Barak Kol Hebrew University - Jerusalem Jun 2009, Crete Outline Definition & Domain of applicability Review of results (caged, EIH) Standing puzzles Renormalization (in progress) Based on BK and M. Smolkin (PRD) – caged (CQG) – PN In progress

Domain of applicability General condition Consider a field theory with two widely separated scales r 0 <<L Seek solutions perturbatively in r 0 /L.

The search for Gravitational waves is on: LIGO (US), VIRGO (Italy), GEO (Hannover), TAMA (Japan) Sources: binary system (steady), collapse, collision Dim’less parameters For periodic motion the latter two are comparable – virial theorem Binary system

Two (equivalent) methods Matched Asymptotic Expansion (MAE) Two zones. Bdry cond. come from matching over overlap. Near: r 0 finite, L invisible. Far: L finite, r 0 point- like. Effective Field Theory (EFT) Replace the near zone by effective interactions of a point particle

Born-Oppenheimer Caged BHs Binary system Post Newtonian (PN) Extreme Mass Ratio (EMR) BHs in Higher dimensions Non-gravitational Applications

Post-Newtonian Small parameter v2v2 Far zone Validity always initially, never at merger Extreme Mass Ratio m/M if initially, then throughout

Non-gravitational Electro-statics of conducting spheres Scattering of long λ waves Boundary layers in fluid dynamics More…

Theoretical aspects Engages the deep concepts of quantum field theory including: –Action rather than EOM approach –Feynman diagrams –Loops –Divergences –Regularization including dimensional reg. –Renormalization and counter-terms The historical hurdles of Quantum Field Theory ( s) could have been met and overcome in classical physics.

Brief review of results Goldberger & Rothstein (9.2004) – Post- Newtonian (PN) including 1PN=Einstein- Infeld-Hoffmann (EIH) Goldberger & Rothstein ( ) BH absorption incorporated through effective BH degrees of freedom Chu, Goldberger & Rothstein (2.2006) caged black holes – asymptotic charges

Caged Black Holes r0r0 Near L Far Effective interaction: field quadrupole at hole’s location induces a deformation and mass quadrupole

Definition of ADM mass in terms of a 0-pt function, rather than 1-pt function as in CGR Rotating black holes CGR US

First Post-Newtonian ≡ Einstein-Infeld-Hoffmann Newtonian two-body action Add corrections in v/c Expect contributions from –Kinetic energy –Potential energy –Retardation

The Post-Newtonian action Post-Newtonian approximation: v<<c – slow motion (CLEFT domain) Start with Stationary case (see caged BHs) Technically – KK reduction over time “Non-Relativistic Gravitation” - NRG fields BK, Smolkin

Physical interpretation of fields Φ – Newtonian potential A – Gravito-magnetic vector potential

EIH in CLEFT Feynman rules Action x φ AiAi

Feynman diagrams PN2 in CLEFT: Gilmore, Ross 0810

Black Hole Effective Action The black hole metric Comments The static limit a=0. Uniqueness Holds all information including: horizon, ergoregion, singularity.

Problem: Determine the motion through slowly curving background r 0 <<L (CLEFT domain) Physical expectations –Geodesic motion –Spin is parallel transported –Finite size effects (including tidal) –backreaction Motion through curved background

Matched Asymptotic expansion (MAE) approach. “Near zone”. Non-Asymptotically flatNeed Non-Asymptotically flat BH solutions.

EFT approach Replace MAE by EFT approach black hole effective actionReplace the BH metric by a black hole effective action Recall that Hawking replaced the black hole by a black body black boxWe shall replace the black hole by a black box.

CLEFT Definition of Eff Action Std definition by integrating out Saddle point approximation Stresses that we can integrate out only given sufficient boundary conditions BK, Smolkin

Goal: Compute the Black hole effective action Comments Universality Perturbative (in background fields, ∂ k g| x ) Non-perturbative Issue: regularize the action, subtract reference background

First terms Point particle Spin (in flat space) Finite size effects, e.g. “Love numbers”, Damour and collab; Poisson Black hole stereotypingBlack hole stereotyping

What is the Full Result?

The Post-Newtonian action (Reminder) Post-Newtonian approximation: v<<c – slow motion (CLEFT domain) Start with Stationary case (see caged BHs) Technically – KK reduction over time “Non-Relativistic Gravitation” - NRG fields BK, Smolkin

Adding time back Generalize the (NRG) field re-definition optimal gaugeChoosing an optimal gauge (especially for t dependent gauge). Optimize for bulk action. Possibly eliminating redundant terms (proportional to EOM) by field re-definition

Goal: Obtain the gauge-fixed action allowing for time dependence - Make Newton happy…

Quadratic level Φ, A sector Proceed to Cubic sector and onward…

What is the full Non-Linear Result?

Renormalization Before considering gravity let us consider Take β=0. The renormalized point charge q(k) or q(r) is defined through k

An integral equation q(k) satisfies Comments: The equation can be solved iteratively, reproducing the diagrammatic expansion of q(k). The equation is classically polynomial for polynomial action

Relation with Φ(r) Φ(r) is defined to be the field due to a point charge It is directly related to q(r) through While q(r) satsifies the above integral equation, Φ(r) satisfies a differential equation – –namely, the equation of motion

Re-organizing the PN expansion These ideas can be applied to PN. For instance at 2PN Can be interpreted through mass renormalization

Comment: The beta function equation

Recap Theory which combines Einstein’s gravity, (Quantum) Field Theory and experiment. Ripe caged black holes 1PN (Einstein-Infeld-Hoffmann) Black hole effective action Post-Newtonian action Renormalization

Darkness and Light in our region ΕΦΧΑΡΙΣΤΟ! Thank you!

Higher dimensional black objects Higher d ring Near zone Emparan, Harmark, Niarchos, Obers, Rodrigues

Born-Oppenheimer approximation (1927) 0+1 Field theory Compute Ψ e w. static nuclei and derive the effective nuclear interactions. In this way the EFT replaces the near zone by effective interactions “Near” “Far”

Caged BH’s and CLEFT CLEFT = CLassical Effective Field Theory, no i’s, no ‘s NRG decompostion (=Non Relativistic Gravitation, which is the same as temporal KK reduction) BK & Smolkin 12.07

Post-Newtonian approx. NRG decompostion terms Reconstructed EIH and following Cardoso-Dias- Figueras generalized to higher dimensions Damour, Blanchet, Schafer BK & Smolkin 12.07b

BH degrees of freedom Physical origin of eff. deg. of freedom? Near horizon fields (notably the metric) delocalized through decomposition to spherical harmonics