1 / 29

General Relativity is Not a Field Theory

General Relativity is Not a Field Theory. Cauchy-Kowalevskaya – Field Theory Has Finite # DOF per Space Point Arnowitt Deser Misner: Space is Dynamical in GR Hawking-Penrose: Generic Initial Conditions Lead to Singularities No Global Existence Theorems

wan
Download Presentation

General Relativity is Not a Field Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. General Relativity is Not a Field Theory • Cauchy-Kowalevskaya – Field Theory Has Finite # DOF per Space Point • Arnowitt Deser Misner: Space is Dynamical in GR • Hawking-Penrose: Generic Initial Conditions Lead to Singularities • No Global Existence Theorems • Cosmic Censorship Conjecture in Various Space-times With Fixed Asymptotic Structure (e.g. Asymptotically Flat): • All Singularities are Black Holes.

  2. Geometry of a Black Hole • Asymptotically Stationary Outside RS • Shrinking d-2 Area and Infinite Length in time ~ RS In Region Inside RS • Interior and Exterior Become Causally Disconnected • For Large Black Holes, Relation Between RS and M Depends on R : • ds2 = - f(r) dt2 + dr2/ f(r) + r2 dW2 • (1 – 2cdM/rd-3 +/- (r/R)2 ) = f(r) • 3R -2 = -/+ L LP2 LP = 10-33 cm h/2p = c = 1

  3. In a Black Hole, Space Stretches and Squeezes Faster Than Light

  4. Particle Scattering at b < 2 Ecm G = RS (E) Leads to Black Hole Formation (Penrose, Amati-Veneziano, Matschull, TB Fischler, Giddings-Eardley etc.) • Suggests that probes of distances smaller than LP ~ 10-33 cm. fail, and instead create larger and larger black holes – the UV/IR correspondence

  5. Black Hole Entropy Formula 4S LPd-2 = A of Bekenstein and Hawking Leads to Association of DOF With Boundary d – 2 Surfaces: Thorn, ‘t Hooft, Susskind, Fischler, Bousso • Feynman & Wilson: Definition of a Quantum Theory Comes From High Energy • Asymptotic Darkness: Black Holes Dominate High Energy Spectrum and Definition of Hamiltonian • Vacua: Different IR Superselection Sectors of Single Local Field Theory • This Concept is Not Applicable in Quantum Theories of Gravity

  6. Trying to Create Other Vacua • Guth Farhi: Create Local Region of Meta-stable dS space, which can Inflate, in Asymptotically flat background. Instead find Black Hole: inflating region separated from exterior by BH singularity. • TB: Same true for large regions of zero c.c. vacuum separated from another by potential barrier. RS = M = T R2 >> R Also true for regions on moduli space separated by f > mP • Matrix Theory and AdS/CFT Confirm Idea that Change of Vacuum Corresponds to Change of Parameters in the Hamiltonian

  7. AdS/CFT • Quantum Theory of AdSd X Y is CFT on Conformal Boundary R X Sd-2 : • ds2 = - (1 + r2 / R2 ) dt2 + dr2/(1 + r2 / R2 ) + r2 dW2 • BH Entropy Formula fits CFT Entropy S = c (TR)d-2 • If c = (RMP)k k = (d2 - 3d – 6)/(d – 1) • C.C. counts # of Degrees of Freedom in CFT

  8. Large RMP Insufficient for Low Curvature Space-time • Scalar field in Euclidean AdSd: • f ~ r - D • D = (1/2) [(d – 1) +/- ((d – 1)2 + (m R)2)1/2] • Zero mass, marginal op. ; negative m2 , real D, Breitenlohner Freedman allowed tachyon. • Normal CFT spectrum of primaries: exponentially growing number of bulk fields. “R is string scale” (generally no weakly coupled string interpretation). • Only known examples with gap in dimension spectrum are exactly supersymmetric CFTs

