Black Holes and Quantum Mechanics Juan Maldacena Institute for Advanced Study Karl Schwarzschild Meeting Frankfurt Institute for Advanced Study, 2017
Letter from Schwarzschild to Einstein. 22 December 1915, It as been very confusing ever since Classically
Extreme slow down of time The coordinate ``singularity Eddington 1924 finds a non singular coordinate system but did not recognize (or comment on its significance). Lemaitre 1933 First published statement that the horizon is not singular. Einstein Rosen 1935 (Still call it a ``singularity)
Szekeres, Kruskal 59-60 coordinates cover the full spacetime Wheeler Fulling 62 It is a wormhole ! Oppenheimer Snyder 1939 Future singularity horizon
star Observer on the surface does not feel anything special at the horizon. Finite time to the singularity It took 45 years to understand a classical solution Why ?
The symmetries are realized in a funny way. The time translation symmetry boost symmetry at the bifurcation surface Future singularity Interior is time dependent for an observer falling in.
It looks like a big crunch. Black holes in astrophysics Quasars (most efficient energy sources) Stellar mass black holes Sources for gravity waves !
Golden age ! Our own supermassive black hole in the Milky Way Interior and singularity Quantum gravity is necessary at the singularity!
What signals ? Singularity Singularity is behind the horizon. interior star It is shielded behind the black hole horizon that acts as a Schwarz-Shild.
Black holes and quantum mechanics Black holes are one of the most surprising predictions of general relativity. Incorporating quantum mechanics leads to a new surprise: Black holes are not black !
Hot black holes Black holes have a temperature Hawking We can have white ``black holes An accelerated expanding universe also has a temperature
Chernikov, Tagirov, Figari, Hoegh-Krohn, Nappi, Gibbons, Hawking, Bunch, Davies, . Very relevant for us! Quantum mechanics is crucial for understanding the large scale geometry of the universe.
Why a temperature ? Consequence of special relativity + quantum mechanics. Flat space first Why a temperature ? t
x Lorentzian Euclidean Accelerated observer energy = boost generator.
Time translation boost transformation Continue to Euclidean space boost becomes rotation. Why a temperature ? t r x
Continue to Euclidean space boost becomes rotation. Angle is periodic temperature Thermische Unruhe Ordinary accelerations are very small, g= 9.8 m/s2 = 1 light year Bisognano Weichman, Unruh Entanglement & temperature
Horizon: accelerated observer only has access to the right wedge. If we only make observations on the right wedge do not see the whole system get a mixed state (finite temperature). General prediction, only special relativity + quantum mechanics + locality Vacuum is highly entangled ! Black hole case singularity
Accelerated observer ! interior horizon r=0 Equivalence principle: region near the horizon is similar to flat space.
star If we stay outside accelerated observer temperature. Black hole from collapse Black holes have a temperature Do they obey the laws of thermodynamics ?
Black hole entropy Special relativity near the black hole horizon Einstein equations 1st Law of thermodynamics Black hole entropy Bekenstein, Hawking
2nd Law area increase from Einstein equations and positive null energy condition. Hawking Including the quantum effects Entanglement entropy of quantum fields across the black hole horizon Has been understood better in quantum field theory 2nd Law extended to include this term.
Wall 2011 Bekenstein bound automatic in relativistic quantum field theory. Casini 2008 Focusing theorems and better understanding of the positivity of energy, Boussos talk
and new ``area increase statements. Bousso, Englehardt, Wall, Faulkner, Bekenstein Casini bound Bekenstein When is this true ? Is it true ? Does it impose a constraint on QFT ?
Casini 2008 It is always true in any relativistic QFT. 2nd Law always satisfied. General relativity and thermodynamics General relativity and thermodynamics
Black hole seen from the outside = thermal system with finite entropy. Is there an exact description where information is preserved ? Yes Gauge/Gravity Duality (or gauge/string duality, AdS/CFT, holography)
Theories of quantum interacting particles (very strongly interacting) Quantum dynamical Space-time (General relativity) string theory
JM 97 Witten, Gubser, Polyakov, Klebanov . Gravity in asymptotically Anti de Sitter Space Duality Gravity, Strings
Quantum interacting particles quantum field theory Black holes in a gravity box Gravity, Black hole
Strings Hot fluid made out of very strongly Interacting particles. Lessons for black holes Black holes as seen from outside (from infinity) are like an ordinary quantum system.
Black hole entropy = ordinary entropy of the quantum system. Absorption into the black hole = thermalization Shenker Stanford Chaos near horizon gravity Kitaev , 2013-2015 Interior ? Mathur, Almheiri, Marolf, Polchinski, Sully
In the meantime Black holes as a source of information Black holes as toy models Used to model strongly interacting systems in high energy physics or condensed matter physics. Black hole
Gravity, Strings Hot fluid made out of strongly Interacting particles. Key insights into the theory of hydrodynamics with anomalies. Damour, Herzog, Son, Kovtun, Starinets, Bhattacharyya, Hubeny, Loganayagam,
Mandal, Minwalla, Morita, Rangamani, Reall, Bredberg, Keeler, Lysov, Strominger Let us go back to chaos Chaos divergence of nearby trajectories Thermal system average over all trajectories Growth Where you are after the perturbation vs. where you would have been.
