Black holes and exotic compact objects
C. Herdeiro Departamento de Matemática and CIDMA, Universidade de Aveiro, Portugal Plan of the lectures:
Lecture 1 Black holes: astrophysical evidence and a theory (brief) timeline Lecture 2 Spherical black holes: the Schwarzschild solution Lecture 3 Spinning black holes: the Kerr solution Lecture 4 Exotic compact objects: the example of bosonic stars Lecture 5 Non-Kerr black holes Timeline 1916: Schwarzschild’s solution
2M dr2 ds2 = 1 dt2 + + r2(d✓2 +sin2 ✓d 2) r 1 2M ✓ ◆ r c =1 G =1 (in the coordinates introduced by Johannes Droste, in 1916) 2M dr2 ds2 = 1 dt2 + + r2(d✓2 +sin2 ✓d 2) r 1 2M ✓ ◆ r Analysis of singularities 1) A metric is said to be singular at some point P if:
either a metric coefficient diverges at P; or the metric determinant vanishes at P.
If neither of these happen, the metric is said to be regular at P.
2) If a metric is singular at some point P, the singularity may be:
a coordinate singularity, if in a different coordinate system P is regular; a physical singularity if there is no coordinate system in which P is regular.
3) Typically (but not always) the diagnosis of physical singularities is done by showing that some curvature invariant diverges at P ; it is then said to be a (scalar polynomial) curvature singularity. 2M dr2 ds2 = 1 dt2 + + r2(d✓2 +sin2 ✓d 2) r 1 2M ✓ ◆ r
Analysis of singularities
For the Schwarzschild metric, the interesting singularities are:
1) r=0, which is a physical curvature singularity, since the Kretschmann scalar diverges
48M 2 Rµ⌫↵ R = µ⌫↵ r6
2) r=2M, which is a coordinate singularity, since there are new coordinates that make this point regular.
Two examples of such coordinate systems are:
- ingoing Eddington-Finkelstein (EF) coordinates (v, r, ✓, ) - outgoing Eddington-Finkelstein (EF) coordinates(u, r, ✓, ) 2M dr2 ds2 = 1 dt2 + + r2(d✓2 +sin2 ✓d 2) r 1 2M ✓ ◆ r Ingoing Eddington-Finkelstein (EF) coordinates (v, r, ✓, )
Idea: follow a radially infalling light ray:
ds2 =0(null) d✓ =0=d (radial)
Then: Regge-Wheeler radial coordinate dr dt = dr⇤ or ±1 2M ⌘± r “tortoise” radial coordinate r =2M r =+ 1
r⇤ =+ r⇤ = 1 1 2M dr2 ds2 = 1 dt2 + + r2(d✓2 +sin2 ✓d 2) r 1 2M ✓ ◆ r Ingoing Eddington-Finkelstein (EF) coordinates (v, r, ✓, )
Idea: follow a radially infalling light ray (set G=1 for simplicity):
ds2 =0(null) d✓ =0=d (radial)
Then: Regge-Wheeler radial coordinate dr dt = dr⇤ or ±1 2M ⌘± r “tortoise” radial coordinate
r =2M r =+ 1
r⇤ =+ r⇤ = 1 1 2M dr2 ds2 = 1 dt2 + + r2(d✓2 +sin2 ✓d 2) r 1 2M ✓ ◆ r Ingoing Eddington-Finkelstein (EF) coordinates (v, r, ✓, )
Idea: follow a radially infalling light ray (set G=1 for simplicity):
ds2 =0(null) d✓ =0=d (radial)
Then: Regge-Wheeler radial coordinate dr dt = dr⇤ or ±1 2M ⌘± r “tortoise” radial coordinate
r =2M r =+ 1
r⇤ =+ r⇤ = 1 1 2M dr2 ds2 = 1 dt2 + + r2(d✓2 +sin2 ✓d 2) r 1 2M ✓ ◆ r Ingoing Eddington-Finkelstein (EF) coordinates (v, r, ✓, )
dr dt = dr⇤ ±1 2M ⌘± r The problematic coordinate at r=2M is actually t; we need to replace it. Following the ingoing, radial light rays suggest the advanced time v:
v t + r⇤ ⌘ v=constant, follows the ingoing, radial light rays. Then, changing from the Schwarzschild coordinates to the ingoing EF coordinates: (t, r, ✓, ) (v, r, ✓, ) ! gives the Schwarzschild metric in the form:
2M Non-singular ds2 = 1 dv2 +2dvdr + r2(d✓2 +sin2 ✓d 2) r at r=2M ✓ ◆ 2M dr2 ds2 = 1 dt2 + + r2(d✓2 +sin2 ✓d 2) r 1 2M ✓ ◆ r Ingoing Eddington-Finkelstein (EF) coordinates (v, r, ✓, ) Outgoing (u, r, ✓, ) dr dt = dr⇤ ±1 2M ⌘± r The problematic coordinate at r=2M is actually t; we need to replace it. Following the ingoing, radial light rays suggest the advanced time v: outgoing outgoing time u v t + r⇤ u=constant outgoing ⌘ u t r⇤ v=constant, follows the ingoing, radial light⌘ rays. Then, changing from the Schwarzschild coordinates to the ingoing EF coordinates: outgoing (t, r, ✓, ) (v, r, ✓, ) ! (u, r, ✓, ) gives the Schwarzschild metric in the form:
2 2M du 2dudr Non-singular ds2 = 1 dv2 +2dvdr + r2(d✓2 +sin2 ✓d 2) r at r=2M ✓ ◆ But is there something special about the (hyper)surface r=2M?
Yes! r=2M is a null hypersurface. And it is also a Killing horizon.
Definition: Let S(xµ) be a smooth function of the spacetimes coordinates xµ and consider the family of hypersurfaces S =constant. They have normal
µ⌫ ` =(g @⌫ S)@µ .
