Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2:...

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Gravedad Cuán,ca Juan Maldacena Ins,tute for Advanced Study 60 mo aniversario del Ins,tuto Balseiro Diciembre 2015

Transcript of Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2:...

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Gravedad  Cuán,ca      

Juan  Maldacena  Ins,tute  for  Advanced  Study  

     60mo    aniversario  del  Ins,tuto  Balseiro  

   Diciembre  2015  

   

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La  rela,vidad  general  produjo  dos  predicciones  sorprendentes:  

•  Agujeros  Negros    •  La  Expansión  del  Universo  “Tu  matemá,ca  está  bien,  pero  tu  Psica  es  un  desastre”      

En  ambos  casos  el  espacio  o  el  ,empo  se  deforman  muchísimo  

(Einstein  to  LeMaitre)  

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Cuan,zación  

Rµ⌫ � 1

2gµ⌫R+ gµ⌫⇤ = 8⇡GNTµ⌫

Materiaà  mecánica  cuán,ca  à  lado  izquierdo  también  deberia  ser    gobernado  por  la  mecánica  cuán,ca      El  tensor  de  energía  impulso  es  cuán,coà  la  geometría  también  !      En  la  mayoría  de  las  circumstancias  podemos  ignorar  las  fluctuaciones      cuán,cas.  In  most  circumstances  we  can  neglect  the  quantum  fluctua,ons.      A  veces  son  cruciales.    Ej.    en  el  principio  del  Big  Bang.    

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Resultados  de  tres  puntos  de  vista  

•  1)    Teoría  de  campos  efec,va.    

•  2)  Teoría  perturba,va  bien  definida.  Teoría  de  cuerdas.    

•  3)  Algunos  resultados  exactos  (no  perturba,vos).    

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Campos  cuán,cos  en  una  geometría  de  fondo  fija.    

Pequeñas  deformaciones  de  la  geometría  à  las  tratamos  como  un  campo  cuán,co  más.        

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•  Incluso  esta  simple  aproximación  produce  resultados  sorprendentes  !      

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Dos  predicciones  sorprendentes  

•  Agujeros  negros  ,enen  una  temperatura  

•  Un  universo  que  se  expande  en  forma  acelerada  ,ene  una  temperatura  

Muy  relevante  para  nosotros  !  

Habría    “agujeros  negros”    blancos    T ⇠ ~

rH

T ⇠ ~H =~RH

Hawking  

Chernikov,  Tagirov,    Figari,  Hoegh-­‐Krohn,  Nappi,    Gibbons,  Hawking,  Bunch,  Davies,  ….    

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Inflación  

•  Período  de  expansión  con  aceleración  casi  constante  

•  Produce  un  gran  universo  homogeneo  •  Fluctuaciones  cuán,cas=  Temperatura  à  pequeñas  fluctuaciones  

Starobinski,  Mukhanov  Guth,  Linde,    Albrecht,  Steinhardt,  …  

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La  mecánica  cuán,ca  es  crucial  para  entender  la    geometría  del  universo  a  gran  escala.  

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¿Por  qué  una  temperatura?  

•  Consecuencia  de  la  rela,vidad  especial  +  la  mecánica  cuán,ca.  

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¿Por  qué  una  temperatura?    t  

x  

θ  

Observador  acelerado  à  energía  =  generador  de  boosts.      

Con,nuando  al  espacio  Euclideo  à  boost  corresponde  a  una  rotación.    

Lorentziano   Euclideo  

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¿Por  qué  una  temperatura?    t  

x  

θ  

Con,nuando  al  espacio  Euclideo  à  boost  corresponde  a  una  rotación.    

Ángulo  periódico  à  temperatura    

r  

Bisognano  Weichman,  Unruh  Aceleraciones  ordinarias  son  muy  chicas,      g=  9.8  m/s2  à  β  =  1  año  luz  

� =1

T= 2⇡r =

2⇡

a

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Entrelazamiento  y  temperatura      

Horizonte:  el  observador  acelerado  solo  ,ene  acceso  al  triangulo    derecho    à    no  ve  todo  el  sistema  à  observa  un  estado  mezclado      (y  una  temperatura  finita).      El  vacío  está  muy  entrelazado  !  