Classical Quantum Quantum General: W, V are two ``simple (initially commuting) observables. Imagine we have a large N system. This is the definition of the quantum Liapunov exponent
For quantum systems that have a gravity dual W(t) horizon Commutator involves the scattering amplitude between these two excitations. Leading order graviton exchange
t Large t large boost between the two particles. V(0) Gravitational interaction has spin 2,
Shapiro time delay proportional to energy. Energy goes as et Can it be different ? Graviton phase shift : Typical size of string (of graviton in string theory)
String phase shift s, t = Mandelstam invariants Radius of curvature of black hole It can be less More ?
In flat space a phase shift has to scale with a power of s less than one in order to have a causal theory Maybe there is a bound Black holes as the most chaotic systems Sekino Susskind JM, Shenker, Stanford
For any large N (small hbar) quantum system. (Strings connect weakly coupled to strongly coupled systems) How do we get order from chaos ? How do we get the vacuum from chaos, or from a chaotic quantum system? horizon
? Example: hydrodynamics, we get something simple for some interactions, but it is more complicated with very small interactions (Boltzman equation). The full Schwarzschild wormhole
No need to postulate any exotic matter horizon left exterior Right exterior
No matter at all ! View it as an entangled state horizon left exterior
Right exterior Entangled Israel 70s JM 00s Symmetry horizon
left exterior Right exterior What is this funny time translation symmetry ?
Exact symmetry Exact boost symmetry! True causal separation If Bob sends a signal , then Alice cannot receive it. These wormholes are not traversable Relay on Integrated null energy condition
(Now a theorem, proven using entanglement methods) Balakrishnan, Faulkner, Khandker, Wang Not good for science fiction. Good for science! Interior is common If they jump in,
they can meet in the interior ! But they cannot tell anyone. Spacetime connectivity from entanglement ER ER = EPR
Van Raamsdonk Verlinde2 Papadodimas Raju JM Susskind Entanglement and geometry Local boundary quantum bits are
highly interacting and very entangled Ryu, Takayanagi, Hubeny, Rangamani Generalization of the Black hole entropy formula.
Interesting connections to quantum information theory Quantum error correction Complexity theory Almheiri, Dong, Harlow, Preskill, Yoshida, Pastawski Harlow, Hayden, Brown, Susskind Conclusions Black holes are extreme objects: most compact, most
efficient energy conversion, most entropic, most chaotic, Most confusing... The process of unravelling these confusions has lead to better understand of gravity, quantum systems, string theory and their interconnections. Black holes are not only in the cosmos, but can also be present in the lab. And there are still very important confusions and
open problems: Interior and singularity ? Thank you! Extra slides Full Schwarzschild solution ER
Eddington, Lemaitre, Einstein, Rosen, Finkelstein Kruskal Simplest spherically symmetric solution of pure Einstein gravity (with no matter)
Wormhole interpretation. Wormhole interpretation. L R Note: If you find two black holes in nature, produced by gravitational collapse, they will not be described by this geometry
No faster than light travel L R Non travesable No signals No causality violation
Fuller, Wheeler, Friedman, Schleich, Witt, Galloway, Wooglar In the exact theory, each black hole is described by a set of microstates from the outside Wormhole is an entangled state EPR Israel JM
Geometric connection from entanglement. ER = EPR Susskind JM Stanford, Shenker, Roberts, Susskind EPR
ER A forbidden meeting Romeo Juliet
String theory String theory started out defined as a perturbative expansion. String theory contains interesting solitons: Dbranes. Polchinski Using D-branes one can ``count the number of states of extremal charged black holes in certain Strominger Vafa superstring theories.
D-branes inspired some non-perturbative definitions of the theory in some cases. Matrix theory: Banks, Fischler, Shenker, Susskind Gauge/gravity duality: JM, Gubser, Klebanov, Polyakov, Witten Entanglement and geometry The entanglement pattern present in the state of the boundary theory can translate into geometrical features of the interior. Van Raamsdonk
Spacetime is closely connected to the entanglement properties of the fundamental degrees of freedom. Slogan: Entanglement is the glue that holds spacetime together Spacetime is the hydrodynamics of entanglement. Questions Black holes look like ordinary thermal systems
if we look at them from the outside. We even have some conjectured exact descriptions. How do we describe the interior within the same framework that we describe the exterior Modern version of the information paradox; ? Mathur, Almheiri, Marolf, Polchinski, Stanford, Sully,.. Once we figure it out: what is the singularity ? What lessons do we learn for cosmology ?
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