Strong-Field Gravitational Lensing by Black Holes

Home , Einstein radius

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I First (unpublished) calculation using corpuscular model of light and Newtonian theory by H. Cavendish (1784).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, |ΦN| = 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. √ I Unstable photon sphere (or light-ring) at rcrit = 3M, bcrit = 3 3M.

I Determines the boundary of the BH shadow.

5 5

0 0

-5 -5

-10 -10 -5 0 5 10 -5 0 5 Lensing by a Kerr black hole Null geodesics

-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

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. Lensing by binary black holes

I What would a binary BH merger look like?

I Problem: No known solution to EFEs describing a dynamical binary.

I Solution: Evolve null geodesic equations on a fully numerical simulation to EFEs. (This is computationally very expensive!)

I Binary shadow is not superposition of two singleton BH shadows. Figure: Binary shadow with eyebrows. [Credit: SXS Lensing I Binary BH shadows look like they have Group, www.black-holes.org.] interesting features, e.g. eyebrows.

I Can we understand qualitative features of lensing by binary BHs by considering a simple imitative model? Toy model: Majumdar–Papapetrou static binary black hole

I Majumdar–Papapetrou (MP) binary BH solution to Einstein–Maxwell equations given in cylindrical coord’s {t, ρ, z, φ} by

2 −2 2 2 2 2 2 2 −1 ds = −U dt + U dρ + dz + ρ dφ ,Aa = [U , 0]. (6)

with

M+ M− ±dM∓ U(ρ, z) = 1 + + , z± = . p 2 2 p 2 2 ρ + (z − z+) ρ + (z − z−) M+ + M−

I Describes two ﬁxed BHs of mass M±(= Q±) in static equilibrium, separated by coordinate distance d.

I Focus on equal-mass case M± = M = 1.

I To analyse lensing eﬀects, consider geodesic motion, governed by Hamiltonian 1 1 1 H = gabp p = p 2 + p 2 − (ρU 2 − p )(ρU 2 + p ) = 0. (7) 2 a b 2 ρ z 2ρ2 φ φ Fate of rays for the static binary black hole

I Planar rays: pφ = 0.

10 I Rays can: 1. fall into the upper BH; 2. fall into the lower BH; 5 3. escape to spatial inﬁnity.

I Open Hamiltonian system with three 0 exits.

I Fate of rays can be visualised using an exit 0 5 10 basin diagram.

I This is actually a 1D binary BH shadow. BH1

I Aim: understand set of non-escaping BH2 orbits – the repellor ΩR. (Corresponds to 0 2 4 6 8 10 boundary of BH shadow.) Non-escaping orbits and the repellor

I There are thee types of unstable, periodic “fundamental orbits”. 10

5 I Fundamental orbits are “dynamically connected”. A ray can transition between asymptotic neighbourhoods of fundamental 0 orbits. -5

There exists . . . -10 I a countable inﬁnity of periodic orbits. -10 -5 0 5 10

I an uncountable inﬁnity of aperiodic orbits.

2 2 2 2

1 1 1 1

0 0 0 0

-1 -1 -1 -1

-2 -2 -2 -2

-2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 Chaotic scattering

Eckhardt’s deﬁnition of chaotic scattering (1988):

I Scattering in a Hamiltonian system is irregular (or chaotic) if there exists on some manifold of initial data an inﬁnity of distinct “scattering singularities” of zero (Lebesgue) measure, typically arranged into a fractal set.

I Scattering singularity: a value of the initial data for which the scattering process is undeﬁned, and some physical quantity (e.g. time-delay, deﬂection angle) becomes singular.

For the Schwarzschild BH, there is a unique scattering singularity, I √ bcrit = 3 3M, r = 3M.

I For the Kerr BH, there are two distinct scattering singularities, b = b±,

r = r±.

I Neither system exhibits chaotic scattering. Binary black holes and chaotic scattering

In our work† ...

I We ﬁnd that static binary BHs exhibit chaotic scattering.

