The Internal Structure of Spinning Black Holes

Edmund Bertschinger Black Holes, Gravitational Waves MIT Department of Physics and and Spacetime Singularities Kavli Institute for Astrophysics and Space Research 9 May 2017 A singularity cloaked by horizons: Is this all?

2 Was Chandrasekhar wrong?

In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein's equations of general relativity provides the absolutely exact representation of untold numbers of black holes that populate the universe.

Subrahmanyan Chandrasekhar, The Nora and Edward Ryerson Lecture, 22 April 1975 Who, then, is right?

A. Astrophysicists: the Kerr solution outside the singularity (inside, quantum happens)

B. Classical relativists: the Kerr solution outside the Cauchy horizon, modified singularities inside

C. String theorists: a “fuzzball,” approaches Kerr asymptotically outside the event horizon

D. Bolder theorists: the event horizon is replaced by a “firewall”

E. None of the above How is this related to Lemaître?

1. Lemaître-Tolman-Bondi model decribes spherical black hole formation (Oppenheimer & Snyder)

2. Curvature terms (beyond L) might modify the Einstein equations in the strong-field regime Outline

A review of Kerr geometry and its limitations

Exploring the event horizon using gravitational waves

Do Kerr black holes imply the possibility of time travel?

Summary and conclusions

6 Charged or spinnning black holes:

Multiple sheets in (r,t) t

r

Image from Wikipedia Kruskal-Szekeres coordinate patches (conformal diagrams)

Outside Inner Horizon r- Inside Outer Horizon r+

r- r- r+ r+

∞ ∞ r- r- r+ r+ 0 0 r+ r+ ∞ ∞ r- r-

r+ r+ r- r- Kerr solution permits travel to another “universe”

2 r+ Timelike

Wormhole r- r=0 r=0

r- r-

r+ r+ r=∞ 1 r=∞ Maximal extension:

Endless ladder of universes connected by wormholes

Image from Wikipedia Past history of a black hole: No black hole

Non-spinning black hole Eternal BH BH that forms r=0 r=∞

r=0 r=∞ Who, then, is right?

A. Astrophysicists: the Kerr solution outside the singularity (inside, quantum happens)

B. Classical relativists: the Kerr solution outside the Cauchy horizon, modified singularities inside

C. String theorists: a “fuzzball,” approaches Kerr asymptotically outside the event horizon

D. Bolder theorists: the event horizon is replaced by a “firewall”

E. None of the above What is the conformal diagram of a spinning black hole formed by gravitational collapse? Relativists’ conjecture: One spacelike, two timelike singularities BKL Belinsky, Khalatnikov & Lifshitz 1970

Poisson & Israel 1990 r=∞ (mass inflation singularity)

Marolf & Ori 2012 (shock Stellar surface singularity) r=0 Cf. Burko, Khanna & r=∞ Zenginoğlu 1601.05120 B. Kerr solution outside the Cauchy horizon, modified singularities inside Realistic black holes evaporate

Quantum fields lead to Hawking radiation

Outgoing blackbody radiation poses information paradox

Resolution of this paradox may imply quantum gravity modifies spacetime on scales >> lPlanck

Fuzzball: Lunin & Mathur 2002 Firewall: Almheiri et al 2013 (AMPS) Outline

A review of Kerr geometry and its limitations

Exploring the event horizon using gravitational waves

Do Kerr black holes imply the possibility of time travel?

Summary and conclusions

16 Exploring the horizon with gravitational waves Space is elastic

Modifications of GR modify the gravitational waves emitted in black hole collisions

LIGO 2016 Objection! Gravitational waves cannot probe what happens inside the horizon!

If the black hole geometry is modified inside the horizon, then Einstein’s equations are modified. This modifies the dynamics outside the horizon and enables an observational test. Long-wavelength echoes from the event horizon?

Abedi, Dykaar & Afshordi 1612.00266

Best-fit echoes from GW150914 2.9s signal for echoes (from 3 BH coalescences combined) Is this evidence for a reflecting wall outside the event horizon?

Regardless, Abedi et al show that gravitational waves can test GR including its predictions for black hole structure.

Fuzzball and fireball models do not predict echoes, but other quantum-gravity inspired models do, e.g. Sibandze et al 1702.04926 Outline

A review of Kerr geometry and its limitations

Exploring the event horizon using gravitational waves

Do Kerr black holes imply the possibility of time travel?

Summary and conclusions

21 Collaborator Kristina Pardo – summer research student, APS minority scholar, current Princeton astrophysics grad student The metric does not fully describe spacetime, even in GR! Topology is crucial: ranges & periodicity of coordinates, connectedness of manifold

Example: Schwarzschild metric. Geodesic completeness requires two copies of every (t,r,q,f). Kerr solution for r < 0: Closed timelike curves

Ring singularity bounds a disk r=0 with (q,f) ranging over S2. r≥0 is geodesically incomplete.

The region r=0+ (q,f) must be analytically continued to a second disk. Conventionally, this is r<0 (Boyer & Lindquist 1967) and results in CTCs (Carter 1968). The disk r=0

Hawking & Ellis 1973 Doubling the circle

Exclude r=0, where f is undefined. x = r cos f, y = r sin f How can the two Euclidean planes be made into one connected surface? 26 Crossing through r=0

r > 0 r < 0 Top sheet Bottom sheet x = r cos f, y = r sin f x and y change sign across r=0, f continuous

27 Is this the right scenario?

r > 0 r < 0 Top sheet Bottom sheet x = r cos f, y = r sin f x and y change sign across r=0, f continuous Boyer & Lindquist 1967 argued this is correct

because of analyticity 28 Alternative scenario

r > 0 r > 0 Top sheet Bottom sheet x = r cos f, y = r sin f x and y change sign across r=0, r>0 on both sides, f jumps by p 29 Does r change sign through the ring singularity? r changes sign r same sign

Second sheet has r < 0 Polar coordinates discontinuous across r = 0 Antigravity f jumps by p, r ≥ 0 before Closed timelike curves and after the jump

f f But x = r cos f, y = r sin f But x = r cos , y = r sin are continuous! are continuous! Both possibilities are mathematically allowed; the second one is simpler and requires no formation of an antigravity universe that violates causality! Summary and conclusions

A review of Kerr geometry and its limitations Case B: the Kerr solution outside the Cauchy horizon, modified singularities inside – but no exact or accurate numerical solution has ever been constructed (unlike Oppenheimer & Snyder 1939)

Exploring the event horizon using gravitational waves Remarkable probe of modifications of GR including reflecting surfaces.

Do Kerr black holes imply the possibility of time travel? No! But this is academic because of Case B.

32 Thank you for your attention! Sibandze et al 1702.04926

Lg= −�[� − 2L + �� ]

Vacuum solution remains Rab=0, however now get scalar gravitational waves 6a� - � =0. Equivalent to infalling massive scalar field, which might generate echoes without reflections.