Is there a magnification paradox in gravitational lensing?

Transcription

1 Is thee a magnification paadox in gavitational ing? Olaf Wucknitz wucknitz@asto.uni-bonn.de Astophysics semina/colloquium, Potsdam, 6 Novembe 7 Is thee a magnification paadox in gavitational ing? gavitational es fomalism equation distotion, magnification, amplification deflection potential, Femat potential magnification theoem magnification paadox fom planes to sphees summay O. Wucknitz 7 Gavitational es What ae they? Light deflection Naive Newtonian calculation: = dz dl = dl c c Φ Φ = GM R Soldne 8 (Newton) = G c M Einstein 95 (geneal elativity) = 4 G c M O. Wucknitz 7 O. Wucknitz 7 3 Gavitational es Galley Cluste ing O. Wucknitz 7 4 O. Wucknitz 7 5 Lens equation Distotion/magnification souce ~ obseve deflection angle tue souce M ed appaent s D s image position souce position s all angles small, tangential plane = D s equation/mapping s = () equation fist deivative s = () d s = d d = M d ( magnification / mapping matix M() = ) (aea) magnification = amplification µ = ± det M O. Wucknitz 7 6 O. Wucknitz 7 7

2 Magnification amplification Magnification amplification suface bightness obseve Ω souce Ω s obseve s view solid angles measue appaent size magnification matix M = D d D s x exchange souce and obseve: D d D sd, D s D so, D do D s souce ~ magnification = Ω Ω s M = ( + z d)( + z s ) D d ( + z d )( + z s ) D s x = M _ Ω s obseve _ Ω souce souce s view flux distibuted ove solid angle amplification = Ω Ω s magnification = amplification suface bightness conseved = x D d, D d obseve D sd = + z s + z d, D so = ( + z s )D s, D do = ( + z d )D d O. Wucknitz 7 8 O. Wucknitz 7 9 Potential Femat potential, time-delay suface potential ψ Poisson equation nomalized suface mass density () = ψ() ψ() =: κ() = σ() σ = Σ Σ c examples point mass ψ() = m ln constant suface mass density ψ() = σ singula isothemal sphee ψ() = 4π Ds σ v c light is delayed by ψ souce obseve light-tavel time fo vitual ay ( s fixed) t = D dd s c φ() Femat-potential φ() = ( s) ψ() O. Wucknitz 7 O. Wucknitz 7 Femat s pinciple light tavel time is stationay equation = φ() [ ( s ) = = s ψ() eal images ae positions of ] ψ() s = () minima, (e.g. unpetubed image) maxima, o saddle-points of φ O. Wucknitz 7 Magnification theoem [ Schneide (984) ] Hessian of Femat-potential is invese magnification matix µ = = φ = κ γ x γ y γ y κ + γ x diagonalise: otate shea, γ = γ x + γ y µ = κ γ κ + γ minimum: both eigenvalues positive Poisson: convegence κ = σ sum: ( κ) > κ < κ < µ = ( κ) γ < µ O. Wucknitz 7 3 Example: point-mass I ve seen this befoe... 4 µ + µ s [ Wambsganss (998), Liv. Rev. Rel., ] [ Einstein (936) ] [ Einstein notebooks 9 9 ] O. Wucknitz 7 4 O. Wucknitz 7 5

3 An appaent paadox The standad explanation amplification > in all diections integate ove complete sphee total flux amplification! consevation of enegy? solution to enegy cisis? ing cannot ceate photons distots geomety aea of suface shinks! have to compae with same mean geomety compae with same mean density in Univese [ Weinbeg (976) ] O. Wucknitz 7 6 O. Wucknitz 7 7 But... equivalent: efaction o Newtonian deflection does not change geomety same fomalism same paadox! Deflection angle fo the sphee calculation fo D s = GM c x = m cot dz [x +(z z ) ] 3/ m so fa: tangential plane 3 m/ (m/) cot (/) no poblem in the plane now: do it on the sphee! s M D d z x π/ π O. Wucknitz 7 8 O. Wucknitz 7 9 Magnification fo the sphee cuvatue of celestial sphee only second-ode effects fo point-mass µ = = + m [ m 4 + O ( )] m ( ) =.35 plana appoximation µ tot µ + µ = m 4 µ tot µ + 3 Lensing on the sphee fa fom optical axis: µ < in this situation: magnification theoem not valid integation ove sphee: mean µ is no paadox modified Poisson equation [ ] ψ() =: κ() = σ() σ µ µ - [ ] not always κ failue of theoem -. s [acsec] s [acsec] -3 field lines decay! O. Wucknitz 7 O. Wucknitz 7 Field lines on the sphee Back to gavitation shot summay flat spacetime with efactive medium (o Newtonian) magnification theoem not valid modified Poisson equation equivalent: gavity with appopiate efeence situation constant coodinate distance (e.g. isotopic o Schwazschild coodinates) constant metic distance magnification theoem invalid, no paadox FIELD LINES not equivalent: inappopiate efeence situation constant affine distance: magnification theoem valid, seeming paadox O. Wucknitz 7 O. Wucknitz 7 3

