ϑ O. Introduce celestial coordinates (θ,φ) at the observer. Relation between celestial coordinates and constants of motion: sin 2 θ = g2 ϕt g ϕϕg tt ( L g ϕt E + g ϕϕ sin 2 φ = ) 2 L 2 E + 2 asin 2 ϑ L2 E 2 + g ϕϕ g ϕϕ r 2 + a 2 cos 2 ϑ asin 2 ϑ L2 E 2 K E 2 K E 2 ( asin 2 ϑ L E sin 2 ϑ ( asin 2 ϑ L E sin 2 ϑ ) 2 ro,ϑ O ) 2 ro,ϑ O Inserting L/E and K/E 2 as functions of r gives the boundary of the shadow parametrised with r.

14 a = 0.4m r O = 6m ϑ O = π/2

15 a = m r O = 6m ϑ O = π/2

16 a = 1.1m r O = 6m ϑ O = π/2

17 a = 2m r O = 8m ϑ O = π/2

18 3. Plebański-Demiański spacetime g = Σ ( dr 2 r + dϑ2 ϑ ) ( (Σ ) 2 ϑ + + aχ sin 2 ϑ r χ 2) dϕ 2 Σ + ( r χ a ( Σ + aχ ) ϑ sin 2 ϑ ) 2dtdϕ Σ ( r a 2 ϑ sin 2 ϑ ) dt2 Σ where Σ = r 2 + ( l + acosϑ ) 2 χ = asin 2 ϑ 2l ( cosϑ + C ) ϑ = 1 + Λ { 4 3 } a2 alcosϑ + 3 cos2 ϑ r = r 2 2mr + a 2 l 2 + q e 2 + q m 2 Λ{ (a 2 l 2) l 2 + ( a l2 ) } r 2 + r4 3 m: mass q e : electric charge l: NUT parameter a: spin q m : magnetic charge C : Manko-Ruiz parameter Λ: cosmological constant

19 Introduction of NUT metric: E. Newman, T. Unti, L. Tamburino, J. Math. Phys. 4, 915 (1963) Interpretation of NUT parameter l as gravitomagnetic charge of a spinning massless string: W. B. Bonnor, Proc. Camb. Philos. Soc. 66, 145 (1969) Introduction of the parameter C: V. Manko, E. Ruiz, Class. Quantum Grav. 22, 3555 (2005)

20 a = 0.99m, l = 0.75m blue: unstable spherical lightlike geodesics green: stable spherical lightlike geodesics orange: causality violation black: region between horizons red solid: singularity red dashed: throat

21 Dependence of the shadow on C

22 Colour: Increasing a

23 Colour: increasing a Left to right: increasing l Top to bottom: Increasing q 2 e + q2 m

24 H. Falcke, F. Melia, E. Agol, Astrophys J. 528, L13 (2000)