  9. Holographic Renormalization Group • Asymptotic values of moduli are lines of fixed points • d dimensional SUGRA (and gauged SUGRA) with scalars has potential with multiple AdS minima. Static domain wall solutions interpolating between minima (Poincare patch) are mapped by AdS/CFT to RG flows between Super-conformal field theories. Two minima have different “c” (c thm becomes area thm). Domain walls interpolate between theories, not vacua. • No examples where lower fixed point is stable SUSY violating CFT. • Non Susic Orbifolds of e.g. SU(N) MSYM are not conformal beyond planar order (cf. Scherk-Schwarz in “flat space”) • Horowitz-Hertog: Coleman de Lucia instanton for “decay of SUSic AdS” corresponds to perturbation of SCFT by SUSY violating marginal operator which is unbounded from below (subtle details of IR b.c.)

  10. The Holographic Principle • In asymptotically flat and AdS (including approximately AdS) space-times, theory is defined on conformal boundary. • Boundary correlators in AdS, S-matrix in AF only gauge invariant objects. • In rigorously established examples, small c.c. realized only with exact (AF) or asymptotically exact (AAdS) SUSY

  11. An Approach to Local Description of Quantum Gravity & Cosmology • Causal Diamond: Intersection of Interior of Forward Light Cone of P and Backward Light-cone of Q in Future of P. Region under experimental control of time-like observer travelling between P and Q. • The Holographic Screen of the Causal Diamond of a Local Observer Has Finite Area: Local Physics Has Inherent Quantum Ambiguity – Quantum Origin of General Covariance

  12. Holographic Cosmology • Hn (x) Hilbert Space of Observer n Time Steps From Big Bang • Dim [Hn (x)] = Dim [K]n K Irrep. Of Pixel Algebra Defined Below • Equal Area Step Time Slicing. • Dynamics Takes Place in Hnmax (x) for Maximal Area Slice, But • H(n,k,x) = Hin (k,k,x) + Hout (n,k,x) • Enforces Concept of Particle Horizon: D.O.F. Inside Horizon Do Not Interact With Those Outside Until Horizon Expands

  13. Degrees of Freedom of Quantum Gravity • Sa (y) Real Components of d – 2 Spinor Determines Orientation of Holoscreen at y via ST gm1 … mk S 1<k<d -2 (Cartan – Penrose) • SaI (m) SbJ (n ) + SbJ (n) SaI (m) = dab dmn MIJ • m,n pixelation of holoscreen. I,J refer to compact dimensions • DOF of Supersymmetric Massless Particles Penetrating Pixels of Holoscreen • 16 Real Components per pixel implies graviton in spectrum

  14. The Dense Black Hole Fluid • A full holographic cosmology introduces a spatial lattice of observers with the topology of d-1 Euclidean space • Nearest neighbor observers have overlap Hilbert space of dimension (dim K)n-1 at Time Step n. Dynamics must agree on overlap. Only known solution of these difficult conditions. Hn (x) is the same random Hamiltonian for each x. Chosen from a distribution with free fermion spectrum for large n. • Gives rise to Emergent Space-Time Geometry : Flat FRW with p = r. • Horizon Filling Black Hole for Every Observer at Every Time

  15. Asymptotically Flat Space Super-Poincare Invariant? • Superstring/M-theory Provides Ample Evidence This is True • Multi-parameter Web of Supersymmetric Theories in SpaceTime d = 4 … 11 • Strange Dualities and Connections (11D Theory Compactified on K3 4-folds = 10D Heterotic String Compactified on 3-torus etc. ) Explanation of Origin of Gauge Theory and Chirality • No Consistent AF Space-time w/o SUSY

  16. These are mathematical theories of quantum gravity, but don’t describe the real world • Exact SUSY, Poincare Invariance, Massless Spin Zero Particles • No Cosmology • Asymptotically Anti-deSitter (negative c.c.) String Theories Lead to Similar Conclusions. • AdS/CFT Gives Rigorous Evidence for UV/IR Connection Between Black Hole Spectrum and c.c..