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Agujero  negro  

 r=0   horizonte  

interior  

estrella  

Agujero  negro  producido  por  el  colapso  de  una  estrella.      

singularidad  

Principio  de  equivalencia:  la  región        cerca  del  horizon  es  como  el      espacio  plano.    Si  nos  quedamos  afuera  à  estamos  acelerando    à  temperatura.  

Accelerated  observer  !  

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Teoría  efec,va  de  bajas  energías  

•  La  fuerza  de  la  gravedad  aumenta  a  altas  energías.    

•  Efectos  cuán,cos  son  pequeños  a  bajas  energías.    

lPlanck ⇠ 10�35m , MPlanck = 1016Tev

g2e↵ ⇠ (energia)2

M2Planck

=l2Planck

(distancia)2

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Limitaciones  de  esta  expansión  

•  Expansión  en  potencias  de  g2eff    .    

•  Infinitos  à  contratérminos  à  nuevos    parametros  

•  No  es  una  teoría  bien  definida.    

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2)  Una  teoría  de  la  gravedad  con  una  expansión  bien  definida  

•  Gravedad:    parucula  con  m=0,  spin  2  .    

•  Agregamos  paruculas  con  m=0,    spin  >  2  à  una  ``extraña’’  theoría        

•  Agregamos  paruculas  con    m>  0  ,    spin  >  2  à  ?                          Ejemplo:  Teoría  de  Cuerdas.    

Vasiliev  

Veneziano,  ….,  Green,  Schwarz,….  

�gµ⌫

⇤ 6= 0

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Teoría  de  Cuerdas  

•  Paruculas  libres  à  Cuerdas  libres    

•  Diferentes  modos  de  oscilación  dan  orígen  a  dis,ntas  paruculas.    

•  Modo  mas  bajo  à  m=0,  spin  =2  à  gravitón.    

•  Se  reduce  a  la  gravedad  de  Einstein  a  bajas  energías.    

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Gravedad  desde  las  cuerdas  

Se  reduce  a  la  gravedad  de  Einstein  

Radio de curvatura

tamano de la cuerda

� 1

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Analogía:  Débil  vs.  gravedad  •  Teoría  de  Fermi  de  las  interacciones  débiles    •     GN    à  GF  =1/MF

2      

MF   Energía  0  

Acoplamiento  1  

MF  Energía  0  

Acoplamiento=g  

1  

MW      MW  =  g  MF  

Fermi   Modelo  Estandar  

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Débil  vs.  gravedad  •  Similar  para  cuerdas.    

MP   Energía  0  

Acoplamiento  1  

MP              Energía  0  

Acoplamiento  1  

Ms      

Ms  =  g  MP  

Ms  =  masa  de  la  parucula  más  liviana  con  spin  >  2  

Gravedad  de  Einstein   Teoría  de  cuerdas  

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El  atrac,vo  de  la  teoría  de  cuerdas  •  No  hay  divergencias  ultravioletas.  Teoría  bien  definida.  

•  Dimensión  del  espacio  (plano)  debe  ser  10.  

•  4  grandes  +  6  chicas  à  podría  explicar  el  resto  de  las  fuerzas    

 •  Incorpora  interacciones  de  gauge  quirales  y  la  gran  

unificación.    

•  Supersymmetría  and  supergravedad  (pequeñas  dimensiones  fermiónicas    

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3)  Descripción  exacta  en  algunos  casos.  

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¿Por  qué  nos  importa?    

•  Problemas  con  agujeros  negros  

•  Big  Bang    y  el  mul,verso.    

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Agujeros  negros  ,enen  temperatura      ¿Obedecen  las  leyes  de  la  termodinámica  ?  

Pregunda  de  Wheeler  a  Bekenstein  

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Entropía  de  los  agujeros  negros    

rH $ M

dM = TdS

T ⇠ ~rH

Ecuación  de  Einstein  

1ra  ley      

Entropía  de  los  agujeros  negros.    

2da  Ley  à  area  aumenta  por  las  ecuaciones  de  Einstein  y  la  condición  de  posi,vidad  (nula)  de  la  energía  .    