2 µ If ` = ` `µ = 0 for a particular hypersurface in the family, ,then is said to be a null surface. N N Exercise 2.1
Show, using (say) ingoing EF coordinates that for the family of hypersurfaces S=r=constant in the Schwarzschild spacetime: 2M `2 =1 r Interpretation:
In Minkowski spacetime: ds2 = dt2 + dr2 + r2(d✓2 +sin2 ✓d 2) The family of hypersurfaces S=r=constant is always timelike: `2 =1
In a spacetime u increasing v increasing diagram: t v = t + r u = t r u = constant
r=constant hypersurfaces are always v = constant timelike r
But in the Schwarzschild spacetime it is different: Ingoing Eddington-Finkelstein coordinates
2M ds2 = 1 dv2 +2dvdr + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ Or, equivalently: 2M 2dvdr = ds2 1 dv2 + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ > 0 for causal curves Ingoing Eddington-Finkelstein coordinates
2M ds2 = 1 dv2 +2dvdr + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ Or, equivalently: 2M 2dvdr = ds2 1 dv2 + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ 0 > > 0 for causal for r 6 2M curves Ingoing Eddington-Finkelstein coordinates
2M ds2 = 1 dv2 +2dvdr + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ Or, equivalently: 2M 2dvdr = ds2 1 dv2 + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ 0 > > 0 > 0 for causal for r 6 2M curves Ingoing Eddington-Finkelstein coordinates
2M ds2 = 1 dv2 +2dvdr + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ Or, equivalently: 2M 2dvdr = ds2 1 dv2 + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ 0 > > 0 > 0 Thus if for causal for r 2M dv > 0(futuredirected) 6 curves then dr < 0 (infalling) particle particle v increasing must fall can fall v r particle can escape Easily Black visualised in Hole Finkelstein diagram region v = constant
r =2M r Outgoing Eddington-Finkelstein coordinates 2M ds2 = 1 du2 2dudr + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ Or, equivalently:
2M 2dudr = ds2 1 du2 + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ > 0 for causal curves Outgoing Eddington-Finkelstein coordinates 2M ds2 = 1 du2 2dudr + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ Or, equivalently:
2M 2dudr = ds2 1 du2 + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ 0 > > 0 for causal for r 6 2M curves Outgoing Eddington-Finkelstein coordinates 2M ds2 = 1 du2 2dudr + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ Or, equivalently:
2M 2dudr = ds2 1 du2 + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ 0 > > 0 > 0 for causal for r 6 2M curves Outgoing Eddington-Finkelstein coordinates 2M ds2 = 1 du2 2dudr + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ Or, equivalently:
2M 2dudr = ds2 1 du2 + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ > 0 0 0 Thus if > > for causal for r 6 2M du > 0(futuredirected) curves then dr > 0 (escaping) particle particle particle u increasing must escape can fall can escape
u + r
White Easily u = constant visualised in Hole Finkelstein region diagram
r =2M r Thus, the Schwarzschild spacetime has a richer structure, (more regions) that one would initially have guessed!
The global structure of a spacetime is determined by its maximal analytic extension:
Definition: The maximal analytic extension of a spacetime is a coordinate system (or set of coordinate systems) in which all geodesics that to not terminate on a physical (irremovable) singularity, can be extended to arbitrary values of their a ne parameters.
For the Schwarzschild spacetime the maximal analytic extension is given by the Kruskal-Szekers coordinates. For the Schwarzschild spacetime the maximal analytic extension is given by the Kruskal-Szekers coordinates.
Exercise 2.2
Introduce the Kruskal-Szekers (KS) coordinates from the Schwarzschild coordinates as: (t, r, ✓, ) (U,V,✓, ) ! v = t + r⇤ u U = e 4M u = t r⇤ v where 4M dr V = e dr⇤ = 1 2M r Show that the Schwarzschild metric becomes: 3 2 32M r 2 2 2 2 ds = e 2M dUdV + r (d✓ +sin ✓d ) r where r is implicitly defined in terms of U,V as:
r 2M r UV = e 2M 2M ✓ ◆ u Initially, the KS coordinates are U = e 4M v only defined for U<0 and V>0: V = e 4M
3 2 32M r 2 2 2 2 But in KS coordinates we can extend ds = e 2M dUdV + r (d✓ +sin ✓d ) r beyond U=0 and V=0 into arbitrary values; r 2M r UV = e 2M the metric is only singular at r=0 i.e. UV=1 2M ✓ ◆
U V Black hole New region asymptotically Initial flat region region White hole region
Finkelstein diagram for the eternal Schwarzschild black hole An even nicer way to visualize the black hole spacetime, and its causal structure is by means of a conformal compactification.
Definition: A conformal compactification is a conformal transformation of the metric, thus preserving the light cone structure,
ds2 ds˜2 =⇤(t, ~x)2ds2 , ! such that all points at infinity in the original metric are at finite a ne parameter in the new metric. Thus, we choose ⇤(t, ~x) such that
⇤(t, ~x) 0 as ~x 0 and/or t 0 . ! | |! | |! U V First introduce Black hole compact coordinates: New region For asymptotically Initial U = tan U˜ Schwarzschild: flat region V = tan V˜ region White ds2 ds˜2 =⇤(t, ~x)2ds2 , ! such that all points at infinity in the original metric are at finite a ne parameter in the new metric. Thus, we choose ⇤(t, ~x) such that ⇤(t, ~x) 0 as ~x 0 and/or t 0 . ! | |! | |! U V First introduce Black hole compact coordinates: New region For asymptotically Initial U = tan U˜ Schwarzschild: flat region V = tan V˜ region White 3 2 1 32M r 2 2 2 2 2 2 ds = e 2M dUd˜ V˜ + r cos U˜ cos V˜ (d✓ +sin ✓d ) 2 ˜ 2 ˜ r cos U cos V And we define: ds˜2 = cos2 U˜ cos2 Vds˜ 2 Infinity is: ⇡ ⇡ ⇤(t, ~x)2 ⇤(t, ~x)=0 U˜ = or V˜ = , ± 2 ± 2 We can now draw the spacetime diagram for the conformal compactification of the Schwarzschild spacetime called the Carter-Penrose diagram. First change to the compact coordinates: U V First introduce Black hole compact coordinates: New region asymptotically Initial U = tan U˜ flat region V = tan V˜ region White Singularity: Infinity: ⇡ U˜ = V˜ ⇡ ⇡ UV =1 tan U˜ tan V˜ =1 2 ⇤(t, ~x)=0 U˜ = or V˜ = ⇡ , ± 2 ± 2 , , U˜ = V˜ 2 U˜ V˜ First introduce Black hole compact coordinates: New region asymptotically Initial U = tan U˜ flat region V = tan V˜ region White Future temporal infinity Black hole Future null infinity New region asymptotically