Strong-field gravitational lensing by black holes Jake O. Shipley School of Mathematics and Statistics University of Sheffield Applied Mathematics Colloquium University of Sheffield 5 December 2018 Outline 1. General relativity and gravitational lensing 2. Black holes 3. Strong-field tests of general relativity 4. Gravitational lensing by black holes 5. Conclusions 1. General relativity and gravitational lensing General relativity I Geometric theory of space, time and gravitation, published by A. Einstein in 1915. I Gravitational field is coded in a metric gab of Lorentzian signature on the curved 4D spacetime manifold M. I GR relates curvature of spacetime to energy and momentum of matter in spacetime through Einstein's field equations, 1 8πG R − R g = T : (1) ab 2 ab c4 ab I Light follows null geodesics on curved spacetime. I Key consequence of GR: light rays are deflected by gravitational fields. Gravitational deflection of light Newtonian prediction I Traces of the idea of light deflection date back to I. Newton: \Do not Bodies act upon Light at a distance, and by their action bend its Rays, and is not this action (cæteris paribus) strongest at the least distance?" | Isaac Newton, Query 1, Opticks, 1704 Gravitational deflection of light Newtonian prediction I First (unpublished) calculation using corpuscular model of light and Newtonian theory by H. Cavendish (1784). I First published calculation by J. G. von Soldner (1804): 2GM δθ = : (2) N c2b I M = mass of body, b = impact parameter, G = gravitational constant, c = speed of light. Gravitational deflection of light Einstein's predictions I Einstein's first prediction (1911) using only the equivalence principle (special relativity) was equal to those of Cavendish and Soldner. I For the Sun, δθN ' 0:87 arcsec. I Attempted observation of light deflection by the Sun during 1914 solar eclipse on Crimean Peninsula by E. Freundlich. I Unfortunately, Freundlich and colleagues were imprisoned. They were eventually released, but missed the eclipse! I However, Einstein's 1911 calculation was incorrect. I In 1916, Einstein found that the deflection angle predicted by GR is twice the Newtonian prediction: 4GM δθ = 2δθ = : (3) GR N c2b Gravitational deflection of light Experimental verification I For a ray which grazes the Sun's surface, the value would be 4GM δθ = 2 ' 1:74 arcsec: (4) c R I In 1919, this was determined to within 20% during A. Eddington's solar eclipse experiment. I More recently, this result has been verified Figure: Solar Eclipse, 1919. experimentally to within 0.0002%. [Credit: F. W. Dyson et al., Philos. Trans. Royal Soc. A 220 (1920).] Weak-field lensing I Gravitational fields affect light in a similar way to an optical lens. I Light deflection by gravitational fields = gravitational lensing. I Weak-field gravitational lensing involves: GM 2 weak gravitational fields, jΦNj = r c ; and small deflection angles δθ. I Theory of weak-field gravitational lensing is well understood. [S. Dodelson, Gravitational Lensing (CUP, 2017).] I Consequences of weak-field lensing: Einstein rings; multiple images; image distortion, . Weak-field lensing phenomena Multiple images and Einstein rings I Multiple rays reach observer from a single source ) multiple images of the source along the tangent to the ray. I Perfect alignment of source, lens and observer results in an Einstein ring. I Einstein radius is r 4GM θE ' 2 : c DL Weak-field lensing phenomena Multiple images and Einstein rings Figure: Einstein ring observed by Sloan Digital Sky Survey & Hubble Space Telescope. [Credit: ESA/Hubble & NASA.] Testing general relativity I Three classical tests of GR proposed by Einstein in 1916: 1. perihelion precession of Mercury's orbit; 2. deflection of light by the Sun; 3. gravitational redshift of light in the solar system. I All established GR as our best theory of gravity. I These are all weak-field tests of GR. I GR also leads to objects with very strong gravitational fields, including black holes and neutron stars. 