Bekenstein,  Hawking  S =

(Area)

4~GN

Hawking,….,  Casini,  Wall:  incluyendo  efectos  cuán,cos  y  entropía  rela,va.    

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Rela,vidad  general  y  termodinámica  

•  Agujero  negro  visto  desde  afuera=  sistema  a  temperatura  finita.    

•  ¿Hay  una  descripción  exacta  donde  ser  preserve  la  información?  

•  Sí,  pero  hay  que  ir  más  alla  de  la  teoría  perturba,va…    

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3)  Descripción  exacta.  

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Teoría  de  Cuerdas:  más  allá  de  la  teoría  perturba,va  

•  Cuerdas  à  originalmenteà  perturba,va  •  Solitones  interesantes:  D-­‐branas.    •  D-­‐branas  inspiraron  ciertas  definiciones  no  perturba,vas.    

Teoría  de  matrices:  Banks,  Fischler,  Shenker,  Susskind  Dualidad  de  gauge-­‐gravedad:  JM,  Gubser,  Klebanov,  Polyakov,  Wi{en  

Polchinski  

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Espacio tiempo dinámico

( gravedad cuántica, cuerdas )

Teorías de partículas cuánticas

interactuantes

Dualidad de gauge-gravedad (o dualidad de gauge/cuerdas, AdS/CFT, holografía)

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Gravedad  en  an,-­‐de-­‐Si{er  

An,  de  Si{er  =  espacio    hiperbólico  con  una  dimensión    temporal  

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Gravedad  en  an,-­‐de-­‐Si{er  

Dualidad  

 paruculas  cuán,cas    interactuantes  Teoría  cuán,ca  de  campos  Gravedad,    

cuerdas  

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Cuerdas  de  la  cromodinámica.    

•  Cromodinámica  à  SU(N,  lineas  de  Faraday  à  cuerdas  dinámicas.    

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Cuerdas  de  la  cromodinámica  Gluón:      color  y  an,-­‐color    N  colores    SU(N)  

     N  grande    t’  Hoo|  ‘74  

Acoplamiento    ~  1/N    

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Evidencia experimental

Cuerda rotante constTJm +max

2 ~

De E. Klempt hep-ex/0101031

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Cuerdas de la cromodinámica son las cuerdas de la “teoría de cuerdas”.

= Gravedad cuántica en una dimensión

más.

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Cuerdas de la cromodinámica supersimétrica en 4d son las cuerdas

en 10 dimensiones

Gravedad  de  Einstein  à    Necesitamos  N  grande                                                                                                y      acoplamiento    grande.    

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   Interior    Frontera  

 Gravedad  de  Einstein  en  el  interior  à  Paruculas  muy  fuertemente      interactuantes  en  la  frontera.          

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   Interior    Boundary  

Agujeros  negros  =    Estados  termalizados  con  energías  bastante  grandes      

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•  Entropía  =  Areadel  horizonte  =Log[  Número  de  estados  en  la  frontera  ]  

•  Caer  en  el  agujero  negro  =  termalización  en  la  frontera.    

•  Unitario  visto  desde  afuera.  

 

Strominger,  Vafa,…  

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Agujeros  negro  como  fuente  de  información!  

 •  Problems  con  paruculas  fuertemente  interactuantesà  problemas  simples  en  la  gravedad.  

 •  Geometrizacíon  de  la  Psica!    

Plasma  de  quarks  y  gluones,  superconductores  de  alta  temperatura  

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Geometría  emergente  

•  Descripción  exacta  à  la  frontera.      •  Espacio-­‐,empo  en  el  interior  ``emerge’’  cuando  N  es  grande  y  las  interacciones  son  fuertes.      

 

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Entrelazamiento  y  geometría  

It  from  qubit    :  colaboración:      Casini  +  JM  +  others    

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Qubits  muy  entrelazados  

S(R) =Amin

4GN

Ryu,  Takayanagi,    Hubbeny,  Rangamani  

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Entrelazamiento  y  geometría  

•  El  patrón  de  entrelazamiento  en  la  frontera  se  traduce  a  aspectos  geometricos  en  el  interior.    