Initial Spatial infinity flat region region White hole region Past null infinity Past temporal infinity For an astrophysically realistic black hole, on the other hand, only the initial region and the black hole region are expected to be present. The surface r=2M in Schwarzschild is the black hole event horizon. Definition: The event horizon of an asymptotically flat spacetime is the boundary of the causal past of future null infinity. This definition is teleological (i.e. global); one needs to know the whole spacetime history to determine where the event horizon is. In a stationary spacetime, however, the event horizon coincides with a local concept: a Killing horizon. Definition: A null hypersurface, , is a Killing horizon of a Killing vector N 2 field, Hawking’s Rigidity Theorem (1972): The event horizon of a stationary, reg- ular, asymptotically flat spacetime is a Killing horizon. For Schwarzschild, using regular coordinates on the horizon, say, ingoing EF coordinates: 2M ds2 = 1 dv2 +2dvdr + r2(d✓2 +sin2 ✓d 2) r ✓ ◆ @ Then, clearly, a Killing vector field is: ⇠ = @v 2 µ ⌫ 2M Its norm is: ⇠ = ⇠ ⇠ gµ⌫ = gvv = 1 r ✓ ◆ Thus, the norm vanishes on the null surface r=2M. Consequently, this surface is a Killing horizon. Consequently it is the spacetime event horizon. Bottom line: The Schwarzschild spacetime represents a black hole. The hypersurface r=2M is the event horizon. Probing the Schwarzschild black hole with light Schwarzschild space-time: BH with mass M (G=1=c) 2 2 2M 2 dr 2 ds = 1 dt + + r d⌦2 r 1 2M ✓ ◆ r Exercise 2.3 Null geodesics obey: dr 2 2M j2 Obtain this equation! = E2 1 d r r2 Vef ✓ ◆ ✓ ◆ Vef Real space motion 10 5 0 3M 10 20 30 40 50 -5 Schwarzschild space-time: BH with mass M 2 2 2M 2 dr 2 ds = 1 dt + + r d⌦2 r 1 2M ✓ ◆ r Null geodesics obey: dr 2 2M j2 = E2 1 d r r2 Vef ✓ ◆ ✓ ◆ Vef Real space motion 10 5 0 3M 10 20 30 40 50 -5 Schwarzschild space-time: BH with mass M 2 2 2M 2 dr 2 ds = 1 dt + + r d⌦2 r 1 2M ✓ ◆ r Null geodesics obey: dr 2 2M j2 = E2 1 d r r2 Vef ✓ ◆ ✓ ◆ Vef Real space motion 10 5 0 3M 10 20 30 40 50 -5 Schwarzschild space-time: BH with mass M 2 2 2M 2 dr 2 ds = 1 dt + + r d⌦2 r 1 2M ✓ ◆ r Null geodesics obey: dr 2 2M j2 = E2 1 d r r2 Vef ✓ ◆ ✓ ◆ Vef Real space motion 10 5 0 3M 10 20 30 40 50 -5 Lesson: The photon escapes if it has an impact parameter greater than a critical value. What is this value? From the maximum of the potential, r=3M; then, from the radial equation: dr 2 2M j2 = E2 1 d r r2 ✓ ◆ ✓ ◆ the circular null orbit occurs for impact parameter j d =33/2M ⌘ E This is the critical value. Consider a “bright” homogeneous background with angular size much larger than the BH 2M Consider a “bright” homogeneous background with angular size much larger than the BH 33/2M 2M As seen by the distant observer the BH will cast a shadow in the middle of the large bright source, larger than the horizon scale The rim of the BH shadow corresponds to the critical impact parameter: j d =33/2M ⌘ E Light source Observer BH Light source BH Observer BH shadow The importance of the light ring The stereotypical light ring arises in the Schwarzschild black hole solution of vacuum general relativity Why is it relevant ? 1) The light ring determines the “shadow” of the black hole. Academic In a more astrophysical scenario: background light is replaced by synchrotron radiation from accretion disk The Astrophysical Journal Letters, 875:L5 (31pp), 2019 April 10 The EHT Collaboration et al. Synthetic images are generated by General Relativistic Magneto-Hydrodynamics (GRMHD) simulations using the Kerr metric Figure 1. Left panel: an EHT2017 image of M87 from Paper IV of this series (see their Figure 15). Middle panel: a simulated image based on a GRMHD model. Right panel: the model image convolved with a 20as FWHM Gaussian beam. Although the most evident features of the model and data are similar, fine features in the model are not resolved by EHT. In Section 6 we combine EHT data with other constraints on the (e.g., Johannsen & Psaltis 2010). For an a* 0 black hole radiative efficiency, X-ray luminosity, and jet power and show of mass M and distance D, the photon ring angular radius on that the latter constraint eliminates all a* 0 models. In the sky is Section 7 we discuss limitations of our models and also briefly discuss alternatives to Kerr black hole models. In Section 8 we 27 GM p summarize our results and discuss how further analysis of existing cD2 1 EHT data, future EHT data, and multiwavelength companion ⎛ M ⎞⎛ D ⎞ observations will sharpen constraints on the models. 18.8 as, 1 ⎜ 9 ⎟⎜ ⎟ () ⎝ 6.2 10M ⎠⎝ 16.9 Mpc ⎠ 2. Review and Estimates where we have scaled to the most likely mass from Gebhardt et al. (2011) and a distance of 16.9 Mpc (see also EHT Collaboration In EHT Collaboration et al. (2019d; hereafter Paper IV) we et al. 2019e, (hereafter Paper VI;Blakesleeetal.2009;Birdetal. present images generated from EHT2017 data (for details on 2010;Cantielloetal.2018).Thephotonringangularradiusfor the array, 2017 observing campaign, correlation, and calibra- other inclinations and values of differs by at most 13% from tion, see Paper II and Paper III). A representative image is a* reproduced in the left panel of Figure 1. Equation (1),andmostofthisvariationoccursat11 ∣∣a* Four features of the image in the left panel of Figure 1 play (e.g., Takahashi 2004;Younsietal.2016).Evidentlytheangular an important role in our analysis: (1) the ring-like geometry, (2) radius of the observed photon ring is approximately 20 as the peak brightness temperature, (3) the total flux density, and (Figure 1 and Paper IV),whichisclosetothepredictionofthe (4) the asymmetry of the ring. We now consider each in turn. black hole model given in Equation (1). (1) The compact source shows a bright ring with a central (2) The observed peak brightness temperature of the ring in 9 dark area without significant extended components. This bears Figure 1 is Tb, pk 6 10 K,whichisconsistentwithpastEHT a remarkable similarity to the long-predicted structure for mm-VLBI measurements at 230 GHz (Doeleman et al. 2012; optically thin emission from a hot plasma surrounding a black Akiyama et al. 2015),andGMVA3 mm-VLBImeasurementsof hole (Falcke et al. 2000). The central hole surrounded by a the