2. Black holes Schwarzschild black hole a I Spacetime geometry specified in coordinates x by line element 2 a b ds = gab dx dx , where gab is a 4D Lorentzian metric. I Solution found by K. Schwarzschild in 1916: 2M 2M −1 ds2 = − 1 − dt2 + 1 − dr2 + r2(dθ2 + sin2 θ dφ2): r r I Describes a spherically symmetric black hole (BH) of mass M. I The BH is characterised by: I an event horizon (\point of no return") at r = 2M; I a curvature singularity at r = 0. Kerr({Newman) black hole I Schwarzschild solution generalised by R. Kerr (1963) to include rotation. I Generalised by E. Newman (1965) to include electric charge. I Kerr{Newman solution describes spacetime of a charged, rotating BH of mass M, angular momentum J = aM, and electric charge Q. I Kerr{Newman solution is given by the line element 2Mr Σ ds2 = − 1 − dt2 + dr2 + Σ dθ2 Σ ∆ 2Ma2r 4Mra sin2 θ + r2 + a2 + sin2 θ sin2 θ dφ − dt dφ, Σ Σ where Σ(r; θ) = r2 + a2 cos2 θ; ∆(r) = r2 − 2Mr + a2 + Q2. I Not spherically symmetric due to rotation. No-hair conjecture Conjecture Stationary isolated BHs in GR are uniquely characterised by three parameters: mass M, angular momentum J, and electric charge Q. I All other parameters are \hair". I Stationary isolated BHs in GR are therefore described by the Kerr{Newman solution. I What about astrophysical BHs in the Universe? Kerr's solution and astrophysical black holes I Q thought to be negligible for astrophysical BHs. I Astrophysical BHs described by Kerr's solution: \The most shattering experience has been the realization that [Kerr's] solution of Einstein's equations of general relativity provides the abso- lutely exact representation of untold numbers of massive black holes that populate the universe." | Subrahmanyan Chandrasekhar, 1975 I Can we test the no-hair conjecture, and other aspects of strong gravity, by observing astrophysical BHs? 3. Strong-field tests of general relativity Gravitational waves I Gravitational waves: ripples in the fabric of spacetime caused by accelerating masses. I GW150914: GW signal from the merger of two BHs detected by LIGO. I Confirmed existence of BHs in Nature. I Nobel Prize (2017): Awarded to R. Weiss, B. Barish and K. Thorne \for decisive contributions to the LIGO detector and the observation of gravitational waves". I Four new detections announced this week (03/12/18)! Multimessenger astronomy I GW170817: \gravity and light". I LIGO{Virgo: GWs from neutron star merger. I Telescopes observed gamma-ray burst (and afterglow) from EM counterpart. I Birth of multimessenger astronomy. Before we could only see the universe, now we can \hear" it. The Event Horizon Telescope Overview I Aims: I Image nearby galactic centres. I Take the first picture of a BH at exquisite angular resolution at event horizon scales. wavelength angular resolution ∼ . I dish size I Employs mm-wavelength very-long-baseline interferometry. Figure: The Event Horizon I Global array of radio telescopes. Telescope. [Credit: EHT.] I Earth-sized virtual telescope. The Event Horizon Telescope Main aims I EHT's targets: 1. Sgr A∗ 6 4 M ∼ 4 × 10 M , D ∼ 8 × 10 pc, θ ∼ 53 µas 2. M87 9 6 M ∼ 6 × 10 M , D ∼ 16 × 10 pc, θ ∼ 22 µas I View the BH shadow: \silhouette" of the BH. I Test GR: deviations from Kerr; no-hair conjecture. I Understand accretion and jets. The Event Horizon Telescope Results I So what does a BH actually look like? I Results from the EHT expected in early 2019! I However, we can answer this question theoretically. 4. Gravitational lensing by black holes Black hole shadow I What would an observer see if they were to look at a BH? I Light from distant stars does not refract off the BH { we can't actually see it . I . but light is deflected by the BH's gravitational field. I Some light is scattered, some is absorbed ) BH will cast a shadow (or \silhouette"). I Shadow = region of the observer's sky which lacks radiation, due to the blockage of the BH. Ray-tracing Undeflected ray Black hole I We simulate BH shadows using Background backwards ray-tracing. Deflected rays Camera plane I Shadow associated with the set of photons which reach the BH event horizon, when traced back in time from the observer. I See Simulating eXtreme Spacetimes for simulations: https://www.black-holes.org/. Lensing by a Schwarzschild black hole I Schwarzschild spacetime is spherically symmetric: consider motion in π equatorial plane (θ = 2 ). I Curvature singularity at r = 0; event horizon at r = 2M. p I Unstable photon sphere (or light-ring) at rcrit = 3M, bcrit = 3 3M. I Determines the boundary of the BH shadow. 10 5 5 0 0 -5 -5 -10 -10 -5 0 5 10 -5 0 5 Lensing by a Kerr black hole Null geodesics 10 5 0 -5 -10 -10 -5 0 5 10 I Kerr BH rotates with angular momentum J = aM (0 < a < 1). I Prograde (+) and retrograde (−) circular photon orbits: 2 a r = 2M 1 + cos arccos ∓ ; b = ±3pMr − a: (5) ± 3 M ± ± Lensing by a Kerr black hole Black hole shadow 10 8 θ = 90 6 5 4 2 0 0 -2 -5 -4 -6 -10 -10 -5 0 5 10 -8 -8 -6 -4 -2 0 2 4 6 8 I Geodesic motion is integrable (due to existence of Carter constant). I Equation for shadow boundary known in closed form. I Shadow deformed due to lack of spherical symmetry. Binary black holes I Pairs of spinning BHs which orbit one another { binary BHs { have recently been discovered by LIGO. I Verification that BHs occur in binary pairs. I LIGO{Virgo since confirmed an abundant binary BH population; 10 GW detections so far.