Barak Kol Hebrew University Jerusalem Jun 2009 Crete

Barak Kol Hebrew University - Jerusalem Jun 2009, Crete Outline Based on BK and M. Smolkin • Definition & Domain of applicability • 0712. 2822 (PRD) – caged • Review of results (caged, EIH) • 0712. 4116 (CQG) – PN • Standing puzzles • In progress • Renormalization (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.

Binary system • 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

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

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

Post-Newtonian Extreme Mass Ratio Small parameter v 2 Far zone m/M Validity always initially, never at merger 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 (1926 -1948 -1970 s) could have been met and overcome in classical physics.

Brief review of results • Goldberger & Rothstein (9. 2004) – Post. Newtonian (PN) including 1 PN=Einstein. Infeld-Hoffmann (EIH) • Goldberger & Rothstein (11. 2005) BH absorption incorporated through effective BH degrees of freedom • Chu, Goldberger & Rothstein (2. 2006) caged black holes – asymptotic charges

Caged Black Holes r 0 L Near 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 US • Rotating black holes

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 0712. 4116 BK, Smolkin

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

EIH in CLEFT Action • Feynman rules φ Ai x

Feynman diagrams PN 2 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.

Motion through curved background • 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

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

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

CLEFT Definition of Eff Action 0712. 2822 BK, Smolkin • Std definition by integrating out Saddle point approximation • Stresses that we can integrate out only given sufficient boundary conditions

Goal: Compute the Black hole effective action Comments • • Universality Perturbative (in background fields, ∂kg|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 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 0712. 4116 BK, Smolkin

Adding time back • Generalize the (NRG) field re-definition • Choosing 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 2 PN 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 • 1 PN (Einstein-Infeld-Hoffmann) • Black hole effective action • Post-Newtonian action • Renormalization

ΕΦΧΑΡΙΣΤΟ! Thank you! Darkness and Light in our region