core region (Kim et al. 2018).Expressedinelectronrest-mass 2 bright ring arises because of strong gravitational lensing (e.g., (me) units, b,B, pk kT b pk() mc e 1,wherekB is Boltzmann’s Hilbert 1917; von Laue 1921; Bardeen 1973; Luminet 1979). constant. The true peak brightness temperature of the source is The so-called “photon ring” corresponds to lines of sight that higher if the ring is unresolved by EHT, as is the case for the pass close to (unstable) photon orbits (see Teo 2003), linger model image in the center panel of Figure 1. near the photon orbit, and therefore have a long path length The 1.3 mm emission from M87 shown in Figure 1 is through the emitting plasma. These lines of sight will appear expected to be generated by the synchrotron process (see Yuan comparatively bright if the emitting plasma is optically thin. & Narayan 2014, and references therein) and thus depends on The central flux depression is the so-called black hole the electron distribution function (eDF). If the emitting plasma “shadow” (Falcke et al. 2000), and corresponds to lines of has a thermal eDF, then it is characterized by an electron 2 sight that terminate on the event horizon. The shadow could be temperature TTeb , or eee kTB () mc 1, because seen in contrast to surrounding emission from the accretion e b, pk if the ring is unresolved or optically thin. flow or lensed counter-jet in M87 (Broderick & Loeb 2009). Is the observed brightness temperature consistent with what The photon ring is nearly circular for all black hole spins and one would expect from phenomenological models of the all inclinations of the black hole spin axis to the line of sight source? Radiatively inefficient accretion flow models of M87 2 Academic A Gaussian Blurring filter is applied to a synthetic image to reproduce real EHT observations TheThe Astrophysical Astrophysical Journal Journal Letters, Letters, 875:L5875:L5 ((31pp31pp)),, 2019 2019 April April 10 10 The EHT Collaboration et al. Synthetic images are generated by General Relativistic Magneto-Hydrodynamics (GRMHD) simulations using the Kerr metric FigureFigure 1. 1. LeftLeft panel: panel: an an EHT2017 EHT2017 image image of of M87 M87 from from Paper Paper IVIV of this series (see their Figure 15).. Middle Middle panel: panel: a a simulated simulated image based on a GRMHD model. Right panel:panel: the the model model image image convolved convolved with with a a 220as0as FWHMFWHM Gaussian Gaussian beam. Although the most evident features of the model and data are similar, fine features in the modelmodel are are not not resolved resolved by by EHT. EHT. In Section 6 we combine EHT data with other constraints on the (e.g., Johannsen & Psaltis 2010). For an a 0 black hole In Section 6 we combine EHT data with other constraints on the (e.g., Johannsen & Psaltis 2010). For an a* 0 black hole radiativeradiative ef effificiency,ciency, X-ray X-ray luminosity, luminosity, and and jet jet power power and show of mass M and distance D, the photon ring angular radius on that the latter constraint eliminates all a 0 models. In the sky is that the latter constraint eliminates all a** 0 models. In the sky is SectionSection 77 wewe discuss discuss limitations limitations of of our our models models and and also briefly discussdiscuss alternatives alternatives to to Kerr Kerr bl blackack hole hole models. models. In In Section 8 we 27 GM p summarizesummarize our our results results and and discuss discuss how how further further analysis analysis of existing cD2 EHT data, future EHT data, and multiwavelength companion 1 EHT data, future EHT data, and multiwavelength companion ⎛ M ⎞⎛ D ⎞ observationsobservations will will sharpen sharpen c constraintsonstraints on on the the models. models. 18.8 as, 1 ⎜ 9 ⎟⎜ ⎟ () ⎝ 6.2 10M ⎠⎝ 16.9 Mpc ⎠ 2.2. Review Review and and Estimates Estimates where we have scaled to the most likely mass from Gebhardt et al. (2011) and a distance of 16.9 Mpc (see also EHT Collaboration In EHT Collaboration et al. (2019d; hereafter Paper IV) we In EHT Collaboration et al. (2019d; hereafter Paper IV) we et al. 2019e, (hereafter Paper VI;Blakesleeetal.2009;Birdetal. present images generated from EHT2017 data (for details on present images generated from EHT2017 data (for details on 2010;Cantielloetal.2018).Thephotonringangularradiusfor thethe array, array, 2017 2017 observing observing campaign, campaign, correlation, correlation, and and calibra- other inclinations and values of a differs by at most 13% from tion,tion, see see Paper Paper IIII andand Paper Paper IIIIII)).. A A representative representative image image is is a* Equation (1),andmostofthisvariationoccursat11 a reproducedreproduced in in the the left left panel panel of of Figure Figure 11.. Equation (1),andmostofthisvariationoccursat11 ∣∣a* FourFour features features of of the the image image in in the the left left panel panel of of Figure 1 play (e.g., Takahashi 2004;Younsietal.2016).Evidentlytheangular anan important important role role in in our our analysis: analysis: ((11)) thethe ring-like ring-like geometry, geometry, (2) radius of the observed photon ring is approximately 20 as thethe peak peak brightness brightness temperature, temperature, ((33)) thethe total total flfluxux density, and (Figure 1 and Paper IV),whichisclosetothepredictionofthe ((44)) thethe asymmetry asymmetry of of the the ring. ring. We We now now consider consider each each in turn. black hole model given in Equation (1). ((11)) TheThe compact compact source source shows shows a a bright bright ring ring with a central (2) The observed peak brightness temperature of the ring in fi 9 darkdark area area without without signi significantcant extended extended components. components. This bears Figure 1 is Tb,, pk 6 10 K,whichisconsistentwithpastEHT aa remarkable remarkable similarity similarity to to the the long-predicted long-predicted structure for mm-VLBI measurements at 230 GHz (Doeleman et al. 2012; opticallyoptically thin thin emission emission from from a a hot hot plasma plasma surrounding surrounding a black Akiyama et al. 2015),andGMVA3 mm-VLBImeasurementsof holehole ((FalckeFalcke et et al. al. 20002000)).. The The central central hole hole surrounded surrounded by by a a the core region (Kim et al. 2018).Expressedinelectronrest-mass 2 ’ brightbright ring ring arises arises because because of of strong strong gravitational gravitational lensing (e.g., (me) units, b,B,,B, pk kT b pk() mc e 1,wherekB is Boltzmann’s HilbertHilbert 19171917;; von von Laue Laue 19211921;; Bardeen Bardeen 19731973;; Luminet Luminet 1979).. constant. The true peak brightness temperature of the source is TheThe so-called so-called ““photonphoton ring ring”” correspondscorresponds to to lines lines of sight that higher if the ring is unresolved by EHT, as is the case for the passpass close close to to ((unstableunstable)) photonphoton orbits orbits ((seesee Teo Teo 2003),, linger linger model image in the center panel of Figure 1. nearnear the the photon photon orbit, orbit, and and therefore therefore have have a a long long path length The 1.3 mm emission from M87 shown in Figure 1 is throughthrough the the emitting emitting plasma. plasma. These These lines lines of of sight sight will will appear expected to be generated by the synchrotron process (see Yuan comparativelycomparatively bright bright if if the the emitting emitting plasma plasma is is optically thin. & Narayan 2014, and references therein) and thus depends on TheThe central central flfluxux depression depression is is the the so-called so-called black hole the electron distribution function (eDF). If the emitting plasma ““shadowshadow”” ((FalckeFalcke et et al. al. 20002000)),, and and corresponds corresponds to to lines lines of of has a thermal eDF, then it is characterized by an electron sight that terminate on the event horizon. The shadow could be temperature TT , or 2 , because sight that terminate on the event horizon. The shadow could be temperature TTeb , or eee kTB () mc 1, because seenseen in in contrast contrast to to surrounding surrounding emission emission from from the accretion e b,, pk if the ring is unresolved or optically thin. flflowow or or lensed lensed counter-jet counter-jet in in M87 M87 ((BroderickBroderick & & Loeb 2009).. Is the observed brightness temperature consistent with what TheThe photon photon ring ring is is nearly nearly circular circular for for all all black black hole spins and one would expect from phenomenological models of the allall inclinations inclinations of of the the black black hole hole spin spin axis axis to to the the line of sight source? Radiatively inefficient accretion flow models of M87 2 Academic The synthetic blurred image is similar do real data, consistent with a Kerr black hole The Astrophysical Journal Letters, 875:L5 (31pp),, 2019 2019 April April 10 10 The The EHT EHT Collaboration Collaboration et et al. al. Figure 1. Left panel: an EHT2017 image of M87 from Paper IV of this series ((see their Figure Figure 15 15)).. Middle Middle panel: panel: a a simulated simulated image image based based on on a a GRMHD GRMHD model. model. Right Right panel: the model image convolved with a 20as FWHM Gaussian beam. Although the the most most evident evident features features of of the the model model and and data data are are similar, similar, fifinene features features in in the the model are not resolved by EHT. model are not resolved by EHT. ApJ Lett. 875 (2019) L5 In Section 6 we combine EHT data with other constraints on the (e.g., Johannsen & Psaltis 2010). For an a 0 black hole In Section 6 we combine EHT data with other constraints on the (e.g., Johannsen & Psaltis 2010). For an a** 0 black hole radiative efficiency, X-ray luminosity, and jet power and show ofof mass mass MM andand distance distance DD,, the the photon photon ring ring angular angular radius radius on on that the latter constraint eliminates all a 0 models. In the sky is that the latter constraint eliminates all a* 0 models. In the sky is Section 7 we discuss limitations of our models and also briefly discuss alternatives to Kerr black hole models. In Section 8 we 2727 GMGM pp summarize our results and discuss how further analysis of existing cDcD22 EHT data, future EHT data, and multiwavelength companion 11 EHT data, future EHT data, and multiwavelength companion ⎛⎛ MM ⎞⎞⎛⎛ DD ⎞⎞ observations will sharpen constraints on the models. 18.818.8 as,as, 1 1 ⎜⎜ 99 ⎟⎟⎜⎜ ⎟⎟ ()() ⎝⎝6.26.2 10 10MM ⎠⎠⎝⎝ 16.9 16.9 Mpc Mpc⎠⎠ 2. Review and Estimates wherewhere we we have have scaled scaled to to the the most most likely likely mass mass from from Gebhardt Gebhardt et et al. al. ((20112011)) andand a a distance distance of of 16.916.9 Mpc Mpc ((seesee also also EHT EHT Collaboration Collaboration In EHT Collaboration et al. (2019d; hereafter Paper IV) we In EHT Collaboration et al. (2019d; hereafter Paper IV) we etet al. al. 2019e2019e,, ((hereafterhereafter Paper Paper VIVI;Blakesleeetal.;Blakesleeetal.20092009;Birdetal.;Birdetal. present images generated from EHT2017 data (for details on present images generated from EHT2017 data (for details on 20102010;Cantielloetal.;Cantielloetal.20182018)).Thephotonringangularradiusfor.Thephotonringangularradiusfor the array, 2017 observing campaign, correlation, and calibra- otherother inclinations inclinations and and values values of of a differsdiffers by by at at most most 13% 13% from from tion, see Paper II and Paper III). A representative image image is is a** Equation (1),andmostofthisvariationoccursat11 a reproduced in the left panel of Figure 1. Equation (1),andmostofthisvariationoccursat11 ∣∣∣∣a** Four features of the image in the left panel of Figure 1 play ((e.g.,e.g., Takahashi Takahashi 20042004;Younsietal.;Younsietal.20162016)).Evidentlytheangular.Evidentlytheangular an important role in our analysis: (1) the ring-like geometry, (2) radiusradius of of the the observed observed photon photon ring ring is is approximately approximately 2020 as as the peak brightness temperature, (3) the total flux density, and ((FigureFigure 11 andand Paper Paper IVIV)),whichisclosetothepredictionofthe,whichisclosetothepredictionofthe (4) the asymmetry of the ring. We now consider each in turn. blackblack hole hole model model given given in in Equation Equation ((11)).. (1) The compact source shows a bright ring with a central ((22)) TheThe observed observed peak peak brightness brightness temperature temperature of of the the ring ring in in fi 99 dark