The Fingerprints of Black Holes - Shadows and their Degeneracies - PowerPoint PPT Presentation

The Fingerprints of Black Holes - Shadows and their Degeneracies Claudio Paganini † joint work with: Marius Oancea † , Marc Mars ‡ † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ ısica Fundamental y Matem´ aticas, Universidad de Salamanca, Salamanca, Spain Frankfurt a.M. ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 1 / 51

( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 2 / 51

( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 3 / 51

Event Horizon Telescope ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 4 / 51

( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 5 / 51

My Goal I want to convince you that in principal an observer can, by measuring the black holes shadow, determine the angular momentum, the charge of the black hole under observation, the observer’s radial position, the angle of elevation above the equatorial plane. Furthermore, his/her relative velocity compared to a standard observer can also be measured. ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 6 / 51

Outline Background 1 Kerr-Newman-Taub-NUT metric Celestial Sphere 2 Shadow Parametrization Degeneracies 3 Definition Continuous Degeneracies Available M¨ obius Transformations Discrete Degeneracies Conclusion & Outlook 4 ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 7 / 51

Background Kerr-Newman-Taub-NUT metric Kerr-Newman-Taub-NUT metric In Boyer-Lindquist coordinates, ( t , r , θ, φ ), the metric is given by � 1 � + 1 d s 2 =Σ ∆ d r 2 + d θ 2 (Σ + a χ ) 2 sin 2 θ − ∆ χ 2 � d φ 2 + � Σ (1) 2 d t d φ − 1 ∆ χ − a (Σ + a χ ) sin 2 θ ∆ − a 2 sin 2 θ d t 2 , � � � � Σ Σ where Σ = r 2 + ( l + a cos θ ) 2 , χ = a sin 2 θ − 2 l (cos θ + C ) , ∆ = r 2 − 2 Mr + a 2 − l 2 + Q 2 . stationary axially symmetric type D spacetimes √ M 2 − a 2 + l 2 − Q 2 , where r + is the event horizon at r + = M + largest root of ∆ = 0 Electro-vac solutions ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 8 / 51

Background Kerr-Newman-Taub-NUT metric Parameters The mass M The charge Q The spin parameter a The NUT parameter l which can be interpreted as a gravitomagnetic charge Manko and Ruiz parameter C Contains Schwarzschild ( a = Q = l = 0), Kerr ( Q = l = 0), Reissner-Nordstr¨ om ( a = l = 0), Kerr-Newman ( l = 0), and Taub-NUT ( a = Q = 0) ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 9 / 51

Background Kerr-Newman-Taub-NUT metric Constants of Motion From metric γ µ ˙ γ ν m = g µν ˙ (2) from Killing vectorfield γ µ , γ µ E = − ( ∂ t ) µ ˙ L z = ( ∂ φ ) µ ˙ (3) from Killing Tensor γ µ ˙ σ µν = Σ(( e 1 ) µ ( e 1 ) ν + ( e 2 ) µ ( e 2 ) ν ) − ( l + a cos θ ) 2 g µν , γ ν . K := σ µν ˙ (4) ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 10 / 51

Background Kerr-Newman-Taub-NUT metric Geodesic Equation The geodesic equation as a system of first order ODEs t = χ ( L z − E χ ) + (Σ + a χ )((Σ + a χ ) E − aL z ) ˙ , (5a) Σ sin 2 θ Σ∆ φ = L z − E χ Σ sin 2 θ + a ((Σ + a χ ) E − aL z ) ˙ , (5b) Σ∆ Σ 2 ˙ r 2 = R ( r , E , L z , K ) := ((Σ + a χ ) E − aL z ) 2 − ∆ K , (5c) θ 2 = Θ( θ, E , L z , Q ) := K − ( χ E − L z ) 2 Σ 2 ˙ . (5d) sin 2 θ System homogeneous in E thus for E � = 0 we have: R ( r , E , L z , Q ) = E 2 R ( r , 1 , L E , K E ) , (6) Θ( θ, E , L z , Q ) = E 2 Θ( r , 1 , L E , K E ) , (7) where L E = L z / E and K E = K / E 2 . ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 11 / 51

Background Kerr-Newman-Taub-NUT metric Trapping The trapped null geodesics are those which stay at a fixed value of r and hence satisfy ˙ r = ¨ r = 0, which corresponds to: R ( r , L E , K E ) = d dr R ( r , L E , K E ) = 0 . (8) These equations can be solved for the constants of motion in terms of the constant value r = r trapp as: K E = 16 r 2 ∆ � � aL E = (Σ + a χ ) − 4 r ∆ � � , , (9) � � (∆ ′ ) 2 ∆ ′ � � r = r trapp r = r trapp ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 12 / 51

Background Kerr-Newman-Taub-NUT metric Area of Trapping, a = 0 . 902 ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 13 / 51

Celestial Sphere Celestial Sphere At any point p in M choose an orthonormal basis ( e 0 , e 1 , e 2 , e 3 ) for the tangent space, with e 0 time-like and future directed. The tangent vector to any past pointing null geodesic at p can be written as: γ ( k | p ) | p = α ( − e 0 + k 1 e 1 + k 2 e 2 + k 3 e 3 ) , ˙ (10) γ, e 0 ) > 0 and k = ( k 1 , k 2 , k 3 ) satisfies | k | 2 = 1, hence where α = g (˙ k ∈ S 2 . Definition Let γ ( k | p ) denote a null geodesic through p for which the tangent vector at p is given by equation (10). ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 14 / 51

Celestial Sphere Sets on the Celestial Sphere We can then define the following sets on S 2 at every point p : Definition The future infalling set: Ω H + ( p ) := { k ∈ S 2 | γ ( k | p ) ∩ H + � = ∅} . The future escaping set: Ω I + ( p ) := { k ∈ S 2 | γ ( k | p ) ∩ I + � = ∅} . The future trapped set: T + ( p ) := { k ∈ S 2 | γ ( k | p ) ∩ ( H + ∪ I + ) = ∅} . The past infalling set: Ω H − ( p ) := { k ∈ S 2 | γ ( k | p ) ∩ H − � = ∅} . The past escaping set: Ω I − ( p ) := { k ∈ S 2 | γ ( k | p ) ∩ I − � = ∅} . The past trapped set: T − ( p ) := { k ∈ S 2 | γ ( k | p ) ∩ ( H − ∪ I − ) = ∅} ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 15 / 51

Celestial Sphere (a) (b) ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 16 / 51

Celestial Sphere The Shadow Definition We refer to the set Ω H − ( p ) ∪ T − ( p ) as the shadow of the black hole. ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 17 / 51

Celestial Sphere ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 18 / 51

Celestial Sphere ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 19 / 51

Celestial Sphere ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 20 / 51

Celestial Sphere ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 21 / 51

Celestial Sphere ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 22 / 51

Celestial Sphere ( † Albert Einstein Institute, Potsdam, Germany ‡ Instituto de F´ C.F. Paganini The Fingerprints of Black Holes - Shadows and their Degeneracies Frankfurt a.M. ısica Fundamental y 23 / 51