   

•  Slogan:  Entrelazamiento  son  los  hilos  que  tejen  el  espacio…    

Van  Raamsdonk,  Swingle  

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Solución  de  Schwarzschild    completa  

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

Solución  más  simple          

Eddington,  Lemaitre,  Einstein,  Rosen,    Finkelstein  Kruskal  

ER  

Page 47: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Agujero  de  gusano  

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

L R

No  se  producen  en  nuestro  universo…  

Page 48: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

No  hay  viajes  más  rápido  que  la  luz  

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

L R

No se viola la causalidad

Fuller, Wheeler, Friedman, Schleich, Witt, Galloway, Wooglar Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

Page 49: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Teoría  exacta:  cada  agujero  se  describe  completamente  mediante  un      número  finito  de  micro-­‐estados  

Agujero  de  gusano:  estado  entrelazado  

| i =X

n

e��En/2|EniL ⇥ |EniR EPR  

Geometría conexa por entrelazamiento ER = EPR

Israel JM

Susskind  JM  Stanford,  Shenker,  Roberts,  Susskind    

Page 50: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

EPR  

ER  

Page 51: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Un  encuentro  prohibido  

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

Romeo   Julieta  

Page 52: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

Page 53: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Misterios  

•  Interior  de  agujeros  negros  

•  Descripción  exacta  para  espacios  sin  frontera  

•  Singularidad  del  Big  bang      

•  Mul,verso.    

Page 54: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

The  Mul,verso  •  Los  efectos  de  la  cuán,ca  pueden  ser  más  importantes  a  distancias  más  grandes  que  el  universo  observable.    

 

Page 55: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

The  Mul,verso  

•  La  teoría  final  puede  tener  muchas  soluciones  posibles.  Ej:  cuerdas  à    muchas  soluciones  porque  hay  muchas  posibilidades  para  el  espacio  de  6  dimensiones.    

•  Inflación  e    inflación  eterna,  pueden  “poblar”  todas  esas  soluciones    

Page 56: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

•  Mul,verso  +  restricciones  anthropicas  podrían  explicar  el  valor  de  la  constante  cosmológica.    

•  No  sabemos  como  calcular…  Hay  mas  propuestas  que  autores…    

•  No  entendemos  bien  la  teoría  aún..      

Page 57: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Conclusiones  

•  Mecánica  cuán,ca  en  espacios  curvos  llevó  la  radiación  de  Hawking  y  a  las  fluctuaciones  primordiales  en  la  inflación.    

•  Estos  efectos  son  cruciales  para  explicar  aspectos  de  nuestro  universo.    

•  La  termodinámica  de  agujeros  negros  da  origen  an  problems  interesantes:  Entropía,  unitariedad,  etc.    

Page 58: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Conclusiones  •  Explorar  estos  problemas  dió  origen  a  una  relación  entre  la  gravedad  y  sistemas  con  interacciones  fuertes.    

•  Rela,vidad  general  aplicada  a  otras  areas  de  la  Psica  (ej.  Materia  condensada)    

•  Conexión  entre  entrelazamiento  y  geometría.    

Page 59: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Preguntas  

•  Interior  de  agujeros  negros  ?  

•  Singularidad  del  Big  bang      ?    

•  Probabilidades  en  el  mul,verso  ?  

Page 60: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Feliz  Cumpleaños  !  

Page 61: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Extra  slides  

Page 62: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Incorpora,ng  Quantum  Mechanics    A  simple  approach  

•  General  rela,vity  à  is  a  classical  field  theory  •  We  should  quan,ze  it  •  It  is  hard  to  change  the  shape  of  space,me  •  For  most  situa,ons  à  quantum  fields  in  a  fixed  geometry  is  a  good  approxima,on  

•  General  rela,vity  as  an  effec,ve  field  theory  à  systema,c  low  energy  approxima,on.    

 

Page 63: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

How  well    established  is  the  gauge/gravity  duality  ?  

•  Lots  of  evidence  in  the  simplest  examples.  •  Large  N:    Techniques  of  integrability  à  computa,ons  at  any  value  of  the  effec,ve  coupling.      

•   No  explicit  change  of  variables  between  bulk  and  boundary  theories    (as  in  a  Fourier  transform).    