I Each distinct perpetual orbit generates a scattering singularity in the initial data (e.g. impact parameter b).

I Any open interval in b contains either zero or inﬁnitely many scattering singularities.

I Set of scattering singularities = strange repellor.

The scattering singularities. . .

I have a Cantor-like distribution in initial data (1D self-similar fractal);

I have zero (Lebesgue) measure;

I can be understood using symbolic dynamics.

† J. O. Shipley & S. R. Dolan, Classical and Quantum Gravity 33, 175001 (2016); arXiv:1603.04469 [gr-qc]. Self-similarity of the one-dimensional shadow

BH1

BH2 5.75 6.10 6.45 6.80 7.15 7.50 5

BH1 0

BH2 6.94 6.97 7.00 7.02 7.05 7.08 0 5 10

BH1 BH1

BH2 BH2 0 2 4 6 8 10 7.040 7.042 7.044 7.047 7.049 7.051 Static binary black hole shadows 5. Conclusions Conclusions

I Weak-ﬁeld gravitational lensing phenomena: multiple images, Einstein rings, . . .

I GR predicts BHs – as conﬁrmed by LIGO–Virgo detections.

I Will soon observe BHs directly with the EHT (early 2019). Images formed will help us test GR in the strong-ﬁeld regime.

I BHs exhibit interesting (strong) lensing eﬀects associated with light rings and event horizons: black hole shadows.

I Geodesic motion on Kerr spacetime is integrable → shadow is “boring”!

I Can understand lensing by binary BHs using a toy model of two ﬁxed BHs.

I Geodesic motion on Majumdar–Papapetrou is non-integrable → chaotic scattering → fractal BH shadows with a Cantor-like structure. Selected references

1. Simulating eXtreme Spacetimes: www.black-holes.org. 2. S. Dodelson, Gravitational Lensing (CUP, 2017). 3. D. Psaltis, arXiv:1806.09740 [astro-ph.HE] (2018). 4. B. Eckhardt, Journal of Physics A 20, 5971 (1987). 5. G. Contopoulos, Proc. Roy. Soc. Lond. A 431, 183 (1990). 6. A. Bohn et al., Class. Quant. Grav. 32, 065002 (2015). 7. J. O. Shipley & S. R. Dolan, Class. Quant. Grav. 33, 175001 (2016). 8. S. R. Dolan & J. O. Shipley, Phys. Rev. D 94, 044038 (2016). 9. A.´ Daza, J. O. Shipley, S. R.Dolan & M. A. F. Sanju´an, Phys. Rev. D 98, 084050 (2018). Appendix Gravitational lensing at home

I Can mimic gravitational lensing using a logarithmic lens (e.g. the foot of a wine glass). Geodesic motion

a I A geodesic is a spacetime path q (λ) which extremises the action

Z dqa S[qa(λ)] = L(qa, q˙a) dλ, q˙a = , λ = aﬃne parameter . (8) dλ

1 a b I Lagrangian: L = 2 gab q˙ q˙ . ∂L Conjugate momenta: p = . I a ∂q˙a

1 ab I Hamiltonian: H = 2 g papb.

a ∂H ∂H I Null geodesics from Hamilton’s equations: q˙ = , p˙a = − a . ∂pa ∂q Symmetries

I Null geodesics ⇒ L = 0 = H.

I Static ⇒ pt = constant; set pt = −1 w.l.o.g.

I Axisymmetric ⇒ pφ = constant. Geodesic motion

I In the MP di-hole spacetime, geodesic motion is governed by a 2D

Hamiltonian with a conserved parameter (pφ) for rays: 1 1 H = gabp p = p 2 + p 2 + V = 0, (9) 2 a b 2 ρ z with geodesic potential 1 V (ρ, z) = − (ρU 2 − p )(ρU 2 + p ). (10) 2ρ2 φ φ

I Motion in (ρ, z) is not separable.

I The system is non-integrable.

I We should anticipate rich dynamics. Static binary black hole shadows Viewing angles

0° 10° 20° 30° 40°

50° 60° 70° 80° 90°