Yi-Zen Chu

BCCS 2008 @ Case Western Reserve University

9 December 2008

The n-body problem in

General Relativityfrom

perturbative field theory

arXiv: 0812.0012 [gr-qc]

• System of n ≥ 2 gravitationally bound compact objects:

• Planets, neutron stars, black holes, etc.

• What is their effective gravitational interaction?

?...||2

1

2

1

1

2n

ba ba

baNn

a

aaxx

MMGvML

eff

?...||2

1

2

1

1

2n

ba ba

baNn

a

aaxx

MMGvML

eff

• Compact objects ≈ point particles• n-body problem: Dynamics for the coordinates of the point particles• Assume non-relativistic motion

• GR corrections to Newtonian gravity: an expansion in (v/c)2

Nomenclature: O[(v/c)2Q] = Q PN

?...||2

1

2

1

1

2n

ba ba

baNn

a

aaxx

MMGvML

eff

• Note that General Relativity is non-linear.

• Superposition does not hold• 2 body lagrangian is not sufficient to obtain n-body lagrangian

Nomenclature: O[(v/c)2Q] = Q PN

• n-body problem known up to O[(v/c)2]: • Einstein-Infeld-Hoffman lagrangian• Eqns of motion used regularly to calculate solar system dynamics, etc.

• Precession of Mercury begins at this order

• O[(v/c)4] only known partially.• Damour, Schafer (1985, 1987)• Compute using field theory? (Goldberger, Rothstein, 2004)

• Solar system probes of GR beginning to go beyond O[(v/c)2]:

• New lunar laser ranging observatory APOLLO; Mars and/or Mercury laser ranging missions?• LATOR, GTDM, BEACON, ASTROD, etc.• See e.g. Turyshev (2008)

• n-body L gives not only dynamics but also geometry.

• Add a test particle, M->0: it moves along geodesic in the spacetime metric generated by the rest of the n masses• Metric can be read off its action

...2

1

2

1

2

11

000

2

dt

dz

dt

dzg

dt

dzggM

dt

zdM

dt

dz

dt

dzgdtM

ji

ij

i

i

gg

• Gravitational wave observatories need the 2 body L beyond O[(v/c)7]:

• LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over O[104] orbital cycles.• GW detection: Raw data integrated against theoretical templates to search for correlations.• Construction of accurate templates requires 2 body dynamics.• Currently, 2 body L known up to O[(v/c)7]• See e.g. Blanchet (2006)

• Starting at 3 PN, O[(v/c)6], GR computations of 2 body L start to give divergences that were eventually handled by dimensional regularization.• Perturbation theory beyond O[(v/c)7] requires systematic, efficient methods.

• Perturbation theory beyond O[(v/c)7] requires systematic, efficient methods.• QFT gives systematic framework for

• Renormalization & regularization• Computational algorithm – Feynman diagrams with appropriate dimensional analysis.

• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line

a

a

a

a

a

aaa

a

n

a

app

dd

plGR

ppGR

Npld

pl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

a

a

a

a

a

aaa

a

n

a

app

dd

plGR

ppGR

Npld

pl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

• –M∫ds describes

structureless point

particle

• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line

a

a

a

a

a

aaa

a

n

a

app

dd

plGR

ppGR

Npld

pl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

• Non-minimal terms

encode information on the

non-trivial structure of

individual objects.

• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line

a

a

a

a

a

aaa

a

n

a

app

dd

plGR

ppGR

Npld

pl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line• Coefficients {cx} have to

be tuned to match physical

observables from full

macroscopic description of

objects.

a

a

a

a

a

aaa

a

n

a

app

dd

plGR

ppGR

Npld

pl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

• For compact objects,

up to O[(v/c)8], only

minimal terms -Ma∫dsa

needed

• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line

• Expand GR and point particle action in powers of graviton fields hμν …

j

a

i

aij

i

ai

d

plaa

n

a

a

dd

pl

d

pl

aaa

d

xxhxhhMxdtM

RgxdMS

M

hg

xxxLL

SSiDhLdti

000

2/12

1

2

1)2/(

1

0

21

2

,...}],,[{

exp

expexp

effeff

classical

gfeff

diagrams connectedFully

• … but some dimensional analysis before computation makes perturbation theory much more systematic• The scales in the n-body problem

• r – typical separation between n bodies.• v – typical speed of point particles• r/v – typical time scale of n-body system

1

00

0

~

~][~

vrxd

r

vxx

dx

d

dd

• Lowest order effective actionn

bad

ba

ba

d

Nn

a

aaxx

MMGxMdtS

3

1)2/(

1

2

||2

1

0

• Schematically, conservative part of effective action is a series:

2

3

1)2/(

321)2/(2

0

0

4

4

2

20

~

~~~

~...,~

vr

MG

MvrrMGdtMvdtS

SSvSvSSS

d

d

N

dd

N

n

eff

• Look at Re[Graviton propagator], non-relativistic limit:

2

2

][||

]2/)3[(

8

0],[],[0Re

;

00

32/)1(

;

00

dP

xxxx

diP

xxhxxhT

dd

cv

][||

]2/)3[(

8

0],[],[0Re

00

32/)1(

;

00

xxxx

diP

xxhxxhT

dd

cv

• Look at Re[Graviton propagator], non-relativistic limit:

2/32/

0

2/12/

2/12/1

~

~

~

vrh

vrh

vrh

d

d

i

d

• n-graviton piece of -Ma∫dsa with χpowers of velocities scales as• n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as

222/1

0

nn vS

422/1

0

nn vS

2220

)()()(

wvw n

Nnn

vS0

• With n(w) world line terms -Ma∫dsa,• With n(v) Einstein-Hilbert action terms,• With N total gravitons,• Every Feynman diagram scales as

• n-graviton piece of -Ma∫dsa with χpowers of velocities scales as• n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as

222/1

0

nn vS

422/1

0

nn vS

2220

)()()(

wvw n

Nnn

vS=1 Q PN

• With n(w) world line terms -Ma∫dsa,• With n(v) Einstein-Hilbert action terms,• With N total gravitons,• Every Feynman diagram scales as

2220

)()()(

wvw n

Nnn

vS

=1 Q PN

• Limited form of superposition holds• At Q PN, i.e. O[(v/c)2Q], max number of distinct terms from -Ma∫dsa is Q+2• 1 PN, O[(v/c)2]: 3 body problem• 2 PN, O[(v/c)4]: 4 body problem• …

0

• Every Feynman diagram scales as

2 bodydiagrams

3 bodydiagrams

Einstein-Infeld-Hoffmand-spacetime dimensions

No gravitonvertices

Gravitonvertices

No gravitonvertices

Gravitonvertices

No gravitonvertices

Gravitonvertices

No gravitonvertices

Gravitonvertices

No gravitonvertices

Gravitonvertices

2121212121

2323

|||||||||| zxzxzyyxyx

zdyd

encjbmai

mnij

• Perturbation theory beyond O[(v/c)7] for 2 body L requires systematic, efficient methods.• QFT gives systematic framework for

• Renormalization & regularization• Computational algorithm – Feynman diagrams with appropriate dimensional analysis.• But the computation is still hard and long – need for new technology.

• Perturbation theory beyond O[(v/c)7] for 2 body L requires systematic, efficient methods.• QFT gives systematic framework for

• Renormalization & regularization• Computational algorithm – Feynman diagrams with appropriate dimensional analysis.• But the computation is still hard and long – need for new technology.• http://www.stargazing.net/yizen/PN.html