  17. The Real World Has(?) Positive L • Evidence From Distant Supernovae, Ages of Globular Clusters/Universe, Large Scale Structure, Cosmic Microwave Background • If True: Holographic Principle Implies Finite Number (ln N = 10120) of Quantum States (TB – Fischler) • (1 – 2cd M/rd-3 -(r/R)2 ) = 0 • No Exact Scattering Theory as in Conventional String Theory

  18. The Sombrero Galaxy: See It Before It’s Too Late

  19. The Quantum Theory of de Sitter (dS) Space • Holographic Principle Implies Finite Number of States (TB Fischler) • Holoscreen Variables ciA : • [ciA , c*Bk ]+ = dikdAB 2N(N + 1) States • Spinor Bundle Over Fuzzy 2 Sphere • N ~ RdS • Static Hamiltonian H : Everything dS d(cays): Spectrum e [0, cT] T = 1/2pR

  20. If H is Random Hamiltonian, then Ergodic Thm Implies Random Initial State Time Averages Are Thermal With Temperature T (Choose c Appropriately) Density of States ~ e – p R2 T, R In Planck Units r = e- 2pR H Is Vacuum Density Matrix. • So What Are Ordinary Energies? : P0 • [P0 , H] ~ f(P0/R) (bounded) : P0 Resolves Degeneracy of H. Low P0 Eigenstates Approximately Stable. Particle and Black Hole Masses, etc. • Semiclassical Result That r Is Thermal In P0 Is Reproduced If P0 Eigenvalue is Related To Entropy Deficit of Eigenspace • This Relation Valid For Small Black Holes S = SdS - 2pRP0

  21. Black Holes as Excitations of Fermionic Pixels • Factor space of states with ciA | BH > = 0 with iBounded by N- < N/31/2 , A by N+ ([N- + N+]2 - N+ N- = N2): right entropy for black hole of Schwarzschild radius ~ N- in Planck units. • P0 = (ln 2 /2 p)1/2 MP (N2 - 2 N )(N2 - N)1/2 • N Fermion Number Operator • <<P0>> = MBH in BH Ensemble, With Small Fluctuations (BH States Not All Eigenstates w/ Same Eigenvale)

  22. Supersymmetric Particles as Excitations of Fermionic Pixels • Heuristic Argument: Maximal Field Theoretic Entropy in Single Horizon Comes from ~ N3/2 Particles With Momenta < N-1/2 and is o(N3/2 ): N1/2 Independent Horizon Volumes • Same Counting Comes From Using Pixel Ops. In Band of Matrix Made of Blocks of Size N1/2 : Matrix Theory Like Counting of Particle Momenta and Explanation of Particle Statistics. • Suggests Corrections [P , Q ] ~ N -1/2 mP ~ L1/4 : m3/2 ~ L1/4

  23. Matrix Theory Decomposition of dS Pixel Operators

  24. Each Band: Field Theory D.O.F. of Single Horizon • Maximal FT Entropy K ~ N1/2 ~ L-1/4 • Permutation Symmetry of Blocks in a Band: Particle Statistics • Exchanges of Bands – Discrete Analog of dS Transformations Not in R X SO(3)?? • TB, Fiol, Morisse: SUSY particles in limit. Chiral mults. Not Graviton. Need More Pixel Ops. Compact Dimensions

  25. Conclusions • Supersymmetric Quantum Theories of Gravity Abound and make Beautiful Quantitative Predictions About Imaginary Worlds, Some of Which Have Properties Tantalizingly Close to Our Own (Heterotic Strings on CY3, 11D SUGRA on G2) • Observables are defined as generalized scattering amplitudes on infinite asymptotic boundaries • The real world is not supersymmetric and may not have such infinite boundaries (acceleration of the universe). • The challenge is to find a consistent quantum gravitational system which violates SUSY in a space-time of low curvature, and to understand the relation between the splitting in supermultiplets and the c.c. • Beginnings of a Quantum Theory of dS Space Which May Solve This Problem Exist. Route to Derivation of m3/2 ~ L1/4 Clear. Need to Understand Compactification

More Related