area without significant extended components. This bears FigureFigure 11 isisTTbb,, pk pk 66 10 10 K K,whichisconsistentwithpastEHT,whichisconsistentwithpastEHT a remarkable similarity to the long-predicted structure for mm-VLBImm-VLBI measurements measurements at at 230 230 GHz GHz ((DoelemanDoeleman et et al. al. 20122012;; optically thin emission from a hot plasma surrounding a black AkiyamaAkiyama et et al. al. 20152015)),andGMVA3,andGMVA3 mm-VLBImeasurementsof mm-VLBImeasurementsof hole (Falcke et al. 2000). The central hole surrounded by by a a thethe core core region region ((KimKim et et al. al. 20182018)).Expressedinelectronrest-mass.Expressedinelectronrest-mass 22 ’ bright ring arises because of strong gravitational lensing (e.g., ((mmee)) units,units, bb,B,,B, pk pk kTkT b b pk pk()() mc mc e e 11,where,wherekkBB isis Boltzmann Boltzmann’ss Hilbert 1917; von Laue 1921; Bardeen 1973; Luminet 1979).. constant.constant. The The true true peak peak brightness brightness temperature temperature of of the the source source is is The so-called “photon ring” corresponds to lines of sight that higherhigher if if the the ring ring is is unresolved unresolved by by EHT, EHT, as as is is the the case case for for the the pass close to (unstable) photon orbits (see Teo 2003),, linger linger modelmodel image image in in the the center center panel panel of of Figure Figure 11.. near the photon orbit, and therefore have a long path length TheThe 1.3 1.3 mm mm emission emission from from M87 M87 shown shown in in Figure Figure 11 isis through the emitting plasma. These lines of sight will appear expectedexpected to to be be generated generated by by the the synchrotron synchrotron process process ((seesee Yuan Yuan comparatively bright if the emitting plasma is optically thin. && Narayan Narayan 20142014,, and and references references therein therein)) andand thus thus depends depends on on The central flux depression is the so-called black hole thethe electron electron distribution distribution function function ((eDFeDF)).. If If the the emitting emitting plasma plasma “shadow” (Falcke et al. 2000), and corresponds to lines lines of of hashas a a thermal thermal eDF, eDF, then then it it is is characterized characterized by by an an electron electron 22 sight that terminate on the event horizon. The shadow could be temperaturetemperature TTTTebeb ,, or or eeeeee kTkTBB ()() mc mc 11,, because because seen in contrast to surrounding emission from the accretion ee b b,, pk pk ifif the the ring ring is is unresolved unresolved or or optically optically thin. thin. flow or lensed counter-jet in M87 (Broderick & Loeb 2009).. IsIs the the observed observed brightness brightness temperature temperature consistent consistent with with what what The photon ring is nearly circular for all black hole spins and oneone would would expect expect from from phenomenological phenomenological models models of of the the all inclinations of the black hole spin axis to the line of sight source?source? Radiatively Radiatively inef ineffificientcient accretion accretion flflowow models models of of M87 M87 2 ...we do not simply assume that the measured emission diameter is that of the photon ring itself. We instead directly calibrate to the emission diameter found in model images from GRMHD simulations. The structure and extent of the emission preferentially from outside the photon ring leads to a < 10% offset between the measured emission diameter in the model images and the size of the photon ring. ApJ Lett. 875 (2019) L6 The Astrophysical Journal Letters, 875:L1 (17pp), 2019 April 10 The EHT Collaboration et al. The Astrophysical Journal Letters, 875:L1 (17pp), 2019 April 10 The EHT Collaboration et al. Table 1 our measurement of the black hole mass in M87* is not Parameters of M87* inconsistent with all of the prior mass measurements, this allows Parameter Estimate us to conclude that the null hypothesis of the Kerr metric a (Psaltis et al. 2015; Johannsen et al. 2016), namely, the Ring diameter d 42±3 μas assumption that the black hole is described by the Kerr metric, Ring width a 20 as has not been violated. Fourth, the observed emission ring Crescent contrast b >10:1 Axial ratio a <4:3 reconstructed in our images is close to circular with an axial Orientation PA 150°–200° east of north ratio 4:3; similarly, the time average images from our 2 c g GM Dc 3.8±0.4 μas GRMHD simulations also show a circular shape. After d 0.5 associating to the shape of the shadow a deviation from the d g 11 0.3 c 9 circularity—measured in terms of root-mean-square distance M (6.5±0.7)×10 Me from an average radius in the image—that is 10%, we can set Parameter Prior Estimate an initial limit of order four on relative deviations of the D e (16.8±0.8) Mpc quadrupole moment from the Kerr value (Johannsen & Psaltis e 1.1 9 M(stars) 6.2 0.6 10 Me 2010). Stated differently, if Q is the quadrupole moment of a e 0.9 9 M(gas) 3.5 0.3 10 Me Kerr black hole and ΔQ the deviation as deduced from circularity, our measurement—and the fact that the inclination Notes. angle is assumed to be small—implies that ΔQ/Q4 a Derived from the image domain. (ΔQ/Q=ε in Johannsen & Psaltis 2010). b Derived from crescent model fitting. Finally, when comparing the visibility amplitudes of M87* c The mass and systematic errors are averages of the three methods (geometric with 2009 and 2012 data(Doeleman et al. 2012; Akiyama et al. models, GRMHD models, and image domain ring extraction). d fl 2015), the overall radio core size at a wavelength of 1.3 mm The exact value depends on the method used to extract d, which is re ected has not changed appreciably, despite variability in total flux in the range given. e Rederived from likelihood distributions (Paper VI). density. This stability is consistent with the expectation that the size of the shadow is a feature tied to the mass of the black hole and not to properties of a variable plasma flow. Figure 4. Top: threeSchwarzschild example models of some ofbased the best-fitting snapshots from the image library of GRMHD simulations for April 11 corresponding to differentIt spin is also straightforward to reject some alternative astrophysical parameters and accretion flows. Bottom: the same theoretical