Polarization Whorls from M87
at the Event Horizon TelescopeAlex Lupsasca
Delilah Gates, Daniel Kapec, Yichen Shi, Andrew Strominger
Harvard University

A thrilling time to study black holes•

LIGO has ushered in a new era of data collection.

•

Black hole physics is now an experimental science!

•The Event Horizon Telescope will soon deliver the first upclose pictures of two black holes: the one at the center ofour Galaxy (Sagittarius A*) and the other in M87.

•This also poses a pressing challenge to theorists! Question:
What will the images look like?
Difficult to predict…

Astrophysics is hard…
•Black holes are typically surrounded by an accretion
disk and a plasma magnetosphere, and may also havea corona, jets, etc.

•

Each of these ingredients obeys messy physics.

•The traditional approach is to run computer simulations.

•

Detailed simulations are time-consuming and costly.

•

They can be incomplete or altogether fail in a special
corner of parameter space: high spin black holes.

…but symmetries are powerful•

A maximally rotating black hole saturates the2Kerr bound: |J| ≤ M•Near extremality, the spacetime geometry near theevent horizon develops an infinitely deep throat.

•In this regime, numerical studies are confronted with aseparation of scales: resolving the near-horizon physics
requires a large grid with an increasingly fine resolution.

•This also implies the existence of a perturbative
expansion in a small parameter: analytics and numerics
are complimentary.

•Here, the small parameter is the Hawking temperature,
which vanishes at extremality.

Application of an old tool (conformal symmetry)
to a new arena (black hole astrophysics).
•At this critical temperature, the region near the horizon
develops local scale invariance, or conformal symmetry:•The global conformal group is geometrically realized inthis “gravitating box” by the emergence of isometries thatenhance time translation symmetry to SL(2,R).

•

The infinite-dimensional local extension arises from an
asymptotic symmetry group analysis (like in AdS3) andrelates black holes of different small temperatures.

•This observation suggests that physics near rapidly spinningblack holes should be constrained by conformal symmetry inmuch the same way as many near-critical condensed mattersystems are governed by conformal field theories.

•Using this insight, we can greatly reduce the number ofparameters, leading to striking displays of universal behavior.

A prediction for the polarization from
M87 from conformal symmetry•

Guided by this symmetry, we analytically compute the polarization
profile of near-horizon emissions from a high spin black hole.

•This result is directly relevant to ongoing observations at the EHTof the black hole at the center of M87.

•Our prediction relies only on three assumptions:
1. The black hole is rapidly spinning: |J| ~ M2.

✓

2. The EHT can observe near-horizon emissions.

✓3. The polarization profile at the source is invariant under theenhanced isometries of the near-horizon geometry.
?

Symmetry assumption•It is natural to expect that a spacetime imparts its symmetries to thephysics that it drives. For instance, accretion onto a black hole isdriven by gravity and as a result, slowly-accreting matter disks tendto relax to a stationary and axisymmetric state.

•We therefore assume that the source polarization is invariant under
the local symmetries, i.e., the source polarization vector field fµcommutes with the four Killing vector fields (stationarity,axisymmetry, dilation invariance & special conformal invariance) thatgenerate the isometries of the near-horizon geometry.

•Because the polarization fµ must also obey parallel transport along
outgoing geodesics, this is an overconstrained problem.

•Surprisingly, a solution nonetheless exists and moreover, it is unique!

Polarimetric images•

A backlit black hole absorbs all incident light up to some impact
parameter and therefore casts a shadow upon an observer’s screen.

•Because gravity bends light, this silhouette depends on the polar
angle θo of the distant observer.

•All the near-horizon emissions turn out to be confined to the shadow.

•I will now show you the following three cases:
1.

“Face-on” observer on the rotation axis (θo = 0°).

2.

“Edge-on” observer in the equatorial plane (θo = 90°).

3.

The intermediate case θo ~ 15° of relevance to M87.

Face-on: The Polarization Whorl
βoαoEdge-on: Double Whorlβoαo

Down the barrel: M87βoαo1) The near-horizon geometry of a high spin black holeis a gravitating box, like AdS.
Unlike AdS, it is realized in nature, and the EHT could detect its signature!

2) The maximally spinning black hole is an example ofcritical point in astronomy.
Emergent (infinite-dimensional) conformal symmetry.
The analogue of the divergent correlation length is the throat’s depth.

3) The critical near-horizon behavior is universal.
The leading behavior near extremality is fixed by conformal symmetry:power law behavior, with critical exponent fixed by the rank of the field.

Thank you

And stay tuned for the images!

Application of an old tool (conformal symmetry)
to a new arena (black hole astrophysics).
•Fields in a (Near-)Extreme Kerr background (NHEK) necessarily
develop self-similarity in the near-horizon limit:
μμx →λx•hF→λ F

Stationary, axisymmetric fields have a surprisingly simple limitingbehavior in NHEK (here, H+ = ∂t and h is the conformal weight):
LH+F = 0

LH0F = h F

L∂φF = 0i.e., they form highest-weight representations of SL(2,R).
•Hence, complicated dynamics near rapidly spinning black holes
reduces to a tractable representation theory problem in NHEK!

Powerful tool with practicalapplications to near-horizon physics
xμ → λ xμ

F → λh F

LH 0 F = h F•The power h is the conformal weight. It labels
representations and also dictates their relative importanceto physics near the critical point.

•F ~ rh so the smaller weights contribute more.

•If F is a tensor field of rank p, then h ≥ -p.

•The leading behavior is obtained by isolating the piece withthe smallest allowed weight. Subleading terms are powersuppressed by their less relevant weight.

Scaling symmetry, self-similarity andpower laws with critical exponents•

This discussion exactly parallels the classification of IR
fixed points in terms of operator scaling dimensions,familiar from the study of critical phenomena incondensed matter systems or quantum field theory.

•In the astrophysical context, the infrared limit arises
geometrically from the scaling to a region of infinite
redshift near the horizon.

•Extremal black holes therefore provide examples ofcritical conformal fixed points in astronomy, with theconformal weights playing the role of critical exponents.