Minahan,  Zarembo,  Beisert,  Eden,  Staudacher  Gromov,  Kazakov,  Vieira  Arutynov,  Frolov  Bombardeli,  Fioravan,,  Tateo  ….    

Page 64: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

 Two  sided  AdS  black  hole  Entangled state in two non-interacting CFT’s.

two interior regions. It is important not to confuse the future interior with the left exterior.

Sometimes the left exterior is referred colloquially as the “interior” of the right black hole,

but we think it is important not to do that. Note that no signal from the future interior

can travel to either of the two exteriors.

InteriorFuture

Past Interior

L R

Left

ExteriorRight

Exterior

Figure 1: Penrose diagram of the eternal black hole in AdS. 1 and 2, or Left and Right,denote the two boundaries and the two CFT’s that the system is dual to.

The system is described by two identical uncoupled CFTs defined on disconnected

boundary spheres. We’ll call them the Left and Right sectors. The energy levels of the

QFT’s En are discrete. The corresponding eigenstates are denoted |n⇤L, |n⇤R. To simplify

the notation the tensor product state |n⇤L ⇥ |m⇤R will be called |n, m⇤.The eternal black hole is described by the entangled state,

|�⇤ =�

n

e��En/2|n, n⇤ (2.1)

where � is the inverse temperature of the black hole. The density matrix of each side is a

pure thermal density matrix.

This state can be interpreted in two ways. The first is that it represents the thermofield

description of a single black hole in thermal equilibrium [6]. In this context the evolution of

the state is usually defined by a fictitious thermofield Hamiltonian which is the di⇥erence

of Hamiltonians of the two CFTs.

Htf = HR �HL. (2.2)

The thermofield hamiltonian (2.2) generates boosts which are translations of the usual

hyperbolic angle ⇥. One can think of the boost as propagating upward on the right side

4

ER  ER  

EPR  

Geometric connection from entanglement

| i =X

n

e��En/2|EniL ⇥ |EniR

Israel JM

Page 65: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Back  to  the  two  sided  Schwarzschild  solu,on  

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

Page 66: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Wormhole  interpreta,on.  

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

L R

Note:  If  you  find  two  black  holes  in  nature,  produced  by  gravita,onal    collapse,    they  will  not  be    described  by  this  geometry    

Page 67: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

No  faster  than  light  travel  

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

L R

Non travesable No signals No causality violation

Fuller, Wheeler, Friedman, Schleich, Witt, Galloway, Wooglar Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

Page 68: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Brane  argument  

Collec,on  of  N      3-­‐branes    

Geometry  of  a  black  3-­‐brane  

Low  energies  

 SU(N)    Super  Quantum    Chromodynamics  in  four  dimensions            

string  theory    on  AdS5  x  S5  =  

JM  1997  

Polchinski  

Horowitz  Strominger  

Page 69: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

ER  =  EPR  

•  Wormhole  =  EPR  pair  of  two  black  holes  in  a  par,cular  entangled  state:      

•  Large  amounts  of  entanglement  can  give  rise  to  a  geometric  connec,on.    

•  We  can  complicate  the  entanglement  of  the  two  sided  black  hole  à  get  longer  wormhole  

 

J.M.,  Susskind  

Stanford,  Shenker,  Roberts,  Susskind    

Page 70: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Black  hole  interior  •  We  do  not  understand  how  to  describe  it  in  the  boundary  theory.    

•  General  rela,vity  tells  us  that  we  have  and  interior  but  it  is  not  clear  that  the  exterior  is  unitary.    

•  Some  paradoxes  arise  in  some  naïve  construc,ons  

•  Ac,vely  explored…  Under  construc,on…    

Hawking,    Mathur,  Almheiri,  Marolf,  Polchinski,  Sully,  Stanford  

Entanglement  

Error  correc,ng  codes  Nonlinear  quantum  mechanics  

Firewalls/Fuzzballs  Non-­‐locality   Final  state  projec,on  

Fuzzballs  

Page 71: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

General  rela,vity  and  thermodynamics  

•  Viewing  the  black  hole  from  outside,  this  suggests  that    that  general  rela,vity  is  giving  us  a  thermodynamic  (approximate)  descrip,on  of  the  system  if  we  stay  outside.    