models,procedure, processed throughwe infer a VLBI values simulation of pipelineθ and with theα samefor schedule, regularized telescope characteristics, fi g interpretations. For instance, the image is unlikely to be produced and weather parametersestimate as in the Aprilof the 11 run light and imaged ring in themaximum same way as Figure likelihood3.2 Note that andp although27CLEAN the3.8reconstructedt toµ theas observations39 images..5 isµ equallyas good in the three cases, they refer to radically different physical scenarios; this highlights that a single good fit does not imply that a model is preferred over others (see Paper V)by. a jet-feature as multi-epoch VLBI observations of the plasma Combining results⇥ from all⇥ methods, we' measure emission sky angle: jet in M87 (Walker et al. 2018) on scales outside the horizon do region diameters of 42 3 μas, angular sizes of the gravita- ± not show circular rings. The same is typically true for AGN jets in tional radius θ =3.8±0.4 μas, and scaling factors in the spin energy through mechanisms akin to the Blandford–Znajek g The diameters of the geometric crescent models measurelarge the VLBI surveys (Lister et al. 2018).Similarly,werethe range α=10.7–11.5, with associated errors of ∼10%. For the process. characteristic sizes of the emitting regions that surroundapparent the ring a random alignment of emission blobs, they should distance of 16.8±0.8 Mpc adopted here, the black hole mass −1 shadows and9 not the sizes of the shadows themselves (see,also e.g., have moved away at relativistic speeds, i.e., at ∼5 μas day is M=(6.5±0.7)×10 M ; the systematic error refers to Psaltis et al. e2015; Johannsen et al. 2016; Kuramochi(Kim et al. et al. 2018b),leadingtomeasurablestructuralchangesand 7. Model Comparison and Parameter Estimationthe 68% confidence level and is much larger than the statistical 20189 , for potential biases). sizes. GRMHD models of hollow jet cones could show under error of 0.2×10 M . Moreover, by tracing the peak of the In Paper VI, the black hole mass is derived from fitting to the Wee model the crescent angular diameter d in termsextreme of the conditions stable ring features (Pu et al. 2017),butthis emission in the ring we can determine the shape of the image2 visibility data of geometric and GRMHD models, as well as gravitational radius and distance, g GM c D,asdeffect=αθg, is included to a certain extent in our Simulation Library for from measurements of the ring diameter in the imageand domain. obtain a ratiowhere betweenα is majora function and of minor spin, inclination, axis of the and ringR that(α;9.6–10.4 is 4:3; this corresponds to a 10% deviation from circularityhigh models with Rhigh>10. Finally, an Einstein ring formed by Our measurements remain consistent across methodologies, corresponds to emission from the lensed photon ring onlygravitational).We lensing of a bright region in the counter-jet would algorithms, data representations, and observed datain sets. terms of root-mean-square distancefi from an average radius. calibrate α by tting the geometric crescent models torequire a large a fine-tuned alignment and a size larger than that measured Motivated by the asymmetric emission ring structuresTable seen1 insummarizesnumber the of visibility measured data parameters generatedfromtheImageLibrary.We of the image features and the inferred black hole properties based on data in 2012 and 2009. the reconstructed images (Section 5) and the similar emission can also fitthemodelvisibilitiesgenerated from the Image Library fi structures seen in the images from GRMHDfrom simulations all bands and all days* combined. The inferred black hole At the same time, it is more dif cult to rule out alternatives to the M87 data, which allows us to measure θg directly.to black holes in GR, because a shadow can be produced by (Section 6),wedevelopedafamilyofgeometriccrescentmass stronglyHowever, favors the such measurement a procedure is complicated based on by stellar the stochastic nature models (see, e.g., Kamruddin & Dexter 2013) to compare directly any compact object with a spacetime characterized by unstable dynamics(Gebhardtof the et emission al. 2011 in). the The accretion size, asymmetry,flow(see, e.g., bright- Kim et al. 2016). to the visibility data. We used two distinct Bayesian-inference circular photon orbits (Mizuno et al. 2018). Indeed, while the ness contrast, andTo circularity account for of this the turbulent reconstructed structure, images we developed and a formalism algorithms and demonstrate that such crescent models are Kerr metric remains a solution in some alternative theories of geometric models,and as multiple well algorithms as the success that estimate of the the statistics GRMHD of the stochastic statistically preferred over other comparably complex geometric gravity (Barausse & Sotiriou 2008; Psaltis et al. 2008), non- fi simulations in describingcomponents the using interferometric ensembles of images data, are from consis- individual GRMHD models that we have explored. We nd that the crescenttent models with the EHT images offi M87* being associated with Kerr black hole solutions do exist in a variety of such modified provide fits to the data that are statistically comparable to those of simulations. We nd that the visibility data are not inconsistent strongly lensed emissionwith being from a realization the vicinity of many of a of Kerr the GRMHDblack hole. simulations.theories We (Berti et al. 2015). Furthermore, exotic alternatives to the reconstructed images presented in Section 5,allowingusto black holes, such as naked singularities(Shaikh et al. 2019), determine the basic parameters of the crescents. The best-fit conclude that the recovered model parameters are consistent boson stars (Kaup 1968; Liebling & Palenzuela 2012), and models for the asymmetric emission ring have diameters of across algorithms.8. Discussion 43±0.9 μas and fractional widths relative to the diameter of Finally, we extract ring diameter, width, and shape directlygravastars (Mazur & Mottola 2004; Chirenti & Rezzolla 2007), <0.5. The emission drops sharply interior to the asymmetricA number of elementsfrom reconstructed reinforce the images robustness(see Section of our5 image). The resultsare