4 Affine distance, light tavel time metic (c = ) ds = ( + Ψ) dt ( Ψ) dx affine distance, light tavel time: measued at obseve s position light tavel time: T = ( + Ψ ) dx( Ψ) affine distance: L = ( Ψ ) dx geneal focusing theoem: µ > fo constant affine distance Lensing by a spheical shell shell of adius (metic) with mass M σ GM/c fo σ limit,m with σ = const no change of global geomety uned situation (e-)move sphee o... affine distance Λ = constant σ focusing theoem: compae with constant Λ focusing O. Wucknitz 7 4 O. Wucknitz 7 5 Suface bightness theoem : magnification µ > theoem : suface bightness is conseved? pevious deivation: did not conside local metic petubation in tems of photon numbe density pe solid angle: µ (A) (s,o) µ (A) (o,s) = + GM ( c ) D d ecipocity theoem [ Etheington (933), Phil. Mag., 5, 76 ] in tems of enegy flux density: F obs = + 4GM F c D yx D xy = + z y + z x ( ) D d O. Wucknitz 7 6 Summay seeming paadox if µ > eveywhee standad solution uned efeence situation with diffeent total suface aea focusing theoem: constant affine distance this talk conside cuvatue of celestial sphee (necessay!) modified deflection angle modified Poisson equation, no field line paadox keep aea unchanged fo compaison no magnification theoem, no paadox suface bightness not conseved bewae of geneal ing theoems! [ Wucknitz (7), A&A submitted ] O. Wucknitz 7 7 Bonus-pages: Exact magnification on the sphee We want to calculate the magnification matix fo abitay functions of the deflection angle. This includes lage deflections and multi-plane es, whee the deflection angle can no longe be witten as the gadient of a potential. This geneality is not necessay fo the main pat of this pape but may seve as the basis fo futue wok. In the plane, the total displacement is not elevant, so that the magnification matix is detemined exclusively by the fist-ode deivatives of the deflection. On the sphee, we have to take into account the cuvatue, and the equation is no longe a vecto equation. To detemine the souce position Θ s fom the image position Θ, we have to move along a geodesic (o geat cicle) in the diection of the negative deflection angle and follow this geodesic fo a length coesponding to the absolute deflection angle. The geodesic equation fo abitay coodinates is ẍ + Γ µνẋ µ ẋ ν =. () The affine paamete λ uns fom at Θ to at Θ s. Deivatives with espect to λ ae witten as dots. In the following, we wite the deflection angle as a µ (with a = a µ a µ) O. Wucknitz 7 8 to avoid confusion with tenso indices. The bounday conditions ae x () = Θ, ẋ () = a, x () = Θ s. () Fo the magnification matix, we have to conside additional geodesics infinitely close to the efeence geodesic. The equation fo the diffeence ǫξ, whee ǫ is infinitely small, is the diffeential equation fo the geodesic deviation: D ξ Dλ = ẋβ ẋ µ ξ ν R µβν (3) The diffeential opeato D denotes covaiant deivatives. The cuvatue tenso R has a paticulaly simple fom fo two-dimensional manifolds. It can be witten in tems of the metic g µν as R µβν := Γ µν,β Γ µβ,ν + Γ ρβγ ρ µν Γ ρνγ ρ µβ (4) = K ( δ β g µν δ ν g µβ ). (5) The cuvatue adius K is constant (K = ) on the sphee. The limit of the tangential plane can be found as K. Eq. (3) is valid in any coodinate system. Fo ou O. Wucknitz 7 9 convenience we use the system defined by the coodinates at Θ, which is then paallel-tanspoted along the geodesic. In this way, the covaiant deivatives become patial deivatives of the components, and ẋ (a,) as well as the cuvatue tenso Eq. (5) have constant components. We use a local Catesian system (with locally vanishing Chistoffel symbols) in which ξ is measued paallel to the negative deflection angle and ξ othogonal to this diection. This leads to With the stating condition ξ =, ξ = ω ξ, ω := a K. (6) ξ = Dξ Dλ = ξµda Dx µ (7) fom the deivative of Eq. (b), we can easily solve the diffeential equation (6) fo the two stating vectos (,) and (,) and in this way wite the tanspot equation fom Θ to Θ s fo abitay vectos ξ µ as ξ() = M ξ(). (8) The invese magnification matix of this mapping in (, ) coodinates eads a M ; a ; = a sin ω ; cos ω a sin ω, (9) ; ω ω in tems of the deivatives (covaiant o patial in these coodinates) of the deflection function a µ. The magnification depends on the deivatives, but also on the deflection angle itself, just as expected. We notice that finite deflection angles intoduce otation even if the deflection field is otation-fee (a ; = a ; ). Futhemoe does the cuvatue of the sphee lead to a magnification of / cos ω in the pependicula diection even fo (locally) constant deflection fields. In the intepetation of this, one should keep in mind that a covaiantly constant deflection does not coespond to a igid otation of the sphee, not even locally. We can decompose the invese magnification matix into a otated convegence and a shea pat, ( ) ( ) A := M cos ϕ sin ϕ γ γ = ( κ) sin ϕ cos ϕ γ γ, () O. Wucknitz 7 3 O. Wucknitz 7 3

5 whee the paametes ae detemined by the following equations: A A a, tanϕ = = a, sin ω/ω A + A + cos ω a, a, sin ω/ω () κ = sign(a + A) (A + A ) + (A A ) () A A γ = A + A γ = = cos ω + a, a, sin ω/ω = a, + a, sin ω/ω Note that the mapping is invaiant unde a sign change of κ with a simultaneous shift of π in ϕ. Equations () and () ae consistent fo π/ < ϕ < π/. In the limit of small ω (coesponding to small deflection angles o K ), the matix educes to the standad fom ( a M ; a ) ; =. (5) a ; a ; (3) (4) O. Wucknitz 7 3