Barak KolHebrew 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

• 0712.2822 (PRD) – caged

• 0712.4116 (CQG) – PN

• In progress

Domain of applicabilityGeneral condition

Consider a field theory with two widely separated scales

r0<<L

Seek solutions perturbatively in r0/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

2

1

2, ,i iv Gmm

m c R

Two (equivalent) methods

Matched Asymptotic Expansion (MAE)

• Two zones. Bdry cond. come from matching over overlap.

• Near: r0 finite, L invisible.

• Far: L finite, r0 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

v2

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 (1926-1948-1970s) 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 (11.2005) BH absorption incorporated through effective BH degrees of freedom

• Chu, Goldberger & Rothstein (2.2006) caged black holes – asymptotic charges

Caged Black Holes

r0

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)

RFdxdtG

S

dxdxedxAdteds

Ah

dd

dd

jiij

dii

iji

2322

412

32

3/2222

exp16

1

,,

• Technically – KK reduction over time

• “Non-Relativistic Gravitation” - NRG fields

0712.4116BK, Smolkin

RFdxdtG

S

dxdxedxAdteds

Ah

dd

dd

jiij

dii

iji

2322

412

32

3/2222

exp16

1

,,

Physical interpretation of fields

• Φ – Newtonian potential

• A – Gravito-magnetic vector potential

EIH in CLEFT

• Feynman rulesAction

4 ij

Gt rG

t r

x 2

Grt

m

mv

φ

Ai

Feynman diagrams

1 2

Gm m

r

1 2 32

21

Gm m

rv

1 21 2

4Gm

rv vm

2

2

1 2

1

2

Gm m

r

12 121 2

2

12

12

, :2

r v rGm m v v

r r

v v

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.

MJa

ar

r

a

r

Mrf

dtadar

ddrrf

dadtrf

ds

:

cos

21

sin

sin

2222

2

2

2222

2

2222

222

2

22

• Problem: Determine the motion through slowly curving background r0<<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”.

• Need Non-Asymptotically flatNon-Asymptotically flat BH solutions.

EFT approach

• Replace MAE by EFT approach

• Replace the BH metric by a black hole black hole effective actioneffective action

• Recall that Hawking replaced the black hole by a black body

• We shall replace the black hole by a black black boxbox.

xijeffeff gJxSS ,,

CLEFT Definition of Eff Action

• Std definition by integrating out

Saddle point approximation

• Stresses that we can integrate out only given sufficient boundary conditions

gJxgSgJxS ijxijeff ,,:,,

0712.2822 BK, Smolkin

ggSiDggSi effeff expexp

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

dxxgm

dmSeff

0

0• Point particle

• Spin (in flat space)

• Finite size effects, e.g. “Love numbers”, Damour and collab; Poisson

• Black hole stereotypingBlack hole stereotyping

0][

41

: jiij

ijij

eff

hF

dtFJS

2/: 00

250

h

dtrS ijeff

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)

RFdxdtG

S

dxdxedxAdteds

Ah

dd

dd

jiij

dii

iji

2322

412

32

3/2222

exp16

1

,,

• Technically – KK reduction over time

• “Non-Relativistic Gravitation” - NRG fields

0712.4116BK, Smolkin

Adding time back

• Generalize the (NRG) field re-definition• Choosing an optimal gaugeoptimal gauge (especially for t

dependent gauge). Optimize for bulk action.

• Possibly eliminating redundant terms (proportional to EOM) by field re-definition

RFxddtG

S

dxdxedxAdteds

Ah

dd

ddd

jiij

dii

iji

2322

412

321

3/2222

exp16

1

,,

Goal: Obtain the gauge-fixed action allowing for time dependence

- Make Newton happy…

Quadratic levelΦ, A sector

232

1

22221

2223161

4

//2

AS

AcAcxddtS

GGF

jGtot

Proceed to Cubic sectorand onward…

What is the full Non-Linear Result?

Renormalization

Before considering gravity let us consider

4 4 41 1 12

2

63

6

0

4

bulk p

bulk

p

S S S

S d x d x d x

S q x d

Take β=0.

The renormalized point charge q(k) or q(r) is defined through

k

3

32

q kd k

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

30 1 22

1

1 1

1

q k qq d k

k

k kq

kk

k

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

r q r

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!