•  Quantum  mechanics  suggests  that  there  should  be  an  exact  descrip,on  where  entropy  does  not  increase.  (As  viewed  from  outside).  And  where  Hawking  radia,on  is  not  mixed.    

•  2nd  law  already  suggests  that  informa,on  is  not  lost  (if  informa,on  were  lost,  why  should  the  2nd  law  be  valid  ?).      

Page 72: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Unitarity  from  outside  ?  •  Iden,fy  the  degrees  of  freedom  that  give  rise  to  black  hole  entropy.    

•  Black  hole  entropy  depends  only  on  gravity  à  fundamental  degrees  of  freedom  of  quantum  gravity.  

•  Should  reveal  the  quantum  structure  of  space,me.    

•  Understand  their  dynamics.    

•  This  seems  to  requires  going  beyond  perturba,on  theory.    

Page 73: Ins,tute’for’Advanced’Study’No’hay’viajes’más’rápido’que’la luz’ Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions. The

Strings  from  gauge  theories  Gluon:      color  and  an,-­‐color  

   Closed  strings    à  glueballs    

Take  N  colors  instead  of  3,    SU(N)  

     Large  N  limit    t’  Hoo|  ‘74  

g2N  =  effec,ve  interac,on  strength.    Keep  it  fixed  when  N  à  infinity  

String  coupling  ~  1/N    

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Unitarity  from  the  outside  

•  We  can  form  a  black  hole  and  predict  what    comes    out  by  using  the  boundary  theory.    

•  If  you  assume  the  duality  à  unitary  evolu,on  for  the  outside  observer,  no  informa,on  loss.    


Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Spacetime Singularities

Spacetime Singularities

SPACETIME SINGULARITIES: The STORY of BLACK HOLES · The Schwarzschild black hole forms a point singularity in spacetime - although, as we will see, one can in theory open this up

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Two Exact Solutions Illustrating the Schwarzschild ...

Two Exact Solutions Illustrating the Schwarzschild ...

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Maximally Localized Wannier Functions: Concepts, Applications

Maximally Localized Wannier Functions: Concepts, Applications

SpaceTime Functions

SpaceTime Functions

Forats negres de schwarzschild

Forats negres de schwarzschild

11 Schwarzschild Geometry

11 Schwarzschild Geometry

Spacetime Constraints

Spacetime Constraints

MAXIMALLY-FLAT FILTERS IN WAVEGUIDE.pdf

MAXIMALLY-FLAT FILTERS IN WAVEGUIDE.pdf

DE SITTER AND SCHWARZSCHILD-DE SITTER ACCORDING TO ... · DE SITTER AND SCHWARZSCHILD-DE SITTER ACCORDING TO SCHWARZSCHILD AND DE SITTER Brett McInnes Department of Mathematics, National

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Spacetime Calculus

Spacetime Calculus

YSICAL VIE% 0 1980 · 2015. 1. 27. · PH YSICAL RK VIE% 0 VOLUME 21, NUMBER 8 Vacuum polarization in Schwarzschild spacetime 15 APRIL 1980 P. Candelas Center for Theoretical Physics,

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Maximally Adaptive Routing

Maximally Adaptive Routing

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Long term stable integration of a maximally sliced Schwarzschild …leo/research/papers/files/lcb02-01.pdf · ADM equations could be applied to a lattice using the Kasner spacetime

int box[]={24,8,8,8}; mdp_lattice spacetime(4,box); fermi_field phi(spacetime,3);

int box[]={24,8,8,8}; mdp_lattice spacetime(4,box); fermi_field phi(spacetime,3);

The schwarzschild spacetime

The schwarzschild spacetime

Class 8 : Schwarzschild Black Holesrichard/ASTRO350/Astro350... · 2020. 3. 3. · I : Schwarzschild n Karl Schwarzschild (1873-1916*) l Solved the Einstein field equations for the

Class 8 : Schwarzschild Black Holesrichard/ASTRO350/Astro350... · 2020. 3. 3. · I : Schwarzschild n Karl Schwarzschild (1873-1916*) l Solved the Einstein field equations for the

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