admissibleare solutions within GR and provide concrete, albeit emission ring with the central depression havingand a brightness the conclusionconsistent that it is with consistent the parameter with the estimates shadow of from a geometriccontrived, models. Some of such exotic compact objects can <10% of the average brightness in the ring. black hole as predictedcrescent by models. GR. First, Following our analysis the same has GRMHD used calibrationalready be shown to be incompatible with our observations multiple independent calibration and imaging techniques, as given our maximum mass prior. For example, the shadows of well as four independent7 data sets taken on four different days naked singularities associated with Kerr spacetimes with in two separate frequency bands. Second, the image structure ∣a*∣ 1 are substantially smaller and very asymmetric matches previous predictions well (Dexter et al. 2012; compared to those of Kerr black holes(Bambi & Freese 2009). Mościbrodzka et al. 2016) and is well reproduced by our Also, some commonly used types of wormholes (Bambi 2013) extensive modeling effort presented in Section 6. Third, because predict much smaller shadows than we have measured. 8 2) The light ring determines the initial “ringdown” of a perturbed black hole Gravitational signals describe an inspiral, merger and ringdown GW150914 Abbot et al., PRL 116 (2016) 061102 1972ApJ...172L..95G black but where ray superposed to unobserved known; where temporary (c) centrated close r these orbit. near of trino) The away small One away Consequently, in 1968), = The ringdownisassociatedtothelightring: © 1972. the The The It mass the optics now Astrophysical 3M; American further is r hole to compared from A well-known from vibrations. l b shown travels The past which = is vibrations only cf. the is M “black-hole l storage, hence are 3M its COMMENTS r University initially the is the becomes is r on are unstable « r inward finds that angular discussed. extremum the consists — described decays — like impact 31f Astronomical description the a of the 3M, to 3M unstable the of angular distribution from Journal, of Short-wavelength the massless Schwarzschild near high-frequency their of + one) r “vibrations only Chicago. University vibration” valid, it with momentum circular with = parameter of a const. It form Received (1) decays by r black light tangential of of m is circular circular ON node = of an time that length All emphasized which 172:L95-L96, the the particles. + Society 3M, the , orbit; X which rights e-folding of THE of ; of 1972 hole (1 number F(r) equations orbits it of “potential” like orbit Wisconsin, e a gravitational last (if 0 and co containing particles orbit black metric. evolves — simplifies - Press reserved. January wavenumbers = = more the described 3 was exp • “VIBRATIONS” light = 2M/r)-'r*^ at radiation C. co that which The X 3M Provided [F(3M)] ABSTRACT hole” of its time 0_2 orbit r 1972 consists and J. 3 [ emitted Printed precisely, — 1 = For a Physics into (c emitted these / L95 + energy motion 4; 2 Goebel in t/(3 ’ 3Af. M, massless radiation. of calculations March are = co V(r) components revised equal extends const. orbits large Press by an is in G 1 which “vibrations” by (gravitational, / The F X asymptotic similarly 2 into U.S.A. Department, (angular) 1/3M. = (at outward-traveling (at Press by = the 15 of 3 = 1972 are it to values 1 = corresponding 1) I X close particle 2 //(3 r a the to orbits is ^ contains finite M)]. NASA gravitational 3 b collapsing = January OF exp a Due r X , - (1971) and latter gravitational whose of = °° X to of provide, The 3 frequency. 2M/r)/r^, A close [//(3 to gravitational ) 1 distance) Madison to °° Astrophysics / 3 ; l is 2 makes the 1 for BLACK Af, frequencies the ). (Press / 27 for electromagnetic, the 2 foregoing an around are M radial For “vibrations” to star X waves, waves circular unstable and not ) r unstable spiraling the gravitational 3 . — wave clear a 1 considered (Ames / a is 2 wavenumbers particle, wave so in M 3M a unstable HOLE source, a the spiral spherical waves results Data )] 1/(3 its orbit the train maximum circular one, « , of of same packet and population orbits X but nature the System a M b in expands with (also or are circular l spinning Thorne 3 = waves 1 . > holds, only orbits orbits / body orbit 2 neu- close con- //co, well M). 20) (2) are (4) (3) (1) an at of a So, a hypothetical horizonless exotic compact object (ECO) with a similar light ring could vibrate similarly, initially... Cardoso, Franzin, Pani, PRL 117 (2016) 089902 Thus, if there are black hole mimickers with similar light rings to black holes but no “event horizon” they could mimic the black hole phenomenology. Additionally, ECOs with or without light rings have been proposed motivated by the singularity problem, the dark matter mystery,… Can there be such speculative exotic compact objects (ECOs)? Black holes have a horizon and (abiding energy conditions) a curvature singularity: issues To avoid this difficulties, models of horizonless ECOs (black hole mimickers) have been considered: a) “geons”, realized by Boson stars (Schunck, Mielke, CQG 20 (2003) R301) and Proca stars (Brito, Cardoso, CH, Radu, PLB752 (2016) 291); can form dynamically (Seidel, Suen, PRL 72 (1994) 2516); Perturbatively stable Gleiser and Watkins, NPB 319 (1989) 733; Lee and Pang, NPB 315 (1989) 477 ; Can be studied dynamically in binaries (Liebling and Palenzuela LRR 20 (2017) 5) b) wormholes c) gravastars (Mazur and Mottola, gr-qc/0109035) d) fuzzballs (Mathur, Fortsch. Phys. 53 (2005) 793) e) anisotropic stars, ... - There are viability issues. But to be able to mimic (up to a fine structure) the GW ringdown the object must be ultracompact, i.e. have a light ring; - Possible issue: for ultracompact ECOs resulting from a smooth, incomplete gravitational collapse, light rings come in pairs and one is stable Cunha, Berti, CH, PRL 119 (2017) 251102 In lecture 4 we will turn our attention to a class of ECOs: bosonic stars (see also Nico’s lectures and Juan’s seminar on wednesday!) Black holes and exotic compact objects C. Herdeiro Departamento de Matemática and CIDMA, Universidade de Aveiro, Portugal Thank you for your attention! Obrigado pela vossa atenção!